Job Scheduling for Hybrid Assembly Differentiation Flowshop to Minimize Total Actual Flow Time

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1 Job Schedulig for Hybrid Assembly Differetiatio Flowshop to Miimize Total Actual Flow Time Rahmi Maulidya, Rachmawati Wagsaputra, Suprayogi, Abdul Hakim Halim Faculty of Idustrial Techology Istitut Tekologi Badug (ITB), Idoesia Tel: (+062) , Abstract. This paper deals with schedulig problems of a hybrid assembly differetiatio flow shop cosistig of parts maufacturig, assembly operatio ad differetiatio stages. I the first stage, a umber of differet parts of a job are processed o (urelated) dedicated machies, ad the, i the secod stage, the completed parts are assembled o a commo machie to produce commo jobs. Fially, i the third stage cosistig dedicated machies, each particullar of the jobs is processed o a dedicated machie. The problem is to fid a job schedule so as to miimize total actual flow time to accommodate that parts should arrive i the shop at the right times ad i the right quatities, ad that the fiished product should be delivered at their due dates. A Mixed Iteger Liear Programmig (MILP) model is proposed to represet the problem ad it ca be solved efficietly for small size problems. However, for larger-size problems, a approach of heuristic algorithms (SPT-based ad NEH-based) for geeratig a iitial solutio, ad meta-heuristic algorithm (Hybrid Geetic Algorithm-Variable Neighborhood Search) for obtaiig the fial solutio is proposed. The approach is tested usig a set of hypothetic data, ad it ca be show that the algorithm ca solve the problems effectively. Keywords: job schedulig, hybrid assembly differetiatio flow shop, total actual flow time 1. INTRODUCTION It ca be observed that there are three stages i a doorlie of car maufacturig, i.e., machiig, assembly ad paitig i this order. I the machiig stage, door parts are maufactured. Those parts are assembled ito a body i the door lie for the ext step to produce variety of fiished products. The stage resultig variety of fiished products is called as a differetiatio stage (Li ad Hwag, 2011, Xiog et al., 2015). Cases for differetiatio as the fiishig stage ca be foud i paitig process (Xiog et al., 2015), ad cuttig process of sticker label (Li ad Liao, 2003). A differetiatio stage will result differet types of fiished products comig from the output of assembly operatio (Xiog et al., 2015). Xiog et al. (2015) have discussed schedulig problems with three stages maufacturig to miimize total flow time ad called a hybrid assembly differetiatio flow shops. The research icludes a fabricatio stage with three parallel machies, assembly stage with a sigle assembly machie, ad differetiatio stage with two dedicated machies each of which produces a type of products. This research is about a arragemet of machies called hybrid flowshop which flows sequecially with a differet umber of parallel machies i every stage (Ruiz ad Vazquez-Rodriguez, 2010). Researchers develop the hybrid flowshop schedulig problem based o variatio of shop cofiguratio, costrait, assumptio ad objective fuctio (Ruiz ad Vazquez-Rodriguez, 2010). The hybrid flow shop could be related to the differetiatio stage whe producig differet type of products ad the shop is arraged due to the product type. Accordig to Wag ad Liu (2013), hybrid flow shop with differetiatio stage has several ames, such as two-stage hybrid flow shop with dedicated machies (Li ad Liao, 2003, Wag ad Liu, 2013), two-stage differetiatio flowshop (Li ad Hwag, 2011) ad two-stage hybrid flow shop with machie eligibility. The hybrid flow shop could also be related to assembly operatio, kow as hybrid assembly flow shops, where parts from the fabricatio stage are assembled ito a assembly part at the assembly stage (Fattahi et al., 2014). The assembly operatio ca be arraged i assembly shop or assembly lie with coveyor (Morto ad Petico, 1993).

