Multi-Criteria Flow-Shop Scheduling Optimization

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1 Multi-Criteria Flow-Shop Schedulig Optimizatio A Seior Project Submitted I Partial Fulfillmet Of the Requiremets for the Degree of Bachelor of Sciece i Idustrial Egieerig Preseted to: The Faculty of Califoria Polytechic State Uiversity, Sa Luis Obispo By: Teja Arikati Project Advisors: Reza Pouraghabagher & Karla Caricher Graded By: Date of Submissio:. Checked by: Approved by:.

2 Ackowledgemets I would like to thak all my professors over the years for givig me the opportuity to grow ad lear at Cal Poly. I would persoally like to thak my project advisors Dr. Pouraghabagher ad Karla Caricher for advisig me durig this project. Also, my iterest i Operatios Research was stemmed from Dr. Freed s classes ad I would like to thak her for helpig me realize my passio for Operatios Research.

3 Table of Cotets Abstract... 1 Itroductio... 2 Backgroud/Literature Review... 3 Mixed Iteger Liear Programmig Formulatio... 6 Desig Requiremets... 9 Implemetatio Results Coclusio Refereces Appedix... 20

4 Abstract A flow-shop is a type of maufacturig job shop where similar jobs follow a similar, liear sequece through the shop. Every day, flow-shops receive several differet orders ad it is up to the scheduler to pla the daily schedule. This schedule should be desiged to prevet bottleecks i the shop, to have o-time delivery of products, ad satisfy several other requiremets. Ofte times, schedulers perform subjective schedulig ad utilize simple heuristics or just ituitio to schedule the jobs. With computer-based schedulig, schedulers ca ow create schedules ad determie quatitatively what sorts of schedules work best. Curretly, much of the computerbased schedules oly try to optimize for oe KPI such as Total Tardiess. This paper cosiders icorporatig multiple-criteria ito computer based schedulig so that schedulers ca have more flexibility ad develop schedules which optimize multiple-criteria; this paper specifically cosiders miimizig Total Tardiess ad maximizig Throughput. Comparisos betwee sigle-criterio models ad the multiple-criteria model are made ad it is discovered the multiple-criteria model provides a great compromise i optimizig both KPIs. A user-friedly program is developed where schedulers of ay flow-shop ca utilize the software to compute schedules for cases up to 10 jobs ad 10 machies. 1

5 Itroductio Flow-shops are maufacturig shops where similar products or jobs follow a specific set of processig steps ad the parts made are geerally quite similar. Also, all the jobs follow a liear order where they do t go ad revisit a previous machie for processig. This is differet from a job-shop where a job ca follow ay order amog the machies ad ca eve visit a machie multiple times. Job schedulig is a huge problem which every flow-shop compay faces. Whe orders come ito a flow-shop, the mai role of the scheduler is to schedule the jobs so that the jobs are built o-time ad as fast as possible. May flow-shops do this maually. Below is a example of a flowshop compared to a job-shop: Figure 1: Compariso betwee Flow-shop ad Job-Shop Sice the jobs follow a liear fashio i a flow-shop, the schedulig of jobs i flow-shops teds to be simpler tha for jobs i job-shops. For this project, the scope will be focused oly o flow-shop schedulig. 2

6 The goal of this project is to develop a geeral-purpose software applicatio which will provide a maufacturig compay with the optimal flow-shop schedule. This geeral-purpose software applicatio should work with ay flow-shop compay. What a optimal flow-shop schedule may vary day to day therefore a multi-criteria formulatio is created where schedulers ca set their prefereces for differet KPIs. A compariso betwee a sigle criterio-formulatio ad the multi-criteria formulatio will be coducted to determie the effectiveess of the ew model. Backgroud/Literature Review As stated above i the itroductio, flow-shop schedulig is a difficult veture which every flow-shop tries to tackle. The complexity of flow-shop schedulig falls i the NP hard classificatio (Hae 1994). What this meas is that it takes expoetially more effort to schedule jobs as more jobs are added. For this reaso, several heuristics ad other methods are used to tackle this problem. Literature for flow-shop schedulig exploded after the itroductio of a heuristic algorithm by S.M. Johso which is ow called Johso s Algorithm (Johso 1954). This algorithm uses a list of rules to develop a flow-shop schedule that optimizes for total elapsed time. For two machies, this algorithm provides a schedule with the miimum Total Elapsed time but for schedules with more tha two machies, this algorithm does t guaratee the best solutio. 70 years after the formulatio of the Johso s Algorithm, the depth of research ito flow-shop schedulig is lackluster. Oly over the last 20 years, with the explosio of computig power, have researchers started cosiderig more iovative ways of solvig this problem. Several differet heuristics have bee developed for flow- 3

