1

INEN 689 – 602 PROJECT Nurse Scheduling Problem

Elif Ilke Gokce

Industrial Engineering Texas A&M University

elifg@tamu.edu

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Abstract

Every hospital needs to produce repeatedly duty rosters for its nursing staff.

Properly scheduling the nursing staff has a great impact on the quality of health care, the

recruitment of nursing staff, the development of nursing budgets, staff and patient safety

and satisfaction, and administrative workload. The scheduling of hospital personnel is

particularly challenging because of different staffing needs on different days and shifts.

The planned personnel schedule usually has to be changed to deal with unforeseen

circumstances such as staff sickness and emergencies.

In this project, a mixed- integer stochastic programming approach to the solution

of the nurse scheduling problem is proposed. The problem is formulated as two-stage

recourse model. The objective is to minimize unsatisfied staff in the first stage and the

expected overtime cost in the second stage. The nurse scheduling problem will be solved

by L2 algorithm. The aim of this study is to help health institutions to determine nurse

schedules which would increase nursing staff, efficiency and satisfaction.

Keywords: L2 algorithm, nurse scheduling, stochastic programming

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1 INTRODUCTION

Every hospital needs to repeatedly produce duty rosters for its nursing staff. The

problem is of critical importance for a variety of reasons. Properly scheduling the nursing

staff has a great impact on the quality of health care, the recruitment of nursing

personnel, the development of nursing budgets, staff and patient safety, staff and patient

satisfaction and administrative workload.

The scheduling of hospital personnel is particularly challenging because of

different staffing needs on different days and shifts. Unlike many other organizations,

healthcare institutions need to be staffed 24 hours a day over seven days a week. In

addition, in many hospitals, nurses are allowed to request preset shifts, while other nurses

are scheduled around these pre-set shifts. Usually, nursing officers spend a substantial

amount of time developing rosters especially when there are many staff requests. Because

of time-consuming manual scheduling, and for various other reasons, the nurse

scheduling problem (NSP) has attracted much attention.

Personnel scheduling problems have been studied extensively over the past three

decades (see survey papers by [3] [22] [7] [5] [12]). Personnel scheduling is the problem

of assigning employees to shifts or duties over a scheduling period so that certain

constraints (organizational and personal) are satisfied. It involves constructing a schedule

for each employee within an organization in order for a set of tasks to be fulfilled. In the

domain of healthcare, this is particularly challenging because of the presence of a range

of different staff requirements on different days and shifts. In addition, unlike many

other organizations, healthcare institutions work twenty-four hours a day for every single

day of the year.

Most nurse scheduling problems are extremely difficult and complex. [22], for

example, say nurse scheduling is more complex than the traveling salesman problem. A

general overview of various approaches for nurse scheduling can be found in [20] [10]

and [8]. Early research ([25] [15][23]) concentrated on the development of mathematical

programming models. To reduce computational complexity, researchers had to restrict

the problem dimensions and consider a smaller size of constraints in their models,

resulting in solutions that are too simple to be applied in real hospital situations.

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The above observations have led to other attempts, trying to solve the real nurse

scheduling problems within reasonable time. Besides heuristic approaches (e.g., [21];

[6]), artificial intelligence approaches such as constraint programming ([16]), expert

systems ([10]) and knowledge based systems ([18]) were investigated with some success.

Since the 1990’s, most papers have tackled the problem with metaheuristic approaches,

including simulated annealing ([17]), tabu search ([11]) and genetic algorithms ([1]).

A key feature of real NSP which is in the subset of the personnel planning

problem is that the planned personnel schedule usually has to be changed (often at very

short notice) to deal with unforeseen circumstances such as staff sickness and

emergencies. Very little study dealing with the uncertainty that is inherent in the real

world has been done and they are mostly based on fuzzy methodologies. However,

stochastic optimization techniques will provide solutions that can be applied to the real

world NSPs.

The NSP involves producing a periodic (weekly, fortnightly, or monthly) duty

roster for nursing staff, subject to a variety of hard/soft constraints such as legal

regulations, personnel policies, nurses preferences and many other requirements that may

be hospital-specific. These constraints can vary from one hospital to another while the

objectives in scheduling can also vary.

In this project, first part gives brief introduction to workforce assignment and

studies in the literature. In the second part, nurse scheduling problem modeling methods

are argued, and then Solution approaches to the nurse scheduling problem which are

optimization approaches, heuristic approaches and AI approaches are described. Third

part contains nurse scheduling problem definition and modeling formulation using

stochastic integer programming formulation. In the fourth part, L2-Binary first stage

algorithm which is used for solving stochastic integer programming model is defined.

Part five gives brief description of exponential design. Data sets and implemented

program code are also explained. In the last part, conclusion and the future work is found.

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2 THE NURSE SCHEDULING PROBLEM

2.1 Modeling the Nurse Scheduling Problem

Parameters in the NSP may include the working shifts per week if night shifts are

worked, preference costs of particular nurses working on particular shift pattern, working

shifts per schedule if day shifts are worked, working shifts per schedule if both day and

night shifts are worked, demand for certain grade of nurses on day and on night shifts.

The NSP is commonly described by a nurse-day, a nurse-task and a nurse-shift

pattern view ([9]).

