research article an optimization method for the...

Research ArticleAn Optimization Method for the Remanufacturing DynamicFacility Layout Problem with Uncertainties

Lingling Li1 Congbo Li1 Huijie Ma1 and Ying Tang2

1State Key Laboratory of Mechanical Transmission Chongqing University Chongqing 400030 China2Department of Electrical and Computer Engineering Rowan University Glassboro NJ 08028 USA

Correspondence should be addressed to Congbo Li cqulcb163com

Received 18 August 2014 Revised 15 October 2014 Accepted 11 November 2014

Academic Editor Zhigang Jiang

Copyright copy 2015 Lingling Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Remanufacturing is a practice of growing importance due to increasing environmental awareness and regulations Facility layoutdesign as the cornerstone of effective facility planning is concerned about resource localization for a well-coordinated workflowthat leads to lowermaterial handling costs and reduced lead times However due to stochastic returns of used productscomponentsand their uncontrollable quality conditions the remanufacturing process exhibits a high level of uncertainty challenging the facilitylayout design for remanufacturing This paper undertakes this problem and presents an optimization method for remanufacturingdynamic facility layout with variable process capacities unequal processing cells and intercell material handling A dynamicmultirow layout model is presented for layout optimization and a modified simulated annealing heuristic is proposed toward thedetermination of optimal layout schemes The approach is demonstrated through a machine tool remanufacturing system

1 Introduction

Remanufacturing as an industrial process of restoring dis-carded productscomponents back to their useful lives isof growing importance due to the emerging pressure oflegislation and increasing awareness of environmental con-servation As the ultimate form of recycling remanufac-turing maintains much of the value added from originalmanufacturing to material and reprocessing conservationleading to lower production costs and improved firm profits[1] A recent Wall Street Journal revealed that large scaleremanufacturing in the United States employs more than500000 people and contributes to approximately $100 bil-lion of goods sold each year [2] More successful industryexamples can be seen in Kodak BMW IBM DEC and Xerox[3] However compared to manufacturing remanufacturingis more complicated in the way that (1) the supply of returnedproducts is unpredictable in timing and quantities (2) thequality and composition of returned products vary and (3)the process routings are not necessarily fixed but rather adaptto the actual conditions of productscomponents [4] Suchcharacteristics have led the facility layout problem (FLP) in

remanufacturing to be much more complicated than that intraditional manufacturing

Facility layout problem (FLP) as the cornerstone ofeffective facility planning is concerned about resource local-ization for a well-coordinated workflow that directly orindirectly leads to lower material handling costs less work-in-process inventory reduced lead times and so forth [5]Research in this area is commonly categorized as a staticor dynamic FLP depending on the nature of the inputrequirements and the time periods under considerationChaieb andKorbaa [6] dealt with the intracellmachine layoutproblem in a circular bidirectional configuration tominimizeboth the movement distance and total transportation timeChan et al [7] incorporated the machine-part grouping deci-sion into the static cellular layout problem to minimize theintercellular movement costs Such efforts on static layout areusually performed for a single time period and assume flowbetweenmachines product demand and level of productmixconstant

Given the nature of manufacturing environments whereuncertainty is inevitable periodic layout reconfigurationwithrespect to changes is necessary as the best facility layout

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 685408 11 pageshttpdxdoiorg1011552015685408

2 Discrete Dynamics in Nature and Society

configuration for one time period may not be efficientfor successive others A relayout or dynamic facility layoutproblem (DFLP) is usually modeled by discretizing thetime into several planning periods where the localizationof an existing facility in each period is a decision variableDharmalingam et al [8] proposed a cellular layout methodfor a manufacturing system with dynamic demands Withthe assumptions of equal size cells and a fixed number ofmachines per cell their objective is to minimize the numberof intercell movements and to perform greatest proportionof part operations within a single cell Sharing the sameassumption that all departments have the same size Bayka-soglu et al [9] developed a dynamic facility layout modelto minimize both material handling and reconfigurationcosts In the authorsrsquo own claim it was the first time thatbudget constraints were considered in DFLP McKendallJr and Hakobyan [10] tackled DFLP with unequal-sizedfree oriental departments However their design focusedon open-field layout design with much loose constraintsBy integrating the decisions of cell formation and grouplayout together Kia et al [11] developed a dynamic layoutmodel that was incorporated with several design featuressuch as alternative process routings operation sequencesmachine capacity lot splitting multirow layout of equalarea facilities and flexible reconfigurations The same groupextended the work to propose a multiobjective model withmore optimization constraints including machine capabilityand capacity part demand satisfaction cell size and locationassignment [12]

Summarizing the findings of the above literatures fewefforts have been made towards the optimization of DFLPwith consideration of remanufacturing natures In fact thehigh level of uncertainty in remanufacturing challenges itslayout design to account for variations For instance thecapacity of each remanufacturing process the dimension ofprocessing cells and the scales of material handing betweentwo cells are nonstationary but change with timeThus thereis a strong case for focusing on DFLP in remanufacturing Inthis paper an attempt has been made to investigate such aproblem In particular a dynamic multirow layout problemis studied with consideration of variable process capacitiesunequal processing cells and intercell material handlingTherest of the paper is organized as follows Section 2 analyzes theuncertainties of DFLP in remanufacturing Section 3 presentsthe dynamic facility layout optimization model Section 4presents a modified simulated annealing heuristic for thedetermination of optimal layout schemes An example isprovided in Section 5 followed by the conclusion in Section 6

2 Uncertainties of FLP in Remanufacturing

In a remanufacturing system there are a number of uncertaincharacteristics that significantly complicate its remanufactur-ing activities [4 13] Firstly the product return process ishighly uncertain with respect to timing when the productsare available for remanufacturing and quantity how manyproducts are available The uncertain characteristic in timingand quantity of returns is a reflection of the uncertain nature

of the life of a product which is commonly influenced by anumber of factors including the life-cycle stage of a productand the rate of technological changes [14]

Productcomponent returned in remanufacturing exhib-its highly uncontrolled variability with respect to productcondition ranging from slightly used with minor blemishesto significantly damaged and requiring extensive repair [14]In a remanufacturing facility used products are firstmanuallydisassembled into what is called ldquocoresrdquo that are disassembledparts or componentsThen inspection and sorting of returnsare usually implemented in the system to identify corersquosquality and organize necessary activities for their recoveryAccording to inspection results the cores are then classifiedinto three groups (ie ldquoreusablerdquo ldquoremanufacturablerdquo andldquomaterial recoverablerdquo) For the remanufacturable cores theirprocess routings are not fixed but rather adapted to theirspecific condition [15ndash17] Figure 1 gives an example ofthe remanufacturing routings for cores (eg spindles guideways worm gears and lead screws) disassembled from aused machine tool For instance if a spindle is detectedwith severe abrasion two typical process routings would beldquogrinding rarr chromium electroplatingrdquo or ldquogrinding rarr coldwelding rarr fine grindrdquo to regain the surface accuracy Amongthese routings some remanufacturing operations may beprobabilistic since the quality attributes of returned cores arecharacterized in the formof a set of classification probabilities[15]

Another uncertain characteristic associated with reman-ufacturing is the high degree of variability in remanufacturingprocessing times For a failed core the amount of timerequired to perform the necessary remanufacturing opera-tion is strongly contingent upon the corersquos quality conditionIn particular the processing time is a function of the corecausing the corersquos failure and the extent of the failure whichis unpredictable in prior

From a facility layout perspective the high level ofuncertainty in remanufacturing challenges its layout designto account for variations For instance the remanufacturingprocesses are subject to extreme fluctuations in workloadsbecause of uncertainty in stochastic returns probabilis-tic process routings and highly variable processing timesAccordingly the required processing capacity of each reman-ufacturing process (ie the number of machines required toperform the specific remanufacturing operation) is nonsta-tionary but changes with time As a result the dimensions ofindividual processing cells to accommodate a certain numberofmachines vary aswell Besides due to the stochastic returnsof used components and their stochastic process routingsthe frequency of intercell material handling between tworemanufacturing processing cells varies constantly over timeGiven somuch stochastic nature in remanufacturing the bestfacility layout configuration obtained in a given time periodmay not be efficient for the successive othersTherefore thereis a strong case for focusing on DFLP in remanufacturing bydiscretizing the time into several planning periodsThis paperundertakes this challenge and incorporates such variability inthe optimization of a remanufacturing DFLP (R-DFLP) withthe aim of achieving overall high efficiency and minimumcost

Discrete Dynamics in Nature and Society 3

4

5

15

6

321

7 11

Guide ways

Leadscrews

Wormgears

Gears

MicrocracksDistortion

Abrasion

Corrosion

Scrape marks

Worm

AbrasionMicrocracks

Spindles

Warehouse for remanufactured

machine tool

(1) Manually disassembly center(2) Automatic cleaning machine(3) Inspection center(4) Manually scraping center

(5) Rough grinding machine(6) Cold welding machine

(7) Precision grinding machine(8) Electrodeposition of chromium

(9) Groove machine(10) Laser cladding(11) Press and straighten(12) Planer machine (16) Reassembly center

(13) Polishing machine(14) Gear hobbing machine

(15) Lathe machine

Slight abrasionSeverer abrasionSlight corrosion

Severe corrosion

Abrasion

GearsSeverer abrasionSlight abrasion

Abrasion

Distortion

External surfaceInternal surface Both surfaces

Warehousefor used machine

tools

Disposal

Reusable components

8

10

12

1314

9

16

Com

pone

nt w

areh

ouse

Figure 1 The remanufacturing process routings of machine tool cores

3 R-DFLP Optimization Model

31 Problem Statement A typical remanufacturing facilityconsidered in the paper is a multirow shop floor each row ofwhich has a set of designated departments with an equal sizeThe facility accommodates a finite set of remanufacturingprocesses to restore components of a specific type of usedproducts A processing cell (ie grouping a number ofmachines to perform a specific remanufacturing operation)is dedicated to each remanufacturing process which occupiesone ormore than one departments according to its dimensionto be determined over time Figure 2 gives an example ofa facility layout configuration with 3 rows 27 departmentsand 6 processing cells The problem is formulated under thefollowing assumptions

(a) A dynamic facility layout in remanufacturing is con-sidered by discretizing the time into several planningperiods (119905 = 1 2 119879) That is the remanufacturingfacility is allowed to be reconfigured at each giventime period according to the dynamics of the system

(b) The required capacity of a processing cell is notfixed but adapted to its workload in a given unit oftime Thus the number of machines assigned to thecell might be different from one period to anotherWhen 119873

119894119905lt 119873119894(119905minus1)

the redundant machine wouldbe removed from the cell When 119873

119894119905gt 119873

119894(119905minus1)

additional machines will be added to the 119894th pro-cessing cell In either case relocation cost would becharged for machine equipment handling regarding

machine removal machine transiting and machineinstallation

(c) There are an infinite number of machines in the facil-ity The purchase of new machines is not necessaryand considered