7 shop schedulig ad these methods iclude geetic algorithms ad at-coloy optimizatio (Pezzella 2008). Most of the research, however, focuses o sigle-criterio optimizatio ad the research that does cover multi-criteria oly covers bi-criteria for machie couts less tha 3 (Dhigra 2010). The purpose of this paper is to fill i the research ad implemetatio gap for multi-criteria flow-shop schedulig. Lookig at flow-shop schedulig formulatios, several differet Mixed-Iteger Liear Programmig (MILP) formulatios are foud i literature. I the literature, three distict formulatios appear. These three differet formulatios described by Wager (Wager 1959), Wilso (Wilso 1989), ad Mae (Mae 1960) are listed uder Appedix A, B, C respectively. To quickly go over the three formulatios: 1. The first formulatio by Wager revolves aroud the relatioships betwee Idle Times, Wait Times, ad Processig times of cosecutive jobs betwee cosecutive machies. 2. The secod formulatio by Wilso revolves aroud the relatioships betwee Start Times ad Processig times of cosecutive jobs betwee cosecutive machies. 3. The third formulatio by Mae revolves aroud the relatioships betwee Completio Times, Processig times, ad precedece requiremets of cosecutive jobs betwee cosecutive machies. I a paper by Rocoi ad Birgi (Rocoi & Birgi 2012), the computatio times for all three methods were measured by solvig sample cases which varied i umber of jobs ad umber of machies. These formulatios were ru through a Simplex Solver which is the same algorithm used by Microsoft Excel s solver. The results showed 4

8 similar results betwee Wager s ad Wilso s formulatios with Wilso s formulatio beig slightly faster. Mae s formulatio was sigificatly slower ad eve ifeasible for large sample sizes (15 jobs ad 10 machies). The exact ru times are listed i Appedix D. Modelig multi-criteria objectives could be doe i several ways. Two methods will be discussed i this literature review. The first method is through applyig a weighted average to each objective directly. Each objective is icluded i the objective fuctio ad a weight from 0-1 is applied to each objective (Dhigra 2010). The sum of all the weights must equal 1. This methodology follows the same priciples as weighted averages ad other weighted calculatios. A disadvatage of this method is that it is difficult to gauge the actual weight give to each objective because of their differet rages. What is meat by this is that each of these objectives has values which lie i differet rages ad differet magitudes. For example: Tardiess could rage from 0 days to 20 hours, Total Elapsed Time from 100 hours to 200 hours, ad Total Flow Time from 300 hours to 500 hours. If equal weightig was applied to each of these objectives, Total Flow Time would have higher priority compared to TET or Tardiess because it has a larger rage tha the other two KPIs. To combat this, there must be a way to ormalize every objective so that the desired priority is applied to each objective. The way to ormalize objectives is through the applicatio of fuzzy set theory. Fuzzy set theory is a part of set theory ad was itroduced by Lotfi Zadeh (Zadeh 1965). The purpose of fuzzy set theory is to apply a cotiuous gradiet to geerally discrete costraits. This paper will be lookig at the ormalizatio part of fuzzy set theory. To ormalize the objectives, the rage of each of the objectives must be kow. 5

9 Usig the previous example, the geeral rage for tardiess is [0,20] hours, for TET is [100,200] hours, ad for TFT is [300,500] hours. To ormalize each objective, we would divide each objective by their total rage. By doig this, each objective is effectively trasformed to a value betwee 0 ad 1. Now, the weighted average method ca be utilized to combie these objectives ito a objective fuctio. A great example of fuzzy set theory applicatio i schedulig is give by Sima Roki (2010). The formulatio is show i Appedix E. Mixed Iteger Liear Programmig Formulatio Based o the literature review, 3 popular formulatios exist for flow-shop schedulig. 2 of these formulatios, the oes by Wager ad Wilso, are more efficiet ad feasible. Wager s ad Wilso s formulatios proved to perform quite similarly. Below is a table comparig both algorithms by listig the total umber of costraits, total biary variables, total cotiuous variables, complexity of code, ad computatioal ru-times. # of Costraits # of Biary Variables # of Cotiuous Variables Ease of Implemetatio Computatioal Ru-times Wager m m Medium-High Great Wilso 2m m 2 2m m Medium Great Table 1: Compariso betwee Wager s ad Wilso s Formulatio Although Wilso s formulatio would geerally have more cotiuous variables ad umber of costraits tha Wager s, the code complexity required to implemet Wilso s algorithm would be simpler tha Wager s. Sice the performace of both 6