In a nurse-day view, the decision variables are defined in two different ways. In the

first case; vij is defined for each nurse on each day, where i=1…N indexes the nurses and

j=1…M indexes the days within a scheduling period. The domains of these variables

consist of on-duty shifts and free shifts. On-duty shifts include any number of shifts per

day. It is common to use only a morning shift (A) of eight working hours, an afternoon

shift (P) of eight working hours, and a night shift (N) of eight working hours. Free shifts

include day-off (O), vacation leave (VL), unpaid leave (UL), etc. In the general, when

there are Z shifts per day, xij can take Z+1 possible values as the following:

⎪⎪⎩

⎪⎪⎨

⎧

=

jdayonZshiftworksinurse

jdayonshiftworksinursejdayondutyoffisinurse

Z

xij

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For 0–1 models, the decision variables are modified as xijk , where i, j are the same

index as before and k=1… Z indexes the Z possible shifts in a day.

⎩⎨⎧

=otherwise0

day on shift works nurse1 jkixijk

In a nurse-task view, the decision variable is defined for each nurse in each shift as

xis, where i=1…N indexes the nurses and s=1…Z indexes the tasks within a scheduling

period. The shift defined in nurse-task view may not necessarily correspond to a day.

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⎩⎨⎧

=otherwise0

s task receives nurse1 ixis

In a nurse-shift pattern view, the decision variable is defined for each nurse and for

each shift pattern as xip, where i=1…N indexes the nurses and p=1…M indexes the shift

patterns.

⎩⎨⎧

=otherwise0

pattern shift works nurse1 pixip

In literature, three types of objective function are used with NSPs. The first type

is ,min!)(

→∑ ∈ iFj ijij xp where pij is the penalty cost of nurse i working on shift pattern j,

xij is the decision variable with a nurse- shift pattern view and F(i) is the set of feasible

shift patterns for nurse i. The objective is to minimize the total penalty cost for all nurses.

Second type purposes to minimize the number of uncovered shifts. Last type purposes

to minimize the cost of schedules and the penalty for violating shift balance.

Two types of constraints commonly occur in NSPs: hard constraints and soft

constraints. Hard constraints include coverage requirements; such as nurse demand per

day per shift type per skill category while soft constraints are usually those involved with

time requirements on personal schedules. The goal is always to schedule resources to

meet the hard constraints while aiming at a high quality result with respect to soft

constraints. Commonly used constraints are as follows:

1. Minimum/maximum nurse workload

2. Consecutive same working shift (minimum/maximum/exact number)

3. Consecutive working shift/days (minimum/maximum/exact number)

4. Nurse skill levels and categories

5. Nurses’ preferences or requirements

6. Nurses free days (minimum/maximum/consecutive free days)

7. Free time between working shifts

8. Shift type(s) assignments (maximum shift type, requirements for each shift types).

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9. Constraints among groups/types of nurses, such as nurses not allowed to work

together or nurses who must work together

10. Shift patterns

11. Other requirements in a shorter or longer time period other than the planning time

period; such as every day in a shift must be assigned

12. Constraints among shifts; such as shifts cannot be assigned to a person at the same

time

13. Requirements of different types of nurses or staff demand for any shift

It is important to note that constraints 1, 3, 5, 6, 8, 11 and 13 are common in NSPs.

Soft constraints can include balance in workload, assigning complete weekends and

patterns enabling specific cyclic constraints.

2.2 Solution Approaches to the Nurse Scheduling Problem

Solution approaches that have been proposed to solve NSPs are classified in three

main categories: optimization-mathematical programming (MP), heuristics and artificial

intelligence.

2.2.1 Optimization Approaches Optimization approaches are usually based on mathematical programming. In

general, optimization using mathematical programming can be classified in three

categories: single-objective mathematical programming, multi-objective mathematical

programming, and mathematical programming-based near-optimal approaches.

Single-objective MP involves maximizing a goal which is preferred by the

decision-maker. The following studies have been done in terms of single-objective MP:

1. An algorithm with three stages is presented for NSP: generate a set of possible

schedules which are seven-tuples of 0–1 depending on whether the day is off or

on, formulate the problem as an IP, and produce a solution ([17]).

2. [26] proposed a model which aims to maximize nurses’ preferences, by

considering the length of a work, rotation patterns, and requests for days off, and

minimum numbers of nursing personnel of each skill class to be assigned to each

day and a 4 to 6-week scheduling period. The problem is solved by a modified

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Balintfy and Blackburns’ algorithm. In this two-phase algorithm, Phase I finds a

feasible solution to meet various constraints, and Phase II improve the Phase I

solution by maximizing individual preferences for various schedule patterns while

maintaining the Phase I solution.

3. NSP is modeled as a large-scale mixed integer quadratic programming problem to

minimize a shortage cost of nursing services for a period of three to four days

subject to nursing skill class requirements, total personnel capacity constraints,

integral assignment, minimum nurse requirements throughout the scheduling

period and other relevant constraints. The problem is decomposed by a primal

resource-directive approach into a 0–1 LP master problem, with smaller quadratic

programming sub-problems ([25]).

4. [15] formulated the problem to minimize an objective function that balances the

trade-off between staffing coverage and schedule preferences of individual nurses,

subject to certain constraints on the nurses schedules. The constraints are divided

into hard and soft constraints. The hard constraints define sets of feasible nurse

schedules, while violation of soft constraints results in a penalty cost that appears

in the objective function. A coordinate descent algorithm was proposed to find

near-optimal solutions.

5. [13] presented a generalized 0–1 column generation model with a resource

constrained shortest path auxiliary problem for NSP. The master problem finds a

configuration of individual schedules to satisfy the demand

coverage constraints while minimizing salary costs and maximizing both nurse

preferences and team balance. A feasible solution of the auxiliary problem is an

acceptable schedule for a given nurse, with respect to collective agreement

requirements such as seniority, workload, rotations and days off. A new resource

structure was defined in the auxiliary problem in order to satisfy complex

collective agreement rules specific to the problem.

6. NSP which considers the case of two consecutive days off per week for each

person.

