PARETO-OPTIMAL SOLUTION OF A SCHEDULING PROBLEM ON A SINGLE MACHINE WITH PERIODIC MAINTENANCE AND NON-PRE-EMPTIVE JOBS

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1 Proceedigs of the Iteratioal Coferece o Mechaical Egieerig 2007 (ICME2007) 2-3 December 2007, Dhaka, Bagladesh ICME07-AM-6 PARETO-OPTIMAL SOLUTION OF A SCHEDULING PROBLEM ON A SINGLE MACHINE WITH PERIODIC MAINTENANCE AND NON-PRE-EMPTIVE JOBS Bhaba R. Sarker, Jufag Yu 2, Deiz Muga 3, Mohammad A. A. Rahma 4, ad Sultaa Parvee 5,3,4 Departmet of Idustrial Egieerig, Louisiaa State Uiversity, Bato Rouge, LA 70803, USA 2 Departmet of Egieerig Maagemet, Iformatio & Systems, Souther Methodist Uiversity Dallas, TX 75205, USA, 5 Departmet of Idustrial ad Productio Egieerig, Bagladesh Uiversity of Egieerig ad Techology, Dhaka, bsarker@lsu.edu, 2 yuj@egr.smu.edu, 5 sparvee@ipe.buet.ac.bd ABSTRACT A Pareto-optimal solutio is developed i this paper for a schedulig problem o a sigle machie with periodic maiteace ad o-preemptive jobs. I literature, most of the schedulig problems address oly oe objective fuctio; while i the real world, such problems are always associated with more tha oe objective. I this paper, both multi-objective fuctios ad multi-maiteace periods are cosidered for a sigle machie schedulig problem. To avoid the complexities ivolved i solvig a explicit multi-objective optimizatio problem, multiple objective fuctios are cosolidated ad combied ito a sigle objective fuctio after they are weighted ad assiged proper weightig factors. O the other had, periodic maiteace schedules are also cosidered i the model. The objective of the model addressed i this paper is to miimize the total job flow time, the maximum tardiess, ad the machie idle time i a sigle machie eviromet. A heuristic algorithm is developed to solve this multiple criterio problem. The parametric aalysis of the trade-offs of all solutios with all possible weighted combiatio of the criterios is performed. Results are provided to explore the best schedule amog all the Pareto-optimality sets. Keywords: Job schedulig, Multi-objective, Periodic maiteace, No-preemptive job. INTRODUCTION It is always challegig to solve multi-criterio schedulig problems due to the lack of suitable solutio techiques. Therefore, such problems are usually trasformed ito a sigle-objective problem. A solutio called Pareto-optimal if it is ot possible to decrease the value of oe objective without icreasig the value of the other (Piedo, 2002). The difficulty that arises with this approach is the rise of a set of Pareto-optimal solutios, istead of a sigle optimum solutio. Traditioally, the most commo way to deal with a multi-objective problem is to cosolidate the multiple criterios ito a sigle scalar value by usig weighted aggregatig fuctios based o a set of predefied prefereces ad the to fid a compromise solutio that reflects these prefereces (Deb, 200). However, i may real cases ivolvig multi-criterio schedulig problems, it is preferable to preset a set of promisig solutios to the decisio-makers so that the most adequate schedule ca be chose. This is why there are icreasig iterests i ivestigatig the applicatio of Pareto-optimizatio techiques to multi-criterio schedulig problems. The focus of Pareto-optimizatio is to fid a set of compromised solutios that represet a good approximatio to the Pareto-optimality (Piedo, 2002). I recet years, several Pareto-optimizatio related algorithms have bee published (Kasprzak ad Lewis, 200, Gupta ad Sivakumar, 2002). This is due to the existig of multi-objective optimizatio problems i may differet domais. I additio to the cosideratio of multiple objectives, periodic maiteace is also cosidered i this schedulig problem. A uexpected breakdow of productio system will make the productio behavior hard to predict, ad thereby will reduce the system efficiecy. Maiteace ca reduce the breakdow rate with mior sacrifices i productio (Liao ad Che, 2003). A modified Pareto-optimality algorithm is developed i this paper for the multi-criterio schedulig problem with periodic maiteace. The algorithm is used to determie the trade-offs betwee total completio time ad maximum lateess. The specific problem cosidered i this paper is to miimize the total flow time, the ICME2007 AM-6

