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Research Article Nurse Scheduling Using Genetic Algorithm Komgrit Leksakul 1,2 and Sukrit Phetsawat 2 1 Excellence Center in Logistics and Supply Chain Management, Chiang Mai University, Chiang Mai 50200, ailand 2 Department of Industrial Engineering, Faculty of Engineering, Chiang Mai University, Chiang Mai 50200, ailand Correspondence should be addressed to Komgrit Leksakul; [email protected] Received 20 May 2014; Revised 20 October 2014; Accepted 23 October 2014; Published 27 November 2014 Academic Editor: Purushothaman Damodaran Copyright © 2014 K. Leksakul and S. Phetsawat. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is study applied engineering techniques to develop a nurse scheduling model that, while maintaining the highest level of service, simultaneously minimized hospital-staffing costs and equitably distributed overtime pay. In the mathematical model, the objective function was the sum of the overtime payment to all nurses and the standard deviation of the total overtime payment that each nurse received. Input data distributions were analyzed in order to formulate a simulation model to determine the optimal demand for nurses that met the hospital’s service standards. To obtain the optimal nurse schedule with the number of nurses acquired from the simulation model, we proposed a genetic algorithm (GA) with two-point crossover and random mutation. Aſter running the algorithm, we compared the expenses and number of nurses between the existing and our proposed nurse schedules. For January 2013, the nurse schedule obtained by GA could save 12% in staffing expenses per month and 13% in number of nurses when compare with the existing schedule, while more equitably distributing overtime pay between all nurses. 1. Introduction In order to succeed in organization management, one of the important factors that should be taken into consideration is human resource management within the organization for maximum efficiency at all times. is will enable the organi- zation to always drive the mission to the target successfully with the highest efficiency and effectiveness. For personnel management within hospitals, scheduling the work of nurses is one factor that is important and difficult to manage for maximum efficiency due to the uncertainty of the number of patients each day, which causes difficulty in managing nurs- ing staff to adequately and appropriately provide services to patients. If there are too many nurses, the hospital will likely unnecessarily waste the budget. However, if there are too few nurses, the hospital will not have enough nurses for all the patients or each nurse may receive or have to take on excessive workload [1]. Due to the high complication associated with nurse scheduling problems, using people to schedule may cause errors easily, such as the task not being done with high effectiveness and the process taking longer times. Approaches in the 1970s and 1980s addressed a number of problem formulations and solution techniques. A goal in many studies was to provide support tools to reduce the need for manual construction of nurse scheduling. Some studies [2–6] addressed the problem of determining staff levels and skills based on the numbers of patients and their medical need. Further advances [7–9] were made in applying linear and mixed integer programming and network optimization techniques for developing nurse scheduling. e numbers of researches have included a mix of heuristic and simulation techniques in an attempt to deal with more complex nurse scheduling. As real world problems are immense and deal with many constraints heuristics and recently metaheuristic such as simulated annealing (SA), tabu search (TA), and genetic algorithms (GA) have been developed to generate high quality nurse schedules in an acceptable computation time. Reference [10] proposed TA for constructing nurse scheduling whose objective is to ensure that enough nurses are on duty at all times while taking account of individual pre- ference and requests for days off in a way that is seen to treat all nurses fairly. Recently, GA has been applied to staff scheduling in different application areas such as transporta- tion systems [11], health care systems [12], emergency ser- vices, ambulance and fire brigade, call centers, and many other service organizations. Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2014, Article ID 246543, 16 pages http://dx.doi.org/10.1155/2014/246543

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Page 1: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

Research ArticleNurse Scheduling Using Genetic Algorithm

Komgrit Leksakul12 and Sukrit Phetsawat2

1 Excellence Center in Logistics and Supply Chain Management Chiang Mai University Chiang Mai 50200 Thailand2Department of Industrial Engineering Faculty of Engineering Chiang Mai University Chiang Mai 50200 Thailand

Correspondence should be addressed to Komgrit Leksakul komgritengcmuacth

Received 20 May 2014 Revised 20 October 2014 Accepted 23 October 2014 Published 27 November 2014

Academic Editor Purushothaman Damodaran

Copyright copy 2014 K Leksakul and S PhetsawatThis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

This study applied engineering techniques to develop a nurse scheduling model that while maintaining the highest level of servicesimultaneously minimized hospital-staffing costs and equitably distributed overtime pay In the mathematical model the objectivefunction was the sum of the overtime payment to all nurses and the standard deviation of the total overtime payment that eachnurse received Input data distributions were analyzed in order to formulate a simulation model to determine the optimal demandfor nurses that met the hospitalrsquos service standards To obtain the optimal nurse schedule with the number of nurses acquired fromthe simulation model we proposed a genetic algorithm (GA) with two-point crossover and random mutation After running thealgorithm we compared the expenses and number of nurses between the existing and our proposed nurse schedules For January2013 the nurse schedule obtained byGA could save 12 in staffing expenses permonth and 13 in number of nurses when comparewith the existing schedule while more equitably distributing overtime pay between all nurses

1 Introduction

In order to succeed in organization management one of theimportant factors that should be taken into considerationis human resource management within the organization formaximum efficiency at all times This will enable the organi-zation to always drive the mission to the target successfullywith the highest efficiency and effectiveness For personnelmanagement within hospitals scheduling the work of nursesis one factor that is important and difficult to manage formaximum efficiency due to the uncertainty of the number ofpatients each day which causes difficulty in managing nurs-ing staff to adequately and appropriately provide services topatients If there are too many nurses the hospital will likelyunnecessarily waste the budget However if there are too fewnurses the hospital will not have enough nurses for all thepatients or each nursemay receive or have to take on excessiveworkload [1] Due to the high complication associated withnurse scheduling problems using people to schedule maycause errors easily such as the task not being done with higheffectiveness and the process taking longer times

Approaches in the 1970s and 1980s addressed a numberof problem formulations and solution techniques A goal in

many studies was to provide support tools to reduce the needfor manual construction of nurse scheduling Some studies[2ndash6] addressed the problem of determining staff levels andskills based on the numbers of patients and their medicalneed Further advances [7ndash9] were made in applying linearand mixed integer programming and network optimizationtechniques for developing nurse scheduling The numbers ofresearches have included a mix of heuristic and simulationtechniques in an attempt to deal with more complex nursescheduling As real world problems are immense and dealwith many constraints heuristics and recently metaheuristicsuch as simulated annealing (SA) tabu search (TA) andgenetic algorithms (GA) have been developed to generatehigh quality nurse schedules in an acceptable computationtime Reference [10] proposed TA for constructing nursescheduling whose objective is to ensure that enough nursesare on duty at all timeswhile taking account of individual pre-ference and requests for days off in a way that is seen totreat all nurses fairly Recently GA has been applied to staffscheduling in different application areas such as transporta-tion systems [11] health care systems [12] emergency ser-vices ambulance and fire brigade call centers and manyother service organizations

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 246543 16 pageshttpdxdoiorg1011552014246543

2 Mathematical Problems in Engineering

In this paper we intended to solve nurse schedulingproblem by combining mathematical model computer sim-ulation and GA Nonlinear integer model was formulatedThe average total overtime payment and fairly payment toall the nurses were considered to minimize which maintainsthe standard of service level When mathematical modelsare applied to solve the problems the objective functionbecomes the sum of the overtime payment of all the nursesand the standard deviation of the total overtime paymentthat each nurse received This study aimed to create themost appropriate nurse scheduling model by applying thesimulation technique to analyze the distribution of the dataThereafter a simulation model was created to determine theappropriate demand for nurses with no more than 15 ofpatients waiting longer than the average service time as wellas those with more than 25 of patients waiting and withthe application of GA coding in Matlab program to calculatethe best work schedule from the most appropriate demandfor nurses acquired from the simulation Then the expensesand the number of nurses from the new work schedule werecompared with the actual expenses of the work schedule andthe previous number of nurses

2 Problem Analysis and Formulation

21 Cause and Problem Analysis of Nurse Scheduling Cur-rently one hospital is using manpower planning for nurseschedulingThis method involves workload and productivityefficiency measure per hour of work on average by having20 nurses in the outpatient department who are rotated forwork in the following five departments (1) department ofsurgery (SUR) (2) department of internal medicine (MED)(3) department of eye ear nose and throat (EENT) (4)department of pediatrics (PED) and (5) department ofobstetrics and gynecology (OBG) There are two types ofnursesrsquo working schedules 8 hours per day (0800ndash1600hours) and 12 hours per day (800ndash2000 hours) with fixedworkload each day and the working time cannot be changedat this stage This causes problems while allocating theholidays in order to meet the needs of the nurses [13 14]

Nurse scheduling at this hospital uses only one formatwhich assigns the same number of nurses to take care ofeach department every weekThis format of scheduling is notconsistent with the number of patients treated each day anddoes not bring into consideration factors related to servicelevels and queuing systems in nurse scheduling It is notpossible to allocate similar rates of overtime payment forall nurses The researcher emphasizes the importance of anappropriate and effective nurse scheduling model for ensur-ing the least cost to the hospital while maintaining the levelof service of the hospital and being fair in paying overtimeto all the nurses at similar rates by using simulation andoptimization method to create the most appropriate nursescheduling

22 Mathematical Modeling and Demand Function We cre-ated the objective and the limitation equations under variousconditions based on the information obtained from the hos-pital by developing an integer programming mathematical

model with assignment problem [15 16] The preliminarydata were set in the following conditions

119909119894119895119896119897

=

1 when nurse 119894 works on date 119895at time 119896 in department 119897

0 otherwise(1)

where 119894 = 1 2 3 20 (nurse) 119895 = 1 2 3 7 (sevenworking days Monday Tuesday Wednesday Sunday)119896 = 1 2 (two working shifts regular working hours (0800ndash1600 hours) and overtime hours (1600ndash2000 hours)) 119897 =1 2 3 4 5 (department of surgery (SUR) department ofinternal medicine (MED) department of eye ear noseand throat (EENT) department of pediatrics (PED) anddepartment of obstetrics and gynecology (OBG)) 119873 = totalnumber of nurses SOT = sum of all the overtime paymentsthat nurses receive SD = standard deviation of the totalovertime payment that each nurse receives AVG = averageof total overtime payment that each nurse receives OTD

119894119895119897

= additional overtime payment that nurse 119894 receives onthe working date 119895 in the department 119897 OTW

119894= overtime

payment that nurse 119894 receives in a week OTS119894= total overtime

payment that nurse 119894 receives and Demand119895119896119897

= demand fornurses on date 119895 at time 119896 in the department 119897

The objective equation includes two purposes

(1) The expenses of the hospital are minimal

(2) The standard deviations of the total overtime pay-ments of the nurses are the most similar

Doing a calculation to find the answers for the problempatterns with multiple objectives would be highly compli-cated and it would be difficult to find an answer To makeit easier to calculate two equations were combined togetherHowever since both the equations are in different units toinclude the two together it is necessary to change them intothe same unit by making these two equations in the form ofa percentage of the total Weighting the importance of thetarget functions to119908 percent and (1minus119908) percent respectivelywillmake up the sumof the hospital cost percentage equationand the standard deviation of the total overtime payment thateach nurse receives can be estimated as follows

Min119885 = 119908 times (SOT

AvgSOT) + (1 minus 119908) times (

SDAvgSD

) (2)

where 119908 = weight adjusted AvgSOT = sum of average totalovertime payment that every nurse receives in that particularmonth and AvgSD = standard deviation of the total overtimepayment that each nurse receives in that particular month

Constraints equation includes the following

(1) The number of nurses must be adequate to meet theneeds of the patients in each working period and day

119873

sum119894=1

119909119894119895119896119897

ge Demand119895119896119897

forall119895forall119896forall119897 (3)

Mathematical Problems in Engineering 3

(2) One nurse must work at least 40 hours per week (4)but fewer than 60 hours per week (5)

8 times

5

sum119897=1

7

sum119895=1

1199091198941198951119897

+ 4 times

5

sum119897=1

7

sum119895=1

1199091198941198952119897

ge 40 forall119894 (4)

8 times

5

sum119897=1

7

sum119895=1

1199091198941198951119897

+ 4 times

5

sum119897=1

7

sum119895=1

1199091198941198952119897

le 60 forall119894 (5)

(3) All nurses must be selected to work (6) and work notover six days per week (7)

7

sum119895=1

119909119894119895119896119897

ge 1 forall119894forall119896forall119897 (6)

7

sum119895=1

119909119894119895119896119897

le 6 forall119894forall119896forall119897 (7)

(4) In a day one nurse can work only one period in onedepartment

11990911989411989511

+ 11990911989411989512

+ 11990911989411989513

+ 11990911989411989514

+ 11990911989411989515

le 1 forall119894forall119895

11990911989411989521

+ 11990911989411989522

+ 11990911989411989523

+ 11990911989411989524

+ 11990911989411989525

le 1 forall119894forall119895

(8)

Equations related to expenses are as followsWhen nurses work formore than 40 hours in a week they

will receive an overtime payment of 650 Baht per eight hoursor 8125 Baht per hour and if within 1600ndash2000 hours theywill receive 50 Baht extra overtime payment per working day

OTS119894= 8125 times [

[

(8 times

5

sum119897=1

7

sum119895=1

1199091198941198951119897

+ 4 times

5

sum119897=1

7

sum119895=1

1199091198941198952119897) minus 40]

]

+

7

sum119895=1

(50 times

5

sum119897=1

1199091198941198952119897) forall

119894

AVG =sum119873

119894=1OTS119894

119873

SOT =

119873

sum119894=1

OTS119894

SD = radicsum119873

119894=1(OTS119894minus AVG)2

119873

(9)

After that the least number of nurses (119894) is determined inorder to be scheduled under existing limitations

Proposed mathematical model with objective function(2) subject to constraints (3)ndash(9) was verified by simpleproblem as shown in Table 1 With the conventional exactsolution approach the nurse scheduling was constructed fortwo departments SUR and MED and reported by Ganttchart as shown in Table 2 In Table 2 lowast and represent theschedule of SUR and MED respectively Lingo software alsoreported that the mathematical model is nonlinear modelwith integer variables

3 Building Simulation to Find AppropriateDemand for Nurses

In this case since a service criterion system has not been inuse in the hospital the demand for nurses as in the presentmay not be an appropriate demand for finding the leastcost work schedule The researcher examined an appropriatedemand for nurses (Demand

119895119896119897) by creating a simulation

using the Arena program to determine the appropriatedemand for nurses in each department and in all the periodsof each day from the original hospital data It was found thaton average 3408 or 5487 patients per week had waitedlonger than 25 of the total average service time Thereforethe researcher aimed to bring about an improvement at theinitial state itself by creating a simulation under the servicecriteria that set the limitation as only 15 of the patients onaverage did not have to wait longer than 25 of the totalaverage service time

31 Data Collection of Patient Service and Nursing ServiceDuration For the data collection of each department at eachperiod of each day (starting from when the patients registeruntil they wait to see a doctor) the researcher chose to use thedata as each day in the month to represent day in that monthCollecting data were done by the quality assurance sectionin case study hospital they defined the outpatient based onthe combination of appointments and walk-ins Interarrivaltime for regular and overtime hour including service timehas been collected for two years in winter season which hasthe largest number of outpatients The interarrival time forregular and overtime periods were separately collected dueto significant static data However no significant differencewas noticed for service timeThen the data were classified byday in a week and analyzed for their statistics distribution forthree categories

32 Data Distribution Analysis The data were analyzed byusing a suitable dispersion input analyzer in the Arenaprogram and by selecting the distribution with the highest 119875value using the significance level (120572) at 005 (95 confidence)as shown in Table 3 The 119875 value is a key concept in theapproach of Ronald Fisher where he uses it to measurethe weight of the data against a specified hypothesis and asa guideline to ignore data that does not reach a specifiedsignificance level

33 Simulation Modeling We created a simulation modelingstarting from when the patient registered at the servicedepartment based on interarrival time and service timestatistics distribution reported in Table 3 Because thesestatistics distributions consider the arrival of appointmentand walk-in patients then this simulation model can takeinto account the dynamics of walk-in patients Beginningwith statistics distributions patientsrsquo arrival patients hadregistered at the service department According to patientrsquossymptom they had moved to required department counterand waited for medical history file To parallel with filesearching and coming patients had queued and waited fornurse service Nurses served the fundamental examination

4 Mathematical Problems in Engineering

Table 1 Demand for nurses verify case study

Department Period Hospital demand (person)Mon Tue Wed Thu Fri Sat Sun

SUR Regular 1 1 2 1 2 1 2Overtime 1 2 1 2 1 1 1

MED Regular 2 1 2 1 1 1 1Overtime 1 1 1 2 1 2 2

such as weight blood pressure temperature measure pre-symptom examined and finally recorded all measure data inthe file after it arrived Service time distributions recordedand fitted by statistic approach in Table 3 were used as thedata set in process box After that we saved the percentageinformation of those patients whose waiting time exceededthe service time limitation as shown in Figure 1

Then we adjusted the number of nurses (resource) inthe capacity column manually as shown in Table 4 until thepercentage of the patients waiting for more than the limit(average and half-width) in each department was less than15 according to what we wanted in every section as shownin Table 5 After that we processed 100 replications

From Table 4 it can be observed that the departments ofSURMED and EENT appointed two people PED appointedthree people andOBG appointed one person to work in theirdepartments

From Table 5 we can see that all the departments withthe percentage of patients waiting for more than the limit(average and half-width) do not exceed 15 The EENTMED OBG PED and SUR departments had the percentagesof patients waiting for longer than the limit (average pluspositive half-width) at 07918 20505 50424 27194and 56773 respectively We considered only the positivehalf-width because we would like to know the maximumpercentage of patients waiting

34 Simulation Model Verification In model verification wecompared the results between real situation which occurredin this hospital and the simulationmodel Average number ofpatients receiving service average number of patients waitinglonger than average service time and average number ofpatients waiting less than average service time were proposedfor preforming simulationmodel efficientThe 100 times witheight-hour simulation were conducted for all five outpatientsdepartments The results were reported in Table 6 In simu-lation row we reported the average number plusminus withtheir corresponding half-width

Table 6 reported that the OBG got the highest error interms of average number of patients receiving service due tothe low 119875 value reported in Table 3 With this error howeverwe can adopt this simulation model generated by the arenaprogram for real case application

After model verification manual adjustment of theresource capacity was done and selected under hospital topmanagement service policy The service level policy was setas no more than 15 of patients waiting longer than the

average service timeThen the appropriate resource (numberof nurses) from simulation was shown for each departmentin Table 7 This number of nurses will be used as the datafor nurses demand (Demand

119895119896119897) in the mathematical model

for obtaining the optimal nurse schedule Even thoughto obtain the optimal nurse schedule with conventionalapproach always consumes computational runtime and hard-ware implementation Genetic algorithm was proposed tosolve and obtain an acceptable solution

4 Programming for Genetic AlgorithmNurse Scheduling

We coded a genetic algorithm program for use in the shiftscheduling of nurses with the Matlab program by using thefollowing steps

41 Chromosome Format and Chromosome Encoding Achromosome format uses a binary number system and it isdivided into three levels as shown in Figure 2 where

level 1 is a nurse at 1 2 3 119873level 2 is the working daysMonday TuesdayWednes-day Sundaylevel 3 is the number of 8 bit code

42 Chromosome Encoding and Chromosome Decoding Toencode the chromosome we used the binary coded decimalsystem which is a system that uses the 8 bit binary codeinstead of the general code we have set up We can encodeand decode the chromosome as shown in Table 8 where

(i) NDA = not working during regular hours(ii) DA1 = working during regular hours at Division 1(iii) DA2 = working during regular hours at Division 2(iv) DA3 = working during regular hours at Division 3(v) DA4 = working during regular hours at Division 4(vi) DA5 = working during regular hours at Division 5(vii) NOT = not working overtime(viii) OT1 = working overtime at Division 1(ix) OT2 = working overtime at Division 2(x) OT3 = working overtime at Division 3(xi) OT4 = working overtime at Division 4(xii) OT5 = working overtime at Division 5

Mathematical Problems in Engineering 5

Table2Th

eGanttcharto

fnursesrsquoworking

schedu

lewas

constructedby

theL

ingo

program

Nurses

Mon

day

Tuesday

Wednesday

Thursday

Friday

Saturday

Sund

ayTo

talh

ours

Regu

larh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

e1

lowastlowast

lowastlowast

402

lowast

lowast

403

lowast

lowast

lowastlowast

404

lowast

lowast

lowast

405

lowast

lowast

406

lowast

lowastlowast

407

lowastlowast

lowastlowast

408

lowast

lowast

409

lowastlowast

4010

lowast

lowastlowast

40

6 Mathematical Problems in Engineering

Table3Distrib

utionpatie

ntsinterarriv

alandservicetim

e

Day

Departm

ent

Regu

larh

our

Overtim

ehou

rServicetim

e119875value

Avgtim

eInterarrivaltim

e119875value

Interarrivaltim

e119875value

Mon

day

SUR

minus05+EX

PO(852

)0333

minus05+EX

PO(107)

0225

05+WEIB(71118

5)gt0750

684

MED

minus05+EX

PO(539

)0654

minus05+EX

PO(768)

0616

POIS(413

)0051

413

PED

minus05+WEIB(95

1114)

0113

minus05+EX

PO(123)

0075

05+EX

PO(65)

0078

700

EENT

minus05+EX

PO(954)

0478

55+EX

PO(165)

0171

05+EX

PO(458)

0615

508

OBG

minus05+WEIB(146154)

0465

115+EX

PO(225)

0228

TRIA(253195

)0244

910

Tuesday

SUR

minus05+WEIB(10512)

0334

minus05+WEIB(98

70925)

0702

05+18lowastBE

TA(0774125)

0481

740

MED

minus05+WEIB(632133

)0510

minus05+EX

PO(913

)0746

05+LO

GN(412

532

)0109

443

PED

minus05+EX

PO(56)

0607

minus05+ER

LA(4982)

0526

15+EX

PO(87)

006

61020

EENT

minus05+GAMM(663111)

0063

minus05+EX

PO(105)

0057

05+18lowastBE

TA(0409123)

0427

498

OBG

minus05+GAMM(75913)

0145

15+EX

PO(301)

0125

TRIA(25596195)

0421

932

Sund

ay

SUR

minus05+LO

GN(97

126)

gt0750

minus05+EX

PO(14

4)

006

805+17lowastBE

TA(076

106)

0581

737

MED

minus05+GAMM(947116)

0283

15+EX

PO(224)

0105

UNIF(05185)

0589

876

PED

minus05+WEIB(5120841)

0197

minus05+EX

PO(103)

0733

05+EX

PO(45)

0210

500

EENT

minus05+LO

GN(105166)

0349

UNIF(minus05195)

0438

05+19lowastBE

TA(0898156)

0489

744

OBG

minus05+LO

GN(131207)

0673

05+EX

PO(175

)0097

POIS(635)

0126

635

Mathematical Problems in Engineering 7

Service LevelOBG

Process OBG Decide 5

MoreMean 5

LowerMean 5 0

0

True

False

0

0

Dispose 5

Service LevelEENT

Process EENT Decide 4

MoreMean 4

LowerMean 4 0

0

True

False

0

0

Dispose 4

Service LevelPED

Process PED Decide 3

MoreMean 3

LowerMean 3 0

0

True

False

0

0

Dispose 3

Service LevelMED

Process MED Decide 2

MoreMean 2

LowerMean 2 0

0

True

False

0

0

Dispose 2

Service LevelSUR

Process SUR Decide 1

MoreMean 1

LowerMean 1 0

0

True

False

0

0

Dispose 1 IT SUR 8hr 02 01

IT MED 8hr 02 01

IT PED 8hr 02 01

IT EENT 8hr 02 01

IT OBG 8hr 02 01

Figure 1 The simulation model created by the Arena program

8 Mathematical Problems in Engineering

Table 4 The example for adjusting the number of nurses in theArena program

Name Type Capacity1 Resource SUR Fixed capacity 22 Resource MED Fixed capacity 23 Resource PED Fixed capacity 34 Resource EENT Fixed capacity 25 Resource OBG Fixed capacity 1

Table 5 Average and half-width values based on number of nursesadjusted

Output Average Half-widthEENT 04418 035MED 12505 080OBG 34024 164PED 16794 104SUR 44273 125

43 Creating Initial Population Each nursersquos chromosomecan be randomly generated for the initial state Then checkwhether it is in the scope of primary goal or not withworkinghours fewer than 60 hours If that chromosome is not in thescope of the primary goal again random search until eachnursersquos chromosome completes for the entire length and allthe nursersquos chromosomes are generated to reach the numberof population specified

44 Crossover and Mutation Crossover is an importantoperator which combines the good properties of both parentsin order to possibly yield new better children chromosomes[17 18] As usual the simple crossover operator (one-pointcrossover) consists of randomly choosing a crossover pointand then recombining the pieces of a pair of chromosomes toform two new chromosomes The simple crossover is com-patible with the random keys encoding though it generallyfails to preserve the permutation when dealing with naturalencoding Hence for natural encoding special crossoveroperators must be used [19] Roulette wheel selection (RWS)ranking selection (RS) tournament selection (TS) partiallymatched crossover (PMX) order crossover (OX) and cyclecrossover (CX) are evaluated by comparing the performanceof GArsquos operators on university course timetabling problem[20] PMX OX and CX operators require two crossoverpoints Given two parent chromosomes 119860 and 119861 childchromosome1198601015840 will inherit form119860 the subsequence betweenthese two points and child chromosome 1198611015840 form 119861 therespective subsequence The elements of 1198601015840 and 1198611015840 outsidethe two points are copied from the other parent chromosomewhile trying to preserve its position under the PMX operatoror by trying to preserve its order under the OX operatorA comparative analysis of PMX CX and OX crossoveroperators for solving travelling salesman problem (TSP) wasreported in [21]The experimental results show that the PMXcrossover outperforms the CX and OX crossover operator inTSP with 25 numbers of cities For our problem simple testwas conducted and we found PMX provided a better solution

Day

1

1

2

2

3

3

4

4 Nmiddot middot middot

middot middot middot

Mo Tu We Th Fr Sa Su

Number of nurses

Main chromosome

1 1 110 000

Binary code

Level 1

Level 2

Level 3

N lowast 7 lowast 8

Figure 2 The chromosome format

than one-point crossover specifically one-point crossovercannot take into account all constraints (3)ndash(8)

After applying crossover the mutation operator actson the pairs of chromosomes Although mutation occursinfrequently in nature it is believed to be an importantdriving force for evolution The mutation is adopted to allowfor the introduction of new chromosome into the populationand is effective to escape from a local optimum The simplemutation operates by randomly changing its value with agiven probability used in our experiments

45 Evaluation of Fitness Value The fitness value can be eva-luated by using two criteria as follows

Criterion one a chromosome is a feasible solution

(i) Fitness value is the objective function that isprocessed by (2)

Criterion two a chromosome is a nonfeasible solu-tion

(i) Fitness value is equal to 1000 units

46 Chromosome Selection We selected the chromosome byusing roulette

47 Elite Preserve Strategy The previous chromosomes withhigh fitness values were replaced by the new chromosomeswith the lower fitness values Then they were used as theinitial population for the next iteration

48 Termination Criteria The condition to stop seeking an-swers was that when the required number of solutions fullymeets the numbers of iteration specified we stop the exami-nation process immediately

5 Experiment Design to Determine OptimalConfiguration in Genetic Programming

In this section we introduced design of experiment (DOE)technique for screening the effected parameters and param-eters setting guideline for genetic algorithm solution search

Mathematical Problems in Engineering 9

Table 6 Comparison between real situation and simulation model

Department Calculated resultAVG number ofpatients receiving

service

AVG number ofpatients waitinglonger than AVG

time

AVG number ofpatients waiting lessthan AVG time

Percentage of patientswaiting longer than

AVG time

SURReal situation 59 02 58 0338Simulation 6163 plusmn 163 019 plusmn 036 6051 plusmn 166 0196 plusmn 032 error 427 526 433 mdash

MEDReal situation 96 0 98 0Simulation 9879 plusmn 191 0 plusmn 0 9794 plusmn 191 0 error 290 0 006 mdash

PEDReal situation 55 0 53 0Simulation 5758 plusmn 148 0 plusmn 0 5669 plusmn 146 0 error 469 0 696 mdash

EENTReal situation 53 182 34 3434Simulation 5446 plusmn 176 1673 plusmn 414 3664 plusmn 346 1999 plusmn 512 error 275 807 776 mdash

OBGReal situation 34 0 35 0Simulation 384 plusmn 085 0 plusmn 0 377 plusmn 084 0 error 1294 0 771 mdash

Table 7 Demand for nurses obtained from simulation model

Department Period Hospital demand (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 3 3 2 2PED 3 3 3 3 3 3 3EENT 2 2 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 2 2EENT 2 2 2 2 2 2 2OBG 1 1 1 2 1 1 1

machine We conducted the experiment by fixing currentnumber of nurses at 18 with the low level nursesrsquo demand ineach department All three expecting parameters were set bytwo levels maximum and minimum

51 Determining Factors and Factor Values in Each LevelFactors can be chosen for the experiment as follows

(1) population number(2) percent crossover rate(3) percent mutation rate

We used the factors in Table 9 to construct the experi-mental design matrix full factorial design (2119896) (8) and thenconducted the experiment by using three center points perblockwith all the three replicatesWe also fixed the number ofmaximum cycles to 400 cyclesThis was because the researchrevealed that the number of the calculated cycles had high

stability The cycles also took a long time to calculate andso may cause inconvenience in their actual use The numberof nurses 18 in total was fixed at the beginning for theexperimental design and the results are shown in Table 10which includes the use of the data for the demand for nursesfrom the first week of January

52 Effect and Coefficient Analysis and Model AdequacyChecking We used the experimental results to analyze theeffects and coefficients by using theMinitab program in orderto find the factors affecting the expected responses

Figure 3 shows that the data were normally and consis-tently distributed without any tendency and can be used toanalyze further results

Table 11 shows that the correlation of the data is linearbecause the center pt is greater than 005 and the 119875 values ofthe 119860 119862 119860119862 119861119862 and 119860119861119862 factors are less than 005 whichis significant for the experiment It was concluded that the 119860

10 Mathematical Problems in Engineering

Table 8 Example for encoding binary coded decimal

Decimal Binary Regular hour Overtime hour Code Type Demand 1 Demand 2 Hours0 00000000 NDA NOT NDA NOT Type 0 0 0 01 00000001 DA1 NOT DA1 NOT Type 1 DA1++ 0 82 00000010 DA2 NOT DA2 NOT Type 2 DA2++ 0 83 00000011 DA3 NOT DA3 NOT Type 3 DA3++ 0 8

255 11111111 NDA NOT NDA NOT Type 255 0 0 0

001000050000

99

90

50

10

1

Residual054052050048046

0010

0005

0000

Fitted value

8

6

4

2

02624222018161412108642

0010

0005

0000

Observation order

Normal probability plot Versus fits

Histogram Versus order

Residual plots for obj

Resid

ual

Resid

ual

minus0010

minus0010

minus0005

minus0005

minus0010

minus0005

001000050000Residual

minus0010 minus0005

Freq

uenc

y(

)

Figure 3 The residual plots for objective function

Table 9 Selected experimental factors and factor values in eachlevel

Factor LevelLow High

(1) Population number (119860) 10 50(2) Percent crossover rate (119861) 5 95(3) Percent mutation rate (119862) 5 95

and 119862 factors were significant at the confidence level of 005as shown in Figure 4

The 119877-Sq value ranges from 0 to 1 or from 0 to 100If the 119877-Sq value is very close to 1 then it indicates that thesimulation can explain the different variables more properlyand accurately Table 12 shows that the 119877-Sq value is equalto 9948 which is very high This means that the data can

explain dependent variables properly and can analyze furtherresponse optimizer functions

53 Analysis for Finding outMost Appropriate Results by UsingResponse Optimizer The results were analyzed by using aresponse optimizer of Minitab as shown in Figures 5 and 6

From the results obtained as shown in Figure 5 this studyaims to determine the lowest target equation As a result wechanged ldquoGoalrdquo to ldquoMinimizerdquo since the target equation wasthe sum of the hospital expense percentage and the standarddeviation of the overtime each nurse received which rangedfrom 0 to 100 As a result the target was set to 0 and the uppertarget to 100

Figure 6 shows that if we want to configure the geneticprogram to obtain the lowest target equation it must beset as shown in Table 13 From the prediction equation thetarget equation should be equal to 04492 (obj = 04492)

Mathematical Problems in Engineering 11

Table 10 Full factorial design (23) experiment with three center points per block type

Run Std order Run order Center pt Blocks 119860 119861 119862 Obj Time (s)1 1 1 1 1 10 5 5 045892 92375212 2 2 1 1 50 5 5 050282 576219023 3 3 1 1 10 95 5 047113 7904851

25 25 25 0 1 30 50 50 048018 31281694

Table 11 Effect and obj coefficient analysis results

Factorial fit obj versus 119860 119861 119862Estimated effects and coefficients for obj (coded units)

Term Effect Coeff SE coeff 119879 119875

Constant 048216 0001318 36575 0000119860 005200 002600 0001318 1972 0000119861 000027 000013 0001318 010 0921119862 minus002134 minus001067 0001318 minus810 0000119860 lowast 119861 minus000285 minus000143 0001318 minus108 0293119860 lowast 119862 minus000850 minus000425 0001318 minus322 0005119861 lowast 119862 minus001395 minus000697 0001318 minus529 0000119860 lowast 119861 lowast 119862 minus000974 minus000487 0001318 minus369 0002Ct Pt minus000198 0003955 minus050 0623

Table 12 119877-square decision-making coefficients

119878 = 208209 PRESS = 152191461119877-Sq = 9948 119877-Sq (pred) = 9899 119877-Sq (adj) = 9925

Table 13 Guideline setting for genetic program from responseoptimizer function

Factor Level(1) Population number (119860) 10(2) Percent crossover rate (119861) 95(3) Percent mutation rate (119862) 95

Possible optimal parameters settingmay occur at the low level mutation we conducted the experiment and reported inTable 14 The reports showed that the objective function isworse when mutation decreased The reports also shownthat the higher mutation the better objective functions

54 Result Confirmation Experiment We experimented tocheck for accuracy by configuring the genetic program asshown in Table 12 together with fixing the maximum itera-tions to the number of nurses at 400 cycles and 18 nursesrespectively as shown in Table 15

Table 15 shows that the actual target value (obj) is actuallybetter than (ie less than) 063 of the predicted value andcan be used practically and actually decreased the target equa-tion (obj)

20151050

99

95908070605040302010

5

1

Standardized effect

Factor NameNot significantSignificant

Effect type

Normal plot of the standardized effects

minus10 minus5

(response is obj 120572 = 005)

()

A

A

B

C

A

B

C

C

BC

ABC

AC

Figure 4 The normal plot of the standardized effects (response isobj alpha = 005)

6 Change in Number of Nurses (119894) in OptimalScheduling under Constraint Equation

We changed the number of nurses (119894) which was an indexin the constraint equation in order to find out the least

12 Mathematical Problems in Engineering

Table 14 Experimental results obtained at low level mutation

Max iteration Population crossover mutation Obj Time (s)1500 10 95 95 044442 80491500 10 95 1 048580 101611500 10 95 01 050286 105231500 10 95 001 053612 10117

Table 15 Target equation (obj) predicted by Minitab program and compared with actual target equation (obj)

Target equation (obj) Comparison of target equation (obj)percentage

Predicted value Actual value Actual value compared with predicted value04492 04429 063

Table 16 Calculation results with reduction in number of nurses (119894)at 119908 = 06

Number of nurses (119894) Target equation(obj)

Calculating time(hour)

21 05241 76720 03959 60319 Infeasible Infeasible18 Infeasible Infeasible

Figure 5 The response optimizer configuration of obj response

number of nurses (119894) who were able to schedule the workunder the restrictions by using genetic programming and bygradually reducing the number of nurses (119894) down to the finalvalue at which the program could calculate the answer Thecalculation was performed at 119908 = 06 in Table 16

Table 16 shows that the least number of nurses (119894) whichcould schedule work under the restrictions was 20 Basedon 20 nurses amount of overtime payment and standarddeviation of overtime payment were calculated and reportedin Table 17 Therefore number of working hours for eachnurse number of nurses in each department and nursersquosschedule planwere generated and reported as shown inTables18 and 19

From Table 17 it can be observed that the target equation(obj) of the responses obtained was equal to 03959The sumof the overtime payment of all nurses was 16600 Baht per

CurHigh

Low099551

Optimal

minimumObj

099551desirabilityComposite

50950

50950

100500

50040004000 100 950 950

y = 04492

d = 099551

A B C D

D

Figure 6 The results of the data analysis using the responseoptimizer of the obj response

week The standard deviation of overtime payment that eachnurse received was 214843 For fair comparison we have toconvert the real overtime payment and standard deviationfor current 18 nurses to 20 nurses Then we can report thatthe sum of the total overtime payment and the standarddeviation of the overtime payment that each nurse receivedwhich is obtained from the model were less than those of theold working schedule at 17991 Baht per week and 882369respectively

Table 18 shows that all the nurses worked according to theconditions specified that is one nurse must work at least 40hours per week as in (4) but must work fewer than 60 hoursper week as in (5) all would be selected to work as in (6)and the duration of work must not exceed six days a week asin (7)

From Table 19 we can see that the number of nursesworking in shifts each day was more than the demand fornurses (Demand

119895119896119897) as shown in Table 6 which is according

to (3) and the specified service criteria (on average 15of thepatients waiting for services should not wait longer than theaverage service time of 25) because the number of nursesworking in shifts was more than the demand

To study the effect ofweight adjusted (119908)more numericalexperiments were conducted by GA coded in Matlab at 400

Mathematical Problems in Engineering 13

Table 17 Calculating results by 20 nurses (119894)

Target equation(obj)

Calculating time(hour)

Sum of overtime paymentof nurses (Baht)

Standard deviation of total overtimepayment that each nurse received

03959 603 16600 214843

Table 18 Number of working hours calculated from 20 nurses (119894)

Number of nurses Hoursperson (hour)Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total

1 12 12 0 4 0 8 12 482 4 8 12 4 8 12 0 483 8 8 8 0 12 12 0 484 12 0 12 12 12 4 0 525 8 12 8 4 0 0 12 44

20 8 12 4 0 12 8 4 48

Table 19 Number of nurses working in shifts calculated by 20 nurses (119894)

Department Period Supply (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 4 3 2 2PED 3 3 4 3 3 4 3EENT 2 3 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 3 3EENT 3 2 2 2 2 2 2OBG 2 2 3 2 1 1 2

Table 20 Effect of weight adjusted to objective function

Weight adjusted119908 = 09 119908 = 06 119908 = 05

Objective function 05090 04753 04823Sum of overtimepayment 24375 24625 26575

Standard deviation 324898 297675 288509

iterations and reported in Table 20 The report was shownas we expected more weight adjusted less overtime paymentsimultaneously with higher standard deviation It means thatif management focuses on reducing the overtime payment itwill lead to unfair payment or bias nursesrsquo schedule plan

7 Comparison of Advantages andDisadvantages of Adaptive Genetic andOptimization Approach

Weprocessed the data to do a comparison between the resultsof the adaptive genetic approach and the results obtained by

the Lingo program using the same data in order to comparethe target equation (obj) and computation runtime as shownin Tables 21 and 22

Table 21 shows the results of the processing experimentafter it was conducted ten times The work schedules using20 and 24 nurses with the calculating cycle of 400 roundsresulted in the best value for the target equations of 04361and 04551 respectively After ten processing experimentsthe work schedules for 20 and 24 nurses with the calculatingcycle at 1500 rounds resulted in the best value for the targetequations of 03959 and 04493 respectively and can besummarized as presented in Table 22

Table 22 shows that the work schedule for 20 nurses at1500 rounds when compared to the Lingo program couldcalculate thework schedulewith the target equation (obj) lessthan the genetic algorithm by around 580 and 108 plusmn 24on average but that the one created by the Lingo programused longer computational runtime by over 86849 and76452 on average We can also notice that the results at400 rounds reported the same direction the best and theaverage solution of genetic algorithm differ about 982 and1497 plusmn 38 from the optimal respectively However theoptimal method used longer time by over 306257 and

14 Mathematical Problems in Engineering

Table 21 Results of genetic program calculated ten times at 400 rounds and 1500 rounds

Target equation (obj)number

Number of 20 nurses Number of 24 nursesCalculating cycle

400Calculating cycle

1500Calculating cycle

400Calculating cycle

15001 04701 04578 05049 047302 04361 04436 05406 045523 04566 04772 04789 044934 04812 04303 04590 051605 04496 04781 04942 045866 04920 04549 05383 048337 04779 03959 05228 046268 05428 04352 05133 045809 05472 04368 05497 0498510 05231 04496 04551 04542Mean 04876 04459 05056 04708Standard deviation 0038 0024 0033 0021

Table 22 Comparison table for target equation (obj) and duration in calculation of GA and Lingo program

Number ofnurses(person)

Number ofcalculatingcycles

GA Lingo

Target equation (obj) Time (hour) Targetequation (obj) Time (hour) Different result

of obj

Different incomputational

run time

201500 Lowest 03959 603 580 86849

Mean 04459 plusmn 0024 685 03379 5237 1080 plusmn 24 76453

400 Lowest 04361 171 982 306257Mean 04876 plusmn 0038 166 1497 plusmn 38 315482

Table 23 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2012

Number of nurses Monthly expense SDOld New Increase Old New Decrease Old New Decrease18 20 2 395964 366400 29564 1089212 214843 8028

Table 24 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2013

Number of nurses Monthly expense SDOld New Decrease Old New Decrease Old New Decrease23 20 3 416548 366400 50148 1966735 214843 8908

315481 on average showing that the genetic program ismore suitable for practical use

8 Conclusion

The findings showed that the genetic program created couldmanage the work schedule of nurses under given conditionsand achieve the desired objective It could create work sche-dules of nurses that were fair in overtime payment to all thenurses and could reduce the hospital expenses by setting thenew coming nursesrsquo minimum salary at 15000 Baht per per-son

Table 25 Target equation and calculation time from geneticprogram

Target equation (obj) Calculation time (hour)03959 603

From Tables 23 24 and 25 the following results canbe concluded the work schedules from proposed methodreduce the waiting time of the number of patients waitinglonger than 125 times of the average length nursing servicefrom 3408 to less than 15 on average (starting from

Mathematical Problems in Engineering 15

when the patients register and wait to see the doctor) Ifwe want to schedule the work of the nurses in January 2012in order to meet the criteria for the service of this studywe have to increase the number of nurses from 18 to 20 sothat the number can perfectly meet the needs of the patientsaccording to the criteriaThework schedules of the nurses canbemost appropriate if the target equation is 03959which usesthe time of calculation as 603 hourswith a standard deviationof less than 8028and the overtimepayment less than 29564Baht per monthWhen this is compared with the actual workschedule in January 2013 the work schedule could reduce thenumber of nurses from 23 to 20 with a decreased standarddeviation that is 8908 and the decrease in the totalovertime paymentwould be 50148 Baht permonthWe couldcalculate the most appropriate work schedule quicker thanby using the conventional approach which results obtainedby Lingo program version 50 86849 with 1500 rounds forthe calculation cycle

As we know hospitals belong to the service businesssector continuous improvement along with response to cus-tomer satisfaction is the key success factor in this businessTo reduce the waiting time in every hospital process highstandards in medical care and highly experienced medicaldoctors are needed Presently Thailand has set its aim tobecome an Asian medical and tourist hub this strategy of thecountry may cause and force hospitals to improve their ser-vice quality for responding well to the high number of outpa-tients Updated data collection needs to be done service-level policies have to be reconsidered Demands of outpa-tient nurses and their schedules may change from time totime Inpatient nurses may be allocated to assist when themanagement wishes to reduce the fluctuation in the numberof outpatient nurses Thus modern management and highefficient tools are necessary this is the reason why GA isinvolved instead of an adoption of the conventional approachHowever regarding the implementation of simulation evenGA is not familiar to the hospital people Optimization capac-ity building is another issue that the hospitalmanagementwillhave to consider

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chiang Mai University (CMU) through the research admin-istration office provided budget to our Excellence Center inLogistics and Supply Chain Management (E-LSCM)

References

[1] T Mongkolrangsarit and W Thamapornpilat Development ofNurse Scheduling Method with Uncertain Workforce DemandDepartment of Industrial Engineering Faculty of EngineeringChulalongkorn University 2007

[2] C M Rothe and H B Wolfe ldquoCyclical scheduling and alloca-tion of nursing staffrdquo Socio-Economic Planning Sciences vol 7no 5 pp 471ndash487 1973

[3] R B Norby L E Freund and B Wagner ldquoA nurse staffingsystem based upon assignment difficultyrdquo Journal of NursingAdministration vol 7 no 9 pp 2ndash24 1977

[4] T Ryan B L Barker and F A Marciante ldquoA system fordetermining appropriate nurse staffingrdquo Journal of NursingAdministration vol 5 no 5 pp 30ndash38 1975

[5] H Wolfe and J P Young ldquoStaffing the nursing unit Part Icontrolled variable staffingrdquo Nursing Research vol 14 pp 236ndash243 1965

[6] HWolfe and J P Young ldquoStaffing the nursing unit IIThemult-iple assignment techniquerdquo Nursing Research vol 14 no 4 pp299ndash303 1965

[7] J L Arthur andA Ravindran ldquoAmultiple objective nurse sche-duling modelrdquo AIIE Transactions vol 13 no 1 pp 55ndash60 1981

