Opimizing Muli-Sie Producion Planning and Scheduling wih a Hybrid Geneic Algorihm 5 Opimizing Muli-Sie Producion Planning and Scheduling wih a Hybrid Geneic Algorihm Chyuan Perng 1, Ping-Teng Chang 1, Shui-Shun Lin 2, Chun-Chia Huang 1 and Zih-Ping Ho 1* 1 Deparmen of Indusrial Engineering & Enerprise Informaion Tunghai Universiy, Taichung Ciy, Taiwan, ROC 2 Deparmen of Business Adminisraion Naional Chin-Yi Universiy of Technology, Taiwan, ROC *Corresponding Auhor: c8880@ms21.hine.ne ABSTRACT As he economic environmen coninues o change rapidly, so does he demand placed on indusries o mee ever expanding orders. In order o mee his demand, he need o expand is mached by he need o uilize exising faciliies, and increase efficiency. As a resul, convenional order managemen programs ofen no longer suffice. Single-sie planning has developed ino muli-sie planning, leading o disance problems beween sies as well as problems regarding informaion ransfers beween hem. This sudy aims a consrucing a decision model of an inegraed muli-sie producion scheduling problem. To suppor muli-sie facories wih heir mass orders, based on he premise ha hey were under oal order managemen sysems, he decision model considered such complicaed facors as he produc marke feaures, due dae, producion scheduling, and order profi and capaciy of each sie. Mos real-world scheduling problems involve muliple objecives which may be conflicing wih each oher. In addiion, he effec facors aken ino accoun by previous muli-objecive scheduling research are essenially quaniaive facors. However, more qualiaive facors also have o be considered relaed o organizaions operaing messages. We propose an inegraed producion scheduling model and use hybrid geneic algorihm mehods as soluion procedures. Keywords: Muli-Sie Scheduling Problem, Assignmen Problem, Geneic Algorihm, Muli-Objecive Scheduling, Allocaion Problem
6 C. Perng, P. Chang, S. Lin, C. Huang, and Z. Ho INTRODUCTION In business, producion aciviies play an imporan role in deermining a company s operaing cos. In modern managemen, organizaions inerac wih he environmen and pursue objecives according o heir specific mission (Tavares, 2000). Scheduling is an imporan ool for manufacuring and engineering, as i can have a major impac on he produciviy of a process. In manufacuring, he purpose of scheduling is o minimize producion ime and coss by direcing a producion faciliy regarding wha o make, when, wih which saff, and on which equipmen. Producion scheduling aims o maximize he efficiency of he operaion and reduce coss (Pinedo, 2005). In general, he producion plan and scheduling of orders are boh very complicaed. Muli-sie scheduling lieraure was firs poined ou in Thierry e al. (1995), who saed ha problems could be solved by improving coordinaion beween differen producion unis in he field of producion planning and conrol for a muli-sie producion. A muli-sie producion akes place when he producion faciliies of an inernaional company are locaed in differen geographical sies. To solve his problem, a cenralized approach is chosen and a muli-sie planning inegraed sysem is buil in relaion o he local planning and conrol sysems of he differen producion sies. Bullinger e al. (1997) proposed an objec-oriened model o projec muli-sie producion. Sauer e al. (1998) saed ha global level daa were normally aggregaed, imprecise, or esimaed. Mos previous mehods focused on local producion sies wihou giving consideraion o he coordinaion issue. They proposed a global view of muli-sie company operaions. During he las en years, various papers have discussed muli-sie scheduling problems. However, here is sill a wide range of problems ha exis due o differen ypes of facories each having is own se of prioriies. The firm in our empirical sudy in is a machine ool company locaed in cenral Taiwan. This sudy aims a consrucing a decision model of an inegraed muli-sie producion scheduling problem, and a novel approach hybrid geneic algorihm is applied ino his combinaorial problem. In his empirical sudy, we demonsrae ha firms can improve heir profis hrough reducing coss such as penalies for job ardiness, ec. The res of he paper is organized as follows: Secion 2 illusraes he lieraure review; secion 3 depics he consrucion of a scheduling model; secion 4 describes he heurisic process and he conclusion is summarized in secion 5. LITERATURE REVIEW During he las decade, a variey of muli-sie scheduling echniques have been developed and applied in pracice. Pirkul and Jayaraman (1998) presened a mixed ineger programming formulaion for he supply chain managemen problem wih capaciaed plans and warehouses. They proposed an efficien heurisic based on Lagrangian relaxaion of he muli-sie scheduling problem. Roux e al. (1999) repored a mehod which alernaed beween solving a planning problem in which lo-sizes were compued for a given sequence of jobs on each machine, and a scheduling problem in which sequences were compued for each sie. The lo-sizing and scheduling problems can also be solved in parallel. Vercellis (1999) proposed he adapaion of maser producion planning (MPS) conceps o muli-sie producion scheduling. Timpe and Kallrah (2000) proposed a
Opimizing Muli-Sie Producion Planning and Scheduling wih a Hybrid Geneic Algorihm 7 mixed linear ineger programming model o a muli-sie scheduling nework. Guine (2001) suggesed a wo-level producion managemen o conrol muli-sie producion sysems divided ino global muli-sie planning and local muli-sie scheduling. Sambasivan and Schmid (2002) presened a heurisic procedure for solving muli-plan, muli-iem, capaciaed lo sizing problems wih iner-plan ransfers. The soluion procedure used he soluion for he non-capaciaed problem as a saring poin. Moon e al. (2002) proposed an inegraed process planning and scheduling (IPPS) model for he muli-plan supply chain (MSC), which behaves liked a single company hrough srong coordinaion and cooperaion oward muual goals. Leung e al. (2003) addressed he problem of aggregae producion planning (APP) for a mulinaional lingerie company in Hong Kong. The muli-sie producion planning problem considered he producion loading plans among manufacuring facories subjec o cerain resricions, such as producion impor/expor quoas imposed by he regulaory requiremens of differen naions, he use of manufacuring facories/locaions wih regard o cusomer preferences, as well as producion capaciy, workforce levels, sorage space and resource condiions of he facories. In ha paper, a muli-objecive model was developed o solve he producion planning problems, in which he profi was maximized bu producion penalies resuling from going over/under quoas and he change in workforce level were minimized. Oher relaed sudies included a muli-sie scheduling sysem proposed by Gnoni e al. (2003), a sysemaic approach o solving he muli-sie resources planning problem proposed by Papageorgiou (2004), and a deerminisic model for solving a muli-sie and muli-warehouse problem repored by Jolayemi and Olorunniwo (2004). A geneic algorihm (GA) is a search echnique used in compuing o find exac or approximae soluions o opimizaion and search problems. Geneic algorihms are caegorized as global search heurisics. Geneic algorihms are a paricular class of evoluionary algorihms ha use echniques inspired by evoluionary biology such as inheriance, muaion, selecion, and crossover (Schmi, 2001). GA was inroduced as early as 1954 by Nils Barricelli. Geneic algorihms are implemened as a compuer simulaion in which a populaion of absrac represenaions (called chromosomes or he genoype or he genome) of candidae soluions (called individuals, creaures, or phenoypes) o an opimizaion problem evolves oward beer soluions (Koza, 1992). The soluions are represened in binary as srings of 0 and 1, bu oher encodings are also possible. The evoluion usually sars from a populaion of randomly generaed individuals and happens over generaions. In each generaion, he finess of every individual in he populaion is evaluaed, muliple individuals are sochasically seleced from he curren populaion (based on heir finess), and modified (recombined and possibly randomly muaed) o form a new populaion. The new populaion is hen used in he nex ieraion of he algorihm (Holland, 1975). The algorihm erminaes when eiher a maximum number of generaions has been produced, or a saisfacory finess level has been reached for he populaion. If he algorihm has erminaed due o a maximum number of generaions, a saisfacory soluion may or may no have been reached. I ypically conains: a geneic represenaion and a finess funcion o be evaluaed (Vose, 1999). Holland (1976) simulaed naural selecion and proceeded wih crossover and muaion. His research has become a popular ciaion. Our sudy call GA module form, he Malab library, is based on his research. In recen years, GA is ofen muaed by oher algorihms o design a new hybrid geneic algorihm (HGA), such as abu search (Diaz e al., 2008; Degerekin e al., 2008; Drezner, 2008). We also design a HGA for our research problem.