2 maximum lateess ad the machie idle time. These multiple objectives are trasformed ito a sigle objective fuctio, cost fuctio, by aggregatig them usig a set of predefied weightig factors. I additio, various sets of Pareto-optimality sequeces are geerated agaist various maiteace plas, which cotai various time itervals betwee two cosecutive maiteaces ad various amout of time to perform the maiteace. Various maiteace plas give the decisio-maker more flexibility i makig his or her decisio. 2. THE PROBLEM We assume that there are idepedet o-preemptive jobs, that is, oce a job is started it must be completed to process them o a sigle machie without iterruptio. Each job j becomes available for processig at ready time zero ad has a due date d j. At every T uit of time, the machie is seized to hold for maiteace. A umber of jobs that are grouped together to fit i every T amout of time is a batch. There could be machie idle time, I b, before the maiteace starts after the last job i a batch is completed. The maiteace period is M which is also a fixed time. The total machie idle time is obtaied by addig the idle time of all the batches. The algorithm preseted here for the problem combies all the criterios together i oe schedule. The ew approach starts with a iitially obtaied set of Pareto-optimal schedule for flow time ad maximum tardiess miimizatio problem. It the icludes machie idle time I ad maiteace time M i each of these iitially foud sequeces. Oce the reschedulig of machie maiteace ad idleess period of machie is completed, it the calculates the ew values of flow time, maximum tardiess ad machie idle time for which the assiged weights are w, w 2, ad w 3, respectively.. All possible weight combiatios, satisfyig w + w 2 + w 3 =, are assiged for criterios i each schedule of Pareto-optimality set. The miimum total cost amog all the Pareto-optimal sequeces accordig to certai weighted parameters gives the best sequece for the problem. It is clear that the problem is NP-hard sice the problem that miimizes the maximum tardiess subject to periodic maiteace period ad o-presumable jobs is NP-hard (Liao ad Che, 2003). For two or more cotradictory criterios, each criterio correspods to a differet optimal solutio, but oe of these trade-off solutios is optimal with respect to all criterios (Deb 200). Thus, multi-criterio optimizatio does ot try to fid oe optimal solutio but a set of trade-off solutios. The fudametal differece is that multi-objective optimizatio deals with a set of Pareto-optimal solutios. The best schedule amog the set that gives the most promisig result for a particular set of weighted criterios is foud. Notatios: Number of jobs for processig at time zero J j Job umber j, (,2,,) p j Processig time of job j Processig time of job j i batch i p ji C j Completio time of job j d j Due date of job j L j Lateess of job j, where L j = C j - d j T j Tardiess of job j, where T j = max{0; L j }, L max = max j {T j } T Time iterval betwee two maiteace periods M Amout of time to perform oe maiteace I bi Machie idle time i batch i, (i=, 2,, r) I Total machie idle time of a schedule T bi Total processig time for scheduled jobs i batch i, (i=, 2,,r) m Iteratio umber. 