(d) The processing cells in the same row have the same119884-coordinate

(e) Each processing cell only occupies departments in thesame row

(f) The dimension of the shop floor is given(g) The dimensions of machines are given but vary

with respect to the processing cells they serve (iethe machines in a processing cell have the samedimensions)

(h) The intracell material handling cost is negligible(i) The width of a department is always larger than the

length of machines in any processing cell(j) The orientations of machines in a processing cell are

kept the same

32 Notations and Variables Involved in R-DFLP The nota-tions and variables used in the model are given below

321 Index

119905 time period index (119905 = 1 2 119879) where 119879 is thetotal number of periods


Main road

Main road

3rd row

2nd row

1st row

1stcolumn

Department

Processing cell

2ndcolumn

3rdcolumn

9thcolumn

middot middot middot middot middot middot middot middot middot middot middot middot

Figure 2 An example of a facility layout configuration with 3 rows 27 departments and 6 processing cells

119894 119895 processing cell index (119894 119895 = 1 2 119885) where119885 is the total number of processing cells in the shopfloor in particular 119894 = 1 2 3 119885 denote disassemblycleaning inspection and reassembly respectivelywhile 119894 = 4 119885 minus 1 represent the remanufacturingoperations119896 core type index119903 remanufacturing processing routing index119891 facility row index (119891 = 1 2 119865) where 119865 is thenumber of rows in the shop floorℎ department index in a given row (ℎ = 1 2 119867)where119867 is the number of departments per row

322 Model Parameters

119863119905 time duration (in days) of the period 119905

119876119896119905 number of the cores of type 119896 in period 119905

119901119896119903119905 probability of the cores of type 119896 being assigned

to the 119903th processing routing in period 119905119888119894 machine availability (ie the available machine

hours per workday of an individual machine) of the119894th processing cell (hoursday)120603119894 machine efficiency (in percentage) of the 119894th

processing cell corresponding to machine failures120595119894119896 average processing time (inminutes) for the cores

of type 119896 in the 119894th processing cell119872119897119894 length of the machines in the 119894th processing cell

(119872119897119894ge 119872119908

119894)

119872119908119894 width of the machines in the 119894th processing cell

119878119897 length of the remanufacturing facility (119878119897 ge 119878119908)119878119908 width of the remanufacturing facility120576 the width of main road designed in the facility120575119894 relocation cost for the machines in the 119894th process-

ing cell120579119896 material handling cost of cores of type 119896 per unit

of distancesΔ119905 total material handling cost in period 119905

119861119905 relocation budget in period 119905

Γ119894119895119896119903

binary variable that has a value of 1 if theprocessing cell 119894 precedes the processing cell 119895 in the119903th processing routing for the cores of type 119896 and 0otherwise120572119894119896119903 binary variable that has a value of 1 if the

119894th processing cell is in the 119903th remanufacturingprocessing routing for the cores of type 119896 and 0otherwise

Among the above-mentioned parameters there are threeuncertain parameters that significantly complicate the R-DFLP and need to be elaborated as follows

(1) The Arrivals of Used Products In a remanufacturingsystem arrivals (119876

119896119905) related to the failures of products

are uncertain in timing and quantities and are commonlymodelled as a Poison process [9]

(2) The Process Routings The process routings are not nec-essarily fixed but rather adapt to the actual conditions ofcores The likelihood (119875

119896119903119905) of a remanufacturing process

routing being assigned to a certain type of cores is randomlygenerated to represent the stochastic features [4]


(3) The Processing TimesThe processing times (120595119894119896) required

to perform the necessary remanufacturing operations for acertain part are highly variable An exponential distributionis used for the processing times in order to simulate the largerange of possible values [9]

323 Decision Variables

119873119894119905 number of machines in the 119894th processing cell in

period 119905120573119894119905 binary variable that has a value of 1 if themachines

in the 119894th processing cell are oriented vertically and 0otherwise119871119894119905 length of the 119894th processing cell in period 119905

120578119894119905 number of departments occupied by the 119894th

processing cell in period 119905120587119894ℎ119891119905

binary variable that has a value of 1 if the 119894thprocessing cell occupies the department in the ℎthrow and 119894th column as its first location in period 119905(119909119894119905 119910119894119905) the centroid of the 119894th processing cell in

period 119905119889119894119895119905 distance (in meters) between the two processing

cells 119894 and 119895 in period 119905120591119894119905 binary variable that has a value of 1 if the centroid

of the processing cell 119894 in period 119905 is different from thatin period 119905 minus 1 and 0 otherwise

As stated earlier the variability in remanufacturingregarding stochastic returns probabilistic process routingsand highly variable processing times complicates its facilitylayout design Thus the decision variables in the R-DFLPincluding the number of machines the dimensions of indi-vidual processing cells and the detailed location (ie thecentroid) of each processing cell in a given time period willbe mathematically elaborated in the following

(1) Number of Machines in Processing Cells The numberof machines dedicated to the 119894th processing cell in period119905 119873119894119905 is a function of the number of cores needed to be

processed at the time the average processing times andmachine availability and efficiency

119873119894119905=

sum119896sum119903119876119896119905sdot 119901119896119903119905

sdot 120595119894119896sdot 120572119894119896119903

120603119894sdot 119888119894sdot 119863119905

(1)

where 119888119894denotes the machine availability (ie the available

machine hours per workday of a certain machine type) ofthe 119894th processing cell (hoursday) and 120603

119894is the machine

efficiency (in percentage) of the 119894th processing cell corre-sponding to machine failures

(2) Dimensions of Processing Cells It is assumed that anindividual processing cell should be equipped with a seriesof machines of the same type to perform a specific reman-ufacturing process The dimension of a processing cell asformulated in (2) is pertinent to the numbers of machines

dwelt in and their orientations The number of departmentsoccupied by the 119894th processing cell is given in (3)

119871119894119905= [119872119897

119894sdot (1 minus 120573

119894119905) + 119872119908

119894sdot 120573119894119905] sdot 119873119894119905 (2)

120578119894119905= ceil [

119871119894119905

119878119897119867

] (3)

where 119878119897119867 (ie dividing 119878119897 by 119867) gets the length of eachdepartment designed in each row of the floor shop and ceiling(119909) returns a value that is the smallest integer not less than 119909

(3) The Centroids of Processing Cells For each processing cellwith the required number of departments (120578

119894119905) and the first

department it occupies (120587119894ℎ119891119905

) being determined its centroidsin period 119905 then can be determined by (4) The distancebetween the centroids of the processing cell 119894 and 119895 in period119905 is given in (5)

119909119894119905= (ℎ minus 05) sdot

119878119897

2119867

+ (120578119894119905minus 1) sdot

119878119897

2119867

119910119894119905=

119878119908

2119865

+ (119891 minus 1) sdot

119878119908

119865

forall120587119894ℎ119891119905

= 1

(4)

119889119894119895119905

=

10038161003816100381610038161003816119909119894119905minus 119909119895119905

10038161003816100381610038161003816+

10038161003816100381610038161003816119910119894119905minus 119910119895119905

10038161003816100381610038161003816 (5)

33 Optimization Model The R-DFLP considered in thispaper aims to balance the trade-off between the intercellmaterial handling cost and the increased relocation cost ofinefficient layouts

(1) Cost of Intercell Material HandingWhile cores go throughvarious processes to be restored back to their useful livesmaterial handling cost is associated with their logistic activ-ities (ie moving from one processing cell to another) Thetotal amount of material handling cost in a given unit of timeis an incremental linear function of the total distances of coresmoving in the facility as defined below

Δ119905= sum

119896

sum

119903

sum

119894

sum

119895

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

sdot 119889119894119895119905sdot 120579119896 (6)

where 120579119896denotes the material handling cost of cores of type

119896 per unit distance

(2) Relocation Cost Since the best facility layout schemefor one time period may not be efficient for the successiveothers the floor shop needs to be reconfigured accordinglyby changing the locations of a series of processing cellsin a given time period In this case machine equipmentin these processing cells needs to be removed transited


and reinstalled which necessarily incur a certain amount ofrelocation costs as formulated in the following

Ω119905= sum

119894

120591119894119905sdot 119873119894119905sdot 120575119894 forall119905 gt 1

120591119894119905=

0 if119909119894119905= 119909119894(119905minus1)

119910119894119905= 119910119894(119905minus1)

1 otherwiseforall119905 gt 1

(7)

where 120591119894119905is a binary variable that has a value of 1 if the centroid

of the processing cell 119894 in period 119905 is different from that inperiod 119905 minus 1 and 0 otherwise 120575

119894is the relocation cost per

machine of the 119894th processing cellThe R-DFLP problem is then formulated as a nonlinear

mixed-integer programming model to minimize the totalcost of intercell material handling and machine relocation

Minimize total cost =119879

sum

119905=1

Δ119905+

119879

sum

119905=2

Ω119905

(8)

Subject to119879

sum

119905=2

sum

119894

120591119894119905sdot 120575119894le 119861119905

(9)

sum

119894

120578119894119905le 119865 sdot 119867 forall119905 (10)

sum

119894=3

119885minus1

sum

119895=4

sum

119896

sum

119903

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

=

119885minus1

sum

119894=4

sum

119895=119885

sum

119896

sum

119903

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

forall119905

(11)

Constraint (9) assures that the total relocation cost in anyperiod does not exceed the given budget Constraint (10)guarantees that the total number of departments occupiedby processing cells is not larger than the total availablein the shop floor Constraint (11) represents material flowconservation between disassembly and reassembly

4 The Simulated Annealing Algorithmfor R-DFLP

Simulated annealing (SA) is a stochastic gradient methodfor global optimization that has been successfully appliedto a wide variety of sophisticated combinatorial problems[18ndash20] While it conducts a local search it is capable ofexploring the solution space stochastically and effectively toprevent being trapped in a local optimumThe SA procedurestarts from an initial basic solution 119904

0and generates a

neighbourhood solution 1199041015840 by a suitable mechanism To

accept 1199041015840 over 1199040 1199041015840 either is a better neighbourhood solution

or has a certain probability to escape from a local minimumthough 1199041015840 is worse than 119904

0[21]The acceptance probability AP

is often defined as follows

AP =

1 ifΛ lt 0

exp minusΛΩ

otherwise(12)

where Λ = 119891(1199041015840) minus 119891(119904

0) Ω is called temperature a

global time-varying parameter and 119891( ) is the optimizationobjective function At each iteration the parameter Ω isreduced by a factor 120599 and the chance of choosing an inferiorsolution decreases as well The search process which isgenerating a neighbour solution rejecting or accepting it withthe above probability continues until the stopping criterion ismet Given SA is a well-established metaheuristic the majorsteps aremodified to suit our specific application followed bythe proposed algorithms