2 Operators of the assembly operatio do their job maually or usig tools or operatig semi automated machie (Groover, 2008). Accordig to a product structure, research has developed o oe assembly level (Fattahi et al., 2014, Halim ad Yusriski, 2009) ad two or more assembly levels (Thiagaraja ad Rajedra, 2005). A hybrid flowshop that is related to differetiatio stage ad assembly stage is kow as a hybrid assembly differetiatio flow shop. It is itroduced by Xiog et.al (2015). The research addresses job schedulig for three productio stages of the hybrid assembly differetiatio flow shop model where the first stage is compoet maufacturig, the secod stage is a assembly operatio ad the third stage is a differetiatio. The approach was the forward schedulig with the objective fuctio of miimizig total flow time. Literature shows that there is aother approach called the backward schedulig i which jobs are scheduled from the due date to the time zero directio. Much research o backward schedulig (Halim, 1993, Halim et.al., 2006) has adopted the criteria of total actual flow time (Halim ad Yusriski, 2009, Halim ad Ohta, 1994) is the criteria of the iterval time of parts i the shop floor. The total actual flow time is defied as the total miimizig iterval time betwee respective product's arrival times ad their commo due date. The actual flow time assumes that the product arrives at the shop floor whe eeded ad the completed product is delivered to the costumer at their due date. A part does ot have to be available at time zero but may arrive at the shop at the time of maufacturig. Therefore, the resultig schedule decisios give impact to the ivetory level ad part receivig time at the productio lie (Halim, 1993). I this paper, we develop a mathematical model ad a algorithm for hybrid assembly differetiatio flow shop to miimize total actual flow time. Sectio 2 describes ad formulates the model cosideratio. Sectio 3 shows the algorithm as a solutio for the problem. Sectio 4 shows the umerical examples ad the result for the problem. Sectio 5 cocludes the paper ad gives suggestios for future research. 2. PROBLEM FORMULATION The problem i this paper is developed for three stages productio system, i.e. machiig, assembly ad differetiatio stage, each of which has differet umber of machies. The system produces G types of products that is similar to the umber of machies at the third stage. Figure 1 shows the system we represet. I the first stage, a umber of differet parts of a job are processed o (urelated) dedicated machies. Parts are processed o each machie i the same order. I the secod stage, the completed parts are assembled o a commo machie to produce commo jobs. I the third stage cosistig dedicated machies, each particullar of the jobs is processed o a dedicated machie to become a certai type of product. All jobs are requested at a commo due date d. Figure 1. Three stages productio system The problem is to fid a job schedule so as to miimize total actual flow time to accommodate that parts should arrive i the shop at the right times i the right quatities, ad the fiished product should be delivered at their due dates. Several assumptios are made as follows: Productio are fiished at their commo due date Each machie ca process at most oe job at a time Each job ca be processed o at most oe machie at a time Setup times ad trasportatio times are eglected Job processig caot be preempted before it is fiished There are ulimited buffers betwee the machies of the stage oe ad two ad the stage two ad three Halim et al. (1994) defie actual flow time of a job as the time that the job speds i the shop from its startig time for processig util its due date. The actual flow time, F a [i], ca be formulated as follows: F a [i] = d B [i], i=1,2,, (1) Where d is a commo due date for jobs ad B [i] is the startig time for processig J [i]. The jobs are scheduled backwardly so the positio i is couted from the due date. Let us modify the formulatio of the actual flow time of a job for the problem cocered i this paper. F a [i] = d mi (B (1) k K [i],k) (2) The geeral formulatio of the total actual flow time (TAFT) is: (B (1) k K [i],k)) TAFT = i=1 (d mi (3)

