A Multi-Neural-Network Learning for Lot Sizing and Sequencing on a Flow-Shop

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1 A Muli-Neural-Nework Learning for Lo Sizing and Sequencing on a Flow-Shop In Lee Jainder N.D. Gupa Amar D. Amar Deparmen of Compuing and Deparmen of Managemen Deparmen of Managemen Decision Sciences Ball Sae Universiy Seon Hall Universiy Seon Hall Universiy Muncie, IN Souh Orange Avenue 400 Souh Orange Avenue Souh Orange, NJ Souh Orange, NJ leein@shu.edu Key Words: Neural Neworks, Sequencing, Lo Sizing, Flow-Shop ABSTRACT This work invesigaes he use of neural neworks o schedule flow shops wih dynamic job arrival. The objecive of such scheduling is o deermine boh sequence and lo-sizes concurrenly ha minimize a makespan. We presen a muli-neural nework archiecure for simulaneous lo-sizing and sequencing neural-nework learning. Our experimenal sudy shows ha our muli-neural neworks can learn boh lo-sizing and sequencing decisions, and ouperform a sequencing-only neural nework. 1. INTRODUCTION In his paper, we consider scheduling problems on an n-job, m-machine flow-shop. Mos research in he area of flow-shop scheduling has concenraed on he developmen of an efficien flowshop sequencing wihou considering lo sizing. However, by no inegraing he sequencing decision wih lo-sizing decision, he sequencing soluion so arrived may be only sub-opimal. Here, we develop wo neural neworks for a real ime manufacuring scheduling sysem wih lo-sizing and sequencing neural neworks. The mehodology presened uses Geneic algorihm (GA) o generae exemplars for he neural nework raining. Embedded algorihm in GA deermines lo sizes and sequences concurrenly. The lo sizing and sequencing neural neworks acquire scheduling knowledge ha is used in real ime scheduling. Permission o make digial or hard copies of par or all of his work or personal or classroom use is graned wihou fee provided ha copies are no made or disribued for profi or commercial advanage and ha copies bear his noice and he full ciaion on he firs page. To copy oherwise, o republish, o pos on servers, or o redisribue o liss, requires prior specific permission and/or a fee. SAC 2001, Las Vegas, NV 2001 ACM /01/02 $5.00 The res of he paper is organized as follows: Secion Two reviews scheduling lieraure relaed o he problem; Secion Three discusses he muli-neural nework archiecure; Secion Four sudies soluion qualiy using problems of varying sizes; and finally, Secion Five concludes he work. 2. LITERATURE REVIEW Neural neworks are modeled based on he srucure of human nerve sysems. Neural neworks, which learn from examples, have proved parallel, oleran o noise, and self-organizing, as well as have been able o approximae funcional relaionships beween cause and effec [11], [15]. Neural neworks are capable of learning incremenally and locally using deerminisic or sochasic acivaion funcions ieraively, which eases he process of updaing knowledge as new examples are added o he sysem. Muli-layered back-propagaion algorihm proposed by Rumelhar e al. [16] is one of he mos popular neural nework models. Typical applicaions of neural neworks include paern recogniion, classificaion, forecasing, ec. Because of he inheren parallel and disribued compuaion, neural neworks are a promising echnique for opimizaion problems especially when a real ime decision is required. During he pas decade, he applicaion of neural neworks o opimizaion problems has been exensively explored by many researchers in many disciplines such as engineering, and operaions managemen [1], [5], [9], [10], [12], [14], [18]. Typically, he problem of scheduling and conrolling jobs in a flow-line is decomposed ino he problems of lo sizing and sequencing. The sudy of lo sizing under dynamic condiions began wih Wagner and Whiin [17] and, now, here is a sizable lieraure in his area exending he basic model o consider capaciy consrains [2], [6]. There have also been 36