[8] I Ozkarahan ldquoA disaggregation model of a flexible nursescheduling support systemrdquo Socio-Economic Planning Sciencesvol 25 no 1 pp 9ndash26 1991

[9] I Ozkarahan and J E Bailey ldquoGoal programming modelsubsystem of a flexible nurse scheduling support systemrdquo IIETransactions vol 20 no 3 pp 306ndash316 1988

[10] K A Dowsland ldquoNurse scheduling with tabu search and stra-tegic oscillationrdquo European Journal of Operational Research vol106 no 2-3 pp 393ndash407 1998

[11] P Tormos A Lova F Barber L Ingolotti M Abril and M ASalido ldquoA genetic algorithm for railway scheduling problemsrdquoinMetaheuristics for Scheduling in Industrial andManufacturingApplications vol 128 of Studies in Computational Intelligencepp 255ndash276 Springer Berlin Germany 2008

[12] H Kawanaka K Yamamoto T Yoshikawa T Shinogi and STsuruoka ldquoGenetic algorithm with the constraints for nursescheduling problemrdquo in Proceedings of the Congress on Evolu-tionary Computation vol 2 pp 1123ndash1130 Seoul Republic ofKorea May 2001

[13] H HMillar and KMona ldquoCyclic and non-cyclic scheduling of12 h shift nurses by network programmingrdquo European Journalof Operational Research vol 104 no 3 pp 582ndash592 1998

[14] I Blochlige ldquoModeling staff scheduling problems A tutorialrdquoEuropean Journal of Operational Research vol 158 no 3 pp533ndash542 2004

[15] B Jaumard F Semet and T Vovor ldquoA generalized linear pro-gramming model for nurse schedulingrdquo European Journal ofOperational Research vol 107 no 1 pp 1ndash18 1998

[16] T Cezik O Gunluk and H Luss ldquoAn integer programmingmodel for the weekly tour scheduling problemrdquoNaval ResearchLogistics vol 48 no 7 pp 607ndash624 2001

[17] Z Wang J-L Liu and X Yu ldquoSelf-fertilization based geneticalgorithm for university timetabling problemrdquo in Proceedingsof the 1st ACMSIGEVO Summit on Genetic and EvolutionaryComputation (GEC rsquo09) pp 1001ndash1004 ACM Shanghai ChinaJune 2009

[18] J Wang and K Chu ldquoAn application of genetic algorithms forthe flexible job-shop scheduling problemrdquo International Journalof Advancements in Computing Technology vol 4 no 3 pp 271ndash278 2012

[19] M Moz and M V Pato ldquoA genetic algorithm approach to anurse rerostering problemrdquo Computers amp Operations Researchvol 34 no 3 pp 667ndash691 2007

[20] C Wotthipong K Soradech and S Nidapan ldquoThe suitablegenetic operators for solving the university course timetablingproblemrdquo Journal of Convergence Information Technology vol8 pp 60ndash65 2013

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

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Page 2: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

2 Mathematical Problems in Engineering

In this paper we intended to solve nurse schedulingproblem by combining mathematical model computer sim-ulation and GA Nonlinear integer model was formulatedThe average total overtime payment and fairly payment toall the nurses were considered to minimize which maintainsthe standard of service level When mathematical modelsare applied to solve the problems the objective functionbecomes the sum of the overtime payment of all the nursesand the standard deviation of the total overtime paymentthat each nurse received This study aimed to create themost appropriate nurse scheduling model by applying thesimulation technique to analyze the distribution of the dataThereafter a simulation model was created to determine theappropriate demand for nurses with no more than 15 ofpatients waiting longer than the average service time as wellas those with more than 25 of patients waiting and withthe application of GA coding in Matlab program to calculatethe best work schedule from the most appropriate demandfor nurses acquired from the simulation Then the expensesand the number of nurses from the new work schedule werecompared with the actual expenses of the work schedule andthe previous number of nurses

2 Problem Analysis and Formulation

21 Cause and Problem Analysis of Nurse Scheduling Cur-rently one hospital is using manpower planning for nurseschedulingThis method involves workload and productivityefficiency measure per hour of work on average by having20 nurses in the outpatient department who are rotated forwork in the following five departments (1) department ofsurgery (SUR) (2) department of internal medicine (MED)(3) department of eye ear nose and throat (EENT) (4)department of pediatrics (PED) and (5) department ofobstetrics and gynecology (OBG) There are two types ofnursesrsquo working schedules 8 hours per day (0800ndash1600hours) and 12 hours per day (800ndash2000 hours) with fixedworkload each day and the working time cannot be changedat this stage This causes problems while allocating theholidays in order to meet the needs of the nurses [13 14]

Nurse scheduling at this hospital uses only one formatwhich assigns the same number of nurses to take care ofeach department every weekThis format of scheduling is notconsistent with the number of patients treated each day anddoes not bring into consideration factors related to servicelevels and queuing systems in nurse scheduling It is notpossible to allocate similar rates of overtime payment forall nurses The researcher emphasizes the importance of anappropriate and effective nurse scheduling model for ensur-ing the least cost to the hospital while maintaining the levelof service of the hospital and being fair in paying overtimeto all the nurses at similar rates by using simulation andoptimization method to create the most appropriate nursescheduling

22 Mathematical Modeling and Demand Function We cre-ated the objective and the limitation equations under variousconditions based on the information obtained from the hos-pital by developing an integer programming mathematical

model with assignment problem [15 16] The preliminarydata were set in the following conditions

119909119894119895119896119897

=

1 when nurse 119894 works on date 119895at time 119896 in department 119897

0 otherwise(1)

where 119894 = 1 2 3 20 (nurse) 119895 = 1 2 3 7 (sevenworking days Monday Tuesday Wednesday Sunday)119896 = 1 2 (two working shifts regular working hours (0800ndash1600 hours) and overtime hours (1600ndash2000 hours)) 119897 =1 2 3 4 5 (department of surgery (SUR) department ofinternal medicine (MED) department of eye ear noseand throat (EENT) department of pediatrics (PED) anddepartment of obstetrics and gynecology (OBG)) 119873 = totalnumber of nurses SOT = sum of all the overtime paymentsthat nurses receive SD = standard deviation of the totalovertime payment that each nurse receives AVG = averageof total overtime payment that each nurse receives OTD

119894119895119897

= additional overtime payment that nurse 119894 receives onthe working date 119895 in the department 119897 OTW

119894= overtime

payment that nurse 119894 receives in a week OTS119894= total overtime

payment that nurse 119894 receives and Demand119895119896119897

= demand fornurses on date 119895 at time 119896 in the department 119897

The objective equation includes two purposes

(1) The expenses of the hospital are minimal

(2) The standard deviations of the total overtime pay-ments of the nurses are the most similar

Doing a calculation to find the answers for the problempatterns with multiple objectives would be highly compli-cated and it would be difficult to find an answer To makeit easier to calculate two equations were combined togetherHowever since both the equations are in different units toinclude the two together it is necessary to change them intothe same unit by making these two equations in the form ofa percentage of the total Weighting the importance of thetarget functions to119908 percent and (1minus119908) percent respectivelywillmake up the sumof the hospital cost percentage equationand the standard deviation of the total overtime payment thateach nurse receives can be estimated as follows

Min119885 = 119908 times (SOT

AvgSOT) + (1 minus 119908) times (

SDAvgSD

) (2)

where 119908 = weight adjusted AvgSOT = sum of average totalovertime payment that every nurse receives in that particularmonth and AvgSD = standard deviation of the total overtimepayment that each nurse receives in that particular month

Constraints equation includes the following

(1) The number of nurses must be adequate to meet theneeds of the patients in each working period and day

119873

sum119894=1

119909119894119895119896119897

ge Demand119895119896119897

forall119895forall119896forall119897 (3)

Mathematical Problems in Engineering 3

(2) One nurse must work at least 40 hours per week (4)but fewer than 60 hours per week (5)

8 times

5

sum119897=1

7

sum119895=1

1199091198941198951119897

+ 4 times

5

sum119897=1

7

sum119895=1

1199091198941198952119897

ge 40 forall119894 (4)

8 times

5

sum119897=1

7

sum119895=1

1199091198941198951119897

+ 4 times

5

sum119897=1

7

sum119895=1

1199091198941198952119897

le 60 forall119894 (5)

(3) All nurses must be selected to work (6) and work notover six days per week (7)

7

sum119895=1

119909119894119895119896119897

ge 1 forall119894forall119896forall119897 (6)

7

sum119895=1

119909119894119895119896119897

le 6 forall119894forall119896forall119897 (7)

(4) In a day one nurse can work only one period in onedepartment

11990911989411989511

+ 11990911989411989512

+ 11990911989411989513

+ 11990911989411989514

+ 11990911989411989515

le 1 forall119894forall119895

11990911989411989521

+ 11990911989411989522

+ 11990911989411989523

+ 11990911989411989524

+ 11990911989411989525

le 1 forall119894forall119895

(8)

Equations related to expenses are as followsWhen nurses work formore than 40 hours in a week they

will receive an overtime payment of 650 Baht per eight hoursor 8125 Baht per hour and if within 1600ndash2000 hours theywill receive 50 Baht extra overtime payment per working day

OTS119894= 8125 times [

[

(8 times

5

sum119897=1

7

sum119895=1

1199091198941198951119897

+ 4 times

5

sum119897=1

7

sum119895=1

1199091198941198952119897) minus 40]

]

+

7

sum119895=1

(50 times

5

sum119897=1

1199091198941198952119897) forall

119894

AVG =sum119873

119894=1OTS119894

119873

SOT =

119873

sum119894=1

OTS119894

SD = radicsum119873

119894=1(OTS119894minus AVG)2

119873

(9)

After that the least number of nurses (119894) is determined inorder to be scheduled under existing limitations

Proposed mathematical model with objective function(2) subject to constraints (3)ndash(9) was verified by simpleproblem as shown in Table 1 With the conventional exactsolution approach the nurse scheduling was constructed fortwo departments SUR and MED and reported by Ganttchart as shown in Table 2 In Table 2 lowast and represent theschedule of SUR and MED respectively Lingo software alsoreported that the mathematical model is nonlinear modelwith integer variables

3 Building Simulation to Find AppropriateDemand for Nurses

In this case since a service criterion system has not been inuse in the hospital the demand for nurses as in the presentmay not be an appropriate demand for finding the leastcost work schedule The researcher examined an appropriatedemand for nurses (Demand

119895119896119897) by creating a simulation

using the Arena program to determine the appropriatedemand for nurses in each department and in all the periodsof each day from the original hospital data It was found thaton average 3408 or 5487 patients per week had waitedlonger than 25 of the total average service time Thereforethe researcher aimed to bring about an improvement at theinitial state itself by creating a simulation under the servicecriteria that set the limitation as only 15 of the patients onaverage did not have to wait longer than 25 of the totalaverage service time

31 Data Collection of Patient Service and Nursing ServiceDuration For the data collection of each department at eachperiod of each day (starting from when the patients registeruntil they wait to see a doctor) the researcher chose to use thedata as each day in the month to represent day in that monthCollecting data were done by the quality assurance sectionin case study hospital they defined the outpatient based onthe combination of appointments and walk-ins Interarrivaltime for regular and overtime hour including service timehas been collected for two years in winter season which hasthe largest number of outpatients The interarrival time forregular and overtime periods were separately collected dueto significant static data However no significant differencewas noticed for service timeThen the data were classified byday in a week and analyzed for their statistics distribution forthree categories

32 Data Distribution Analysis The data were analyzed byusing a suitable dispersion input analyzer in the Arenaprogram and by selecting the distribution with the highest 119875value using the significance level (120572) at 005 (95 confidence)as shown in Table 3 The 119875 value is a key concept in theapproach of Ronald Fisher where he uses it to measurethe weight of the data against a specified hypothesis and asa guideline to ignore data that does not reach a specifiedsignificance level

33 Simulation Modeling We created a simulation modelingstarting from when the patient registered at the servicedepartment based on interarrival time and service timestatistics distribution reported in Table 3 Because thesestatistics distributions consider the arrival of appointmentand walk-in patients then this simulation model can takeinto account the dynamics of walk-in patients Beginningwith statistics distributions patientsrsquo arrival patients hadregistered at the service department According to patientrsquossymptom they had moved to required department counterand waited for medical history file To parallel with filesearching and coming patients had queued and waited fornurse service Nurses served the fundamental examination

4 Mathematical Problems in Engineering

Table 1 Demand for nurses verify case study

Department Period Hospital demand (person)Mon Tue Wed Thu Fri Sat Sun

SUR Regular 1 1 2 1 2 1 2Overtime 1 2 1 2 1 1 1

MED Regular 2 1 2 1 1 1 1Overtime 1 1 1 2 1 2 2

such as weight blood pressure temperature measure pre-symptom examined and finally recorded all measure data inthe file after it arrived Service time distributions recordedand fitted by statistic approach in Table 3 were used as thedata set in process box After that we saved the percentageinformation of those patients whose waiting time exceededthe service time limitation as shown in Figure 1

Then we adjusted the number of nurses (resource) inthe capacity column manually as shown in Table 4 until thepercentage of the patients waiting for more than the limit(average and half-width) in each department was less than15 according to what we wanted in every section as shownin Table 5 After that we processed 100 replications

From Table 4 it can be observed that the departments ofSURMED and EENT appointed two people PED appointedthree people andOBG appointed one person to work in theirdepartments

From Table 5 we can see that all the departments withthe percentage of patients waiting for more than the limit(average and half-width) do not exceed 15 The EENTMED OBG PED and SUR departments had the percentagesof patients waiting for longer than the limit (average pluspositive half-width) at 07918 20505 50424 27194and 56773 respectively We considered only the positivehalf-width because we would like to know the maximumpercentage of patients waiting

34 Simulation Model Verification In model verification wecompared the results between real situation which occurredin this hospital and the simulationmodel Average number ofpatients receiving service average number of patients waitinglonger than average service time and average number ofpatients waiting less than average service time were proposedfor preforming simulationmodel efficientThe 100 times witheight-hour simulation were conducted for all five outpatientsdepartments The results were reported in Table 6 In simu-lation row we reported the average number plusminus withtheir corresponding half-width

Table 6 reported that the OBG got the highest error interms of average number of patients receiving service due tothe low 119875 value reported in Table 3 With this error howeverwe can adopt this simulation model generated by the arenaprogram for real case application

After model verification manual adjustment of theresource capacity was done and selected under hospital topmanagement service policy The service level policy was setas no more than 15 of patients waiting longer than the

average service timeThen the appropriate resource (numberof nurses) from simulation was shown for each departmentin Table 7 This number of nurses will be used as the datafor nurses demand (Demand

119895119896119897) in the mathematical model

for obtaining the optimal nurse schedule Even thoughto obtain the optimal nurse schedule with conventionalapproach always consumes computational runtime and hard-ware implementation Genetic algorithm was proposed tosolve and obtain an acceptable solution

4 Programming for Genetic AlgorithmNurse Scheduling

We coded a genetic algorithm program for use in the shiftscheduling of nurses with the Matlab program by using thefollowing steps

41 Chromosome Format and Chromosome Encoding Achromosome format uses a binary number system and it isdivided into three levels as shown in Figure 2 where

level 1 is a nurse at 1 2 3 119873level 2 is the working daysMonday TuesdayWednes-day Sundaylevel 3 is the number of 8 bit code

42 Chromosome Encoding and Chromosome Decoding Toencode the chromosome we used the binary coded decimalsystem which is a system that uses the 8 bit binary codeinstead of the general code we have set up We can encodeand decode the chromosome as shown in Table 8 where

(i) NDA = not working during regular hours(ii) DA1 = working during regular hours at Division 1(iii) DA2 = working during regular hours at Division 2(iv) DA3 = working during regular hours at Division 3(v) DA4 = working during regular hours at Division 4(vi) DA5 = working during regular hours at Division 5(vii) NOT = not working overtime(viii) OT1 = working overtime at Division 1(ix) OT2 = working overtime at Division 2(x) OT3 = working overtime at Division 3(xi) OT4 = working overtime at Division 4(xii) OT5 = working overtime at Division 5

Mathematical Problems in Engineering 5

Table2Th

eGanttcharto

fnursesrsquoworking

schedu

lewas

constructedby

theL

ingo

program

Nurses

Mon

day

Tuesday

Wednesday

Thursday

Friday

Saturday

Sund

ayTo

talh

ours

Regu

larh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

e1

lowastlowast

lowastlowast

402

lowast

lowast

403

lowast

lowast

lowastlowast

404

lowast

lowast

lowast

405

lowast

lowast

406

lowast

lowastlowast

407

lowastlowast

lowastlowast

408

lowast

lowast

409

lowastlowast

4010

lowast

lowastlowast

40

6 Mathematical Problems in Engineering

Table3Distrib

utionpatie

ntsinterarriv

alandservicetim

e

Day

Departm

ent

Regu

larh

our

Overtim

ehou

rServicetim

e119875value

Avgtim

eInterarrivaltim

e119875value

Interarrivaltim

e119875value

Mon

day

SUR

minus05+EX

PO(852

)0333

minus05+EX

PO(107)

0225

05+WEIB(71118

5)gt0750

684

MED

minus05+EX

PO(539

)0654

minus05+EX

PO(768)

0616

POIS(413

)0051

413

PED

minus05+WEIB(95

1114)

0113

minus05+EX

PO(123)

0075

05+EX

PO(65)

0078

700

EENT

minus05+EX

PO(954)

0478

55+EX

PO(165)

0171

05+EX

PO(458)

0615

508

OBG

minus05+WEIB(146154)

0465

115+EX

PO(225)

0228

TRIA(253195

)0244

910

Tuesday

SUR

minus05+WEIB(10512)

0334

minus05+WEIB(98

70925)

0702

05+18lowastBE

TA(0774125)

0481

740

MED

minus05+WEIB(632133

)0510

minus05+EX

PO(913

)0746

05+LO

GN(412

532

)0109

443

PED

minus05+EX

PO(56)

0607

minus05+ER

LA(4982)

0526

15+EX

PO(87)

006

61020

EENT

minus05+GAMM(663111)

0063

minus05+EX

PO(105)

0057

05+18lowastBE

TA(0409123)

0427

498

OBG

minus05+GAMM(75913)

0145

15+EX

PO(301)

0125

TRIA(25596195)

0421

932

Sund

ay

SUR

minus05+LO

GN(97

126)

gt0750

minus05+EX

PO(14

4)

006

805+17lowastBE

TA(076

106)

0581

737

MED

minus05+GAMM(947116)

0283

15+EX

PO(224)

0105

UNIF(05185)

0589

876

PED

minus05+WEIB(5120841)

0197

minus05+EX

PO(103)

0733

05+EX

PO(45)

0210

500

EENT

minus05+LO

GN(105166)

0349

UNIF(minus05195)

0438

05+19lowastBE

TA(0898156)

0489

744

OBG

minus05+LO

GN(131207)

0673

05+EX

PO(175

)0097

POIS(635)

0126

635

Mathematical Problems in Engineering 7

Service LevelOBG

Process OBG Decide 5

MoreMean 5

LowerMean 5 0

0

True

False

0

0

Dispose 5

Service LevelEENT

Process EENT Decide 4

MoreMean 4

LowerMean 4 0

0

True

False

0

0

Dispose 4

Service LevelPED

Process PED Decide 3

MoreMean 3

LowerMean 3 0

0

True

False

0

0

Dispose 3

Service LevelMED

Process MED Decide 2

MoreMean 2

LowerMean 2 0

0

True

False

0

0

Dispose 2

Service LevelSUR

Process SUR Decide 1

MoreMean 1

LowerMean 1 0

0

True

False

0

0

Dispose 1 IT SUR 8hr 02 01

IT MED 8hr 02 01

IT PED 8hr 02 01

IT EENT 8hr 02 01

IT OBG 8hr 02 01

Figure 1 The simulation model created by the Arena program

8 Mathematical Problems in Engineering

Table 4 The example for adjusting the number of nurses in theArena program

Name Type Capacity1 Resource SUR Fixed capacity 22 Resource MED Fixed capacity 23 Resource PED Fixed capacity 34 Resource EENT Fixed capacity 25 Resource OBG Fixed capacity 1

Table 5 Average and half-width values based on number of nursesadjusted

Output Average Half-widthEENT 04418 035MED 12505 080OBG 34024 164PED 16794 104SUR 44273 125

43 Creating Initial Population Each nursersquos chromosomecan be randomly generated for the initial state Then checkwhether it is in the scope of primary goal or not withworkinghours fewer than 60 hours If that chromosome is not in thescope of the primary goal again random search until eachnursersquos chromosome completes for the entire length and allthe nursersquos chromosomes are generated to reach the numberof population specified

44 Crossover and Mutation Crossover is an importantoperator which combines the good properties of both parentsin order to possibly yield new better children chromosomes[17 18] As usual the simple crossover operator (one-pointcrossover) consists of randomly choosing a crossover pointand then recombining the pieces of a pair of chromosomes toform two new chromosomes The simple crossover is com-patible with the random keys encoding though it generallyfails to preserve the permutation when dealing with naturalencoding Hence for natural encoding special crossoveroperators must be used [19] Roulette wheel selection (RWS)ranking selection (RS) tournament selection (TS) partiallymatched crossover (PMX) order crossover (OX) and cyclecrossover (CX) are evaluated by comparing the performanceof GArsquos operators on university course timetabling problem[20] PMX OX and CX operators require two crossoverpoints Given two parent chromosomes 119860 and 119861 childchromosome1198601015840 will inherit form119860 the subsequence betweenthese two points and child chromosome 1198611015840 form 119861 therespective subsequence The elements of 1198601015840 and 1198611015840 outsidethe two points are copied from the other parent chromosomewhile trying to preserve its position under the PMX operatoror by trying to preserve its order under the OX operatorA comparative analysis of PMX CX and OX crossoveroperators for solving travelling salesman problem (TSP) wasreported in [21]The experimental results show that the PMXcrossover outperforms the CX and OX crossover operator inTSP with 25 numbers of cities For our problem simple testwas conducted and we found PMX provided a better solution

Day

1

1

2

2

3

3

4

4 Nmiddot middot middot

middot middot middot

Mo Tu We Th Fr Sa Su

Number of nurses

Main chromosome

1 1 110 000

Binary code

Level 1

Level 2

Level 3

N lowast 7 lowast 8

Figure 2 The chromosome format

than one-point crossover specifically one-point crossovercannot take into account all constraints (3)ndash(8)

After applying crossover the mutation operator actson the pairs of chromosomes Although mutation occursinfrequently in nature it is believed to be an importantdriving force for evolution The mutation is adopted to allowfor the introduction of new chromosome into the populationand is effective to escape from a local optimum The simplemutation operates by randomly changing its value with agiven probability used in our experiments

45 Evaluation of Fitness Value The fitness value can be eva-luated by using two criteria as follows

Criterion one a chromosome is a feasible solution

(i) Fitness value is the objective function that isprocessed by (2)

Criterion two a chromosome is a nonfeasible solu-tion

(i) Fitness value is equal to 1000 units

46 Chromosome Selection We selected the chromosome byusing roulette

47 Elite Preserve Strategy The previous chromosomes withhigh fitness values were replaced by the new chromosomeswith the lower fitness values Then they were used as theinitial population for the next iteration

48 Termination Criteria The condition to stop seeking an-swers was that when the required number of solutions fullymeets the numbers of iteration specified we stop the exami-nation process immediately

5 Experiment Design to Determine OptimalConfiguration in Genetic Programming

In this section we introduced design of experiment (DOE)technique for screening the effected parameters and param-eters setting guideline for genetic algorithm solution search

Mathematical Problems in Engineering 9

Table 6 Comparison between real situation and simulation model

Department Calculated resultAVG number ofpatients receiving

service

AVG number ofpatients waitinglonger than AVG

time

AVG number ofpatients waiting lessthan AVG time

Percentage of patientswaiting longer than

AVG time

SURReal situation 59 02 58 0338Simulation 6163 plusmn 163 019 plusmn 036 6051 plusmn 166 0196 plusmn 032 error 427 526 433 mdash

MEDReal situation 96 0 98 0Simulation 9879 plusmn 191 0 plusmn 0 9794 plusmn 191 0 error 290 0 006 mdash

PEDReal situation 55 0 53 0Simulation 5758 plusmn 148 0 plusmn 0 5669 plusmn 146 0 error 469 0 696 mdash

EENTReal situation 53 182 34 3434Simulation 5446 plusmn 176 1673 plusmn 414 3664 plusmn 346 1999 plusmn 512 error 275 807 776 mdash

OBGReal situation 34 0 35 0Simulation 384 plusmn 085 0 plusmn 0 377 plusmn 084 0 error 1294 0 771 mdash

Table 7 Demand for nurses obtained from simulation model

Department Period Hospital demand (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 3 3 2 2PED 3 3 3 3 3 3 3EENT 2 2 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 2 2EENT 2 2 2 2 2 2 2OBG 1 1 1 2 1 1 1

machine We conducted the experiment by fixing currentnumber of nurses at 18 with the low level nursesrsquo demand ineach department All three expecting parameters were set bytwo levels maximum and minimum

51 Determining Factors and Factor Values in Each LevelFactors can be chosen for the experiment as follows

(1) population number(2) percent crossover rate(3) percent mutation rate

We used the factors in Table 9 to construct the experi-mental design matrix full factorial design (2119896) (8) and thenconducted the experiment by using three center points perblockwith all the three replicatesWe also fixed the number ofmaximum cycles to 400 cyclesThis was because the researchrevealed that the number of the calculated cycles had high

stability The cycles also took a long time to calculate andso may cause inconvenience in their actual use The numberof nurses 18 in total was fixed at the beginning for theexperimental design and the results are shown in Table 10which includes the use of the data for the demand for nursesfrom the first week of January

52 Effect and Coefficient Analysis and Model AdequacyChecking We used the experimental results to analyze theeffects and coefficients by using theMinitab program in orderto find the factors affecting the expected responses

Figure 3 shows that the data were normally and consis-tently distributed without any tendency and can be used toanalyze further results

Table 11 shows that the correlation of the data is linearbecause the center pt is greater than 005 and the 119875 values ofthe 119860 119862 119860119862 119861119862 and 119860119861119862 factors are less than 005 whichis significant for the experiment It was concluded that the 119860

10 Mathematical Problems in Engineering

Table 8 Example for encoding binary coded decimal

Decimal Binary Regular hour Overtime hour Code Type Demand 1 Demand 2 Hours0 00000000 NDA NOT NDA NOT Type 0 0 0 01 00000001 DA1 NOT DA1 NOT Type 1 DA1++ 0 82 00000010 DA2 NOT DA2 NOT Type 2 DA2++ 0 83 00000011 DA3 NOT DA3 NOT Type 3 DA3++ 0 8

255 11111111 NDA NOT NDA NOT Type 255 0 0 0

001000050000

99

90

50

10

1

Residual054052050048046

0010

0005

0000

Fitted value

8

6

4

2

02624222018161412108642

0010

0005

0000

Observation order

Normal probability plot Versus fits

Histogram Versus order

Residual plots for obj

Resid

ual

Resid

ual

minus0010

minus0010

minus0005

minus0005

minus0010

minus0005

001000050000Residual

minus0010 minus0005

Freq

uenc

y(

)

Figure 3 The residual plots for objective function

Table 9 Selected experimental factors and factor values in eachlevel

Factor LevelLow High

(1) Population number (119860) 10 50(2) Percent crossover rate (119861) 5 95(3) Percent mutation rate (119862) 5 95

and 119862 factors were significant at the confidence level of 005as shown in Figure 4

The 119877-Sq value ranges from 0 to 1 or from 0 to 100If the 119877-Sq value is very close to 1 then it indicates that thesimulation can explain the different variables more properlyand accurately Table 12 shows that the 119877-Sq value is equalto 9948 which is very high This means that the data can

explain dependent variables properly and can analyze furtherresponse optimizer functions

53 Analysis for Finding outMost Appropriate Results by UsingResponse Optimizer The results were analyzed by using aresponse optimizer of Minitab as shown in Figures 5 and 6

From the results obtained as shown in Figure 5 this studyaims to determine the lowest target equation As a result wechanged ldquoGoalrdquo to ldquoMinimizerdquo since the target equation wasthe sum of the hospital expense percentage and the standarddeviation of the overtime each nurse received which rangedfrom 0 to 100 As a result the target was set to 0 and the uppertarget to 100

Figure 6 shows that if we want to configure the geneticprogram to obtain the lowest target equation it must beset as shown in Table 13 From the prediction equation thetarget equation should be equal to 04492 (obj = 04492)

Mathematical Problems in Engineering 11

Table 10 Full factorial design (23) experiment with three center points per block type

Run Std order Run order Center pt Blocks 119860 119861 119862 Obj Time (s)1 1 1 1 1 10 5 5 045892 92375212 2 2 1 1 50 5 5 050282 576219023 3 3 1 1 10 95 5 047113 7904851

25 25 25 0 1 30 50 50 048018 31281694

Table 11 Effect and obj coefficient analysis results

Factorial fit obj versus 119860 119861 119862Estimated effects and coefficients for obj (coded units)

Term Effect Coeff SE coeff 119879 119875

Constant 048216 0001318 36575 0000119860 005200 002600 0001318 1972 0000119861 000027 000013 0001318 010 0921119862 minus002134 minus001067 0001318 minus810 0000119860 lowast 119861 minus000285 minus000143 0001318 minus108 0293119860 lowast 119862 minus000850 minus000425 0001318 minus322 0005119861 lowast 119862 minus001395 minus000697 0001318 minus529 0000119860 lowast 119861 lowast 119862 minus000974 minus000487 0001318 minus369 0002Ct Pt minus000198 0003955 minus050 0623

Table 12 119877-square decision-making coefficients

119878 = 208209 PRESS = 152191461119877-Sq = 9948 119877-Sq (pred) = 9899 119877-Sq (adj) = 9925

Table 13 Guideline setting for genetic program from responseoptimizer function

Factor Level(1) Population number (119860) 10(2) Percent crossover rate (119861) 95(3) Percent mutation rate (119862) 95

Possible optimal parameters settingmay occur at the low level mutation we conducted the experiment and reported inTable 14 The reports showed that the objective function isworse when mutation decreased The reports also shownthat the higher mutation the better objective functions

54 Result Confirmation Experiment We experimented tocheck for accuracy by configuring the genetic program asshown in Table 12 together with fixing the maximum itera-tions to the number of nurses at 400 cycles and 18 nursesrespectively as shown in Table 15

Table 15 shows that the actual target value (obj) is actuallybetter than (ie less than) 063 of the predicted value andcan be used practically and actually decreased the target equa-tion (obj)

20151050

99

95908070605040302010

5

1

Standardized effect

Factor NameNot significantSignificant

Effect type

Normal plot of the standardized effects

minus10 minus5

(response is obj 120572 = 005)

()

A

A

B

C

A

B

C

C

BC

ABC

AC

Figure 4 The normal plot of the standardized effects (response isobj alpha = 005)

6 Change in Number of Nurses (119894) in OptimalScheduling under Constraint Equation

We changed the number of nurses (119894) which was an indexin the constraint equation in order to find out the least

12 Mathematical Problems in Engineering

Table 14 Experimental results obtained at low level mutation

Max iteration Population crossover mutation Obj Time (s)1500 10 95 95 044442 80491500 10 95 1 048580 101611500 10 95 01 050286 105231500 10 95 001 053612 10117

Table 15 Target equation (obj) predicted by Minitab program and compared with actual target equation (obj)

Target equation (obj) Comparison of target equation (obj)percentage

Predicted value Actual value Actual value compared with predicted value04492 04429 063

Table 16 Calculation results with reduction in number of nurses (119894)at 119908 = 06

Number of nurses (119894) Target equation(obj)

Calculating time(hour)

21 05241 76720 03959 60319 Infeasible Infeasible18 Infeasible Infeasible

Figure 5 The response optimizer configuration of obj response

number of nurses (119894) who were able to schedule the workunder the restrictions by using genetic programming and bygradually reducing the number of nurses (119894) down to the finalvalue at which the program could calculate the answer Thecalculation was performed at 119908 = 06 in Table 16

Table 16 shows that the least number of nurses (119894) whichcould schedule work under the restrictions was 20 Basedon 20 nurses amount of overtime payment and standarddeviation of overtime payment were calculated and reportedin Table 17 Therefore number of working hours for eachnurse number of nurses in each department and nursersquosschedule planwere generated and reported as shown inTables18 and 19

From Table 17 it can be observed that the target equation(obj) of the responses obtained was equal to 03959The sumof the overtime payment of all nurses was 16600 Baht per

CurHigh

Low099551

Optimal

minimumObj

099551desirabilityComposite

50950

50950

100500

50040004000 100 950 950

y = 04492

d = 099551

A B C D

D

Figure 6 The results of the data analysis using the responseoptimizer of the obj response

week The standard deviation of overtime payment that eachnurse received was 214843 For fair comparison we have toconvert the real overtime payment and standard deviationfor current 18 nurses to 20 nurses Then we can report thatthe sum of the total overtime payment and the standarddeviation of the overtime payment that each nurse receivedwhich is obtained from the model were less than those of theold working schedule at 17991 Baht per week and 882369respectively

Table 18 shows that all the nurses worked according to theconditions specified that is one nurse must work at least 40hours per week as in (4) but must work fewer than 60 hoursper week as in (5) all would be selected to work as in (6)and the duration of work must not exceed six days a week asin (7)

From Table 19 we can see that the number of nursesworking in shifts each day was more than the demand fornurses (Demand

119895119896119897) as shown in Table 6 which is according

to (3) and the specified service criteria (on average 15of thepatients waiting for services should not wait longer than theaverage service time of 25) because the number of nursesworking in shifts was more than the demand

To study the effect ofweight adjusted (119908)more numericalexperiments were conducted by GA coded in Matlab at 400

Mathematical Problems in Engineering 13

Table 17 Calculating results by 20 nurses (119894)

Target equation(obj)

Calculating time(hour)

Sum of overtime paymentof nurses (Baht)

Standard deviation of total overtimepayment that each nurse received

03959 603 16600 214843

Table 18 Number of working hours calculated from 20 nurses (119894)

Number of nurses Hoursperson (hour)Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total

1 12 12 0 4 0 8 12 482 4 8 12 4 8 12 0 483 8 8 8 0 12 12 0 484 12 0 12 12 12 4 0 525 8 12 8 4 0 0 12 44

20 8 12 4 0 12 8 4 48

Table 19 Number of nurses working in shifts calculated by 20 nurses (119894)

Department Period Supply (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 4 3 2 2PED 3 3 4 3 3 4 3EENT 2 3 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 3 3EENT 3 2 2 2 2 2 2OBG 2 2 3 2 1 1 2

Table 20 Effect of weight adjusted to objective function

Weight adjusted119908 = 09 119908 = 06 119908 = 05

Objective function 05090 04753 04823Sum of overtimepayment 24375 24625 26575

Standard deviation 324898 297675 288509

iterations and reported in Table 20 The report was shownas we expected more weight adjusted less overtime paymentsimultaneously with higher standard deviation It means thatif management focuses on reducing the overtime payment itwill lead to unfair payment or bias nursesrsquo schedule plan

7 Comparison of Advantages andDisadvantages of Adaptive Genetic andOptimization Approach

Weprocessed the data to do a comparison between the resultsof the adaptive genetic approach and the results obtained by

the Lingo program using the same data in order to comparethe target equation (obj) and computation runtime as shownin Tables 21 and 22

Table 21 shows the results of the processing experimentafter it was conducted ten times The work schedules using20 and 24 nurses with the calculating cycle of 400 roundsresulted in the best value for the target equations of 04361and 04551 respectively After ten processing experimentsthe work schedules for 20 and 24 nurses with the calculatingcycle at 1500 rounds resulted in the best value for the targetequations of 03959 and 04493 respectively and can besummarized as presented in Table 22

Table 22 shows that the work schedule for 20 nurses at1500 rounds when compared to the Lingo program couldcalculate thework schedulewith the target equation (obj) lessthan the genetic algorithm by around 580 and 108 plusmn 24on average but that the one created by the Lingo programused longer computational runtime by over 86849 and76452 on average We can also notice that the results at400 rounds reported the same direction the best and theaverage solution of genetic algorithm differ about 982 and1497 plusmn 38 from the optimal respectively However theoptimal method used longer time by over 306257 and

14 Mathematical Problems in Engineering

Table 21 Results of genetic program calculated ten times at 400 rounds and 1500 rounds

Target equation (obj)number

Number of 20 nurses Number of 24 nursesCalculating cycle

400Calculating cycle

1500Calculating cycle

400Calculating cycle

15001 04701 04578 05049 047302 04361 04436 05406 045523 04566 04772 04789 044934 04812 04303 04590 051605 04496 04781 04942 045866 04920 04549 05383 048337 04779 03959 05228 046268 05428 04352 05133 045809 05472 04368 05497 0498510 05231 04496 04551 04542Mean 04876 04459 05056 04708Standard deviation 0038 0024 0033 0021

Table 22 Comparison table for target equation (obj) and duration in calculation of GA and Lingo program

Number ofnurses(person)

Number ofcalculatingcycles

GA Lingo

Target equation (obj) Time (hour) Targetequation (obj) Time (hour) Different result

of obj

Different incomputational

run time

201500 Lowest 03959 603 580 86849

Mean 04459 plusmn 0024 685 03379 5237 1080 plusmn 24 76453

400 Lowest 04361 171 982 306257Mean 04876 plusmn 0038 166 1497 plusmn 38 315482

Table 23 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2012

Number of nurses Monthly expense SDOld New Increase Old New Decrease Old New Decrease18 20 2 395964 366400 29564 1089212 214843 8028

Table 24 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2013

Number of nurses Monthly expense SDOld New Decrease Old New Decrease Old New Decrease23 20 3 416548 366400 50148 1966735 214843 8908

315481 on average showing that the genetic program ismore suitable for practical use

8 Conclusion

The findings showed that the genetic program created couldmanage the work schedule of nurses under given conditionsand achieve the desired objective It could create work sche-dules of nurses that were fair in overtime payment to all thenurses and could reduce the hospital expenses by setting thenew coming nursesrsquo minimum salary at 15000 Baht per per-son

Table 25 Target equation and calculation time from geneticprogram

Target equation (obj) Calculation time (hour)03959 603

From Tables 23 24 and 25 the following results canbe concluded the work schedules from proposed methodreduce the waiting time of the number of patients waitinglonger than 125 times of the average length nursing servicefrom 3408 to less than 15 on average (starting from

Mathematical Problems in Engineering 15

when the patients register and wait to see the doctor) Ifwe want to schedule the work of the nurses in January 2012in order to meet the criteria for the service of this studywe have to increase the number of nurses from 18 to 20 sothat the number can perfectly meet the needs of the patientsaccording to the criteriaThework schedules of the nurses canbemost appropriate if the target equation is 03959which usesthe time of calculation as 603 hourswith a standard deviationof less than 8028and the overtimepayment less than 29564Baht per monthWhen this is compared with the actual workschedule in January 2013 the work schedule could reduce thenumber of nurses from 23 to 20 with a decreased standarddeviation that is 8908 and the decrease in the totalovertime paymentwould be 50148 Baht permonthWe couldcalculate the most appropriate work schedule quicker thanby using the conventional approach which results obtainedby Lingo program version 50 86849 with 1500 rounds forthe calculation cycle

As we know hospitals belong to the service businesssector continuous improvement along with response to cus-tomer satisfaction is the key success factor in this businessTo reduce the waiting time in every hospital process highstandards in medical care and highly experienced medicaldoctors are needed Presently Thailand has set its aim tobecome an Asian medical and tourist hub this strategy of thecountry may cause and force hospitals to improve their ser-vice quality for responding well to the high number of outpa-tients Updated data collection needs to be done service-level policies have to be reconsidered Demands of outpa-tient nurses and their schedules may change from time totime Inpatient nurses may be allocated to assist when themanagement wishes to reduce the fluctuation in the numberof outpatient nurses Thus modern management and highefficient tools are necessary this is the reason why GA isinvolved instead of an adoption of the conventional approachHowever regarding the implementation of simulation evenGA is not familiar to the hospital people Optimization capac-ity building is another issue that the hospitalmanagementwillhave to consider

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chiang Mai University (CMU) through the research admin-istration office provided budget to our Excellence Center inLogistics and Supply Chain Management (E-LSCM)

References

[1] T Mongkolrangsarit and W Thamapornpilat Development ofNurse Scheduling Method with Uncertain Workforce DemandDepartment of Industrial Engineering Faculty of EngineeringChulalongkorn University 2007

[2] C M Rothe and H B Wolfe ldquoCyclical scheduling and alloca-tion of nursing staffrdquo Socio-Economic Planning Sciences vol 7no 5 pp 471ndash487 1973

[3] R B Norby L E Freund and B Wagner ldquoA nurse staffingsystem based upon assignment difficultyrdquo Journal of NursingAdministration vol 7 no 9 pp 2ndash24 1977

[4] T Ryan B L Barker and F A Marciante ldquoA system fordetermining appropriate nurse staffingrdquo Journal of NursingAdministration vol 5 no 5 pp 30ndash38 1975

[5] H Wolfe and J P Young ldquoStaffing the nursing unit Part Icontrolled variable staffingrdquo Nursing Research vol 14 pp 236ndash243 1965

[6] HWolfe and J P Young ldquoStaffing the nursing unit IIThemult-iple assignment techniquerdquo Nursing Research vol 14 no 4 pp299ndash303 1965

[7] J L Arthur andA Ravindran ldquoAmultiple objective nurse sche-duling modelrdquo AIIE Transactions vol 13 no 1 pp 55ndash60 1981

[8] I Ozkarahan ldquoA disaggregation model of a flexible nursescheduling support systemrdquo Socio-Economic Planning Sciencesvol 25 no 1 pp 9ndash26 1991

[9] I Ozkarahan and J E Bailey ldquoGoal programming modelsubsystem of a flexible nurse scheduling support systemrdquo IIETransactions vol 20 no 3 pp 306ndash316 1988

[10] K A Dowsland ldquoNurse scheduling with tabu search and stra-tegic oscillationrdquo European Journal of Operational Research vol106 no 2-3 pp 393ndash407 1998

[11] P Tormos A Lova F Barber L Ingolotti M Abril and M ASalido ldquoA genetic algorithm for railway scheduling problemsrdquoinMetaheuristics for Scheduling in Industrial andManufacturingApplications vol 128 of Studies in Computational Intelligencepp 255ndash276 Springer Berlin Germany 2008

[12] H Kawanaka K Yamamoto T Yoshikawa T Shinogi and STsuruoka ldquoGenetic algorithm with the constraints for nursescheduling problemrdquo in Proceedings of the Congress on Evolu-tionary Computation vol 2 pp 1123ndash1130 Seoul Republic ofKorea May 2001

[13] H HMillar and KMona ldquoCyclic and non-cyclic scheduling of12 h shift nurses by network programmingrdquo European Journalof Operational Research vol 104 no 3 pp 582ndash592 1998

[14] I Blochlige ldquoModeling staff scheduling problems A tutorialrdquoEuropean Journal of Operational Research vol 158 no 3 pp533ndash542 2004

[15] B Jaumard F Semet and T Vovor ldquoA generalized linear pro-gramming model for nurse schedulingrdquo European Journal ofOperational Research vol 107 no 1 pp 1ndash18 1998

[16] T Cezik O Gunluk and H Luss ldquoAn integer programmingmodel for the weekly tour scheduling problemrdquoNaval ResearchLogistics vol 48 no 7 pp 607ndash624 2001

[17] Z Wang J-L Liu and X Yu ldquoSelf-fertilization based geneticalgorithm for university timetabling problemrdquo in Proceedingsof the 1st ACMSIGEVO Summit on Genetic and EvolutionaryComputation (GEC rsquo09) pp 1001ndash1004 ACM Shanghai ChinaJune 2009

[18] J Wang and K Chu ldquoAn application of genetic algorithms forthe flexible job-shop scheduling problemrdquo International Journalof Advancements in Computing Technology vol 4 no 3 pp 271ndash278 2012

[19] M Moz and M V Pato ldquoA genetic algorithm approach to anurse rerostering problemrdquo Computers amp Operations Researchvol 34 no 3 pp 667ndash691 2007

[20] C Wotthipong K Soradech and S Nidapan ldquoThe suitablegenetic operators for solving the university course timetablingproblemrdquo Journal of Convergence Information Technology vol8 pp 60ndash65 2013

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

Page 3: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

Mathematical Problems in Engineering 3

(2) One nurse must work at least 40 hours per week (4)but fewer than 60 hours per week (5)

8 times

5

sum119897=1

7

sum119895=1

1199091198941198951119897

+ 4 times

5

sum119897=1

7

sum119895=1

1199091198941198952119897

ge 40 forall119894 (4)

8 times

5

sum119897=1

7

sum119895=1

1199091198941198951119897

+ 4 times

5

sum119897=1

7

sum119895=1

1199091198941198952119897

le 60 forall119894 (5)

(3) All nurses must be selected to work (6) and work notover six days per week (7)

7

sum119895=1

119909119894119895119896119897

ge 1 forall119894forall119896forall119897 (6)

7

sum119895=1

119909119894119895119896119897

le 6 forall119894forall119896forall119897 (7)

(4) In a day one nurse can work only one period in onedepartment

11990911989411989511

+ 11990911989411989512

+ 11990911989411989513

+ 11990911989411989514

+ 11990911989411989515

le 1 forall119894forall119895

11990911989411989521

+ 11990911989411989522

+ 11990911989411989523

+ 11990911989411989524

+ 11990911989411989525

le 1 forall119894forall119895

(8)