3 This paper adopts the followig otatios: Idex j = idex of job, j=1, 2,, i = idex of job positios, i=1, 2,, k = idex of machies at the first stage, k=1, 2,...,K h = idex of machies at the third stage, also state the product type, h=1, 2,,G Parameter K = umber of machies at the firt stage G = umber of machies at the third stage = umber of processed job h = umber of job processed i machie h N = set of processed job N h = set of job that is processed i machie h d = commo due date M 1k = machie for processig part k of all the jobs at the first stage, k = 1, 2,,K M 2 = assembly machie i the secod stage M 3h = differetiatio machie for processig types of products i the third stage, h = 1, 2,,G (1) = the processig time of part k job j i machie M 1k at p j,k the first stage p j (2) = the processig time of assembly operatio of job j o machie M 2 at the secod stage p (3) j,h = the processig time of job j type h i machie M 3h at the third stage. p (3) j,h > 0 if j N h ; 0, otherwise. A j,h = the biary variable equals 1 if job j is assiged i machie h with the processig time p (3) j,h > 0 ; otherwise 0 Decisio variables = the biary variable equals 1 if job j is assiged to X j[i] positio i; otherwise 0 Y [i],h = the biary variable for the third stage equals 1 if (1) B [i],k (2) B [i] (3) B [i],h job i positio i is processed i machie h; otherwise 0; = the startig time of job i positio i i machie M 1k at the first stage = the startig time of assembly job i positio i i machie M 2 at the secod stage = the startig time of job i positio i i machie M 3h at the third stage Objective fuctio TAFT = total actual flow time The problem ca be formulated as follows. (B (1) k K [i],k)) Mi TAFT = i=1 (d mi (4) Subject to: i i =1 (3) B [i],h (3) ) = Y [i],h (d j=1 (X j[i ]. p j,h ), i i, i, h (5) (2) + (X j[i]. p (2) j=1 j ) d (Y [i],h B [i] B [i],k (1) B [i+1],k i i =1 (3) ) j=1 (X j[i ]. p j,h ), i i, i, h (6) (2) B [i+1] B [i] (X j[i+1]. p (2) j=1 j ), i <, i (7) (1) + (X j[i]. p (2) j=1 j,k ) B [i] i, k (8) B [i],k (X j[i+1]. p j=1 j,k ), i <, i, k (9) A j,h = { 1, if p (3) j,h > 0 0, if p (3) j,h = 0 j = 1,2.., h (10) j=1 (X j[i]. A j,h ) = Y [i],h, i, h (11) j=1 X j[i] = 1, i = 1,2.. (12) i=1 X j[i] = 1, j = 1,2.. (13) X j[i] {0,1} (14) B (1) [i],k, B (2) [i], B (3) [i],h 0, ad iteger (15) The objective fuctio (4) shows the total actual flow time that is calculated from the actual flow time of all positio where the actual flow time of each positio is the logest iterval betwee the part beig processed i the first stage util its commo due date. Costrait (5) defie the startig time of job i positio i i machie h at the third stage from its commo due date cosiderig the decisio that positio i is assiged i machie h. Costrait (6) defie the startig time of job i positio i at the secod stage from backward based o job i positio i at the third stage, with coditio where i i. Costrait (7) ad (9) esure that each positio ca oly be processed at the same stage after the ext positio is fiished. Costrait (8) defie the startig time of job i positio i o machie k at the first stage based o the startig time of the same job i positio i at the secod stage. Costrait (10) defie the job with the processig time at the third stage greater tha zero, is processed i machie h at the third stage. Costrait (11) decide the job i positio i that is assiged o machie h at the third stage oly if the value is 1. Costrait (12) ad (13) esure that each positio i the sequece of job is assiged to oly oe job, ad each job is assiged oly to oe positio i the sequece of job. Costrait (14) ad (15) defie the domai of the decisio variables.