2 several proposals for heurisic lo-sizing mehods [3], [7], [8], [16] give a comprehensive review and caegorizaion of he various lo-sizing mehods available in he lieraure. The general problem of sequencing jobs in a flow-shop has been shown o be NP-complee [4], [8]. Alhough much research has been done over several decades, only a few pracical algorihms exis o solve large problems. Exac mehods such as complee enumeraion, ineger programming, and branch-andbound can handle only small sequencing problems. For a problem where he number of jobs and machines are large, here is usually a radeoff beween he soluion qualiy and he compuaional ime. The problem of sequencing is more complicaed when he sequencedependen seup imes and he buffer space beween he machines are considered. Because of he combinaorial complexiy of such problems, a more viable mehod for solving such problems is use of heurisics [13]. There are generally wo kinds of improvemen heurisics used for he sequencing problem: local search heurisics, which sar wih an iniial soluion and incremenally ry o improve i, and guided search heurisics, which use some kind of guided search o sample promising porions of he search space. Improvemen heurisics can be applied o he sequence obained from consrucive heurisics. Two examples of local search heurisics are hill climbing and seepes descen mehods. The hill climbing mehod chooses a nearby poin randomly and moves here if he new sae is beer han or equal o he curren sae. (In he conex of minimizaion, he hill is invered so ha he move is downward). The seepes descen mehod considers each sae in he neighborhood of he curren sae and selecs he bes one as he nex sae. Some of he guided search heurisics ha have been used successfully are Simulaed Annealing (SA), Tabu Search (TS), and Geneic Algorihms (GA). One of he drawbacks of he above-menioned heurisic mehods for lo-sizing and sequencing is ha hey end o use differen objecives for lo-sizing and sequencing problems. For insance, he objecive of losizing is o reduce he oal invenory cos, and he objecives of sequencing are minimizaion of ardiness, makespan, or flow ime. However, he lo-sizing decision has imporan implicaions for he objecive of he sequence. Conversely, he sequencing decisions affec he opimal lo-sizing decisions. Due o he ineracive naure of he lo-sizes and sequences, he schedules generaed by eiher one of he mehods may be sub-opimal. 3. A MULTI-NEURAL-NETWORK ARCHITECTURE The muli-neural-nework archiecure consiss of four modules ha inerac wih each oher. The firs module is a GA module. Since he flow-shop scheduling problem is an NP-hard problem, GA module generaes opimal/near-opimal sequence, and lo sizes for a given scheduling problem. Figure 1 describes he muli-neural-nework archiecure for simulaneous lo-sizing and sequencing. Sequencing Exemplars GA Module Job Characerisics And Lo-Sizing Parameers Figure 1. A Muli-Neural-Nework Archiecure for Lo-Sizing and Sequencing The following describes he GA procedure for concurren lo-sizing and sequencing decisions. Suppose scheduling is needed for he following se of jobs and iniial lo-sizes (in parenhesis) ha arrive a a flow-shop: A(20), B(14), C(12), D(10), and E(12). In deermining a lo-size for each job, we use he following formula. Le i = 1,...,n index iems; Pi = processing rae for iem i (unis/ime); Qi = bach size for iem i (unis); τ i = seup ime for iem i; ψ i = parameer for lo-sizing decision (0 ψ i 1) P Q i = ψ i τ i, Sequencing Jobs Neural-Nework Module Sequencing Decision Lo-Sizing Neural-Nework Module If processing rae, P i, increases, he lo size, Q i, will also increase. Likewise, if seup ime, τ i, increases, he lo size, Q, will increase. GA embeds a i Lo-Sizing Decision Real-Time User Inerface Job Characerisics search procedure o idenify he opimal value of losizing parameer ψ ( 0 ψ 1) while simulaneously searching for he sequence ha minimizes he makespan: C(6)-D(5)-E(4)-B(7)-A(10)-C(6)-D(5)-E(4)- B(7)-A(10)-E(4) Since differen job characerisics may require differen lo sizing parameers and sequences, 37

