Expt'n Systems WJth Applic,UIOns 37 {2010) 6919-6930

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Ying hua, c. (2010): adopting co-evolution and constraint-satisfaction concept on genetic algorithms to solve supply chain network design problems

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Ying hua, c. (2010): adopting co-evolution and constraint-satisfaction concept on genetic algorithms to solve supply chain network design problems

  1. 1. Expt'n Systems WJth Applic,UIOns 37 {2010) 6919-6930 Contents lists available at ScienceOirect Expert Systems with Applications jou rna I hom ep age: www. elsevier .co m/locate /e swa Adopting co-evolution and constraint-satisfaction concept on genetic algorithms to solve supply chain network design problems Chang Ying-Hua • Deparrmenr of /nformarian Management. TomKang Vnrversrry, 151 Ymg-fhuan Rood. Tamsui. Taipei County 251 J7, Taiwan, ROC ART ICLE INFO Keywords: Supply ch<ain network design Genetic Algorithms Co-evolution concept Consrraint~sat i sf<action concept Op11miution ABSTRACT With the r.apid glob.lliz.ation of m.arkets, integrating supply chain technology h.as become increasingly complex. That is. most supply chains are no longer limited to a particular region. Because the numbers of branch nodes of supply chains have increased, products and raw materials vary and resource constrainrs differ. Thus, integrating planning mechanisms should include the capacity to respond to change. In the past. mathematical programming <md a general heuristics algorithm were used to so lve globalized supply chain network design problems. When mathematical programming is used to solve a problem and the number of decision variables is too high or constraint conditions are too complex, computation time is long, resulting in low efficiency, and can easily become trapped in partia l optimum solution. When a general heuristics algorithm is used and the number of variables and constraints is too high, the degree of complexity incre.1ses. This usu.ally results in .an in.ability of people to think about resource constraints of enterprises .and obtain an optimum so Iutton. Therefore. this study uses genetic algorithms with optimum search features. This work combines the co-evolutionary mode. which is in accordance with various criteria and evolves dynamically. and consrrainH<Hisfaction mode capacity to narrow the search space, which helps in finding rapidly a solution that, solves supply chain integration network design problems. Additionally, via mathematical programming, .a si mple genetic .J:Igorithm. co-evolutionary genetic .algorithm. constr.aint-satisfaction generic algorithm and co-evolut ionary constraint generic algorithm are used to compare the experiments result and processing time to con finn the perform.mce of the proposed method. e 2010 Elsevier ltd. All rights resetved. 1. Introduction With the rapid development of a global economy, info rmation and manufacturing technology. the needs of consumers. production and sales have changed. Enterprises must now think globally. Independent operation modes are no longer suitable for markets undergoing upheaval s. The integration of ind ustries and the division of work has become the best so lution for enterprise suJVival. To create an efficiem globa l supply chai n. resources. the supply cha in and all fa ctories must be tightly integrated. Additionally, firms must respond to customer requirements efficiently, offer high-quality products, reduce operational cos ts and increase customer sa ti sfaction. Beca use most supply chain nerwork designs have multiple layers. members. periods and products. and a comparative resource constraint exists between different layers. For example, a factory may have constra ints on its productivity. and a distribution center may have limited storage ca pacity. These problems typically in• Tel.: +886 911 23IS3G: f<ilc:: +886 2 26209737 E-mail address: yhch;mgGJ'mdii.fku.('du.rw crease as the numbe r of supply chain layers increase, the time period increases. and the number of prod ucts and purchase orders increase. These cause the network search space and time required to obtain a sol ution to increase markedly. Thus, the supply cha in network design problem is an NP-complere problem (lbaraki & Katoh. 1988). Many studi es adopted mathema tica l programming or a heuristics algorithm to solve such problems. For example. Robinson .1nd S<1tterficld ( 1998) establ ished a mixed- imeger program mi ng solution for a two-echelon, multip le products storage problem. Hou and Ch~ng (2 003 ) use the evo lution principle to so lve a production and distribution problem. When faced with a simple supply chain system, mathematical programming ca n easily find the optimum solution; however. mathematical program mi ng is not suitable for large mixed -in teger programming problems. By adopt in g a single poi nt r.:tndom sedfch, the number of decision variables increases, which extends the rime required to find the solution and the method can easily become trapped in a local optimum solution (Wu. 2002). When adopting a general heuristics algorithm, such as simulated .:tnneali ng, the search for the optimum so lution cannot be reached completely and efficiently (Ait ipannak. Gen. & Lin. 2005; Wang, 2002). 0957-4174{S- see front m<atter ~ 2010 Elsevier Lid. All rights reserved. doi: 10.10 6 /J.CSW<a.201 O.oJ.o30
