FLOWSHOP SCHEDULING USING A NETWORK APPROACH

Transcription

1 ,*, A. M. H. Oladeide 1,*, A. I. Momodu 2 ad C. A. Oladeide 3 1, 2, 3DEPARTMENT OF PRODUCTION ENGINEERING, FACULTY OF ENGINEERING, UNIVERSITY OF BENIN, NIGERIA. addresses: 1 moladeide@uibe.edu, 2 oloagbaye.momodu@uibe.edu, 3 kallayama@yahoo.com ABSTRACT I this paper, a etwork based formulatio of a permutatio flow shop problem is preseted. Two uaces of flow shop problems with differet levels of complexity are solved usig differet approaches to the liear programmig formulatio. Key flow shop parameters iclosig makespa of the flow shop problems were obtaied without recourse to the traditioal approach of usig Gatt charts. The liear programmig models of the flow shop problems cosidered were solved usig LINGO 7.0. The preset techique has bee show to be very effective ad efficiet. Keywords: Flow shop, etwork, liear programmig, makespa, Gatt Chart, LINGO Nigeria Joural of Techology (NIJOTECH) Vol. 34 No. 2, April 2015, pp Copyright Faculty of Egieerig, Uiversity of Nigeria, Nsukka, ISSN: INTRODUCTION The traditioal flow shop schedulig problem, i which a fiite umber of jobs require processig o a specified umber of machies, is a fudametal productio schedulig research problem. Each uique job cosists of a umber of tasks ad is processed by routig it through each machie i a prescribed order. The objective is to fid a permutatio of jobs which miimizes some objective of iterest. The most frequetly studied objective is makespa, which is the time whe the last job completes o the last machie. The objective may be machie related or job related. Miimizatio of the overall completio times of all the jobs i the flow shop is the most commo objective documeted i literature. A complete survey of the flow shop schedulig problems has bee reported by [1]. Miimizatios of both tardiess ad flow time of jobs have also bee reported i literature. The flow shop schedulig problem has bee show to be NP-hard by [2]. A umber of uaces of the flow shop schedulig problem exist. I some istaces, all the jobs i the flow shop are processed through the machie i the same order. This costitutes a subclass of problems kow as permutatio flow shop problem. From a applicatio stad poit, oe or more jobs i a permutatio flow shop schedulig problem may ot require processig i oe or more machies resultig i what is kow as a bypass flow shop schedulig problem. A hybrid flow shop is differet from the traditioal flow shop i the sese that it cosists of several maufacturig stages with several urelated parallel machies istead of a sequece of sigle machies [3].A extesive review o research carried out o Hybrid Flow Shop has bee preseted by [4]. A umber of techiques [5] have bee proposed to obtai ear optimal solutios to flow shop schedulig problems. They iclude geetic algorithm ([6], [7]), At Coloy Optimizatio [8], NEH algorithm [9], CDS algorithm [10], Simulated Aealig [11], ad Tabu Search [12]. The differet exact techiques ad heuristics for flow shop schedulig ca oly be used to fid the sequece of the jobs which optimizes the objective of iterest but does ot provide the value of the objective. The Gatt chart is commoly used for obtaiig the objective value after the optimal sequece has bee obtaied. A Gatt chart is a bar chart that is a visual represetatio of the sequecig ad duratio of activities o ay give project [13]. The use of Gatt charts for presetig the start ad fiish time of jobs o differet machies as well as the completio time of all jobs ca become complicated to develop ad uderstad whe the umber of jobs ad machies icreases. To address this challege, a etwork represetatio of a flow shop schedulig problem is developed to facilitate the determiatio of the makespa of a permutatio flow shop. The developed etwork represetatio of the flow shop is solved usig iteger liear programmig approach. * Correspodig Author, Tel: +234 (0)