3 sufficien number of exemplars is obained from differen se of machine and job characerisics. Once enough exemplars are obained, he neural nework rainings are performed and used o idenify he relaionship beween job characerisics and he qualiy lo size. The following describes he lo-sizing neural nework approach. Lo-Sizing Neural-Nework To capure he characerisics of a se of jobs ha are processed by he manufacuring sysem, four job parameers on each machine were used as inpus o he neural nework: (1) average processing ime, (2) sandard deviaion of processing ime, (3) average seup ime, and (4) sandard deviaion of he seup imes. Noe ha he number of inpu nodes is independen of he job size, bu dependen on he number of machines in he flow-shop. For example, he number of inpu nodes for four machines is 16 (4*4). An opimal lo-sizing parameer idenified in he GA module is used as a arge oupu for each exemplar. Figure 2 shows he neural nework archiecure for he lo-sizing applicaion. Oupu Oupu Layer Hidden Layer Inpu Layer Inpus ϖ φ ϕ σ m m ϖ m1 φ m1 ϕ m1 Figure 2. A Three-Layer Backpropagaion Neural Nework Archiecure for Lo-Sizing Sequencing Neural-Nework For raining of he sequencing neural nework, each elemen of he inpu se is composed of jobs o be processed. Oupu for he corresponding inpu is he desired job sequencing. Inpus (i.e. sequencing exemplars) o he sequencing neuralnework are generaed by he GA module. In order o build a neural nework ha sores a sequencing knowledge, a neural weigh marix is inroduced. Weighs are associaed wih he desirabiliy of he σ m1 Average processing ime on machine n O Sandard deviaion of processing ime on machine n Average seup ime on machine n Sandard deviaion of seup ime on machine n ϖ mn φ ϕ mn σ m m sequences beween jobs. The acquired sequencing knowledge is sored in long-erm memory in he form of a maser neural weigh marix. In he real ime sequencing sage, he relevan knowledge for a sequencing decision is exraced from he long-erm memory. The sequencing neural-ne uilizes he concep of an associaive memory. The neural weigh marix associaes he desirabiliy of sequences beween jobs wih he magniude of weighs beween jobs. We calculae he neural weigh marix by: w = n w(, i= 1 ( y( 1 x( 1 + y( 2 x( w = y( x(. Where, w is he maser weigh marix; n is he number of raining exemplars; w( is he weigh marix for a raining exemplar i; x(k is a vecor for preceding job a posiion k in he job sequence, and y(k is a vecor for succeeding job a posiion k+1 in he sequence for he exemplar i. L is he number of vecor pairs ha represen job sequencing knowledge for he exemplar i. To illusrae he consrucion of he neural weigh marix wih a simple example, suppose ha for a job se i, {1, 2, 3, 4, 5}, he opimal sequence is ( ). In his case, 4 vecor pairs ha represen he desirabiliy of job sequences are idenified from 4 pairs of adjacen jobs, (2 5), (5 3), (3 1), and (1 4). Vecor x1 for job 2 is represened as (0, 1, 0, 0, 0), and he corresponding vecor y1 for succeeding job 5 is (0, 0, 0, 0, 1). Similarly, vecor x2 for job 5 is represened as (0, 0, 0, 0, 1), and he corresponding vecor y2 for succeeding job 3 is (0, 0, 1, 0, 0). To vecor x3 for job 3 of (0, 0, 1, 0, 0), he corresponding vecor y3 for succeeding job 1 is (1, 0, 0, 0, 0). To vecor x4 for job 1 of (1, 0, 0, 0, 0), he corresponding vecor y4 for succeeding job 4 is (0, 0, 0, 1, 0). As a resul of he weigh marix calculaion, he values for w(25, w(53, w(31, and w(14 become 1, respecively. The maser neural weigh marix, w, hus represens he accumulaed job sequencing knowledge (i.e., generalized sequencing paerns) gained from he raining exemplars. Figure 3 shows he fully conneced wo-layer neural nework used for a real-ime sequencing decision. Once he neural weigh marix has been consruced, he neural nework shown in Figure 3 uilizes he marix o make real-ime sequencing decisions. The neural nework consiss of wo layers wih an equivalen number of processing elemens, and each processing elemen is fully conneced o all processing elemens in he oher layer. The L L 38

