COMPARISON BETWEEN MIXED INTEGER PROGRAMMING WITH HEURISTIC METHOD FOR JOB SHOP SCHEDULING WITH SEPARABLE SEQUENCE-DEPENDENT SETUPS

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1 COMPARISON BETWEEN MIXED INTEGER PROGRAMMING WITH HEURISTIC (I Gede Agus Wdyadana) COMPARISON BETWEEN MIXED INTEGER PROGRAMMING WITH HEURISTIC METHOD FOR JOB SHOP SCHEDULING WITH SEPARABLE SEQUENCE-DEPENDENT SETUPS I Gede Agus Wdyadana Lecturer at the Department of Industral Engneerng, Faculty of Industral Technology Petra Chrstan Unversty ABSTRACT The decsons to choose approprate tools for solvng ndustral problems are not ust tools that acheve optmal soluton only but t should consder computaton tme too. One of ndustral problems that stll dffcult to acheve both crtera s schedulng problem. Ths paper dscuss comparson between mxed nteger programmng whch result optmal soluton and heurstc method to solve ob shop schedulng problem wth separable sequence-dependent setup. The problems are generated and the result shows that the heurstc methods stll cannot satsfy optmal soluton. Keywords: ob shop, sequence dependent setup, Mxed Integer Programmng, heurstc. 1. INTRODUCTION Today many researches are focused n schedulng problem because ths problems are occur practcally and stll dffcult to fnd some methods that are satsfy computaton tme and optmal soluton. The ob shop schedulng problem can be descrbed as the problem of assgnng a set of obs wth predetermned machne operaton sequences to a set of machnes so as to mnmze the tme requred to complete all obs n the systems nown as maespan. The sequence dependent setups tme means that setup of partcular ob n the partcular machne are depend on ob that have been processed before. The ob shop schedulng problem wth sequence dependent setups appears frequently n the real lfe stuatons, but manly because ts complexty, t has receved lttle attenton from researchers untl recently. Studes of schedulng problems nvolvng sequence dependent setup have mostly focused on sngle and parallel machne cases. For the ob shop schedulng problem wth sequence dependent setups, Zhou and Egbelu presents a heurstc procedure for mnmzng the maespan and Ovac and Uzsoy descrbe a heurstc, based on the shftng bottlenec procedure, for mnmzng the maxmum lateness obs. In ther paper, Cho and Kormaz bult a new heurstc method for ob shop schedulng wth separable sequence dependent setups and then compare ther heurstc method wth Zhou and Egbelu method and Earlest-Start-Tme-Frst wth shortestprocessng-tme te breang rule (EST/SPT) by generated some problems. The result of ther comparson shows that ther heurstc method s the most powerful n general. But they do not compare ther heurstc method wth mxed nteger programmng formulaton that guarantees best soluton. 9

2 JURNAL TEKNIK INDUSTRI VOL. 3, NO. 1, JUNI 2001: METHODS There are two methods that are used to solve Job Shop Schedulng wth Separable Sequence-Dependent Setups, and then they are compared accordng good soluton and computaton tme. 2.1 Mxed Integer Programmng Formulaton The ob shop-schedulng problem we consder has the followng usual assumptons. There are m machnes and n obs n the system. Each ob has a fxed machne (operaton) sequence σ = (σ 1, σ 2,, σ m ), where σ represents the th machne that the ob s proceed on. Each machne can process one ob at a tme and each ob can be processed by only one machne at a tme. Preempton of obs s not allowed and a separable, sequence-dependent setup s necessary for each ob before t can be processed on each machne. The followng notaton s used throughout the paper. Unless stated otherwse, ndces and wll denoted obs and except for the case when used as subscrpt of σ, ndces and h wll be used for denotng machnes. Also, n ths paper, we wll dstngush an operaton from a ob. A ob conssts of seres of operaton and an operaton (O ) s defned as the processng of ob on a partcular machne. n = number of obs n the system; m = number of machnes n the system; A = set of ordered pars of obs, {(,) = 1,,n, = 1,,n, }; σ = machne operaton sequence of ob, σ,=( σ 1,, σ m ) = 1,,n; O = operaton correspondng to ob on machne, =1..,m, = 1..,n; p = processng tme of operaton O, =1..,m, = 1..,n; S = requred setup tme for ob on machne, O, when ob on machne, O, s processed mmedately before O, = 1,,n, = 1,,n, ; t = completon tme of latest scheduled operaton on machne, = 1, m q = lower bound estmate of the remanng tme that ob has to stay n the system, = 1,,n; V = set of obs avalable for processng on machne at or after t, = 1,,m; V = unon of V; C max = maespan of a schedule; = scheduled completon tme of operaton O ( CT(O )), =1..,m, = 1..,n; t Wth the above notatons, Job Shop Schedulng wth Separable Sequence-Dependent Setups can be formulated as mxed nteger programmng problem as follows. Wthout loss of generalty, t s assumed that dummy ob wth zero processng tme and zero sequence-dependent setup tme s added on each machne,.e. S 0 = 0, for = 1, n, and p 0 = 0 for = 1,,m. Let x 1, = 0, f ob mmedatel yfollows ob onmachne or equvalent ly,o mmedatel yfollows O, O t her w se., 10