7. NSP is modeled as an integer programming with 0–1 constraint matrix, and the

IP was solved parametrically as a bounded series of network problems.

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8. NSP which considers the case of 10 working days in a 14-day period with

variable demands.

However, multi-objective models appear to be more realistic. [2] proposed a goal

programming model. Goals are minimizing staffing requirements, minimizing desired

staffing requirements, meeting nurses’ preferences, and nurses’ special requests. This

approach works in two phases. In the first phase, the nurses are assigned their day-

on/day-off pattern for the two-week scheduling horizon by a goal programming model

that allows for consideration of the multiple conflicting objectives inherent in the

scheduling of a staff of nurses. The second phase makes specific shift assignment through

the use of a heuristic procedure. The major advantage of the goal programming

formulation of the nurse scheduling problem is the flexibility it permits in choosing

priorities; it can take into account such factors as nurses' preferences and desired staffing

requirements.

In MP-based near-optimal methods is aimed to combine the MP and AI

approaches. The problem was formulated as an approximate IP model. The IP problem is

first solved and then its solution improved.

2.2.2 Heuristic Approaches For combinatorial problems, exact optimization usually requires large

computational times to produce optimal solutions. In contrast, heuristic approaches can

produce satisfactory results in reasonably short times. Heuristics used to solve NSPs is

divided into two categories: classical heuristics and metaheuristics. In the recent years,

metaheuristics such as Tabu Search (TS), Genetic Algorithm (GA) and Simulated

Annealing (SA), have been proved to be very efficient in obtaining near-optimal

solutions for a variety of hard combinatorial problems including the NRP ([3]).

Classical heuristic approaches which have been widely studied in nursing

literature were straightforward automation of manual practices. For instance, Greedy

Shuffling. First of all, the problem is solved for the worst schedule and then it is

improved by exchanging a part of this schedule with a part from another persons’

schedule. Many human-inspired approaches can be found in Greedy Shuffling type

algorithms which work by calculating all the shuffles for all personnel and listing them

with the highest cost benefit first. This is repeated as many times as possible.

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Classical heuristic approaches have been proposed to help decision-makers select

work patterns to provide the needed coverage for given skilled nurse classes on each

shift, develop a basic pattern to meet shift and coverage constraints and meet required

staffing levels.

TS approaches have been proposed to solve the NRP. TS is a search that moves

iteratively from one solution to another in a neighborhood space with the assistance of an

adaptive memory that forbids solution attribute changes recorded in the short-term

memory to be reused. In TS, a move, for example, can take on an assigned shift type

from one nurse to another on the same day and a move not allowed if the person does not

belong to the skill category required or if there is already an assignment for that shift

type. In TS, hard constraints remained fulfilled, while solutions move. TS approaches

have been proposed to ensure enough nurses are on duty at all times while taking account

of individual preferences and requests for days off.

GAs, which are stochastic meta-heuristics, have also been used to solve the NRP.

In GA, the basic idea is to find a genetic representation of the problem. Starting with a

population of randomly created solutions, better solutions are more likely to be selected

for recombination into new solutions. In addition, new solutions may be formed by

mutating or randomly changing old ones. For example, in NRP, for crossover and

mutation, the best personal schedule from each of the parents can be selected, a random

selection from the personal schedule of parents can be selected, or we can select the best

events in a schedule. Some of the best solutions in each generation are kept while others

are replaced by newly formed solutions. [13] used GA for a problem with multiple

criteria where the concept of a Pareto optimality scheme is used for the evaluation of the

multi-criteria objective function. [1] developed a GA approach to solve an NRP. Instead

of working directly with populations of potential solutions and handling the constraints

using penalty functions or repairs, they proposed an indirect approach in which the task

of balancing optimization and constraint satisfaction is shared between a greedy heuristic

and the GA. Individuals are represented by permutations of the available nurses and the

heuristic is used to build schedules by allocating the nurses to their shifts in the given

order. Memetic algorithms, which are viewed as hybrid GA, are a population-based

approach for heuristic search in optimization problems. Basically, they combine local

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search heuristics with crossover operators. [8] described a memetic algorithm that

incorporates TS into a GA, using steepest descent for each individual. The results

reported for the NRP are better than those obtained by a hybrid TS approach by [8]. This

work has gone further in combining hybrid TS with evolutionary approaches. There has

been some use of simulated annealing techniques for the NRP. For example, [22]

presented a SA heuristic for shift-scheduling using non-continuously available

employees.

2.2.3. AI approaches

AI techniques have been used to solve NRPs modelled as a CSP [10] modeled the

NRP as a CSP which was solved by a combined approach of look-ahead and intelligent

scoring which determines which nurse is to be scheduled next and which shift satis.es

most of the soft constraints. [1] adopt a PCSP model for the NRP. INTERDIP, which is

their prototype system, supports semi-automatic creation of duty rosters and imitates

certain aspects of manual planning to improve on the theoretical complexity of the

problem, using a constraint package based on CHIP. The package includes linear

equations, constraints over definite domains and boolean constraints. [15] modeled the

NRP as a HCSP, where legal regulations are hard constraints and nurses_ preferences are

usually lower level soft constraints. [15] reported a commercial system ORBIS which

models the NRP as a HCSP with fuzzy constraints and inferred control strategies. ORBIS

uses a B&B algorithm with constraint propagation and variable/value-ordering

techniques to solve problems involving 250–1200 variables with on few minutes [10]

Constraint logic programming languages have the advantage of describing constraint

logic easily. [12] presented a non-cyclic scheduling system, namely Horoplan, whose

algorithm is a constraint-based arti.cial intelligence approach implemented with Charme,

which is a constraint-based programming language. [16] discussed an approach, which

takes advantage of the declarative ability of Prolog language for the description of

constraints, for incorporating the constraints to generalize the NRP. [18] combined

constraint logic programming with case-based reasoning to reduce the search spaces

further. As a commercial constraint-based package for the powerful C++ programming

language, ILOG SOLVER has been widely used to solve the NRP, with the help of

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heuristic techniques It should be noted that [10] used redundant modeling which

increases constraint propagation through cooperation among different models for the

same problem via channeling constraints. Knowledge-based search approaches have also

been used to solve the NRP by Lukman et al. [14] and [10].