Equations related to expenses are as followsWhen nurses work formore than 40 hours in a week they

will receive an overtime payment of 650 Baht per eight hoursor 8125 Baht per hour and if within 1600ndash2000 hours theywill receive 50 Baht extra overtime payment per working day

OTS119894= 8125 times [

[

(8 times

5

sum119897=1

7

sum119895=1

1199091198941198951119897

+ 4 times

5

sum119897=1

7

sum119895=1

1199091198941198952119897) minus 40]

]

+

7

sum119895=1

(50 times

5

sum119897=1

1199091198941198952119897) forall

119894

AVG =sum119873

119894=1OTS119894

119873

SOT =

119873

sum119894=1

OTS119894

SD = radicsum119873

119894=1(OTS119894minus AVG)2

119873

(9)

After that the least number of nurses (119894) is determined inorder to be scheduled under existing limitations

Proposed mathematical model with objective function(2) subject to constraints (3)ndash(9) was verified by simpleproblem as shown in Table 1 With the conventional exactsolution approach the nurse scheduling was constructed fortwo departments SUR and MED and reported by Ganttchart as shown in Table 2 In Table 2 lowast and represent theschedule of SUR and MED respectively Lingo software alsoreported that the mathematical model is nonlinear modelwith integer variables

3 Building Simulation to Find AppropriateDemand for Nurses

In this case since a service criterion system has not been inuse in the hospital the demand for nurses as in the presentmay not be an appropriate demand for finding the leastcost work schedule The researcher examined an appropriatedemand for nurses (Demand

119895119896119897) by creating a simulation

using the Arena program to determine the appropriatedemand for nurses in each department and in all the periodsof each day from the original hospital data It was found thaton average 3408 or 5487 patients per week had waitedlonger than 25 of the total average service time Thereforethe researcher aimed to bring about an improvement at theinitial state itself by creating a simulation under the servicecriteria that set the limitation as only 15 of the patients onaverage did not have to wait longer than 25 of the totalaverage service time

31 Data Collection of Patient Service and Nursing ServiceDuration For the data collection of each department at eachperiod of each day (starting from when the patients registeruntil they wait to see a doctor) the researcher chose to use thedata as each day in the month to represent day in that monthCollecting data were done by the quality assurance sectionin case study hospital they defined the outpatient based onthe combination of appointments and walk-ins Interarrivaltime for regular and overtime hour including service timehas been collected for two years in winter season which hasthe largest number of outpatients The interarrival time forregular and overtime periods were separately collected dueto significant static data However no significant differencewas noticed for service timeThen the data were classified byday in a week and analyzed for their statistics distribution forthree categories

32 Data Distribution Analysis The data were analyzed byusing a suitable dispersion input analyzer in the Arenaprogram and by selecting the distribution with the highest 119875value using the significance level (120572) at 005 (95 confidence)as shown in Table 3 The 119875 value is a key concept in theapproach of Ronald Fisher where he uses it to measurethe weight of the data against a specified hypothesis and asa guideline to ignore data that does not reach a specifiedsignificance level

33 Simulation Modeling We created a simulation modelingstarting from when the patient registered at the servicedepartment based on interarrival time and service timestatistics distribution reported in Table 3 Because thesestatistics distributions consider the arrival of appointmentand walk-in patients then this simulation model can takeinto account the dynamics of walk-in patients Beginningwith statistics distributions patientsrsquo arrival patients hadregistered at the service department According to patientrsquossymptom they had moved to required department counterand waited for medical history file To parallel with filesearching and coming patients had queued and waited fornurse service Nurses served the fundamental examination

4 Mathematical Problems in Engineering

Table 1 Demand for nurses verify case study

Department Period Hospital demand (person)Mon Tue Wed Thu Fri Sat Sun

SUR Regular 1 1 2 1 2 1 2Overtime 1 2 1 2 1 1 1

MED Regular 2 1 2 1 1 1 1Overtime 1 1 1 2 1 2 2

such as weight blood pressure temperature measure pre-symptom examined and finally recorded all measure data inthe file after it arrived Service time distributions recordedand fitted by statistic approach in Table 3 were used as thedata set in process box After that we saved the percentageinformation of those patients whose waiting time exceededthe service time limitation as shown in Figure 1

Then we adjusted the number of nurses (resource) inthe capacity column manually as shown in Table 4 until thepercentage of the patients waiting for more than the limit(average and half-width) in each department was less than15 according to what we wanted in every section as shownin Table 5 After that we processed 100 replications

From Table 4 it can be observed that the departments ofSURMED and EENT appointed two people PED appointedthree people andOBG appointed one person to work in theirdepartments

From Table 5 we can see that all the departments withthe percentage of patients waiting for more than the limit(average and half-width) do not exceed 15 The EENTMED OBG PED and SUR departments had the percentagesof patients waiting for longer than the limit (average pluspositive half-width) at 07918 20505 50424 27194and 56773 respectively We considered only the positivehalf-width because we would like to know the maximumpercentage of patients waiting

34 Simulation Model Verification In model verification wecompared the results between real situation which occurredin this hospital and the simulationmodel Average number ofpatients receiving service average number of patients waitinglonger than average service time and average number ofpatients waiting less than average service time were proposedfor preforming simulationmodel efficientThe 100 times witheight-hour simulation were conducted for all five outpatientsdepartments The results were reported in Table 6 In simu-lation row we reported the average number plusminus withtheir corresponding half-width

Table 6 reported that the OBG got the highest error interms of average number of patients receiving service due tothe low 119875 value reported in Table 3 With this error howeverwe can adopt this simulation model generated by the arenaprogram for real case application

After model verification manual adjustment of theresource capacity was done and selected under hospital topmanagement service policy The service level policy was setas no more than 15 of patients waiting longer than the

average service timeThen the appropriate resource (numberof nurses) from simulation was shown for each departmentin Table 7 This number of nurses will be used as the datafor nurses demand (Demand

119895119896119897) in the mathematical model

for obtaining the optimal nurse schedule Even thoughto obtain the optimal nurse schedule with conventionalapproach always consumes computational runtime and hard-ware implementation Genetic algorithm was proposed tosolve and obtain an acceptable solution

4 Programming for Genetic AlgorithmNurse Scheduling

We coded a genetic algorithm program for use in the shiftscheduling of nurses with the Matlab program by using thefollowing steps

41 Chromosome Format and Chromosome Encoding Achromosome format uses a binary number system and it isdivided into three levels as shown in Figure 2 where

level 1 is a nurse at 1 2 3 119873level 2 is the working daysMonday TuesdayWednes-day Sundaylevel 3 is the number of 8 bit code

42 Chromosome Encoding and Chromosome Decoding Toencode the chromosome we used the binary coded decimalsystem which is a system that uses the 8 bit binary codeinstead of the general code we have set up We can encodeand decode the chromosome as shown in Table 8 where

(i) NDA = not working during regular hours(ii) DA1 = working during regular hours at Division 1(iii) DA2 = working during regular hours at Division 2(iv) DA3 = working during regular hours at Division 3(v) DA4 = working during regular hours at Division 4(vi) DA5 = working during regular hours at Division 5(vii) NOT = not working overtime(viii) OT1 = working overtime at Division 1(ix) OT2 = working overtime at Division 2(x) OT3 = working overtime at Division 3(xi) OT4 = working overtime at Division 4(xii) OT5 = working overtime at Division 5

Mathematical Problems in Engineering 5

Table2Th

eGanttcharto

fnursesrsquoworking

schedu

lewas

constructedby

theL

ingo

program

Nurses

Mon

day

Tuesday

Wednesday

Thursday

Friday

Saturday

Sund

ayTo

talh

ours

Regu

larh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

e1

lowastlowast

lowastlowast

402

lowast

lowast

403

lowast

lowast

lowastlowast

404

lowast

lowast

lowast

405

lowast

lowast

406

lowast

lowastlowast

407

lowastlowast

lowastlowast

408

lowast

lowast

409

lowastlowast

4010

lowast

lowastlowast

40

6 Mathematical Problems in Engineering

Table3Distrib

utionpatie

ntsinterarriv

alandservicetim

e

Day

Departm

ent

Regu

larh

our

Overtim

ehou

rServicetim

e119875value

Avgtim

eInterarrivaltim

e119875value

Interarrivaltim

e119875value

Mon

day

SUR

minus05+EX

PO(852

)0333

minus05+EX

PO(107)

0225

05+WEIB(71118

5)gt0750

684

MED

minus05+EX

PO(539

)0654

minus05+EX

PO(768)

0616

POIS(413

)0051

413

PED

minus05+WEIB(95

1114)

0113

minus05+EX

PO(123)

0075

05+EX

PO(65)

0078

700

EENT

minus05+EX

PO(954)

0478

55+EX

PO(165)

0171

05+EX

PO(458)

0615

508

OBG

minus05+WEIB(146154)

0465

115+EX

PO(225)

0228

TRIA(253195

)0244

910

Tuesday

SUR

minus05+WEIB(10512)

0334

minus05+WEIB(98

70925)

0702

05+18lowastBE

TA(0774125)

0481

740

MED

minus05+WEIB(632133

)0510

minus05+EX

PO(913

)0746

05+LO

GN(412

532

)0109

443

PED

minus05+EX

PO(56)

0607

minus05+ER

LA(4982)

0526

15+EX

PO(87)

006

61020

EENT

minus05+GAMM(663111)

0063

minus05+EX

PO(105)

0057

05+18lowastBE

TA(0409123)

0427

498

OBG

minus05+GAMM(75913)

0145

15+EX

PO(301)

0125

TRIA(25596195)

0421

932

Sund

ay

SUR

minus05+LO

GN(97

126)

gt0750

minus05+EX

PO(14

4)

006

805+17lowastBE

TA(076

106)

0581

737

MED

minus05+GAMM(947116)

0283

15+EX

PO(224)

0105

UNIF(05185)

0589

876

PED

minus05+WEIB(5120841)

0197

minus05+EX

PO(103)

0733

05+EX

PO(45)

0210

500

EENT

minus05+LO

GN(105166)

0349

UNIF(minus05195)

0438

05+19lowastBE

TA(0898156)

0489

744

OBG

minus05+LO

GN(131207)

0673

05+EX

PO(175

)0097

POIS(635)

0126

635

Mathematical Problems in Engineering 7

Service LevelOBG

Process OBG Decide 5

MoreMean 5

LowerMean 5 0

0

True

False

0

0

Dispose 5

Service LevelEENT

Process EENT Decide 4

MoreMean 4

LowerMean 4 0

0

True

False

0

0

Dispose 4

Service LevelPED

Process PED Decide 3

MoreMean 3

LowerMean 3 0

0

True

False

0

0

Dispose 3

Service LevelMED

Process MED Decide 2

MoreMean 2

LowerMean 2 0

0

True

False

0

0

Dispose 2

Service LevelSUR

Process SUR Decide 1

MoreMean 1

LowerMean 1 0

0

True

False

0

0

Dispose 1 IT SUR 8hr 02 01

IT MED 8hr 02 01

IT PED 8hr 02 01

IT EENT 8hr 02 01

IT OBG 8hr 02 01

Figure 1 The simulation model created by the Arena program

8 Mathematical Problems in Engineering

Table 4 The example for adjusting the number of nurses in theArena program

Name Type Capacity1 Resource SUR Fixed capacity 22 Resource MED Fixed capacity 23 Resource PED Fixed capacity 34 Resource EENT Fixed capacity 25 Resource OBG Fixed capacity 1

Table 5 Average and half-width values based on number of nursesadjusted

Output Average Half-widthEENT 04418 035MED 12505 080OBG 34024 164PED 16794 104SUR 44273 125

43 Creating Initial Population Each nursersquos chromosomecan be randomly generated for the initial state Then checkwhether it is in the scope of primary goal or not withworkinghours fewer than 60 hours If that chromosome is not in thescope of the primary goal again random search until eachnursersquos chromosome completes for the entire length and allthe nursersquos chromosomes are generated to reach the numberof population specified

44 Crossover and Mutation Crossover is an importantoperator which combines the good properties of both parentsin order to possibly yield new better children chromosomes[17 18] As usual the simple crossover operator (one-pointcrossover) consists of randomly choosing a crossover pointand then recombining the pieces of a pair of chromosomes toform two new chromosomes The simple crossover is com-patible with the random keys encoding though it generallyfails to preserve the permutation when dealing with naturalencoding Hence for natural encoding special crossoveroperators must be used [19] Roulette wheel selection (RWS)ranking selection (RS) tournament selection (TS) partiallymatched crossover (PMX) order crossover (OX) and cyclecrossover (CX) are evaluated by comparing the performanceof GArsquos operators on university course timetabling problem[20] PMX OX and CX operators require two crossoverpoints Given two parent chromosomes 119860 and 119861 childchromosome1198601015840 will inherit form119860 the subsequence betweenthese two points and child chromosome 1198611015840 form 119861 therespective subsequence The elements of 1198601015840 and 1198611015840 outsidethe two points are copied from the other parent chromosomewhile trying to preserve its position under the PMX operatoror by trying to preserve its order under the OX operatorA comparative analysis of PMX CX and OX crossoveroperators for solving travelling salesman problem (TSP) wasreported in [21]The experimental results show that the PMXcrossover outperforms the CX and OX crossover operator inTSP with 25 numbers of cities For our problem simple testwas conducted and we found PMX provided a better solution

Day

1

1

2

2

3

3

4

4 Nmiddot middot middot

middot middot middot

Mo Tu We Th Fr Sa Su

Number of nurses

Main chromosome

1 1 110 000

Binary code

Level 1

Level 2

Level 3

N lowast 7 lowast 8

Figure 2 The chromosome format

than one-point crossover specifically one-point crossovercannot take into account all constraints (3)ndash(8)

After applying crossover the mutation operator actson the pairs of chromosomes Although mutation occursinfrequently in nature it is believed to be an importantdriving force for evolution The mutation is adopted to allowfor the introduction of new chromosome into the populationand is effective to escape from a local optimum The simplemutation operates by randomly changing its value with agiven probability used in our experiments

45 Evaluation of Fitness Value The fitness value can be eva-luated by using two criteria as follows

Criterion one a chromosome is a feasible solution

(i) Fitness value is the objective function that isprocessed by (2)

Criterion two a chromosome is a nonfeasible solu-tion

(i) Fitness value is equal to 1000 units

46 Chromosome Selection We selected the chromosome byusing roulette

47 Elite Preserve Strategy The previous chromosomes withhigh fitness values were replaced by the new chromosomeswith the lower fitness values Then they were used as theinitial population for the next iteration

48 Termination Criteria The condition to stop seeking an-swers was that when the required number of solutions fullymeets the numbers of iteration specified we stop the exami-nation process immediately

5 Experiment Design to Determine OptimalConfiguration in Genetic Programming

In this section we introduced design of experiment (DOE)technique for screening the effected parameters and param-eters setting guideline for genetic algorithm solution search

Mathematical Problems in Engineering 9

Table 6 Comparison between real situation and simulation model

Department Calculated resultAVG number ofpatients receiving

service

AVG number ofpatients waitinglonger than AVG

time

AVG number ofpatients waiting lessthan AVG time

Percentage of patientswaiting longer than

AVG time

SURReal situation 59 02 58 0338Simulation 6163 plusmn 163 019 plusmn 036 6051 plusmn 166 0196 plusmn 032 error 427 526 433 mdash

MEDReal situation 96 0 98 0Simulation 9879 plusmn 191 0 plusmn 0 9794 plusmn 191 0 error 290 0 006 mdash

PEDReal situation 55 0 53 0Simulation 5758 plusmn 148 0 plusmn 0 5669 plusmn 146 0 error 469 0 696 mdash

EENTReal situation 53 182 34 3434Simulation 5446 plusmn 176 1673 plusmn 414 3664 plusmn 346 1999 plusmn 512 error 275 807 776 mdash

OBGReal situation 34 0 35 0Simulation 384 plusmn 085 0 plusmn 0 377 plusmn 084 0 error 1294 0 771 mdash

Table 7 Demand for nurses obtained from simulation model

Department Period Hospital demand (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 3 3 2 2PED 3 3 3 3 3 3 3EENT 2 2 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 2 2EENT 2 2 2 2 2 2 2OBG 1 1 1 2 1 1 1

machine We conducted the experiment by fixing currentnumber of nurses at 18 with the low level nursesrsquo demand ineach department All three expecting parameters were set bytwo levels maximum and minimum

51 Determining Factors and Factor Values in Each LevelFactors can be chosen for the experiment as follows

(1) population number(2) percent crossover rate(3) percent mutation rate

We used the factors in Table 9 to construct the experi-mental design matrix full factorial design (2119896) (8) and thenconducted the experiment by using three center points perblockwith all the three replicatesWe also fixed the number ofmaximum cycles to 400 cyclesThis was because the researchrevealed that the number of the calculated cycles had high

stability The cycles also took a long time to calculate andso may cause inconvenience in their actual use The numberof nurses 18 in total was fixed at the beginning for theexperimental design and the results are shown in Table 10which includes the use of the data for the demand for nursesfrom the first week of January

52 Effect and Coefficient Analysis and Model AdequacyChecking We used the experimental results to analyze theeffects and coefficients by using theMinitab program in orderto find the factors affecting the expected responses

Figure 3 shows that the data were normally and consis-tently distributed without any tendency and can be used toanalyze further results

Table 11 shows that the correlation of the data is linearbecause the center pt is greater than 005 and the 119875 values ofthe 119860 119862 119860119862 119861119862 and 119860119861119862 factors are less than 005 whichis significant for the experiment It was concluded that the 119860

10 Mathematical Problems in Engineering

Table 8 Example for encoding binary coded decimal

Decimal Binary Regular hour Overtime hour Code Type Demand 1 Demand 2 Hours0 00000000 NDA NOT NDA NOT Type 0 0 0 01 00000001 DA1 NOT DA1 NOT Type 1 DA1++ 0 82 00000010 DA2 NOT DA2 NOT Type 2 DA2++ 0 83 00000011 DA3 NOT DA3 NOT Type 3 DA3++ 0 8

255 11111111 NDA NOT NDA NOT Type 255 0 0 0

001000050000

99

90

50

10

1

Residual054052050048046

0010

0005

0000

Fitted value

8

6

4

2

02624222018161412108642

0010

0005

0000

Observation order

Normal probability plot Versus fits

Histogram Versus order

Residual plots for obj

Resid

ual

Resid

ual

minus0010

minus0010

minus0005

minus0005

minus0010

minus0005

001000050000Residual

minus0010 minus0005

Freq

uenc

y(

)

Figure 3 The residual plots for objective function

Table 9 Selected experimental factors and factor values in eachlevel

Factor LevelLow High

(1) Population number (119860) 10 50(2) Percent crossover rate (119861) 5 95(3) Percent mutation rate (119862) 5 95

and 119862 factors were significant at the confidence level of 005as shown in Figure 4

The 119877-Sq value ranges from 0 to 1 or from 0 to 100If the 119877-Sq value is very close to 1 then it indicates that thesimulation can explain the different variables more properlyand accurately Table 12 shows that the 119877-Sq value is equalto 9948 which is very high This means that the data can

explain dependent variables properly and can analyze furtherresponse optimizer functions

53 Analysis for Finding outMost Appropriate Results by UsingResponse Optimizer The results were analyzed by using aresponse optimizer of Minitab as shown in Figures 5 and 6

From the results obtained as shown in Figure 5 this studyaims to determine the lowest target equation As a result wechanged ldquoGoalrdquo to ldquoMinimizerdquo since the target equation wasthe sum of the hospital expense percentage and the standarddeviation of the overtime each nurse received which rangedfrom 0 to 100 As a result the target was set to 0 and the uppertarget to 100

Figure 6 shows that if we want to configure the geneticprogram to obtain the lowest target equation it must beset as shown in Table 13 From the prediction equation thetarget equation should be equal to 04492 (obj = 04492)

Mathematical Problems in Engineering 11

Table 10 Full factorial design (23) experiment with three center points per block type

Run Std order Run order Center pt Blocks 119860 119861 119862 Obj Time (s)1 1 1 1 1 10 5 5 045892 92375212 2 2 1 1 50 5 5 050282 576219023 3 3 1 1 10 95 5 047113 7904851

25 25 25 0 1 30 50 50 048018 31281694

Table 11 Effect and obj coefficient analysis results

Factorial fit obj versus 119860 119861 119862Estimated effects and coefficients for obj (coded units)

Term Effect Coeff SE coeff 119879 119875

Constant 048216 0001318 36575 0000119860 005200 002600 0001318 1972 0000119861 000027 000013 0001318 010 0921119862 minus002134 minus001067 0001318 minus810 0000119860 lowast 119861 minus000285 minus000143 0001318 minus108 0293119860 lowast 119862 minus000850 minus000425 0001318 minus322 0005119861 lowast 119862 minus001395 minus000697 0001318 minus529 0000119860 lowast 119861 lowast 119862 minus000974 minus000487 0001318 minus369 0002Ct Pt minus000198 0003955 minus050 0623

Table 12 119877-square decision-making coefficients

119878 = 208209 PRESS = 152191461119877-Sq = 9948 119877-Sq (pred) = 9899 119877-Sq (adj) = 9925

Table 13 Guideline setting for genetic program from responseoptimizer function

Factor Level(1) Population number (119860) 10(2) Percent crossover rate (119861) 95(3) Percent mutation rate (119862) 95

Possible optimal parameters settingmay occur at the low level mutation we conducted the experiment and reported inTable 14 The reports showed that the objective function isworse when mutation decreased The reports also shownthat the higher mutation the better objective functions

54 Result Confirmation Experiment We experimented tocheck for accuracy by configuring the genetic program asshown in Table 12 together with fixing the maximum itera-tions to the number of nurses at 400 cycles and 18 nursesrespectively as shown in Table 15

Table 15 shows that the actual target value (obj) is actuallybetter than (ie less than) 063 of the predicted value andcan be used practically and actually decreased the target equa-tion (obj)

20151050

99

95908070605040302010

5

1

Standardized effect

Factor NameNot significantSignificant

Effect type

Normal plot of the standardized effects

minus10 minus5

(response is obj 120572 = 005)

()

A

A

B

C

A

B

C

C

BC

ABC

AC

Figure 4 The normal plot of the standardized effects (response isobj alpha = 005)

6 Change in Number of Nurses (119894) in OptimalScheduling under Constraint Equation

We changed the number of nurses (119894) which was an indexin the constraint equation in order to find out the least

12 Mathematical Problems in Engineering

Table 14 Experimental results obtained at low level mutation

Max iteration Population crossover mutation Obj Time (s)1500 10 95 95 044442 80491500 10 95 1 048580 101611500 10 95 01 050286 105231500 10 95 001 053612 10117

Table 15 Target equation (obj) predicted by Minitab program and compared with actual target equation (obj)

Target equation (obj) Comparison of target equation (obj)percentage

Predicted value Actual value Actual value compared with predicted value04492 04429 063

Table 16 Calculation results with reduction in number of nurses (119894)at 119908 = 06

Number of nurses (119894) Target equation(obj)

Calculating time(hour)

21 05241 76720 03959 60319 Infeasible Infeasible18 Infeasible Infeasible

Figure 5 The response optimizer configuration of obj response

number of nurses (119894) who were able to schedule the workunder the restrictions by using genetic programming and bygradually reducing the number of nurses (119894) down to the finalvalue at which the program could calculate the answer Thecalculation was performed at 119908 = 06 in Table 16

Table 16 shows that the least number of nurses (119894) whichcould schedule work under the restrictions was 20 Basedon 20 nurses amount of overtime payment and standarddeviation of overtime payment were calculated and reportedin Table 17 Therefore number of working hours for eachnurse number of nurses in each department and nursersquosschedule planwere generated and reported as shown inTables18 and 19

From Table 17 it can be observed that the target equation(obj) of the responses obtained was equal to 03959The sumof the overtime payment of all nurses was 16600 Baht per

CurHigh

Low099551

Optimal

minimumObj

099551desirabilityComposite

50950

50950

100500

50040004000 100 950 950

y = 04492

d = 099551

A B C D

D

Figure 6 The results of the data analysis using the responseoptimizer of the obj response

week The standard deviation of overtime payment that eachnurse received was 214843 For fair comparison we have toconvert the real overtime payment and standard deviationfor current 18 nurses to 20 nurses Then we can report thatthe sum of the total overtime payment and the standarddeviation of the overtime payment that each nurse receivedwhich is obtained from the model were less than those of theold working schedule at 17991 Baht per week and 882369respectively

Table 18 shows that all the nurses worked according to theconditions specified that is one nurse must work at least 40hours per week as in (4) but must work fewer than 60 hoursper week as in (5) all would be selected to work as in (6)and the duration of work must not exceed six days a week asin (7)

From Table 19 we can see that the number of nursesworking in shifts each day was more than the demand fornurses (Demand

119895119896119897) as shown in Table 6 which is according

to (3) and the specified service criteria (on average 15of thepatients waiting for services should not wait longer than theaverage service time of 25) because the number of nursesworking in shifts was more than the demand

To study the effect ofweight adjusted (119908)more numericalexperiments were conducted by GA coded in Matlab at 400

Mathematical Problems in Engineering 13

Table 17 Calculating results by 20 nurses (119894)

Target equation(obj)

Calculating time(hour)

Sum of overtime paymentof nurses (Baht)

Standard deviation of total overtimepayment that each nurse received

03959 603 16600 214843

Table 18 Number of working hours calculated from 20 nurses (119894)

Number of nurses Hoursperson (hour)Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total

1 12 12 0 4 0 8 12 482 4 8 12 4 8 12 0 483 8 8 8 0 12 12 0 484 12 0 12 12 12 4 0 525 8 12 8 4 0 0 12 44

20 8 12 4 0 12 8 4 48

Table 19 Number of nurses working in shifts calculated by 20 nurses (119894)

Department Period Supply (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 4 3 2 2PED 3 3 4 3 3 4 3EENT 2 3 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 3 3EENT 3 2 2 2 2 2 2OBG 2 2 3 2 1 1 2

Table 20 Effect of weight adjusted to objective function

Weight adjusted119908 = 09 119908 = 06 119908 = 05

Objective function 05090 04753 04823Sum of overtimepayment 24375 24625 26575

Standard deviation 324898 297675 288509

iterations and reported in Table 20 The report was shownas we expected more weight adjusted less overtime paymentsimultaneously with higher standard deviation It means thatif management focuses on reducing the overtime payment itwill lead to unfair payment or bias nursesrsquo schedule plan

7 Comparison of Advantages andDisadvantages of Adaptive Genetic andOptimization Approach

Weprocessed the data to do a comparison between the resultsof the adaptive genetic approach and the results obtained by

the Lingo program using the same data in order to comparethe target equation (obj) and computation runtime as shownin Tables 21 and 22

Table 21 shows the results of the processing experimentafter it was conducted ten times The work schedules using20 and 24 nurses with the calculating cycle of 400 roundsresulted in the best value for the target equations of 04361and 04551 respectively After ten processing experimentsthe work schedules for 20 and 24 nurses with the calculatingcycle at 1500 rounds resulted in the best value for the targetequations of 03959 and 04493 respectively and can besummarized as presented in Table 22

Table 22 shows that the work schedule for 20 nurses at1500 rounds when compared to the Lingo program couldcalculate thework schedulewith the target equation (obj) lessthan the genetic algorithm by around 580 and 108 plusmn 24on average but that the one created by the Lingo programused longer computational runtime by over 86849 and76452 on average We can also notice that the results at400 rounds reported the same direction the best and theaverage solution of genetic algorithm differ about 982 and1497 plusmn 38 from the optimal respectively However theoptimal method used longer time by over 306257 and

14 Mathematical Problems in Engineering

Table 21 Results of genetic program calculated ten times at 400 rounds and 1500 rounds

Target equation (obj)number

Number of 20 nurses Number of 24 nursesCalculating cycle

400Calculating cycle

1500Calculating cycle

400Calculating cycle

15001 04701 04578 05049 047302 04361 04436 05406 045523 04566 04772 04789 044934 04812 04303 04590 051605 04496 04781 04942 045866 04920 04549 05383 048337 04779 03959 05228 046268 05428 04352 05133 045809 05472 04368 05497 0498510 05231 04496 04551 04542Mean 04876 04459 05056 04708Standard deviation 0038 0024 0033 0021

Table 22 Comparison table for target equation (obj) and duration in calculation of GA and Lingo program

Number ofnurses(person)

Number ofcalculatingcycles

GA Lingo

Target equation (obj) Time (hour) Targetequation (obj) Time (hour) Different result

of obj

Different incomputational

run time

201500 Lowest 03959 603 580 86849

Mean 04459 plusmn 0024 685 03379 5237 1080 plusmn 24 76453

400 Lowest 04361 171 982 306257Mean 04876 plusmn 0038 166 1497 plusmn 38 315482

Table 23 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2012

Number of nurses Monthly expense SDOld New Increase Old New Decrease Old New Decrease18 20 2 395964 366400 29564 1089212 214843 8028

Table 24 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2013

Number of nurses Monthly expense SDOld New Decrease Old New Decrease Old New Decrease23 20 3 416548 366400 50148 1966735 214843 8908

315481 on average showing that the genetic program ismore suitable for practical use

8 Conclusion

The findings showed that the genetic program created couldmanage the work schedule of nurses under given conditionsand achieve the desired objective It could create work sche-dules of nurses that were fair in overtime payment to all thenurses and could reduce the hospital expenses by setting thenew coming nursesrsquo minimum salary at 15000 Baht per per-son

Table 25 Target equation and calculation time from geneticprogram

Target equation (obj) Calculation time (hour)03959 603

From Tables 23 24 and 25 the following results canbe concluded the work schedules from proposed methodreduce the waiting time of the number of patients waitinglonger than 125 times of the average length nursing servicefrom 3408 to less than 15 on average (starting from

Mathematical Problems in Engineering 15

when the patients register and wait to see the doctor) Ifwe want to schedule the work of the nurses in January 2012in order to meet the criteria for the service of this studywe have to increase the number of nurses from 18 to 20 sothat the number can perfectly meet the needs of the patientsaccording to the criteriaThework schedules of the nurses canbemost appropriate if the target equation is 03959which usesthe time of calculation as 603 hourswith a standard deviationof less than 8028and the overtimepayment less than 29564Baht per monthWhen this is compared with the actual workschedule in January 2013 the work schedule could reduce thenumber of nurses from 23 to 20 with a decreased standarddeviation that is 8908 and the decrease in the totalovertime paymentwould be 50148 Baht permonthWe couldcalculate the most appropriate work schedule quicker thanby using the conventional approach which results obtainedby Lingo program version 50 86849 with 1500 rounds forthe calculation cycle

As we know hospitals belong to the service businesssector continuous improvement along with response to cus-tomer satisfaction is the key success factor in this businessTo reduce the waiting time in every hospital process highstandards in medical care and highly experienced medicaldoctors are needed Presently Thailand has set its aim tobecome an Asian medical and tourist hub this strategy of thecountry may cause and force hospitals to improve their ser-vice quality for responding well to the high number of outpa-tients Updated data collection needs to be done service-level policies have to be reconsidered Demands of outpa-tient nurses and their schedules may change from time totime Inpatient nurses may be allocated to assist when themanagement wishes to reduce the fluctuation in the numberof outpatient nurses Thus modern management and highefficient tools are necessary this is the reason why GA isinvolved instead of an adoption of the conventional approachHowever regarding the implementation of simulation evenGA is not familiar to the hospital people Optimization capac-ity building is another issue that the hospitalmanagementwillhave to consider

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chiang Mai University (CMU) through the research admin-istration office provided budget to our Excellence Center inLogistics and Supply Chain Management (E-LSCM)

References

[1] T Mongkolrangsarit and W Thamapornpilat Development ofNurse Scheduling Method with Uncertain Workforce DemandDepartment of Industrial Engineering Faculty of EngineeringChulalongkorn University 2007

[2] C M Rothe and H B Wolfe ldquoCyclical scheduling and alloca-tion of nursing staffrdquo Socio-Economic Planning Sciences vol 7no 5 pp 471ndash487 1973

[3] R B Norby L E Freund and B Wagner ldquoA nurse staffingsystem based upon assignment difficultyrdquo Journal of NursingAdministration vol 7 no 9 pp 2ndash24 1977

[4] T Ryan B L Barker and F A Marciante ldquoA system fordetermining appropriate nurse staffingrdquo Journal of NursingAdministration vol 5 no 5 pp 30ndash38 1975

[5] H Wolfe and J P Young ldquoStaffing the nursing unit Part Icontrolled variable staffingrdquo Nursing Research vol 14 pp 236ndash243 1965

[6] HWolfe and J P Young ldquoStaffing the nursing unit IIThemult-iple assignment techniquerdquo Nursing Research vol 14 no 4 pp299ndash303 1965

[7] J L Arthur andA Ravindran ldquoAmultiple objective nurse sche-duling modelrdquo AIIE Transactions vol 13 no 1 pp 55ndash60 1981

[8] I Ozkarahan ldquoA disaggregation model of a flexible nursescheduling support systemrdquo Socio-Economic Planning Sciencesvol 25 no 1 pp 9ndash26 1991

[9] I Ozkarahan and J E Bailey ldquoGoal programming modelsubsystem of a flexible nurse scheduling support systemrdquo IIETransactions vol 20 no 3 pp 306ndash316 1988

[10] K A Dowsland ldquoNurse scheduling with tabu search and stra-tegic oscillationrdquo European Journal of Operational Research vol106 no 2-3 pp 393ndash407 1998

[11] P Tormos A Lova F Barber L Ingolotti M Abril and M ASalido ldquoA genetic algorithm for railway scheduling problemsrdquoinMetaheuristics for Scheduling in Industrial andManufacturingApplications vol 128 of Studies in Computational Intelligencepp 255ndash276 Springer Berlin Germany 2008

[12] H Kawanaka K Yamamoto T Yoshikawa T Shinogi and STsuruoka ldquoGenetic algorithm with the constraints for nursescheduling problemrdquo in Proceedings of the Congress on Evolu-tionary Computation vol 2 pp 1123ndash1130 Seoul Republic ofKorea May 2001

[13] H HMillar and KMona ldquoCyclic and non-cyclic scheduling of12 h shift nurses by network programmingrdquo European Journalof Operational Research vol 104 no 3 pp 582ndash592 1998

[14] I Blochlige ldquoModeling staff scheduling problems A tutorialrdquoEuropean Journal of Operational Research vol 158 no 3 pp533ndash542 2004

[15] B Jaumard F Semet and T Vovor ldquoA generalized linear pro-gramming model for nurse schedulingrdquo European Journal ofOperational Research vol 107 no 1 pp 1ndash18 1998

[16] T Cezik O Gunluk and H Luss ldquoAn integer programmingmodel for the weekly tour scheduling problemrdquoNaval ResearchLogistics vol 48 no 7 pp 607ndash624 2001

[17] Z Wang J-L Liu and X Yu ldquoSelf-fertilization based geneticalgorithm for university timetabling problemrdquo in Proceedingsof the 1st ACMSIGEVO Summit on Genetic and EvolutionaryComputation (GEC rsquo09) pp 1001ndash1004 ACM Shanghai ChinaJune 2009

[18] J Wang and K Chu ldquoAn application of genetic algorithms forthe flexible job-shop scheduling problemrdquo International Journalof Advancements in Computing Technology vol 4 no 3 pp 271ndash278 2012

[19] M Moz and M V Pato ldquoA genetic algorithm approach to anurse rerostering problemrdquo Computers amp Operations Researchvol 34 no 3 pp 667ndash691 2007

[20] C Wotthipong K Soradech and S Nidapan ldquoThe suitablegenetic operators for solving the university course timetablingproblemrdquo Journal of Convergence Information Technology vol8 pp 60ndash65 2013

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Page 4: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

4 Mathematical Problems in Engineering

Table 1 Demand for nurses verify case study

Department Period Hospital demand (person)Mon Tue Wed Thu Fri Sat Sun

SUR Regular 1 1 2 1 2 1 2Overtime 1 2 1 2 1 1 1

MED Regular 2 1 2 1 1 1 1Overtime 1 1 1 2 1 2 2

such as weight blood pressure temperature measure pre-symptom examined and finally recorded all measure data inthe file after it arrived Service time distributions recordedand fitted by statistic approach in Table 3 were used as thedata set in process box After that we saved the percentageinformation of those patients whose waiting time exceededthe service time limitation as shown in Figure 1

Then we adjusted the number of nurses (resource) inthe capacity column manually as shown in Table 4 until thepercentage of the patients waiting for more than the limit(average and half-width) in each department was less than15 according to what we wanted in every section as shownin Table 5 After that we processed 100 replications

From Table 4 it can be observed that the departments ofSURMED and EENT appointed two people PED appointedthree people andOBG appointed one person to work in theirdepartments

From Table 5 we can see that all the departments withthe percentage of patients waiting for more than the limit(average and half-width) do not exceed 15 The EENTMED OBG PED and SUR departments had the percentagesof patients waiting for longer than the limit (average pluspositive half-width) at 07918 20505 50424 27194and 56773 respectively We considered only the positivehalf-width because we would like to know the maximumpercentage of patients waiting

34 Simulation Model Verification In model verification wecompared the results between real situation which occurredin this hospital and the simulationmodel Average number ofpatients receiving service average number of patients waitinglonger than average service time and average number ofpatients waiting less than average service time were proposedfor preforming simulationmodel efficientThe 100 times witheight-hour simulation were conducted for all five outpatientsdepartments The results were reported in Table 6 In simu-lation row we reported the average number plusminus withtheir corresponding half-width

Table 6 reported that the OBG got the highest error interms of average number of patients receiving service due tothe low 119875 value reported in Table 3 With this error howeverwe can adopt this simulation model generated by the arenaprogram for real case application

After model verification manual adjustment of theresource capacity was done and selected under hospital topmanagement service policy The service level policy was setas no more than 15 of patients waiting longer than the

average service timeThen the appropriate resource (numberof nurses) from simulation was shown for each departmentin Table 7 This number of nurses will be used as the datafor nurses demand (Demand

119895119896119897) in the mathematical model

for obtaining the optimal nurse schedule Even thoughto obtain the optimal nurse schedule with conventionalapproach always consumes computational runtime and hard-ware implementation Genetic algorithm was proposed tosolve and obtain an acceptable solution

4 Programming for Genetic AlgorithmNurse Scheduling

We coded a genetic algorithm program for use in the shiftscheduling of nurses with the Matlab program by using thefollowing steps

41 Chromosome Format and Chromosome Encoding Achromosome format uses a binary number system and it isdivided into three levels as shown in Figure 2 where

level 1 is a nurse at 1 2 3 119873level 2 is the working daysMonday TuesdayWednes-day Sundaylevel 3 is the number of 8 bit code

42 Chromosome Encoding and Chromosome Decoding Toencode the chromosome we used the binary coded decimalsystem which is a system that uses the 8 bit binary codeinstead of the general code we have set up We can encodeand decode the chromosome as shown in Table 8 where

(i) NDA = not working during regular hours(ii) DA1 = working during regular hours at Division 1(iii) DA2 = working during regular hours at Division 2(iv) DA3 = working during regular hours at Division 3(v) DA4 = working during regular hours at Division 4(vi) DA5 = working during regular hours at Division 5(vii) NOT = not working overtime(viii) OT1 = working overtime at Division 1(ix) OT2 = working overtime at Division 2(x) OT3 = working overtime at Division 3(xi) OT4 = working overtime at Division 4(xii) OT5 = working overtime at Division 5

Mathematical Problems in Engineering 5

Table2Th

eGanttcharto

fnursesrsquoworking

schedu

lewas

constructedby

theL

ingo

program

Nurses

Mon

day

Tuesday

Wednesday

Thursday

Friday

Saturday

Sund

ayTo

talh

ours

Regu

larh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

e1

lowastlowast

lowastlowast

402

lowast

lowast

403

lowast

lowast

lowastlowast

404

lowast

lowast

lowast

405

lowast

lowast

406

lowast

lowastlowast

407

lowastlowast

lowastlowast

408

lowast

lowast

409

lowastlowast

4010

lowast

lowastlowast

40

6 Mathematical Problems in Engineering

Table3Distrib

utionpatie

ntsinterarriv

alandservicetim

e

Day

Departm

ent

Regu

larh

our

Overtim

ehou

rServicetim

e119875value

Avgtim

eInterarrivaltim

e119875value

Interarrivaltim

e119875value

Mon

day

SUR

minus05+EX

PO(852

)0333

minus05+EX

PO(107)

0225

05+WEIB(71118

5)gt0750

684

MED

minus05+EX

PO(539

)0654

minus05+EX

PO(768)

0616

POIS(413

)0051

413

PED

minus05+WEIB(95

1114)

0113

minus05+EX

PO(123)

0075

05+EX

PO(65)

0078

700

EENT

minus05+EX

PO(954)

0478

55+EX

PO(165)

0171

05+EX

PO(458)

0615

508

OBG

minus05+WEIB(146154)

0465

115+EX

PO(225)

0228

TRIA(253195

)0244

910

Tuesday

SUR

minus05+WEIB(10512)

0334

minus05+WEIB(98

70925)

0702

05+18lowastBE

TA(0774125)

0481

740

MED

minus05+WEIB(632133

)0510

minus05+EX

PO(913

)0746

05+LO

GN(412

532

)0109

443

PED

minus05+EX

PO(56)

0607

minus05+ER

LA(4982)

0526

15+EX

PO(87)

006

61020

EENT

minus05+GAMM(663111)

0063

minus05+EX

PO(105)

0057

05+18lowastBE

TA(0409123)

0427

498

OBG

minus05+GAMM(75913)

0145

15+EX

PO(301)

0125

TRIA(25596195)

0421

932

Sund

ay

SUR

minus05+LO

GN(97

126)

gt0750

minus05+EX

PO(14

4)

006

805+17lowastBE

TA(076

106)

0581

737

MED

minus05+GAMM(947116)

0283

15+EX

PO(224)

0105

UNIF(05185)

0589

876

PED

minus05+WEIB(5120841)

0197

minus05+EX

PO(103)

0733

05+EX

PO(45)

0210

500

EENT

minus05+LO

GN(105166)

0349

UNIF(minus05195)

0438

05+19lowastBE

TA(0898156)

0489

744

OBG

minus05+LO

GN(131207)

0673

05+EX

PO(175

)0097

POIS(635)

0126

635

Mathematical Problems in Engineering 7

Service LevelOBG

Process OBG Decide 5

MoreMean 5

LowerMean 5 0

0

True

False

0

0

Dispose 5

Service LevelEENT

Process EENT Decide 4

MoreMean 4

LowerMean 4 0

0

True

False

0

0

Dispose 4

Service LevelPED

Process PED Decide 3

MoreMean 3

LowerMean 3 0

0

True

False

0

0

Dispose 3

Service LevelMED

Process MED Decide 2

MoreMean 2

LowerMean 2 0

0

True

False

0

0

Dispose 2

Service LevelSUR

Process SUR Decide 1

MoreMean 1

LowerMean 1 0

0

True

False

0

0

Dispose 1 IT SUR 8hr 02 01

IT MED 8hr 02 01

IT PED 8hr 02 01

IT EENT 8hr 02 01

IT OBG 8hr 02 01

Figure 1 The simulation model created by the Arena program

8 Mathematical Problems in Engineering

Table 4 The example for adjusting the number of nurses in theArena program

Name Type Capacity1 Resource SUR Fixed capacity 22 Resource MED Fixed capacity 23 Resource PED Fixed capacity 34 Resource EENT Fixed capacity 25 Resource OBG Fixed capacity 1

Table 5 Average and half-width values based on number of nursesadjusted

Output Average Half-widthEENT 04418 035MED 12505 080OBG 34024 164PED 16794 104SUR 44273 125

43 Creating Initial Population Each nursersquos chromosomecan be randomly generated for the initial state Then checkwhether it is in the scope of primary goal or not withworkinghours fewer than 60 hours If that chromosome is not in thescope of the primary goal again random search until eachnursersquos chromosome completes for the entire length and allthe nursersquos chromosomes are generated to reach the numberof population specified

44 Crossover and Mutation Crossover is an importantoperator which combines the good properties of both parentsin order to possibly yield new better children chromosomes[17 18] As usual the simple crossover operator (one-pointcrossover) consists of randomly choosing a crossover pointand then recombining the pieces of a pair of chromosomes toform two new chromosomes The simple crossover is com-patible with the random keys encoding though it generallyfails to preserve the permutation when dealing with naturalencoding Hence for natural encoding special crossoveroperators must be used [19] Roulette wheel selection (RWS)ranking selection (RS) tournament selection (TS) partiallymatched crossover (PMX) order crossover (OX) and cyclecrossover (CX) are evaluated by comparing the performanceof GArsquos operators on university course timetabling problem[20] PMX OX and CX operators require two crossoverpoints Given two parent chromosomes 119860 and 119861 childchromosome1198601015840 will inherit form119860 the subsequence betweenthese two points and child chromosome 1198611015840 form 119861 therespective subsequence The elements of 1198601015840 and 1198611015840 outsidethe two points are copied from the other parent chromosomewhile trying to preserve its position under the PMX operatoror by trying to preserve its order under the OX operatorA comparative analysis of PMX CX and OX crossoveroperators for solving travelling salesman problem (TSP) wasreported in [21]The experimental results show that the PMXcrossover outperforms the CX and OX crossover operator inTSP with 25 numbers of cities For our problem simple testwas conducted and we found PMX provided a better solution

Day

1

1

2

2

3

3

4

4 Nmiddot middot middot

middot middot middot

Mo Tu We Th Fr Sa Su

Number of nurses

Main chromosome

1 1 110 000

Binary code

Level 1

Level 2

Level 3

N lowast 7 lowast 8

Figure 2 The chromosome format

than one-point crossover specifically one-point crossovercannot take into account all constraints (3)ndash(8)