2 2. FLOW SHOP A flow shop schedulig problem ivolves schedulig jobs o m machies. A job cosists of m tasks or operatios ad the processed o the th j operatio of each job must be th j machie. A job ca oly be processed o machie j if it has bee completely processed o machie j 1 ad machie j is idle. A job j i ca be viewed as a set of tasks, with each of the tasks carried out o a specific machie. The time it takes to process a job o a machie may be costat, o-egative or eve zero. I the case where the processig time of a job o a machie is zero, it implies that the job is ot processed o the machie i questio. A umber of assumptios are made i flow shop schedulig. All jobs are ready to be processed at time zero, each job has a predetermied operatio time o all machies, machies ca be kept idle, always available ad ca process oly oe operatio at a time, jobs are o preemptable ad set up times of jobs o the machies are sequece idepedet ad the set up times as well as trasportatio times of a job are icluded i the processig time for jobs.cosider the followig sequece of jobs i a flow shop: j { j, j,... j } = ad p ij = processig i i1 i 2 im time of task i o machie j. Accordig to the omeclature, the permutatio flow shop is depicted as F / Permu / C, where the objective to be max miimized is the makespa. 3. CONVERTING A FLOW SHOP TO A CPM NETWORK I geeral, the procedure for geeratig the precedece relatioships amogst the differet tasks of the jobs ca be summarized below. Cosider a flow shop with jobs ad m machies. A job i cosists of m tasks, amely, J i1, J i2,..j im. Let the positio the job occupies be represeted by a vector P =(1,2,3,.). Let m represet the umber of machies i the flow shop, which is equal to the umber of tasks required to completely process a job: 1. The first task for the first Job (Job 1) o machie 1 has o predecessor. 2. The first task for the first Job(Job 1) o machie m>1 is preceded by the first task of the first Job (Job 1) o machie m-1 or the (m-1)th machie. 3. The first task for every Job >1 o machie m=1 is preceded by the first task o Job Every other task of a Job >1 for machie m>1 has two predecessors the task of Job (-1) o machie m ad the task of Job o machie m LINEAR PROGRAMMING Liear programmig is a optimizatio techique. It is used i Operatios research to fid optimal values of decisio variables for problems which ca be represeted i form of objective fuctio(s) ad costraits. The liear programmig techique ca be used to fid the logest path i the CPM etwork formulated from the flow shop problem. The logest path correspods to the critical path i the etwork. Two approaches are available for critical path aalysis of project etworks. The choice of the formulatio approach depeds o the compoet(s) of the output of the solutio that is/are of iterest to us. However, i the preset paper, we preset the two methods of formulatio ad illustrate each with a umerical example. 5. LONGEST PATH FORMULATION Assume that a uit flow eters a project etwork at the source (start ode) ad leaves at the sik (fiish ode) of the project etwork. Give that there are odes i the etwork, we adopt the followig otatio. x pq= A biary variable which represets the amout of flow through a path betwee odes p ad q i a project etwork. If the value of the variable assumes 1, it implies that there is a uit flow through the path ad 0 otherwise. The duratio of task betwee odes p ad q is give as t pq The logest path of the etwork ca be foud usig the followig iteger LP formulatio Max = pq p= 1 q= 1 Subject to x1 q= 1 pq q= 1 k = 1 k= 1 t x q pq = 1 x = x p = 2,..., 1 x k x pq = = 1 0 or 1 kp (1) (2) (3) (4) I the liear programmig model described i Eqs 1 4, Eq (1) esures that the uit flow goes through the logest path i the etwork; Eq (2) guaratees that Nigeria Joural of Techology, Vol. 34, No. 1, Jauary

3 oly a uit flow eters the etwork through the source (startig ode); Eq (3) forces coservatio of flow at iteral ode i the project etwork. Specifically, it esures that a uit of flow goes ito a ode ad a uit of flow leaves the ode. 6. LINEAR PROGRAM MODEL Cosider a project etwork with activity time t ij where the vertices ( V ) represet the set of ode umbers ad the direct edges represet the activities. A activity is represeted by oe ad oly oe arrow with its tail evet at ode iad its head evet at ode j ( i, j V ) where i < j.. Let x jdeote the earliest time that the evet correspodig to j occurs. Sice, i < j, it implies that for each pair of directed edge (, ) i j, x j xi + tij. Assumig the project has odes, our objective is to miimize the time required to complete the project. The liear program to fid the earliest evet times ad the project completio time ca be stated formally as: Miimize x x 1 (5) Subject to x j xi tij (6) x, x 0 i V i j + ( i, j) A I the liear program, A ad V represet the set of directed edges ad odes respectively. I the liear programmig model preseted, the critical activities correspod to the costraits which are satisfied as equatios at the optimal solutio. 7. NUMERICAL SOLUTION I this sectio, the applicatios of the two approaches for schedulig flow shop problems are preseted. I the first example, our iterest is to fid the makespa of the flow shop whose optimal sequece has bee obtaied a priori. The secod example illustrates the case where we are also iterested i the start, fiish, job waitig ad machie idle times i additio to the makespa of the flow shop problem. 8 NUMERICAL EXAMPLE 1 I the followig 7 job 2 machie problems, the processig times (hours) for each job o each machie is preseted i Table 1: Table 1: Jobs ad processig times of example 1 Job A B C D E F G Machie Machie The optimal sequece which miimizes the makespa of the flow shop problem is A-D-E-C-B-G-F. The optimal sequece is obtaied usig Johso s rule. Johso s rule gives the optimal sequece but ot the makespa of the flow shop problem. Usig the methodology described previously, the followig etwork is developed from the flow shop problem. J D1 J A1 J E1 J C1 J B J G1 J F1 7 8 J A2 J D2 J E2 J C2 J B2 J G2 J F Figure 1: Network diagram of flow shop problem The liear program i LINGO is show below: (Objective fuctio) max=6*x12+12*x23+20*x34+30*x45+24*x56+18*x67+22*x78+16*x29+13*x *x *x *x1617+6*x1819+2*x2021+0*(x310+x910+x1112+x412+x1314+x514+x1516+x616+x1718+x718+x192 0+x820); (etwork costraits) x12=1;x12=x23+x29; x23=x34+x310; x29=x910; x910+x310=x1011; x34=x45+x412; x1011=x1112; x1112+x412=x1213; x45=x56+x514; x1213=x1314; x1314+x514=x1415; x56=x67+x616; x1415=x1516; x1516+x616=x1617; x67=x78+x718; x1617=x1718; Nigeria Joural of Techology, Vol. 34, No. 1, Jauary