  2. 2. 6920 Y.-H. Chang/Ex~rr Systems ""'irh App/icalions 37 (1010) 6919- 6930 This study uses the features of ge netic algorithms (parallel searc hing and can evolve in an environment ) combined with the co-evolutionary mode (cons idering multiple criteria. search speed .:~nd avoiding converge before the right time) and the constraintsatisfaction mode (narrow the sea rch space accelerate the speed at which the optimum solution is acquired ) ro establ ish sup ply chain system for all manufacturing branches. The minimum cost of a supply chain network is adopted as the most suitable plan for developing an algorithm for a supp ly chain network distribution. The proposed method is a rapid way of find ing a solution and obtaining optimum searc h capacity. To verify the accuracy and efficacy of the proposed algo rithm. LINGO software was used in mathematica l programming mode. The experimental result was compared with the algorithmic result to determine the difference in cost and calculation time. This comparison c.:m verify the efficiency of the proposed algo rithm such that a nexib le and efficient supply chain integration network mode can be obtained. This result can function as a reference for man ufacturers in daily operations and for management decisions. 2. Literature review This study focuses on supply chain integration and its network design problem. Relrtted literature is discussed in the following sequence: the sup pl y chain integration model: genetic algorithms; co-evo lutionary mode; and. co nstraint-satisfaction mode. 2. 1. Supply chain integration mode Supply chain nerworks obtain ra w materials, supplies, manufactures and distributes product to customers. All these planning, organizing and controlling from the supply point to the demand point include cash flows, information flows and materials circulation (M aloni & Benton. 1997}. By combining strategies to ach ieve integration of ideality and rea li ry of different enterprises to form a reticulation supply chain network. Bec.:wse different enterprises in a supply chain may have different or conflicting targets, and supply chains must adju st to changes. integration of a supply chain is not easily achieved (Hayashi. Koguro, & Murakami, 2005; liu & Hsieh. 2005; Su. 2000; Sue, 1999; Vaidyanathan & Hasan. 2001; Viswanathan & Piplani, 2001 }. Thus. supply chain members must be coordinated and cooperate to achieve a common target. Additionally. a dynamic integration mode must be established that can respond effectively to changes in global markets. As supply chain management is a complex decision problem, many rese.uchers adopted mathematica l programming or a heuristics algorithm to solve this problem. For instance, Robinson and Sa tterfield ( 1998) established a mixed-integer programming model that maximizes profit and is designed to find the solution for a single period and single product in a supply chain network. Koray and Marc (1999) proposed a multiple products mixed-integer programming mode for minimum cost that included manufacturing cost. storage cost and transport cost. They used Bender's decomposition of integer programming to find the solution. Melachrinoudis et al. (2000) used mult iple objective approaches to find a solution. Ross {2000) employed a pertormance-based strategic resource allocation to solve su pply nerwork design problems. Syam (2002 ) applied Lagrangian relaxation and simulated annealing to a supply chain network with multiple levels and settings; minimize cost was the target. Syarif. Yun, and Gen (2002} used genetic algorithms and a spanni ng tree to find the solution for su pply chain network with three levels and single product with the aim of minimizing cost. Altipannak. Gen. Lin. and Karaoglan {2007) employed steady-state genetic algorithms to obtain the solution for the design of a supply chain nerwork with multiple products and phases, and compare their algorithm with linear programming, Lagrangian relaxation and simulated annealing. This study aims to minimize purchasing cost, transportation cost. fixed cost and manufacturing cost. Based on the above