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3 PROBLEM STATEMENT AND MODEL FORMULATION

Nurse scheduling problem is assigning nurse workforce to the shifts subject to a

number of constraints such as time, demand and preference constrains. The objective is to

minimize the total cost which consists of preference cost, overtime cost and unsatisfied

demand cost. Inputs of the problem are the number of nurse types, number of each type

of nurses. In this project, it is assumed that there are three types of nurses, and the

number of these nurses are N1, N2 and N3 respectively. There are 3 kinds of shift on a

day, 7am-3pm, 3pm-11pm and 11pm-7am. Each nurse has to take one day off in a week.

This obligation requires that each nurse can work at 6 shifts in a week. Demand is in

terms of number of nurses. The output is a feasible schedule which minimizes the total

cost.

The problem is formulated as two-stage recourse model as follows:

Indices:

k = 1, 2… N1 (nurse grade 1 index)

l = 1, 2… N2 (nurse grade 2 index)

m = 1, 2… N3 (nurse grade 3 index)

j = 1, 2…21 (shift index)

j = 1, 4, 7, 10, 13, 16, 19 corresponds to 7am-3pm shifts

j = 2, 5, 8, 11, 14, 17, 20 corresponds to 3pm-11pm shifts

j = 3, 6, 9, 12, 15, 18, 21 corresponds to 11pm-7am shifts

g = 1, 2, 3 (nurse grade index)

First Stage Parameters:

N1: number of first grade nurses

N2: number of second grade nurses

N3: number of third grade nurses

P1kj: preference cost of first grade nurse k working at shift j

P2lj: preference cost of second grade nurse l working at shift j

P3mj: preference cost of third grade nurse m working at shift j

ADjg: average demand for g th grade nurses at shift j

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First Stage Decision Variable:

⎩⎨⎧

=o/w0

nurse grade a as shift toassigned is nurse gradefirst theif1 gjkxkjg

⎩⎨⎧

=o/w0

nurse grade a as shift toassigned is nurse grade second theif1 gjlyljg

⎩⎨⎧

=o/w0

shift toassigned is nurse grade third theif1 jizmj

Second Stage Parameters:

ocg: over time cost for nurse grade type g or higher in a shift

ucg: unsatisfied demand cost for nurse grade type g or higher in a shift

jgD~ : demand realization for nurse grade type g or higher at shift j

Second Stage Decision Variable:

⎩⎨⎧

=o/w0

nurse grade a as shift toassigned is nurse gradefirst theif1 gjioxkjg

⎩⎨⎧

=o/w0

nurse grade a as shift toassigned is nurse grade second theif1 gjioyljg

⎩⎨⎧

=o/w0

shift toassigned is nurse grade third theif1 jiozmj

ujg: unsatisfied demand for nurse grade type g at shift j

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))~,,,((321Min 1 2 3

1

21

1 1

21

1 1

21

1

3

2

3

1∑∑ ∑∑ ∑∑∑∑= = = = = ===

+++N

k j

N

l j

N

m jmjmj

gljglj

gkjgkj DzyxfEzPyPxP

Subject to:

21...2,111

1

1

=∀≥∑=

jADx j

N

kkj (1)

21...2,121

21

2

21

=∀≥+∑∑==

jADyx j

N

llj

N

kkj (2)

21...2,1311

31

3

321

=∀≥++ ∑∑∑===

jADzyx j

N

mmj

N

llj

N

kkj (3)

1

21

1

3

1

...2,16 Nkxj g

kjg =∀=∑∑= =

(4)

2

21

1

3

2...2,16 Nly

j gljg =∀=∑∑

= =

(5)

3

21

1

...2,16 Nmzj

mj =∀=∑=

(6)

19,16,13,10,7,4,1

...2,11 13)2(3)1(32)2(2)1(21)2(1)1(1

=

=∀≤++++++++ ++++++

j

Nkxxxxxxxxx jkjkkjjkjkkjjkjkkj

(7)

19,16,13,10,7,4,1...2,11 23)2(3)1(32)2(2)1(2 ==∀≤+++++ ++++ jNlyyyyyy jljlljjljllj (8)

19,16,13,10,7,4,1...2,11 2)2()1( ==∀≤++ ++ jNmzzz jmjmmj (9)

3,2,121,...2,1...2,11,0 1 ==∀=∀∈ gjNkxkjg (10)

3,221,...2,1...2,11,0 2 ==∀=∀∈ gjNmymjg (11)

21,...2,1...2,11,0 3 =∀=∀∈ jNmzmj (12)

Where for each scenario:

∑∑ ∑∑ ∑∑ ∑∑∑∑= = = = = = = ===

+++=1 2 3

1

21

1 1

21

1 1

21

1

21

1

3

13

3

22

3

11Min )~,((

N

k j

N

l j

N

m j j gjggmj

gljg

gkjg uucozocoyocoxocDxfE

Subject to:

16

21...2,1111

11

1

11

=∀≥++∑∑==

jDuoxx jj

N

kkj

N

kkj (13)