3 COMPARISON BETWEEN MIXED INTEGER PROGRAMMING WITH HEURISTIC (I Gede Agus Wdyadana) Then: Mnmze C max (1) Subect to t + p t h = 1,,m-1, = 1,,n; (2) ó h ó h + p m σ m ó h C max t σ = 1,,n; (3) t n n + p + S x t + (1 x )M = 1,,m, = 0,1,,n, (4) x = 1 = 1,,m, = 0,1,,n; (5) x = 1 = 1,,m, = 0,1,,n; (6) U 1 (, ) U; A U x x 2 U (n + 1) 1; = 1,..m, (7) {0,1},t 0. Constrant (2) and (3), whch also appear n a regular ob shop schedulng problem formulaton, enforce the specfed machne operaton sequence of each ob. Constrant (4) prevents operatons wthn each machne from overlappng. Wth ths constrant, ob mmedately followng ob on machne can start only after both the processng of ob and the setup for ob are completed. Ths constrant becomes redundant f there s no adacency between obs and. Observe that ob ndex starts form 1 rather than 0. By gnorng the case for = 0, constrant (4) assocated wth the last scheduled ob an each machne and the dummy ob s nullfed. Constrant (5) (8) defne the crcular permutatons of operaton on each machne. Constrant (5) pcs exactly one ob mmedately followng ob on machne and constrant (6) pcs exactly one ob mmedately precedng ob on machne. Constrant (7), whch s equvalent to the subtour elmnaton constrant n travelng salesman problem, guarantees the contnuty of the adacency between obs n each machne,.e. t prohbts the partal groupng of obs n each machne. It should be mentoned here that constraned (7) s redundant wth respect to constran (4) n the sense that ts removal from the formulaton does not affect the solutons of above mxed nteger program. However, the ncluson of constrant (7) n the formulaton provdes stronger mxed nteger programmng formulaton for equvalent related to travelng salesman problem. 2.2 Heurstc Procedure Three defntons for the descrpton of the heurstc are: 1. EST(O ) s defned as earlest possble process start tme of ob on machne. = 1,,n, = 1,,m. Let us assume wthout loss of generalty that the h th operaton of ob, Oσ s performed on machne, = σ. Then EST(O ) s computed as: h h (8) 11

4 JURNAL TEKNIK INDUSTRI VOL. 3, NO. 1, JUNI 2001: 9-17 max{ct(oó h ),t + S f } EST(O ) max{est(oó h 1 ) + póh 1,t + S f f Oσ f Oσ h 1 h 1 s scheduled s not scheduled where f, fnshng ts operaton O f at t, s the last scheduled ob on machne. It should be note that EST(O, ) ncludes both setup tme and dle tme and that t s dynamcally updated as t changes. 2. The tal of ob, denoted by q, represents a lower bound estmate of the remanng tme that the ob has to stay n the system. Ideally, ths estmate would nclude lower bounds of the forced dle tmes and setup tmes. In ths algorthm, q s estmated by a smple dfference between the lower bound estmate of the completon tme of ob, EST ( σ m ) + p σ m, and a proected completon tme of operaton O. Thus f ob s avalable for processng on machne, then ts tal s computed as q = EST( σm ) + pσm ( EST ( O ) + p ) Intally, the tal of ob s computed as q = = p Mae use of n ths heurstc s the mnmum local maespan of a par of operatons, whch s defned as the mnmum maespan obtaned by solvng a two-ob/m machne problem wth release tme, where the two obs are those assocated wth the par of operatons under consderaton. Instead of fndng the mnmum local maespan of a par of operatons, they use the lower bound on t. For two operatons (obs) on the same machne, the lower bound on the mnmum local maespan s computed as, for, and, V EST ( O ) + p + q, LB( O, O ) max EST ( O ) + p + q, (11) EST ( O ) + p + q + S + p + q, then the lower bound on the completon tme of ob at the state of system (t 1,,t m ),, = 1,,m, can be obtaned as V LB( O ) = mn { LB( O, O )} V The lower bound n equaton (3) represents an optmstc completon tme of ob f operaton O s followed by operaton O. The lower bound n equaton (4) s the best optmstc completon tme of ob at a gven state of the system (t 1,,t m ). In each teraton of algorthm below, they choose an operaton from two canddate operatons: one wth the earlest start tme and the other wth mnmum LB(O ) at a gven state of the system, (t 1,,t m ). Algorthm SDJSS Inputs: m, n, σ, p, S Outputs: CT(O ) (=t ) STEP 1. t 1 = 0, EST(O ) = 0, and CT(O ) = 0 for = 1,,m, = 1, n. STEP 2. For all O not yet scheduled, compute EST(O ) usng equaton (9). m (9) (10) (12) 12