After applying crossover the mutation operator actson the pairs of chromosomes Although mutation occursinfrequently in nature it is believed to be an importantdriving force for evolution The mutation is adopted to allowfor the introduction of new chromosome into the populationand is effective to escape from a local optimum The simplemutation operates by randomly changing its value with agiven probability used in our experiments

45 Evaluation of Fitness Value The fitness value can be eva-luated by using two criteria as follows

Criterion one a chromosome is a feasible solution

(i) Fitness value is the objective function that isprocessed by (2)

Criterion two a chromosome is a nonfeasible solu-tion

(i) Fitness value is equal to 1000 units

46 Chromosome Selection We selected the chromosome byusing roulette

47 Elite Preserve Strategy The previous chromosomes withhigh fitness values were replaced by the new chromosomeswith the lower fitness values Then they were used as theinitial population for the next iteration

48 Termination Criteria The condition to stop seeking an-swers was that when the required number of solutions fullymeets the numbers of iteration specified we stop the exami-nation process immediately

5 Experiment Design to Determine OptimalConfiguration in Genetic Programming

In this section we introduced design of experiment (DOE)technique for screening the effected parameters and param-eters setting guideline for genetic algorithm solution search

Mathematical Problems in Engineering 9

Table 6 Comparison between real situation and simulation model

Department Calculated resultAVG number ofpatients receiving

service

AVG number ofpatients waitinglonger than AVG

time

AVG number ofpatients waiting lessthan AVG time

Percentage of patientswaiting longer than

AVG time

SURReal situation 59 02 58 0338Simulation 6163 plusmn 163 019 plusmn 036 6051 plusmn 166 0196 plusmn 032 error 427 526 433 mdash

MEDReal situation 96 0 98 0Simulation 9879 plusmn 191 0 plusmn 0 9794 plusmn 191 0 error 290 0 006 mdash

PEDReal situation 55 0 53 0Simulation 5758 plusmn 148 0 plusmn 0 5669 plusmn 146 0 error 469 0 696 mdash

EENTReal situation 53 182 34 3434Simulation 5446 plusmn 176 1673 plusmn 414 3664 plusmn 346 1999 plusmn 512 error 275 807 776 mdash

OBGReal situation 34 0 35 0Simulation 384 plusmn 085 0 plusmn 0 377 plusmn 084 0 error 1294 0 771 mdash

Table 7 Demand for nurses obtained from simulation model

Department Period Hospital demand (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 3 3 2 2PED 3 3 3 3 3 3 3EENT 2 2 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 2 2EENT 2 2 2 2 2 2 2OBG 1 1 1 2 1 1 1

machine We conducted the experiment by fixing currentnumber of nurses at 18 with the low level nursesrsquo demand ineach department All three expecting parameters were set bytwo levels maximum and minimum

51 Determining Factors and Factor Values in Each LevelFactors can be chosen for the experiment as follows

(1) population number(2) percent crossover rate(3) percent mutation rate

We used the factors in Table 9 to construct the experi-mental design matrix full factorial design (2119896) (8) and thenconducted the experiment by using three center points perblockwith all the three replicatesWe also fixed the number ofmaximum cycles to 400 cyclesThis was because the researchrevealed that the number of the calculated cycles had high

stability The cycles also took a long time to calculate andso may cause inconvenience in their actual use The numberof nurses 18 in total was fixed at the beginning for theexperimental design and the results are shown in Table 10which includes the use of the data for the demand for nursesfrom the first week of January

52 Effect and Coefficient Analysis and Model AdequacyChecking We used the experimental results to analyze theeffects and coefficients by using theMinitab program in orderto find the factors affecting the expected responses

Figure 3 shows that the data were normally and consis-tently distributed without any tendency and can be used toanalyze further results

Table 11 shows that the correlation of the data is linearbecause the center pt is greater than 005 and the 119875 values ofthe 119860 119862 119860119862 119861119862 and 119860119861119862 factors are less than 005 whichis significant for the experiment It was concluded that the 119860

10 Mathematical Problems in Engineering

Table 8 Example for encoding binary coded decimal

Decimal Binary Regular hour Overtime hour Code Type Demand 1 Demand 2 Hours0 00000000 NDA NOT NDA NOT Type 0 0 0 01 00000001 DA1 NOT DA1 NOT Type 1 DA1++ 0 82 00000010 DA2 NOT DA2 NOT Type 2 DA2++ 0 83 00000011 DA3 NOT DA3 NOT Type 3 DA3++ 0 8

255 11111111 NDA NOT NDA NOT Type 255 0 0 0

001000050000

99

90

50

10

1

Residual054052050048046

0010

0005

0000

Fitted value

8

6

4

2

02624222018161412108642

0010

0005

0000

Observation order

Normal probability plot Versus fits

Histogram Versus order

Residual plots for obj

Resid

ual

Resid

ual

minus0010

minus0010

minus0005

minus0005

minus0010

minus0005

001000050000Residual

minus0010 minus0005

Freq

uenc

y(

)

Figure 3 The residual plots for objective function

Table 9 Selected experimental factors and factor values in eachlevel

Factor LevelLow High

(1) Population number (119860) 10 50(2) Percent crossover rate (119861) 5 95(3) Percent mutation rate (119862) 5 95

and 119862 factors were significant at the confidence level of 005as shown in Figure 4

The 119877-Sq value ranges from 0 to 1 or from 0 to 100If the 119877-Sq value is very close to 1 then it indicates that thesimulation can explain the different variables more properlyand accurately Table 12 shows that the 119877-Sq value is equalto 9948 which is very high This means that the data can

explain dependent variables properly and can analyze furtherresponse optimizer functions

53 Analysis for Finding outMost Appropriate Results by UsingResponse Optimizer The results were analyzed by using aresponse optimizer of Minitab as shown in Figures 5 and 6

From the results obtained as shown in Figure 5 this studyaims to determine the lowest target equation As a result wechanged ldquoGoalrdquo to ldquoMinimizerdquo since the target equation wasthe sum of the hospital expense percentage and the standarddeviation of the overtime each nurse received which rangedfrom 0 to 100 As a result the target was set to 0 and the uppertarget to 100

Figure 6 shows that if we want to configure the geneticprogram to obtain the lowest target equation it must beset as shown in Table 13 From the prediction equation thetarget equation should be equal to 04492 (obj = 04492)

Mathematical Problems in Engineering 11

Table 10 Full factorial design (23) experiment with three center points per block type

Run Std order Run order Center pt Blocks 119860 119861 119862 Obj Time (s)1 1 1 1 1 10 5 5 045892 92375212 2 2 1 1 50 5 5 050282 576219023 3 3 1 1 10 95 5 047113 7904851

25 25 25 0 1 30 50 50 048018 31281694

Table 11 Effect and obj coefficient analysis results

Factorial fit obj versus 119860 119861 119862Estimated effects and coefficients for obj (coded units)

Term Effect Coeff SE coeff 119879 119875

Constant 048216 0001318 36575 0000119860 005200 002600 0001318 1972 0000119861 000027 000013 0001318 010 0921119862 minus002134 minus001067 0001318 minus810 0000119860 lowast 119861 minus000285 minus000143 0001318 minus108 0293119860 lowast 119862 minus000850 minus000425 0001318 minus322 0005119861 lowast 119862 minus001395 minus000697 0001318 minus529 0000119860 lowast 119861 lowast 119862 minus000974 minus000487 0001318 minus369 0002Ct Pt minus000198 0003955 minus050 0623

Table 12 119877-square decision-making coefficients

119878 = 208209 PRESS = 152191461119877-Sq = 9948 119877-Sq (pred) = 9899 119877-Sq (adj) = 9925

Table 13 Guideline setting for genetic program from responseoptimizer function

Factor Level(1) Population number (119860) 10(2) Percent crossover rate (119861) 95(3) Percent mutation rate (119862) 95

Possible optimal parameters settingmay occur at the low level mutation we conducted the experiment and reported inTable 14 The reports showed that the objective function isworse when mutation decreased The reports also shownthat the higher mutation the better objective functions

54 Result Confirmation Experiment We experimented tocheck for accuracy by configuring the genetic program asshown in Table 12 together with fixing the maximum itera-tions to the number of nurses at 400 cycles and 18 nursesrespectively as shown in Table 15

Table 15 shows that the actual target value (obj) is actuallybetter than (ie less than) 063 of the predicted value andcan be used practically and actually decreased the target equa-tion (obj)

20151050

99

95908070605040302010

5

1

Standardized effect

Factor NameNot significantSignificant

Effect type

Normal plot of the standardized effects

minus10 minus5

(response is obj 120572 = 005)

()

A

A

B

C

A

B

C

C

BC

ABC

AC

Figure 4 The normal plot of the standardized effects (response isobj alpha = 005)

6 Change in Number of Nurses (119894) in OptimalScheduling under Constraint Equation

We changed the number of nurses (119894) which was an indexin the constraint equation in order to find out the least

12 Mathematical Problems in Engineering

Table 14 Experimental results obtained at low level mutation

Max iteration Population crossover mutation Obj Time (s)1500 10 95 95 044442 80491500 10 95 1 048580 101611500 10 95 01 050286 105231500 10 95 001 053612 10117

Table 15 Target equation (obj) predicted by Minitab program and compared with actual target equation (obj)

Target equation (obj) Comparison of target equation (obj)percentage

Predicted value Actual value Actual value compared with predicted value04492 04429 063

Table 16 Calculation results with reduction in number of nurses (119894)at 119908 = 06

Number of nurses (119894) Target equation(obj)

Calculating time(hour)

21 05241 76720 03959 60319 Infeasible Infeasible18 Infeasible Infeasible

Figure 5 The response optimizer configuration of obj response

number of nurses (119894) who were able to schedule the workunder the restrictions by using genetic programming and bygradually reducing the number of nurses (119894) down to the finalvalue at which the program could calculate the answer Thecalculation was performed at 119908 = 06 in Table 16

Table 16 shows that the least number of nurses (119894) whichcould schedule work under the restrictions was 20 Basedon 20 nurses amount of overtime payment and standarddeviation of overtime payment were calculated and reportedin Table 17 Therefore number of working hours for eachnurse number of nurses in each department and nursersquosschedule planwere generated and reported as shown inTables18 and 19

From Table 17 it can be observed that the target equation(obj) of the responses obtained was equal to 03959The sumof the overtime payment of all nurses was 16600 Baht per

CurHigh

Low099551

Optimal

minimumObj

099551desirabilityComposite

50950

50950

100500

50040004000 100 950 950

y = 04492

d = 099551

A B C D

D

Figure 6 The results of the data analysis using the responseoptimizer of the obj response

week The standard deviation of overtime payment that eachnurse received was 214843 For fair comparison we have toconvert the real overtime payment and standard deviationfor current 18 nurses to 20 nurses Then we can report thatthe sum of the total overtime payment and the standarddeviation of the overtime payment that each nurse receivedwhich is obtained from the model were less than those of theold working schedule at 17991 Baht per week and 882369respectively

Table 18 shows that all the nurses worked according to theconditions specified that is one nurse must work at least 40hours per week as in (4) but must work fewer than 60 hoursper week as in (5) all would be selected to work as in (6)and the duration of work must not exceed six days a week asin (7)

From Table 19 we can see that the number of nursesworking in shifts each day was more than the demand fornurses (Demand

119895119896119897) as shown in Table 6 which is according

to (3) and the specified service criteria (on average 15of thepatients waiting for services should not wait longer than theaverage service time of 25) because the number of nursesworking in shifts was more than the demand

To study the effect ofweight adjusted (119908)more numericalexperiments were conducted by GA coded in Matlab at 400

Mathematical Problems in Engineering 13

Table 17 Calculating results by 20 nurses (119894)

Target equation(obj)

Calculating time(hour)

Sum of overtime paymentof nurses (Baht)

Standard deviation of total overtimepayment that each nurse received

03959 603 16600 214843

Table 18 Number of working hours calculated from 20 nurses (119894)

Number of nurses Hoursperson (hour)Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total

1 12 12 0 4 0 8 12 482 4 8 12 4 8 12 0 483 8 8 8 0 12 12 0 484 12 0 12 12 12 4 0 525 8 12 8 4 0 0 12 44

20 8 12 4 0 12 8 4 48

Table 19 Number of nurses working in shifts calculated by 20 nurses (119894)

Department Period Supply (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 4 3 2 2PED 3 3 4 3 3 4 3EENT 2 3 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 3 3EENT 3 2 2 2 2 2 2OBG 2 2 3 2 1 1 2

Table 20 Effect of weight adjusted to objective function

Weight adjusted119908 = 09 119908 = 06 119908 = 05

Objective function 05090 04753 04823Sum of overtimepayment 24375 24625 26575

Standard deviation 324898 297675 288509

iterations and reported in Table 20 The report was shownas we expected more weight adjusted less overtime paymentsimultaneously with higher standard deviation It means thatif management focuses on reducing the overtime payment itwill lead to unfair payment or bias nursesrsquo schedule plan

7 Comparison of Advantages andDisadvantages of Adaptive Genetic andOptimization Approach

Weprocessed the data to do a comparison between the resultsof the adaptive genetic approach and the results obtained by

the Lingo program using the same data in order to comparethe target equation (obj) and computation runtime as shownin Tables 21 and 22

Table 21 shows the results of the processing experimentafter it was conducted ten times The work schedules using20 and 24 nurses with the calculating cycle of 400 roundsresulted in the best value for the target equations of 04361and 04551 respectively After ten processing experimentsthe work schedules for 20 and 24 nurses with the calculatingcycle at 1500 rounds resulted in the best value for the targetequations of 03959 and 04493 respectively and can besummarized as presented in Table 22

Table 22 shows that the work schedule for 20 nurses at1500 rounds when compared to the Lingo program couldcalculate thework schedulewith the target equation (obj) lessthan the genetic algorithm by around 580 and 108 plusmn 24on average but that the one created by the Lingo programused longer computational runtime by over 86849 and76452 on average We can also notice that the results at400 rounds reported the same direction the best and theaverage solution of genetic algorithm differ about 982 and1497 plusmn 38 from the optimal respectively However theoptimal method used longer time by over 306257 and

14 Mathematical Problems in Engineering

Table 21 Results of genetic program calculated ten times at 400 rounds and 1500 rounds

Target equation (obj)number

Number of 20 nurses Number of 24 nursesCalculating cycle

400Calculating cycle

1500Calculating cycle

400Calculating cycle

15001 04701 04578 05049 047302 04361 04436 05406 045523 04566 04772 04789 044934 04812 04303 04590 051605 04496 04781 04942 045866 04920 04549 05383 048337 04779 03959 05228 046268 05428 04352 05133 045809 05472 04368 05497 0498510 05231 04496 04551 04542Mean 04876 04459 05056 04708Standard deviation 0038 0024 0033 0021

Table 22 Comparison table for target equation (obj) and duration in calculation of GA and Lingo program

Number ofnurses(person)

Number ofcalculatingcycles

GA Lingo

Target equation (obj) Time (hour) Targetequation (obj) Time (hour) Different result

of obj

Different incomputational

run time

201500 Lowest 03959 603 580 86849

Mean 04459 plusmn 0024 685 03379 5237 1080 plusmn 24 76453

400 Lowest 04361 171 982 306257Mean 04876 plusmn 0038 166 1497 plusmn 38 315482

Table 23 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2012

Number of nurses Monthly expense SDOld New Increase Old New Decrease Old New Decrease18 20 2 395964 366400 29564 1089212 214843 8028

Table 24 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2013

Number of nurses Monthly expense SDOld New Decrease Old New Decrease Old New Decrease23 20 3 416548 366400 50148 1966735 214843 8908

315481 on average showing that the genetic program ismore suitable for practical use

8 Conclusion

The findings showed that the genetic program created couldmanage the work schedule of nurses under given conditionsand achieve the desired objective It could create work sche-dules of nurses that were fair in overtime payment to all thenurses and could reduce the hospital expenses by setting thenew coming nursesrsquo minimum salary at 15000 Baht per per-son

Table 25 Target equation and calculation time from geneticprogram

Target equation (obj) Calculation time (hour)03959 603

From Tables 23 24 and 25 the following results canbe concluded the work schedules from proposed methodreduce the waiting time of the number of patients waitinglonger than 125 times of the average length nursing servicefrom 3408 to less than 15 on average (starting from

Mathematical Problems in Engineering 15

when the patients register and wait to see the doctor) Ifwe want to schedule the work of the nurses in January 2012in order to meet the criteria for the service of this studywe have to increase the number of nurses from 18 to 20 sothat the number can perfectly meet the needs of the patientsaccording to the criteriaThework schedules of the nurses canbemost appropriate if the target equation is 03959which usesthe time of calculation as 603 hourswith a standard deviationof less than 8028and the overtimepayment less than 29564Baht per monthWhen this is compared with the actual workschedule in January 2013 the work schedule could reduce thenumber of nurses from 23 to 20 with a decreased standarddeviation that is 8908 and the decrease in the totalovertime paymentwould be 50148 Baht permonthWe couldcalculate the most appropriate work schedule quicker thanby using the conventional approach which results obtainedby Lingo program version 50 86849 with 1500 rounds forthe calculation cycle

As we know hospitals belong to the service businesssector continuous improvement along with response to cus-tomer satisfaction is the key success factor in this businessTo reduce the waiting time in every hospital process highstandards in medical care and highly experienced medicaldoctors are needed Presently Thailand has set its aim tobecome an Asian medical and tourist hub this strategy of thecountry may cause and force hospitals to improve their ser-vice quality for responding well to the high number of outpa-tients Updated data collection needs to be done service-level policies have to be reconsidered Demands of outpa-tient nurses and their schedules may change from time totime Inpatient nurses may be allocated to assist when themanagement wishes to reduce the fluctuation in the numberof outpatient nurses Thus modern management and highefficient tools are necessary this is the reason why GA isinvolved instead of an adoption of the conventional approachHowever regarding the implementation of simulation evenGA is not familiar to the hospital people Optimization capac-ity building is another issue that the hospitalmanagementwillhave to consider

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chiang Mai University (CMU) through the research admin-istration office provided budget to our Excellence Center inLogistics and Supply Chain Management (E-LSCM)

References

[1] T Mongkolrangsarit and W Thamapornpilat Development ofNurse Scheduling Method with Uncertain Workforce DemandDepartment of Industrial Engineering Faculty of EngineeringChulalongkorn University 2007

[2] C M Rothe and H B Wolfe ldquoCyclical scheduling and alloca-tion of nursing staffrdquo Socio-Economic Planning Sciences vol 7no 5 pp 471ndash487 1973

[3] R B Norby L E Freund and B Wagner ldquoA nurse staffingsystem based upon assignment difficultyrdquo Journal of NursingAdministration vol 7 no 9 pp 2ndash24 1977

[4] T Ryan B L Barker and F A Marciante ldquoA system fordetermining appropriate nurse staffingrdquo Journal of NursingAdministration vol 5 no 5 pp 30ndash38 1975

[5] H Wolfe and J P Young ldquoStaffing the nursing unit Part Icontrolled variable staffingrdquo Nursing Research vol 14 pp 236ndash243 1965

[6] HWolfe and J P Young ldquoStaffing the nursing unit IIThemult-iple assignment techniquerdquo Nursing Research vol 14 no 4 pp299ndash303 1965

[7] J L Arthur andA Ravindran ldquoAmultiple objective nurse sche-duling modelrdquo AIIE Transactions vol 13 no 1 pp 55ndash60 1981

[8] I Ozkarahan ldquoA disaggregation model of a flexible nursescheduling support systemrdquo Socio-Economic Planning Sciencesvol 25 no 1 pp 9ndash26 1991

[9] I Ozkarahan and J E Bailey ldquoGoal programming modelsubsystem of a flexible nurse scheduling support systemrdquo IIETransactions vol 20 no 3 pp 306ndash316 1988

[10] K A Dowsland ldquoNurse scheduling with tabu search and stra-tegic oscillationrdquo European Journal of Operational Research vol106 no 2-3 pp 393ndash407 1998

[11] P Tormos A Lova F Barber L Ingolotti M Abril and M ASalido ldquoA genetic algorithm for railway scheduling problemsrdquoinMetaheuristics for Scheduling in Industrial andManufacturingApplications vol 128 of Studies in Computational Intelligencepp 255ndash276 Springer Berlin Germany 2008

[12] H Kawanaka K Yamamoto T Yoshikawa T Shinogi and STsuruoka ldquoGenetic algorithm with the constraints for nursescheduling problemrdquo in Proceedings of the Congress on Evolu-tionary Computation vol 2 pp 1123ndash1130 Seoul Republic ofKorea May 2001

[13] H HMillar and KMona ldquoCyclic and non-cyclic scheduling of12 h shift nurses by network programmingrdquo European Journalof Operational Research vol 104 no 3 pp 582ndash592 1998

[14] I Blochlige ldquoModeling staff scheduling problems A tutorialrdquoEuropean Journal of Operational Research vol 158 no 3 pp533ndash542 2004

[15] B Jaumard F Semet and T Vovor ldquoA generalized linear pro-gramming model for nurse schedulingrdquo European Journal ofOperational Research vol 107 no 1 pp 1ndash18 1998

[16] T Cezik O Gunluk and H Luss ldquoAn integer programmingmodel for the weekly tour scheduling problemrdquoNaval ResearchLogistics vol 48 no 7 pp 607ndash624 2001

[17] Z Wang J-L Liu and X Yu ldquoSelf-fertilization based geneticalgorithm for university timetabling problemrdquo in Proceedingsof the 1st ACMSIGEVO Summit on Genetic and EvolutionaryComputation (GEC rsquo09) pp 1001ndash1004 ACM Shanghai ChinaJune 2009

[18] J Wang and K Chu ldquoAn application of genetic algorithms forthe flexible job-shop scheduling problemrdquo International Journalof Advancements in Computing Technology vol 4 no 3 pp 271ndash278 2012

[19] M Moz and M V Pato ldquoA genetic algorithm approach to anurse rerostering problemrdquo Computers amp Operations Researchvol 34 no 3 pp 667ndash691 2007

[20] C Wotthipong K Soradech and S Nidapan ldquoThe suitablegenetic operators for solving the university course timetablingproblemrdquo Journal of Convergence Information Technology vol8 pp 60ndash65 2013

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

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Mathematical Problems in Engineering

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Page 5: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

Mathematical Problems in Engineering 5

Table2Th

eGanttcharto

fnursesrsquoworking

schedu

lewas

constructedby

theL

ingo

program

Nurses

Mon

day

Tuesday

Wednesday

Thursday

Friday

Saturday

Sund

ayTo

talh

ours

Regu

larh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

eRe

gularh

ours

Overtim

e1

lowastlowast

lowastlowast

402

lowast

lowast

403

lowast

lowast

lowastlowast

404

lowast

lowast

lowast

405

lowast

lowast

406

lowast

lowastlowast

407

lowastlowast

lowastlowast

408

lowast

lowast

409

lowastlowast

4010

lowast

lowastlowast

40

6 Mathematical Problems in Engineering

Table3Distrib

utionpatie

ntsinterarriv

alandservicetim

e

Day

Departm

ent

Regu

larh

our

Overtim

ehou

rServicetim

e119875value

Avgtim

eInterarrivaltim

e119875value

Interarrivaltim

e119875value

Mon

day

SUR

minus05+EX

PO(852

)0333

minus05+EX

PO(107)

0225

05+WEIB(71118

5)gt0750

684

MED

minus05+EX

PO(539

)0654

minus05+EX

PO(768)

0616

POIS(413

)0051

413

PED

minus05+WEIB(95

1114)

0113

minus05+EX

PO(123)

0075

05+EX

PO(65)

0078

700

EENT

minus05+EX

PO(954)

0478

55+EX

PO(165)

0171

05+EX

PO(458)

0615

508

OBG

minus05+WEIB(146154)

0465

115+EX

PO(225)

0228

TRIA(253195

)0244

910

Tuesday

SUR

minus05+WEIB(10512)

0334

minus05+WEIB(98

70925)

0702

05+18lowastBE

TA(0774125)

0481

740

MED

minus05+WEIB(632133

)0510

minus05+EX

PO(913

)0746

05+LO

GN(412

532

)0109

443

PED

minus05+EX

PO(56)

0607

minus05+ER

LA(4982)

0526

15+EX

PO(87)

006

61020

EENT

minus05+GAMM(663111)

0063

minus05+EX

PO(105)

0057

05+18lowastBE

TA(0409123)

0427

498

OBG

minus05+GAMM(75913)

0145

15+EX

PO(301)

0125

TRIA(25596195)

0421

932

Sund

ay

SUR

minus05+LO

GN(97

126)

gt0750

minus05+EX

PO(14

4)

006

805+17lowastBE

TA(076

106)

0581

737

MED

minus05+GAMM(947116)

0283

15+EX

PO(224)

0105

UNIF(05185)

0589

876

PED

minus05+WEIB(5120841)

0197

minus05+EX

PO(103)

0733

05+EX

PO(45)

0210

500

EENT

minus05+LO

GN(105166)

0349

UNIF(minus05195)

0438

05+19lowastBE

TA(0898156)

0489

744

OBG

minus05+LO

GN(131207)

0673

05+EX

PO(175

)0097

POIS(635)

0126

635

Mathematical Problems in Engineering 7

Service LevelOBG

Process OBG Decide 5

MoreMean 5

LowerMean 5 0

0

True

False

0

0

Dispose 5

Service LevelEENT

Process EENT Decide 4

MoreMean 4

LowerMean 4 0

0

True

False

0

0

Dispose 4

Service LevelPED

Process PED Decide 3

MoreMean 3

LowerMean 3 0

0

True

False

0

0

Dispose 3

Service LevelMED

Process MED Decide 2

MoreMean 2

LowerMean 2 0

0

True

False

0

0

Dispose 2

Service LevelSUR

Process SUR Decide 1

MoreMean 1

LowerMean 1 0

0

True

False

0

0

Dispose 1 IT SUR 8hr 02 01

IT MED 8hr 02 01

IT PED 8hr 02 01

IT EENT 8hr 02 01

IT OBG 8hr 02 01

Figure 1 The simulation model created by the Arena program

8 Mathematical Problems in Engineering

Table 4 The example for adjusting the number of nurses in theArena program

Name Type Capacity1 Resource SUR Fixed capacity 22 Resource MED Fixed capacity 23 Resource PED Fixed capacity 34 Resource EENT Fixed capacity 25 Resource OBG Fixed capacity 1

Table 5 Average and half-width values based on number of nursesadjusted

Output Average Half-widthEENT 04418 035MED 12505 080OBG 34024 164PED 16794 104SUR 44273 125

43 Creating Initial Population Each nursersquos chromosomecan be randomly generated for the initial state Then checkwhether it is in the scope of primary goal or not withworkinghours fewer than 60 hours If that chromosome is not in thescope of the primary goal again random search until eachnursersquos chromosome completes for the entire length and allthe nursersquos chromosomes are generated to reach the numberof population specified

44 Crossover and Mutation Crossover is an importantoperator which combines the good properties of both parentsin order to possibly yield new better children chromosomes[17 18] As usual the simple crossover operator (one-pointcrossover) consists of randomly choosing a crossover pointand then recombining the pieces of a pair of chromosomes toform two new chromosomes The simple crossover is com-patible with the random keys encoding though it generallyfails to preserve the permutation when dealing with naturalencoding Hence for natural encoding special crossoveroperators must be used [19] Roulette wheel selection (RWS)ranking selection (RS) tournament selection (TS) partiallymatched crossover (PMX) order crossover (OX) and cyclecrossover (CX) are evaluated by comparing the performanceof GArsquos operators on university course timetabling problem[20] PMX OX and CX operators require two crossoverpoints Given two parent chromosomes 119860 and 119861 childchromosome1198601015840 will inherit form119860 the subsequence betweenthese two points and child chromosome 1198611015840 form 119861 therespective subsequence The elements of 1198601015840 and 1198611015840 outsidethe two points are copied from the other parent chromosomewhile trying to preserve its position under the PMX operatoror by trying to preserve its order under the OX operatorA comparative analysis of PMX CX and OX crossoveroperators for solving travelling salesman problem (TSP) wasreported in [21]The experimental results show that the PMXcrossover outperforms the CX and OX crossover operator inTSP with 25 numbers of cities For our problem simple testwas conducted and we found PMX provided a better solution

Day

1

1

2

2

3

3

4

4 Nmiddot middot middot

middot middot middot

Mo Tu We Th Fr Sa Su

Number of nurses

Main chromosome

1 1 110 000

Binary code

Level 1

Level 2

Level 3

N lowast 7 lowast 8

Figure 2 The chromosome format

than one-point crossover specifically one-point crossovercannot take into account all constraints (3)ndash(8)

After applying crossover the mutation operator actson the pairs of chromosomes Although mutation occursinfrequently in nature it is believed to be an importantdriving force for evolution The mutation is adopted to allowfor the introduction of new chromosome into the populationand is effective to escape from a local optimum The simplemutation operates by randomly changing its value with agiven probability used in our experiments

45 Evaluation of Fitness Value The fitness value can be eva-luated by using two criteria as follows

Criterion one a chromosome is a feasible solution

(i) Fitness value is the objective function that isprocessed by (2)

Criterion two a chromosome is a nonfeasible solu-tion

(i) Fitness value is equal to 1000 units

46 Chromosome Selection We selected the chromosome byusing roulette

47 Elite Preserve Strategy The previous chromosomes withhigh fitness values were replaced by the new chromosomeswith the lower fitness values Then they were used as theinitial population for the next iteration

48 Termination Criteria The condition to stop seeking an-swers was that when the required number of solutions fullymeets the numbers of iteration specified we stop the exami-nation process immediately

5 Experiment Design to Determine OptimalConfiguration in Genetic Programming

In this section we introduced design of experiment (DOE)technique for screening the effected parameters and param-eters setting guideline for genetic algorithm solution search

Mathematical Problems in Engineering 9

Table 6 Comparison between real situation and simulation model

Department Calculated resultAVG number ofpatients receiving

service

AVG number ofpatients waitinglonger than AVG

time

AVG number ofpatients waiting lessthan AVG time

Percentage of patientswaiting longer than

AVG time

SURReal situation 59 02 58 0338Simulation 6163 plusmn 163 019 plusmn 036 6051 plusmn 166 0196 plusmn 032 error 427 526 433 mdash

MEDReal situation 96 0 98 0Simulation 9879 plusmn 191 0 plusmn 0 9794 plusmn 191 0 error 290 0 006 mdash

PEDReal situation 55 0 53 0Simulation 5758 plusmn 148 0 plusmn 0 5669 plusmn 146 0 error 469 0 696 mdash

EENTReal situation 53 182 34 3434Simulation 5446 plusmn 176 1673 plusmn 414 3664 plusmn 346 1999 plusmn 512 error 275 807 776 mdash

OBGReal situation 34 0 35 0Simulation 384 plusmn 085 0 plusmn 0 377 plusmn 084 0 error 1294 0 771 mdash

Table 7 Demand for nurses obtained from simulation model

Department Period Hospital demand (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 3 3 2 2PED 3 3 3 3 3 3 3EENT 2 2 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 2 2EENT 2 2 2 2 2 2 2OBG 1 1 1 2 1 1 1

machine We conducted the experiment by fixing currentnumber of nurses at 18 with the low level nursesrsquo demand ineach department All three expecting parameters were set bytwo levels maximum and minimum

51 Determining Factors and Factor Values in Each LevelFactors can be chosen for the experiment as follows

(1) population number(2) percent crossover rate(3) percent mutation rate

We used the factors in Table 9 to construct the experi-mental design matrix full factorial design (2119896) (8) and thenconducted the experiment by using three center points perblockwith all the three replicatesWe also fixed the number ofmaximum cycles to 400 cyclesThis was because the researchrevealed that the number of the calculated cycles had high

stability The cycles also took a long time to calculate andso may cause inconvenience in their actual use The numberof nurses 18 in total was fixed at the beginning for theexperimental design and the results are shown in Table 10which includes the use of the data for the demand for nursesfrom the first week of January

52 Effect and Coefficient Analysis and Model AdequacyChecking We used the experimental results to analyze theeffects and coefficients by using theMinitab program in orderto find the factors affecting the expected responses

Figure 3 shows that the data were normally and consis-tently distributed without any tendency and can be used toanalyze further results

Table 11 shows that the correlation of the data is linearbecause the center pt is greater than 005 and the 119875 values ofthe 119860 119862 119860119862 119861119862 and 119860119861119862 factors are less than 005 whichis significant for the experiment It was concluded that the 119860

10 Mathematical Problems in Engineering

Table 8 Example for encoding binary coded decimal

Decimal Binary Regular hour Overtime hour Code Type Demand 1 Demand 2 Hours0 00000000 NDA NOT NDA NOT Type 0 0 0 01 00000001 DA1 NOT DA1 NOT Type 1 DA1++ 0 82 00000010 DA2 NOT DA2 NOT Type 2 DA2++ 0 83 00000011 DA3 NOT DA3 NOT Type 3 DA3++ 0 8

255 11111111 NDA NOT NDA NOT Type 255 0 0 0

001000050000

99

90

50

10

1

Residual054052050048046

0010

0005

0000

Fitted value

8

6

4

2

02624222018161412108642

0010

0005

0000

Observation order

Normal probability plot Versus fits

Histogram Versus order

Residual plots for obj

Resid

ual

Resid

ual

minus0010

minus0010

minus0005

minus0005

minus0010

minus0005

001000050000Residual

minus0010 minus0005

Freq

uenc

y(

)

Figure 3 The residual plots for objective function

Table 9 Selected experimental factors and factor values in eachlevel

Factor LevelLow High

(1) Population number (119860) 10 50(2) Percent crossover rate (119861) 5 95(3) Percent mutation rate (119862) 5 95

and 119862 factors were significant at the confidence level of 005as shown in Figure 4

The 119877-Sq value ranges from 0 to 1 or from 0 to 100If the 119877-Sq value is very close to 1 then it indicates that thesimulation can explain the different variables more properlyand accurately Table 12 shows that the 119877-Sq value is equalto 9948 which is very high This means that the data can

explain dependent variables properly and can analyze furtherresponse optimizer functions

53 Analysis for Finding outMost Appropriate Results by UsingResponse Optimizer The results were analyzed by using aresponse optimizer of Minitab as shown in Figures 5 and 6

From the results obtained as shown in Figure 5 this studyaims to determine the lowest target equation As a result wechanged ldquoGoalrdquo to ldquoMinimizerdquo since the target equation wasthe sum of the hospital expense percentage and the standarddeviation of the overtime each nurse received which rangedfrom 0 to 100 As a result the target was set to 0 and the uppertarget to 100

Figure 6 shows that if we want to configure the geneticprogram to obtain the lowest target equation it must beset as shown in Table 13 From the prediction equation thetarget equation should be equal to 04492 (obj = 04492)

Mathematical Problems in Engineering 11

Table 10 Full factorial design (23) experiment with three center points per block type

Run Std order Run order Center pt Blocks 119860 119861 119862 Obj Time (s)1 1 1 1 1 10 5 5 045892 92375212 2 2 1 1 50 5 5 050282 576219023 3 3 1 1 10 95 5 047113 7904851

25 25 25 0 1 30 50 50 048018 31281694

Table 11 Effect and obj coefficient analysis results

Factorial fit obj versus 119860 119861 119862Estimated effects and coefficients for obj (coded units)

Term Effect Coeff SE coeff 119879 119875

Constant 048216 0001318 36575 0000119860 005200 002600 0001318 1972 0000119861 000027 000013 0001318 010 0921119862 minus002134 minus001067 0001318 minus810 0000119860 lowast 119861 minus000285 minus000143 0001318 minus108 0293119860 lowast 119862 minus000850 minus000425 0001318 minus322 0005119861 lowast 119862 minus001395 minus000697 0001318 minus529 0000119860 lowast 119861 lowast 119862 minus000974 minus000487 0001318 minus369 0002Ct Pt minus000198 0003955 minus050 0623

Table 12 119877-square decision-making coefficients

119878 = 208209 PRESS = 152191461119877-Sq = 9948 119877-Sq (pred) = 9899 119877-Sq (adj) = 9925

Table 13 Guideline setting for genetic program from responseoptimizer function

Factor Level(1) Population number (119860) 10(2) Percent crossover rate (119861) 95(3) Percent mutation rate (119862) 95

Possible optimal parameters settingmay occur at the low level mutation we conducted the experiment and reported inTable 14 The reports showed that the objective function isworse when mutation decreased The reports also shownthat the higher mutation the better objective functions

54 Result Confirmation Experiment We experimented tocheck for accuracy by configuring the genetic program asshown in Table 12 together with fixing the maximum itera-tions to the number of nurses at 400 cycles and 18 nursesrespectively as shown in Table 15

Table 15 shows that the actual target value (obj) is actuallybetter than (ie less than) 063 of the predicted value andcan be used practically and actually decreased the target equa-tion (obj)

20151050

99

95908070605040302010

5

1

Standardized effect

Factor NameNot significantSignificant

Effect type

Normal plot of the standardized effects

minus10 minus5

(response is obj 120572 = 005)

()

A

A

B

C

A

B

C

C

BC

ABC

AC

Figure 4 The normal plot of the standardized effects (response isobj alpha = 005)

6 Change in Number of Nurses (119894) in OptimalScheduling under Constraint Equation

We changed the number of nurses (119894) which was an indexin the constraint equation in order to find out the least

12 Mathematical Problems in Engineering

Table 14 Experimental results obtained at low level mutation

Max iteration Population crossover mutation Obj Time (s)1500 10 95 95 044442 80491500 10 95 1 048580 101611500 10 95 01 050286 105231500 10 95 001 053612 10117

Table 15 Target equation (obj) predicted by Minitab program and compared with actual target equation (obj)

Target equation (obj) Comparison of target equation (obj)percentage

Predicted value Actual value Actual value compared with predicted value04492 04429 063

Table 16 Calculation results with reduction in number of nurses (119894)at 119908 = 06

Number of nurses (119894) Target equation(obj)

Calculating time(hour)

21 05241 76720 03959 60319 Infeasible Infeasible18 Infeasible Infeasible

Figure 5 The response optimizer configuration of obj response

number of nurses (119894) who were able to schedule the workunder the restrictions by using genetic programming and bygradually reducing the number of nurses (119894) down to the finalvalue at which the program could calculate the answer Thecalculation was performed at 119908 = 06 in Table 16

Table 16 shows that the least number of nurses (119894) whichcould schedule work under the restrictions was 20 Basedon 20 nurses amount of overtime payment and standarddeviation of overtime payment were calculated and reportedin Table 17 Therefore number of working hours for eachnurse number of nurses in each department and nursersquosschedule planwere generated and reported as shown inTables18 and 19

From Table 17 it can be observed that the target equation(obj) of the responses obtained was equal to 03959The sumof the overtime payment of all nurses was 16600 Baht per

CurHigh

Low099551

Optimal

minimumObj

099551desirabilityComposite

50950

50950

100500

50040004000 100 950 950

y = 04492

d = 099551

A B C D

D

Figure 6 The results of the data analysis using the responseoptimizer of the obj response

week The standard deviation of overtime payment that eachnurse received was 214843 For fair comparison we have toconvert the real overtime payment and standard deviationfor current 18 nurses to 20 nurses Then we can report thatthe sum of the total overtime payment and the standarddeviation of the overtime payment that each nurse receivedwhich is obtained from the model were less than those of theold working schedule at 17991 Baht per week and 882369respectively

Table 18 shows that all the nurses worked according to theconditions specified that is one nurse must work at least 40hours per week as in (4) but must work fewer than 60 hoursper week as in (5) all would be selected to work as in (6)and the duration of work must not exceed six days a week asin (7)

From Table 19 we can see that the number of nursesworking in shifts each day was more than the demand fornurses (Demand

119895119896119897) as shown in Table 6 which is according

to (3) and the specified service criteria (on average 15of thepatients waiting for services should not wait longer than theaverage service time of 25) because the number of nursesworking in shifts was more than the demand

To study the effect ofweight adjusted (119908)more numericalexperiments were conducted by GA coded in Matlab at 400

Mathematical Problems in Engineering 13

Table 17 Calculating results by 20 nurses (119894)

Target equation(obj)

Calculating time(hour)

Sum of overtime paymentof nurses (Baht)

Standard deviation of total overtimepayment that each nurse received

03959 603 16600 214843

Table 18 Number of working hours calculated from 20 nurses (119894)

Number of nurses Hoursperson (hour)Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total

1 12 12 0 4 0 8 12 482 4 8 12 4 8 12 0 483 8 8 8 0 12 12 0 484 12 0 12 12 12 4 0 525 8 12 8 4 0 0 12 44

20 8 12 4 0 12 8 4 48

Table 19 Number of nurses working in shifts calculated by 20 nurses (119894)

Department Period Supply (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 4 3 2 2PED 3 3 4 3 3 4 3EENT 2 3 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 3 3EENT 3 2 2 2 2 2 2OBG 2 2 3 2 1 1 2

Table 20 Effect of weight adjusted to objective function

Weight adjusted119908 = 09 119908 = 06 119908 = 05

Objective function 05090 04753 04823Sum of overtimepayment 24375 24625 26575

Standard deviation 324898 297675 288509

iterations and reported in Table 20 The report was shownas we expected more weight adjusted less overtime paymentsimultaneously with higher standard deviation It means thatif management focuses on reducing the overtime payment itwill lead to unfair payment or bias nursesrsquo schedule plan

7 Comparison of Advantages andDisadvantages of Adaptive Genetic andOptimization Approach

Weprocessed the data to do a comparison between the resultsof the adaptive genetic approach and the results obtained by

the Lingo program using the same data in order to comparethe target equation (obj) and computation runtime as shownin Tables 21 and 22

Table 21 shows the results of the processing experimentafter it was conducted ten times The work schedules using20 and 24 nurses with the calculating cycle of 400 roundsresulted in the best value for the target equations of 04361and 04551 respectively After ten processing experimentsthe work schedules for 20 and 24 nurses with the calculatingcycle at 1500 rounds resulted in the best value for the targetequations of 03959 and 04493 respectively and can besummarized as presented in Table 22

Table 22 shows that the work schedule for 20 nurses at1500 rounds when compared to the Lingo program couldcalculate thework schedulewith the target equation (obj) lessthan the genetic algorithm by around 580 and 108 plusmn 24on average but that the one created by the Lingo programused longer computational runtime by over 86849 and76452 on average We can also notice that the results at400 rounds reported the same direction the best and theaverage solution of genetic algorithm differ about 982 and1497 plusmn 38 from the optimal respectively However theoptimal method used longer time by over 306257 and

14 Mathematical Problems in Engineering

Table 21 Results of genetic program calculated ten times at 400 rounds and 1500 rounds

Target equation (obj)number

Number of 20 nurses Number of 24 nursesCalculating cycle

400Calculating cycle

1500Calculating cycle

400Calculating cycle

15001 04701 04578 05049 047302 04361 04436 05406 045523 04566 04772 04789 044934 04812 04303 04590 051605 04496 04781 04942 045866 04920 04549 05383 048337 04779 03959 05228 046268 05428 04352 05133 045809 05472 04368 05497 0498510 05231 04496 04551 04542Mean 04876 04459 05056 04708Standard deviation 0038 0024 0033 0021

Table 22 Comparison table for target equation (obj) and duration in calculation of GA and Lingo program

Number ofnurses(person)

Number ofcalculatingcycles

GA Lingo

Target equation (obj) Time (hour) Targetequation (obj) Time (hour) Different result

of obj

Different incomputational

run time

201500 Lowest 03959 603 580 86849

Mean 04459 plusmn 0024 685 03379 5237 1080 plusmn 24 76453

400 Lowest 04361 171 982 306257Mean 04876 plusmn 0038 166 1497 plusmn 38 315482

Table 23 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2012

Number of nurses Monthly expense SDOld New Increase Old New Decrease Old New Decrease18 20 2 395964 366400 29564 1089212 214843 8028

Table 24 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2013

Number of nurses Monthly expense SDOld New Decrease Old New Decrease Old New Decrease23 20 3 416548 366400 50148 1966735 214843 8908

315481 on average showing that the genetic program ismore suitable for practical use

8 Conclusion

The findings showed that the genetic program created couldmanage the work schedule of nurses under given conditionsand achieve the desired objective It could create work sche-dules of nurses that were fair in overtime payment to all thenurses and could reduce the hospital expenses by setting thenew coming nursesrsquo minimum salary at 15000 Baht per per-son

Table 25 Target equation and calculation time from geneticprogram

Target equation (obj) Calculation time (hour)03959 603

From Tables 23 24 and 25 the following results canbe concluded the work schedules from proposed methodreduce the waiting time of the number of patients waitinglonger than 125 times of the average length nursing servicefrom 3408 to less than 15 on average (starting from

Mathematical Problems in Engineering 15

when the patients register and wait to see the doctor) Ifwe want to schedule the work of the nurses in January 2012in order to meet the criteria for the service of this studywe have to increase the number of nurses from 18 to 20 sothat the number can perfectly meet the needs of the patientsaccording to the criteriaThework schedules of the nurses canbemost appropriate if the target equation is 03959which usesthe time of calculation as 603 hourswith a standard deviationof less than 8028and the overtimepayment less than 29564Baht per monthWhen this is compared with the actual workschedule in January 2013 the work schedule could reduce thenumber of nurses from 23 to 20 with a decreased standarddeviation that is 8908 and the decrease in the totalovertime paymentwould be 50148 Baht permonthWe couldcalculate the most appropriate work schedule quicker thanby using the conventional approach which results obtainedby Lingo program version 50 86849 with 1500 rounds forthe calculation cycle