literature. some studies rake maximize profit as thei r objective, whe reas most supply chain network models take minimize cost as their measure of performance and consider many costs. including purchasing cost, trans portation cos t and storage cost Altiparmak et al. (2007 ) proposed a supply chain network model that covered almost all of these costs. Thus. this study adopts Alti parmak's model as the nonn in the mathematical mode. 2.2. Genetic algorithms Genet ic algorith ms first developed by Holland ( 1975 ). use computer programs to simulate the evo lutionary process with the ch romosome as the solution to the solved problem. Based on the environmental adaptation of chromosomes, researchers identified a fitness val ue such that a researcher could determine whether a chromosome wo uld survive to the next generation. The evolutionary process continues umil the ta rget has been mer. via self-adaptation and an iteration threshold. the algorithm has the ability to evolve to the optimum solution for a problem. The parallel search feature is hidden in generic algorithms. This feature is suita ble for handling mu ltip le space dimensions, and non-linear complex NP-IMrd problems (Goldberg, 1989). Genetic algorithms is faster than the simp le point search function of traditional algorithms and has a wide application in such areas as stock market analysis (Szeto & Fang. 2000), vehicle routing problems (Ting & Huang. 2005), production allocation problems (Hou and Chang, 2002), tourism itinerary planning {Christophe & Hugues, 2007), communications network design {Chou, Premkumar. & Chu. 200 1), weather prediction (Wong, Yip. & Li, 2008). locationall ocatio n problems Uaramillo. Bhadury. & Batta. 2002: Zhou & Liu. 2003). personnel training (Juang, Li n. & Kao, 2007), and bus network optimization (Bielli. Caramia. & Carotenuto. 2002). However, genetic algorithms have two shortcomings. If the fitness value is set improperly, convergence can occur prematurely. The second is that it is not suitable for highly non -linear problems (Hi lli s. 1992). 2.3. Co-evolutionary mode To overcome these shortcomings. Hillis developed the co-evolutionary scheme in 1992. In this scheme a living being (chromosome ) and environment {m ultiple criteria ) interact and c<H."volve. The genes are improved continuously for survival and the environment changes with the living being. For instance, when an eagle hunts a rabbit, the rabbit must run fast to suiVive. and the eagle must also ny fast to catch the rabbit. This means that both the chromosome and multiple criteria constraint conditions must evolve. Restated, as the chromosome must ma tch the fitness value, criteria constraint conditions must be evaluated by a fitness function. The evalua ti on cri teria for next evaluation are selected and based on th e degree of fitness of criteria. In th is way, the shortcomings of traditional genetic algorithms can be overcome and search speed accelerated. In the operation mode. all chromosomes must be evaluated to determine their degree of march with the mu ltiple criteria constraint cond itions. The ease with which these constra int conditions are met is directly related to the size of the fitness values of the criteria. Conversely. the fitness value can be large such that the criteria have the opportunity to be the evaluation function for the next evaluation. Fig. 1 presents this co-evolutionary mode. After this process, the evolutionary di recrion of a chromosome will be that for which conditions are not easily met. Thus. rhe near optimum
  3. 3. 692 1 Y. -H Chang/ Expert 5ysrtms with Applications 37 (20 10) 6919-6930 3. Supply chain integratio n mode Chromosome I Chromosome .! AJapti v.: Chromo..;ome n Evaluate Criterion I 3.1. Supply chain archirecrure Cntcrion 2 Cnte 1i o nm ~ Fig. 1. Co-evolution.uy modt-. solution can be acquired rapidly and convergent immaturity can be avoided. 2.4. Conscraint-sacis[action mode Ge net ic algorithms are re liable. but do not have capac ity to process complex constraint conditions. Thus. generic algorit hms should be integrated with another mechanism to solve the satisfactio n problem of constraint conditions. Gen and Cheng (2000) proposed four strategies to improve a genetic algorithm 's efficiency-reject strategy, repair strategy. mod ify the genetic operator strategy and penalty strategy. The reject strategy is used when a result falls into a non-feasible solution region: the chromosome can then be discarded directly. The repair strategy is used when a gene disobeys constra int conditions such that it should be modified to feasible solution region. The modify strategy means according to the different problem: the chromosomes and fundam ental operat ions sho uld be modified to