10 algorithms is similar, the easier implemetatio of Wilso s formulatio was chose for the foudatio of the model preseted i this report. Variables & iputs of the followig model with their correspodig boud costraits are listed below: x ij is biary where x ij {0,1}, T j 0, C jm, S jk i = 1,,, j = 1,,, k = 1,, m, p ik, d i What each of these variables meas are as follows: x ij is equal to 1 if job i (there are umber of jobs ad these jobs are labelled betwee 1 ad ) is i the j-th positio of the sequece, equal to 0 otherwise. T j is the Tardiess of the j-th job i the sequece (remember here that this is ot related to i which is the origial label of the jobs, the j oly correspods to the sequece order). C jm correspods to the completio of the j-th job of the sequece at machie m (the last machie). S jk correspods to the start time of the j-th job at machie k. i is ordered from 1 to where is the umber of jobs i the flow-shop. j is ordered from 1 to where is the umber of jobs i the flow-shop (the legth of the sequece is the same as the umber of jobs). k is ordered from 1 to m where m is the umber of machies i the flow-shop. p ik is the processig time of job i at machie k. d i is the due date of job i. The followig formulatio is for the sigle-criterio flow-shop model which aims to miimize total Tardiess of the jobs: 7

11 Miimize T j subject to j=1 T j C jm x ij d i, for all j i=1 C jm = S jm + x ij p im, for all j i=1 S j+1,k S jk + x ij p ik, for all j except ad for all k i=1 S j,k+1 S jk + x ij p ik, for all j ad for all k except m S 11 0 i=1 x ij = 1, for all j i=1 x ij = 1, for all i j=1 The sigle-criterio optio for maximizig throughput requires just chagig the objective fuctio as follows: Miimize C jm j=1 The multi-criteria optio which optimizes for both throughput ad tardiess requires the additio of fuzzy costraits as discussed before durig the literature review. What this requires is the additio of two user iputs f α ad f β which are the ormalizig factors which will effectively covert two factors ito a rage betwee 0 ad 8

12 1. Also two more iputs idicatig importace must be icluded ad these iputs will be labelled as P α ad P β. The costraits will remai the same as before from the siglecostrait formulatio. The revised objective fuctio is as follows: Miimize ( P α C jm j=1 f α + P β T j f β ) Desig Requiremets Program Iput Requiremets Sice there is o specific customer i this project, the parameters ad costraits were set for ease ad practicality so that virtually ay flow-shop could use the program. The basic expected iputs from the flow-shop will be as follows. 1. Jobs & Machies A list of jobs that eed to be completed is provided. This should also iclude the umber of jobs ad the umber of machies i the flow-shop. 2. Maufacturig Ru Times: The maufacturig lead time for each of these jobs icludig wait time, queue time, setup time, ad ru time should be provided. However, a ru time for each job at each machie is sufficiet if other specific data is ot available. 3. KPIs or Goals to Achieve: The scheduler eeds to iput what KPIs or goals are to be optimized as well as a raked priority for each of these. For example, a scheduler could choose to miimize Tardiess ad maximize Throughput with raks of 1 ad 2 respectively. 9