3. THE SOLUTION ALGORITHM Whe there are multiple objectives, the cocept of Pareto-optimality plays a role i schedulig. A schedule is Pareto-optimal if it is impossible to improve o oe of the objectives without makig at least oe other objective worse. The scheduler may wat to view a set of Pareto-optimal schedules before decidig which schedule to select, whe there are multiple objectives. I this paper, the algorithm of determiig trade-offs betwee total completio time ad maximum lateess (Piedo, 2002) is modified by icludig periodic maiteace. I additio to the total completio time ad the maximum lateess, the machie idle time is also cosidered as the third objective. A set of Pareto-optimal schedules represet the trade-offs betwee total completio time, maximum lateess ad machie idle time. There are may sequecig rules that ca be applied to the jobs through the machies i a job shop accordig to the prefereces. Two of those basic sequecig rules, shortest processig time (SPT) ad earliest due date (EDD) are adapted i the modified Pareto-optimality algorithm. For explaatory coveiece, we defie two terms that are eeded i the algorithm. Defiitio. The machie idle time of a batch, I bi, is defied as the time by subtractig the total processig time for scheduled jobs i a batch, T bi, from the time iterval betwee two maiteace periods T (i.e., I bi = T T bi ). Defiitio 2. The total machie idle time of all machie i a schedule is defied as I = k I bi for all batches. i= Pareto-optimality algorithm determies the trade-offs betwee total completio time ad maximum lateess oly (Piedo, 2002). A third objective, machie idle time, is added ad the stated algorithm is modified accordigly. The steps of the modified Pareto-optimality algorithm are outlied as follows: Algorithm : Modified Pareto-Optimality Algorithm Step : Set m = (umber of iteratio) a) Schedule the jobs by SPT rule ad apply EDD rule to the jobs with same processig time as schedule S SPT/EDD b) Compute L max (SPT/EDD) ICME AM-6

3 c) Go to Step 8 to fid machie idle time i the schedule S SPT/EDD ad the revised S SPT/EDD is ow called S SPT/EDD whe maiteace time is icluded. Step 2: Set m = 2 a) First schedule the jobs by EDD rule, ad apply SPT rule to the jobs with same due date, as schedule S EDD/SPT b) Compute L max (EDD/SPT) c) Go to Step 8 to fid machie idle time i the schedule S EDD/SPT ad, o iclusio of maiteace time the revised S EDD/SPT is called S EDD/SPT. Step 3: Iteratio m = 3. Set L max = Lmax ( EDD) ad d j = d j + Lmax. Step 4: Set k =, J c = {,..., }, τ = p j ad δ = τ. Step 5: c Fid j i J such that d j τ, ad p j pl for c all jobs l i J such that d l τ. Put job j i positio k of the sequece. Step 6: If there is o job l such that d l < τ ad p l > p j, go to Step 7. Otherwise fid j such that τ d j = mi( τ dl ) l for all l such that d l < τ ad p l > p j, Set δ = τ d j. Ifδ < δ, theδ = δ. Step 7: Set k k -, ad τ τ - p j. c c Update the set as J = J j. If k go to Step 5. Step 8: Geerate a batch by groupig a set of jobs such that p j T Repeat groupig of the remaiig jobs to form other batches. Set b i = umber of batches i oe schedule, where i =, 2,,r. Fid machie idle time for oe batch Ibi = T p ji Fid machie idle time for oe schedule, I = r I bi i= Revise the schedule by addig the amout of time to perform maiteace, M, to the ed of each batch. Compute j = C j, L max, ad I. Step : Set L max = L max + δ. If Lmax Lmax ( SPT / EDD) set m = m +, d d + δ, ad go to Step 4. j = j Otherwise STOP. I Step, the algorithm starts with sequecig the jobs i SPT order. If two jobs have the same processig time, the job with smaller due date is placed earlier. The L max (SPT/EDD) is calculated for this geerated SPT/EDD schedule. This L max (SPT/EDD) value idicate whe to stop the iteratios i the algorithm. For the schedule of SPT/EDD (S SPT/EDD ), batches are geerated by groupig sets of jobs accordig to p j T. The first Pareto-optimal schedule (S SPT/EDD) is obtaied after addig the maiteace time to the geerated batches i S SPT/EDD. The secod Pareto-optimal schedule (S EDD/SPT) is obtaied i Step 2 which is similar to Step. The oly differece is that istead of startig with SPT order, the procedure starts with EDD order ad follows the same idea as i Step. I Step 3, due dates of the jobs are icreased by L max (EDD/SPT) for the ext iteratio. Step 4 calculates the total processig time for jobs ad assigs that value to δ. Step 5 geerates a Pareto-optimal schedule that miimizes C j i which job k is scheduled last, if ad oly if (i) d k p j, ad c (ii) p k pl for all jobs l i J such that d l p j. Step 6 determies the miimum icremet δ i the L max that would allow for a decrease i the miimum C j from the ew geerated Pareto-optimal schedule. Maiteace time is icluded to the geerated Pareto-optimal schedule after formig the batches i Step 8. Three objective values ( C j, Lmax, Im ) are also calculated at this poit for all the Pareto-optimal schedules with periodic maiteace. 4. COMPUTATIONAL RESULTS Cosider a sigle-machie schedulig problem with ie jobs, as give i Table (take from Liao ad Che, 2003). The time iterval betwee two cosecutive maiteaces, T, is 8 hours ad the amout of time to perform oe maiteace, M, is 2 hours. Now, all possible Pareto-optimal schedules are geerated to determie average flow time of jobs, F (= C j / ), job tardiess L max, ad machie idle time I for ie jobs. Startig with Step, a optimal schedule S ( SPT / EDD) : < > is foud by arragig jobs i SPT order ad followed by d[ j] d[ k ] if P j = Pk ad job j ad k are adjacet (see Table 2). The maximum lateess, L max ( SPT / EDD), equals to correspodig to job 2. Now Step 8 is applied to fid the machie idle ICME AM-6

4 time after the isertio of maiteace. Table : The processig time ad due dates for -job problem (i hour) Jobs P j D j Liao ad Che (2003). Table 2: Pareto-optimal schedule, ( SPT / EDD) S Jobs P j d j C j L j Cotiue with the rest of the steps ad iteratios. Table 3 summarizes the results from all the iteratios. Table 3: All the iteratios of the algorithm Iteratio Schedule Pareto-Optimal Sequece m C j, L max, I m Curret d + δ δ SPT/EDD < > 5, 22, 8 30, 20, 4, 3, 3, 2, 0, 2, - 2 EDD/SPT < > 82, 20, 8 30, 20, 4, 3, 3, 2, 0, 2, 7 3 S < > 72, 20, 8 37, 27, 2, 20, 20,, 7,, 8 4 S 2 < > 37, 2, 0 38, 28, 22, 2, 2, 20, 8, 0, 2 5 S 3 < > 5, 20, 8 40, 30, 24, 23, 23, 22, 20, 2, 6 S 4 < > 5, 22, 8 4, 3, 25, 24, 24, 23, 2, 3,2 Stop j Table 4: A Pareto-Optimality Set Iteratio, m Schedule Pareto-Optimal Sequece C j, L max, I m c ( F, Lmax, I SPT/EDD < > 5, 22, EDD/SPT < > 82, 20, S < > 72, 20, S 2 < > 37, 2, S 3 < > 5, 20, S 4 < > 5, 22, The best schedule with the miimum weighted fuctio ) After geeratig all possible Pareto-optimal schedules with respect to job completio time F, job tardiess L max, ad machie idle time I for ie jobs, the total weighted fuctio, c( F, L max, I ) = w ( Cj /) + w2lmax + w3i, where w = 0.5, w 2 = 0.4, w 3 = 0. (arbitrarily chose) for 6 schedules is calculated [see Table 4]. The miimum-weighted schedule is S = S 2 : < > correspodig to c ( F, L, I ) = 2.4. max To compare the modified Pareto-optimality algorithm with the eighborhood search heuristic (Baker, 8), the same istace with parameters T = 8, M = 2, w = 0.5, w 2 = 0.4 ad w 3 = 0. is used. The eighborhood search heuristic gives the best ear-optimal schedule as S: < > with the miimum weighted fuctio equals to O the other had, the modified Pareto-optimality algorithm gives the best