8 C. Perng, P. Chang, S. Lin, C. Huang, and Z. Ho Alhough some lieraures are available on muli-sie scheduling problems, here is only a small amoun of his lieraure focused on machine ool indusries. This is also he case wih regard o he lieraure available on discussing hybrid geneic algorihm applied o muli-sie scheduling problem. Besides, our objecive is derived from real-world scheduling problems. This sudy would offer muli-sie firms furher informaion for reducing he ardy penalies. A SCHEDULING MODEL This research esablishes a non-linear mahemaical program which simulaes he srucure of muli-sie producion scheduling. Through spliing orders of pars of each generaion in differen saellie plans, he evoluion of sequence of order assignmen of pars in each generaion reached opimized performance. The model is described as follows: Noaions p : order of pars number ( p1,, P); : producion sage ( 1,, T ); i : producion sage number of order of pars ( i1,, I ); i j : sie number of producion sage of order of pars ( j 1,, J ); m : machine number ( m1,, M J i ) Parameers P : processing ime a order of pars p, order of par sage i, sie j, machine m, sage S : seup ime a order of pars p, order of par sage i, sie j, machine m, sage Q : quaniy a seup ime a order of pars p, order of par sage i, sie j, machine m, sage cap : capaciy a order of par sage i, sie j, sage cap m : capaciy a order of par sage i, sie j, machine m DD p : due dae of order of pars number p 1 : weighed facor of machine uilizaion : weighed facor of due dae 2 : weighed facor of makespan 3 W : weighed facor of quaniaive impac 1 W : weighed facor of qualiaive impac 2 ; W 1 2 3 1 1W2 1
Opimizing Muli-Sie Producion Planning and Scheduling wih a Hybrid Geneic Algorihm 9 Decision Variables L l p np l np : size a order of pars p, order of par sage i, sie j : spliing numbers of size a order of pars p, order of par sage i, sie j Q p ) L p T : sar ime a order of pars p, order of par sage i, sie j, machine m, sage C : compleion ime a order of pars p, order of par sage i, sie j, machine m, sage UC ( ): saisfacion in compleion ime a order of pars p, order of par sage i, sie j, machine m, sage Objecive Funcion i I J M J i N P S L p l np P(, ) 1 i j 1 M i 1 1 1 1 1 J T i j m n p C Max W1 m m1 1 i I J M J i i P M i UC ( min (, ) ) P i j C T I J J 2 3 i1 j1 m1 p1 p i1 j1 m1 p1 C W2 (1 p( x)) A simplified version of he objecive funcion can be saed as: Max W 1 ( 1 (average uilizaion of machine)+ 2 (average saisfacion of compleion ime)+ 3 (makespan performance)+ W 2 (1-penaly funcion) Consrains (1) Due dae consrains C T DD i, j, m, p, p (2) Consrains of bach processing ime C T S T L l P i, j, m, p, p np MAX (3) The same operaion wih differen machines T S T L l P i, j, m, p, ( i, j ) p np (4) Differen operaions wih he same machine T S T L l P i, j, m, p, p np
10 C. Perng, P. Chang, S. Lin, C. Huang, and Z. Ho (5) Consrains of Capaciies S + T L l P cap m i, j, m, p, p np m (6) Consrains of lo sizing L l Q i, j, m, p, p np (7) Non-negaive consrains L i, j, p p 0 T i, j, m, p, 0 C i, j, m, p, 0 UC ( ) 0 i, j, m, p, Descripion of objecive funcion The objecive funcion designed in his sudy is esablished across wo pars: a muli-objecive funcion and a penaly funcion. The muli-objecive funcion is he performance indicaor, o evaluae he allocaion of he sie, while he penaly funcion is obained from he sequence. The weighs in objecive funcion, W1 and W 2, were derived by he Analyic Hierarchy Process (AHP). Descripion of consrains Equaion (1) illusraes ha compleion ime subracs he release ime, should be shorer han due dae. Eq. (2) describes ha when one manufacures he order of pars, due o he resricion of bach processing ime of machines, one mus saisfy is consrains. Eqs. (3) and (4) sae he consrain of sequences of manufacure of orders of pars. Eq. (5) describes manufacure bach of order of pars should saisfy he consrain of plan capaciy. Eq. (6) saes ha lo-size should saisfy he machine capaciies. Eq. (7) illusraes non-negaiviy. Heurisic Soluion Process The heurisic soluion process is described as shown in Figure 1. The GA (Holland,1976) was described in Figure 1 wihou abu search. I spends oo much compuaion ime, and does no produce a good opimal soluion. Tabu is used for exclusion of some populaion (Cvijovic and Klinowski, 1995). In our hybrid geneic algorihm, we successfully add some resricions of abu consideraion in order o accelerae he process and o generae a beer opimal soluion. Our Hybrid Geneic Algorihm (HGA) is saed as follows: Sep 1: 1.1 Randomly selecs an operaion o proceed. 1.2 The se of sequence defines as S. Sep 2: 2.1 The size of he sequence is M, and is searched Z imes. 2.2 The opimal is sequence S*. 2.3 The objecive funcion is G(S*)=G(I). 2.4 Se S 1 =S and Z=1.