13 This meas the program will try to achieve the Tardiess goal with a higher priority compared to the Throughput goal. User Iterface Requiremets: The program will have a clea ad ituitive layout which a tester ca easily iterface with. This will be preseted through a Pytho iteractive iterface. The scheduler ca the type the metioed iputs above through this applicatio. There will be a optio to ru radom jobs through a flow-shop or to iput actual processig times. After ruig the program, a clea ad clear output of the job schedule should be displayed. Aother key aspect to the User Iterface will be clear graphical outputs of the results. Bar graphs ad statistical comparisos will be displayed to show the differeces betwee various program desigs (sigle-criterio vs multi-criteria). Techical Aspects/Backed Desig Requiremets: This program will be built from groud up i Pytho. There will be two programs built; the first will be a implemetatio of the sigle-criterio flow-shop schedulig program ad the secod will be a implemetatio of the multi-criteria flow-shop schedulig program. Wilso s formulatio proved to be oe of the fastest computatioally ad the easiest to implemet therefore this formulatio will be used. For the multi-criteria desig, the use of fuzzy costraits will be icorporated so that the schedulers ca accurately assig priorities to differet KPIs. The optimizatio add-o, which will be the workhorse for computig the results, is Gurobi Optimizatio Versio 7. This software package is idustry leadig i terms of havig the fastest ru-times for 10

14 Mixed Iteger Liear Programmig Formulatios. A free academic licese was acquired for Gurobi. Implemetatio I this sectio, the back-ed code used to icorporate the costraits ad the objectives will be described. The etire code is i Appedix F. Followig this explaatio, a overview of the User Iterface will be described. Code Firstly, all the variables eed to be iitialized. The biary x ij variables, the cotiuous start time variables S jk, ad the cotiuous Tardiess variables T j are iitialized i the code below: Figure 2: Code for variable iitializatio Secodly, tardiess ad completio time costraits eed to be specified. These are the lies i the formulatio: T j C jm i=1 x ij d i, for all j; C jm = S jm + i=1 x ij p im, for all j. The code is listed below: Figure 3: Code for tardiess ad completio time costraits 11

15 Thirdly, the start time costraits eed to be specified. These are the lies i the formulatio: S j+1,k S jk + i=1 x ij p ik, for all j except ad for all k; S j,k+1 S jk + i=1 x ij p ik, for all j ad for all k except m; S The code is listed below: Figure 4: Code for start time costraits Fourthly, the biary costraits eed to be specified. These are the correspodig lies i the formulatio: i=1 x ij = 1, for all j ; x ij = 1, for all i. code is listed below: j=1 The Figure 5: Code for biary costraits 12

16 Lastly, the objective fuctios eed to be defied. These three objective fuctios are defied: Miimize j=1 T j ; Miimize j=1 C jm ; Miimize ( P α C jm j=1 f α + P β T j f β ). The code is listed below: Figure 6: Code for Objective fuctio defiitios User Iterface I the User Iterface, the user ca chage the umber of jobs to ay umber from 1 to 5 ad the umber of machies to ay umber from 1 to 5. There is also a choice to choose which sceario (objective) fuctio to ru. The first two scearios are the sigle-criterio objective fuctios while the third sceario is the multi-criteria objective fuctio. After this, the throughput ad total tardiess of the flow-shop is displayed. A graph displayig the differece i throughput ad tardiess of the chose sceario to the other two scearios is also visualized at the bottom. 13

17 Figure 7: Sample User Iterface Results To test the effectiveess of each of the scearios, a experimet was coducted comparig each sceario s tardiess ad throughput results. I this experimet, the flow-shop was scheduled to have 10 jobs with each of these jobs eedig to be processed at 10 machies. The ru times of each of these jobs at each of the machies was radomly created by the computer followig a uiform distributio betwee 4 ad 16. The due dates of each of these jobs were radomly created by the computer 14

18 followig a uiform distributio betwee 90 ad 200. The throughput ad tardiess results were recorded for each of the scearios. The experimet was ru 20 times by the computer ad all the data was stored ito Pytho arrays. The experimet took 10 miutes to complete o a Itel i5-6300u CPU. Based o this experimet, the average tardiess ad average throughput of each of the scearios are as follows: Average Tardiess Average Throughput Sigle-Criterio: Miimize Tardiess Sigle-Criterio: Maximize Throughput Multi-Criteria: Optimize Both Criteria Table 2: Average Tardiess ad Average Throughput of each Model Lookig at the data, it is ecessary to perform statistical aalyses to determie sigificace. To compare the values, oe-way ANOVAs were performed i Pytho to compare tardiess values ad throughput values. Coductig ANOVA to compare the tardiess values ad throughput values amog the programs resulted i a p-value of for tardiess ad a p-value of.0038 for throughput. This meas at a 95% cofidece level; the values of tardiess ad throughput are statistically differet. To visualize the data sets, bar graphs ad box-plots are preseted below comparig the tardiess ad throughput performace of each of the three models: 15