21...2,1221

21

21

21

2

2121

=∀≥++++ ∑∑∑∑====

jDuoyoxyx jj

N

llj

N

kkj

N

llj

N

kkj (14)

21...2,13311

31

311

31

3

321321

=∀≥++++++ ∑∑∑∑∑∑======

jDuozoyoxzyx jj

N

mmj

N

llj

N

kkj

N

mmj

N

llj

N

kkj

(15)

21...3,2...2,10 13)1(2)1(1)1(321 =∀=∀≤−−−++ −−− jNkxxxoxoxox jkjkjkkjkjkj

(16)

21...3,2...2,10 23)1(2)1(32 =∀=∀≤+−+ −− jNlyyoyoy jljlljlj (17)

21...3,2...2,10 3)1( =∀=∀≤− − jNmzoz jmmj (18)

1

21

1

3

1...2,11 Nkox

j gkjg =∀≤∑∑

= =

(19)

2

21

1

3

2...2,11 Nloy

j gljg =∀≤∑∑

= =

(20)

3

21

1...2,11 Nmoz

jmj =∀≤∑

=

(21)

3,2,121,...2,1...2,11,0 1 ==∀=∀∈ gjNkoxkjg (22)

3,221,...2,1...2,11,0 2 ==∀=∀∈ gjNmoymjg (23)

21,...2,1...2,11,0 3 =∀=∀∈ jNmozmj (24)

3,2,121...2,1int,0 ==∀≥ gju jg (25)

(1), (2) and (3) are demand constraints. (4), (5) and (6) provides that each nurse

has to assign 6 shifts in a week. (7), (8) and (9) provides that each nurse has to assign as

one type of nurse and one shift in a day. (13), (14) and (15) are demand constraints. (16),

(17) and (18) provides that a nurse has to work on previous shift in order to assign as an

overtime nurse. (19), (20) and (21) are the constraints of assigning each nurse only once

as overtime nurse. (10), (11), (12), (22), (23) and (24) are binary constraints.

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4 ALGORITHM 4.1. Definition

L2-Binary first stage algorithm is used to solve the stochastic integer programming

problem. The algorithm is as the following;

Step 0: Initialization

• Let 0ε given

• 1,0,|min 11 nT xbAxxcx ∈==

• LB −∞← , UB ∞← , and k 1←

• Compute L:

o Alternative 1: At point x1 solve the linear relaxation of the second

stage and take the expected value of linear relaxation objective values

o Alternative 2: Observe the objective function of the second stage to

come up with a valid lower bound for the expected value of the

recourse.

Step 1: Solve subproblems for all Ωw∈

• Solve all subproblems and obtain F(xk)= )]~,([ wxfE k

• soln= )]~,([ wxfExc kkT +

• Update upper bound by UB=max(soln, UB)

Step 2: Update and solve master problem

• Use L, xk and F(xk) to derive L2 optimality cut which is:

LSxxLxF k

Sjj

Sjj

k

kk

++−−−≥ ∑∑∉∈

)1)()((η where Sk is the set of variables

in solution xk whose value is 1.

Compute kk β and α such that the above cut is represented as

kT

k x αηβ ≥+

• Add this cut to the master problem and solve master again and obtain

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1,0 ...2,1,

,|minarg1

∈=≥+

=+=+

xktx

bAxxcx

tT

t

Tk

αηβ

η

Step 3: Termination

If ε≤− LBUB STOP,

Else set k=k+1 and go back to step 1.

4.2. L2 Code Validation

The code is tested with three data instance from literature. The instances are

SSLP_5_25_50, SSLP_5_25_100 and SSLP_15_45_5. In these calculations, L value is

taken as -50,000.

Solution CPU Time (sec) Number of Iterations

SSLP_5_25_50 -121.6 1.625000 33

SSLP_5_25_100 -127.37 5.969000 33

SSLP_15_45_5 -262.4 199.167 327

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SELECT ALTERNATIVE

Alternative 1 Alternative 2

SOLVE SUBPROBLEMS

UPDATE AND SOLVE MASTER PROBLEM

START

INITIALIZATION

ε≤− LBUB?

Y

N

STOP

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5 COMPUTATIONAL RESULTS

5.1 Experimental Design

Experimental methods are widely used in research as well as in industrial settings,

however, sometimes for very different purposes. The primary goal in scientific research

is usually to show the statistical significance of an effect that a particular factor exerts on

the dependent variable of interest.

A well designed experiment will have the following properties;

1) A well defined objective,

2) The ability to estimate error,

3) Sufficient precision,

4) The ability to distinguish various effects by randomization and factorial design.

In this work, the objective is to determine the change in total cost under different

parameters. If the proposed stochastic programming model works right, as the parameters

in the model change, the solution always becomes consistent. In addition to this, at the

end of the experiments for different parameters, it is found that which parameter has the

vital effect on the cost. Based on this result, modifications for better design can be done

on the hospital system.

The experimental design has the ability to estimate error. Experimental error is

defined as observed – expected (find using model). In this work, the expected results such

as total cost, nurse schedules are found as a result of stochastic model solution. If the

model is applied on the real hospital, the observed values can be easily determined.

Therefore, error estimate is calculated easily.

The most common experimental design employed in this study is called a

completely randomized design. This experiment involves a comparison of the means for

a number, say k, of treatments, based on independent random samples of n1, n2, . . ., nk

observations, drawn from populations associated with treatments 1, 2, k, respectively.

After collecting the data from a completely randomized design, our goal is to make

inferences about k population means where m i is the mean of the population of

measurements associated with treatment i, for i = 1, 2. . . k. The null hypothesis to be

tested is that the k treatment means are equal, i.e.,

21

H0: m 1 = m 2 = . . . = m k

and the alternative hypothesis is that at least two of the treatment means differ.