41 Solution Representation To address the challenge of R-DFLP with unequal processing cells this paper proposes aspecial solution representation that uses two-stage matricesfor remanufacturing layout schemes

The first matrix 119860 = [119886119905119908]119879times119882

is called sequence matrixthat defines the sequence of processing cells to be allocateddepartments in any given period where 119882 = 119885 + 119865 sdot 119867 minus

sum120578119894119905 119886119905119908

is a nonnegative integer indicating the index of aprocessing cell that is the 119908th in sequence to be assigneddepartments in period 119905 If 119886

119905119908= 0 a department is

left unoccupied in the 119908th action For each unoccupieddepartment an intangiblemachine is assumed to be allocatedin it That is forall119886

119905119908= 0 and there exists 119873

[119886119905119908]119905

= 1119872119897[119886119905119908]= 119878119871119867 and119872119908

[119886119905119908]= [119878119882minus(119865minus1)120576]119865 Figure 3(a)

gives an example of119860 that represents allocation sequences infour time periods For instance in the first time period theallocation order is the processing cells 1 3 2 and 4 followedby an unoccupied department

The secondmatrix119864 = [119890119905119908]119879times119882

has the same dimensionsas matrix 119860 that defines the orientation of processing cells inperiod 119905 119890

119905119908takes a binary value of zero or one indicating

the processing cell in the 119908th allocation sequence is orientedvertically (1) or horizontally (0) In this case matrix 119864 isdetermined according tomatrix119860 as defined in (13) It shouldbe noted that if 119886

119905119908gt 0 then the machine can be located

either vertically or horizontally (ie 119890119905119908

isin 0 1) If 119886119905119908

= 0then 119890

119905119908= 1 which denotes that the intangible machine can

only be oriented vertically in the unoccupied departmentFigure 3(b) gives an example of matrix 119864 that aligns to matrix119860 in Figure 3(b)

119890119905119908

=

120573[119886119905119908]119905isin 0 1 119886

119905119908gt 0

1 119886119905119908

= 0

(13)

To validate the generated solutions another matrix 119866 =

[119892119891ℎ119905

]119865times119867times119879

is defined to provide the detailed layout schemefor all processing cells in remanufacturing facility 119892

119891ℎ119905is the

index of the processing cell that occupies the department inthe 119891th row and ℎth column in period 119905 If 119892

119891ℎ119905= 0 the

department in the119891th row and the ℎth column is unoccupiedMatrix 119866 is not part of the solution representation but can bedirectly derived from matrices 119860 and 119864 Given the matrices119860 and 119864 119871

119894119905and 120578119894119905can be first calculated and then matrix119866

can be determined according to Procedure 1 A sample layoutmatrix is provided in Figure 3(b) that is directly derived fromthe sample matrix 119860 in Figure 3(a)


A =

1 3 2 4 0

2 1 3 0 4

1 0 2 4 3

1 2 0 3 4lfloor

lceilrceilrceilrceil

lfloor


(a)

E =

1 0 0 0 1

1 0 1 1 0

1 1 0 1 0

1 0 1 0 1lfloor


lfloor


(b)

2nd period

1st row

1st column

2nd column

3rd column

1st period

3rd period

4th period

2nd row

1 1 02 4 3

2 1 13 0 4

1 1 32 4 0

21 10 3 4

Matrix G

(c)

Figure 3 (a) Sample matrix 119860 (b) sample matrix 119864 (c) sample matrix 119866 derived from 119860 and 119864

Input119860 Sequence matrix119864 Orientation matrix

OutputSet ℎ = 1 119891 = 1for 119905 = 1 to 119879for 119908 = 1 to 119899

if (119886119905119908

= 0) 119892119891ℎ119905

= 119886119905119908

if (ℎ le 119867) ℎ = ℎ + 1else ℎ = 1 119891 = 119891 + 1

else

120585 = 120578[119886119905119908]119905

while (120585 = 0) do

120585 = 120585 minus 1119892119891ℎ119905

= 119886119905119908

if (ℎ le 119867) ℎ = ℎ + 1else ℎ = 1 119891 = 119891 + 1

Procedure 1 Determination of the layout matrix 119866

For each processing cell with its dimension (ie 119871119894119905

120578119894119905) and allocation sequence (ie in matrix 119860) determined

Procedure 1 first selects a corresponding department as itsstarting location The selection continues with the rightneighbours of the departments from the left to the right untilthe dimension of the processing cell is met With the way ofallocating departments to processing cells and the assump-tion that each processing cell only occupies departments inthe same row the layout matrix should meet the followingcondition

119892119891119867119905

= 119892(119891+1)1119905

forall119891 = 1 2 119865 minus 1 (14)

With a feasible matric 119866 obtained the first departmentthat the 119894th processing cell occupies can be determined

by (15) Then using (4)ndash(8) the total cost can be calculatedaccordingly

120587119894ℎ119891119905

=

1 forall119891 = minforall119892119891ℎ119905=119894

119891

0 otherwiseforall119894 119905 (15)

42 Initial Solution Generation The initial solution is impor-tant to the performance of SA as the quality of the finalsolution relies heavily on that of the initial solution [22 23]The procedure of generating an initial solution is given below

43 Procedure SA Initialization

(1) Generate two sets the acceptable solution set AS =

(119860V 119864V) V = 1 2 119874 and the invalid solution setIS = (119860

119906 119864119906) Initialize AS and IS to be empty

(2) Randomly generate a solution 2-tuple (119860lowast 119864lowast) If(119860lowast 119864lowast) notin IS go to (3) otherwise restart (2)

(3) Derive the layout matrix 119866lowast based on (119860

lowast 119864lowast) If 119866

satisfies constraints (6) and (10) AS = AS cup (119860lowast 119864lowast)

If |AS| = 119874 go to (4) otherwise IS = IS cup (119860lowast 119864lowast)

go back to (2)(4) Calculate the total cost of each feasible solution in AS

and choose the one with the minimal total cost as theinitial solution 119904

0for the SA algorithm (Algorithm 1)

44 Neighbourhood Solution Generation In order to obtainthe neighbourhood solution from the current solution thestandard SA neighbourhood structure that features varioustypes ofmove in work [23] is adoptedThe detailed procedureis given below

45 Procedure Neighbour 119878(119904)

(1) Randomly select a row 119905 and a column 119908 in matrix119860 and use one type of moves (ie 119898-opt 119898 isin

1 2 119899 minus 119908) to exchange 119886119905119908

with 119886119905(119908+119898)

andformulate a new matrix 119860

1015840 Then randomly select a119890119905119908

gt 0 in matrix 119864 and update 119890119905119908

= 1 minus 119890119905119908


Step 1 Set the annealing parametersΩ and 120599 and initialize the outer loop counter (ol) and the inner loop counter (il) zeroolmax ilmax and 120590max are maximum outer loop inner loop and identical solution respectively

Step 2 Apply Procedure SA initialization to generate an initial solution 1199040 Assign 119904ol = 119904 = 119904

0

Step 3 While (ol le olmax and 120590 le 120590max) dofor (il = 0 il le ilmax il = il + 1)

Apply Procedure Neighbour S(s)Calculate Λ = 119891(119904

1015840) minus 119891(119904)

If (Λ lt 0) or (expminus ΛΩ ge Random(0 1))119904 = 1199041015840

else reject 1199041015840ol = ol + 1119904ol = 119904if (119904ol = 119904

(olminus1)) 120590 = 120590 + 1else 120590 = 0

Step 4 Output the matrix 119866 based on 119904ol

Algorithm 1 Proposed SA

to formulate a new matrix 1198641015840 A potential neighboursolution 119904

1015840= (1198601015840 1198641015840) is then formulated

(2) Output 1199041015840 as the neighbourhood solution if it satisfiesthe constraints (6) and (10) otherwise go back to (1)

5 An Example

In this section a sample machine tool remanufacturing facil-ity with 2 rows and 16 processing cells is used to demonstratethe application of the proposed model and algorithm Theworkflows of cores in this remanufacturing shop are shownin Figure 2 In order to verify the efficiency of the proposedmethod two cases are considered in the simulation for a12-month period In the static case the system treats the 12months as a single period and no reconfiguration is allowedIn the dynamic case the facility layout is reconfigured at every4 months based on the changes in the supplier The proposedSA algorithm is applied to both cases using Matlab 2009During the implementation of the simulation the input datais populated as follows The facility data is shown in Table 1

(1) The arrival of used machine tools satisfies a Pois-son distribution Thus the arrival of each type ofremanufacturable cores in period 119905 follows a Poissondistribution with the arrival rate 120582 isin 119880[10 15] lowast 120587where 120587 isin 10 11 12 13 14 15 is the arrival rateexpansion factor

(2) The probabilities of each type of cores going throughvarious process routings in a certain period arerandomly generated 119901

119896119903119905isin [0 1] and sum

119903119901119896119903119905

= 1(3) The average processing time for the cores of type

119896 in the 119894th processing cell follows an exponentialdistribution 120595

119894119896sim Γ(1 120582) where 120582 isin [008 010]

Figure 4 compares the total cost of the dynamic layoutscheme to that of the static one via twenty-four runs of the SAwhere the relocation cost 120575 = 100 and 120587 = 10 It is clear thatthe dynamic layout scheme outperforms the static one withabout 20 cost reduction Figure 5 shows the optimal layout

Table 1 Other data involved in the dynamic facility layout problem

119878119897 = 85m 119878119908 = 20m119865 = 2 119867 = 16

119888119894= 8 hoursday 120603

119868= 95

1198611= 1198612= 1198613= 1198614= 1000 120579

1= 1205793= 1205794= 1205795= 1 120579

2= 5

120575119894= 100 forall119894 isin 1 2 16 120603 = 80

1205951119896= 1205952119896= 15min 120595

3119896= 5min 120595

16119896= 20min

1198721198971= 1198721198972= 11987211989716=

11987211989714= 11987211989712= 1198721198976=

1198721198978= 11987211989710= 6m

119872119897119894= 3m forall119894 isin 4 5 6

1198721199081= 119872119908

2= 119872119908

16=

11987211990814= 119872119908

12= 3m

1198721199086= 119872119908

8= 119872119908

10= 5m

119872119908119894= 2m forall119894 isin 4 5 6

Dynamic layoutStatic layout

Number of SA implementation processes

Tota

l cos

t

times104

14

12

10

8

6

4

2 4 6 8 10 12 14 16 18 20 22 24

Figure 4 Performance comparison of the dynamic layout versus thestatic one (120575 = 100 120587 = 10)

schemes for both dynamic layout and static one derived fromthe 8th simulation Table 2 gives the detailed indication of theletters AndashT in Figures 5 and 6

In order to determine any significance difference betweenperformances of the dynamic layout and the static one a setof simulations are implemented for different arrival rate level