4 3. SOLUTION The decisio variable of this model are the startig time of jobs o each stage ad the job sequece. Followig is the algorithm developed i Xiog et.al (2015) that cosists of two heuristics (SPT-based algorithm ad NEHbased algorithm) as a a iitial solutio ad the metaheuristic algorithm (Hybrid Geetic Algorithm-Variable Neighborhood Search) for fidig the best solutio. I this paper, we modify the algorithm by schedulig the jobs backwardly from the due date to miimize the total actual flow time. The algorithms to determie iitial solutio for the Hybrid Assembly Differetiatio Flow Shop-Total Actual Flow Time are as follows. SPT-based heuristic Step 1: Geerate six job sequeces S 1, S 2, S 3, S 4, S 5, ad S 6 i icreasig orders of max max 1 k 3 (p j,k (1) ) + p (2) j, p (2) (3) j + p j,h 1 k 3 (p j,k ad (1) ), p j (2), p j,h max 1 k 3 (p j,k (3), (1) ) + p (2) j +p (3) j,h, respectively. Schedule the jobs from due date. Step 2: select the best oe of the six job sequeces as the solutio. The output it ad its TAFT value. NEH-based heuristic Step 1: obtai a job sequece S 0 i ascedig order of max {p (1) k=1,2,,k j,k } + p (2) j + max {p (3) h=1,2,..g j,h } Step 2: set r=2. Select the first two jobs from S 0 ad schedule jobs from due date to miimize the total actual flow time as if there are oly two jobs. Set the best oe as a curret solutio S 1. Step 3: while r<+1 do Set rr+1. Geerate r cadidate sequece by isertig the r th job i the job sequece S 0 i each slot of the curret solutio. Select the best oe with the least partial total actual flow time. Update the best oe as a curret solutio S 1. Ed while Step 4: output the curret solutio S 1 ad its TAFT value. The best iitial solutio is selected from the six job sequeces of SPT-based ad oe job sequece of NEHbased heuristic. The best iitial solutio will be the iput of the Hybrid Geetic Algorithm-Variable Neighborhood Search show as follows. The Hybrid Geetic Algorithm-Variable Neighborhood Search (HGA-VNS) Step 1: Iput the best iitial solutio for a istace ad set algorithm parameters. Step 2: Geerate Populatio Size idividuals as the iitial populatio. Set maximum rutime t max. Step 3: Is t* > t max? If Yes, Output the best solutio If No, go to step 4. Step 4: perform procedure VNSL-I o each idividual as follows. Iput iitial solutio x ad set maximum iteratio Radomly choose two positios u ad v, where u<v. Move the job i positio u to positio v, whereas all jobs i positio k, with k = u+1,,v, are shifted oe positio forward alog solutio x. The a ew solutio will be x. If TAFT (x ) < TAFT (x), the x x. else swap the job i positio u ad the job i positio v of solutio x. The a ew solutio will be x. If TAFT (x ) < TAFT (x), the x x. ed. Repeat this procedure util the maximum iteratio. Step 5: Fid the best idividual of the populatio ad improved it by procedure VNSL-II as follows. Iput iitial solutio x ad set maximum iteratio Radomly choose two positios u ad v, where u<v. Move the job radomly i positio u to positio v, whereas all jobs i positio k, with k = u+1,,v, are shifted oe positio forward alog solutio x. The a ew solutio will be x. If TAFT (x ) < TAFT (x), the x x. else Swap the job i positio u ad the job i positio v of solutio x. The a ew solutio will be x. If TAFT (x ) < TAFT (x), the x x. else Iverse the jobs betwee positios u ad v of solutio x. The a ew solutio will be x. If TAFT (x ) < TAFT (x), the x x. else Isert the job i positio u ad the job i positio u+1 betwee positio v ad v+1, whereas all jobs i positio k, with k = u+2,,v, are shifted two positios forward alog solutio x. The a ew solutio will be x. If TAFT (x ) < TAFT (x), the x x. ed. Repeat this procedure util the maximum iteratio. Step 6: Perform Geetic algorithm Step 6.1. Select the offsprig to the ext geeratio Select pairs of strigs from the curret populatio. Step 6.2. Perform crossover operatios Apply two poit crossover to selected pairs to geerate solutio. Step 6.3. Perform mutatio operatios Mutatio is performed o sigle idividuals like isert move, swap move ad iverse move operatio.