An analysis of variance provides an easy way to analyze the data from a

completely randomized design. The analysis partitions SS (Total) into two components,

SST and SSE. These two quantities are defined in general term as follows:

Recall that the quantity SST denotes the sum of squares for treatments and

measures the variation explained by the differences between the treatment means. The

sum of squares for error, SSE, is a measure of the unexplained variability, obtained by

calculating a pooled measure of the variability within the k samples. If the treatment

means truly differ, then SSE should be substantially smaller than SST. We compare the

two sources of variability by forming an F statistic:

where n is the total number of measurements. Under certain conditions, the F statistic has

a repeated sampling distribution known as the F distribution that the F distribution

depends on n 1 numerator degrees of freedom and n 2, denominator degrees of freedom.

For the completely randomized design, F is based on n 1 = (k - 1) and n 2 = (n - k) degrees

of freedom. If the computed value of F exceeds the upper critical value, F¥ we reject H0

and conclude that at least two of the treatment means differ.

22

5.1.1 Test to Compare k Population Means for a Completely Randomized Design

H0: m 1 = m 2 . . . = m k [i.e., there is no difference in the treatment (population)

means]

Ha: At least two treatment means differ

Test statistic: F = MST/MSE

Rejection region: F > Fa

where the distribution of F is based on (k - 1) numerator df and (n - k) denominator df,

and Fa is the F value found in the table such that P(F > Fa ) = a .

Assumptions: 1. All k population probability distributions are normal.

2. The k population variances are equal.

3. The samples from each population are random and

independent.

The results of an analysis of variance are usually summarized and presented in an

analysis of variance (ANOVA) table. Such a table shows the sources of variation, their

respective degrees of freedom, sums of squares, mean squares, and computed F statistic.

Source of Variation

Sum of Squares Degrees of Freedom

Mean of Squares

F

Between groups SST k - 1 MST/(k - 1) Within groups SSE n - k SSE/(n - k)

F = MST/MSE

Total SS(Total) n -1

Analysis of Variance Table for Completely Random Design

23

5.1.2 Data Used in Experimental Design

As it stated above, the data used in experimental design is created randomly.

There are three kinds of data instances. Each data set has two different kinds of scenarios.

In the first data set, the total number of nurses in each data set is 20, 25, and 30

respectively. Classification of each nurses in each data set are as the flowing;

N1 N2 N3 Total

Data set 1 6 8 6 20

Data set 2 8 9 8 25

Data set 3 9 12 9 30

Preference costs for each nurse is an integer between 0 and 10. These values are

generates randomly. For each of these values, one independent random variable is

generated and it is multiplied by 10 and rounded to the nearest integer. The formula is as

the following;

⎡ ⎤10*Rand() |Pikj| =

Data sets also contain weekly data for average demand which is denoted by ADjg.

This is a 3x21 matrix, each column denote nurse type, and each row denotes data for a

shift. As it is known, each day contains three shifts. Therefore there are 21 rows for

weekly data. Average demand data is created randomly considering feasible constraints,

such as each nurse should work 6 days, total number of nurse for each type should not be

exceeded.

Over time cost is determined as 15 for the first grade, 10 for the second grade and

6 for the third grade. Unsatisfied demand cost is the same as overtime cost for each nurse

type, and these two cost values are fixed for each the data set.

As it is mentioned above, for each three data set type, two scenarios sets designed.

In the first of these scenarios, there are three kinds of scenarios. Second one contains five

kinds of scenarios. Each of these scenarios are directly created from AD, such that: First

of all, uniformly distributed random number between 0 and 1 are generated. If this

24

number is between 0-0.3, the value is decreased by 1. If it is between 0.3-0.7, the number

stays the same. If it is between 0-7-1.0, it is increased by one.

Each data set is named as nurse_a_b, a denotes the total number of nurse, b

denotes number of scenarios. (See appendix for the data used in experiments)

5.2 Computational Results The experiments are run on P4 2.8Mhz 512MB ram computer. L value of the L2

algorithm is taken as the LP relaxation solution of the problem. First of all, each three

problem with only one scenario is solved both using L2 Algoruthm and Cplex interactive

Optimizer 9.0.0. Solutions are as the following;

Total Number of

Nurse

Cplex Cost Cplex Time (sec) L2 L2 Time

(Sec)

20 3.5200000000e+002 0.09 3.52 0.226000

25 3.3300000000e+002 0.03 3.33 0.452000

30 4.8300000000e+002 0.05 4.83 0.567000

The first graph shows that as the nurse number increases cost value found by

Cplex and L2 stays the same. Second graph shows that L2 Computational time is higher

than Cplex computational time. This difference comes from the data set and algorithm

properties. Problem data are not dense, so Cplex can solve it.

Cost

0.00E+00

1.00E+02

2.00E+02

3.00E+02

4.00E+02

5.00E+02

6.00E+02

20 25 30

Nurse

$

L2Cplex Cost

25

Time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

20 25 30

Nurse

Sec L2 Time (Sec)

Cplex Time (sec)

In general, it can be seen hat L2 gives solution which is very close to the optimal.

In LP Relaxation master problem is solved as MIP problem, and subproblems are

solved as LP Problem.