Opimizing Muli-Sie Producion Planning and Scheduling wih a Hybrid Geneic Algorihm 11 Sep 3: Search for S z s neighborhood, S 1, S 2,, S N-1 and filling S 1, S 2,,S N-1, back in se. Check if he above violaes he resricion of sequence; oherwise, proceed wih adjusmen. If no, hen calculaes is objecive funcion values G(S 1 ), G(S 2 ),, G(S N-1 ), where N is he job number. Sep 4: 4.1 From G(S 1 ), G(S 2 ),, G(S N-1 ), selecs he opimal value G(S z * ), where i does no belong o he abu lis. 4.2 If G(S*)< G(S z * ), hen S* = S z *. Sep 5: 5.1 Updaes he abu lis by FIFO (firs-in-firs-ou); 5.2 Z++; 5.3 If z = Z, hen Sop. Sep 6: 6.1 Replaces iniial se S by S *, and proceeds wih he GA procedure. Figure 1: Flow Char of a Hybrid Geneic Algorihm
12 C. Perng, P. Chang, S. Lin, C. Huang, and Z. Ho RESULTS AND DISCUSSION The experimens are processed by Malab 6. The GA is creaed by calling he GA module from he Malab library. The HGA is coded from Malab programming and based on he seps as menioned in secion 4. Afer inensive calculaions using a Penium 4 PC, he opimal scheduling resul is compued. The muli-objecive values are shown in Tables 1 o 6, and plo a Figure 2. To compare he uilizaion of a machine using GA (in Table 1) and HGA (in Table 2), HGA oally ouperforms GA. The managerial meaning is increasing he uilizaion of he machine. When comparing he saisfacion of compleion ime using GA (in Table 3) and HGA (in Table 4), once again he HGA ouperforms GA. The managerial meaning for his is an increase in overall cusomer saisfacion due o a reducion of ardiness. To compare he makespan of GA (in Table 5) and HGA (in Table 6), he HGA ouperforms GA on a consisen basis. The managerial meaning is a reducion in he overall operaional cos (such as elecric power or manpower fees). Therefore, in Figure 2, he objecive value of HGA is beer han GA. Table 1: Uilizaion of Machine (GA) M1 M2 M3 M4 M5 M6 M7 M8 Sie 1 0.6753 0.7716 0.4991 0.4776 0.7609 0.6574 0.6549 0.6470 Sie 2 0.6489 0.5806 0.5294 0.3461 0.5635 0.3910 0.5672 0.6898 Sie 3 0.5924 0.5161 0.4293 0.3051 0.4171 0 0.4425 0.5882 Table 2: Uilizaion of Machine (HGA) M1 M2 M3 M4 M5 M6 M7 M8 Sie 1 0.6148 0.8117 0.9050 0.4811 0.6272 0.5948 0.5617 0.8467 Sie 2 0.7963 0.5806 0.5865 0.8267 0.5665 0.4607 0.5332 0.5400 Sie 3 0.3889 0.5161 0.7747 0.3051 0.4496 0.6632 0.4307 0.5882 Table 3: Saisfacion of Compleion Time (GA) Order No. 1 2 3 4 5 6 7 8 9 10 Avg Due Dae 55 100 104 121 60 72 80 110 77 50 Compleion Time 86.1 85.5 81.5 77 48 70.3 81.3 93.5 93.1 52.5 Due Dae Saisfacion 0 1 1 1 1 1 0.87 1 0 0.75 0.762 Table 4: Saisfacion of Compleion Time (HGA) Order No. 1 2 3 4 5 6 7 8 9 10 Avg Due Dae 55 100 104 121 60 72 80 110 77 50 Compleion Time 106.5 85.5 80.8 78.8 61.8 64.8 80.3 90.6 61.6 54.5 Due Dae Saisfacion 0 1 1 1 0.82 1 0.97 1 1 0.55 0.834
Opimizing Muli-Sie Producion Planning and Scheduling wih a Hybrid Geneic Algorihm 13 Table 5: Makespan Performance (GA) RT: Release ime CT: Compleion ime