5 Step 6.4. Add the best strig to the curret populatio ad remove the other strig. Step 6.5. Go to step 6.1. for aother pairs of strigs util the maximum iteratio is achieved. Step 7: Collect rutime ad add the rutime ito t*. Go the step 3. Table 1. Processig time of =8, K=3, G=2 The best iitial solutio is obtaied with the sequece of J 4,J 8,J 1,J 5,J 3,J 7,J 2,J 6 ad TAFT=266 from the SPT-based heuristic algorithm. The, it is improved with the Hybrid Geetic Algorithm-Variable Neighborhood Search. Table 3. shows the result for the procedure of VNSL-I o step 4. Table 2. Result of the procedure of VNSL-I o step 4 Figure 2. Flowchart of the algorithm 4. NUMERICAL EXPERIENCE Suppose there is a istace of 8 jobs (=8) to be processed i the system. Table 1 shows the processig time of 2 product types (Type 1 ad Type 2). Type 1 cosists of jobs J 1, J 2, J 3 ad J 4, writte as N 1={J 1,J 2,J 3,J 4} ad type 2 cosists of jobs J 5, J 6,J 7 ad J 8 writte as N 2={J 5,J 6,J 7,J 8}. This illustratio is for the umber of jobs ()=8, the umber of machie i the first stage (K)=3, ad the umber of machie i the third stage (G)=2. The completed products should be delivered at commo due date d=1000. I Table 3, the best iitial solutio of J 4,J 8,J 1,J 5,J 3,J 7,J 2,J 6 is become the iitial solutio x ad the first isert move operatio is movig the job i positio u=2 (J 8) to positio v=6. The result of sequece is J 4,J 1,J 5,J 3,J 7,J 8,J 2,J 6 with TAFT=289. Because the solutio x is ot improved, the the ext procedure is swapig the job i positio u=3 (J 1) ad positio v=5 (J 3) ad the result of sequece is J 4,J 8,J 3,J 5,J 1,J 7,J 2,J 6 with TAFT=277. The best result is i the sequece of J 4,J 5,J 1,J 8,J 3,J 7,J 2,J 6 with TAFT=264 from step 4. Next step is procedure of VNSL-II o step 5. The procedure of isert move ad swap move are the same as i step 5. For the procedure of iverse move, the jobs are iversed betwee positio u=3 ad positio v=7. The result of sequece is J 4,J 5,J 2,J 7,J 3,J 8,J 1,J 6 with TAFT=317. For the procedure of Or-Opt move, the jobs are shifted two positios forward alog solutio x. The job i positio u=3 ad u+1=4 are iserted betwee positio v=6 ad v+1=7. The result of sequece is J 4,J 5,J 3,J 7,J 2,J 1,J 8,J 6 with TAFT=285. Table 4 shows the result for step 5 ad the best idividual populatio is the sequece of

6 J 4,J 5,J 1,J 8,J 3,J 7,J 2,J 6 with TAFT=264. I step 6.1, offsprig are geerated from two parets. The first paret is the sequece from step 5 with TAFT=264 ad the secod paret is the sequece from the best iitial solutio with TAFT=266. I step 6.2, Crossover is applied to geerate offsprig by exchagig some gees of the two parets. The first paret is J 4,J 5,J 1,J 8,J 3,J 7,J 2,J 6 ad the secod paret is J 4,J 8,J 1,J 5,J 3,J 7,J 2,J 6. Those are the paret chromosomes for two poit crossover. Table 3. Result for step 5 (procedure of VNSL-II) The two poit crossover are developed as follows. The paret chromosome A1 ad B1 are: The best solutio for the crossover procedure is J 4,J 5,J 1,J 8,J 3,J 7,J 2,J 6 with TAFT=264. O step 6.3, mutatio is performed o sigle idividuals like isert move, swap move ad iverse move operatio. Example for mutatio with iverse move ca be see as follows. The offsprig chromosome A1 ad B1 are: The repaired offsprig chromosome A1 ad B1 are: The mutatio operatio is performed i iversig the jobs betwee u=3 ad v=6, ad repeated util the maximum iteratio is achieved. The best solutio for the mutatio procedure is J 4,J 5,J 1,J 8,J 3,J 7,J 2,J 6 with TAFT=264. Gatt chart for the resultig schedule ca be see i Figure 4.