C

A

B

D1 D2 D3

E1

E2

F1 S1 S2 S3 S4

T1

R1

T2

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1

(a) The dynamic layout scheme for 1ndash4 months (material handling cost = 23224)

C

A

B

D1 D2 D3 D4

E1

E2

F1 S1 S2 S3 S4R1

T1

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1

(b) The dynamic layout scheme for 5ndash8 months (total cost = 32148)

C

A

B

D1 D2 D3 D4

E1

E2

F1S1 S2 S3 S4

R1

R2

T1

L 1

L 2

L 3

H1 H2 I1 O1N1K2K1P1

M1 M2

G1G2

J1

Q1

(c) The dynamic layout scheme for 9ndash12 months (total cost = 27663)

Figure 5 The dynamic layout scheme for a whole year (total cost = 83035)

C

A

B

D1 D2 D3

E1

E2

F1 K2

G1

S1 S2 S3N1

H1

J1

T1 O1 O3O2M1 I1 I2 Q1

K1 L 1 J2

P1P2

R1

T2

Figure 6 The static layout scheme for a whole year (total cost = 113250)

Table 2 The detailed indication of the letters AndashT in Figures 5 and 6

AndashE Indication FndashJ Indication KndashO Indication PndashT Indication

A Warehouse for usedmachine tool F Inspection center K Chromium electroplating P Polishing machine

B Workshop office G Manually scraping center L Groove machine Q Gear hobbing machine

CWarehouse forremanufacturedmachine tool

H Rough grinding machine M Laser cladding R Lathe machine

D Manually disassemblycenter I Cold welding machine N Press and straighten S Reassembly center

E Automatic cleaningmachine J Precision grinding machine O Planer machine T Unoccupied department


Table 3 Comparison of the average total cost of two cases

Arrival rate levelThe average total cost Cost difference

1206012

Cost reduction ratio(1206011minus 1206012)1206011

The static layout1206011

The dynamic layout1206012

V = 10 100915 78084 28831 0226239V = 11 124947 90250 34697 0277694V = 12 148068 109396 38672 0261177V = 13 178295 135050 43245 0242547V = 14 196572 147063 49509 0251862V = 15 233200 178090 55110 0236321Sample mean 1636662 1229888 4167733 0249307Sample standard deviation 486044 3750791 9664517 0018437

The average total cost is obtained and compared among thedynamic layout and the static one as shown in Table 3 It isclear to see that the dynamic layout scheme outperforms thestatic one in terms of generating less cost A 95 confidenceinterval for the cost reduction ratio of the dynamic layout isalso conducted using the data in Table 3 The approximate100(1 minus 120572) confidence interval for 120601 is defined as

120601 plusmn 1199051minus1205722 (

119899 minus 1)

119878

radic119899

or 120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

le 120601 le120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

(16)

where 120601 is the sample mean of 120601 based on a sample of size 119899

119878 is the sample standard deviation 1199051minus1205722

(119899minus1) is the 100(1minus120572) percentage point of a 119905-distributed with 119899 minus 1 degree offreedom

A 95 confidence interval for the cost reduction ratio ofthe dynamic layout is given by

0249307 plusmn (2571 times

0018437

radic6

)

0229955 le

1206011minus 1206012

1206011

le 0268659

(17)

The 95 confidence interval for the cost reduction ratiolies completely above zero that is (120601

1minus 1206012)1206011gt 0 which

provides strong evidence that at 95 confidence there issignificant difference between performances of the dynamiclayout and the static one

The similar comparison is conducted with the relocationcost changing from 100 to 1900 Figure 7 shows that whenrelocation cost is less than 1200 relocation of machines indynamic layout would lead to much reduction in materialhandling and is significant to lower the overall cost Howeverwhen relocation becomes extremely expensive (ie higherthan 1200) the dynamic layout starts to lose its advantageThetrends in Figure 7 indicate that the dynamic layout is muchsuperior to the static one especially for the facilities withhigher flexibility inmachine equipment handling However ifthe facility has gainedmuch resistance tomachine equipmenthandling the static layout would outperform the dynamicone with less cost


times105

16

15

14

13

12

11

1

09

08

07

06

Tota

l cos

t

0 200 400 600 800 1000 1200 1400 1600 1800 2000


Figure 7 Performance of the dynamic layout schemes and the staticones with relocation cost ranging from 100 to 1900

6 Conclusion

The remanufacturing process exhibits a high level of uncer-tainty due to stochastic returns of used productscomponentsand their uncontrollable quality Such uncertainty chal-lenges the facility layout design for remanufacturing Thispaper undertakes this problem and presents an optimizationmethod for remanufacturing dynamic layout design withvariable process capacities unequal processing cells andintercell material handling The uncertainties inherent inthe remanufacturing facility layout are explicitly analyzedA dynamic multirow layout model is present for layoutoptimization with minimization of both material handlingcost and machine relocation cost A modified simulatedannealing heuristic is proposed towards the determinationof optimal layout schemes with unequal processing cellsThe approach is validated through a layout design for amachine tool remanufacturing system and its effectivenessis demonstrated through simulations Through simulationsit is shown that while relocation cost incurs in dynamiclayout reconfiguration the reduction in material handlingis significant to lower the overall cost However whenrelocation becomes extremely expensive the dynamic layoutstarts to lose its advantage In the future factory date will


be used to test our methodology through more complicatedindustrial cases

Conflict of Interests

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

Acknowledgments

This work was supported in part by the National KeyTechnology RampDProgramofChina (2012BAF02B03) and theNational Natural Science Foundation of China (51105395)

References

[1] T Guidat A P Barquet H Widera H Rozenfeld and GSeliger ldquoGuidelines for the definition of innovative industrialproduct-service systems (PSS) business models for remanufac-turingrdquo in Proceedings of the 6th CIRP Conference on IndustrialProduct Service Systems pp 193ndash198 Ontario Canada May2014

[2] A AtasuM Sarvary and L N VWassenhove ldquoRemanufactur-ing as amarketing strategyrdquoManagement Science vol 54 no 10pp 1731ndash1746 2008

[3] J Ostlin E Sundin and M Bjorkman ldquoProduct life-cycleimplications for remanufacturing strategiesrdquo Journal of CleanerProduction vol 17 no 11 pp 999ndash1009 2009

[4] Y Tang and C B Li ldquoUncertainty management in remanufac-turing a reviewrdquo in Proceedings of the 8th IEEE InternationalConference on Automation Science and Engineering pp 52ndash57Seoul Republic of Korea August 2012

[5] R Sahin K Ertogral and O Turkbey ldquoA simulated annealingheuristic for the dynamic layout problem with budget controlrdquoComputers and Operations Research vol 59 no 2 pp 308ndash3132010

[6] I Chaieb and O Korbaa ldquoIntra-cell machine layout associatedwith flexible production and transport systemsrdquo Proceedingsof the Institution of Mechanical Engineers Part B Journal ofEngineering Manufacture vol 217 no 7 pp 883ndash892 2003

[7] F T S Chan K W Lau P L Y Chan and K L ChoyldquoTwo-stage approach formachine-part grouping and cell layoutproblemsrdquo Robotics and Computer-Integrated Manufacturingvol 22 no 3 pp 217ndash238 2006

[8] P Dharmalingam K Kanthavel R Sathiyamoorthy M Sak-thivel R Krishnaraj and C Elango ldquoOptimization of cellularlayout under dynamic demand environment by simulatedannealingrdquo International Journal of Scientific and EngineeringResearch vol 3 no 10 pp 91ndash97 2012

[9] A Baykasoglu T Dereli and I Sabuncu ldquoAn ant colonyalgorithm for solving budget constrained and unconstraineddynamic facility layout problemsrdquo Omega vol 34 no 4 pp385ndash396 2006

[10] A R McKendall Jr and A Hakobyan ldquoHeuristics for thedynamic facility layout problem with unequal-area depart-mentsrdquo European Journal of Operational Research vol 201 no1 pp 171ndash182 2010

[11] R Kia A Baboli N Javadian R Tavakkoli-Moghaddam MKazemi and J Khorrami ldquoSolving a group layout design modelof a dynamic cellular manufacturing system with alternative

process routings lot splitting and flexible reconfiguration bysimulated annealingrdquoComputersampOperations Research vol 39no 11 pp 2642ndash2658 2012

[12] R Kia H Shirazi N Javadian and R TMoghaddam ldquoAmulti-objective model for designing a group layout of a dynamic cel-lular manufacturing systemrdquo International Journal of IndustrialEngineering vol 9 no 1 pp 1ndash14 2013

[13] Z Jiang S Zhu H Zhang and Y Wang ldquoOptimal acquisitionand inventory control for a remanufacturing systemrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 120256 7pages 2013

[14] V D R Guide Jr G C Souza and E van der LaanldquoPerformance of static priority rules for shared facilities ina remanufacturing shop with disassembly and reassemblyrdquoEuropean Journal of Operational Research vol 164 no 2 pp341ndash353 2005

[15] C Li Y Tang C Li and L Li ldquoA modeling approach toanalyze variability of remanufacturing process routingrdquo IEEETransactions onAutomation Science and Engineering vol 10 no1 pp 86ndash98 2013

[16] Z Jiang H Zhang and J W Sutherland ldquoDevelopment ofmulti-criteria decision making model for remanufacturingtechnology portfolio selectionrdquo Journal of Cleaner Productionvol 19 no 17-18 pp 1939ndash1945 2011

[17] G Tian Y Liu H Ke and J Chu ldquoEnergy evaluationmethod and its optimization models for process planning withstochastic characteristics a case study in disassembly decision-makingrdquo Computers and Industrial Engineering vol 63 no 3pp 553ndash563 2012

[18] S M Mousavi and R T Moghaddam ldquoA hybrid simulatedannealing algorithm for location and routing scheduling prob-lems with cross-docking in the supply chainrdquo Journal of Manu-facturing Systems vol 32 no 2 pp 335ndash347 2013

[19] T Chen and R Xiao ldquoEnhancing artificial bee colony algorithmwith self-adaptive searching strategy and artificial immunenetwork operators for global optimizationrdquoThe ScientificWorldJournal vol 2014 Article ID 438260 12 pages 2014

[20] T Chen and R Xiao ldquoA dynamic intelligent decision approachto dependency modeling of project tasks in complex engineer-ing system optimizationrdquo Mathematical Problems in Engineer-ing vol 2013 Article ID 398123 12 pages 2013

[21] R Zhang ldquoA simulated annealing-based heuristic algorithm forjob shop scheduling tominimize latenessrdquo International Journalof Advanced Robotic Systems vol 10 article 214 2013

[22] N Manavizadeh N-S Hosseini M Rabbani and F Jolai ldquoASimulated Annealing algorithm for a mixed model assemblyU-line balancing type-I problem considering human efficiencyand Just-In-Time approachrdquo Computers amp Industrial Engineer-ing vol 64 no 2 pp 669ndash685 2013