14 C. Perng, P. Chang, S. Lin, C. Huang, and Z. Ho Table 6 Makespan Performance (HGA) RT: Release ime CT: Compleion ime
Opimizing Muli-Sie Producion Planning and Scheduling wih a Hybrid Geneic Algorihm 15 0.88 HGA vs GA HGA GA 0.86 Max. Finess Funcion Value 0.84 0.82 0.80 0.78 0.76 0.74 0.72 0 10 20 30 40 50 60 70 80 90 100 Generaion Figure 2: The Comparison of HGA and GA (Objecive Value) In a scheduling problem, i represens a sequence. 100 generaions represen 100 differen sequence resuls. In GA, he same sequence may exis wihin 100 differen sequences. However, he same sequence will no exis in HGA due o he hybrid s abu lis. In Figure 3, under he same number of generaions (100), HGA is beer han GA in each bach of orders (10, 20, 30). We force he program o sop under a pre-defined compuaion ime, and HGA is beer han GA in each bach of orders (10, 20, 30) in Figure 4. 1.0 0.9 MAX. Finess Funcion Value 0.8 0.7 0.6 0.5 3-GA 3-HGA 4-GA 4-HGA Number of Plan-Algorihm 10 20 30 Figure 3: Performance Comparison of Differen Number of Orders (under he same generaions)
16 C. Perng, P. Chang, S. Lin, C. Huang, and Z. Ho 1.0 0.9 MAX. Finess Funcion Value 0.8 0.7 0.6 0.5 3-GA 3-HGA 4-GA 4-HGA Number of Plan-Algorihm Figure 4: Performance Comparison of Differen Number of Orders (under he same compuaion ime) 10 20 30 In Table 7, we show an empirical applicaion of scheduling. Based on his recommendaion, he firm can easily deal wih heir orders. Table 7: Orders Sequence Order No. 1 2 3 Operaion Number of Release Compleion Sie No. Baches Time Time 1 2 2, 3 0 11 2 1 2 11 19 3 2 1, 3 48 58.5 4 1 3 58.5 73.5 5 1 1 73.5 86.5 6 1 1 86.5 106.5 1 3 1, 2, 3 0 7.6 2 2 2, 3 11 19.5 3 3 1, 2, 3 22 26.6 4 2 1, 2 54.1 62.1 5 1 2 62.1 79.1 6 3 1, 2, 3 79.1 85.5 1 1 1 0 21 2 1 3 21 39 3 1 1 39 46 4 1 1 52.8 68.8 5 2 1, 2 68.8 80.8
Opimizing Muli-Sie Producion Planning and Scheduling wih a Hybrid Geneic Algorihm 17 4 5 6 7 8 9 10 1 1 1 0 11 2 1 1 11 28 3 1 1 28 38 4 1 3 39 58 5 2 1, 2 58.5 73.5 6 3 1, 2, 3 73.5 78.8 1 1 3 0 15 2 2 2, 3 19.5 29 3 1 3 29 48 4 2 2, 3 52.8 61.8 1 1 2 0 17 2 1 1 17 26 3 2 1, 2 26 34.5 4 2 1, 2 34.5 46.5 5 3 1, 2, 3 46.5 52.8 6 1 1 52.8 64.8 1 3 1, 2, 3 3 8 2 1 2 28.6 48.6 3 2 1, 2 48.6 59.6 4 3 1, 2, 3 59.6 66.6 5 2 1, 3 68.8 80.3 1 2 1, 3 0 7 2 3 1, 2, 3 14 19 3 1 1 28.6 45.6 4 2 1, 2 45.6 54.1 5 2 1, 3 61.6 68.6 6 1 3 68.6 90.6 1 3 1, 2, 3 0 3 2 1 1 3 22 3 3 1, 2, 3 22 28.6 4 1 1 28.6 35.6 5 1 2 35.6 50.6 6 2 1, 2 50.6 61.6 1 1 1, 2, 3 7.6 12 2 1 2 12 23 3 2 1, 2 29 36.5 4 1 1 36.5 44.5 5 2 2, 3 44.5 54.5 CONCLUSION Producion aciviies play an imporan role in deermining a company s operaing cos. There are more and more orders being made-o-order. During he las decade, a variey of muli-sie scheduling echniques have been developed and applied in pracice. However, here are few lieraures focused on machine ool indusries as well as lile