7 d = 1000 J6-1 J2-1 J6-2 J2-2 J7-1 J7-2 J3-2 J3-1 J8-2 J8-1 J1-2 J1-1 J5-2 J5-1 J4-2 J4-1 M11 M12 Machiig J6-3 J2-3 J7-3 J3-3 J8-3 J1-3 J5-3 J4-3 M13 J6 J2 J7 J3 J8 J1 J5 J4 M2 Assembly J2 J6 J3 J7 J8 J1 J5 J4 M31 M32 Differetiatio Figure 4. Gatt chart for the resultig schedule of =8, K=3, G=2 I order to shows the behaviour of the solutios, the umerical experiece is doe uder the combiatio of umber of jobs ( = 4, 6 ad 8), umber of machies o machiig stage (K = 3 ad 4), ad umber of machies o differetiatio stage (G = 2, 3, ad 4). Total problem sizes is obtaied from xkxg = 3x2x3 = 18 problem sizes. The istace for each umber of job is obtaied as a part of Table 1. For =4, the processig time ca be see from the itace of J 1, J 2, J 3 ad J 4, ad for =6, the processig time ca be see from the itace of J 1, J 2, J 3, J 4, J 5, ad J 6. All combiatio were ru o a PC with Itel Core i CPU, 3,10 GHz ad 8 GB RAM. The result of algorithm is compared to optimized model. Table 5 shows the compariso of optimized model ad algorithm. I Table 5, the symbol * shows that the result of optimized model is still o local optimal after the rutime is iterupted at 100 hours. It is probably because the istace is cosisted of duplicatio data ad ca ot get rid of loopig coditio. For problem size =4 ad =6, the algorithm solutio ca achieve the optimal solutio. The result ca be compared from the average gap. The SPTbased algorithm is the best iitial solutio with the average gap 1,9 compared to NEH-based algorithm. The HGA-VNS has the lowest average gap of 0,5. It meas that the HGA VNS ca effectively give the solutio for the medium ad large size problem. *local optimal, due to the completio time limitatio Table 4. Result of optimized model ad algorithm Model SPT NEH HGA-VNS K G TAFT TAFT Gap TAFT Gap TAFT Gap * * * * * Average CONCLUDING REMARKS This paper shows the problem of job schedulig for three stages productio system called the hybrid assembly

8 differetiatio flow shop, processig G differet types of products to miimize total actual flow time. The approach for this problem shows that fial schedule ca be obtaied from heuristics algorithms ad the fialized by metaheuristic algorithm. The umerical examples for this approach are limited to 4, 6 ad 8 jobs. The umber of iteratios ad the maximum rutime must be set higher to get better result. The illustratios have show the improvemet of the best solutio. The future research is to fid the proposed algorithm that ca give optimal solutio i short time compare to the curret approach. REFERENCES Fattahi, P., Hosseii, S.M.H., Jolai, F. ad Tavakkoli- Moghaddam, R. (2014) A Brach ad Boud Algorithm for Hybrid Flow Shop Schedulig Problem with Setup Time ad Assembly Operatio. Applied Mathematical Modellig, 38, Groover, M.P. (2008) Automatio, Productio Systems, ad Computer Itegrated Maufacturig, 3rd editio, Pearso Iteratioal Editio, NJ. Halim, A.H. (1993) Batch Schedulig for Productio System uder Just i Time Eviromet. Dissertatio. Osaka Perfecture Uiversity Jepag. Halim, A.H. ad Ohta, H. (1994) Batch Schedulig Problem to Miimize Ivetory Cost i the Shop with Both Receivig ad Delivery Just I Times. Iteratioal Joural of Productios Ecoomics, 33, Halim, A.H, Suryadhii, P.P ad Toha, I.S. (2006) Batch Schedulig to Miimize Total Actual Flow Time i a Two-stage Flow Shop with Dedicated Machies i the Secod Stage, Proc. of 7th Asia Pasific Idustrial Egieerig & Maagemet System (APIEMS) Cof