[23] C Ju and T Chen ldquoSimplifying multiproject scheduling prob-lem based on design structure matrix and its solution by animproved aiNet algorithmrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 713740 22 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


configuration for one time period may not be efficientfor successive others A relayout or dynamic facility layoutproblem (DFLP) is usually modeled by discretizing thetime into several planning periods where the localizationof an existing facility in each period is a decision variableDharmalingam et al [8] proposed a cellular layout methodfor a manufacturing system with dynamic demands Withthe assumptions of equal size cells and a fixed number ofmachines per cell their objective is to minimize the numberof intercell movements and to perform greatest proportionof part operations within a single cell Sharing the sameassumption that all departments have the same size Bayka-soglu et al [9] developed a dynamic facility layout modelto minimize both material handling and reconfigurationcosts In the authorsrsquo own claim it was the first time thatbudget constraints were considered in DFLP McKendallJr and Hakobyan [10] tackled DFLP with unequal-sizedfree oriental departments However their design focusedon open-field layout design with much loose constraintsBy integrating the decisions of cell formation and grouplayout together Kia et al [11] developed a dynamic layoutmodel that was incorporated with several design featuressuch as alternative process routings operation sequencesmachine capacity lot splitting multirow layout of equalarea facilities and flexible reconfigurations The same groupextended the work to propose a multiobjective model withmore optimization constraints including machine capabilityand capacity part demand satisfaction cell size and locationassignment [12]

Summarizing the findings of the above literatures fewefforts have been made towards the optimization of DFLPwith consideration of remanufacturing natures In fact thehigh level of uncertainty in remanufacturing challenges itslayout design to account for variations For instance thecapacity of each remanufacturing process the dimension ofprocessing cells and the scales of material handing betweentwo cells are nonstationary but change with timeThus thereis a strong case for focusing on DFLP in remanufacturing Inthis paper an attempt has been made to investigate such aproblem In particular a dynamic multirow layout problemis studied with consideration of variable process capacitiesunequal processing cells and intercell material handlingTherest of the paper is organized as follows Section 2 analyzes theuncertainties of DFLP in remanufacturing Section 3 presentsthe dynamic facility layout optimization model Section 4presents a modified simulated annealing heuristic for thedetermination of optimal layout schemes An example isprovided in Section 5 followed by the conclusion in Section 6

2 Uncertainties of FLP in Remanufacturing

In a remanufacturing system there are a number of uncertaincharacteristics that significantly complicate its remanufactur-ing activities [4 13] Firstly the product return process ishighly uncertain with respect to timing when the productsare available for remanufacturing and quantity how manyproducts are available The uncertain characteristic in timingand quantity of returns is a reflection of the uncertain nature

of the life of a product which is commonly influenced by anumber of factors including the life-cycle stage of a productand the rate of technological changes [14]

Productcomponent returned in remanufacturing exhib-its highly uncontrolled variability with respect to productcondition ranging from slightly used with minor blemishesto significantly damaged and requiring extensive repair [14]In a remanufacturing facility used products are firstmanuallydisassembled into what is called ldquocoresrdquo that are disassembledparts or componentsThen inspection and sorting of returnsare usually implemented in the system to identify corersquosquality and organize necessary activities for their recoveryAccording to inspection results the cores are then classifiedinto three groups (ie ldquoreusablerdquo ldquoremanufacturablerdquo andldquomaterial recoverablerdquo) For the remanufacturable cores theirprocess routings are not fixed but rather adapted to theirspecific condition [15ndash17] Figure 1 gives an example ofthe remanufacturing routings for cores (eg spindles guideways worm gears and lead screws) disassembled from aused machine tool For instance if a spindle is detectedwith severe abrasion two typical process routings would beldquogrinding rarr chromium electroplatingrdquo or ldquogrinding rarr coldwelding rarr fine grindrdquo to regain the surface accuracy Amongthese routings some remanufacturing operations may beprobabilistic since the quality attributes of returned cores arecharacterized in the formof a set of classification probabilities[15]

Another uncertain characteristic associated with reman-ufacturing is the high degree of variability in remanufacturingprocessing times For a failed core the amount of timerequired to perform the necessary remanufacturing opera-tion is strongly contingent upon the corersquos quality conditionIn particular the processing time is a function of the corecausing the corersquos failure and the extent of the failure whichis unpredictable in prior

From a facility layout perspective the high level ofuncertainty in remanufacturing challenges its layout designto account for variations For instance the remanufacturingprocesses are subject to extreme fluctuations in workloadsbecause of uncertainty in stochastic returns probabilis-tic process routings and highly variable processing timesAccordingly the required processing capacity of each reman-ufacturing process (ie the number of machines required toperform the specific remanufacturing operation) is nonsta-tionary but changes with time As a result the dimensions ofindividual processing cells to accommodate a certain numberofmachines vary aswell Besides due to the stochastic returnsof used components and their stochastic process routingsthe frequency of intercell material handling between tworemanufacturing processing cells varies constantly over timeGiven somuch stochastic nature in remanufacturing the bestfacility layout configuration obtained in a given time periodmay not be efficient for the successive othersTherefore thereis a strong case for focusing on DFLP in remanufacturing bydiscretizing the time into several planning periodsThis paperundertakes this challenge and incorporates such variability inthe optimization of a remanufacturing DFLP (R-DFLP) withthe aim of achieving overall high efficiency and minimumcost


4

5

15

6

321

7 11

Guide ways

Leadscrews

Wormgears

Gears


Abrasion

Corrosion

Scrape marks

Worm

AbrasionMicrocracks

Spindles


machine tool






(15) Lathe machine


Severe corrosion

Abrasion


Abrasion

Distortion



tools

Disposal

Reusable components

8

10

12

1314

9

16

Com

pone

nt w

areh

ouse






119894119905lt 119873119894(119905minus1)


119894119905gt 119873

119894(119905minus1)










kept the same


321 Index



Main road

Main road

3rd row

2nd row

1st row

1stcolumn

Department

Processing cell

2ndcolumn

3rdcolumn

9thcolumn












(119872119897119894ge 119872119908

119894)






Γ119894119895119896119903


























119873119894119905=

sum119896sum119903119876119896119905sdot 119901119896119903119905

sdot 120595119894119896sdot 120572119894119896119903

120603119894sdot 119888119894sdot 119863119905

(1)







119871119894119905= [119872119897

119894sdot (1 minus 120573

119894119905) + 119872119908

119894sdot 120573119894119905] sdot 119873119894119905 (2)

120578119894119905= ceil [

119871119894119905

119878119897119867

] (3)






119909119894119905= (ℎ minus 05) sdot

119878119897

2119867

+ (120578119894119905minus 1) sdot

119878119897

2119867

119910119894119905=

119878119908

2119865

+ (119891 minus 1) sdot

119878119908

119865

forall120587119894ℎ119891119905

= 1

(4)

119889119894119895119905

=

10038161003816100381610038161003816119909119894119905minus 119909119895119905

10038161003816100381610038161003816+

10038161003816100381610038161003816119910119894119905minus 119910119895119905

10038161003816100381610038161003816 (5)



Δ119905= sum

119896

sum

119903

sum

119894

sum

119895

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

sdot 119889119894119895119905sdot 120579119896 (6)






Ω119905= sum

119894


120591119894119905=

0 if119909119894119905= 119909119894(119905minus1)

119910119894119905= 119910119894(119905minus1)


(7)







sum

119905=1

Δ119905+

119879

sum

119905=2

Ω119905

(8)

Subject to119879

sum

119905=2

sum

119894

120591119894119905sdot 120575119894le 119861119905

(9)

sum

119894

120578119894119905le 119865 sdot 119867 forall119905 (10)

sum

119894=3

119885minus1

sum

119895=4

sum

119896

sum

119903

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

=

119885minus1

sum

119894=4

sum

119895=119885

sum

119896

sum

119903

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

forall119905

(11)




0and generates a






AP =

1 ifΛ lt 0

exp minusΛΩ

otherwise(12)

where Λ = 119891(1199041015840) minus 119891(119904






sum120578119894119905 119886119905119908




119905119908= 0 and there exists 119873

[119886119905119908]119905

= 1119872119897[119886119905119908]= 119878119871119867 and119872119908









isin 0 1) If 119886119905119908

= 0then 119890



119890119905119908

=

120573[119886119905119908]119905isin 0 1 119886

119905119908gt 0

1 119886119905119908

= 0

(13)


[119892119891ℎ119905

]119865times119867times119879


119891ℎ119905is the


119891ℎ119905= 0 the





A =

1 3 2 4 0

2 1 3 0 4

1 0 2 4 3

1 2 0 3 4lfloor


lfloor


(a)

E =

1 0 0 0 1

1 0 1 1 0

1 1 0 1 0

1 0 1 0 1lfloor


lfloor


(b)

2nd period

1st row

1st column

2nd column

3rd column

1st period

3rd period

4th period

2nd row

1 1 02 4 3

2 1 13 0 4

1 1 32 4 0

21 10 3 4

Matrix G

(c)




if (119886119905119908

= 0) 119892119891ℎ119905

= 119886119905119908

if (ℎ le 119867) ℎ = ℎ + 1else ℎ = 1 119891 = 119891 + 1

else

120585 = 120578[119886119905119908]119905

while (120585 = 0) do

120585 = 120585 minus 1119892119891ℎ119905

= 119886119905119908

if (ℎ le 119867) ℎ = ℎ + 1else ℎ = 1 119891 = 119891 + 1





119892119891119867119905

= 119892(119891+1)1119905

forall119891 = 1 2 119865 minus 1 (14)



120587119894ℎ119891119905

=


119891



















with 119886119905(119908+119898)




= 1 minus 119890119905119908




0



1015840) minus 119891(119904)



(olminus1)) 120590 = 120590 + 1else 120590 = 0






5 An Example




119896119903119905isin [0 1] and sum

119903119901119896119903119905



119894119896sim Γ(1 120582) where 120582 isin [008 010]