4 connecions beween x-layer for preceding jobs and y- layer for succeeding jobs hold he sored weigh informaion. For example, w 12 represens he desirabiliy of sequence from job 1 o job 2. To guaranee a feasible sequence of differen jobs, each job is no allowed o choose more han one oher job each as is successor and predecessor. Even hough he same se of jobs were no used in he iniial raining, he neural nework can consruc he sequence of jobs using he generalized sequencing paern. y-layer (Succeeding Jobs) W xy x-layer (Preceding Jobs) n W n Figure 3. A Neural Nework applied in he Real- Time Sequencing 4. EXPERIMENTAL STUDY We performed experimenal sudy o compare muli-neural-nework for simulaneous lo-sizing and sequencing wih sequencing-only neural nework. For he daa generaion, a maximum of 50 differen jobs were considered. Buffer size beween machines was fixed a 20. Processing imes for jobs on each machine were generaed randomly from a uniform disribuion wih a range [0.01, 1]. Sequence independen seup imes for jobs on each machine were generaed randomly from a uniform disribuion wih a range [0.01, 1]. Iniial lo sizes were generaed randomly [50, 120]. A oal of 9 problem ses were generaed, wih he number of machines se a 2, 3, and 4, and he number of jobs a 6, 10, and 14. For each problem se, 300 raining and 30 es problems were generaed. We evaluaed he consisency of he soluion qualiy using an average relaive percenage deviaion (ARPD) from he bes known soluion reurned in all he runs. ARPD and was calculaed by For he lo-sizing neural nework, parameers values are se as follows. The number of nodes in he hidden layer was se o 60. One bias uni is used for each layer. Weigh values have been uniformly generaed beween 0.3 and 0.3. Generalized dela rule is used for he updae of weigh value a each epoch. A each epoch, he sequence of raining examples has been randomly reshuffled. Maximum allowable oal error is se o zero and maximum epoch is se o Error of less han 0.2 was reaed as zero. One bias uni was included in each layer. The number of oupu node is se o one. W nn The soluion qualiy produced by each mehod is shown in Table 1. Table 1. Summary of Soluion Qualiy m-n Random Sequence Sequencing- Only Neural Nework Muli- Neural- Neworks m: number of machines in a flow-shop n: number of jobs The resuls show ha he soluion qualiy of he muli-neural-neworks was consisenly beer han hose of he oher wo mehods across all problem ses. The soluion qualiy of he randomly generaed schedule deerioraes as he number of machines increases. The soluion qualiy of he sequencing-only neural nework was consisenly beer han he randomly generaed sequence. (See comparison of sequencing-only neural nework and oher well-known heurisics in Lee and Shaw [12]). The soluion qualiy of he sequencing-only neural neworks also deerioraes as he number of machines increases. However, he soluion qualiy of he muli-neuralneworks is consisen across all problem ses. The number of jobs does no affec he soluion qualiy of all hree mehods. 5. CONCLUSION This paper considered he problem of simulaneously deermining sequence and lo-sizes of jobs ha arrive dynamically a a flow-shop. We showed ha our muli-neural-nework archiecure could be used as a framework for auomaed manufacuring decision suppor sysem. The mulineural-nework archiecure consiss of four ineracive modules: GA module, lo-sizing neural-ne, sequencing neural-nework, and real-ime user inerface. GA module is used o generae opimal/near opimal losizes and sequences for a given scheduling problem. Large number of exemplars are generaed from he GA module and fed ino wo kinds of neural neworks each of which learns lo-sizing and sequencing paerns, respecively. The sequence learning neural nework learns generalized sequencing paern for a flow-shop. In he 39