As we know hospitals belong to the service businesssector continuous improvement along with response to cus-tomer satisfaction is the key success factor in this businessTo reduce the waiting time in every hospital process highstandards in medical care and highly experienced medicaldoctors are needed Presently Thailand has set its aim tobecome an Asian medical and tourist hub this strategy of thecountry may cause and force hospitals to improve their ser-vice quality for responding well to the high number of outpa-tients Updated data collection needs to be done service-level policies have to be reconsidered Demands of outpa-tient nurses and their schedules may change from time totime Inpatient nurses may be allocated to assist when themanagement wishes to reduce the fluctuation in the numberof outpatient nurses Thus modern management and highefficient tools are necessary this is the reason why GA isinvolved instead of an adoption of the conventional approachHowever regarding the implementation of simulation evenGA is not familiar to the hospital people Optimization capac-ity building is another issue that the hospitalmanagementwillhave to consider

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chiang Mai University (CMU) through the research admin-istration office provided budget to our Excellence Center inLogistics and Supply Chain Management (E-LSCM)

References

[1] T Mongkolrangsarit and W Thamapornpilat Development ofNurse Scheduling Method with Uncertain Workforce DemandDepartment of Industrial Engineering Faculty of EngineeringChulalongkorn University 2007

[2] C M Rothe and H B Wolfe ldquoCyclical scheduling and alloca-tion of nursing staffrdquo Socio-Economic Planning Sciences vol 7no 5 pp 471ndash487 1973

[3] R B Norby L E Freund and B Wagner ldquoA nurse staffingsystem based upon assignment difficultyrdquo Journal of NursingAdministration vol 7 no 9 pp 2ndash24 1977

[4] T Ryan B L Barker and F A Marciante ldquoA system fordetermining appropriate nurse staffingrdquo Journal of NursingAdministration vol 5 no 5 pp 30ndash38 1975

[5] H Wolfe and J P Young ldquoStaffing the nursing unit Part Icontrolled variable staffingrdquo Nursing Research vol 14 pp 236ndash243 1965

[6] HWolfe and J P Young ldquoStaffing the nursing unit IIThemult-iple assignment techniquerdquo Nursing Research vol 14 no 4 pp299ndash303 1965

[7] J L Arthur andA Ravindran ldquoAmultiple objective nurse sche-duling modelrdquo AIIE Transactions vol 13 no 1 pp 55ndash60 1981

[8] I Ozkarahan ldquoA disaggregation model of a flexible nursescheduling support systemrdquo Socio-Economic Planning Sciencesvol 25 no 1 pp 9ndash26 1991

[9] I Ozkarahan and J E Bailey ldquoGoal programming modelsubsystem of a flexible nurse scheduling support systemrdquo IIETransactions vol 20 no 3 pp 306ndash316 1988

[10] K A Dowsland ldquoNurse scheduling with tabu search and stra-tegic oscillationrdquo European Journal of Operational Research vol106 no 2-3 pp 393ndash407 1998

[11] P Tormos A Lova F Barber L Ingolotti M Abril and M ASalido ldquoA genetic algorithm for railway scheduling problemsrdquoinMetaheuristics for Scheduling in Industrial andManufacturingApplications vol 128 of Studies in Computational Intelligencepp 255ndash276 Springer Berlin Germany 2008

[12] H Kawanaka K Yamamoto T Yoshikawa T Shinogi and STsuruoka ldquoGenetic algorithm with the constraints for nursescheduling problemrdquo in Proceedings of the Congress on Evolu-tionary Computation vol 2 pp 1123ndash1130 Seoul Republic ofKorea May 2001

[13] H HMillar and KMona ldquoCyclic and non-cyclic scheduling of12 h shift nurses by network programmingrdquo European Journalof Operational Research vol 104 no 3 pp 582ndash592 1998

[14] I Blochlige ldquoModeling staff scheduling problems A tutorialrdquoEuropean Journal of Operational Research vol 158 no 3 pp533ndash542 2004

[15] B Jaumard F Semet and T Vovor ldquoA generalized linear pro-gramming model for nurse schedulingrdquo European Journal ofOperational Research vol 107 no 1 pp 1ndash18 1998

[16] T Cezik O Gunluk and H Luss ldquoAn integer programmingmodel for the weekly tour scheduling problemrdquoNaval ResearchLogistics vol 48 no 7 pp 607ndash624 2001

[17] Z Wang J-L Liu and X Yu ldquoSelf-fertilization based geneticalgorithm for university timetabling problemrdquo in Proceedingsof the 1st ACMSIGEVO Summit on Genetic and EvolutionaryComputation (GEC rsquo09) pp 1001ndash1004 ACM Shanghai ChinaJune 2009

[18] J Wang and K Chu ldquoAn application of genetic algorithms forthe flexible job-shop scheduling problemrdquo International Journalof Advancements in Computing Technology vol 4 no 3 pp 271ndash278 2012

[19] M Moz and M V Pato ldquoA genetic algorithm approach to anurse rerostering problemrdquo Computers amp Operations Researchvol 34 no 3 pp 667ndash691 2007

[20] C Wotthipong K Soradech and S Nidapan ldquoThe suitablegenetic operators for solving the university course timetablingproblemrdquo Journal of Convergence Information Technology vol8 pp 60ndash65 2013

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

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Page 6: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

6 Mathematical Problems in Engineering

Table3Distrib

utionpatie

ntsinterarriv

alandservicetim

e

Day

Departm

ent

Regu

larh

our

Overtim

ehou

rServicetim

e119875value

Avgtim

eInterarrivaltim

e119875value

Interarrivaltim

e119875value

Mon

day

SUR

minus05+EX

PO(852

)0333

minus05+EX

PO(107)

0225

05+WEIB(71118

5)gt0750

684

MED

minus05+EX

PO(539

)0654

minus05+EX

PO(768)

0616

POIS(413

)0051

413

PED

minus05+WEIB(95

1114)

0113

minus05+EX

PO(123)

0075

05+EX

PO(65)

0078

700

EENT

minus05+EX

PO(954)

0478

55+EX

PO(165)

0171

05+EX

PO(458)

0615

508

OBG

minus05+WEIB(146154)

0465

115+EX

PO(225)

0228

TRIA(253195

)0244

910

Tuesday

SUR

minus05+WEIB(10512)

0334

minus05+WEIB(98

70925)

0702

05+18lowastBE

TA(0774125)

0481

740

MED

minus05+WEIB(632133

)0510

minus05+EX

PO(913

)0746

05+LO

GN(412

532

)0109

443

PED

minus05+EX

PO(56)

0607

minus05+ER

LA(4982)

0526

15+EX

PO(87)

006

61020

EENT

minus05+GAMM(663111)

0063

minus05+EX

PO(105)

0057

05+18lowastBE

TA(0409123)

0427

498

OBG

minus05+GAMM(75913)

0145

15+EX

PO(301)

0125

TRIA(25596195)

0421

932

Sund

ay

SUR

minus05+LO

GN(97

126)

gt0750

minus05+EX

PO(14

4)

006

805+17lowastBE

TA(076

106)

0581

737

MED

minus05+GAMM(947116)

0283

15+EX

PO(224)

0105

UNIF(05185)

0589

876

PED

minus05+WEIB(5120841)

0197

minus05+EX

PO(103)

0733

05+EX

PO(45)

0210

500

EENT

minus05+LO

GN(105166)

0349

UNIF(minus05195)

0438

05+19lowastBE

TA(0898156)

0489

744

OBG

minus05+LO

GN(131207)

0673

05+EX

PO(175

)0097

POIS(635)

0126

635

Mathematical Problems in Engineering 7

Service LevelOBG

Process OBG Decide 5

MoreMean 5

LowerMean 5 0

0

True

False

0

0

Dispose 5

Service LevelEENT

Process EENT Decide 4

MoreMean 4

LowerMean 4 0

0

True

False

0

0

Dispose 4

Service LevelPED

Process PED Decide 3

MoreMean 3

LowerMean 3 0

0

True

False

0

0

Dispose 3

Service LevelMED

Process MED Decide 2

MoreMean 2

LowerMean 2 0

0

True

False

0

0

Dispose 2

Service LevelSUR

Process SUR Decide 1

MoreMean 1

LowerMean 1 0

0

True

False

0

0

Dispose 1 IT SUR 8hr 02 01

IT MED 8hr 02 01

IT PED 8hr 02 01

IT EENT 8hr 02 01

IT OBG 8hr 02 01

Figure 1 The simulation model created by the Arena program

8 Mathematical Problems in Engineering

Table 4 The example for adjusting the number of nurses in theArena program

Name Type Capacity1 Resource SUR Fixed capacity 22 Resource MED Fixed capacity 23 Resource PED Fixed capacity 34 Resource EENT Fixed capacity 25 Resource OBG Fixed capacity 1

Table 5 Average and half-width values based on number of nursesadjusted

Output Average Half-widthEENT 04418 035MED 12505 080OBG 34024 164PED 16794 104SUR 44273 125

43 Creating Initial Population Each nursersquos chromosomecan be randomly generated for the initial state Then checkwhether it is in the scope of primary goal or not withworkinghours fewer than 60 hours If that chromosome is not in thescope of the primary goal again random search until eachnursersquos chromosome completes for the entire length and allthe nursersquos chromosomes are generated to reach the numberof population specified

44 Crossover and Mutation Crossover is an importantoperator which combines the good properties of both parentsin order to possibly yield new better children chromosomes[17 18] As usual the simple crossover operator (one-pointcrossover) consists of randomly choosing a crossover pointand then recombining the pieces of a pair of chromosomes toform two new chromosomes The simple crossover is com-patible with the random keys encoding though it generallyfails to preserve the permutation when dealing with naturalencoding Hence for natural encoding special crossoveroperators must be used [19] Roulette wheel selection (RWS)ranking selection (RS) tournament selection (TS) partiallymatched crossover (PMX) order crossover (OX) and cyclecrossover (CX) are evaluated by comparing the performanceof GArsquos operators on university course timetabling problem[20] PMX OX and CX operators require two crossoverpoints Given two parent chromosomes 119860 and 119861 childchromosome1198601015840 will inherit form119860 the subsequence betweenthese two points and child chromosome 1198611015840 form 119861 therespective subsequence The elements of 1198601015840 and 1198611015840 outsidethe two points are copied from the other parent chromosomewhile trying to preserve its position under the PMX operatoror by trying to preserve its order under the OX operatorA comparative analysis of PMX CX and OX crossoveroperators for solving travelling salesman problem (TSP) wasreported in [21]The experimental results show that the PMXcrossover outperforms the CX and OX crossover operator inTSP with 25 numbers of cities For our problem simple testwas conducted and we found PMX provided a better solution

Day

1

1

2

2

3

3

4

4 Nmiddot middot middot

middot middot middot

Mo Tu We Th Fr Sa Su

Number of nurses

Main chromosome

1 1 110 000

Binary code

Level 1

Level 2

Level 3

N lowast 7 lowast 8

Figure 2 The chromosome format

than one-point crossover specifically one-point crossovercannot take into account all constraints (3)ndash(8)

After applying crossover the mutation operator actson the pairs of chromosomes Although mutation occursinfrequently in nature it is believed to be an importantdriving force for evolution The mutation is adopted to allowfor the introduction of new chromosome into the populationand is effective to escape from a local optimum The simplemutation operates by randomly changing its value with agiven probability used in our experiments

45 Evaluation of Fitness Value The fitness value can be eva-luated by using two criteria as follows

Criterion one a chromosome is a feasible solution

(i) Fitness value is the objective function that isprocessed by (2)

Criterion two a chromosome is a nonfeasible solu-tion

(i) Fitness value is equal to 1000 units

46 Chromosome Selection We selected the chromosome byusing roulette

47 Elite Preserve Strategy The previous chromosomes withhigh fitness values were replaced by the new chromosomeswith the lower fitness values Then they were used as theinitial population for the next iteration

48 Termination Criteria The condition to stop seeking an-swers was that when the required number of solutions fullymeets the numbers of iteration specified we stop the exami-nation process immediately

5 Experiment Design to Determine OptimalConfiguration in Genetic Programming

In this section we introduced design of experiment (DOE)technique for screening the effected parameters and param-eters setting guideline for genetic algorithm solution search

Mathematical Problems in Engineering 9

Table 6 Comparison between real situation and simulation model

Department Calculated resultAVG number ofpatients receiving

service

AVG number ofpatients waitinglonger than AVG

time

AVG number ofpatients waiting lessthan AVG time

Percentage of patientswaiting longer than

AVG time

SURReal situation 59 02 58 0338Simulation 6163 plusmn 163 019 plusmn 036 6051 plusmn 166 0196 plusmn 032 error 427 526 433 mdash

MEDReal situation 96 0 98 0Simulation 9879 plusmn 191 0 plusmn 0 9794 plusmn 191 0 error 290 0 006 mdash

PEDReal situation 55 0 53 0Simulation 5758 plusmn 148 0 plusmn 0 5669 plusmn 146 0 error 469 0 696 mdash

EENTReal situation 53 182 34 3434Simulation 5446 plusmn 176 1673 plusmn 414 3664 plusmn 346 1999 plusmn 512 error 275 807 776 mdash

OBGReal situation 34 0 35 0Simulation 384 plusmn 085 0 plusmn 0 377 plusmn 084 0 error 1294 0 771 mdash

Table 7 Demand for nurses obtained from simulation model

Department Period Hospital demand (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 3 3 2 2PED 3 3 3 3 3 3 3EENT 2 2 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 2 2EENT 2 2 2 2 2 2 2OBG 1 1 1 2 1 1 1

machine We conducted the experiment by fixing currentnumber of nurses at 18 with the low level nursesrsquo demand ineach department All three expecting parameters were set bytwo levels maximum and minimum

51 Determining Factors and Factor Values in Each LevelFactors can be chosen for the experiment as follows

(1) population number(2) percent crossover rate(3) percent mutation rate

We used the factors in Table 9 to construct the experi-mental design matrix full factorial design (2119896) (8) and thenconducted the experiment by using three center points perblockwith all the three replicatesWe also fixed the number ofmaximum cycles to 400 cyclesThis was because the researchrevealed that the number of the calculated cycles had high

stability The cycles also took a long time to calculate andso may cause inconvenience in their actual use The numberof nurses 18 in total was fixed at the beginning for theexperimental design and the results are shown in Table 10which includes the use of the data for the demand for nursesfrom the first week of January

52 Effect and Coefficient Analysis and Model AdequacyChecking We used the experimental results to analyze theeffects and coefficients by using theMinitab program in orderto find the factors affecting the expected responses

Figure 3 shows that the data were normally and consis-tently distributed without any tendency and can be used toanalyze further results

Table 11 shows that the correlation of the data is linearbecause the center pt is greater than 005 and the 119875 values ofthe 119860 119862 119860119862 119861119862 and 119860119861119862 factors are less than 005 whichis significant for the experiment It was concluded that the 119860

10 Mathematical Problems in Engineering

Table 8 Example for encoding binary coded decimal

Decimal Binary Regular hour Overtime hour Code Type Demand 1 Demand 2 Hours0 00000000 NDA NOT NDA NOT Type 0 0 0 01 00000001 DA1 NOT DA1 NOT Type 1 DA1++ 0 82 00000010 DA2 NOT DA2 NOT Type 2 DA2++ 0 83 00000011 DA3 NOT DA3 NOT Type 3 DA3++ 0 8

255 11111111 NDA NOT NDA NOT Type 255 0 0 0

001000050000

99

90

50

10

1

Residual054052050048046

0010

0005

0000

Fitted value

8

6

4

2

02624222018161412108642

0010

0005

0000

Observation order

Normal probability plot Versus fits

Histogram Versus order

Residual plots for obj

Resid

ual

Resid

ual

minus0010

minus0010

minus0005

minus0005

minus0010

minus0005

001000050000Residual

minus0010 minus0005

Freq

uenc

y(

)

Figure 3 The residual plots for objective function

Table 9 Selected experimental factors and factor values in eachlevel

Factor LevelLow High

(1) Population number (119860) 10 50(2) Percent crossover rate (119861) 5 95(3) Percent mutation rate (119862) 5 95

and 119862 factors were significant at the confidence level of 005as shown in Figure 4

The 119877-Sq value ranges from 0 to 1 or from 0 to 100If the 119877-Sq value is very close to 1 then it indicates that thesimulation can explain the different variables more properlyand accurately Table 12 shows that the 119877-Sq value is equalto 9948 which is very high This means that the data can

explain dependent variables properly and can analyze furtherresponse optimizer functions

53 Analysis for Finding outMost Appropriate Results by UsingResponse Optimizer The results were analyzed by using aresponse optimizer of Minitab as shown in Figures 5 and 6

From the results obtained as shown in Figure 5 this studyaims to determine the lowest target equation As a result wechanged ldquoGoalrdquo to ldquoMinimizerdquo since the target equation wasthe sum of the hospital expense percentage and the standarddeviation of the overtime each nurse received which rangedfrom 0 to 100 As a result the target was set to 0 and the uppertarget to 100

Figure 6 shows that if we want to configure the geneticprogram to obtain the lowest target equation it must beset as shown in Table 13 From the prediction equation thetarget equation should be equal to 04492 (obj = 04492)

Mathematical Problems in Engineering 11

Table 10 Full factorial design (23) experiment with three center points per block type

Run Std order Run order Center pt Blocks 119860 119861 119862 Obj Time (s)1 1 1 1 1 10 5 5 045892 92375212 2 2 1 1 50 5 5 050282 576219023 3 3 1 1 10 95 5 047113 7904851

25 25 25 0 1 30 50 50 048018 31281694

Table 11 Effect and obj coefficient analysis results

Factorial fit obj versus 119860 119861 119862Estimated effects and coefficients for obj (coded units)

Term Effect Coeff SE coeff 119879 119875

Constant 048216 0001318 36575 0000119860 005200 002600 0001318 1972 0000119861 000027 000013 0001318 010 0921119862 minus002134 minus001067 0001318 minus810 0000119860 lowast 119861 minus000285 minus000143 0001318 minus108 0293119860 lowast 119862 minus000850 minus000425 0001318 minus322 0005119861 lowast 119862 minus001395 minus000697 0001318 minus529 0000119860 lowast 119861 lowast 119862 minus000974 minus000487 0001318 minus369 0002Ct Pt minus000198 0003955 minus050 0623

Table 12 119877-square decision-making coefficients

119878 = 208209 PRESS = 152191461119877-Sq = 9948 119877-Sq (pred) = 9899 119877-Sq (adj) = 9925

Table 13 Guideline setting for genetic program from responseoptimizer function

Factor Level(1) Population number (119860) 10(2) Percent crossover rate (119861) 95(3) Percent mutation rate (119862) 95

Possible optimal parameters settingmay occur at the low level mutation we conducted the experiment and reported inTable 14 The reports showed that the objective function isworse when mutation decreased The reports also shownthat the higher mutation the better objective functions

54 Result Confirmation Experiment We experimented tocheck for accuracy by configuring the genetic program asshown in Table 12 together with fixing the maximum itera-tions to the number of nurses at 400 cycles and 18 nursesrespectively as shown in Table 15

Table 15 shows that the actual target value (obj) is actuallybetter than (ie less than) 063 of the predicted value andcan be used practically and actually decreased the target equa-tion (obj)

20151050

99

95908070605040302010

5

1

Standardized effect

Factor NameNot significantSignificant

Effect type

Normal plot of the standardized effects

minus10 minus5

(response is obj 120572 = 005)

()

A

A

B

C

A

B

C

C

BC

ABC

AC

Figure 4 The normal plot of the standardized effects (response isobj alpha = 005)

6 Change in Number of Nurses (119894) in OptimalScheduling under Constraint Equation

We changed the number of nurses (119894) which was an indexin the constraint equation in order to find out the least

12 Mathematical Problems in Engineering

Table 14 Experimental results obtained at low level mutation

Max iteration Population crossover mutation Obj Time (s)1500 10 95 95 044442 80491500 10 95 1 048580 101611500 10 95 01 050286 105231500 10 95 001 053612 10117

Table 15 Target equation (obj) predicted by Minitab program and compared with actual target equation (obj)

Target equation (obj) Comparison of target equation (obj)percentage

Predicted value Actual value Actual value compared with predicted value04492 04429 063

Table 16 Calculation results with reduction in number of nurses (119894)at 119908 = 06

Number of nurses (119894) Target equation(obj)

Calculating time(hour)

21 05241 76720 03959 60319 Infeasible Infeasible18 Infeasible Infeasible

Figure 5 The response optimizer configuration of obj response

number of nurses (119894) who were able to schedule the workunder the restrictions by using genetic programming and bygradually reducing the number of nurses (119894) down to the finalvalue at which the program could calculate the answer Thecalculation was performed at 119908 = 06 in Table 16

Table 16 shows that the least number of nurses (119894) whichcould schedule work under the restrictions was 20 Basedon 20 nurses amount of overtime payment and standarddeviation of overtime payment were calculated and reportedin Table 17 Therefore number of working hours for eachnurse number of nurses in each department and nursersquosschedule planwere generated and reported as shown inTables18 and 19

From Table 17 it can be observed that the target equation(obj) of the responses obtained was equal to 03959The sumof the overtime payment of all nurses was 16600 Baht per

CurHigh

Low099551

Optimal

minimumObj

099551desirabilityComposite

50950

50950

100500

50040004000 100 950 950

y = 04492

d = 099551

A B C D

D

Figure 6 The results of the data analysis using the responseoptimizer of the obj response

week The standard deviation of overtime payment that eachnurse received was 214843 For fair comparison we have toconvert the real overtime payment and standard deviationfor current 18 nurses to 20 nurses Then we can report thatthe sum of the total overtime payment and the standarddeviation of the overtime payment that each nurse receivedwhich is obtained from the model were less than those of theold working schedule at 17991 Baht per week and 882369respectively

Table 18 shows that all the nurses worked according to theconditions specified that is one nurse must work at least 40hours per week as in (4) but must work fewer than 60 hoursper week as in (5) all would be selected to work as in (6)and the duration of work must not exceed six days a week asin (7)

From Table 19 we can see that the number of nursesworking in shifts each day was more than the demand fornurses (Demand

119895119896119897) as shown in Table 6 which is according

to (3) and the specified service criteria (on average 15of thepatients waiting for services should not wait longer than theaverage service time of 25) because the number of nursesworking in shifts was more than the demand

To study the effect ofweight adjusted (119908)more numericalexperiments were conducted by GA coded in Matlab at 400

Mathematical Problems in Engineering 13

Table 17 Calculating results by 20 nurses (119894)

Target equation(obj)

Calculating time(hour)

Sum of overtime paymentof nurses (Baht)

Standard deviation of total overtimepayment that each nurse received

03959 603 16600 214843

Table 18 Number of working hours calculated from 20 nurses (119894)

Number of nurses Hoursperson (hour)Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total

1 12 12 0 4 0 8 12 482 4 8 12 4 8 12 0 483 8 8 8 0 12 12 0 484 12 0 12 12 12 4 0 525 8 12 8 4 0 0 12 44

20 8 12 4 0 12 8 4 48

Table 19 Number of nurses working in shifts calculated by 20 nurses (119894)

Department Period Supply (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 4 3 2 2PED 3 3 4 3 3 4 3EENT 2 3 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 3 3EENT 3 2 2 2 2 2 2OBG 2 2 3 2 1 1 2

Table 20 Effect of weight adjusted to objective function

Weight adjusted119908 = 09 119908 = 06 119908 = 05

Objective function 05090 04753 04823Sum of overtimepayment 24375 24625 26575

Standard deviation 324898 297675 288509

iterations and reported in Table 20 The report was shownas we expected more weight adjusted less overtime paymentsimultaneously with higher standard deviation It means thatif management focuses on reducing the overtime payment itwill lead to unfair payment or bias nursesrsquo schedule plan

7 Comparison of Advantages andDisadvantages of Adaptive Genetic andOptimization Approach

Weprocessed the data to do a comparison between the resultsof the adaptive genetic approach and the results obtained by

the Lingo program using the same data in order to comparethe target equation (obj) and computation runtime as shownin Tables 21 and 22

Table 21 shows the results of the processing experimentafter it was conducted ten times The work schedules using20 and 24 nurses with the calculating cycle of 400 roundsresulted in the best value for the target equations of 04361and 04551 respectively After ten processing experimentsthe work schedules for 20 and 24 nurses with the calculatingcycle at 1500 rounds resulted in the best value for the targetequations of 03959 and 04493 respectively and can besummarized as presented in Table 22

Table 22 shows that the work schedule for 20 nurses at1500 rounds when compared to the Lingo program couldcalculate thework schedulewith the target equation (obj) lessthan the genetic algorithm by around 580 and 108 plusmn 24on average but that the one created by the Lingo programused longer computational runtime by over 86849 and76452 on average We can also notice that the results at400 rounds reported the same direction the best and theaverage solution of genetic algorithm differ about 982 and1497 plusmn 38 from the optimal respectively However theoptimal method used longer time by over 306257 and

14 Mathematical Problems in Engineering

Table 21 Results of genetic program calculated ten times at 400 rounds and 1500 rounds

Target equation (obj)number

Number of 20 nurses Number of 24 nursesCalculating cycle

400Calculating cycle

1500Calculating cycle

400Calculating cycle

15001 04701 04578 05049 047302 04361 04436 05406 045523 04566 04772 04789 044934 04812 04303 04590 051605 04496 04781 04942 045866 04920 04549 05383 048337 04779 03959 05228 046268 05428 04352 05133 045809 05472 04368 05497 0498510 05231 04496 04551 04542Mean 04876 04459 05056 04708Standard deviation 0038 0024 0033 0021

Table 22 Comparison table for target equation (obj) and duration in calculation of GA and Lingo program

Number ofnurses(person)

Number ofcalculatingcycles

GA Lingo

Target equation (obj) Time (hour) Targetequation (obj) Time (hour) Different result

of obj

Different incomputational

run time

201500 Lowest 03959 603 580 86849

Mean 04459 plusmn 0024 685 03379 5237 1080 plusmn 24 76453

400 Lowest 04361 171 982 306257Mean 04876 plusmn 0038 166 1497 plusmn 38 315482

Table 23 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2012

Number of nurses Monthly expense SDOld New Increase Old New Decrease Old New Decrease18 20 2 395964 366400 29564 1089212 214843 8028

Table 24 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2013

Number of nurses Monthly expense SDOld New Decrease Old New Decrease Old New Decrease23 20 3 416548 366400 50148 1966735 214843 8908

315481 on average showing that the genetic program ismore suitable for practical use

8 Conclusion

The findings showed that the genetic program created couldmanage the work schedule of nurses under given conditionsand achieve the desired objective It could create work sche-dules of nurses that were fair in overtime payment to all thenurses and could reduce the hospital expenses by setting thenew coming nursesrsquo minimum salary at 15000 Baht per per-son

Table 25 Target equation and calculation time from geneticprogram

Target equation (obj) Calculation time (hour)03959 603

From Tables 23 24 and 25 the following results canbe concluded the work schedules from proposed methodreduce the waiting time of the number of patients waitinglonger than 125 times of the average length nursing servicefrom 3408 to less than 15 on average (starting from

Mathematical Problems in Engineering 15

when the patients register and wait to see the doctor) Ifwe want to schedule the work of the nurses in January 2012in order to meet the criteria for the service of this studywe have to increase the number of nurses from 18 to 20 sothat the number can perfectly meet the needs of the patientsaccording to the criteriaThework schedules of the nurses canbemost appropriate if the target equation is 03959which usesthe time of calculation as 603 hourswith a standard deviationof less than 8028and the overtimepayment less than 29564Baht per monthWhen this is compared with the actual workschedule in January 2013 the work schedule could reduce thenumber of nurses from 23 to 20 with a decreased standarddeviation that is 8908 and the decrease in the totalovertime paymentwould be 50148 Baht permonthWe couldcalculate the most appropriate work schedule quicker thanby using the conventional approach which results obtainedby Lingo program version 50 86849 with 1500 rounds forthe calculation cycle

As we know hospitals belong to the service businesssector continuous improvement along with response to cus-tomer satisfaction is the key success factor in this businessTo reduce the waiting time in every hospital process highstandards in medical care and highly experienced medicaldoctors are needed Presently Thailand has set its aim tobecome an Asian medical and tourist hub this strategy of thecountry may cause and force hospitals to improve their ser-vice quality for responding well to the high number of outpa-tients Updated data collection needs to be done service-level policies have to be reconsidered Demands of outpa-tient nurses and their schedules may change from time totime Inpatient nurses may be allocated to assist when themanagement wishes to reduce the fluctuation in the numberof outpatient nurses Thus modern management and highefficient tools are necessary this is the reason why GA isinvolved instead of an adoption of the conventional approachHowever regarding the implementation of simulation evenGA is not familiar to the hospital people Optimization capac-ity building is another issue that the hospitalmanagementwillhave to consider

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chiang Mai University (CMU) through the research admin-istration office provided budget to our Excellence Center inLogistics and Supply Chain Management (E-LSCM)

References

[1] T Mongkolrangsarit and W Thamapornpilat Development ofNurse Scheduling Method with Uncertain Workforce DemandDepartment of Industrial Engineering Faculty of EngineeringChulalongkorn University 2007

[2] C M Rothe and H B Wolfe ldquoCyclical scheduling and alloca-tion of nursing staffrdquo Socio-Economic Planning Sciences vol 7no 5 pp 471ndash487 1973

[3] R B Norby L E Freund and B Wagner ldquoA nurse staffingsystem based upon assignment difficultyrdquo Journal of NursingAdministration vol 7 no 9 pp 2ndash24 1977

[4] T Ryan B L Barker and F A Marciante ldquoA system fordetermining appropriate nurse staffingrdquo Journal of NursingAdministration vol 5 no 5 pp 30ndash38 1975

[5] H Wolfe and J P Young ldquoStaffing the nursing unit Part Icontrolled variable staffingrdquo Nursing Research vol 14 pp 236ndash243 1965

[6] HWolfe and J P Young ldquoStaffing the nursing unit IIThemult-iple assignment techniquerdquo Nursing Research vol 14 no 4 pp299ndash303 1965

[7] J L Arthur andA Ravindran ldquoAmultiple objective nurse sche-duling modelrdquo AIIE Transactions vol 13 no 1 pp 55ndash60 1981

[8] I Ozkarahan ldquoA disaggregation model of a flexible nursescheduling support systemrdquo Socio-Economic Planning Sciencesvol 25 no 1 pp 9ndash26 1991

[9] I Ozkarahan and J E Bailey ldquoGoal programming modelsubsystem of a flexible nurse scheduling support systemrdquo IIETransactions vol 20 no 3 pp 306ndash316 1988

[10] K A Dowsland ldquoNurse scheduling with tabu search and stra-tegic oscillationrdquo European Journal of Operational Research vol106 no 2-3 pp 393ndash407 1998

[11] P Tormos A Lova F Barber L Ingolotti M Abril and M ASalido ldquoA genetic algorithm for railway scheduling problemsrdquoinMetaheuristics for Scheduling in Industrial andManufacturingApplications vol 128 of Studies in Computational Intelligencepp 255ndash276 Springer Berlin Germany 2008

[12] H Kawanaka K Yamamoto T Yoshikawa T Shinogi and STsuruoka ldquoGenetic algorithm with the constraints for nursescheduling problemrdquo in Proceedings of the Congress on Evolu-tionary Computation vol 2 pp 1123ndash1130 Seoul Republic ofKorea May 2001

[13] H HMillar and KMona ldquoCyclic and non-cyclic scheduling of12 h shift nurses by network programmingrdquo European Journalof Operational Research vol 104 no 3 pp 582ndash592 1998

[14] I Blochlige ldquoModeling staff scheduling problems A tutorialrdquoEuropean Journal of Operational Research vol 158 no 3 pp533ndash542 2004

[15] B Jaumard F Semet and T Vovor ldquoA generalized linear pro-gramming model for nurse schedulingrdquo European Journal ofOperational Research vol 107 no 1 pp 1ndash18 1998

[16] T Cezik O Gunluk and H Luss ldquoAn integer programmingmodel for the weekly tour scheduling problemrdquoNaval ResearchLogistics vol 48 no 7 pp 607ndash624 2001

[17] Z Wang J-L Liu and X Yu ldquoSelf-fertilization based geneticalgorithm for university timetabling problemrdquo in Proceedingsof the 1st ACMSIGEVO Summit on Genetic and EvolutionaryComputation (GEC rsquo09) pp 1001ndash1004 ACM Shanghai ChinaJune 2009

[18] J Wang and K Chu ldquoAn application of genetic algorithms forthe flexible job-shop scheduling problemrdquo International Journalof Advancements in Computing Technology vol 4 no 3 pp 271ndash278 2012

[19] M Moz and M V Pato ldquoA genetic algorithm approach to anurse rerostering problemrdquo Computers amp Operations Researchvol 34 no 3 pp 667ndash691 2007

[20] C Wotthipong K Soradech and S Nidapan ldquoThe suitablegenetic operators for solving the university course timetablingproblemrdquo Journal of Convergence Information Technology vol8 pp 60ndash65 2013

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

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Mathematical Problems in Engineering

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

Mathematical Problems in Engineering 7

Service LevelOBG

Process OBG Decide 5

MoreMean 5

LowerMean 5 0

0

True

False

0

0

Dispose 5

Service LevelEENT

Process EENT Decide 4

MoreMean 4

LowerMean 4 0

0

True

False

0

0

Dispose 4

Service LevelPED

Process PED Decide 3

MoreMean 3

LowerMean 3 0

0

True

False

0

0

Dispose 3

Service LevelMED

Process MED Decide 2

MoreMean 2

LowerMean 2 0

0

True

False

0

0

Dispose 2

Service LevelSUR

Process SUR Decide 1

MoreMean 1

LowerMean 1 0

0

True

False

0

0

Dispose 1 IT SUR 8hr 02 01

IT MED 8hr 02 01

IT PED 8hr 02 01

IT EENT 8hr 02 01

IT OBG 8hr 02 01

Figure 1 The simulation model created by the Arena program

8 Mathematical Problems in Engineering

Table 4 The example for adjusting the number of nurses in theArena program

Name Type Capacity1 Resource SUR Fixed capacity 22 Resource MED Fixed capacity 23 Resource PED Fixed capacity 34 Resource EENT Fixed capacity 25 Resource OBG Fixed capacity 1

Table 5 Average and half-width values based on number of nursesadjusted

Output Average Half-widthEENT 04418 035MED 12505 080OBG 34024 164PED 16794 104SUR 44273 125

43 Creating Initial Population Each nursersquos chromosomecan be randomly generated for the initial state Then checkwhether it is in the scope of primary goal or not withworkinghours fewer than 60 hours If that chromosome is not in thescope of the primary goal again random search until eachnursersquos chromosome completes for the entire length and allthe nursersquos chromosomes are generated to reach the numberof population specified

44 Crossover and Mutation Crossover is an importantoperator which combines the good properties of both parentsin order to possibly yield new better children chromosomes[17 18] As usual the simple crossover operator (one-pointcrossover) consists of randomly choosing a crossover pointand then recombining the pieces of a pair of chromosomes toform two new chromosomes The simple crossover is com-patible with the random keys encoding though it generallyfails to preserve the permutation when dealing with naturalencoding Hence for natural encoding special crossoveroperators must be used [19] Roulette wheel selection (RWS)ranking selection (RS) tournament selection (TS) partiallymatched crossover (PMX) order crossover (OX) and cyclecrossover (CX) are evaluated by comparing the performanceof GArsquos operators on university course timetabling problem[20] PMX OX and CX operators require two crossoverpoints Given two parent chromosomes 119860 and 119861 childchromosome1198601015840 will inherit form119860 the subsequence betweenthese two points and child chromosome 1198611015840 form 119861 therespective subsequence The elements of 1198601015840 and 1198611015840 outsidethe two points are copied from the other parent chromosomewhile trying to preserve its position under the PMX operatoror by trying to preserve its order under the OX operatorA comparative analysis of PMX CX and OX crossoveroperators for solving travelling salesman problem (TSP) wasreported in [21]The experimental results show that the PMXcrossover outperforms the CX and OX crossover operator inTSP with 25 numbers of cities For our problem simple testwas conducted and we found PMX provided a better solution

Day

1

1

2

2

3

3

4

4 Nmiddot middot middot

middot middot middot

Mo Tu We Th Fr Sa Su

Number of nurses

Main chromosome

1 1 110 000

Binary code

Level 1

Level 2

Level 3

N lowast 7 lowast 8

Figure 2 The chromosome format

than one-point crossover specifically one-point crossovercannot take into account all constraints (3)ndash(8)

After applying crossover the mutation operator actson the pairs of chromosomes Although mutation occursinfrequently in nature it is believed to be an importantdriving force for evolution The mutation is adopted to allowfor the introduction of new chromosome into the populationand is effective to escape from a local optimum The simplemutation operates by randomly changing its value with agiven probability used in our experiments

45 Evaluation of Fitness Value The fitness value can be eva-luated by using two criteria as follows

Criterion one a chromosome is a feasible solution

(i) Fitness value is the objective function that isprocessed by (2)

Criterion two a chromosome is a nonfeasible solu-tion

(i) Fitness value is equal to 1000 units

46 Chromosome Selection We selected the chromosome byusing roulette

47 Elite Preserve Strategy The previous chromosomes withhigh fitness values were replaced by the new chromosomeswith the lower fitness values Then they were used as theinitial population for the next iteration

48 Termination Criteria The condition to stop seeking an-swers was that when the required number of solutions fullymeets the numbers of iteration specified we stop the exami-nation process immediately

5 Experiment Design to Determine OptimalConfiguration in Genetic Programming

In this section we introduced design of experiment (DOE)technique for screening the effected parameters and param-eters setting guideline for genetic algorithm solution search

Mathematical Problems in Engineering 9

Table 6 Comparison between real situation and simulation model

Department Calculated resultAVG number ofpatients receiving

service

AVG number ofpatients waitinglonger than AVG

time

AVG number ofpatients waiting lessthan AVG time

Percentage of patientswaiting longer than

AVG time

SURReal situation 59 02 58 0338Simulation 6163 plusmn 163 019 plusmn 036 6051 plusmn 166 0196 plusmn 032 error 427 526 433 mdash

MEDReal situation 96 0 98 0Simulation 9879 plusmn 191 0 plusmn 0 9794 plusmn 191 0 error 290 0 006 mdash

PEDReal situation 55 0 53 0Simulation 5758 plusmn 148 0 plusmn 0 5669 plusmn 146 0 error 469 0 696 mdash

EENTReal situation 53 182 34 3434Simulation 5446 plusmn 176 1673 plusmn 414 3664 plusmn 346 1999 plusmn 512 error 275 807 776 mdash

OBGReal situation 34 0 35 0Simulation 384 plusmn 085 0 plusmn 0 377 plusmn 084 0 error 1294 0 771 mdash

Table 7 Demand for nurses obtained from simulation model

Department Period Hospital demand (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 3 3 2 2PED 3 3 3 3 3 3 3EENT 2 2 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 2 2EENT 2 2 2 2 2 2 2OBG 1 1 1 2 1 1 1

machine We conducted the experiment by fixing currentnumber of nurses at 18 with the low level nursesrsquo demand ineach department All three expecting parameters were set bytwo levels maximum and minimum

51 Determining Factors and Factor Values in Each LevelFactors can be chosen for the experiment as follows

(1) population number(2) percent crossover rate(3) percent mutation rate

We used the factors in Table 9 to construct the experi-mental design matrix full factorial design (2119896) (8) and thenconducted the experiment by using three center points perblockwith all the three replicatesWe also fixed the number ofmaximum cycles to 400 cyclesThis was because the researchrevealed that the number of the calculated cycles had high

stability The cycles also took a long time to calculate andso may cause inconvenience in their actual use The numberof nurses 18 in total was fixed at the beginning for theexperimental design and the results are shown in Table 10which includes the use of the data for the demand for nursesfrom the first week of January

52 Effect and Coefficient Analysis and Model AdequacyChecking We used the experimental results to analyze theeffects and coefficients by using theMinitab program in orderto find the factors affecting the expected responses

Figure 3 shows that the data were normally and consis-tently distributed without any tendency and can be used toanalyze further results

Table 11 shows that the correlation of the data is linearbecause the center pt is greater than 005 and the 119875 values ofthe 119860 119862 119860119862 119861119862 and 119860119861119862 factors are less than 005 whichis significant for the experiment It was concluded that the 119860

10 Mathematical Problems in Engineering

Table 8 Example for encoding binary coded decimal

Decimal Binary Regular hour Overtime hour Code Type Demand 1 Demand 2 Hours0 00000000 NDA NOT NDA NOT Type 0 0 0 01 00000001 DA1 NOT DA1 NOT Type 1 DA1++ 0 82 00000010 DA2 NOT DA2 NOT Type 2 DA2++ 0 83 00000011 DA3 NOT DA3 NOT Type 3 DA3++ 0 8

255 11111111 NDA NOT NDA NOT Type 255 0 0 0

001000050000

99

90

50

10

1

Residual054052050048046

0010

0005

0000

Fitted value

8

6

4

2

02624222018161412108642

0010

0005

0000

Observation order

Normal probability plot Versus fits

Histogram Versus order

Residual plots for obj

Resid

ual

Resid

ual

minus0010

minus0010

minus0005

minus0005

minus0010

minus0005

001000050000Residual

minus0010 minus0005

Freq

uenc

y(

)

Figure 3 The residual plots for objective function

Table 9 Selected experimental factors and factor values in eachlevel

Factor LevelLow High

(1) Population number (119860) 10 50(2) Percent crossover rate (119861) 5 95(3) Percent mutation rate (119862) 5 95

and 119862 factors were significant at the confidence level of 005as shown in Figure 4

The 119877-Sq value ranges from 0 to 1 or from 0 to 100If the 119877-Sq value is very close to 1 then it indicates that thesimulation can explain the different variables more properlyand accurately Table 12 shows that the 119877-Sq value is equalto 9948 which is very high This means that the data can

explain dependent variables properly and can analyze furtherresponse optimizer functions

53 Analysis for Finding outMost Appropriate Results by UsingResponse Optimizer The results were analyzed by using aresponse optimizer of Minitab as shown in Figures 5 and 6

From the results obtained as shown in Figure 5 this studyaims to determine the lowest target equation As a result wechanged ldquoGoalrdquo to ldquoMinimizerdquo since the target equation wasthe sum of the hospital expense percentage and the standarddeviation of the overtime each nurse received which rangedfrom 0 to 100 As a result the target was set to 0 and the uppertarget to 100

Figure 6 shows that if we want to configure the geneticprogram to obtain the lowest target equation it must beset as shown in Table 13 From the prediction equation thetarget equation should be equal to 04492 (obj = 04492)

Mathematical Problems in Engineering 11

Table 10 Full factorial design (23) experiment with three center points per block type

Run Std order Run order Center pt Blocks 119860 119861 119862 Obj Time (s)1 1 1 1 1 10 5 5 045892 92375212 2 2 1 1 50 5 5 050282 576219023 3 3 1 1 10 95 5 047113 7904851

25 25 25 0 1 30 50 50 048018 31281694

Table 11 Effect and obj coefficient analysis results

Factorial fit obj versus 119860 119861 119862Estimated effects and coefficients for obj (coded units)

Term Effect Coeff SE coeff 119879 119875

Constant 048216 0001318 36575 0000119860 005200 002600 0001318 1972 0000119861 000027 000013 0001318 010 0921119862 minus002134 minus001067 0001318 minus810 0000119860 lowast 119861 minus000285 minus000143 0001318 minus108 0293119860 lowast 119862 minus000850 minus000425 0001318 minus322 0005119861 lowast 119862 minus001395 minus000697 0001318 minus529 0000119860 lowast 119861 lowast 119862 minus000974 minus000487 0001318 minus369 0002Ct Pt minus000198 0003955 minus050 0623

Table 12 119877-square decision-making coefficients

119878 = 208209 PRESS = 152191461119877-Sq = 9948 119877-Sq (pred) = 9899 119877-Sq (adj) = 9925

Table 13 Guideline setting for genetic program from responseoptimizer function

Factor Level(1) Population number (119860) 10(2) Percent crossover rate (119861) 95(3) Percent mutation rate (119862) 95

Possible optimal parameters settingmay occur at the low level mutation we conducted the experiment and reported inTable 14 The reports showed that the objective function isworse when mutation decreased The reports also shownthat the higher mutation the better objective functions

54 Result Confirmation Experiment We experimented tocheck for accuracy by configuring the genetic program asshown in Table 12 together with fixing the maximum itera-tions to the number of nurses at 400 cycles and 18 nursesrespectively as shown in Table 15

Table 15 shows that the actual target value (obj) is actuallybetter than (ie less than) 063 of the predicted value andcan be used practically and actually decreased the target equa-tion (obj)

20151050

99

95908070605040302010

5

1

Standardized effect

Factor NameNot significantSignificant

Effect type

Normal plot of the standardized effects

minus10 minus5

(response is obj 120572 = 005)

()

A

A

B

C

A

B

C

C

BC

ABC

AC

Figure 4 The normal plot of the standardized effects (response isobj alpha = 005)

6 Change in Number of Nurses (119894) in OptimalScheduling under Constraint Equation

We changed the number of nurses (119894) which was an indexin the constraint equation in order to find out the least

12 Mathematical Problems in Engineering

Table 14 Experimental results obtained at low level mutation

Max iteration Population crossover mutation Obj Time (s)1500 10 95 95 044442 80491500 10 95 1 048580 101611500 10 95 01 050286 105231500 10 95 001 053612 10117

Table 15 Target equation (obj) predicted by Minitab program and compared with actual target equation (obj)