5 COMPARISON BETWEEN MIXED INTEGER PROGRAMMING WITH HEURISTIC (I Gede Agus Wdyadana) For all V, I = 1,,m, compute q I usng equaton (10). STEP 3. Let Q 1 = {(,) EST(O ) EST(O h ), V, V h,, h = 1,,m}, R 1 = {(,) Q 1 p p h, (h,) Q 1 }. Choose an operaton O 1 1 such that ( 1, 1 ) R 1. STEP 4. Compute LB(O ) usng equaton (12), for V and = 1,,m. Let P = {(,) LB(O ) LB(O h ), V, V h,, h = 1,,m}, Q 2 = {(,) P EST(O ) EST(O h ), (h,) P}, R 2 = {(,) Q 2 p p h, (h,) Q 2 }. Choose an operaton O 2 2 such that ( 2, 2 ) R 2. STEP 5. If 1 = 2, then 2, 2 ; Else (f 1 2 ), 1, 1 STEP 6. t EST ( O ) + p ; CT O t ; V ( ) { }, V V { }, where s the ndex of machne on V whch the ob s processed mmedately after machne. If unscheduled operatons reman, go to step COMPARISON Ths paper only compare two problem due to tme restrctons, both of the problems have 3 machnes and 5 obs wth same operaton. 3.1 Frst Problem The frst problem be generated wth randomze processng tme and setup tmes have a lower lmt of 1 and upper lmt of 5. Detal of the problem s: Table 1. Processng Tme Job () Machne seq. (s) Proc. Tme (p) Setup Tmes (S ) Table 2. Machne

6 JURNAL TEKNIK INDUSTRI VOL. 3, NO. 1, JUNI 2001: 9-17 Table 3. Machne Table 4. Machne The optmal soluton s calculated usng softare LINGO and run n computer wth Pentum II/400 specfcaton. The result s: Maespan : 33 No. of teratons : Elapsed tme : 13 mnutes, 39 seconds Table 5. Schedule Machne Job Sequences Machne Job Start setup Fnsh proc Machne Job Start setup Fnsh proc tme tme tme tme

7 COMPARISON BETWEEN MIXED INTEGER PROGRAMMING WITH HEURISTIC (I Gede Agus Wdyadana) The maespan of heurstc methods are 34 wth sequences: Table 6. Job Sequences Machne Job Sequences The frst problem show that the result from mxed nteger programmng s better one unt tme than heurstc method. 3.2 Second Problem In the second problem, setup tme s greater than processng tme. Ths stuaton s common n real condton for ths problem; so the second problem s generated randomly usng spreadsheet wth lower lmt of 1 and an upper lmt of 5 for processng tme and lower lmt of 11 and upper lmt of 31 for setup tme. Table 7. The sequence dan processng tme Job () Machne seq. (s ) Proc. Tme (p) Setup Tmes (S ) Table 8. Machne Table 9. Machne

8 JURNAL TEKNIK INDUSTRI VOL. 3, NO. 1, JUNI 2001: 9-17 Table 10. Machne The optmal soluton from mxed nteger programmng model s: Maespan : 115 No. of teratons : Elapsed tme : 19 mnutes, 54 seconds Table 11. Schedule Machne Job Sequences Machne Job Start setup Fnsh proc Machne Job Start setup Fnsh proc tme tme tme tme The maespan of heurstc methods are 125 wth sequences: Table 12. Job Sequences Machne Job Sequences It can be seen from the result that maespan from heurstc method are 125, so mxed nteger programmng n ths case are better 10 unt tmes or 8% than heurstc method. 16

9 COMPARISON BETWEEN MIXED INTEGER PROGRAMMING WITH HEURISTIC (I Gede Agus Wdyadana) 4. CONCLUSION Calculaton result from two case show that mxed nteger programmng gve better result than heurstc method whch be buld by Cho & Kormaz although ther method are better than other heurstc method that showed n ther paper. It can be seen that wth same problem sze and operaton, but wth wder devaton and greater setup tme, performance of heurstc method become worst compare wth mxed nteger-programmng method. Although mxed nteger programmng method wll gve optmal result for ob shop schedulng wth separable sequence-dependent setups, but computaton tme ncrease fast when problem sze are bgger, and there s possblty that software can t solve the problem because of lmtaton capacty. The result of two cases s not enough to mae satsfy concluson, so t s better to mae more cases by generate t wth varaton n problem sze, operaton route, set up and operaton tme. Because performances of heurstc methods n ths problem have bg possblty far from optmal soluton, so there s area for research to fnd another heurstc method, whch can gve better performance. Research n meta-heurstc method le genetc algorthm, smulated annealng and tabu search maybe can gve better result. REFERENCES Murty Katta G., Operatons Research Determnstc Optmzaton Models, Prentce Hall: Upper Saddle Rver, New Jersey. Pnendo Mchael, Schedulng Theory, Adgorthms and System, Prentce hall: Eaglewood Clffs New Jersey. LINGO, User s gude, Lndo Systems Inc. 17