19 Figure 8: Bar graphs comparig Throughputs ad Tardiess of each Model Figure 9: Box plots comparig Throughputs ad Tardiess of each Model Based o the results, there is a 3.2% greater throughput with the multi-criteria model compared to the sigle-criterio tardiess model. Also there is a 49.3% improvemet i Total Tardiess with the multi-criteria model compared with the siglecriterio throughput model. The multi-criteria model, as would be expected, resulted i total tardiess ad throughput values i betwee correspodig tardiess ad throughput values i both sigle-criterio models. This ca be visually see i the bar 16

20 graphs ad the box-plots. Sample ru cases with differet machie ad job couts with all three differet scearios display similar results to the results foud with the 20 ru experimet coducted with 10 jobs ad 10 machies. Coclusios The results from the experimet prove that the multi-criteria program performs as plaed. The multi-criteria program proved to provide solutios which were compromises betwee the sigle-criterio models. The feasibility of the multi-criteria program was of cocer but all programs fiished withi 20 secods for eve 10 job ad 10 machie scearios. This program proves to become ifeasible as machie ad job couts go past 20 where ru times take a hour or more but eve the sigle-criterio program started to become ifeasible at these scales. Sice there was o idustry sposor, it was difficult to fid a way to do a ecoomic aalysis. If a idustry sposor was available for this project, comparisos of throughput ad tardiess betwee subjective schedulig ad computer-based schedulig could be made. Based o the time saved, a ecoomic aalysis could be coducted to see how much moey could be saved. Schedulig is a problem that flow-shops face o a daily problem. With various priorities, such as customer deadlies ad productio efficiecy, it ca be difficult to develop a schedule which ca satisfy every goal ad objective. As discussed i this paper, computer-based schedulig ca help schedulers create schedules that satisfy differet KPI requiremets. With the iclusio of multi-criteria objectives ad the developmet of a user-friedly iterface for schedulers to iteract with, the proposed program i this paper offers a simplistic tool that schedulers ca utilize to create more 17

21 optimum schedules. The requiremets each day may chage ad perhaps priorities for each differet KPI may chage as well. Havig low tardiess may be importat oe day while havig high throughput may be importat aother day. The ability to alter priorities gives schedulers higher flexibility ad flexibility is extremely importat i a flow-shop where requiremets ca chage at a momet s otice. 18

22 Refereces: 1. A. Dhigra, P. Chada, "Multi objective flow-shop schedulig usig hybrid simulated aealig", Measurig Busiess Excellece, Vol. 14 Iss: 3 (2010), pp C. Hae. "Study of a NP-hard cyclic schedulig problem: The recurret jobshop." Europea joural of operatioal research 72.1 (1994): S.M. Johso, Optimal two- ad three-stage productio schedules with setup times icluded, Naval Res. Log. Quart. I (1954) A. S. Mae, O the job-shop schedulig problem, Operatios Research 8 (1960), pp , 5. F. Pezzella, G. Morgati, G. Ciaschetti, A geetic algorithm for the Flexible Jobshop Schedulig Problem, Computers & Operatios Research, Volume 35, Issue 10 (2008), Pages S. Roki Optimizatio of idustrial shop schedulig usig simulatio ad fuzzy logic, Uiversity of Alberta M.S. Thesis (2010) 7. D. Rocoi, E. Birgi, Mixed-iteger programmig models for flowshop schedulig problems miimizig the total earliess ad tardiess, Just-i-Time Systems (2012), pp H. M. Wager, A iteger liear-programmig model for machie schedulig, Naval Research Logistic 6 (1959), pp , 9. J. M. Wilso, Alterative formulatios of a flow-shop schedulig problem, Joural of the Operatioal Research Society 40 (1989), pp , 10. L. A. Zadeh. Fuzzy sets. Iform. Cotr. 8 (1965):338-53, 19

23 Appedix A. Wager s MILP Formulatio for Flow-Shop Schedulig 20

24 B. Wilso s MILP Formulatio for Flow-Shop Schedulig C. Mae s MILP Formulatio for Flow-Shop Schedulig 21

25 D. Ru Times for each formulatio. (MUB1,MUB2,MUB3 are Wager, Wilso, ad Mae Respectively) E. Fuzzy Costrait Formulatio 22

26 F. Complete Back-Ed Code 23

27 24

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