ear-optimal schedule as S: < > with the miimum weighted fuctio equals to It ca be cocluded that the modified Pareto-optimality algorithm provides a better result tha the eighborhood search heuristic for this istace. The same istace show i the example is repeated for four levels of T (7, 8,, 0) ad two levels of M (2, 3) as give i Liao ad Che (2002) to show the performace of various sets of Pareto-optimality schedules for differet maiteace plas. Various maiteace plas give more flexibility to the scheduler (or decisio-maker) to make a decisio accordig to the preferece ad available maiteace alteratives. It ca be cocluded that the objective c ( F, L max, I ) is depeded o T ad M, ad this shows the importace of a good maiteace pla. As the time to perform maiteace, M icreases, both the completio time ad the maximum lateess icreases too. O the other had, as the time iterval betwee two maiteace periods, T chages, all three objectives are chagig as well because of formig differet batches. 5. CONCLUSION A multi-criterio, o-preemptive, ad periodically maitaied sigle machie schedulig problem is studied i this paper. Three criterios are cosidered: reductio of flow time, maximum tardiess ad machie idle time. ICME AM-6

5 The trade-offs betwee the flow time ad maximum tardiess is comparatively simple. Sometimes a sigle sequece (with o trade-off) or a set of Pareto-optimal sequeces (with trade-off) miimizes the flow time ad maximum tardiess, but the trade-off betwee miimum flow time, maximum tardiess ad machie idle time is a complex problem. I this study a ew kid of approach that allows the use of simple recombiatio of the criterios is preseted. The ew approach started with a iitially obtaied set of Pareto-optimal schedule for flow time ad maximum tardiess miimizatio problem. It the itroduces machie idle time ad maiteace time i each of these iitially foud sequeces. Oce the reschedulig of machie maiteace ad idleess period of machie is completed, it the calculates the ew values of flow time, maximum tardiess ad machie idle time. The search for the miimum total cost amog all the Pareto-optimal schedules with the assiged weights o criterios is obtaied. Fially, a promisig sequece is chose that gives the miimum total cost for a particular set of weights o the criterios. The computatioal results have show that the modified Pareto-optimality algorithm provides a better solutio tha the eighborhood search heuristic (Sarker et al., 2007) ad this shows the efficiecy of the modified Pareto-optimality algorithm. Direct applicatio of this study may be applied to the idustries where performace of machie maiteace is a routie work ad worthwhile as well. Chemical processig equipmets, boilers, furaces, mechaical machieries etc. are the examples of such implicatios. 6. REFERENCES. Baker, K. R. (8), Elemets of Sequecig ad Schedulig, Joh Wiley, Ic., NY. 2. Gupta, A. K. ad Sivakumar, A. I. (2002), Simulatio based multiobjective schedule optimizatio i semicoductor maufacturig, Proceedigs of the 2002 Witer Simulatio Coferece, Sigapore, Kasprzak, E.M. ad Lewis, K.E. (200), Pareto aalysis i multiobjective optimizatio usig the coliearity theorem ad scalig method, Structural ad Multidiscipliary Optimizatio, 22(3): Liao, C.J. ad Che, W.J. (2003), Sigle-machie schedulig with periodic maiteace ad oresumable jobs, Computers & Operatios Research, 30: Piedo, M. (2002), Schedulig, Theory, Algorithms, ad Systems (2d editio), Pretice Hall, Ic., Upper Saddle River, New Jersey. 6. Sarker, B.R., Yu, J., Muga, D. ad Mohammad, A.A.R. (2007), A Multiple Criterio Schedulig o a Sigle Machie with Periodic Maiteace ad No-preemptive Jobs, Workig Paper, Departmet of Idustrial Egieerig, Louisiaa State Uiversity, Bato Rouge, LA ICME AM-6