5 raining sage, a variey of inpus and desired oupus are given o he neural nework. The acquired sequencing knowledge is sored in he form of he neural weigh marix. The neural weigh marix associaes he desirabiliy of sequences beween jobs wih he magniude of weighs beween jobs. For he real-ime sequencing, he wo-layer neural nework uilizes he sequencing knowledge exraced from he neural weigh marix. The lo-sizing neural nework uses a radiional backpropagaion feedforward neural nework. To capure he knowledge, i uses aggregae four job characerisics on each machine. Afer rained by exemplars, he neural nework generaes he necessary esimaes of he lo-sizing parameer values ha are fed ino a user inerface module. The user inerface module combines boh sequences and losizes o make a final scheduling decision on a real ime basis. The resuls show ha he soluion qualiy of he muli-neural-neworks was consisenly beer han hose of he oher wo mehods across all problem ses. The soluion qualiy of he randomly generaed schedule rapidly deerioraes as he number of machines increases. The soluion qualiy of he sequencing-only neural neworks also deerioraes as he number of machines increases. However, he soluion qualiy of he muli-neural-neworks is consisen across all problem ses. 6. REFERENCES [1] Angeniol, B., de La Croix, G. Vaubois, and J. LeTexier, Self-Organizing Feaure Maps and he Traveling Salesman Problem, Neural Neworks, 1, 4, 1988, pp [2] Aras, D. A. and L. A. Swanson, "A Lo Sizing and Sequencing Algorihm for Dynamic Demands upon a Single Faciliy," Journal of Operaions Managemen, 2, 1982, pp [3] Bahl, H. C., L. P. Rizman, and J. N. D. Gupa, "Deermining Lo Sizes and Resource Requiremens: A Review," Operaions Research, Vol. 35, No. 3, May-June 1987, pp [4] Baker, K. R., Inroducion o Sequencing and Scheduling, Wiley and Sons Inc., New York, [5] Burke, L. I., "Neural Mehods for he Traveling Salesman Problem: Insighs From Operaions Research," Neural Neworks, Vol. 7, No. 4, 1994, pp [6] Eisenhu, P. S., "A Dynamic Lo-Sizing Algorihm wih Capaciy Consrains," AIIE Trans., 7, 1975, pp [7] Gorham, T., "Dynamic Order Quaniies," Producion and Invenory Managemen, 10, 1968, pp [8] Graves, S. C., "Muli-Sage Lo Sizing: An Ieraive Procedure," In Muli-Level Producion/Invenory Conrol Sysem: Theory and Pracice, L. B. Schwarz (Ed.), Norh-Holland, Amserdam, 1981, pp [9] Gupa, J., R. Sexon, and E. Tunc, Selecing Scheduling Heurisics Using Neural Neworks, INFORMS Journal on Compuing, Vol. 12, No. 3, Spring 2000, pp [10] Hopfield, J. J. and D. W. Tank, Neural Compuaion of Decisions in Opimizaion Problems, Biological Cyberneics, 52, 3, 1985, pp [11] Kohonen, T., Self-Organizaion and Associaive Memory, Springer-Verlag, Berlin [12] Lee, I. And M. Shaw,, A Neural-Ne Approach o Real Time Flow-Shop Sequencing, Compuers and Indusrial Engineering, 38, 2000, pp [13] Moron, T. and D. W. Penico, Heurisic Scheduling Sysems wih Applicaion o Producion Sysem and Projec Managemen, John Wiley & Sons Inc., New York, [14] Park, Y., S. Kim, and Y. Lee, Scheduling Jobs on Parallel Machines Applying Neural Nework and Heurisics Rules, Compuers and Indusrial Engineering, 38, 2000, pp [15] Rumelhar, D. and J. L. McClelland and PDP Research Group, Parallel Disribued Processing - Exploraions in he microsrucure of Cogniion, Volume I: Foundaions, The MIT Press, [16] Silver, E. A. and H. C. Meal, "A Heurisic Selecing Lo Size Requiremen for he Case of a Deerminisic Time Varying Demand Rae and Discree Opporuniies for Replenishmen," Producion and Invenory Managemen, 14, 1973, pp [17] Wagner, H. M. and T. M. Whiin, "Dynamic Version of he Economic Lo Size Model," Managemen Science, 5, 1958, pp [18] Wang, J., "A Deerminisic Connecionis Machine for he Traveling Salesman Problem," Proceedings of IEEE Inernaional Conference on Sysems, Man, and Cyberneics, 1990, pp