18 C. Perng, P. Chang, S. Lin, C. Huang, and Z. Ho discussing he hybrid geneic algorihm applied o muli-sie scheduling problem. In his sudy, our objecive is derived from real-world scheduling problems. This sudy would offer muli-sie firms an increase in machine uilizaion, increasing cusomer saisfacion, and decreasing operaional cos. A decision model of an inegraed muli-sie producion scheduling problem is proposed, and a novel heurisic (HGA) approach is also proposed o his problem. Based on he above experimens, he objecive value of he HGA consisenly ouperforms he GA. This sudy conribues in academic research of managemen and also in empirical form. We propose ha for fuure sudies, he indicaor of scheduling performance, such as he number of ardy jobs or oal flow ime, could be added ino his muli-objecive model. The objecive value of his sudy can be furher improved by oher approaches if he resuls are used as he basis for a comparison in fuure sudies. REFERENCES Barricelli, N.A. (1954). Esempi numerici di processi di evoluzione. Mehodos, 45-68. Bullinger, H.J., Faehnrich, K.P. & Laubscher, H.P. (1997). Planning and muli-sie producion An objec-oriened model. Inernaional Journal of Producion Economics, 51, 19-35. Cvijovic, D. & Klinowski, J. (1995). Taboo search - an approach o he muliple minima problem. Science, 267, 664-666. Diaz, E., Tuya, J., Blanco, R. & Dolado, J.J. (2008). A abu search algorihm for srucural sofware esing. Compuers and Operaions Research, 35, 3052-3072. Degerekin, S.O., Saka & Hayalioglu, M.S. (2008). Opimal load and resisance facor design of geomerically nonlinear seel space frames via abu search and geneic algorihm. Engineering Srucures, 30(1), 197-205. Drezner, Z. (2008). Exensive experimens wih hybrid geneic algorihms for he soluion of he quadraic assignmen problem. Compuers and Operaions Research, 35(3), 717-736. Gnoni, M.G., Iavagnilio, R., Mossa, G., Mummolo, G. & Di Leva, A. (2003). Producion planning of a muli-sie manufacuring sysem by hybrid modelling: A case sudy from he auomoive indusry. Inernaional Journal of Producion Economics, 85, 251-262. Guine, A. (2001). Muli-sie planning: A ransshipmen problem. Inernaional Journal of Producion Economics. 74, 21-32. Holland, J.H. (1975). Adapaion in Naural and Arificial Sysems. Universiy of Michigan Press. Holland, J.H., (1976). Geneic algorihm opimal allocaion of rails. SIAM Journal of Compuing, 5, 704-714. Jolayemi, J.K. & Olorunniwo, F.O. (2004). A deerminisic model for planning producion quaniies in a muli-plan, muli-warehouse environmen wih exensible capaciies. Inernaional Journal of Producion Economics, 87, 99 113. Koza, J. (1992). Geneic Programming: On he Programming of Compuers by Means of Naural Selecion, MIT Press. Levis, A.A. & Papageorgiou, L.G. (2004). A hierarchical soluion approach for muli-sie capaciy planning under uncerainy in he pharmaceuical indusry. Compuers and Chemical Engineering, 28, 707 725.