Total

Number of

Nurse

L2 L2 # of

Iterations

L2 CPU

Time

LP_RELAX LP_RELAX

# of Iterations

LP_RELAX

CPU Time

nurse_20_3 244 2 0.265000 244 2 0.195000

nurse_20_5 259 2 0.328000 259 2 0.268000

nurse_25_3 414 3 0.563000 412.4 2 0.392000

nurse_25_5 438 3 0.875000 418.8 3 0.568000

nurse_30_3 523 2 0.687000 484 2 0.386

nurse_30_5 560 3 0.735000 558.4 2 0.534

It is important to note that, in L2 algorithm, C++ or Cplex sometimes takes

0.99999999 instead of 1. This affects the solution. Also, in L2 algorithm lower bound is

important. If you choose different lower bound, you can find different solutions.

26

Cost

0100200300400500600

nurse

_20_

3

nurse

_20_

5

nurse

_25_

3

nurse

_25_

5

nurse

_30_

3

nurse

_30_

5

Data set

$L2 LP_RELAX

CPU Time

0

0.2

0.4

0.6

0.8

1

nurse

_20_

3

nurse

_20_

5

nurse

_25_

3

nurse

_25_

5

nurse

_30_

3

nurse

_30_

5

Data

Sec

L2 CPU Time

LP_RELAX CPUTime

For nurse number 20, for these two scenario scenarios sets, L2 and LP_RELAX

costs are the same, and CPU time is very close to each other. When the nurse number

increases, difference between L2 and LP_RELAX CPU times increases. In addition this,

cost values start to increase. The same result can be found for nurse number 30. However,

increment rate of the CPU times stays the same.

Each scenario is examined in the inconsistent environment. When the scenario

number increases, the total cost value and CPU times increase. From the second graph

above it can also be determined that if the nurse number increases, the algorithm becomes

more complex.

27

6 CONCLUSION AND FUTURE RESEARCH

In this project, a mixed- integer stochastic programming approach to the solution of

the nurse scheduling problem is proposed. The problem is formulated as two-stage

recourse model. The objective is to minimize unsatisfied staff in the first stage and the

expected overtime cost in the second stage. The nurse scheduling problem will be solved

by L2 algorithm. The nurse scheduling problem with one scenario is solved by both Cplex

and L2 algorithms. The same solution is obtained, but as the problem is not complex,

Cplex can solve it easily.

Six data sets are designed and solved by both L2 algorithm and LP Relaxation. It is

observed that the solution of L2 algorithm is closely related to the lower bound chosen. If

a good lower bound is chosen, the solution close to the optimal solution can be found in a

very short time. As the number of nurse increases the CPU time requires for solving

problem is increases. In addition, when the number of scenarios increases, the problem

becomes more complex. Therefore CPU time also increases.

The following future studies can be done;

• Finding the best lowerbound for L2 algorithm,

• Solve the problem with the data instances from real world,

• Add more constraints to the model,

• The cut in L2 algorithm can be developed.

28

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23. Trivedi VM, and Warner M A branch and bound algorithm for optimum allocation of

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856

31

8 APPENDIX *.mod file param N1; param N2; param N3; set K:=k in 1..N1 ; #nurse grade 1 index set L:=l in 1..N2; #nurse grade 2 index set M:=l in 1..n3; #nurse grade 3 index set J:= j in 1..21; #shift index set G;#:=1,2,3 #grade index set H; param P1K,J; param P2L,J; param P3M,J; param ADJ,1,2,3; param ocG; param ucH; param DJ,G; var xk in K, j in J, g in 1,2,3>=0,<=1, integer; var yl in L, j in J, g in 2,3>=0,<=1, integer; var zm in M, j in J>=0,<=1, integer; var oxk in K, j in J, g in 1,2,3>=0,<=1, integer; var oyl in L, j in J, g in 2,3>=0,<=1, integer; var ozm in M, j in J>=0,<=1, integer; minimize PenaltyCost: sumk in K, j in J, g in 1,2,3 P1[k,j]*x[k,j,g]+suml in L, j in J, g in 2,3 P2[l,j]*y[l,j,g]+summ in M, j in J P3[m,j]*z[m,j]+sumk in K, j in J, g in 1,2,3 oc[1]*ox[k,j,g]+suml in L, j in J, g in 2,3 oc[2]*oy[l,j,g]+summ in M, j in J oc[3]*oz[m,j]; subject to AverageFirstNurseDemand j in J: sumk in K x[k,j,1]>=AD[j,1]; subject to AverageSecondNurseDemand j in J: sumk in K x[k,j,2]+suml in L y[l,j,2]>=AD[j,2]; subject to AverageThirdNurseDemand j in J: sumk in K x[k,j,3]+suml in L y[l,j,3]+summ in M z[m,j]>=AD[j,3];

32

subject to WorkingDayForFirstNurse k in K: sumj in J , g in 1,2,3 x[k,j,g]=6; subject to WorkingDayForSecondNUrse l in L: sumj in J , g in 2,3 y[l,j,g]=6; subject to WorkingDayForThirdNurse m in M: sumj in J z[m,j]=6; subject to shiftPerDayConstraintForFirstNurse k in K, j in 1,4,7,10,13,16,19: x[k,j,1]+x[k,j+1,1]+x[k,j+2,1]+x[k,j,2]+x[k,j+1,2]+x[k,j+2,2]+x[k,j,3]+x[k,j+1,3]+x[k,j+2,3]<=1; subject to shiftPerDayConstraintForSecondNurse l in L, j in 1,4,7,10,13,16,19: y[l,j,2]+y[l,j+1,2]+y[l,j+2,2]+y[l,j,3]+y[l,j+1,3]+y[l,j+2,3]<=1; subject to shiftPerDayConstraintForThirdNurse m in M, j in 1,4,7,10,13,16,19: z[m,j]+z[m,j+1]+z[m,j+2]<=1; subject to ExcessFirstNurseDemand j in J: sumk in K x[k,j,1]+sumk in K ox[k,j,1]>=D[j,1]; subject to ExcessSecondNurseDemand j in J: sumk in K x[k,j,2]+suml in L y[l,j,2]+sumk in K ox[k,j,2]+suml in L oy[l,j,2]>=D[j,2]; subject to ExcessThirdNurseDemand j in J: sumk in K x[k,j,3]+suml in L y[l,j,3]+summ in M z[m,j]+sumk in K ox[k,j,3]+suml in L oy[l,j,3]+summ in M oz[m,j]>=D[j,3]; subject to OverTimeRequirementForFirstNurse k in K, j in 2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21: ox[k,j,1]+ox[k,j,2]+ox[k,j,3]-x[k,j-1,1]-x[k,j-1,2]-x[k,j-1,3]<=0; subject to OverTimeRequirementForSecondNurse l in L, j in 2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21: oy[l,j,2]+oy[l,j,3]-y[l,j-1,2]-y[l,j-1,3]<=0; subject to OverTimeRequirementForThirdNurse m in M, j in 2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21: oz[m,j]-z[m,j-1]<=0;