avoid violating co nstraint condit ions during the calcu lation process. The penalty strategy adds a pena lty item to the origina l target function. The degree of the penalty depends on degree to chromosome violates constraints. This concept, which is used to find the so lution. is fast and efficie nt in findi ng the near optimum solu tion. Simple genetic algorithms adopt multipoint parallel search modes and find th e optimum solution gradually. Once the search space becomes too large. the time spent finding the solution in creases, or the algorithm falls inro the local optimum solution left graph in Fig. 2. However, the constraint-satisfact ion genetic algorithm n<~.rrow s down the soluti on spetce to feasible so lution region. The chromosome solution that meets the constraint condition criteria is then created. Because it obta ins the near optimum solution from feasible solutio n directly, its sea rch region is very sma ll (right graph in Fi g. 2). Thus. the constrain t-satisfaction gene tic algorithm is faster at finding the optimum solution than the sim ple genetic algorithm (Hou and Chang. 2002 ). This research adopts the model developed by Alti parmak e r a!. (2007 ) co establish multiple products supply chai n network model with multiple levels. Fig. 3 shows this supply chain system. The supply chain model has four leve ls-raw material suppliers. manufacturers. distribution centers and retailers. The costs are purchasing cost. transportation cost. manufacturing cost and storage cosr. The manufacturer and distribution center are fixed and non-dynamic invocation points. The same retailer can be controlled by mu ltiple distribution centers. Thus, if productivity is defic ient, other di st ribution centers can offer support. Fig. 4 shows the research mode. The mode takes the minimum coral cos t as the solu tion goal and assumes that all levels are under resource constraints. and the production demand of a retailer can be satisfied. The fundamental assumptions in this study are as follows. ( 1) The demand and productivity of retailers and raw material suppliers are known. (2) The supply balance must meet demand such that no our-ofstock situation wi ll occur. (3) Quantity an d productiv ity constraints of the raw material supplier. manufacturer and distribution center are known. Definition of the mathematic model in this study is as follows . ~~,~~, ·:· ' .... ~ ~)i _:il.ll" ~. . m.:. nu facturers '~ retailers ~upp l i.::rs -~ rn.:.tcrial distributi on center:. - pmducts Fig. l . Supply chonn network system for an mdusrry. soluuon space ' "'ib"~'"'"""''"' ~ Fig. 2. Se.Hch mode of sLmple genetic ~lgorithm And comtr~int-sAtlSfAction generic ~lgorirhm
  4. 4. 6922 Y.-H. Chang/Expert Sysrtms wich Applicacions 37 (2010) 6919-6930 ( 1) Model Symbols quantity of raw material sup pliers (s E S) production capacity of manufacturers (J E F) D number of distribution centers (d E D) M number of retailers (r E R) qu.mtity of raw material (m E M) number of products (p t P) (2) Model Parameters suppliers. unit purchase cost of raw material m unit rransporrarion cost of manufacturer f that transport products to distribution cen ter d unit rrans ponation cost of distribution cemer d that transports products ro retailer r unit manufacturing cost of manufacturer f for produ ct p w, BOMpm D"' Ld unit distribution cost of distribution center d quantity of raw material m that product p needs demand of retailer r for product p upper quan tity bound of raw material m that can be provided by raw material suppliers productivity upper lim it of manufaclUrer Jfor product p distribution upper limit for distribution center d (3) Decision Variables ~pd quantity of product p that is transported from distribution center d to retailer r ~PI quantity of product p that is transported from manufacturer f to distribution center d £2tms quantity of raw material m that is purchased by manufacturer f from suppliers (4 ) Objective Function Min Z = L L L B,m QJ~- L L LMw l2.pJ f - L L LT/,Qdpf+ L L L W,Q,., f -LLLT,,Q,., I m d J d ' P r p p p d d (5) Constraints LQ,., =D"' (1) LC2.,r = LQ,., (2) d I ' L LQ!m> = L LQ,PIBOM,. 111 5 d (3) p (4) L:C2.,r <= L w (5) d LQfm> <= l,m (6) I Q,., >= 0, Q,,, >= 0, Qfm, .