Opimizing Muli-Sie Producion Planning and Scheduling wih a Hybrid Geneic Algorihm 19 Leung, S.C.H., Wu, Y. & Lai, K.K. (2003). Muli-sie aggregae producion planning wih muliple objecives: A goal programming approach. Producion Planning & Conrol, 14 (5), 425 436. Moon, C., Kim, J. & Hur, S. (2002). Inegraed process planning and scheduling wih minimizing oal ardiness in muli-plans supply chain. Compuers & Indusrial Engineering, 43, 331-349. Pirkul, H. & Jayaraman, V. (1998). A muli-commodiy, muli-plan, capaciaed faciliy locaion problem: formulaion and efficien heurisic soluion. Compuers and Operaions Research, 25 (10), 869-878. Roux, W., Dauzere-Pereas, S. & Lasserre, J.B. (1999). Planning and scheduling in a muli-sie environmen, Producion Planning & Conrol, 10 (1), 19-28. Sauer, J., Suelmamn, G. & Appelrah, H.J. (1998). Muli-sie scheduling wih fuzzy conceps. Inernaional Journal of Approximae Reasoning, 19, 145-160. Sambasivan, M. & Schmid, C.P. (2002). A heurisic procedure for solving muli-plan, muli-iem, muli-period capaciaed lo-sizing problems. Asia-Pacific Journal of Operaional Research, 87-105. Schmi, L.M. (2001) Theory of Geneic Algorihms. Theoreical Compuer Science, 259, 1-61. Thierry, C., Besnard, P., Ghaas, D. & Bel, G. (1995). Muli-sie planning: non flexible producion unis and se-up ime reamen, Emerging Technologies and Facory Auomaion. ETFA '95 Proceedings., 1995 INRIA/IEEE Symposium, 3, 261 269. Timpe, C.H. & Kallrah, J. (2000). Opimal planning in large muli-sie producion neworks. European Journal of Operaional Research, 126, 422-435. Vercellis, C. (1999). Muli-plan producion planning in capaciaed self-configuring wo-sage serial sysems. European Journal of Operaional Research, 119, 451-460. Vose, M.D. (1999). The Simple Geneic Algorihm: Foundaions and Theory, MIT Press. Chyuan Perng is an associae professor in he Deparmen of Indusrial Engineering and Enerprise Informaion a Tunghai Universiy, Taiwan. He received his Ph.D. degree in Indusrial Engineering from Texas Tech Universiy, USA. He has also paricipaed in several indusrial and governmenal projecs in Taiwan. Ping-Teng Chang is a Professor in he Deparmen of Indusrial Engineering and Enerprise Informaion a Tunghai Universiy, Taiwan. He received his Ph.D. degree in Indusrial Engineering from Kansas Sae Universiy, USA. He was head of he same deparmen from 2004 o 2007, and currenly Edior-in-Chief of he Inernaional Journal of Operaions Research. Shui-Shun Lin is an Associae Professor in he Deparmen of Business Adminisraion, Naional Chin-Yi Universiy of Technology, Taiwan. He received his Maser of Science degree from he Universiy of Wisconsin-Madison, and Ph.D. degree in Indusrial Engineering from he Florida Sae Universiy, USA. He is he founder of he Susainabiliy and Innovaion Managemen Laboraory and has paricipaed in numerous projecs sponsored by indusry. His curren research ineress include Design for Susainabiliy, Sof Compuing Techniques, and Knowledge Managemen.
20 C. Perng, P. Chang, S. Lin, C. Huang, and Z. Ho Chun-Chia Huang graduaed from he Deparmen of Indusrial Engineering and Enerprise Informaion a Tunghai Universiy, Taiwan. He is currenly serving his duy of Naional Service in he Taiwanese armed forces. Zih-Ping Ho is currenly a Ph.D. suden in he Deparmen of Indusrial Engineering and Enerprise Informaion a Tunghai Universiy, Taiwan. He is also a lecurer in he same deparmen.