Target equation (obj) Comparison of target equation (obj)percentage

Predicted value Actual value Actual value compared with predicted value04492 04429 063

Table 16 Calculation results with reduction in number of nurses (119894)at 119908 = 06

Number of nurses (119894) Target equation(obj)

Calculating time(hour)

21 05241 76720 03959 60319 Infeasible Infeasible18 Infeasible Infeasible

Figure 5 The response optimizer configuration of obj response

number of nurses (119894) who were able to schedule the workunder the restrictions by using genetic programming and bygradually reducing the number of nurses (119894) down to the finalvalue at which the program could calculate the answer Thecalculation was performed at 119908 = 06 in Table 16

Table 16 shows that the least number of nurses (119894) whichcould schedule work under the restrictions was 20 Basedon 20 nurses amount of overtime payment and standarddeviation of overtime payment were calculated and reportedin Table 17 Therefore number of working hours for eachnurse number of nurses in each department and nursersquosschedule planwere generated and reported as shown inTables18 and 19

From Table 17 it can be observed that the target equation(obj) of the responses obtained was equal to 03959The sumof the overtime payment of all nurses was 16600 Baht per

CurHigh

Low099551

Optimal

minimumObj

099551desirabilityComposite

50950

50950

100500

50040004000 100 950 950

y = 04492

d = 099551

A B C D

D

Figure 6 The results of the data analysis using the responseoptimizer of the obj response

week The standard deviation of overtime payment that eachnurse received was 214843 For fair comparison we have toconvert the real overtime payment and standard deviationfor current 18 nurses to 20 nurses Then we can report thatthe sum of the total overtime payment and the standarddeviation of the overtime payment that each nurse receivedwhich is obtained from the model were less than those of theold working schedule at 17991 Baht per week and 882369respectively

Table 18 shows that all the nurses worked according to theconditions specified that is one nurse must work at least 40hours per week as in (4) but must work fewer than 60 hoursper week as in (5) all would be selected to work as in (6)and the duration of work must not exceed six days a week asin (7)

From Table 19 we can see that the number of nursesworking in shifts each day was more than the demand fornurses (Demand

119895119896119897) as shown in Table 6 which is according

to (3) and the specified service criteria (on average 15of thepatients waiting for services should not wait longer than theaverage service time of 25) because the number of nursesworking in shifts was more than the demand

To study the effect ofweight adjusted (119908)more numericalexperiments were conducted by GA coded in Matlab at 400

Mathematical Problems in Engineering 13

Table 17 Calculating results by 20 nurses (119894)

Target equation(obj)

Calculating time(hour)

Sum of overtime paymentof nurses (Baht)

Standard deviation of total overtimepayment that each nurse received

03959 603 16600 214843

Table 18 Number of working hours calculated from 20 nurses (119894)

Number of nurses Hoursperson (hour)Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total

1 12 12 0 4 0 8 12 482 4 8 12 4 8 12 0 483 8 8 8 0 12 12 0 484 12 0 12 12 12 4 0 525 8 12 8 4 0 0 12 44

20 8 12 4 0 12 8 4 48

Table 19 Number of nurses working in shifts calculated by 20 nurses (119894)

Department Period Supply (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 4 3 2 2PED 3 3 4 3 3 4 3EENT 2 3 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 3 3EENT 3 2 2 2 2 2 2OBG 2 2 3 2 1 1 2

Table 20 Effect of weight adjusted to objective function

Weight adjusted119908 = 09 119908 = 06 119908 = 05

Objective function 05090 04753 04823Sum of overtimepayment 24375 24625 26575

Standard deviation 324898 297675 288509

iterations and reported in Table 20 The report was shownas we expected more weight adjusted less overtime paymentsimultaneously with higher standard deviation It means thatif management focuses on reducing the overtime payment itwill lead to unfair payment or bias nursesrsquo schedule plan

7 Comparison of Advantages andDisadvantages of Adaptive Genetic andOptimization Approach

Weprocessed the data to do a comparison between the resultsof the adaptive genetic approach and the results obtained by

the Lingo program using the same data in order to comparethe target equation (obj) and computation runtime as shownin Tables 21 and 22

Table 21 shows the results of the processing experimentafter it was conducted ten times The work schedules using20 and 24 nurses with the calculating cycle of 400 roundsresulted in the best value for the target equations of 04361and 04551 respectively After ten processing experimentsthe work schedules for 20 and 24 nurses with the calculatingcycle at 1500 rounds resulted in the best value for the targetequations of 03959 and 04493 respectively and can besummarized as presented in Table 22

Table 22 shows that the work schedule for 20 nurses at1500 rounds when compared to the Lingo program couldcalculate thework schedulewith the target equation (obj) lessthan the genetic algorithm by around 580 and 108 plusmn 24on average but that the one created by the Lingo programused longer computational runtime by over 86849 and76452 on average We can also notice that the results at400 rounds reported the same direction the best and theaverage solution of genetic algorithm differ about 982 and1497 plusmn 38 from the optimal respectively However theoptimal method used longer time by over 306257 and

14 Mathematical Problems in Engineering

Table 21 Results of genetic program calculated ten times at 400 rounds and 1500 rounds

Target equation (obj)number

Number of 20 nurses Number of 24 nursesCalculating cycle

400Calculating cycle

1500Calculating cycle

400Calculating cycle

15001 04701 04578 05049 047302 04361 04436 05406 045523 04566 04772 04789 044934 04812 04303 04590 051605 04496 04781 04942 045866 04920 04549 05383 048337 04779 03959 05228 046268 05428 04352 05133 045809 05472 04368 05497 0498510 05231 04496 04551 04542Mean 04876 04459 05056 04708Standard deviation 0038 0024 0033 0021

Table 22 Comparison table for target equation (obj) and duration in calculation of GA and Lingo program

Number ofnurses(person)

Number ofcalculatingcycles

GA Lingo

Target equation (obj) Time (hour) Targetequation (obj) Time (hour) Different result

of obj

Different incomputational

run time

201500 Lowest 03959 603 580 86849

Mean 04459 plusmn 0024 685 03379 5237 1080 plusmn 24 76453

400 Lowest 04361 171 982 306257Mean 04876 plusmn 0038 166 1497 plusmn 38 315482

Table 23 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2012

Number of nurses Monthly expense SDOld New Increase Old New Decrease Old New Decrease18 20 2 395964 366400 29564 1089212 214843 8028

Table 24 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2013

Number of nurses Monthly expense SDOld New Decrease Old New Decrease Old New Decrease23 20 3 416548 366400 50148 1966735 214843 8908

315481 on average showing that the genetic program ismore suitable for practical use

8 Conclusion

The findings showed that the genetic program created couldmanage the work schedule of nurses under given conditionsand achieve the desired objective It could create work sche-dules of nurses that were fair in overtime payment to all thenurses and could reduce the hospital expenses by setting thenew coming nursesrsquo minimum salary at 15000 Baht per per-son

Table 25 Target equation and calculation time from geneticprogram

Target equation (obj) Calculation time (hour)03959 603

From Tables 23 24 and 25 the following results canbe concluded the work schedules from proposed methodreduce the waiting time of the number of patients waitinglonger than 125 times of the average length nursing servicefrom 3408 to less than 15 on average (starting from

Mathematical Problems in Engineering 15

when the patients register and wait to see the doctor) Ifwe want to schedule the work of the nurses in January 2012in order to meet the criteria for the service of this studywe have to increase the number of nurses from 18 to 20 sothat the number can perfectly meet the needs of the patientsaccording to the criteriaThework schedules of the nurses canbemost appropriate if the target equation is 03959which usesthe time of calculation as 603 hourswith a standard deviationof less than 8028and the overtimepayment less than 29564Baht per monthWhen this is compared with the actual workschedule in January 2013 the work schedule could reduce thenumber of nurses from 23 to 20 with a decreased standarddeviation that is 8908 and the decrease in the totalovertime paymentwould be 50148 Baht permonthWe couldcalculate the most appropriate work schedule quicker thanby using the conventional approach which results obtainedby Lingo program version 50 86849 with 1500 rounds forthe calculation cycle

As we know hospitals belong to the service businesssector continuous improvement along with response to cus-tomer satisfaction is the key success factor in this businessTo reduce the waiting time in every hospital process highstandards in medical care and highly experienced medicaldoctors are needed Presently Thailand has set its aim tobecome an Asian medical and tourist hub this strategy of thecountry may cause and force hospitals to improve their ser-vice quality for responding well to the high number of outpa-tients Updated data collection needs to be done service-level policies have to be reconsidered Demands of outpa-tient nurses and their schedules may change from time totime Inpatient nurses may be allocated to assist when themanagement wishes to reduce the fluctuation in the numberof outpatient nurses Thus modern management and highefficient tools are necessary this is the reason why GA isinvolved instead of an adoption of the conventional approachHowever regarding the implementation of simulation evenGA is not familiar to the hospital people Optimization capac-ity building is another issue that the hospitalmanagementwillhave to consider

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chiang Mai University (CMU) through the research admin-istration office provided budget to our Excellence Center inLogistics and Supply Chain Management (E-LSCM)

References

[1] T Mongkolrangsarit and W Thamapornpilat Development ofNurse Scheduling Method with Uncertain Workforce DemandDepartment of Industrial Engineering Faculty of EngineeringChulalongkorn University 2007

[2] C M Rothe and H B Wolfe ldquoCyclical scheduling and alloca-tion of nursing staffrdquo Socio-Economic Planning Sciences vol 7no 5 pp 471ndash487 1973

[3] R B Norby L E Freund and B Wagner ldquoA nurse staffingsystem based upon assignment difficultyrdquo Journal of NursingAdministration vol 7 no 9 pp 2ndash24 1977

[4] T Ryan B L Barker and F A Marciante ldquoA system fordetermining appropriate nurse staffingrdquo Journal of NursingAdministration vol 5 no 5 pp 30ndash38 1975

[5] H Wolfe and J P Young ldquoStaffing the nursing unit Part Icontrolled variable staffingrdquo Nursing Research vol 14 pp 236ndash243 1965

[6] HWolfe and J P Young ldquoStaffing the nursing unit IIThemult-iple assignment techniquerdquo Nursing Research vol 14 no 4 pp299ndash303 1965

[7] J L Arthur andA Ravindran ldquoAmultiple objective nurse sche-duling modelrdquo AIIE Transactions vol 13 no 1 pp 55ndash60 1981

[8] I Ozkarahan ldquoA disaggregation model of a flexible nursescheduling support systemrdquo Socio-Economic Planning Sciencesvol 25 no 1 pp 9ndash26 1991

[9] I Ozkarahan and J E Bailey ldquoGoal programming modelsubsystem of a flexible nurse scheduling support systemrdquo IIETransactions vol 20 no 3 pp 306ndash316 1988

[10] K A Dowsland ldquoNurse scheduling with tabu search and stra-tegic oscillationrdquo European Journal of Operational Research vol106 no 2-3 pp 393ndash407 1998

[11] P Tormos A Lova F Barber L Ingolotti M Abril and M ASalido ldquoA genetic algorithm for railway scheduling problemsrdquoinMetaheuristics for Scheduling in Industrial andManufacturingApplications vol 128 of Studies in Computational Intelligencepp 255ndash276 Springer Berlin Germany 2008

[12] H Kawanaka K Yamamoto T Yoshikawa T Shinogi and STsuruoka ldquoGenetic algorithm with the constraints for nursescheduling problemrdquo in Proceedings of the Congress on Evolu-tionary Computation vol 2 pp 1123ndash1130 Seoul Republic ofKorea May 2001

[13] H HMillar and KMona ldquoCyclic and non-cyclic scheduling of12 h shift nurses by network programmingrdquo European Journalof Operational Research vol 104 no 3 pp 582ndash592 1998

[14] I Blochlige ldquoModeling staff scheduling problems A tutorialrdquoEuropean Journal of Operational Research vol 158 no 3 pp533ndash542 2004

[15] B Jaumard F Semet and T Vovor ldquoA generalized linear pro-gramming model for nurse schedulingrdquo European Journal ofOperational Research vol 107 no 1 pp 1ndash18 1998

[16] T Cezik O Gunluk and H Luss ldquoAn integer programmingmodel for the weekly tour scheduling problemrdquoNaval ResearchLogistics vol 48 no 7 pp 607ndash624 2001

[17] Z Wang J-L Liu and X Yu ldquoSelf-fertilization based geneticalgorithm for university timetabling problemrdquo in Proceedingsof the 1st ACMSIGEVO Summit on Genetic and EvolutionaryComputation (GEC rsquo09) pp 1001ndash1004 ACM Shanghai ChinaJune 2009

[18] J Wang and K Chu ldquoAn application of genetic algorithms forthe flexible job-shop scheduling problemrdquo International Journalof Advancements in Computing Technology vol 4 no 3 pp 271ndash278 2012

[19] M Moz and M V Pato ldquoA genetic algorithm approach to anurse rerostering problemrdquo Computers amp Operations Researchvol 34 no 3 pp 667ndash691 2007

[20] C Wotthipong K Soradech and S Nidapan ldquoThe suitablegenetic operators for solving the university course timetablingproblemrdquo Journal of Convergence Information Technology vol8 pp 60ndash65 2013

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

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Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of

Page 8: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

8 Mathematical Problems in Engineering

Table 4 The example for adjusting the number of nurses in theArena program

Name Type Capacity1 Resource SUR Fixed capacity 22 Resource MED Fixed capacity 23 Resource PED Fixed capacity 34 Resource EENT Fixed capacity 25 Resource OBG Fixed capacity 1

Table 5 Average and half-width values based on number of nursesadjusted

Output Average Half-widthEENT 04418 035MED 12505 080OBG 34024 164PED 16794 104SUR 44273 125

43 Creating Initial Population Each nursersquos chromosomecan be randomly generated for the initial state Then checkwhether it is in the scope of primary goal or not withworkinghours fewer than 60 hours If that chromosome is not in thescope of the primary goal again random search until eachnursersquos chromosome completes for the entire length and allthe nursersquos chromosomes are generated to reach the numberof population specified

44 Crossover and Mutation Crossover is an importantoperator which combines the good properties of both parentsin order to possibly yield new better children chromosomes[17 18] As usual the simple crossover operator (one-pointcrossover) consists of randomly choosing a crossover pointand then recombining the pieces of a pair of chromosomes toform two new chromosomes The simple crossover is com-patible with the random keys encoding though it generallyfails to preserve the permutation when dealing with naturalencoding Hence for natural encoding special crossoveroperators must be used [19] Roulette wheel selection (RWS)ranking selection (RS) tournament selection (TS) partiallymatched crossover (PMX) order crossover (OX) and cyclecrossover (CX) are evaluated by comparing the performanceof GArsquos operators on university course timetabling problem[20] PMX OX and CX operators require two crossoverpoints Given two parent chromosomes 119860 and 119861 childchromosome1198601015840 will inherit form119860 the subsequence betweenthese two points and child chromosome 1198611015840 form 119861 therespective subsequence The elements of 1198601015840 and 1198611015840 outsidethe two points are copied from the other parent chromosomewhile trying to preserve its position under the PMX operatoror by trying to preserve its order under the OX operatorA comparative analysis of PMX CX and OX crossoveroperators for solving travelling salesman problem (TSP) wasreported in [21]The experimental results show that the PMXcrossover outperforms the CX and OX crossover operator inTSP with 25 numbers of cities For our problem simple testwas conducted and we found PMX provided a better solution

Day

1

1

2

2

3

3

4

4 Nmiddot middot middot

middot middot middot

Mo Tu We Th Fr Sa Su

Number of nurses

Main chromosome

1 1 110 000

Binary code

Level 1

Level 2

Level 3

N lowast 7 lowast 8

Figure 2 The chromosome format

than one-point crossover specifically one-point crossovercannot take into account all constraints (3)ndash(8)

After applying crossover the mutation operator actson the pairs of chromosomes Although mutation occursinfrequently in nature it is believed to be an importantdriving force for evolution The mutation is adopted to allowfor the introduction of new chromosome into the populationand is effective to escape from a local optimum The simplemutation operates by randomly changing its value with agiven probability used in our experiments

45 Evaluation of Fitness Value The fitness value can be eva-luated by using two criteria as follows

Criterion one a chromosome is a feasible solution

(i) Fitness value is the objective function that isprocessed by (2)

Criterion two a chromosome is a nonfeasible solu-tion

(i) Fitness value is equal to 1000 units

46 Chromosome Selection We selected the chromosome byusing roulette

47 Elite Preserve Strategy The previous chromosomes withhigh fitness values were replaced by the new chromosomeswith the lower fitness values Then they were used as theinitial population for the next iteration

48 Termination Criteria The condition to stop seeking an-swers was that when the required number of solutions fullymeets the numbers of iteration specified we stop the exami-nation process immediately

5 Experiment Design to Determine OptimalConfiguration in Genetic Programming

In this section we introduced design of experiment (DOE)technique for screening the effected parameters and param-eters setting guideline for genetic algorithm solution search

Mathematical Problems in Engineering 9

Table 6 Comparison between real situation and simulation model

Department Calculated resultAVG number ofpatients receiving

service

AVG number ofpatients waitinglonger than AVG

time

AVG number ofpatients waiting lessthan AVG time

Percentage of patientswaiting longer than

AVG time

SURReal situation 59 02 58 0338Simulation 6163 plusmn 163 019 plusmn 036 6051 plusmn 166 0196 plusmn 032 error 427 526 433 mdash

MEDReal situation 96 0 98 0Simulation 9879 plusmn 191 0 plusmn 0 9794 plusmn 191 0 error 290 0 006 mdash

PEDReal situation 55 0 53 0Simulation 5758 plusmn 148 0 plusmn 0 5669 plusmn 146 0 error 469 0 696 mdash

EENTReal situation 53 182 34 3434Simulation 5446 plusmn 176 1673 plusmn 414 3664 plusmn 346 1999 plusmn 512 error 275 807 776 mdash

OBGReal situation 34 0 35 0Simulation 384 plusmn 085 0 plusmn 0 377 plusmn 084 0 error 1294 0 771 mdash

Table 7 Demand for nurses obtained from simulation model

Department Period Hospital demand (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 3 3 2 2PED 3 3 3 3 3 3 3EENT 2 2 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 2 2EENT 2 2 2 2 2 2 2OBG 1 1 1 2 1 1 1

machine We conducted the experiment by fixing currentnumber of nurses at 18 with the low level nursesrsquo demand ineach department All three expecting parameters were set bytwo levels maximum and minimum

51 Determining Factors and Factor Values in Each LevelFactors can be chosen for the experiment as follows

(1) population number(2) percent crossover rate(3) percent mutation rate

We used the factors in Table 9 to construct the experi-mental design matrix full factorial design (2119896) (8) and thenconducted the experiment by using three center points perblockwith all the three replicatesWe also fixed the number ofmaximum cycles to 400 cyclesThis was because the researchrevealed that the number of the calculated cycles had high

stability The cycles also took a long time to calculate andso may cause inconvenience in their actual use The numberof nurses 18 in total was fixed at the beginning for theexperimental design and the results are shown in Table 10which includes the use of the data for the demand for nursesfrom the first week of January

52 Effect and Coefficient Analysis and Model AdequacyChecking We used the experimental results to analyze theeffects and coefficients by using theMinitab program in orderto find the factors affecting the expected responses

Figure 3 shows that the data were normally and consis-tently distributed without any tendency and can be used toanalyze further results

Table 11 shows that the correlation of the data is linearbecause the center pt is greater than 005 and the 119875 values ofthe 119860 119862 119860119862 119861119862 and 119860119861119862 factors are less than 005 whichis significant for the experiment It was concluded that the 119860

10 Mathematical Problems in Engineering

Table 8 Example for encoding binary coded decimal

Decimal Binary Regular hour Overtime hour Code Type Demand 1 Demand 2 Hours0 00000000 NDA NOT NDA NOT Type 0 0 0 01 00000001 DA1 NOT DA1 NOT Type 1 DA1++ 0 82 00000010 DA2 NOT DA2 NOT Type 2 DA2++ 0 83 00000011 DA3 NOT DA3 NOT Type 3 DA3++ 0 8

255 11111111 NDA NOT NDA NOT Type 255 0 0 0

001000050000

99

90

50

10

1

Residual054052050048046

0010

0005

0000

Fitted value

8

6

4

2

02624222018161412108642

0010

0005

0000

Observation order

Normal probability plot Versus fits

Histogram Versus order

Residual plots for obj

Resid

ual

Resid

ual

minus0010

minus0010

minus0005

minus0005

minus0010

minus0005

001000050000Residual

minus0010 minus0005

Freq

uenc

y(

)

Figure 3 The residual plots for objective function

Table 9 Selected experimental factors and factor values in eachlevel

Factor LevelLow High

(1) Population number (119860) 10 50(2) Percent crossover rate (119861) 5 95(3) Percent mutation rate (119862) 5 95

and 119862 factors were significant at the confidence level of 005as shown in Figure 4

The 119877-Sq value ranges from 0 to 1 or from 0 to 100If the 119877-Sq value is very close to 1 then it indicates that thesimulation can explain the different variables more properlyand accurately Table 12 shows that the 119877-Sq value is equalto 9948 which is very high This means that the data can

explain dependent variables properly and can analyze furtherresponse optimizer functions

53 Analysis for Finding outMost Appropriate Results by UsingResponse Optimizer The results were analyzed by using aresponse optimizer of Minitab as shown in Figures 5 and 6

From the results obtained as shown in Figure 5 this studyaims to determine the lowest target equation As a result wechanged ldquoGoalrdquo to ldquoMinimizerdquo since the target equation wasthe sum of the hospital expense percentage and the standarddeviation of the overtime each nurse received which rangedfrom 0 to 100 As a result the target was set to 0 and the uppertarget to 100

Figure 6 shows that if we want to configure the geneticprogram to obtain the lowest target equation it must beset as shown in Table 13 From the prediction equation thetarget equation should be equal to 04492 (obj = 04492)

Mathematical Problems in Engineering 11

Table 10 Full factorial design (23) experiment with three center points per block type

Run Std order Run order Center pt Blocks 119860 119861 119862 Obj Time (s)1 1 1 1 1 10 5 5 045892 92375212 2 2 1 1 50 5 5 050282 576219023 3 3 1 1 10 95 5 047113 7904851

25 25 25 0 1 30 50 50 048018 31281694

Table 11 Effect and obj coefficient analysis results

Factorial fit obj versus 119860 119861 119862Estimated effects and coefficients for obj (coded units)

Term Effect Coeff SE coeff 119879 119875

Constant 048216 0001318 36575 0000119860 005200 002600 0001318 1972 0000119861 000027 000013 0001318 010 0921119862 minus002134 minus001067 0001318 minus810 0000119860 lowast 119861 minus000285 minus000143 0001318 minus108 0293119860 lowast 119862 minus000850 minus000425 0001318 minus322 0005119861 lowast 119862 minus001395 minus000697 0001318 minus529 0000119860 lowast 119861 lowast 119862 minus000974 minus000487 0001318 minus369 0002Ct Pt minus000198 0003955 minus050 0623

Table 12 119877-square decision-making coefficients

119878 = 208209 PRESS = 152191461119877-Sq = 9948 119877-Sq (pred) = 9899 119877-Sq (adj) = 9925

Table 13 Guideline setting for genetic program from responseoptimizer function

Factor Level(1) Population number (119860) 10(2) Percent crossover rate (119861) 95(3) Percent mutation rate (119862) 95

Possible optimal parameters settingmay occur at the low level mutation we conducted the experiment and reported inTable 14 The reports showed that the objective function isworse when mutation decreased The reports also shownthat the higher mutation the better objective functions

54 Result Confirmation Experiment We experimented tocheck for accuracy by configuring the genetic program asshown in Table 12 together with fixing the maximum itera-tions to the number of nurses at 400 cycles and 18 nursesrespectively as shown in Table 15

Table 15 shows that the actual target value (obj) is actuallybetter than (ie less than) 063 of the predicted value andcan be used practically and actually decreased the target equa-tion (obj)

20151050

99

95908070605040302010

5

1

Standardized effect

Factor NameNot significantSignificant

Effect type

Normal plot of the standardized effects

minus10 minus5

(response is obj 120572 = 005)

()

A

A

B

C

A

B

C

C

BC

ABC

AC

Figure 4 The normal plot of the standardized effects (response isobj alpha = 005)

6 Change in Number of Nurses (119894) in OptimalScheduling under Constraint Equation

We changed the number of nurses (119894) which was an indexin the constraint equation in order to find out the least

12 Mathematical Problems in Engineering

Table 14 Experimental results obtained at low level mutation

Max iteration Population crossover mutation Obj Time (s)1500 10 95 95 044442 80491500 10 95 1 048580 101611500 10 95 01 050286 105231500 10 95 001 053612 10117

Table 15 Target equation (obj) predicted by Minitab program and compared with actual target equation (obj)

Target equation (obj) Comparison of target equation (obj)percentage

Predicted value Actual value Actual value compared with predicted value04492 04429 063

Table 16 Calculation results with reduction in number of nurses (119894)at 119908 = 06

Number of nurses (119894) Target equation(obj)

Calculating time(hour)

21 05241 76720 03959 60319 Infeasible Infeasible18 Infeasible Infeasible

Figure 5 The response optimizer configuration of obj response

number of nurses (119894) who were able to schedule the workunder the restrictions by using genetic programming and bygradually reducing the number of nurses (119894) down to the finalvalue at which the program could calculate the answer Thecalculation was performed at 119908 = 06 in Table 16

Table 16 shows that the least number of nurses (119894) whichcould schedule work under the restrictions was 20 Basedon 20 nurses amount of overtime payment and standarddeviation of overtime payment were calculated and reportedin Table 17 Therefore number of working hours for eachnurse number of nurses in each department and nursersquosschedule planwere generated and reported as shown inTables18 and 19

From Table 17 it can be observed that the target equation(obj) of the responses obtained was equal to 03959The sumof the overtime payment of all nurses was 16600 Baht per

CurHigh

Low099551

Optimal

minimumObj

099551desirabilityComposite

50950

50950

100500

50040004000 100 950 950

y = 04492

d = 099551

A B C D

D

Figure 6 The results of the data analysis using the responseoptimizer of the obj response

week The standard deviation of overtime payment that eachnurse received was 214843 For fair comparison we have toconvert the real overtime payment and standard deviationfor current 18 nurses to 20 nurses Then we can report thatthe sum of the total overtime payment and the standarddeviation of the overtime payment that each nurse receivedwhich is obtained from the model were less than those of theold working schedule at 17991 Baht per week and 882369respectively

Table 18 shows that all the nurses worked according to theconditions specified that is one nurse must work at least 40hours per week as in (4) but must work fewer than 60 hoursper week as in (5) all would be selected to work as in (6)and the duration of work must not exceed six days a week asin (7)

From Table 19 we can see that the number of nursesworking in shifts each day was more than the demand fornurses (Demand

119895119896119897) as shown in Table 6 which is according

to (3) and the specified service criteria (on average 15of thepatients waiting for services should not wait longer than theaverage service time of 25) because the number of nursesworking in shifts was more than the demand

To study the effect ofweight adjusted (119908)more numericalexperiments were conducted by GA coded in Matlab at 400

Mathematical Problems in Engineering 13

Table 17 Calculating results by 20 nurses (119894)

Target equation(obj)

Calculating time(hour)

Sum of overtime paymentof nurses (Baht)

Standard deviation of total overtimepayment that each nurse received

03959 603 16600 214843

Table 18 Number of working hours calculated from 20 nurses (119894)

Number of nurses Hoursperson (hour)Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total

1 12 12 0 4 0 8 12 482 4 8 12 4 8 12 0 483 8 8 8 0 12 12 0 484 12 0 12 12 12 4 0 525 8 12 8 4 0 0 12 44

20 8 12 4 0 12 8 4 48

Table 19 Number of nurses working in shifts calculated by 20 nurses (119894)

Department Period Supply (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 4 3 2 2PED 3 3 4 3 3 4 3EENT 2 3 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 3 3EENT 3 2 2 2 2 2 2OBG 2 2 3 2 1 1 2

Table 20 Effect of weight adjusted to objective function

Weight adjusted119908 = 09 119908 = 06 119908 = 05

Objective function 05090 04753 04823Sum of overtimepayment 24375 24625 26575

Standard deviation 324898 297675 288509

iterations and reported in Table 20 The report was shownas we expected more weight adjusted less overtime paymentsimultaneously with higher standard deviation It means thatif management focuses on reducing the overtime payment itwill lead to unfair payment or bias nursesrsquo schedule plan

7 Comparison of Advantages andDisadvantages of Adaptive Genetic andOptimization Approach

Weprocessed the data to do a comparison between the resultsof the adaptive genetic approach and the results obtained by

the Lingo program using the same data in order to comparethe target equation (obj) and computation runtime as shownin Tables 21 and 22

Table 21 shows the results of the processing experimentafter it was conducted ten times The work schedules using20 and 24 nurses with the calculating cycle of 400 roundsresulted in the best value for the target equations of 04361and 04551 respectively After ten processing experimentsthe work schedules for 20 and 24 nurses with the calculatingcycle at 1500 rounds resulted in the best value for the targetequations of 03959 and 04493 respectively and can besummarized as presented in Table 22

Table 22 shows that the work schedule for 20 nurses at1500 rounds when compared to the Lingo program couldcalculate thework schedulewith the target equation (obj) lessthan the genetic algorithm by around 580 and 108 plusmn 24on average but that the one created by the Lingo programused longer computational runtime by over 86849 and76452 on average We can also notice that the results at400 rounds reported the same direction the best and theaverage solution of genetic algorithm differ about 982 and1497 plusmn 38 from the optimal respectively However theoptimal method used longer time by over 306257 and

14 Mathematical Problems in Engineering

Table 21 Results of genetic program calculated ten times at 400 rounds and 1500 rounds

Target equation (obj)number

Number of 20 nurses Number of 24 nursesCalculating cycle

400Calculating cycle

1500Calculating cycle

400Calculating cycle

15001 04701 04578 05049 047302 04361 04436 05406 045523 04566 04772 04789 044934 04812 04303 04590 051605 04496 04781 04942 045866 04920 04549 05383 048337 04779 03959 05228 046268 05428 04352 05133 045809 05472 04368 05497 0498510 05231 04496 04551 04542Mean 04876 04459 05056 04708Standard deviation 0038 0024 0033 0021

Table 22 Comparison table for target equation (obj) and duration in calculation of GA and Lingo program

Number ofnurses(person)

Number ofcalculatingcycles

GA Lingo

Target equation (obj) Time (hour) Targetequation (obj) Time (hour) Different result

of obj

Different incomputational

run time

201500 Lowest 03959 603 580 86849

Mean 04459 plusmn 0024 685 03379 5237 1080 plusmn 24 76453

400 Lowest 04361 171 982 306257Mean 04876 plusmn 0038 166 1497 plusmn 38 315482

Table 23 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2012

Number of nurses Monthly expense SDOld New Increase Old New Decrease Old New Decrease18 20 2 395964 366400 29564 1089212 214843 8028

Table 24 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2013

Number of nurses Monthly expense SDOld New Decrease Old New Decrease Old New Decrease23 20 3 416548 366400 50148 1966735 214843 8908

315481 on average showing that the genetic program ismore suitable for practical use

8 Conclusion

The findings showed that the genetic program created couldmanage the work schedule of nurses under given conditionsand achieve the desired objective It could create work sche-dules of nurses that were fair in overtime payment to all thenurses and could reduce the hospital expenses by setting thenew coming nursesrsquo minimum salary at 15000 Baht per per-son

Table 25 Target equation and calculation time from geneticprogram

Target equation (obj) Calculation time (hour)03959 603

From Tables 23 24 and 25 the following results canbe concluded the work schedules from proposed methodreduce the waiting time of the number of patients waitinglonger than 125 times of the average length nursing servicefrom 3408 to less than 15 on average (starting from

Mathematical Problems in Engineering 15

when the patients register and wait to see the doctor) Ifwe want to schedule the work of the nurses in January 2012in order to meet the criteria for the service of this studywe have to increase the number of nurses from 18 to 20 sothat the number can perfectly meet the needs of the patientsaccording to the criteriaThework schedules of the nurses canbemost appropriate if the target equation is 03959which usesthe time of calculation as 603 hourswith a standard deviationof less than 8028and the overtimepayment less than 29564Baht per monthWhen this is compared with the actual workschedule in January 2013 the work schedule could reduce thenumber of nurses from 23 to 20 with a decreased standarddeviation that is 8908 and the decrease in the totalovertime paymentwould be 50148 Baht permonthWe couldcalculate the most appropriate work schedule quicker thanby using the conventional approach which results obtainedby Lingo program version 50 86849 with 1500 rounds forthe calculation cycle

As we know hospitals belong to the service businesssector continuous improvement along with response to cus-tomer satisfaction is the key success factor in this businessTo reduce the waiting time in every hospital process highstandards in medical care and highly experienced medicaldoctors are needed Presently Thailand has set its aim tobecome an Asian medical and tourist hub this strategy of thecountry may cause and force hospitals to improve their ser-vice quality for responding well to the high number of outpa-tients Updated data collection needs to be done service-level policies have to be reconsidered Demands of outpa-tient nurses and their schedules may change from time totime Inpatient nurses may be allocated to assist when themanagement wishes to reduce the fluctuation in the numberof outpatient nurses Thus modern management and highefficient tools are necessary this is the reason why GA isinvolved instead of an adoption of the conventional approachHowever regarding the implementation of simulation evenGA is not familiar to the hospital people Optimization capac-ity building is another issue that the hospitalmanagementwillhave to consider

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chiang Mai University (CMU) through the research admin-istration office provided budget to our Excellence Center inLogistics and Supply Chain Management (E-LSCM)

References

[1] T Mongkolrangsarit and W Thamapornpilat Development ofNurse Scheduling Method with Uncertain Workforce DemandDepartment of Industrial Engineering Faculty of EngineeringChulalongkorn University 2007

[2] C M Rothe and H B Wolfe ldquoCyclical scheduling and alloca-tion of nursing staffrdquo Socio-Economic Planning Sciences vol 7no 5 pp 471ndash487 1973

[3] R B Norby L E Freund and B Wagner ldquoA nurse staffingsystem based upon assignment difficultyrdquo Journal of NursingAdministration vol 7 no 9 pp 2ndash24 1977

[4] T Ryan B L Barker and F A Marciante ldquoA system fordetermining appropriate nurse staffingrdquo Journal of NursingAdministration vol 5 no 5 pp 30ndash38 1975

[5] H Wolfe and J P Young ldquoStaffing the nursing unit Part Icontrolled variable staffingrdquo Nursing Research vol 14 pp 236ndash243 1965

[6] HWolfe and J P Young ldquoStaffing the nursing unit IIThemult-iple assignment techniquerdquo Nursing Research vol 14 no 4 pp299ndash303 1965

[7] J L Arthur andA Ravindran ldquoAmultiple objective nurse sche-duling modelrdquo AIIE Transactions vol 13 no 1 pp 55ndash60 1981

[8] I Ozkarahan ldquoA disaggregation model of a flexible nursescheduling support systemrdquo Socio-Economic Planning Sciencesvol 25 no 1 pp 9ndash26 1991

[9] I Ozkarahan and J E Bailey ldquoGoal programming modelsubsystem of a flexible nurse scheduling support systemrdquo IIETransactions vol 20 no 3 pp 306ndash316 1988

[10] K A Dowsland ldquoNurse scheduling with tabu search and stra-tegic oscillationrdquo European Journal of Operational Research vol106 no 2-3 pp 393ndash407 1998

[11] P Tormos A Lova F Barber L Ingolotti M Abril and M ASalido ldquoA genetic algorithm for railway scheduling problemsrdquoinMetaheuristics for Scheduling in Industrial andManufacturingApplications vol 128 of Studies in Computational Intelligencepp 255ndash276 Springer Berlin Germany 2008

[12] H Kawanaka K Yamamoto T Yoshikawa T Shinogi and STsuruoka ldquoGenetic algorithm with the constraints for nursescheduling problemrdquo in Proceedings of the Congress on Evolu-tionary Computation vol 2 pp 1123ndash1130 Seoul Republic ofKorea May 2001

[13] H HMillar and KMona ldquoCyclic and non-cyclic scheduling of12 h shift nurses by network programmingrdquo European Journalof Operational Research vol 104 no 3 pp 582ndash592 1998

[14] I Blochlige ldquoModeling staff scheduling problems A tutorialrdquoEuropean Journal of Operational Research vol 158 no 3 pp533ndash542 2004

[15] B Jaumard F Semet and T Vovor ldquoA generalized linear pro-gramming model for nurse schedulingrdquo European Journal ofOperational Research vol 107 no 1 pp 1ndash18 1998

[16] T Cezik O Gunluk and H Luss ldquoAn integer programmingmodel for the weekly tour scheduling problemrdquoNaval ResearchLogistics vol 48 no 7 pp 607ndash624 2001

[17] Z Wang J-L Liu and X Yu ldquoSelf-fertilization based geneticalgorithm for university timetabling problemrdquo in Proceedingsof the 1st ACMSIGEVO Summit on Genetic and EvolutionaryComputation (GEC rsquo09) pp 1001ndash1004 ACM Shanghai ChinaJune 2009

[18] J Wang and K Chu ldquoAn application of genetic algorithms forthe flexible job-shop scheduling problemrdquo International Journalof Advancements in Computing Technology vol 4 no 3 pp 271ndash278 2012

[19] M Moz and M V Pato ldquoA genetic algorithm approach to anurse rerostering problemrdquo Computers amp Operations Researchvol 34 no 3 pp 667ndash691 2007

[20] C Wotthipong K Soradech and S Nidapan ldquoThe suitablegenetic operators for solving the university course timetablingproblemrdquo Journal of Convergence Information Technology vol8 pp 60ndash65 2013

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

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Page 9: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

Mathematical Problems in Engineering 9

Table 6 Comparison between real situation and simulation model

Department Calculated resultAVG number ofpatients receiving

service

AVG number ofpatients waitinglonger than AVG

time

AVG number ofpatients waiting lessthan AVG time

Percentage of patientswaiting longer than

AVG time

SURReal situation 59 02 58 0338Simulation 6163 plusmn 163 019 plusmn 036 6051 plusmn 166 0196 plusmn 032 error 427 526 433 mdash

MEDReal situation 96 0 98 0Simulation 9879 plusmn 191 0 plusmn 0 9794 plusmn 191 0 error 290 0 006 mdash

PEDReal situation 55 0 53 0Simulation 5758 plusmn 148 0 plusmn 0 5669 plusmn 146 0 error 469 0 696 mdash

EENTReal situation 53 182 34 3434Simulation 5446 plusmn 176 1673 plusmn 414 3664 plusmn 346 1999 plusmn 512 error 275 807 776 mdash

OBGReal situation 34 0 35 0Simulation 384 plusmn 085 0 plusmn 0 377 plusmn 084 0 error 1294 0 771 mdash

Table 7 Demand for nurses obtained from simulation model

Department Period Hospital demand (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 3 3 2 2PED 3 3 3 3 3 3 3EENT 2 2 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 2 2EENT 2 2 2 2 2 2 2OBG 1 1 1 2 1 1 1

machine We conducted the experiment by fixing currentnumber of nurses at 18 with the low level nursesrsquo demand ineach department All three expecting parameters were set bytwo levels maximum and minimum

51 Determining Factors and Factor Values in Each LevelFactors can be chosen for the experiment as follows

(1) population number(2) percent crossover rate(3) percent mutation rate

We used the factors in Table 9 to construct the experi-mental design matrix full factorial design (2119896) (8) and thenconducted the experiment by using three center points perblockwith all the three replicatesWe also fixed the number ofmaximum cycles to 400 cyclesThis was because the researchrevealed that the number of the calculated cycles had high

stability The cycles also took a long time to calculate andso may cause inconvenience in their actual use The numberof nurses 18 in total was fixed at the beginning for theexperimental design and the results are shown in Table 10which includes the use of the data for the demand for nursesfrom the first week of January

52 Effect and Coefficient Analysis and Model AdequacyChecking We used the experimental results to analyze theeffects and coefficients by using theMinitab program in orderto find the factors affecting the expected responses

Figure 3 shows that the data were normally and consis-tently distributed without any tendency and can be used toanalyze further results

Table 11 shows that the correlation of the data is linearbecause the center pt is greater than 005 and the 119875 values ofthe 119860 119862 119860119862 119861119862 and 119860119861119862 factors are less than 005 whichis significant for the experiment It was concluded that the 119860

10 Mathematical Problems in Engineering

Table 8 Example for encoding binary coded decimal

Decimal Binary Regular hour Overtime hour Code Type Demand 1 Demand 2 Hours0 00000000 NDA NOT NDA NOT Type 0 0 0 01 00000001 DA1 NOT DA1 NOT Type 1 DA1++ 0 82 00000010 DA2 NOT DA2 NOT Type 2 DA2++ 0 83 00000011 DA3 NOT DA3 NOT Type 3 DA3++ 0 8

255 11111111 NDA NOT NDA NOT Type 255 0 0 0

001000050000

99

90

50

10

1

Residual054052050048046

0010

0005

0000

Fitted value

8

6

4

2

02624222018161412108642

0010

0005

0000

Observation order

Normal probability plot Versus fits

Histogram Versus order

Residual plots for obj

Resid

ual

Resid

ual

minus0010

minus0010

minus0005

minus0005

minus0010

minus0005

001000050000Residual

minus0010 minus0005

Freq

uenc

y(

)

Figure 3 The residual plots for objective function

Table 9 Selected experimental factors and factor values in eachlevel

Factor LevelLow High

(1) Population number (119860) 10 50(2) Percent crossover rate (119861) 5 95(3) Percent mutation rate (119862) 5 95

and 119862 factors were significant at the confidence level of 005as shown in Figure 4

The 119877-Sq value ranges from 0 to 1 or from 0 to 100If the 119877-Sq value is very close to 1 then it indicates that thesimulation can explain the different variables more properlyand accurately Table 12 shows that the 119877-Sq value is equalto 9948 which is very high This means that the data can

explain dependent variables properly and can analyze furtherresponse optimizer functions

53 Analysis for Finding outMost Appropriate Results by UsingResponse Optimizer The results were analyzed by using aresponse optimizer of Minitab as shown in Figures 5 and 6

From the results obtained as shown in Figure 5 this studyaims to determine the lowest target equation As a result wechanged ldquoGoalrdquo to ldquoMinimizerdquo since the target equation wasthe sum of the hospital expense percentage and the standarddeviation of the overtime each nurse received which rangedfrom 0 to 100 As a result the target was set to 0 and the uppertarget to 100

Figure 6 shows that if we want to configure the geneticprogram to obtain the lowest target equation it must beset as shown in Table 13 From the prediction equation thetarget equation should be equal to 04492 (obj = 04492)

Mathematical Problems in Engineering 11

Table 10 Full factorial design (23) experiment with three center points per block type

Run Std order Run order Center pt Blocks 119860 119861 119862 Obj Time (s)1 1 1 1 1 10 5 5 045892 92375212 2 2 1 1 50 5 5 050282 576219023 3 3 1 1 10 95 5 047113 7904851

25 25 25 0 1 30 50 50 048018 31281694

Table 11 Effect and obj coefficient analysis results

Factorial fit obj versus 119860 119861 119862Estimated effects and coefficients for obj (coded units)

Term Effect Coeff SE coeff 119879 119875

Constant 048216 0001318 36575 0000119860 005200 002600 0001318 1972 0000119861 000027 000013 0001318 010 0921119862 minus002134 minus001067 0001318 minus810 0000119860 lowast 119861 minus000285 minus000143 0001318 minus108 0293119860 lowast 119862 minus000850 minus000425 0001318 minus322 0005119861 lowast 119862 minus001395 minus000697 0001318 minus529 0000119860 lowast 119861 lowast 119862 minus000974 minus000487 0001318 minus369 0002Ct Pt minus000198 0003955 minus050 0623

Table 12 119877-square decision-making coefficients

119878 = 208209 PRESS = 152191461119877-Sq = 9948 119877-Sq (pred) = 9899 119877-Sq (adj) = 9925

Table 13 Guideline setting for genetic program from responseoptimizer function

Factor Level(1) Population number (119860) 10(2) Percent crossover rate (119861) 95(3) Percent mutation rate (119862) 95

Possible optimal parameters settingmay occur at the low level mutation we conducted the experiment and reported inTable 14 The reports showed that the objective function isworse when mutation decreased The reports also shownthat the higher mutation the better objective functions

54 Result Confirmation Experiment We experimented tocheck for accuracy by configuring the genetic program asshown in Table 12 together with fixing the maximum itera-tions to the number of nurses at 400 cycles and 18 nursesrespectively as shown in Table 15

Table 15 shows that the actual target value (obj) is actuallybetter than (ie less than) 063 of the predicted value andcan be used practically and actually decreased the target equa-tion (obj)

20151050

99

95908070605040302010

5

1

Standardized effect

Factor NameNot significantSignificant

Effect type

Normal plot of the standardized effects

minus10 minus5

(response is obj 120572 = 005)

()

A

A

B

C

A

B

C

C

BC

ABC

AC

Figure 4 The normal plot of the standardized effects (response isobj alpha = 005)

6 Change in Number of Nurses (119894) in OptimalScheduling under Constraint Equation

We changed the number of nurses (119894) which was an indexin the constraint equation in order to find out the least

12 Mathematical Problems in Engineering

Table 14 Experimental results obtained at low level mutation

Max iteration Population crossover mutation Obj Time (s)1500 10 95 95 044442 80491500 10 95 1 048580 101611500 10 95 01 050286 105231500 10 95 001 053612 10117