4 x1718+x718=x1819; x78=x820; x1819=x1920; x1920+x820=x2021; x2021=1; (biary The solutio to the etwork problem is show i Table 2 Table 2: Solutio to example 1 Variable Value X12 1 X23 1 X34 1 X45 1 X56 1 X67 1 X78 1 X29 0 X X X X X X X310 0 X910 0 X X412 0 X X514 0 X X616 0 X X718 0 X X820 1 The objective fuctio value is equal to NUMERICAL EXAMPLE 2 Cosider the followig five jobs three machies flow shop schedulig problem. The optimal sequece which miimizes the makespa has bee foud usig Johso s rule ad is ADEBC. The task is to determie the makespa of the flow shop Table 3: Jobs ad processig times of example 2 Job A B C D E Machie Machie Machie The flow shop problem with the sequece ADEBC is equivalet to the etwork model show i Figure 2.0. The project etwork data for the umerical example is preseted i Table 4. Table 4: Project data for etwork model of flow shop Node Activity(job) Predecessor Duratio 1-2 J A J A2 J A J D1 J A J A3 J A J D2 J A2, J D J E1 J D J E2 J E1, J D J D3 J D2, J A J B1 J E J B2 J B1, J E J E3 J E2, J D J C1 J B J C2 J C1, J B J B3 J B2, J E J C3 J C2, J B3 6 The etwork model of the flow shop problem is show i Figure 2. The jobs i the flow shop i the respective machies have bee represeted as activities o the project etwork. A activity is deoted by a directed lie betwee two odes. The activities have bee carefully ordered i the etwork diagram to retai the techological costraits. The uses of dummy activities have bee used to preserve the precedece requiremets amogst the differet activities i the project etwork. Dummy activities have bee deoted by directed dashed lies. Based o the developed etwork diagram, a liear programmig model was formulated for the flow shop problem usig the method adopted by [14]. The decisio variables used i the formulatio of the liear program are the start ad fiish times of the activities (jobs o machies). The liear program is preseted here uder i Figure 2. The liear program was ru usig LINGO 7.0. A global optimum result was obtaied alog with the associated values of the decisio variables. The output of LINGO 7.0 for the twety (20) decisio variables is preseted i Table 5. Nigeria Joural of Techology, Vol. 34, No. 1, Jauary

5 J A1 1 2 J D1 4 J E1 6 J B1 12 J C1 16 J A2 3 J D J E2 J B2 14 J C J A3 8 J D3 11 J E3 15 J B3 J C Figure 2: Network represetatio of flow shop problem (objective fuctio) MIN =T20; (etwork costraits) T2-T1>=4; T4-T2>=6; T6-T4>5; T12-T6>=9; T16-T12>=8; T3-T2>=5; T5-T4>=0; T9-T6>=0; T13-T12>=0; T17-T16>=0; T5-T3>=0; T7-T5>=3; T9-T7>=0; T10-T9>=4; T13-T10>=0; T14-T13>=6; T17-T14>=0; T18-T17>=2; T8-T3>=8; T11-T8>=7; T15-T11>=11; T19-T15>=10; T20-T19>=6; T8-T7>=0; T11-T10>=0; T15-T14>=0; T19-T18>=0; Table 5: solutio to example 2 Variable Value T20 51 T2 4 T1 0 T4 10 T6 15 T12 24 T16 32 T3 9 T5 10 T9 15 T13 24 T17 32 T7 13 T10 19 T14 30 T18 34 T8 17 T11 24 T15 35 T19 45 The result of the flow shop schedulig problem is summarized i Table 6. The Table shows the start, fiish ad the waitig times of the jobs o each machie i the flow shop problem. Table 6: Summary of result of umerical example 2 Job Machie Job Start Fiish waitig time A D Machie 1 E B C A D Machie 2 E B C A D Machie 3 E B C Machie Idle time 10. DISCUSSION OF RESULTS The result shows for umerical example I shows that a uit flow passes through the paths The variables represetig paths which coect the pair wise ode umbers assumed a value of uity from the solutio to the liear programmig model. The additio of the duratio of the tasks alog this path amouts to 134 hours. Therefore the makespa of the flow shop problem for the sequece ADECBGF is equal to 134 hours. Table 6 shows a summary of the result extracted from the LINGO 7.0 output for umerical example 2. The Nigeria Joural of Techology, Vol. 34, No. 1, Jauary