119878119897 = 85m 119878119908 = 20m119865 = 2 119867 = 16

119888119894= 8 hoursday 120603

119868= 95

1198611= 1198612= 1198613= 1198614= 1000 120579

1= 1205793= 1205794= 1205795= 1 120579

2= 5

120575119894= 100 forall119894 isin 1 2 16 120603 = 80

1205951119896= 1205952119896= 15min 120595

3119896= 5min 120595

16119896= 20min

1198721198971= 1198721198972= 11987211989716=

11987211989714= 11987211989712= 1198721198976=

1198721198978= 11987211989710= 6m

119872119897119894= 3m forall119894 isin 4 5 6

1198721199081= 119872119908

2= 119872119908

16=

11987211990814= 119872119908

12= 3m

1198721199086= 119872119908

8= 119872119908

10= 5m

119872119908119894= 2m forall119894 isin 4 5 6



Tota

l cos

t

times104

14

12

10

8

6

4

2 4 6 8 10 12 14 16 18 20 22 24





C

A

B

D1 D2 D3

E1

E2

F1 S1 S2 S3 S4

T1

R1

T2

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1 S1 S2 S3 S4R1

T1

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1S1 S2 S3 S4

R1

R2

T1

L 1

L 2

L 3

H1 H2 I1 O1N1K2K1P1

M1 M2

G1G2

J1

Q1



C

A

B

D1 D2 D3

E1

E2

F1 K2

G1

S1 S2 S3N1

H1

J1

T1 O1 O3O2M1 I1 I2 Q1

K1 L 1 J2

P1P2

R1

T2













1206012






120601 plusmn 1199051minus1205722 (

119899 minus 1)

119878

radic119899

or 120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

le 120601 le120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

(16)






0018437

radic6

)

0229955 le

1206011minus 1206012

1206011

le 0268659

(17)


1minus 1206012)1206011gt 0 which




times105

16

15

14

13

12

11

1

09

08

07

06

Tota

l cos

t

0 200 400 600 800 1000 1200 1400 1600 1800 2000



6 Conclusion






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4

5

15

6

321

7 11

Guide ways

Leadscrews

Wormgears

Gears


Abrasion

Corrosion

Scrape marks

Worm

AbrasionMicrocracks

Spindles


machine tool






(15) Lathe machine


Severe corrosion

Abrasion


Abrasion

Distortion



tools

Disposal

Reusable components

8

10

12

1314

9

16

Com

pone

nt w

areh

ouse






119894119905lt 119873119894(119905minus1)


119894119905gt 119873

119894(119905minus1)










kept the same


321 Index



Main road

Main road

3rd row

2nd row

1st row

1stcolumn

Department

Processing cell

2ndcolumn

3rdcolumn

9thcolumn












(119872119897119894ge 119872119908

119894)






Γ119894119895119896119903


























119873119894119905=

sum119896sum119903119876119896119905sdot 119901119896119903119905

sdot 120595119894119896sdot 120572119894119896119903

120603119894sdot 119888119894sdot 119863119905

(1)







119871119894119905= [119872119897

119894sdot (1 minus 120573

119894119905) + 119872119908

119894sdot 120573119894119905] sdot 119873119894119905 (2)

120578119894119905= ceil [

119871119894119905

119878119897119867

] (3)






119909119894119905= (ℎ minus 05) sdot

119878119897

2119867

+ (120578119894119905minus 1) sdot

119878119897

2119867

119910119894119905=

119878119908

2119865

+ (119891 minus 1) sdot

119878119908

119865

forall120587119894ℎ119891119905

= 1

(4)

119889119894119895119905

=

10038161003816100381610038161003816119909119894119905minus 119909119895119905

10038161003816100381610038161003816+

10038161003816100381610038161003816119910119894119905minus 119910119895119905

10038161003816100381610038161003816 (5)



Δ119905= sum

119896

sum

119903

sum

119894

sum

119895

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

sdot 119889119894119895119905sdot 120579119896 (6)






Ω119905= sum

119894


120591119894119905=

0 if119909119894119905= 119909119894(119905minus1)

119910119894119905= 119910119894(119905minus1)


(7)







sum

119905=1

Δ119905+

119879

sum

119905=2

Ω119905

(8)

Subject to119879

sum

119905=2

sum

119894

120591119894119905sdot 120575119894le 119861119905

(9)

sum

119894

120578119894119905le 119865 sdot 119867 forall119905 (10)

sum

119894=3

119885minus1

sum

119895=4

sum

119896

sum

119903

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

=

119885minus1

sum

119894=4

sum

119895=119885

sum

119896

sum

119903

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

forall119905

(11)




0and generates a






AP =

1 ifΛ lt 0

exp minusΛΩ

otherwise(12)

where Λ = 119891(1199041015840) minus 119891(119904






sum120578119894119905 119886119905119908




119905119908= 0 and there exists 119873

[119886119905119908]119905

= 1119872119897[119886119905119908]= 119878119871119867 and119872119908









isin 0 1) If 119886119905119908

= 0then 119890



119890119905119908

=

120573[119886119905119908]119905isin 0 1 119886

119905119908gt 0

1 119886119905119908

= 0

(13)


[119892119891ℎ119905

]119865times119867times119879


119891ℎ119905is the


119891ℎ119905= 0 the





A =

1 3 2 4 0

2 1 3 0 4

1 0 2 4 3

1 2 0 3 4lfloor


lfloor


(a)

E =

1 0 0 0 1

1 0 1 1 0

1 1 0 1 0

1 0 1 0 1lfloor


lfloor


(b)

2nd period

1st row

1st column

2nd column

3rd column

1st period

3rd period

4th period

2nd row

1 1 02 4 3

2 1 13 0 4

1 1 32 4 0

21 10 3 4

Matrix G

(c)




if (119886119905119908

= 0) 119892119891ℎ119905

= 119886119905119908

if (ℎ le 119867) ℎ = ℎ + 1else ℎ = 1 119891 = 119891 + 1

else

120585 = 120578[119886119905119908]119905

while (120585 = 0) do

120585 = 120585 minus 1119892119891ℎ119905

= 119886119905119908

if (ℎ le 119867) ℎ = ℎ + 1else ℎ = 1 119891 = 119891 + 1





119892119891119867119905

= 119892(119891+1)1119905

forall119891 = 1 2 119865 minus 1 (14)



120587119894ℎ119891119905

=


119891



















with 119886119905(119908+119898)




= 1 minus 119890119905119908




0



1015840) minus 119891(119904)



(olminus1)) 120590 = 120590 + 1else 120590 = 0






5 An Example




119896119903119905isin [0 1] and sum

119903119901119896119903119905



119894119896sim Γ(1 120582) where 120582 isin [008 010]



119878119897 = 85m 119878119908 = 20m119865 = 2 119867 = 16

119888119894= 8 hoursday 120603

119868= 95

1198611= 1198612= 1198613= 1198614= 1000 120579

1= 1205793= 1205794= 1205795= 1 120579

2= 5

120575119894= 100 forall119894 isin 1 2 16 120603 = 80

1205951119896= 1205952119896= 15min 120595

3119896= 5min 120595

16119896= 20min

1198721198971= 1198721198972= 11987211989716=

11987211989714= 11987211989712= 1198721198976=

1198721198978= 11987211989710= 6m

119872119897119894= 3m forall119894 isin 4 5 6

1198721199081= 119872119908

2= 119872119908

16=

11987211990814= 119872119908

12= 3m

1198721199086= 119872119908

8= 119872119908

10= 5m

119872119908119894= 2m forall119894 isin 4 5 6



Tota

l cos

t

times104

14

12

10

8

6

4

2 4 6 8 10 12 14 16 18 20 22 24





C

A

B

D1 D2 D3

E1

E2

F1 S1 S2 S3 S4

T1

R1

T2

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1 S1 S2 S3 S4R1

T1

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1S1 S2 S3 S4

R1

R2

T1

L 1

L 2

L 3

H1 H2 I1 O1N1K2K1P1

M1 M2

G1G2

J1

Q1



C

A

B

D1 D2 D3

E1

E2

F1 K2

G1

S1 S2 S3N1

H1

J1

T1 O1 O3O2M1 I1 I2 Q1

K1 L 1 J2

P1P2

R1

T2













1206012






120601 plusmn 1199051minus1205722 (

119899 minus 1)

119878

radic119899

or 120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

le 120601 le120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

(16)






0018437

radic6

)

0229955 le

1206011minus 1206012

1206011

le 0268659

(17)


1minus 1206012)1206011gt 0 which




times105

16

15

14

13

12

11

1

09

08

07

06

Tota

l cos

t

0 200 400 600 800 1000 1200 1400 1600 1800 2000



6 Conclusion






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Main road

Main road

3rd row

2nd row

1st row

1stcolumn

Department

Processing cell

2ndcolumn

3rdcolumn

9thcolumn












(119872119897119894ge 119872119908

119894)






Γ119894119895119896119903


























119873119894119905=

sum119896sum119903119876119896119905sdot 119901119896119903119905

sdot 120595119894119896sdot 120572119894119896119903

120603119894sdot 119888119894sdot 119863119905

(1)







119871119894119905= [119872119897

119894sdot (1 minus 120573

119894119905) + 119872119908

119894sdot 120573119894119905] sdot 119873119894119905 (2)

120578119894119905= ceil [

119871119894119905

119878119897119867

] (3)






119909119894119905= (ℎ minus 05) sdot

119878119897

2119867

+ (120578119894119905minus 1) sdot

119878119897

2119867

119910119894119905=

119878119908

2119865

+ (119891 minus 1) sdot

119878119908

119865

forall120587119894ℎ119891119905

= 1

(4)

119889119894119895119905

=

10038161003816100381610038161003816119909119894119905minus 119909119895119905

10038161003816100381610038161003816+

10038161003816100381610038161003816119910119894119905minus 119910119895119905

10038161003816100381610038161003816 (5)



Δ119905= sum

119896

sum

119903

sum

119894

sum

119895

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

sdot 119889119894119895119905sdot 120579119896 (6)






Ω119905= sum

119894


120591119894119905=

0 if119909119894119905= 119909119894(119905minus1)

119910119894119905= 119910119894(119905minus1)


(7)







sum

119905=1

Δ119905+

119879

sum

119905=2

Ω119905

(8)

Subject to119879

sum

119905=2

sum

119894

120591119894119905sdot 120575119894le 119861119905

(9)

sum

119894

120578119894119905le 119865 sdot 119867 forall119905 (10)

sum

119894=3

119885minus1

sum

119895=4

sum

119896

sum

119903

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

=

119885minus1

sum

119894=4

sum

119895=119885

sum

119896

sum

119903

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

forall119905

(11)




0and generates a






AP =

1 ifΛ lt 0

exp minusΛΩ

otherwise(12)

where Λ = 119891(1199041015840) minus 119891(119904






sum120578119894119905 119886119905119908




119905119908= 0 and there exists 119873

[119886119905119908]119905

= 1119872119897[119886119905119908]= 119878119871119867 and119872119908









isin 0 1) If 119886119905119908

= 0then 119890



119890119905119908

=

120573[119886119905119908]119905isin 0 1 119886

119905119908gt 0

1 119886119905119908

= 0

(13)


[119892119891ℎ119905

]119865times119867times119879


119891ℎ119905is the


119891ℎ119905= 0 the





A =

1 3 2 4 0

2 1 3 0 4

1 0 2 4 3

1 2 0 3 4lfloor


lfloor


(a)