Table 15 Target equation (obj) predicted by Minitab program and compared with actual target equation (obj)

Target equation (obj) Comparison of target equation (obj)percentage

Predicted value Actual value Actual value compared with predicted value04492 04429 063

Table 16 Calculation results with reduction in number of nurses (119894)at 119908 = 06

Number of nurses (119894) Target equation(obj)

Calculating time(hour)

21 05241 76720 03959 60319 Infeasible Infeasible18 Infeasible Infeasible

Figure 5 The response optimizer configuration of obj response

number of nurses (119894) who were able to schedule the workunder the restrictions by using genetic programming and bygradually reducing the number of nurses (119894) down to the finalvalue at which the program could calculate the answer Thecalculation was performed at 119908 = 06 in Table 16

Table 16 shows that the least number of nurses (119894) whichcould schedule work under the restrictions was 20 Basedon 20 nurses amount of overtime payment and standarddeviation of overtime payment were calculated and reportedin Table 17 Therefore number of working hours for eachnurse number of nurses in each department and nursersquosschedule planwere generated and reported as shown inTables18 and 19

From Table 17 it can be observed that the target equation(obj) of the responses obtained was equal to 03959The sumof the overtime payment of all nurses was 16600 Baht per

CurHigh

Low099551

Optimal

minimumObj

099551desirabilityComposite

50950

50950

100500

50040004000 100 950 950

y = 04492

d = 099551

A B C D

D

Figure 6 The results of the data analysis using the responseoptimizer of the obj response

week The standard deviation of overtime payment that eachnurse received was 214843 For fair comparison we have toconvert the real overtime payment and standard deviationfor current 18 nurses to 20 nurses Then we can report thatthe sum of the total overtime payment and the standarddeviation of the overtime payment that each nurse receivedwhich is obtained from the model were less than those of theold working schedule at 17991 Baht per week and 882369respectively

Table 18 shows that all the nurses worked according to theconditions specified that is one nurse must work at least 40hours per week as in (4) but must work fewer than 60 hoursper week as in (5) all would be selected to work as in (6)and the duration of work must not exceed six days a week asin (7)

From Table 19 we can see that the number of nursesworking in shifts each day was more than the demand fornurses (Demand

119895119896119897) as shown in Table 6 which is according

to (3) and the specified service criteria (on average 15of thepatients waiting for services should not wait longer than theaverage service time of 25) because the number of nursesworking in shifts was more than the demand

To study the effect ofweight adjusted (119908)more numericalexperiments were conducted by GA coded in Matlab at 400

Mathematical Problems in Engineering 13

Table 17 Calculating results by 20 nurses (119894)

Target equation(obj)

Calculating time(hour)

Sum of overtime paymentof nurses (Baht)

Standard deviation of total overtimepayment that each nurse received

03959 603 16600 214843

Table 18 Number of working hours calculated from 20 nurses (119894)

Number of nurses Hoursperson (hour)Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total

1 12 12 0 4 0 8 12 482 4 8 12 4 8 12 0 483 8 8 8 0 12 12 0 484 12 0 12 12 12 4 0 525 8 12 8 4 0 0 12 44

20 8 12 4 0 12 8 4 48

Table 19 Number of nurses working in shifts calculated by 20 nurses (119894)

Department Period Supply (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 4 3 2 2PED 3 3 4 3 3 4 3EENT 2 3 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 3 3EENT 3 2 2 2 2 2 2OBG 2 2 3 2 1 1 2

Table 20 Effect of weight adjusted to objective function

Weight adjusted119908 = 09 119908 = 06 119908 = 05

Objective function 05090 04753 04823Sum of overtimepayment 24375 24625 26575

Standard deviation 324898 297675 288509

iterations and reported in Table 20 The report was shownas we expected more weight adjusted less overtime paymentsimultaneously with higher standard deviation It means thatif management focuses on reducing the overtime payment itwill lead to unfair payment or bias nursesrsquo schedule plan

7 Comparison of Advantages andDisadvantages of Adaptive Genetic andOptimization Approach

Weprocessed the data to do a comparison between the resultsof the adaptive genetic approach and the results obtained by

the Lingo program using the same data in order to comparethe target equation (obj) and computation runtime as shownin Tables 21 and 22

Table 21 shows the results of the processing experimentafter it was conducted ten times The work schedules using20 and 24 nurses with the calculating cycle of 400 roundsresulted in the best value for the target equations of 04361and 04551 respectively After ten processing experimentsthe work schedules for 20 and 24 nurses with the calculatingcycle at 1500 rounds resulted in the best value for the targetequations of 03959 and 04493 respectively and can besummarized as presented in Table 22

Table 22 shows that the work schedule for 20 nurses at1500 rounds when compared to the Lingo program couldcalculate thework schedulewith the target equation (obj) lessthan the genetic algorithm by around 580 and 108 plusmn 24on average but that the one created by the Lingo programused longer computational runtime by over 86849 and76452 on average We can also notice that the results at400 rounds reported the same direction the best and theaverage solution of genetic algorithm differ about 982 and1497 plusmn 38 from the optimal respectively However theoptimal method used longer time by over 306257 and

14 Mathematical Problems in Engineering

Table 21 Results of genetic program calculated ten times at 400 rounds and 1500 rounds

Target equation (obj)number

Number of 20 nurses Number of 24 nursesCalculating cycle

400Calculating cycle

1500Calculating cycle

400Calculating cycle

15001 04701 04578 05049 047302 04361 04436 05406 045523 04566 04772 04789 044934 04812 04303 04590 051605 04496 04781 04942 045866 04920 04549 05383 048337 04779 03959 05228 046268 05428 04352 05133 045809 05472 04368 05497 0498510 05231 04496 04551 04542Mean 04876 04459 05056 04708Standard deviation 0038 0024 0033 0021

Table 22 Comparison table for target equation (obj) and duration in calculation of GA and Lingo program

Number ofnurses(person)

Number ofcalculatingcycles

GA Lingo

Target equation (obj) Time (hour) Targetequation (obj) Time (hour) Different result

of obj

Different incomputational

run time

201500 Lowest 03959 603 580 86849

Mean 04459 plusmn 0024 685 03379 5237 1080 plusmn 24 76453

400 Lowest 04361 171 982 306257Mean 04876 plusmn 0038 166 1497 plusmn 38 315482

Table 23 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2012

Number of nurses Monthly expense SDOld New Increase Old New Decrease Old New Decrease18 20 2 395964 366400 29564 1089212 214843 8028

Table 24 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2013

Number of nurses Monthly expense SDOld New Decrease Old New Decrease Old New Decrease23 20 3 416548 366400 50148 1966735 214843 8908

315481 on average showing that the genetic program ismore suitable for practical use

8 Conclusion

The findings showed that the genetic program created couldmanage the work schedule of nurses under given conditionsand achieve the desired objective It could create work sche-dules of nurses that were fair in overtime payment to all thenurses and could reduce the hospital expenses by setting thenew coming nursesrsquo minimum salary at 15000 Baht per per-son

Table 25 Target equation and calculation time from geneticprogram

Target equation (obj) Calculation time (hour)03959 603

From Tables 23 24 and 25 the following results canbe concluded the work schedules from proposed methodreduce the waiting time of the number of patients waitinglonger than 125 times of the average length nursing servicefrom 3408 to less than 15 on average (starting from

Mathematical Problems in Engineering 15

when the patients register and wait to see the doctor) Ifwe want to schedule the work of the nurses in January 2012in order to meet the criteria for the service of this studywe have to increase the number of nurses from 18 to 20 sothat the number can perfectly meet the needs of the patientsaccording to the criteriaThework schedules of the nurses canbemost appropriate if the target equation is 03959which usesthe time of calculation as 603 hourswith a standard deviationof less than 8028and the overtimepayment less than 29564Baht per monthWhen this is compared with the actual workschedule in January 2013 the work schedule could reduce thenumber of nurses from 23 to 20 with a decreased standarddeviation that is 8908 and the decrease in the totalovertime paymentwould be 50148 Baht permonthWe couldcalculate the most appropriate work schedule quicker thanby using the conventional approach which results obtainedby Lingo program version 50 86849 with 1500 rounds forthe calculation cycle

As we know hospitals belong to the service businesssector continuous improvement along with response to cus-tomer satisfaction is the key success factor in this businessTo reduce the waiting time in every hospital process highstandards in medical care and highly experienced medicaldoctors are needed Presently Thailand has set its aim tobecome an Asian medical and tourist hub this strategy of thecountry may cause and force hospitals to improve their ser-vice quality for responding well to the high number of outpa-tients Updated data collection needs to be done service-level policies have to be reconsidered Demands of outpa-tient nurses and their schedules may change from time totime Inpatient nurses may be allocated to assist when themanagement wishes to reduce the fluctuation in the numberof outpatient nurses Thus modern management and highefficient tools are necessary this is the reason why GA isinvolved instead of an adoption of the conventional approachHowever regarding the implementation of simulation evenGA is not familiar to the hospital people Optimization capac-ity building is another issue that the hospitalmanagementwillhave to consider

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chiang Mai University (CMU) through the research admin-istration office provided budget to our Excellence Center inLogistics and Supply Chain Management (E-LSCM)

References

[1] T Mongkolrangsarit and W Thamapornpilat Development ofNurse Scheduling Method with Uncertain Workforce DemandDepartment of Industrial Engineering Faculty of EngineeringChulalongkorn University 2007

[2] C M Rothe and H B Wolfe ldquoCyclical scheduling and alloca-tion of nursing staffrdquo Socio-Economic Planning Sciences vol 7no 5 pp 471ndash487 1973

[3] R B Norby L E Freund and B Wagner ldquoA nurse staffingsystem based upon assignment difficultyrdquo Journal of NursingAdministration vol 7 no 9 pp 2ndash24 1977

[4] T Ryan B L Barker and F A Marciante ldquoA system fordetermining appropriate nurse staffingrdquo Journal of NursingAdministration vol 5 no 5 pp 30ndash38 1975

[5] H Wolfe and J P Young ldquoStaffing the nursing unit Part Icontrolled variable staffingrdquo Nursing Research vol 14 pp 236ndash243 1965

[6] HWolfe and J P Young ldquoStaffing the nursing unit IIThemult-iple assignment techniquerdquo Nursing Research vol 14 no 4 pp299ndash303 1965

[7] J L Arthur andA Ravindran ldquoAmultiple objective nurse sche-duling modelrdquo AIIE Transactions vol 13 no 1 pp 55ndash60 1981

[8] I Ozkarahan ldquoA disaggregation model of a flexible nursescheduling support systemrdquo Socio-Economic Planning Sciencesvol 25 no 1 pp 9ndash26 1991

[9] I Ozkarahan and J E Bailey ldquoGoal programming modelsubsystem of a flexible nurse scheduling support systemrdquo IIETransactions vol 20 no 3 pp 306ndash316 1988

[10] K A Dowsland ldquoNurse scheduling with tabu search and stra-tegic oscillationrdquo European Journal of Operational Research vol106 no 2-3 pp 393ndash407 1998

[11] P Tormos A Lova F Barber L Ingolotti M Abril and M ASalido ldquoA genetic algorithm for railway scheduling problemsrdquoinMetaheuristics for Scheduling in Industrial andManufacturingApplications vol 128 of Studies in Computational Intelligencepp 255ndash276 Springer Berlin Germany 2008

[12] H Kawanaka K Yamamoto T Yoshikawa T Shinogi and STsuruoka ldquoGenetic algorithm with the constraints for nursescheduling problemrdquo in Proceedings of the Congress on Evolu-tionary Computation vol 2 pp 1123ndash1130 Seoul Republic ofKorea May 2001

[13] H HMillar and KMona ldquoCyclic and non-cyclic scheduling of12 h shift nurses by network programmingrdquo European Journalof Operational Research vol 104 no 3 pp 582ndash592 1998

[14] I Blochlige ldquoModeling staff scheduling problems A tutorialrdquoEuropean Journal of Operational Research vol 158 no 3 pp533ndash542 2004

[15] B Jaumard F Semet and T Vovor ldquoA generalized linear pro-gramming model for nurse schedulingrdquo European Journal ofOperational Research vol 107 no 1 pp 1ndash18 1998

[16] T Cezik O Gunluk and H Luss ldquoAn integer programmingmodel for the weekly tour scheduling problemrdquoNaval ResearchLogistics vol 48 no 7 pp 607ndash624 2001

[17] Z Wang J-L Liu and X Yu ldquoSelf-fertilization based geneticalgorithm for university timetabling problemrdquo in Proceedingsof the 1st ACMSIGEVO Summit on Genetic and EvolutionaryComputation (GEC rsquo09) pp 1001ndash1004 ACM Shanghai ChinaJune 2009

[18] J Wang and K Chu ldquoAn application of genetic algorithms forthe flexible job-shop scheduling problemrdquo International Journalof Advancements in Computing Technology vol 4 no 3 pp 271ndash278 2012

[19] M Moz and M V Pato ldquoA genetic algorithm approach to anurse rerostering problemrdquo Computers amp Operations Researchvol 34 no 3 pp 667ndash691 2007

[20] C Wotthipong K Soradech and S Nidapan ldquoThe suitablegenetic operators for solving the university course timetablingproblemrdquo Journal of Convergence Information Technology vol8 pp 60ndash65 2013

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

Page 10: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

10 Mathematical Problems in Engineering

Table 8 Example for encoding binary coded decimal

Decimal Binary Regular hour Overtime hour Code Type Demand 1 Demand 2 Hours0 00000000 NDA NOT NDA NOT Type 0 0 0 01 00000001 DA1 NOT DA1 NOT Type 1 DA1++ 0 82 00000010 DA2 NOT DA2 NOT Type 2 DA2++ 0 83 00000011 DA3 NOT DA3 NOT Type 3 DA3++ 0 8

255 11111111 NDA NOT NDA NOT Type 255 0 0 0

001000050000

99

90

50

10

1

Residual054052050048046

0010

0005

0000

Fitted value

8

6

4

2

02624222018161412108642

0010

0005

0000

Observation order

Normal probability plot Versus fits

Histogram Versus order

Residual plots for obj

Resid

ual

Resid

ual

minus0010

minus0010

minus0005

minus0005

minus0010

minus0005

001000050000Residual

minus0010 minus0005

Freq

uenc

y(

)

Figure 3 The residual plots for objective function

Table 9 Selected experimental factors and factor values in eachlevel

Factor LevelLow High

(1) Population number (119860) 10 50(2) Percent crossover rate (119861) 5 95(3) Percent mutation rate (119862) 5 95

and 119862 factors were significant at the confidence level of 005as shown in Figure 4

The 119877-Sq value ranges from 0 to 1 or from 0 to 100If the 119877-Sq value is very close to 1 then it indicates that thesimulation can explain the different variables more properlyand accurately Table 12 shows that the 119877-Sq value is equalto 9948 which is very high This means that the data can

explain dependent variables properly and can analyze furtherresponse optimizer functions

53 Analysis for Finding outMost Appropriate Results by UsingResponse Optimizer The results were analyzed by using aresponse optimizer of Minitab as shown in Figures 5 and 6

From the results obtained as shown in Figure 5 this studyaims to determine the lowest target equation As a result wechanged ldquoGoalrdquo to ldquoMinimizerdquo since the target equation wasthe sum of the hospital expense percentage and the standarddeviation of the overtime each nurse received which rangedfrom 0 to 100 As a result the target was set to 0 and the uppertarget to 100

Figure 6 shows that if we want to configure the geneticprogram to obtain the lowest target equation it must beset as shown in Table 13 From the prediction equation thetarget equation should be equal to 04492 (obj = 04492)

Mathematical Problems in Engineering 11

Table 10 Full factorial design (23) experiment with three center points per block type

Run Std order Run order Center pt Blocks 119860 119861 119862 Obj Time (s)1 1 1 1 1 10 5 5 045892 92375212 2 2 1 1 50 5 5 050282 576219023 3 3 1 1 10 95 5 047113 7904851

25 25 25 0 1 30 50 50 048018 31281694

Table 11 Effect and obj coefficient analysis results

Factorial fit obj versus 119860 119861 119862Estimated effects and coefficients for obj (coded units)

Term Effect Coeff SE coeff 119879 119875

Constant 048216 0001318 36575 0000119860 005200 002600 0001318 1972 0000119861 000027 000013 0001318 010 0921119862 minus002134 minus001067 0001318 minus810 0000119860 lowast 119861 minus000285 minus000143 0001318 minus108 0293119860 lowast 119862 minus000850 minus000425 0001318 minus322 0005119861 lowast 119862 minus001395 minus000697 0001318 minus529 0000119860 lowast 119861 lowast 119862 minus000974 minus000487 0001318 minus369 0002Ct Pt minus000198 0003955 minus050 0623

Table 12 119877-square decision-making coefficients

119878 = 208209 PRESS = 152191461119877-Sq = 9948 119877-Sq (pred) = 9899 119877-Sq (adj) = 9925

Table 13 Guideline setting for genetic program from responseoptimizer function

Factor Level(1) Population number (119860) 10(2) Percent crossover rate (119861) 95(3) Percent mutation rate (119862) 95

Possible optimal parameters settingmay occur at the low level mutation we conducted the experiment and reported inTable 14 The reports showed that the objective function isworse when mutation decreased The reports also shownthat the higher mutation the better objective functions

54 Result Confirmation Experiment We experimented tocheck for accuracy by configuring the genetic program asshown in Table 12 together with fixing the maximum itera-tions to the number of nurses at 400 cycles and 18 nursesrespectively as shown in Table 15

Table 15 shows that the actual target value (obj) is actuallybetter than (ie less than) 063 of the predicted value andcan be used practically and actually decreased the target equa-tion (obj)

20151050

99

95908070605040302010

5

1

Standardized effect

Factor NameNot significantSignificant

Effect type

Normal plot of the standardized effects

minus10 minus5

(response is obj 120572 = 005)

()

A

A

B

C

A

B

C

C

BC

ABC

AC

Figure 4 The normal plot of the standardized effects (response isobj alpha = 005)

6 Change in Number of Nurses (119894) in OptimalScheduling under Constraint Equation

We changed the number of nurses (119894) which was an indexin the constraint equation in order to find out the least

12 Mathematical Problems in Engineering

Table 14 Experimental results obtained at low level mutation

Max iteration Population crossover mutation Obj Time (s)1500 10 95 95 044442 80491500 10 95 1 048580 101611500 10 95 01 050286 105231500 10 95 001 053612 10117

Table 15 Target equation (obj) predicted by Minitab program and compared with actual target equation (obj)

Target equation (obj) Comparison of target equation (obj)percentage

Predicted value Actual value Actual value compared with predicted value04492 04429 063

Table 16 Calculation results with reduction in number of nurses (119894)at 119908 = 06

Number of nurses (119894) Target equation(obj)

Calculating time(hour)

21 05241 76720 03959 60319 Infeasible Infeasible18 Infeasible Infeasible

Figure 5 The response optimizer configuration of obj response

number of nurses (119894) who were able to schedule the workunder the restrictions by using genetic programming and bygradually reducing the number of nurses (119894) down to the finalvalue at which the program could calculate the answer Thecalculation was performed at 119908 = 06 in Table 16

Table 16 shows that the least number of nurses (119894) whichcould schedule work under the restrictions was 20 Basedon 20 nurses amount of overtime payment and standarddeviation of overtime payment were calculated and reportedin Table 17 Therefore number of working hours for eachnurse number of nurses in each department and nursersquosschedule planwere generated and reported as shown inTables18 and 19

From Table 17 it can be observed that the target equation(obj) of the responses obtained was equal to 03959The sumof the overtime payment of all nurses was 16600 Baht per

CurHigh

Low099551

Optimal

minimumObj

099551desirabilityComposite

50950

50950

100500

50040004000 100 950 950

y = 04492

d = 099551

A B C D

D

Figure 6 The results of the data analysis using the responseoptimizer of the obj response

week The standard deviation of overtime payment that eachnurse received was 214843 For fair comparison we have toconvert the real overtime payment and standard deviationfor current 18 nurses to 20 nurses Then we can report thatthe sum of the total overtime payment and the standarddeviation of the overtime payment that each nurse receivedwhich is obtained from the model were less than those of theold working schedule at 17991 Baht per week and 882369respectively

Table 18 shows that all the nurses worked according to theconditions specified that is one nurse must work at least 40hours per week as in (4) but must work fewer than 60 hoursper week as in (5) all would be selected to work as in (6)and the duration of work must not exceed six days a week asin (7)

From Table 19 we can see that the number of nursesworking in shifts each day was more than the demand fornurses (Demand

119895119896119897) as shown in Table 6 which is according

to (3) and the specified service criteria (on average 15of thepatients waiting for services should not wait longer than theaverage service time of 25) because the number of nursesworking in shifts was more than the demand

To study the effect ofweight adjusted (119908)more numericalexperiments were conducted by GA coded in Matlab at 400

Mathematical Problems in Engineering 13

Table 17 Calculating results by 20 nurses (119894)

Target equation(obj)

Calculating time(hour)

Sum of overtime paymentof nurses (Baht)

Standard deviation of total overtimepayment that each nurse received

03959 603 16600 214843

Table 18 Number of working hours calculated from 20 nurses (119894)

Number of nurses Hoursperson (hour)Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total

1 12 12 0 4 0 8 12 482 4 8 12 4 8 12 0 483 8 8 8 0 12 12 0 484 12 0 12 12 12 4 0 525 8 12 8 4 0 0 12 44

20 8 12 4 0 12 8 4 48

Table 19 Number of nurses working in shifts calculated by 20 nurses (119894)

Department Period Supply (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 4 3 2 2PED 3 3 4 3 3 4 3EENT 2 3 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 3 3EENT 3 2 2 2 2 2 2OBG 2 2 3 2 1 1 2

Table 20 Effect of weight adjusted to objective function

Weight adjusted119908 = 09 119908 = 06 119908 = 05

Objective function 05090 04753 04823Sum of overtimepayment 24375 24625 26575

Standard deviation 324898 297675 288509

iterations and reported in Table 20 The report was shownas we expected more weight adjusted less overtime paymentsimultaneously with higher standard deviation It means thatif management focuses on reducing the overtime payment itwill lead to unfair payment or bias nursesrsquo schedule plan

7 Comparison of Advantages andDisadvantages of Adaptive Genetic andOptimization Approach

Weprocessed the data to do a comparison between the resultsof the adaptive genetic approach and the results obtained by

the Lingo program using the same data in order to comparethe target equation (obj) and computation runtime as shownin Tables 21 and 22

Table 21 shows the results of the processing experimentafter it was conducted ten times The work schedules using20 and 24 nurses with the calculating cycle of 400 roundsresulted in the best value for the target equations of 04361and 04551 respectively After ten processing experimentsthe work schedules for 20 and 24 nurses with the calculatingcycle at 1500 rounds resulted in the best value for the targetequations of 03959 and 04493 respectively and can besummarized as presented in Table 22

Table 22 shows that the work schedule for 20 nurses at1500 rounds when compared to the Lingo program couldcalculate thework schedulewith the target equation (obj) lessthan the genetic algorithm by around 580 and 108 plusmn 24on average but that the one created by the Lingo programused longer computational runtime by over 86849 and76452 on average We can also notice that the results at400 rounds reported the same direction the best and theaverage solution of genetic algorithm differ about 982 and1497 plusmn 38 from the optimal respectively However theoptimal method used longer time by over 306257 and

14 Mathematical Problems in Engineering

Table 21 Results of genetic program calculated ten times at 400 rounds and 1500 rounds

Target equation (obj)number

Number of 20 nurses Number of 24 nursesCalculating cycle

400Calculating cycle

1500Calculating cycle

400Calculating cycle

15001 04701 04578 05049 047302 04361 04436 05406 045523 04566 04772 04789 044934 04812 04303 04590 051605 04496 04781 04942 045866 04920 04549 05383 048337 04779 03959 05228 046268 05428 04352 05133 045809 05472 04368 05497 0498510 05231 04496 04551 04542Mean 04876 04459 05056 04708Standard deviation 0038 0024 0033 0021

Table 22 Comparison table for target equation (obj) and duration in calculation of GA and Lingo program

Number ofnurses(person)

Number ofcalculatingcycles

GA Lingo

Target equation (obj) Time (hour) Targetequation (obj) Time (hour) Different result

of obj

Different incomputational

run time

201500 Lowest 03959 603 580 86849

Mean 04459 plusmn 0024 685 03379 5237 1080 plusmn 24 76453

400 Lowest 04361 171 982 306257Mean 04876 plusmn 0038 166 1497 plusmn 38 315482

Table 23 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2012

Number of nurses Monthly expense SDOld New Increase Old New Decrease Old New Decrease18 20 2 395964 366400 29564 1089212 214843 8028

Table 24 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2013

Number of nurses Monthly expense SDOld New Decrease Old New Decrease Old New Decrease23 20 3 416548 366400 50148 1966735 214843 8908

315481 on average showing that the genetic program ismore suitable for practical use

8 Conclusion

The findings showed that the genetic program created couldmanage the work schedule of nurses under given conditionsand achieve the desired objective It could create work sche-dules of nurses that were fair in overtime payment to all thenurses and could reduce the hospital expenses by setting thenew coming nursesrsquo minimum salary at 15000 Baht per per-son

Table 25 Target equation and calculation time from geneticprogram

Target equation (obj) Calculation time (hour)03959 603

From Tables 23 24 and 25 the following results canbe concluded the work schedules from proposed methodreduce the waiting time of the number of patients waitinglonger than 125 times of the average length nursing servicefrom 3408 to less than 15 on average (starting from

Mathematical Problems in Engineering 15

when the patients register and wait to see the doctor) Ifwe want to schedule the work of the nurses in January 2012in order to meet the criteria for the service of this studywe have to increase the number of nurses from 18 to 20 sothat the number can perfectly meet the needs of the patientsaccording to the criteriaThework schedules of the nurses canbemost appropriate if the target equation is 03959which usesthe time of calculation as 603 hourswith a standard deviationof less than 8028and the overtimepayment less than 29564Baht per monthWhen this is compared with the actual workschedule in January 2013 the work schedule could reduce thenumber of nurses from 23 to 20 with a decreased standarddeviation that is 8908 and the decrease in the totalovertime paymentwould be 50148 Baht permonthWe couldcalculate the most appropriate work schedule quicker thanby using the conventional approach which results obtainedby Lingo program version 50 86849 with 1500 rounds forthe calculation cycle

As we know hospitals belong to the service businesssector continuous improvement along with response to cus-tomer satisfaction is the key success factor in this businessTo reduce the waiting time in every hospital process highstandards in medical care and highly experienced medicaldoctors are needed Presently Thailand has set its aim tobecome an Asian medical and tourist hub this strategy of thecountry may cause and force hospitals to improve their ser-vice quality for responding well to the high number of outpa-tients Updated data collection needs to be done service-level policies have to be reconsidered Demands of outpa-tient nurses and their schedules may change from time totime Inpatient nurses may be allocated to assist when themanagement wishes to reduce the fluctuation in the numberof outpatient nurses Thus modern management and highefficient tools are necessary this is the reason why GA isinvolved instead of an adoption of the conventional approachHowever regarding the implementation of simulation evenGA is not familiar to the hospital people Optimization capac-ity building is another issue that the hospitalmanagementwillhave to consider

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chiang Mai University (CMU) through the research admin-istration office provided budget to our Excellence Center inLogistics and Supply Chain Management (E-LSCM)

References

[1] T Mongkolrangsarit and W Thamapornpilat Development ofNurse Scheduling Method with Uncertain Workforce DemandDepartment of Industrial Engineering Faculty of EngineeringChulalongkorn University 2007

[2] C M Rothe and H B Wolfe ldquoCyclical scheduling and alloca-tion of nursing staffrdquo Socio-Economic Planning Sciences vol 7no 5 pp 471ndash487 1973

[3] R B Norby L E Freund and B Wagner ldquoA nurse staffingsystem based upon assignment difficultyrdquo Journal of NursingAdministration vol 7 no 9 pp 2ndash24 1977

[4] T Ryan B L Barker and F A Marciante ldquoA system fordetermining appropriate nurse staffingrdquo Journal of NursingAdministration vol 5 no 5 pp 30ndash38 1975

[5] H Wolfe and J P Young ldquoStaffing the nursing unit Part Icontrolled variable staffingrdquo Nursing Research vol 14 pp 236ndash243 1965

[6] HWolfe and J P Young ldquoStaffing the nursing unit IIThemult-iple assignment techniquerdquo Nursing Research vol 14 no 4 pp299ndash303 1965

[7] J L Arthur andA Ravindran ldquoAmultiple objective nurse sche-duling modelrdquo AIIE Transactions vol 13 no 1 pp 55ndash60 1981

[8] I Ozkarahan ldquoA disaggregation model of a flexible nursescheduling support systemrdquo Socio-Economic Planning Sciencesvol 25 no 1 pp 9ndash26 1991

[9] I Ozkarahan and J E Bailey ldquoGoal programming modelsubsystem of a flexible nurse scheduling support systemrdquo IIETransactions vol 20 no 3 pp 306ndash316 1988

[10] K A Dowsland ldquoNurse scheduling with tabu search and stra-tegic oscillationrdquo European Journal of Operational Research vol106 no 2-3 pp 393ndash407 1998

[11] P Tormos A Lova F Barber L Ingolotti M Abril and M ASalido ldquoA genetic algorithm for railway scheduling problemsrdquoinMetaheuristics for Scheduling in Industrial andManufacturingApplications vol 128 of Studies in Computational Intelligencepp 255ndash276 Springer Berlin Germany 2008

[12] H Kawanaka K Yamamoto T Yoshikawa T Shinogi and STsuruoka ldquoGenetic algorithm with the constraints for nursescheduling problemrdquo in Proceedings of the Congress on Evolu-tionary Computation vol 2 pp 1123ndash1130 Seoul Republic ofKorea May 2001

[13] H HMillar and KMona ldquoCyclic and non-cyclic scheduling of12 h shift nurses by network programmingrdquo European Journalof Operational Research vol 104 no 3 pp 582ndash592 1998

[14] I Blochlige ldquoModeling staff scheduling problems A tutorialrdquoEuropean Journal of Operational Research vol 158 no 3 pp533ndash542 2004

[15] B Jaumard F Semet and T Vovor ldquoA generalized linear pro-gramming model for nurse schedulingrdquo European Journal ofOperational Research vol 107 no 1 pp 1ndash18 1998

[16] T Cezik O Gunluk and H Luss ldquoAn integer programmingmodel for the weekly tour scheduling problemrdquoNaval ResearchLogistics vol 48 no 7 pp 607ndash624 2001

[17] Z Wang J-L Liu and X Yu ldquoSelf-fertilization based geneticalgorithm for university timetabling problemrdquo in Proceedingsof the 1st ACMSIGEVO Summit on Genetic and EvolutionaryComputation (GEC rsquo09) pp 1001ndash1004 ACM Shanghai ChinaJune 2009

[18] J Wang and K Chu ldquoAn application of genetic algorithms forthe flexible job-shop scheduling problemrdquo International Journalof Advancements in Computing Technology vol 4 no 3 pp 271ndash278 2012

[19] M Moz and M V Pato ldquoA genetic algorithm approach to anurse rerostering problemrdquo Computers amp Operations Researchvol 34 no 3 pp 667ndash691 2007

[20] C Wotthipong K Soradech and S Nidapan ldquoThe suitablegenetic operators for solving the university course timetablingproblemrdquo Journal of Convergence Information Technology vol8 pp 60ndash65 2013

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 11: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

Mathematical Problems in Engineering 11

Table 10 Full factorial design (23) experiment with three center points per block type

Run Std order Run order Center pt Blocks 119860 119861 119862 Obj Time (s)1 1 1 1 1 10 5 5 045892 92375212 2 2 1 1 50 5 5 050282 576219023 3 3 1 1 10 95 5 047113 7904851

25 25 25 0 1 30 50 50 048018 31281694

Table 11 Effect and obj coefficient analysis results

Factorial fit obj versus 119860 119861 119862Estimated effects and coefficients for obj (coded units)

Term Effect Coeff SE coeff 119879 119875

Constant 048216 0001318 36575 0000119860 005200 002600 0001318 1972 0000119861 000027 000013 0001318 010 0921119862 minus002134 minus001067 0001318 minus810 0000119860 lowast 119861 minus000285 minus000143 0001318 minus108 0293119860 lowast 119862 minus000850 minus000425 0001318 minus322 0005119861 lowast 119862 minus001395 minus000697 0001318 minus529 0000119860 lowast 119861 lowast 119862 minus000974 minus000487 0001318 minus369 0002Ct Pt minus000198 0003955 minus050 0623

Table 12 119877-square decision-making coefficients

119878 = 208209 PRESS = 152191461119877-Sq = 9948 119877-Sq (pred) = 9899 119877-Sq (adj) = 9925

Table 13 Guideline setting for genetic program from responseoptimizer function

Factor Level(1) Population number (119860) 10(2) Percent crossover rate (119861) 95(3) Percent mutation rate (119862) 95

Possible optimal parameters settingmay occur at the low level mutation we conducted the experiment and reported inTable 14 The reports showed that the objective function isworse when mutation decreased The reports also shownthat the higher mutation the better objective functions

54 Result Confirmation Experiment We experimented tocheck for accuracy by configuring the genetic program asshown in Table 12 together with fixing the maximum itera-tions to the number of nurses at 400 cycles and 18 nursesrespectively as shown in Table 15

Table 15 shows that the actual target value (obj) is actuallybetter than (ie less than) 063 of the predicted value andcan be used practically and actually decreased the target equa-tion (obj)

20151050

99

95908070605040302010

5

1

Standardized effect

Factor NameNot significantSignificant

Effect type

Normal plot of the standardized effects

minus10 minus5

(response is obj 120572 = 005)

()

A

A

B

C

A

B

C

C

BC

ABC

AC

Figure 4 The normal plot of the standardized effects (response isobj alpha = 005)

6 Change in Number of Nurses (119894) in OptimalScheduling under Constraint Equation

We changed the number of nurses (119894) which was an indexin the constraint equation in order to find out the least

12 Mathematical Problems in Engineering

Table 14 Experimental results obtained at low level mutation

Max iteration Population crossover mutation Obj Time (s)1500 10 95 95 044442 80491500 10 95 1 048580 101611500 10 95 01 050286 105231500 10 95 001 053612 10117

Table 15 Target equation (obj) predicted by Minitab program and compared with actual target equation (obj)

Target equation (obj) Comparison of target equation (obj)percentage

Predicted value Actual value Actual value compared with predicted value04492 04429 063

Table 16 Calculation results with reduction in number of nurses (119894)at 119908 = 06

Number of nurses (119894) Target equation(obj)

Calculating time(hour)

21 05241 76720 03959 60319 Infeasible Infeasible18 Infeasible Infeasible

Figure 5 The response optimizer configuration of obj response

number of nurses (119894) who were able to schedule the workunder the restrictions by using genetic programming and bygradually reducing the number of nurses (119894) down to the finalvalue at which the program could calculate the answer Thecalculation was performed at 119908 = 06 in Table 16

Table 16 shows that the least number of nurses (119894) whichcould schedule work under the restrictions was 20 Basedon 20 nurses amount of overtime payment and standarddeviation of overtime payment were calculated and reportedin Table 17 Therefore number of working hours for eachnurse number of nurses in each department and nursersquosschedule planwere generated and reported as shown inTables18 and 19

From Table 17 it can be observed that the target equation(obj) of the responses obtained was equal to 03959The sumof the overtime payment of all nurses was 16600 Baht per

CurHigh

Low099551

Optimal

minimumObj

099551desirabilityComposite

50950

50950

100500

50040004000 100 950 950

y = 04492

d = 099551

A B C D

D

Figure 6 The results of the data analysis using the responseoptimizer of the obj response

week The standard deviation of overtime payment that eachnurse received was 214843 For fair comparison we have toconvert the real overtime payment and standard deviationfor current 18 nurses to 20 nurses Then we can report thatthe sum of the total overtime payment and the standarddeviation of the overtime payment that each nurse receivedwhich is obtained from the model were less than those of theold working schedule at 17991 Baht per week and 882369respectively

Table 18 shows that all the nurses worked according to theconditions specified that is one nurse must work at least 40hours per week as in (4) but must work fewer than 60 hoursper week as in (5) all would be selected to work as in (6)and the duration of work must not exceed six days a week asin (7)

From Table 19 we can see that the number of nursesworking in shifts each day was more than the demand fornurses (Demand

119895119896119897) as shown in Table 6 which is according

to (3) and the specified service criteria (on average 15of thepatients waiting for services should not wait longer than theaverage service time of 25) because the number of nursesworking in shifts was more than the demand

To study the effect ofweight adjusted (119908)more numericalexperiments were conducted by GA coded in Matlab at 400

Mathematical Problems in Engineering 13

Table 17 Calculating results by 20 nurses (119894)

Target equation(obj)

Calculating time(hour)

Sum of overtime paymentof nurses (Baht)

Standard deviation of total overtimepayment that each nurse received

03959 603 16600 214843

Table 18 Number of working hours calculated from 20 nurses (119894)

Number of nurses Hoursperson (hour)Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total

1 12 12 0 4 0 8 12 482 4 8 12 4 8 12 0 483 8 8 8 0 12 12 0 484 12 0 12 12 12 4 0 525 8 12 8 4 0 0 12 44

20 8 12 4 0 12 8 4 48

Table 19 Number of nurses working in shifts calculated by 20 nurses (119894)

Department Period Supply (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 4 3 2 2PED 3 3 4 3 3 4 3EENT 2 3 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 3 3EENT 3 2 2 2 2 2 2OBG 2 2 3 2 1 1 2

Table 20 Effect of weight adjusted to objective function

Weight adjusted119908 = 09 119908 = 06 119908 = 05

Objective function 05090 04753 04823Sum of overtimepayment 24375 24625 26575

Standard deviation 324898 297675 288509

iterations and reported in Table 20 The report was shownas we expected more weight adjusted less overtime paymentsimultaneously with higher standard deviation It means thatif management focuses on reducing the overtime payment itwill lead to unfair payment or bias nursesrsquo schedule plan

7 Comparison of Advantages andDisadvantages of Adaptive Genetic andOptimization Approach

Weprocessed the data to do a comparison between the resultsof the adaptive genetic approach and the results obtained by

the Lingo program using the same data in order to comparethe target equation (obj) and computation runtime as shownin Tables 21 and 22

Table 21 shows the results of the processing experimentafter it was conducted ten times The work schedules using20 and 24 nurses with the calculating cycle of 400 roundsresulted in the best value for the target equations of 04361and 04551 respectively After ten processing experimentsthe work schedules for 20 and 24 nurses with the calculatingcycle at 1500 rounds resulted in the best value for the targetequations of 03959 and 04493 respectively and can besummarized as presented in Table 22

Table 22 shows that the work schedule for 20 nurses at1500 rounds when compared to the Lingo program couldcalculate thework schedulewith the target equation (obj) lessthan the genetic algorithm by around 580 and 108 plusmn 24on average but that the one created by the Lingo programused longer computational runtime by over 86849 and76452 on average We can also notice that the results at400 rounds reported the same direction the best and theaverage solution of genetic algorithm differ about 982 and1497 plusmn 38 from the optimal respectively However theoptimal method used longer time by over 306257 and

14 Mathematical Problems in Engineering

Table 21 Results of genetic program calculated ten times at 400 rounds and 1500 rounds

Target equation (obj)number

Number of 20 nurses Number of 24 nursesCalculating cycle

400Calculating cycle

1500Calculating cycle

400Calculating cycle

15001 04701 04578 05049 047302 04361 04436 05406 045523 04566 04772 04789 044934 04812 04303 04590 051605 04496 04781 04942 045866 04920 04549 05383 048337 04779 03959 05228 046268 05428 04352 05133 045809 05472 04368 05497 0498510 05231 04496 04551 04542Mean 04876 04459 05056 04708Standard deviation 0038 0024 0033 0021

Table 22 Comparison table for target equation (obj) and duration in calculation of GA and Lingo program

Number ofnurses(person)

Number ofcalculatingcycles

GA Lingo

Target equation (obj) Time (hour) Targetequation (obj) Time (hour) Different result

of obj

Different incomputational

run time

201500 Lowest 03959 603 580 86849

Mean 04459 plusmn 0024 685 03379 5237 1080 plusmn 24 76453

400 Lowest 04361 171 982 306257Mean 04876 plusmn 0038 166 1497 plusmn 38 315482

Table 23 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2012

Number of nurses Monthly expense SDOld New Increase Old New Decrease Old New Decrease18 20 2 395964 366400 29564 1089212 214843 8028

Table 24 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2013

Number of nurses Monthly expense SDOld New Decrease Old New Decrease Old New Decrease23 20 3 416548 366400 50148 1966735 214843 8908

315481 on average showing that the genetic program ismore suitable for practical use

8 Conclusion

The findings showed that the genetic program created couldmanage the work schedule of nurses under given conditionsand achieve the desired objective It could create work sche-dules of nurses that were fair in overtime payment to all thenurses and could reduce the hospital expenses by setting thenew coming nursesrsquo minimum salary at 15000 Baht per per-son

Table 25 Target equation and calculation time from geneticprogram

Target equation (obj) Calculation time (hour)03959 603

From Tables 23 24 and 25 the following results canbe concluded the work schedules from proposed methodreduce the waiting time of the number of patients waitinglonger than 125 times of the average length nursing servicefrom 3408 to less than 15 on average (starting from

Mathematical Problems in Engineering 15

when the patients register and wait to see the doctor) Ifwe want to schedule the work of the nurses in January 2012in order to meet the criteria for the service of this studywe have to increase the number of nurses from 18 to 20 sothat the number can perfectly meet the needs of the patientsaccording to the criteriaThework schedules of the nurses canbemost appropriate if the target equation is 03959which usesthe time of calculation as 603 hourswith a standard deviationof less than 8028and the overtimepayment less than 29564Baht per monthWhen this is compared with the actual workschedule in January 2013 the work schedule could reduce thenumber of nurses from 23 to 20 with a decreased standarddeviation that is 8908 and the decrease in the totalovertime paymentwould be 50148 Baht permonthWe couldcalculate the most appropriate work schedule quicker thanby using the conventional approach which results obtainedby Lingo program version 50 86849 with 1500 rounds forthe calculation cycle

As we know hospitals belong to the service businesssector continuous improvement along with response to cus-tomer satisfaction is the key success factor in this businessTo reduce the waiting time in every hospital process highstandards in medical care and highly experienced medicaldoctors are needed Presently Thailand has set its aim tobecome an Asian medical and tourist hub this strategy of thecountry may cause and force hospitals to improve their ser-vice quality for responding well to the high number of outpa-tients Updated data collection needs to be done service-level policies have to be reconsidered Demands of outpa-tient nurses and their schedules may change from time totime Inpatient nurses may be allocated to assist when themanagement wishes to reduce the fluctuation in the numberof outpatient nurses Thus modern management and highefficient tools are necessary this is the reason why GA isinvolved instead of an adoption of the conventional approachHowever regarding the implementation of simulation evenGA is not familiar to the hospital people Optimization capac-ity building is another issue that the hospitalmanagementwillhave to consider

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chiang Mai University (CMU) through the research admin-istration office provided budget to our Excellence Center inLogistics and Supply Chain Management (E-LSCM)

References

[1] T Mongkolrangsarit and W Thamapornpilat Development ofNurse Scheduling Method with Uncertain Workforce DemandDepartment of Industrial Engineering Faculty of EngineeringChulalongkorn University 2007

[2] C M Rothe and H B Wolfe ldquoCyclical scheduling and alloca-tion of nursing staffrdquo Socio-Economic Planning Sciences vol 7no 5 pp 471ndash487 1973

[3] R B Norby L E Freund and B Wagner ldquoA nurse staffingsystem based upon assignment difficultyrdquo Journal of NursingAdministration vol 7 no 9 pp 2ndash24 1977

[4] T Ryan B L Barker and F A Marciante ldquoA system fordetermining appropriate nurse staffingrdquo Journal of NursingAdministration vol 5 no 5 pp 30ndash38 1975

[5] H Wolfe and J P Young ldquoStaffing the nursing unit Part Icontrolled variable staffingrdquo Nursing Research vol 14 pp 236ndash243 1965

[6] HWolfe and J P Young ldquoStaffing the nursing unit IIThemult-iple assignment techniquerdquo Nursing Research vol 14 no 4 pp299ndash303 1965

[7] J L Arthur andA Ravindran ldquoAmultiple objective nurse sche-duling modelrdquo AIIE Transactions vol 13 no 1 pp 55ndash60 1981

[8] I Ozkarahan ldquoA disaggregation model of a flexible nursescheduling support systemrdquo Socio-Economic Planning Sciencesvol 25 no 1 pp 9ndash26 1991

[9] I Ozkarahan and J E Bailey ldquoGoal programming modelsubsystem of a flexible nurse scheduling support systemrdquo IIETransactions vol 20 no 3 pp 306ndash316 1988

[10] K A Dowsland ldquoNurse scheduling with tabu search and stra-tegic oscillationrdquo European Journal of Operational Research vol106 no 2-3 pp 393ndash407 1998

[11] P Tormos A Lova F Barber L Ingolotti M Abril and M ASalido ldquoA genetic algorithm for railway scheduling problemsrdquoinMetaheuristics for Scheduling in Industrial andManufacturingApplications vol 128 of Studies in Computational Intelligencepp 255ndash276 Springer Berlin Germany 2008

[12] H Kawanaka K Yamamoto T Yoshikawa T Shinogi and STsuruoka ldquoGenetic algorithm with the constraints for nursescheduling problemrdquo in Proceedings of the Congress on Evolu-tionary Computation vol 2 pp 1123ndash1130 Seoul Republic ofKorea May 2001