>= (7) 0 rurchase .. supphcl"i In the above mathematic model, item 1 in the objective function is the purchase cost of the m.1nufacturer that purchases raw material from the raw material supplie r. Item 2 is the transportation cost of raw material that is transported by the raw material supplier to the manufacturer. Item 3 is the manufacturing cost for products made by the manufacturer. Item 4 is transportation cost of products tran sported from the manufacturer to the distribution center. Item 5 is storage cost of products stored at the distribution center. Item 6 is transportation cost of products transported from the distribution center to the retailer. Constrai nt I ensures that the products transported from the distribution center to the retailer meet retailer demands. Constraint 2 is the limit for products transported from the manufacturer to the distribution center. such that the quantity of products is the same as that transported from the distribution center to the retailer and storage in distribution center is controlled. Constraint 3 is the limit for products transported to the retaile r: the number of products should be the same as that manufactured. Constraint 4 is explains the distribution constraints on the distribution center. Constra int 5 is the capacity constrai nt for the manufacturer. Constraint 6 is the quantity constraint of raw material the material supplier purchases. Constraint 7 is the numerical value constraint of the decision variable. 3.2. Co-Evolutionary and constrainr-sacisfacrion genetic algorithms ( 1) Chromosome coding and decoding Because the supply chain network has many decision variables and these va ri ables are constrained as a positive integer. the chromosome is encoding in integer. This supp ly ch.1in network has four levels {r.1w material supplier. manufacture r. distribution center .1nd retaile r). Thus. this study uses three sections of chromosomes to demonstrate the relationship among the four levels. The research go<~. l is to find the purchase decision solution for various supply chai n members. such that the coding uses purchase decisions of all members as the decode value of chromosomes. Suppose there are 3 ret.1ilers. 2 distribution centers, 3 manufacturers, 2 raw material suppliers. 2 products and 2 raw materials. Table 1 presents the parameters for this case. Fig. 5 shows the corresponding supply chain network mode. where R, is a retailer. D, is a distribution center, F, is the m.a.nufac[Urers, S, is a sup plier.P, is a product, and M, is the raw m.a.terials. The relations among the four levels in the supply chain network are mapped in a tree structure. Fig. 6 shows the chromosome coding form . Phase I shows the demands of various retailers. and the quantity delivered to all distribution centers (Fig. 6). The first two nodes (0 1 . D2 ) (last row in Fig. 6) are th e demand of retailer R1 for product P1 (total. 250 yards (69 + 181 )): distribution center D1 provides 69 yards and distribution center D2 provides 181 yards. Phase 2 shows that after all distribution centers receive purchase orders from retailers. they gather th is information and make purchase orders to manufac[Urers ( phase 2 in Fig. 6). The first three nodes (F1 . F2 and F3 ) of Phase 2's last row mean distribution center purchases mate- ~ _lraMpo~auon., 1;::-:~_tr..msponauun ., l.1!1.!.D manuf!K'tun:rs lmanuf~cturecost) co~t c;o-:; co~t di stri huuun c:c:nh:rs (di.>tributioncost l Fig. 4. Mode gr01ph of I he supply ch01in nerwork 111 I his srudy. retailc:r
  5. 5. 6923 Y.-H. Chang / Expert Sys tl."ms with Applications 37 (2010) 6919 - 6930 To~b le ers already meet the needs of the distriburion center. no further purchase is made from manufacturerf 3 • Ph.:1.se 3 in Fi g. 6. the first two nodes (5 1 • S~ ) purchases raw mate rial M 1 ( 14 kg) from supplier 51 and purchases 5 kg from supplierS~. The full length of the chromosome is the sum of the lengths of these three phases. Take tree schematic graph of a chromosome (Fig. 6) as an example, the total length of the chromo- 1 Pnameteruble. Mode parameter Symbol Retailer Distribution center Manuf.lcturer Raw material supplier Product R,(i• l . 2. 3) D,(i•1.2) f, (i• l. 2. 3) S;{i•1,2) P, (i•l. 2) Raw material M,(i•.2 ) Number some is (3 • 2 • 2) + (2 • 2 "3 ) + (3 "2 "2)-36 (2) Evaluation rial P1 (58 yards in total) from manufacturerf 1 : manufacturer F~ the purchases 132 yards. Because these two manufactur- ma~enal (M,) product~( P'~h<>>< ~l /?1'~1COSt P) product~(P,) "'"' """"" •,....~ "'~'P"""';""~ "' CO~ manufacmrc:~F1 1 ~uppliers(S) ~ COSI d1stribut10n (;C:ntc:n.f0 ,) (manufacmrccosr) (dJstrihutioncosl) Fig. S. The relation graph for the three levels in the supply chain network. Phase I : Distnbutwn Centc-t Quamit} Quantify (ll<1} (58) (1!11) (13::!) (163) fO) (17) (45) (l~::!J rl::!1) ( 155) 175) (95f 1 76) (39-1) (5 ) (155) {0) (3) (3) (724) (j) (5) (:~5()) (li J ( I) (50) (31} (62) {IJ {::!::!5) :::~::::'"'"~ s""'''"'~~ Qu~ntity (14) (5) (lJ (JO) ( I S) genes. The objective of this study is ro minimize cost. Thus, the objective val ue and fitness value have an inverse relationship. That is. as cost increases, the fitness value decreases. !ote: the grey part is a chromosome Fig. 6. Tree schemouc groph of o chromosome.