6 computatio reveals that the makespa is equal to 51 miutes. This is equal to the evet time of the termial ode i the project etwork. As expected for a permutatio flow shop problem, there is o job waitig time o the first machie i the flow shop. The start ad fiish times of the jobs o the machies shows that a total of 10 miute idle time is obtaied o machie 2 while o iserted idle times are icluded for the third machie. 11. CONCLUSION A etwork methodology has bee applied flow shop problem. Liear programmig has bee applied to the etwork equivalet of two uaces of flow shop problem. The method has facilitated the determiatio of key flow shop parameters without costructio of Gatt chart. The curret method is iheretly flexible as it allows chages i problem parameters to be hadled efficietly. 12. REFERENCES [1] Hejazi S. R. ad Saghafia S, Flow shop Schedulig problems with makespa criterio: A review, Iteratioal Joural of Productio Research, Vol. 43, No. 14, 2005, pp [2] Kise, H, O a automated two-machie Flowshop schedulig problem with ifiite buffer, J. Oper. Res. Soc. Japa 34, 1991, pp [3] Havill, J.T ad Mao,W, O-lie Algorithms for Hybrid Flow Shop Schedulig, Proceedigs of the Fourth Joit Coferece o Iformatio Scieces, Research Triagle Park, North Carolia, 1998, pp [4] Riuz,.R. ad Vazquez-Rodriguez, J.A. The Hybrid Flow Shop Schedulig Problem, Europea Joural of Operatioal Research, 205(1),2010, pp [5] Oyeagoro, E.O, Sequecig for Batch Productio i a Shop Flowlie Machie Shop, Nigeria Joural of Techology, Vo 6, Number 1, 1982, pp [6] Ruiz, R ad Maroto. C. A geetic algorithm for hybrid flow shops with sequece depedet setup times ad machie eligibility, Europea Joural of Operatioal Research 169 (3); 2006,pp [7] Reeves, C.R Geetic Algorithms, Path Relikig ad the Flowshop Sequecig Problem Evolutioary Computatio Joural, Vol.6 No.1, 1998, pp [8] Shyu, S.J ad Li, B.M.T ad Yi, P.Y Applicatio of at coloy optimizatio for o-wait flowshop schedulig problem to miimize the total completio time, Computers ad Idustrial Egieerig, Volume 47, issue 2-3, 2004, pp [9] Nawaz, M., Escore Jr, E. ad Ham, I., A heuristic algorithm for them-machie,-job flow-shop sequecig problem.it. J. Maag. Sci., 11, 1983, pp [10] Campbell, H.G., Dudek, R.A. ad Smith, M.L, A heuristic algorithm of the-job,m-machie sequecig problem, Maag. Sci., 16, 1970, pp [11] Osma, I.H ad Potts, C.N Simulated aealig for permutatio flow-shop schedulig, Omega vol. 17, issue 6, 1989, pp [12] Hooda N, ad Dhigra A. K Flow Shop Schedulig usig Simulated Aealig: A Review, Iteratioal Joural Of Applied Egieerig Research,Volume 2, No 1,2011, pp [13 Elmabrouk, O.M A Liear Programmig Techique for the Optimizatio of the Activities i Maiteace Projects, Iteratioal Joural of Egieerig ad Computig, vol 11, o 1, 2011, pp [14] Oladeide M.H ad Oladeide.C.A, Modelig Critical Path Aalysis usig liear programmig, Joural of Mathematics ad Techology, 3(2), 2012, pp Nigeria Joural of Techology, Vol. 34, No. 1, Jauary