[13] H HMillar and KMona ldquoCyclic and non-cyclic scheduling of12 h shift nurses by network programmingrdquo European Journalof Operational Research vol 104 no 3 pp 582ndash592 1998

[14] I Blochlige ldquoModeling staff scheduling problems A tutorialrdquoEuropean Journal of Operational Research vol 158 no 3 pp533ndash542 2004

[15] B Jaumard F Semet and T Vovor ldquoA generalized linear pro-gramming model for nurse schedulingrdquo European Journal ofOperational Research vol 107 no 1 pp 1ndash18 1998

[16] T Cezik O Gunluk and H Luss ldquoAn integer programmingmodel for the weekly tour scheduling problemrdquoNaval ResearchLogistics vol 48 no 7 pp 607ndash624 2001

[17] Z Wang J-L Liu and X Yu ldquoSelf-fertilization based geneticalgorithm for university timetabling problemrdquo in Proceedingsof the 1st ACMSIGEVO Summit on Genetic and EvolutionaryComputation (GEC rsquo09) pp 1001ndash1004 ACM Shanghai ChinaJune 2009

[18] J Wang and K Chu ldquoAn application of genetic algorithms forthe flexible job-shop scheduling problemrdquo International Journalof Advancements in Computing Technology vol 4 no 3 pp 271ndash278 2012

[19] M Moz and M V Pato ldquoA genetic algorithm approach to anurse rerostering problemrdquo Computers amp Operations Researchvol 34 no 3 pp 667ndash691 2007

[20] C Wotthipong K Soradech and S Nidapan ldquoThe suitablegenetic operators for solving the university course timetablingproblemrdquo Journal of Convergence Information Technology vol8 pp 60ndash65 2013

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 12: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

12 Mathematical Problems in Engineering

Table 14 Experimental results obtained at low level mutation

Max iteration Population crossover mutation Obj Time (s)1500 10 95 95 044442 80491500 10 95 1 048580 101611500 10 95 01 050286 105231500 10 95 001 053612 10117

Table 15 Target equation (obj) predicted by Minitab program and compared with actual target equation (obj)

Target equation (obj) Comparison of target equation (obj)percentage

Predicted value Actual value Actual value compared with predicted value04492 04429 063

Table 16 Calculation results with reduction in number of nurses (119894)at 119908 = 06

Number of nurses (119894) Target equation(obj)

Calculating time(hour)

21 05241 76720 03959 60319 Infeasible Infeasible18 Infeasible Infeasible

Figure 5 The response optimizer configuration of obj response

number of nurses (119894) who were able to schedule the workunder the restrictions by using genetic programming and bygradually reducing the number of nurses (119894) down to the finalvalue at which the program could calculate the answer Thecalculation was performed at 119908 = 06 in Table 16

Table 16 shows that the least number of nurses (119894) whichcould schedule work under the restrictions was 20 Basedon 20 nurses amount of overtime payment and standarddeviation of overtime payment were calculated and reportedin Table 17 Therefore number of working hours for eachnurse number of nurses in each department and nursersquosschedule planwere generated and reported as shown inTables18 and 19

From Table 17 it can be observed that the target equation(obj) of the responses obtained was equal to 03959The sumof the overtime payment of all nurses was 16600 Baht per

CurHigh

Low099551

Optimal

minimumObj

099551desirabilityComposite

50950

50950

100500

50040004000 100 950 950

y = 04492

d = 099551

A B C D

D

Figure 6 The results of the data analysis using the responseoptimizer of the obj response

week The standard deviation of overtime payment that eachnurse received was 214843 For fair comparison we have toconvert the real overtime payment and standard deviationfor current 18 nurses to 20 nurses Then we can report thatthe sum of the total overtime payment and the standarddeviation of the overtime payment that each nurse receivedwhich is obtained from the model were less than those of theold working schedule at 17991 Baht per week and 882369respectively

Table 18 shows that all the nurses worked according to theconditions specified that is one nurse must work at least 40hours per week as in (4) but must work fewer than 60 hoursper week as in (5) all would be selected to work as in (6)and the duration of work must not exceed six days a week asin (7)

From Table 19 we can see that the number of nursesworking in shifts each day was more than the demand fornurses (Demand

119895119896119897) as shown in Table 6 which is according

to (3) and the specified service criteria (on average 15of thepatients waiting for services should not wait longer than theaverage service time of 25) because the number of nursesworking in shifts was more than the demand

To study the effect ofweight adjusted (119908)more numericalexperiments were conducted by GA coded in Matlab at 400

Mathematical Problems in Engineering 13

Table 17 Calculating results by 20 nurses (119894)

Target equation(obj)

Calculating time(hour)

Sum of overtime paymentof nurses (Baht)

Standard deviation of total overtimepayment that each nurse received

03959 603 16600 214843

Table 18 Number of working hours calculated from 20 nurses (119894)

Number of nurses Hoursperson (hour)Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total

1 12 12 0 4 0 8 12 482 4 8 12 4 8 12 0 483 8 8 8 0 12 12 0 484 12 0 12 12 12 4 0 525 8 12 8 4 0 0 12 44

20 8 12 4 0 12 8 4 48

Table 19 Number of nurses working in shifts calculated by 20 nurses (119894)

Department Period Supply (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 4 3 2 2PED 3 3 4 3 3 4 3EENT 2 3 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 3 3EENT 3 2 2 2 2 2 2OBG 2 2 3 2 1 1 2

Table 20 Effect of weight adjusted to objective function

Weight adjusted119908 = 09 119908 = 06 119908 = 05

Objective function 05090 04753 04823Sum of overtimepayment 24375 24625 26575

Standard deviation 324898 297675 288509

iterations and reported in Table 20 The report was shownas we expected more weight adjusted less overtime paymentsimultaneously with higher standard deviation It means thatif management focuses on reducing the overtime payment itwill lead to unfair payment or bias nursesrsquo schedule plan

7 Comparison of Advantages andDisadvantages of Adaptive Genetic andOptimization Approach

Weprocessed the data to do a comparison between the resultsof the adaptive genetic approach and the results obtained by

the Lingo program using the same data in order to comparethe target equation (obj) and computation runtime as shownin Tables 21 and 22

Table 21 shows the results of the processing experimentafter it was conducted ten times The work schedules using20 and 24 nurses with the calculating cycle of 400 roundsresulted in the best value for the target equations of 04361and 04551 respectively After ten processing experimentsthe work schedules for 20 and 24 nurses with the calculatingcycle at 1500 rounds resulted in the best value for the targetequations of 03959 and 04493 respectively and can besummarized as presented in Table 22

Table 22 shows that the work schedule for 20 nurses at1500 rounds when compared to the Lingo program couldcalculate thework schedulewith the target equation (obj) lessthan the genetic algorithm by around 580 and 108 plusmn 24on average but that the one created by the Lingo programused longer computational runtime by over 86849 and76452 on average We can also notice that the results at400 rounds reported the same direction the best and theaverage solution of genetic algorithm differ about 982 and1497 plusmn 38 from the optimal respectively However theoptimal method used longer time by over 306257 and

14 Mathematical Problems in Engineering

Table 21 Results of genetic program calculated ten times at 400 rounds and 1500 rounds

Target equation (obj)number

Number of 20 nurses Number of 24 nursesCalculating cycle

400Calculating cycle

1500Calculating cycle

400Calculating cycle

15001 04701 04578 05049 047302 04361 04436 05406 045523 04566 04772 04789 044934 04812 04303 04590 051605 04496 04781 04942 045866 04920 04549 05383 048337 04779 03959 05228 046268 05428 04352 05133 045809 05472 04368 05497 0498510 05231 04496 04551 04542Mean 04876 04459 05056 04708Standard deviation 0038 0024 0033 0021

Table 22 Comparison table for target equation (obj) and duration in calculation of GA and Lingo program

Number ofnurses(person)

Number ofcalculatingcycles

GA Lingo

Target equation (obj) Time (hour) Targetequation (obj) Time (hour) Different result

of obj

Different incomputational

run time

201500 Lowest 03959 603 580 86849

Mean 04459 plusmn 0024 685 03379 5237 1080 plusmn 24 76453

400 Lowest 04361 171 982 306257Mean 04876 plusmn 0038 166 1497 plusmn 38 315482

Table 23 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2012

Number of nurses Monthly expense SDOld New Increase Old New Decrease Old New Decrease18 20 2 395964 366400 29564 1089212 214843 8028

Table 24 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2013

Number of nurses Monthly expense SDOld New Decrease Old New Decrease Old New Decrease23 20 3 416548 366400 50148 1966735 214843 8908

315481 on average showing that the genetic program ismore suitable for practical use

8 Conclusion

The findings showed that the genetic program created couldmanage the work schedule of nurses under given conditionsand achieve the desired objective It could create work sche-dules of nurses that were fair in overtime payment to all thenurses and could reduce the hospital expenses by setting thenew coming nursesrsquo minimum salary at 15000 Baht per per-son

Table 25 Target equation and calculation time from geneticprogram

Target equation (obj) Calculation time (hour)03959 603

From Tables 23 24 and 25 the following results canbe concluded the work schedules from proposed methodreduce the waiting time of the number of patients waitinglonger than 125 times of the average length nursing servicefrom 3408 to less than 15 on average (starting from

Mathematical Problems in Engineering 15

when the patients register and wait to see the doctor) Ifwe want to schedule the work of the nurses in January 2012in order to meet the criteria for the service of this studywe have to increase the number of nurses from 18 to 20 sothat the number can perfectly meet the needs of the patientsaccording to the criteriaThework schedules of the nurses canbemost appropriate if the target equation is 03959which usesthe time of calculation as 603 hourswith a standard deviationof less than 8028and the overtimepayment less than 29564Baht per monthWhen this is compared with the actual workschedule in January 2013 the work schedule could reduce thenumber of nurses from 23 to 20 with a decreased standarddeviation that is 8908 and the decrease in the totalovertime paymentwould be 50148 Baht permonthWe couldcalculate the most appropriate work schedule quicker thanby using the conventional approach which results obtainedby Lingo program version 50 86849 with 1500 rounds forthe calculation cycle

As we know hospitals belong to the service businesssector continuous improvement along with response to cus-tomer satisfaction is the key success factor in this businessTo reduce the waiting time in every hospital process highstandards in medical care and highly experienced medicaldoctors are needed Presently Thailand has set its aim tobecome an Asian medical and tourist hub this strategy of thecountry may cause and force hospitals to improve their ser-vice quality for responding well to the high number of outpa-tients Updated data collection needs to be done service-level policies have to be reconsidered Demands of outpa-tient nurses and their schedules may change from time totime Inpatient nurses may be allocated to assist when themanagement wishes to reduce the fluctuation in the numberof outpatient nurses Thus modern management and highefficient tools are necessary this is the reason why GA isinvolved instead of an adoption of the conventional approachHowever regarding the implementation of simulation evenGA is not familiar to the hospital people Optimization capac-ity building is another issue that the hospitalmanagementwillhave to consider

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chiang Mai University (CMU) through the research admin-istration office provided budget to our Excellence Center inLogistics and Supply Chain Management (E-LSCM)

References

[1] T Mongkolrangsarit and W Thamapornpilat Development ofNurse Scheduling Method with Uncertain Workforce DemandDepartment of Industrial Engineering Faculty of EngineeringChulalongkorn University 2007

[2] C M Rothe and H B Wolfe ldquoCyclical scheduling and alloca-tion of nursing staffrdquo Socio-Economic Planning Sciences vol 7no 5 pp 471ndash487 1973

[3] R B Norby L E Freund and B Wagner ldquoA nurse staffingsystem based upon assignment difficultyrdquo Journal of NursingAdministration vol 7 no 9 pp 2ndash24 1977

[4] T Ryan B L Barker and F A Marciante ldquoA system fordetermining appropriate nurse staffingrdquo Journal of NursingAdministration vol 5 no 5 pp 30ndash38 1975

[5] H Wolfe and J P Young ldquoStaffing the nursing unit Part Icontrolled variable staffingrdquo Nursing Research vol 14 pp 236ndash243 1965

[6] HWolfe and J P Young ldquoStaffing the nursing unit IIThemult-iple assignment techniquerdquo Nursing Research vol 14 no 4 pp299ndash303 1965

[7] J L Arthur andA Ravindran ldquoAmultiple objective nurse sche-duling modelrdquo AIIE Transactions vol 13 no 1 pp 55ndash60 1981

[8] I Ozkarahan ldquoA disaggregation model of a flexible nursescheduling support systemrdquo Socio-Economic Planning Sciencesvol 25 no 1 pp 9ndash26 1991

[9] I Ozkarahan and J E Bailey ldquoGoal programming modelsubsystem of a flexible nurse scheduling support systemrdquo IIETransactions vol 20 no 3 pp 306ndash316 1988

[10] K A Dowsland ldquoNurse scheduling with tabu search and stra-tegic oscillationrdquo European Journal of Operational Research vol106 no 2-3 pp 393ndash407 1998

[11] P Tormos A Lova F Barber L Ingolotti M Abril and M ASalido ldquoA genetic algorithm for railway scheduling problemsrdquoinMetaheuristics for Scheduling in Industrial andManufacturingApplications vol 128 of Studies in Computational Intelligencepp 255ndash276 Springer Berlin Germany 2008

[12] H Kawanaka K Yamamoto T Yoshikawa T Shinogi and STsuruoka ldquoGenetic algorithm with the constraints for nursescheduling problemrdquo in Proceedings of the Congress on Evolu-tionary Computation vol 2 pp 1123ndash1130 Seoul Republic ofKorea May 2001

[13] H HMillar and KMona ldquoCyclic and non-cyclic scheduling of12 h shift nurses by network programmingrdquo European Journalof Operational Research vol 104 no 3 pp 582ndash592 1998

[14] I Blochlige ldquoModeling staff scheduling problems A tutorialrdquoEuropean Journal of Operational Research vol 158 no 3 pp533ndash542 2004

[15] B Jaumard F Semet and T Vovor ldquoA generalized linear pro-gramming model for nurse schedulingrdquo European Journal ofOperational Research vol 107 no 1 pp 1ndash18 1998

[16] T Cezik O Gunluk and H Luss ldquoAn integer programmingmodel for the weekly tour scheduling problemrdquoNaval ResearchLogistics vol 48 no 7 pp 607ndash624 2001

[17] Z Wang J-L Liu and X Yu ldquoSelf-fertilization based geneticalgorithm for university timetabling problemrdquo in Proceedingsof the 1st ACMSIGEVO Summit on Genetic and EvolutionaryComputation (GEC rsquo09) pp 1001ndash1004 ACM Shanghai ChinaJune 2009

[18] J Wang and K Chu ldquoAn application of genetic algorithms forthe flexible job-shop scheduling problemrdquo International Journalof Advancements in Computing Technology vol 4 no 3 pp 271ndash278 2012

[19] M Moz and M V Pato ldquoA genetic algorithm approach to anurse rerostering problemrdquo Computers amp Operations Researchvol 34 no 3 pp 667ndash691 2007

[20] C Wotthipong K Soradech and S Nidapan ldquoThe suitablegenetic operators for solving the university course timetablingproblemrdquo Journal of Convergence Information Technology vol8 pp 60ndash65 2013

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 13: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

Mathematical Problems in Engineering 13

Table 17 Calculating results by 20 nurses (119894)

Target equation(obj)

Calculating time(hour)

Sum of overtime paymentof nurses (Baht)

Standard deviation of total overtimepayment that each nurse received

03959 603 16600 214843

Table 18 Number of working hours calculated from 20 nurses (119894)

Number of nurses Hoursperson (hour)Monday Tuesday Wednesday Thursday Friday Saturday Sunday Total

1 12 12 0 4 0 8 12 482 4 8 12 4 8 12 0 483 8 8 8 0 12 12 0 484 12 0 12 12 12 4 0 525 8 12 8 4 0 0 12 44

20 8 12 4 0 12 8 4 48

Table 19 Number of nurses working in shifts calculated by 20 nurses (119894)

Department Period Supply (person)Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SUR

Regular

2 2 2 2 2 2 2MED 3 3 3 4 3 2 2PED 3 3 4 3 3 4 3EENT 2 3 2 2 2 2 2OBG 2 2 2 2 2 2 2SUR

Overtime

2 2 2 2 2 1 2MED 3 2 2 2 2 2 2PED 2 2 2 2 3 3 3EENT 3 2 2 2 2 2 2OBG 2 2 3 2 1 1 2

Table 20 Effect of weight adjusted to objective function

Weight adjusted119908 = 09 119908 = 06 119908 = 05

Objective function 05090 04753 04823Sum of overtimepayment 24375 24625 26575

Standard deviation 324898 297675 288509

iterations and reported in Table 20 The report was shownas we expected more weight adjusted less overtime paymentsimultaneously with higher standard deviation It means thatif management focuses on reducing the overtime payment itwill lead to unfair payment or bias nursesrsquo schedule plan

7 Comparison of Advantages andDisadvantages of Adaptive Genetic andOptimization Approach

Weprocessed the data to do a comparison between the resultsof the adaptive genetic approach and the results obtained by

the Lingo program using the same data in order to comparethe target equation (obj) and computation runtime as shownin Tables 21 and 22

Table 21 shows the results of the processing experimentafter it was conducted ten times The work schedules using20 and 24 nurses with the calculating cycle of 400 roundsresulted in the best value for the target equations of 04361and 04551 respectively After ten processing experimentsthe work schedules for 20 and 24 nurses with the calculatingcycle at 1500 rounds resulted in the best value for the targetequations of 03959 and 04493 respectively and can besummarized as presented in Table 22

Table 22 shows that the work schedule for 20 nurses at1500 rounds when compared to the Lingo program couldcalculate thework schedulewith the target equation (obj) lessthan the genetic algorithm by around 580 and 108 plusmn 24on average but that the one created by the Lingo programused longer computational runtime by over 86849 and76452 on average We can also notice that the results at400 rounds reported the same direction the best and theaverage solution of genetic algorithm differ about 982 and1497 plusmn 38 from the optimal respectively However theoptimal method used longer time by over 306257 and

14 Mathematical Problems in Engineering

Table 21 Results of genetic program calculated ten times at 400 rounds and 1500 rounds

Target equation (obj)number

Number of 20 nurses Number of 24 nursesCalculating cycle

400Calculating cycle

1500Calculating cycle

400Calculating cycle

15001 04701 04578 05049 047302 04361 04436 05406 045523 04566 04772 04789 044934 04812 04303 04590 051605 04496 04781 04942 045866 04920 04549 05383 048337 04779 03959 05228 046268 05428 04352 05133 045809 05472 04368 05497 0498510 05231 04496 04551 04542Mean 04876 04459 05056 04708Standard deviation 0038 0024 0033 0021

Table 22 Comparison table for target equation (obj) and duration in calculation of GA and Lingo program

Number ofnurses(person)

Number ofcalculatingcycles

GA Lingo

Target equation (obj) Time (hour) Targetequation (obj) Time (hour) Different result

of obj

Different incomputational

run time

201500 Lowest 03959 603 580 86849

Mean 04459 plusmn 0024 685 03379 5237 1080 plusmn 24 76453

400 Lowest 04361 171 982 306257Mean 04876 plusmn 0038 166 1497 plusmn 38 315482

Table 23 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2012

Number of nurses Monthly expense SDOld New Increase Old New Decrease Old New Decrease18 20 2 395964 366400 29564 1089212 214843 8028

Table 24 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2013

Number of nurses Monthly expense SDOld New Decrease Old New Decrease Old New Decrease23 20 3 416548 366400 50148 1966735 214843 8908

315481 on average showing that the genetic program ismore suitable for practical use

8 Conclusion

The findings showed that the genetic program created couldmanage the work schedule of nurses under given conditionsand achieve the desired objective It could create work sche-dules of nurses that were fair in overtime payment to all thenurses and could reduce the hospital expenses by setting thenew coming nursesrsquo minimum salary at 15000 Baht per per-son

Table 25 Target equation and calculation time from geneticprogram

Target equation (obj) Calculation time (hour)03959 603

From Tables 23 24 and 25 the following results canbe concluded the work schedules from proposed methodreduce the waiting time of the number of patients waitinglonger than 125 times of the average length nursing servicefrom 3408 to less than 15 on average (starting from

Mathematical Problems in Engineering 15

when the patients register and wait to see the doctor) Ifwe want to schedule the work of the nurses in January 2012in order to meet the criteria for the service of this studywe have to increase the number of nurses from 18 to 20 sothat the number can perfectly meet the needs of the patientsaccording to the criteriaThework schedules of the nurses canbemost appropriate if the target equation is 03959which usesthe time of calculation as 603 hourswith a standard deviationof less than 8028and the overtimepayment less than 29564Baht per monthWhen this is compared with the actual workschedule in January 2013 the work schedule could reduce thenumber of nurses from 23 to 20 with a decreased standarddeviation that is 8908 and the decrease in the totalovertime paymentwould be 50148 Baht permonthWe couldcalculate the most appropriate work schedule quicker thanby using the conventional approach which results obtainedby Lingo program version 50 86849 with 1500 rounds forthe calculation cycle

As we know hospitals belong to the service businesssector continuous improvement along with response to cus-tomer satisfaction is the key success factor in this businessTo reduce the waiting time in every hospital process highstandards in medical care and highly experienced medicaldoctors are needed Presently Thailand has set its aim tobecome an Asian medical and tourist hub this strategy of thecountry may cause and force hospitals to improve their ser-vice quality for responding well to the high number of outpa-tients Updated data collection needs to be done service-level policies have to be reconsidered Demands of outpa-tient nurses and their schedules may change from time totime Inpatient nurses may be allocated to assist when themanagement wishes to reduce the fluctuation in the numberof outpatient nurses Thus modern management and highefficient tools are necessary this is the reason why GA isinvolved instead of an adoption of the conventional approachHowever regarding the implementation of simulation evenGA is not familiar to the hospital people Optimization capac-ity building is another issue that the hospitalmanagementwillhave to consider

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chiang Mai University (CMU) through the research admin-istration office provided budget to our Excellence Center inLogistics and Supply Chain Management (E-LSCM)

References

[1] T Mongkolrangsarit and W Thamapornpilat Development ofNurse Scheduling Method with Uncertain Workforce DemandDepartment of Industrial Engineering Faculty of EngineeringChulalongkorn University 2007

[2] C M Rothe and H B Wolfe ldquoCyclical scheduling and alloca-tion of nursing staffrdquo Socio-Economic Planning Sciences vol 7no 5 pp 471ndash487 1973

[3] R B Norby L E Freund and B Wagner ldquoA nurse staffingsystem based upon assignment difficultyrdquo Journal of NursingAdministration vol 7 no 9 pp 2ndash24 1977

[4] T Ryan B L Barker and F A Marciante ldquoA system fordetermining appropriate nurse staffingrdquo Journal of NursingAdministration vol 5 no 5 pp 30ndash38 1975

[5] H Wolfe and J P Young ldquoStaffing the nursing unit Part Icontrolled variable staffingrdquo Nursing Research vol 14 pp 236ndash243 1965

[6] HWolfe and J P Young ldquoStaffing the nursing unit IIThemult-iple assignment techniquerdquo Nursing Research vol 14 no 4 pp299ndash303 1965

[7] J L Arthur andA Ravindran ldquoAmultiple objective nurse sche-duling modelrdquo AIIE Transactions vol 13 no 1 pp 55ndash60 1981

[8] I Ozkarahan ldquoA disaggregation model of a flexible nursescheduling support systemrdquo Socio-Economic Planning Sciencesvol 25 no 1 pp 9ndash26 1991

[9] I Ozkarahan and J E Bailey ldquoGoal programming modelsubsystem of a flexible nurse scheduling support systemrdquo IIETransactions vol 20 no 3 pp 306ndash316 1988

[10] K A Dowsland ldquoNurse scheduling with tabu search and stra-tegic oscillationrdquo European Journal of Operational Research vol106 no 2-3 pp 393ndash407 1998

[11] P Tormos A Lova F Barber L Ingolotti M Abril and M ASalido ldquoA genetic algorithm for railway scheduling problemsrdquoinMetaheuristics for Scheduling in Industrial andManufacturingApplications vol 128 of Studies in Computational Intelligencepp 255ndash276 Springer Berlin Germany 2008

[12] H Kawanaka K Yamamoto T Yoshikawa T Shinogi and STsuruoka ldquoGenetic algorithm with the constraints for nursescheduling problemrdquo in Proceedings of the Congress on Evolu-tionary Computation vol 2 pp 1123ndash1130 Seoul Republic ofKorea May 2001

[13] H HMillar and KMona ldquoCyclic and non-cyclic scheduling of12 h shift nurses by network programmingrdquo European Journalof Operational Research vol 104 no 3 pp 582ndash592 1998

[14] I Blochlige ldquoModeling staff scheduling problems A tutorialrdquoEuropean Journal of Operational Research vol 158 no 3 pp533ndash542 2004

[15] B Jaumard F Semet and T Vovor ldquoA generalized linear pro-gramming model for nurse schedulingrdquo European Journal ofOperational Research vol 107 no 1 pp 1ndash18 1998

[16] T Cezik O Gunluk and H Luss ldquoAn integer programmingmodel for the weekly tour scheduling problemrdquoNaval ResearchLogistics vol 48 no 7 pp 607ndash624 2001

[17] Z Wang J-L Liu and X Yu ldquoSelf-fertilization based geneticalgorithm for university timetabling problemrdquo in Proceedingsof the 1st ACMSIGEVO Summit on Genetic and EvolutionaryComputation (GEC rsquo09) pp 1001ndash1004 ACM Shanghai ChinaJune 2009

[18] J Wang and K Chu ldquoAn application of genetic algorithms forthe flexible job-shop scheduling problemrdquo International Journalof Advancements in Computing Technology vol 4 no 3 pp 271ndash278 2012

[19] M Moz and M V Pato ldquoA genetic algorithm approach to anurse rerostering problemrdquo Computers amp Operations Researchvol 34 no 3 pp 667ndash691 2007

[20] C Wotthipong K Soradech and S Nidapan ldquoThe suitablegenetic operators for solving the university course timetablingproblemrdquo Journal of Convergence Information Technology vol8 pp 60ndash65 2013

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 14: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

14 Mathematical Problems in Engineering

Table 21 Results of genetic program calculated ten times at 400 rounds and 1500 rounds

Target equation (obj)number

Number of 20 nurses Number of 24 nursesCalculating cycle

400Calculating cycle

1500Calculating cycle

400Calculating cycle

15001 04701 04578 05049 047302 04361 04436 05406 045523 04566 04772 04789 044934 04812 04303 04590 051605 04496 04781 04942 045866 04920 04549 05383 048337 04779 03959 05228 046268 05428 04352 05133 045809 05472 04368 05497 0498510 05231 04496 04551 04542Mean 04876 04459 05056 04708Standard deviation 0038 0024 0033 0021

Table 22 Comparison table for target equation (obj) and duration in calculation of GA and Lingo program

Number ofnurses(person)

Number ofcalculatingcycles

GA Lingo

Target equation (obj) Time (hour) Targetequation (obj) Time (hour) Different result

of obj

Different incomputational

run time

201500 Lowest 03959 603 580 86849

Mean 04459 plusmn 0024 685 03379 5237 1080 plusmn 24 76453

400 Lowest 04361 171 982 306257Mean 04876 plusmn 0038 166 1497 plusmn 38 315482

Table 23 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2012

Number of nurses Monthly expense SDOld New Increase Old New Decrease Old New Decrease18 20 2 395964 366400 29564 1089212 214843 8028

Table 24 Comparison of differences in number of nurses monthly expenses and SD of work schedule created and actual work schedule inJanuary 2013

Number of nurses Monthly expense SDOld New Decrease Old New Decrease Old New Decrease23 20 3 416548 366400 50148 1966735 214843 8908

315481 on average showing that the genetic program ismore suitable for practical use

8 Conclusion

The findings showed that the genetic program created couldmanage the work schedule of nurses under given conditionsand achieve the desired objective It could create work sche-dules of nurses that were fair in overtime payment to all thenurses and could reduce the hospital expenses by setting thenew coming nursesrsquo minimum salary at 15000 Baht per per-son

Table 25 Target equation and calculation time from geneticprogram

Target equation (obj) Calculation time (hour)03959 603

From Tables 23 24 and 25 the following results canbe concluded the work schedules from proposed methodreduce the waiting time of the number of patients waitinglonger than 125 times of the average length nursing servicefrom 3408 to less than 15 on average (starting from

Mathematical Problems in Engineering 15

when the patients register and wait to see the doctor) Ifwe want to schedule the work of the nurses in January 2012in order to meet the criteria for the service of this studywe have to increase the number of nurses from 18 to 20 sothat the number can perfectly meet the needs of the patientsaccording to the criteriaThework schedules of the nurses canbemost appropriate if the target equation is 03959which usesthe time of calculation as 603 hourswith a standard deviationof less than 8028and the overtimepayment less than 29564Baht per monthWhen this is compared with the actual workschedule in January 2013 the work schedule could reduce thenumber of nurses from 23 to 20 with a decreased standarddeviation that is 8908 and the decrease in the totalovertime paymentwould be 50148 Baht permonthWe couldcalculate the most appropriate work schedule quicker thanby using the conventional approach which results obtainedby Lingo program version 50 86849 with 1500 rounds forthe calculation cycle

As we know hospitals belong to the service businesssector continuous improvement along with response to cus-tomer satisfaction is the key success factor in this businessTo reduce the waiting time in every hospital process highstandards in medical care and highly experienced medicaldoctors are needed Presently Thailand has set its aim tobecome an Asian medical and tourist hub this strategy of thecountry may cause and force hospitals to improve their ser-vice quality for responding well to the high number of outpa-tients Updated data collection needs to be done service-level policies have to be reconsidered Demands of outpa-tient nurses and their schedules may change from time totime Inpatient nurses may be allocated to assist when themanagement wishes to reduce the fluctuation in the numberof outpatient nurses Thus modern management and highefficient tools are necessary this is the reason why GA isinvolved instead of an adoption of the conventional approachHowever regarding the implementation of simulation evenGA is not familiar to the hospital people Optimization capac-ity building is another issue that the hospitalmanagementwillhave to consider

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chiang Mai University (CMU) through the research admin-istration office provided budget to our Excellence Center inLogistics and Supply Chain Management (E-LSCM)

References

[1] T Mongkolrangsarit and W Thamapornpilat Development ofNurse Scheduling Method with Uncertain Workforce DemandDepartment of Industrial Engineering Faculty of EngineeringChulalongkorn University 2007

[2] C M Rothe and H B Wolfe ldquoCyclical scheduling and alloca-tion of nursing staffrdquo Socio-Economic Planning Sciences vol 7no 5 pp 471ndash487 1973

[3] R B Norby L E Freund and B Wagner ldquoA nurse staffingsystem based upon assignment difficultyrdquo Journal of NursingAdministration vol 7 no 9 pp 2ndash24 1977

[4] T Ryan B L Barker and F A Marciante ldquoA system fordetermining appropriate nurse staffingrdquo Journal of NursingAdministration vol 5 no 5 pp 30ndash38 1975

[5] H Wolfe and J P Young ldquoStaffing the nursing unit Part Icontrolled variable staffingrdquo Nursing Research vol 14 pp 236ndash243 1965

[6] HWolfe and J P Young ldquoStaffing the nursing unit IIThemult-iple assignment techniquerdquo Nursing Research vol 14 no 4 pp299ndash303 1965

[7] J L Arthur andA Ravindran ldquoAmultiple objective nurse sche-duling modelrdquo AIIE Transactions vol 13 no 1 pp 55ndash60 1981

[8] I Ozkarahan ldquoA disaggregation model of a flexible nursescheduling support systemrdquo Socio-Economic Planning Sciencesvol 25 no 1 pp 9ndash26 1991

[9] I Ozkarahan and J E Bailey ldquoGoal programming modelsubsystem of a flexible nurse scheduling support systemrdquo IIETransactions vol 20 no 3 pp 306ndash316 1988

[10] K A Dowsland ldquoNurse scheduling with tabu search and stra-tegic oscillationrdquo European Journal of Operational Research vol106 no 2-3 pp 393ndash407 1998

[11] P Tormos A Lova F Barber L Ingolotti M Abril and M ASalido ldquoA genetic algorithm for railway scheduling problemsrdquoinMetaheuristics for Scheduling in Industrial andManufacturingApplications vol 128 of Studies in Computational Intelligencepp 255ndash276 Springer Berlin Germany 2008

[12] H Kawanaka K Yamamoto T Yoshikawa T Shinogi and STsuruoka ldquoGenetic algorithm with the constraints for nursescheduling problemrdquo in Proceedings of the Congress on Evolu-tionary Computation vol 2 pp 1123ndash1130 Seoul Republic ofKorea May 2001

[13] H HMillar and KMona ldquoCyclic and non-cyclic scheduling of12 h shift nurses by network programmingrdquo European Journalof Operational Research vol 104 no 3 pp 582ndash592 1998

[14] I Blochlige ldquoModeling staff scheduling problems A tutorialrdquoEuropean Journal of Operational Research vol 158 no 3 pp533ndash542 2004

[15] B Jaumard F Semet and T Vovor ldquoA generalized linear pro-gramming model for nurse schedulingrdquo European Journal ofOperational Research vol 107 no 1 pp 1ndash18 1998

[16] T Cezik O Gunluk and H Luss ldquoAn integer programmingmodel for the weekly tour scheduling problemrdquoNaval ResearchLogistics vol 48 no 7 pp 607ndash624 2001

[17] Z Wang J-L Liu and X Yu ldquoSelf-fertilization based geneticalgorithm for university timetabling problemrdquo in Proceedingsof the 1st ACMSIGEVO Summit on Genetic and EvolutionaryComputation (GEC rsquo09) pp 1001ndash1004 ACM Shanghai ChinaJune 2009

[18] J Wang and K Chu ldquoAn application of genetic algorithms forthe flexible job-shop scheduling problemrdquo International Journalof Advancements in Computing Technology vol 4 no 3 pp 271ndash278 2012

[19] M Moz and M V Pato ldquoA genetic algorithm approach to anurse rerostering problemrdquo Computers amp Operations Researchvol 34 no 3 pp 667ndash691 2007

[20] C Wotthipong K Soradech and S Nidapan ldquoThe suitablegenetic operators for solving the university course timetablingproblemrdquo Journal of Convergence Information Technology vol8 pp 60ndash65 2013

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 15: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

Mathematical Problems in Engineering 15

when the patients register and wait to see the doctor) Ifwe want to schedule the work of the nurses in January 2012in order to meet the criteria for the service of this studywe have to increase the number of nurses from 18 to 20 sothat the number can perfectly meet the needs of the patientsaccording to the criteriaThework schedules of the nurses canbemost appropriate if the target equation is 03959which usesthe time of calculation as 603 hourswith a standard deviationof less than 8028and the overtimepayment less than 29564Baht per monthWhen this is compared with the actual workschedule in January 2013 the work schedule could reduce thenumber of nurses from 23 to 20 with a decreased standarddeviation that is 8908 and the decrease in the totalovertime paymentwould be 50148 Baht permonthWe couldcalculate the most appropriate work schedule quicker thanby using the conventional approach which results obtainedby Lingo program version 50 86849 with 1500 rounds forthe calculation cycle

As we know hospitals belong to the service businesssector continuous improvement along with response to cus-tomer satisfaction is the key success factor in this businessTo reduce the waiting time in every hospital process highstandards in medical care and highly experienced medicaldoctors are needed Presently Thailand has set its aim tobecome an Asian medical and tourist hub this strategy of thecountry may cause and force hospitals to improve their ser-vice quality for responding well to the high number of outpa-tients Updated data collection needs to be done service-level policies have to be reconsidered Demands of outpa-tient nurses and their schedules may change from time totime Inpatient nurses may be allocated to assist when themanagement wishes to reduce the fluctuation in the numberof outpatient nurses Thus modern management and highefficient tools are necessary this is the reason why GA isinvolved instead of an adoption of the conventional approachHowever regarding the implementation of simulation evenGA is not familiar to the hospital people Optimization capac-ity building is another issue that the hospitalmanagementwillhave to consider

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

Chiang Mai University (CMU) through the research admin-istration office provided budget to our Excellence Center inLogistics and Supply Chain Management (E-LSCM)

References

[1] T Mongkolrangsarit and W Thamapornpilat Development ofNurse Scheduling Method with Uncertain Workforce DemandDepartment of Industrial Engineering Faculty of EngineeringChulalongkorn University 2007

[2] C M Rothe and H B Wolfe ldquoCyclical scheduling and alloca-tion of nursing staffrdquo Socio-Economic Planning Sciences vol 7no 5 pp 471ndash487 1973

[3] R B Norby L E Freund and B Wagner ldquoA nurse staffingsystem based upon assignment difficultyrdquo Journal of NursingAdministration vol 7 no 9 pp 2ndash24 1977

[4] T Ryan B L Barker and F A Marciante ldquoA system fordetermining appropriate nurse staffingrdquo Journal of NursingAdministration vol 5 no 5 pp 30ndash38 1975

[5] H Wolfe and J P Young ldquoStaffing the nursing unit Part Icontrolled variable staffingrdquo Nursing Research vol 14 pp 236ndash243 1965

[6] HWolfe and J P Young ldquoStaffing the nursing unit IIThemult-iple assignment techniquerdquo Nursing Research vol 14 no 4 pp299ndash303 1965

[7] J L Arthur andA Ravindran ldquoAmultiple objective nurse sche-duling modelrdquo AIIE Transactions vol 13 no 1 pp 55ndash60 1981

[8] I Ozkarahan ldquoA disaggregation model of a flexible nursescheduling support systemrdquo Socio-Economic Planning Sciencesvol 25 no 1 pp 9ndash26 1991

[9] I Ozkarahan and J E Bailey ldquoGoal programming modelsubsystem of a flexible nurse scheduling support systemrdquo IIETransactions vol 20 no 3 pp 306ndash316 1988

[10] K A Dowsland ldquoNurse scheduling with tabu search and stra-tegic oscillationrdquo European Journal of Operational Research vol106 no 2-3 pp 393ndash407 1998

[11] P Tormos A Lova F Barber L Ingolotti M Abril and M ASalido ldquoA genetic algorithm for railway scheduling problemsrdquoinMetaheuristics for Scheduling in Industrial andManufacturingApplications vol 128 of Studies in Computational Intelligencepp 255ndash276 Springer Berlin Germany 2008

[12] H Kawanaka K Yamamoto T Yoshikawa T Shinogi and STsuruoka ldquoGenetic algorithm with the constraints for nursescheduling problemrdquo in Proceedings of the Congress on Evolu-tionary Computation vol 2 pp 1123ndash1130 Seoul Republic ofKorea May 2001

[13] H HMillar and KMona ldquoCyclic and non-cyclic scheduling of12 h shift nurses by network programmingrdquo European Journalof Operational Research vol 104 no 3 pp 582ndash592 1998

[14] I Blochlige ldquoModeling staff scheduling problems A tutorialrdquoEuropean Journal of Operational Research vol 158 no 3 pp533ndash542 2004

[15] B Jaumard F Semet and T Vovor ldquoA generalized linear pro-gramming model for nurse schedulingrdquo European Journal ofOperational Research vol 107 no 1 pp 1ndash18 1998

[16] T Cezik O Gunluk and H Luss ldquoAn integer programmingmodel for the weekly tour scheduling problemrdquoNaval ResearchLogistics vol 48 no 7 pp 607ndash624 2001

[17] Z Wang J-L Liu and X Yu ldquoSelf-fertilization based geneticalgorithm for university timetabling problemrdquo in Proceedingsof the 1st ACMSIGEVO Summit on Genetic and EvolutionaryComputation (GEC rsquo09) pp 1001ndash1004 ACM Shanghai ChinaJune 2009

[18] J Wang and K Chu ldquoAn application of genetic algorithms forthe flexible job-shop scheduling problemrdquo International Journalof Advancements in Computing Technology vol 4 no 3 pp 271ndash278 2012

[19] M Moz and M V Pato ldquoA genetic algorithm approach to anurse rerostering problemrdquo Computers amp Operations Researchvol 34 no 3 pp 667ndash691 2007

[20] C Wotthipong K Soradech and S Nidapan ldquoThe suitablegenetic operators for solving the university course timetablingproblemrdquo Journal of Convergence Information Technology vol8 pp 60ndash65 2013

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 16: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

16 Mathematical Problems in Engineering

[21] K Naveen Karambir and K Rajiv ldquoA comparative analysis ofPMX CX and OX crossover operators for solving travellingsalesman problemrdquo International Journal of Latest Research inScience and Technology vol 1 no 2 pp 98ndash101 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 17: Research Article Nurse Scheduling Using Genetic Algorithmdownloads.hindawi.com › journals › mpe › 2014 › 246543.pdf · 2019-07-31 · Research Article Nurse Scheduling Using

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


CPU Scheduling. Basic Concepts Scheduling Criteria Scheduling Algorithms Multiple-Processor Scheduling Real-Time Scheduling Thread Scheduling Operating

Exact and heuristic approaches to nurse scheduling …data.math.au.dk/publications/phd/2013/math-phd-2013-jb.pdf · Exact and heuristic approaches to nurse scheduling Jonas Bæklund

NURSE SCHEDULING AND RESCHEDULING PROBLEM UNDER ...etd.lib.metu.edu.tr/upload/12619545/index.pdf · Approval of the thesis NURSE SCHEDULING AND RESCHEDULING PROBLEM UNDER UNCERTAINTY

Implementing a Self-Scheduling Model to Decrease Nurse

Pre-Conference Staffing Workshop Morning Session · Pre-Conference Staffing Workshop Morning Session ... Nurse Manager of Scheduling and Staff Deployment ... • Review/Create Master

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Scheduling Scheduling (cont.)

Integrating Matheuristics and Metaheuristics for Timetabling · ow shop scheduling, and nurse rostering respectively. Sorensen and Stidsen [15] presented some preliminary results

OPTIMASI NURSE SCHEDULING PROBLEM (STUDI KASUS …

Infor Healthcare Workforce Management Suite · Infor Healthcare Workforce Management Suite Brochure 5 Increase nurse satisfaction with self-scheduling The Infor Healthcare Workforce

An Online Stochastic Algorithm for a Dynamic Nurse ...€¦ · An Online Stochastic Algorithm for a Dynamic Nurse Scheduling Problem Antoine Legraina,b, J´er´emy Omerc,d, Samuel

Chapter 5 CPU Scheduling. CPU Scheduling Topics: Basic Concepts Scheduling Criteria Scheduling Algorithms Multiple-Processor Scheduling Real-Time Scheduling

Why become a nurse? - Welcome to WACAC · PDF fileWhy become a nurse? Lucrative profession Caring profession Many opportunities Flexible scheduling Portable skills Opportunities for

Scheduling Algorithems First Come First Serve Scheduling Shortest Job First Scheduling Priority Scheduling Round-Robin Scheduling Multilevel Queue Scheduling

1 Nurse scheduling with tabu search and strategic oscillation Jia-Xian Zhu

Nurse scheduling with lunch break assignments in operating ...socl.egr.uh.edu/sites/socl/files/files/publications/OR-HCJuly2016.pdfNurse Scheduling with Lunch Break Assignments in

Infor Healthcare Workforce Management Suite...Infor Healthcare Workforce Management Suite Brochure 5 Increase nurse satisfaction with self-scheduling The Infor Healthcare Workforce

Providence Hospital Nurse Scheduling

Scheduling Practices of Acute Care RN’s and Nurse Fatigue

The State of the Art of Nurse Rostering · scheduling process is a very considerable ... distinguishes 3 major areas of manpower decision research: staffing, scheduling and reallocation

Scheduling Scheduling

BRINGING YOUR NURSE STAFFING COMMITTEE TO LIFE: Staffing, Scheduling & Budgeting for the Bedside Clinician Cathe Clapp, RN, MN Barbara Hyland-Hill, RN,

Line Balancing, Baggage Handling, Nurse Scheduling

Nurse anesthetist CRNA Nurse midwife CNM Nurse practitioner NP Clinical Nurse Specialist CNS

© J. Christopher Beck 20081 Lecture 26: Nurse Scheduling

White Paper: Nurse Scheduling and Fatigue in the Acute Care 24

Acuity-based nurse assignment and patient scheduling …aturkcan/Publications_files/HCMS_… ·  · 2015-02-24Acuity-based nurse assignment and patient ... Northeastern University,

Staffing And Scheduling. Definitions Nurse to patient ratio: Number of patients cared for by one nurse typically specified by job category. This varies

Chapter 5: CPU Scheduling. Basic Concepts Scheduling Criteria Scheduling Algorithms Multiple-Processor Scheduling Real-Time Scheduling Thread Scheduling

“A nurse is a nurse is a nurse?” - EPSU · “A nurse is a nurse is a nurse? ... New definition (Machteld Huber):

NURSE SCHEDULING DECISION SUPPORT SYSTEMS AS A …

TRICARE NON-NETWORK REGISTERED NURSE (RN)/LICENSED PRACTICAL NURSE (LPN)/ NURSE ... · 2020. 8. 12. · REGISTERED NURSE (RN)/LICENSED PRACTICAL NURSE (LPN)/ NURSE PRACTITIONER (NP)

An ACO algorithm for a dynamic regional nurse-scheduling ... · 1 An ACO algorithm for a dynamic regional nurse-scheduling problem in Austria Abstract To the best of our knowledge,

Why become a nurse? Lucrative profession Caring profession Many opportunities Flexible scheduling Portable skills Opportunities for advancement

Lean Six Sigma for Nurse Scheduling