E =

1 0 0 0 1

1 0 1 1 0

1 1 0 1 0

1 0 1 0 1lfloor


lfloor


(b)

2nd period

1st row

1st column

2nd column

3rd column

1st period

3rd period

4th period

2nd row

1 1 02 4 3

2 1 13 0 4

1 1 32 4 0

21 10 3 4

Matrix G

(c)




if (119886119905119908

= 0) 119892119891ℎ119905

= 119886119905119908

if (ℎ le 119867) ℎ = ℎ + 1else ℎ = 1 119891 = 119891 + 1

else

120585 = 120578[119886119905119908]119905

while (120585 = 0) do

120585 = 120585 minus 1119892119891ℎ119905

= 119886119905119908

if (ℎ le 119867) ℎ = ℎ + 1else ℎ = 1 119891 = 119891 + 1





119892119891119867119905

= 119892(119891+1)1119905

forall119891 = 1 2 119865 minus 1 (14)



120587119894ℎ119891119905

=


119891



















with 119886119905(119908+119898)




= 1 minus 119890119905119908




0



1015840) minus 119891(119904)



(olminus1)) 120590 = 120590 + 1else 120590 = 0






5 An Example




119896119903119905isin [0 1] and sum

119903119901119896119903119905



119894119896sim Γ(1 120582) where 120582 isin [008 010]



119878119897 = 85m 119878119908 = 20m119865 = 2 119867 = 16

119888119894= 8 hoursday 120603

119868= 95

1198611= 1198612= 1198613= 1198614= 1000 120579

1= 1205793= 1205794= 1205795= 1 120579

2= 5

120575119894= 100 forall119894 isin 1 2 16 120603 = 80

1205951119896= 1205952119896= 15min 120595

3119896= 5min 120595

16119896= 20min

1198721198971= 1198721198972= 11987211989716=

11987211989714= 11987211989712= 1198721198976=

1198721198978= 11987211989710= 6m

119872119897119894= 3m forall119894 isin 4 5 6

1198721199081= 119872119908

2= 119872119908

16=

11987211990814= 119872119908

12= 3m

1198721199086= 119872119908

8= 119872119908

10= 5m

119872119908119894= 2m forall119894 isin 4 5 6



Tota

l cos

t

times104

14

12

10

8

6

4

2 4 6 8 10 12 14 16 18 20 22 24





C

A

B

D1 D2 D3

E1

E2

F1 S1 S2 S3 S4

T1

R1

T2

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1 S1 S2 S3 S4R1

T1

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1S1 S2 S3 S4

R1

R2

T1

L 1

L 2

L 3

H1 H2 I1 O1N1K2K1P1

M1 M2

G1G2

J1

Q1



C

A

B

D1 D2 D3

E1

E2

F1 K2

G1

S1 S2 S3N1

H1

J1

T1 O1 O3O2M1 I1 I2 Q1

K1 L 1 J2

P1P2

R1

T2













1206012






120601 plusmn 1199051minus1205722 (

119899 minus 1)

119878

radic119899

or 120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

le 120601 le120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

(16)






0018437

radic6

)

0229955 le

1206011minus 1206012

1206011

le 0268659

(17)


1minus 1206012)1206011gt 0 which




times105

16

15

14

13

12

11

1

09

08

07

06

Tota

l cos

t

0 200 400 600 800 1000 1200 1400 1600 1800 2000



6 Conclusion






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























119873119894119905=

sum119896sum119903119876119896119905sdot 119901119896119903119905

sdot 120595119894119896sdot 120572119894119896119903

120603119894sdot 119888119894sdot 119863119905

(1)







119871119894119905= [119872119897

119894sdot (1 minus 120573

119894119905) + 119872119908

119894sdot 120573119894119905] sdot 119873119894119905 (2)

120578119894119905= ceil [

119871119894119905

119878119897119867

] (3)






119909119894119905= (ℎ minus 05) sdot

119878119897

2119867

+ (120578119894119905minus 1) sdot

119878119897

2119867

119910119894119905=

119878119908

2119865

+ (119891 minus 1) sdot

119878119908

119865

forall120587119894ℎ119891119905

= 1

(4)

119889119894119895119905

=

10038161003816100381610038161003816119909119894119905minus 119909119895119905

10038161003816100381610038161003816+

10038161003816100381610038161003816119910119894119905minus 119910119895119905

10038161003816100381610038161003816 (5)



Δ119905= sum

119896

sum

119903

sum

119894

sum

119895

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

sdot 119889119894119895119905sdot 120579119896 (6)






Ω119905= sum

119894


120591119894119905=

0 if119909119894119905= 119909119894(119905minus1)

119910119894119905= 119910119894(119905minus1)


(7)







sum

119905=1

Δ119905+

119879

sum

119905=2

Ω119905

(8)

Subject to119879

sum

119905=2

sum

119894

120591119894119905sdot 120575119894le 119861119905

(9)

sum

119894

120578119894119905le 119865 sdot 119867 forall119905 (10)

sum

119894=3

119885minus1

sum

119895=4

sum

119896

sum

119903

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

=

119885minus1

sum

119894=4

sum

119895=119885

sum

119896

sum

119903

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

forall119905

(11)




0and generates a






AP =

1 ifΛ lt 0

exp minusΛΩ

otherwise(12)

where Λ = 119891(1199041015840) minus 119891(119904






sum120578119894119905 119886119905119908




119905119908= 0 and there exists 119873

[119886119905119908]119905

= 1119872119897[119886119905119908]= 119878119871119867 and119872119908









isin 0 1) If 119886119905119908

= 0then 119890



119890119905119908

=

120573[119886119905119908]119905isin 0 1 119886

119905119908gt 0

1 119886119905119908

= 0

(13)


[119892119891ℎ119905

]119865times119867times119879


119891ℎ119905is the


119891ℎ119905= 0 the





A =

1 3 2 4 0

2 1 3 0 4

1 0 2 4 3

1 2 0 3 4lfloor


lfloor


(a)

E =

1 0 0 0 1

1 0 1 1 0

1 1 0 1 0

1 0 1 0 1lfloor


lfloor


(b)

2nd period

1st row

1st column

2nd column

3rd column

1st period

3rd period

4th period

2nd row

1 1 02 4 3

2 1 13 0 4

1 1 32 4 0

21 10 3 4

Matrix G

(c)




if (119886119905119908

= 0) 119892119891ℎ119905

= 119886119905119908

if (ℎ le 119867) ℎ = ℎ + 1else ℎ = 1 119891 = 119891 + 1

else

120585 = 120578[119886119905119908]119905

while (120585 = 0) do

120585 = 120585 minus 1119892119891ℎ119905

= 119886119905119908

if (ℎ le 119867) ℎ = ℎ + 1else ℎ = 1 119891 = 119891 + 1





119892119891119867119905

= 119892(119891+1)1119905

forall119891 = 1 2 119865 minus 1 (14)



120587119894ℎ119891119905

=


119891



















with 119886119905(119908+119898)




= 1 minus 119890119905119908




0



1015840) minus 119891(119904)



(olminus1)) 120590 = 120590 + 1else 120590 = 0






5 An Example




119896119903119905isin [0 1] and sum

119903119901119896119903119905



119894119896sim Γ(1 120582) where 120582 isin [008 010]



119878119897 = 85m 119878119908 = 20m119865 = 2 119867 = 16

119888119894= 8 hoursday 120603

119868= 95

1198611= 1198612= 1198613= 1198614= 1000 120579

1= 1205793= 1205794= 1205795= 1 120579

2= 5

120575119894= 100 forall119894 isin 1 2 16 120603 = 80

1205951119896= 1205952119896= 15min 120595

3119896= 5min 120595

16119896= 20min

1198721198971= 1198721198972= 11987211989716=

11987211989714= 11987211989712= 1198721198976=

1198721198978= 11987211989710= 6m

119872119897119894= 3m forall119894 isin 4 5 6

1198721199081= 119872119908

2= 119872119908

16=

11987211990814= 119872119908

12= 3m

1198721199086= 119872119908

8= 119872119908

10= 5m

119872119908119894= 2m forall119894 isin 4 5 6



Tota

l cos

t

times104

14

12

10

8

6

4

2 4 6 8 10 12 14 16 18 20 22 24





C

A

B

D1 D2 D3

E1

E2

F1 S1 S2 S3 S4

T1

R1

T2

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1 S1 S2 S3 S4R1

T1

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1S1 S2 S3 S4

R1

R2

T1

L 1

L 2

L 3

H1 H2 I1 O1N1K2K1P1

M1 M2

G1G2

J1

Q1



C

A

B

D1 D2 D3

E1

E2

F1 K2

G1

S1 S2 S3N1

H1

J1

T1 O1 O3O2M1 I1 I2 Q1

K1 L 1 J2

P1P2

R1

T2













1206012






120601 plusmn 1199051minus1205722 (

119899 minus 1)

119878

radic119899

or 120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

le 120601 le120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

(16)






0018437

radic6

)

0229955 le

1206011minus 1206012

1206011

le 0268659

(17)


1minus 1206012)1206011gt 0 which




times105

16

15

14

13

12

11

1

09

08

07

06

Tota

l cos

t

0 200 400 600 800 1000 1200 1400 1600 1800 2000



6 Conclusion






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Ω119905= sum

119894


120591119894119905=

0 if119909119894119905= 119909119894(119905minus1)

119910119894119905= 119910119894(119905minus1)


(7)







sum

119905=1

Δ119905+

119879

sum

119905=2

Ω119905

(8)

Subject to119879

sum

119905=2

sum

119894

120591119894119905sdot 120575119894le 119861119905

(9)

sum

119894

120578119894119905le 119865 sdot 119867 forall119905 (10)

sum

119894=3

119885minus1

sum

119895=4

sum

119896

sum

119903

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

=

119885minus1

sum

119894=4

sum

119895=119885

sum

119896

sum

119903

119876119896119905sdot 119901119896119903119905

sdot Γ119894119895119896119903

forall119905

(11)




0and generates a






AP =

1 ifΛ lt 0

exp minusΛΩ

otherwise(12)

where Λ = 119891(1199041015840) minus 119891(119904






sum120578119894119905 119886119905119908




119905119908= 0 and there exists 119873

[119886119905119908]119905

= 1119872119897[119886119905119908]= 119878119871119867 and119872119908









isin 0 1) If 119886119905119908

= 0then 119890



119890119905119908

=

120573[119886119905119908]119905isin 0 1 119886

119905119908gt 0

1 119886119905119908

= 0

(13)


[119892119891ℎ119905

]119865times119867times119879


119891ℎ119905is the


119891ℎ119905= 0 the





A =

1 3 2 4 0

2 1 3 0 4

1 0 2 4 3

1 2 0 3 4lfloor


lfloor


(a)