December 2006 (Bagkok), Halim, A.H. ad Yusriski, R. (2009) Batch Schedulig for the Two-stage Assembly Model to Miimize Total Actual Flow Time. Proc. of 7th Asia Pasific Idustrial Egieerig & Maagemet System (APIEMS) Cof (Kitakyushu). Li, B.M.T. ad Hwag, F.J. (2011) Total Completio Time Miimizatio i a 2-Stage Differetiatio Flowshop with Fixed Sequeces per Job Type. Iformatio Processig Letters, 111, Li, H.T. ad Liao, C.J. (2003) A Case Study i a Twostage Hybrid Flowshop with Setup Time ad Dedicated Machies. Iteratioal Joural of Productio Ecoomics, 86, Morto, T.E. ad Petico, D.W. (1993) Heuristic Schedulig System with Applicatio i Productio System ad Project Maagemet. Joh Wiley & sos, New York. Ruiz, R. ad Vazquez-Rodriguez, J.A. (2010) The Hybrid Flow Shop Schedulig Problem. Europea Joural of Operatio Research, 205, Thiagaraja, S. ad Rajedra, C. (2005) Schedulig i Dyamic Assembly Job-Shops to Miimize the Sum of Weighted Earliess, Weighted Tardiess, ad Weighted Flow Time of Jobs. Computer & Idustrial Egieerig, 49, Wag, S. ad Liu, M. (2013) A heuristic method for twostage hybrid flow shop with dedicated machies. Computers & Operatios Research, 40, Xiog, F., Xig, K. ad Wag, F. (2015) Schedulig a Hybrid Assembly-Differetiatio Flow Shop to Miimize Total Flow Time. Europea Joural of Operatio Research, 240, BIOGRAPHY Rahmi Maulidya is a lecturer at Idustrial Egieerig Departmet, Trisakti Uiversity i Jakarta, Idoesia. She received her udergraduate o Idustrial Egieerig, Trisakti Uiversity i 1998, ad her Master degree from Idustrial Egieerig ad Maagemet Istitut Tekologi Badug (ITB) i She is doctoral studet i Idustrial Egieerig ad Maagemet at Istitut Tekologi Badug (ITB). Her research iterests are Productio Plaig ad Schedulig. Her address is rachmi@trisakti.ac.id ad rahmimaulidya@gmail.com. Rachmawati Wagsaputra is a lecturer at the Departmet of Idustrial Egieerig, Istitut Tekologi Badug (ITB). She received her doctoral degree from the Departmet of Idustrial Egieerig, New Mexico State Uiversity, USA i Her teachig ad research iterest iclude dyamic optimizatio, maufacturig systems desig, visio ad robotic system, maufacturig automatio ad artificial itelliget with special emphasis o itelliget visio system. Her address is rwagsap@mail.ti.itb.ac.id ad rachmawati_wagsaputra@yahoo.com. Suprayogi is a Associate Professor at Faculty of Idustrial Techology, Istitut Tekologi Badug (ITB). He received his udergraduate ad master degree i Idustrial Egieerig from ITB ad his Doctorate degree from Uiversity of Tokyo, Japa. His research iterests are Trasportatio ad Distributio, Maritime Logistic, Operatio Research, ad Soft Computig ad Algorithm Desig. His address is yogi@mail.ti.itb.ac.id

9 Abdul Hakim Halim is Professor of Idustrial Egieerig at Idustrial Egieerig Departmet, Istitut Tekologi Badug (ITB), Badug, Idoesia. He received his udergraduate ad master degree i Idustrial Egieerig from ITB, ad his Doctorate degree from Idustrial Egieerig Departmet, Osaka Prefecture Uiversity, Osaka, Japa. His research iterests are Productio Schedulig, Ivetory cotrol, FMS, ad JIT systems. He published his papers i Iteratioal Joural of Productio Research, Europea Joural of Operatioal Research, Iteratioal Joural of Productio Ecoomics, ad Productio Plaig ad Cotrol, as well as i Joural of Japa ad several atioal jourals o productio system writte i Idoesia. His address is ahakimhalim@mail.ti.itb.ac.id ad ahakimhalim@gmail.com.