33

subject to OvertimeDayForFirstNurse k in K: sumj in J, g in 1,2,3 ox[k,j,g]<=1; subject to OvertimeDayForSecondNurse l in L: sumj in J, g in 2,3 oy[l,j,g]<=1; subject to OvertimeDayForThirdNurse m in M: sumj in J oz[m,j]<=1;

*.dat

param N1:=9;

param N2:=12;

param N3:=9;

param P1: 1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20 21 :=

1 0 5 0 8 7 2 4 3 9 9

9 5 7 8 1 3 3 9 3 4 9

2 7 3 9 1 5 4 5 2 7 0

8 6 0 8 8 3 2 2 3 7 7

3 8 2 4 9 9 1 2 5 2 7

6 1 5 2 1 1 8 4 10 7 8

4 1 6 3 4 0 7 2 1 9 5

7 10 7 10 7 4 8 7 3 1 1

5 8 3 2 3 1 8 5 7 4 6

4 4 8 1 2 0 1 5 6 1 5

6 6 4 5 9 9 9 6 1 9 1

9 6 4 2 7 3 1 10 9 9 5

7 5 8 5 6 2 1 5 2 10 7

6 8 8 8 2 4 9 7 1 6 5

34

8 8 8 1 5 5 9 1 8 9 7

2 8 2 7 5 5 8 3 2 1 2;

9 7 3 9 9 6 3 9 8 0 7

param P2: 1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20 21 :=

1 5 8 9 7 9 4 6 0 6 9

4 1 7 4 3 1 4 0 3 2 6

2 1 7 1 8 7 9 8 8 7 2

10 5 5 2 1 10 1 6 1 3 8

3 8 1 6 8 6 3 1 8 3 0

1 7 5 0 3 10 8 1 5 1 5

4 0 9 8 3 2 5 4 8 6 8

7 9 3 3 6 5 9 5 0 4 0

5 5 10 7 2 5 5 1 9 8 4

6 1 4 7 4 4 10 10 4 0 0

6 6 1 3 2 0 3 7 6 4 8

1 2 10 0 10 5 1 0 1 2 6

7 5 0 1 3 8 7 2 3 1 1

1 9 3 1 2 4 3 6 3 6 10

8 2 8 0 5 5 9 9 0 5 2

5 5 2 10 2 6 7 10 10 3 9

9 5 8 6 1 9 7 6 3 5 3

1 8 5 5 1 4 2 2 6 5 8;

10 7 10 3 0 9 9 1 7 2 2

11 7 5 2 3 6 6 1 6 7 6

12 2 3 9 5 8 8 6 0 3 6

35

param P3: 1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20 21 :=

1 3 5 3 3 2 6 1 2 0 2

1 0 5 3 2 6 3 8 4 2 0

2 3 6 5 10 2 2 0 1 0 3

10 1 4 2 6 3 3 1 2 4 3

3 8 7 2 5 2 1 1 5 8 1

4 10 2 9 2 3 9 5 2 7 7

4 7 1 2 5 9 0 6 9 10 10

7 8 3 3 1 4 8 6 2 5 7

5 2 5 4 3 7 9 6 5 3 0

6 4 8 2 9 3 3 6 10 2 3

6 9 7 9 8 3 5 7 1 8 6

8 4 9 10 2 2 4 5 8 7 4

7 8 10 7 5 7 5 9 2 3 8

3 3 2 7 2 1 9 2 4 1 5

8 0 5 4 9 0 9 9 9 s 4

8 6 9 9 1 5 3 3 8 3 2;

9 4 0 4 0 8 3 8 6 3 9

param AD: 1 2 3:=

1 2 4 2

2 2 3 3

3 1 2 2

36

4 3 4 3

5 3 4 2

6 2 3 1

7 3 4 2

8 3 4 3

9 3 3 2

10 3 4 3

11 3 4 2

12 2 3 2

13 3 4 2

14 3 3 2

15 2 3 3

16 3 3 3

17 2 3 3

18 3 3 3

19 2 4 3

20 2 3 2

21 2 2 3;

param: G: oc :=

1 15

2 10

3 6;

37

param: H: uc :=

1 15

2 10

3 6;

param D: 1 2 3:=

1 2 4 3

2 3 3 2

3 1 4 3

4 3 4 3

5 4 3 1

6 2 2 2

7 3 4 4

8 3 4 3

9 2 3 2

10 4 5 4

11 3 3 3

12 2 4 2

13 4 4 3

14 3 3 2

15 2 5 2

16 4 4 4

17 2 3 2

18 2 2 3

19 2 3 3

20 2 2 2

21 1 2 1;