E =

1 0 0 0 1

1 0 1 1 0

1 1 0 1 0

1 0 1 0 1lfloor


lfloor


(b)

2nd period

1st row

1st column

2nd column

3rd column

1st period

3rd period

4th period

2nd row

1 1 02 4 3

2 1 13 0 4

1 1 32 4 0

21 10 3 4

Matrix G

(c)




if (119886119905119908

= 0) 119892119891ℎ119905

= 119886119905119908

if (ℎ le 119867) ℎ = ℎ + 1else ℎ = 1 119891 = 119891 + 1

else

120585 = 120578[119886119905119908]119905

while (120585 = 0) do

120585 = 120585 minus 1119892119891ℎ119905

= 119886119905119908

if (ℎ le 119867) ℎ = ℎ + 1else ℎ = 1 119891 = 119891 + 1





119892119891119867119905

= 119892(119891+1)1119905

forall119891 = 1 2 119865 minus 1 (14)



120587119894ℎ119891119905

=


119891



















with 119886119905(119908+119898)




= 1 minus 119890119905119908




0



1015840) minus 119891(119904)



(olminus1)) 120590 = 120590 + 1else 120590 = 0






5 An Example




119896119903119905isin [0 1] and sum

119903119901119896119903119905



119894119896sim Γ(1 120582) where 120582 isin [008 010]



119878119897 = 85m 119878119908 = 20m119865 = 2 119867 = 16

119888119894= 8 hoursday 120603

119868= 95

1198611= 1198612= 1198613= 1198614= 1000 120579

1= 1205793= 1205794= 1205795= 1 120579

2= 5

120575119894= 100 forall119894 isin 1 2 16 120603 = 80

1205951119896= 1205952119896= 15min 120595

3119896= 5min 120595

16119896= 20min

1198721198971= 1198721198972= 11987211989716=

11987211989714= 11987211989712= 1198721198976=

1198721198978= 11987211989710= 6m

119872119897119894= 3m forall119894 isin 4 5 6

1198721199081= 119872119908

2= 119872119908

16=

11987211990814= 119872119908

12= 3m

1198721199086= 119872119908

8= 119872119908

10= 5m

119872119908119894= 2m forall119894 isin 4 5 6



Tota

l cos

t

times104

14

12

10

8

6

4

2 4 6 8 10 12 14 16 18 20 22 24





C

A

B

D1 D2 D3

E1

E2

F1 S1 S2 S3 S4

T1

R1

T2

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1 S1 S2 S3 S4R1

T1

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1S1 S2 S3 S4

R1

R2

T1

L 1

L 2

L 3

H1 H2 I1 O1N1K2K1P1

M1 M2

G1G2

J1

Q1



C

A

B

D1 D2 D3

E1

E2

F1 K2

G1

S1 S2 S3N1

H1

J1

T1 O1 O3O2M1 I1 I2 Q1

K1 L 1 J2

P1P2

R1

T2













1206012






120601 plusmn 1199051minus1205722 (

119899 minus 1)

119878

radic119899

or 120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

le 120601 le120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

(16)






0018437

radic6

)

0229955 le

1206011minus 1206012

1206011

le 0268659

(17)


1minus 1206012)1206011gt 0 which




times105

16

15

14

13

12

11

1

09

08

07

06

Tota

l cos

t

0 200 400 600 800 1000 1200 1400 1600 1800 2000



6 Conclusion






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A =

1 3 2 4 0

2 1 3 0 4

1 0 2 4 3

1 2 0 3 4lfloor


lfloor


(a)

E =

1 0 0 0 1

1 0 1 1 0

1 1 0 1 0

1 0 1 0 1lfloor


lfloor


(b)

2nd period

1st row

1st column

2nd column

3rd column

1st period

3rd period

4th period

2nd row

1 1 02 4 3

2 1 13 0 4

1 1 32 4 0

21 10 3 4

Matrix G

(c)




if (119886119905119908

= 0) 119892119891ℎ119905

= 119886119905119908

if (ℎ le 119867) ℎ = ℎ + 1else ℎ = 1 119891 = 119891 + 1

else

120585 = 120578[119886119905119908]119905

while (120585 = 0) do

120585 = 120585 minus 1119892119891ℎ119905

= 119886119905119908

if (ℎ le 119867) ℎ = ℎ + 1else ℎ = 1 119891 = 119891 + 1





119892119891119867119905

= 119892(119891+1)1119905

forall119891 = 1 2 119865 minus 1 (14)



120587119894ℎ119891119905

=


119891



















with 119886119905(119908+119898)




= 1 minus 119890119905119908




0



1015840) minus 119891(119904)



(olminus1)) 120590 = 120590 + 1else 120590 = 0






5 An Example




119896119903119905isin [0 1] and sum

119903119901119896119903119905



119894119896sim Γ(1 120582) where 120582 isin [008 010]



119878119897 = 85m 119878119908 = 20m119865 = 2 119867 = 16

119888119894= 8 hoursday 120603

119868= 95

1198611= 1198612= 1198613= 1198614= 1000 120579

1= 1205793= 1205794= 1205795= 1 120579

2= 5

120575119894= 100 forall119894 isin 1 2 16 120603 = 80

1205951119896= 1205952119896= 15min 120595

3119896= 5min 120595

16119896= 20min

1198721198971= 1198721198972= 11987211989716=

11987211989714= 11987211989712= 1198721198976=

1198721198978= 11987211989710= 6m

119872119897119894= 3m forall119894 isin 4 5 6

1198721199081= 119872119908

2= 119872119908

16=

11987211990814= 119872119908

12= 3m

1198721199086= 119872119908

8= 119872119908

10= 5m

119872119908119894= 2m forall119894 isin 4 5 6



Tota

l cos

t

times104

14

12

10

8

6

4

2 4 6 8 10 12 14 16 18 20 22 24





C

A

B

D1 D2 D3

E1

E2

F1 S1 S2 S3 S4

T1

R1

T2

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1 S1 S2 S3 S4R1

T1

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1S1 S2 S3 S4

R1

R2

T1

L 1

L 2

L 3

H1 H2 I1 O1N1K2K1P1

M1 M2

G1G2

J1

Q1



C

A

B

D1 D2 D3

E1

E2

F1 K2

G1

S1 S2 S3N1

H1

J1

T1 O1 O3O2M1 I1 I2 Q1

K1 L 1 J2

P1P2

R1

T2













1206012






120601 plusmn 1199051minus1205722 (

119899 minus 1)

119878

radic119899

or 120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

le 120601 le120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

(16)






0018437

radic6

)

0229955 le

1206011minus 1206012

1206011

le 0268659

(17)


1minus 1206012)1206011gt 0 which




times105

16

15

14

13

12

11

1

09

08

07

06

Tota

l cos

t

0 200 400 600 800 1000 1200 1400 1600 1800 2000



6 Conclusion






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0



1015840) minus 119891(119904)



(olminus1)) 120590 = 120590 + 1else 120590 = 0






5 An Example




119896119903119905isin [0 1] and sum

119903119901119896119903119905



119894119896sim Γ(1 120582) where 120582 isin [008 010]



119878119897 = 85m 119878119908 = 20m119865 = 2 119867 = 16

119888119894= 8 hoursday 120603

119868= 95

1198611= 1198612= 1198613= 1198614= 1000 120579

1= 1205793= 1205794= 1205795= 1 120579

2= 5

120575119894= 100 forall119894 isin 1 2 16 120603 = 80

1205951119896= 1205952119896= 15min 120595

3119896= 5min 120595

16119896= 20min

1198721198971= 1198721198972= 11987211989716=

11987211989714= 11987211989712= 1198721198976=

1198721198978= 11987211989710= 6m

119872119897119894= 3m forall119894 isin 4 5 6

1198721199081= 119872119908

2= 119872119908

16=

11987211990814= 119872119908

12= 3m

1198721199086= 119872119908

8= 119872119908

10= 5m

119872119908119894= 2m forall119894 isin 4 5 6



Tota

l cos

t

times104

14

12

10

8

6

4

2 4 6 8 10 12 14 16 18 20 22 24





C

A

B

D1 D2 D3

E1

E2

F1 S1 S2 S3 S4

T1

R1

T2

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1 S1 S2 S3 S4R1

T1

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1S1 S2 S3 S4

R1

R2

T1

L 1

L 2

L 3

H1 H2 I1 O1N1K2K1P1

M1 M2

G1G2

J1

Q1



C

A

B

D1 D2 D3

E1

E2

F1 K2

G1

S1 S2 S3N1

H1

J1

T1 O1 O3O2M1 I1 I2 Q1

K1 L 1 J2

P1P2

R1

T2













1206012






120601 plusmn 1199051minus1205722 (

119899 minus 1)

119878

radic119899

or 120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

le 120601 le120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

(16)






0018437

radic6

)

0229955 le

1206011minus 1206012

1206011

le 0268659

(17)


1minus 1206012)1206011gt 0 which




times105

16

15

14

13

12

11

1

09

08

07

06

Tota

l cos

t

0 200 400 600 800 1000 1200 1400 1600 1800 2000



6 Conclusion






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










C

A

B

D1 D2 D3

E1

E2

F1 S1 S2 S3 S4

T1

R1

T2

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1 S1 S2 S3 S4R1

T1

L 1

L 2

L 3

H1 H2 I1O1

N1

N2

K2K1P1

M1 M2

G1

G2

J1Q1


C

A

B

D1 D2 D3 D4

E1

E2

F1S1 S2 S3 S4

R1

R2

T1

L 1

L 2

L 3

H1 H2 I1 O1N1K2K1P1

M1 M2

G1G2

J1

Q1



C

A

B

D1 D2 D3

E1

E2

F1 K2

G1

S1 S2 S3N1

H1

J1

T1 O1 O3O2M1 I1 I2 Q1

K1 L 1 J2

P1P2

R1

T2













1206012






120601 plusmn 1199051minus1205722 (

119899 minus 1)

119878

radic119899

or 120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

le 120601 le120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

(16)






0018437

radic6

)

0229955 le

1206011minus 1206012

1206011

le 0268659

(17)


1minus 1206012)1206011gt 0 which




times105

16

15

14

13

12

11

1

09

08

07

06

Tota

l cos

t

0 200 400 600 800 1000 1200 1400 1600 1800 2000



6 Conclusion






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1206012






120601 plusmn 1199051minus1205722 (

119899 minus 1)

119878

radic119899

or 120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

le 120601 le120601 minus 1199051minus1205722 (

119899 minus 1)

119878

radic119899

(16)






0018437

radic6

)

0229955 le

1206011minus 1206012

1206011

le 0268659

(17)


1minus 1206012)1206011gt 0 which




times105

16

15

14

13

12

11

1

09

08

07

06

Tota

l cos

t

0 200 400 600 800 1000 1200 1400 1600 1800 2000



6 Conclusion






Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article an optimization method for the...

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