  6. 6. Y.-H. Chang/ Expm Systems wirh Applicarions 37 (2010) 6919 - 6930 6924 To express the relationship between the fitness value and objective value. this study defines the formula for fitness value as the maximum cost of this generation's chromosome plus a certain integer value, and deducts the cost of the chromosome such that the solution quality can be determined by the final fitness value. The near optimum solution can then be obtained. To make the final evolution result match the supply upper bound of various levels and meet the demands of manufacture. a penalty function is added to force the result to evolve according to a constraint criterion. The rules of the penalty formula are as follows. a. When the value is lower than one level's demand, the cost should plus the quantity of insufficient and multiply the value of one level's highest unit cost. If demand exceeds a level's demand. the penalty is waived because it has <~!ready been included in the cost calculation. b. When the value exceeds the upper bound, the cost should plus the surplus qu<~ntities and multiply the unit cost of all levels (e.g., distribution cos t, manufacturing cost and purchasing cost). (3) Selection. crossover and mutation Selection me<ms th.lt only some chromosomes smvive to the next generation from the parent generation based on a fitness value. Chromosomes with high fitness va lues Me most like ly to su1vive. This study uses the roulette wheel method for selection. which takes the fitness value of a chromosome as the probability value; thi s method can be used to choose the next gene ration fairly, and is the most widely used reproduction method (Gen & Cheng, 2000). Crossover switches some genes in two chromosome groups to create a new filial generation of chromosomes. The goa l is to enlarge the se.:uch space and increase the speed at which the opti mu m solution is acquired. The crossover method in this study adopts the common two-point crossover method. Two tangent points are made first and the gene values of two chromosomes are switched between two tangent points (Fig. 7). The purpose of mutation is to increase the diversity of chromosomes and prevent premature convergence and arriving at a partial optimum solu tion during the search process. The study adopts the single point mutation method (Fig. 8). The system chooses a chromosome to mutate based on mutation probab ility. The chromosome selected can mutate again based on the gene picked randomly from chromosome genes by the system and add to, or deduct the muta tion value which randomly generated by according to the protoge ne value. For instance. the protogene value is 122. The system picks the change value randomly (0 <change value :; :;: protogene v<~lue ). If ch.mge value is the deduction value and the change value is 62. the gene value after mutation is 60 ( 122 - 62). (4) Co-evolutionary mode The difference between a co-evolutionary and simple genetic algorithm is that the co-evolutionary genetic algorithm takes the fitness value as the evaluation criterion and. based on the interactive evaluation of chromosomes and criteria, which make the chromosome of each generation evolve toward S.ltisfy the difficult contented criterion. Thus, the evolutionary speed can be increased and the system can obtain the optimum solution. Table 2 li sts the study criteria. The co-evolutionary steps are as follows. a. Take the number of chromosomes that match the criterion as the fitness value of the criterion. A criterion easily met has a low fitness. The fitness value of a criterion equals the value that the number of population size minus the number inside that matches criterion. b. Ta ke the fitness value as the probability val ue in the rou lette wheel method and select the criterion. If a criterion is selected more than once. it should be neglected. For instance, if there .are 6 crite ria and the selection res ult is 1, 3, 2, 3, 1, 4, the selection result should be the first. second, third and fourth criteria. c. Based on above selec tion criteria, all chromosomes are ev.alu.ated. (Before Crossov er Chromosome A I ss Chromosome B I 20 I 36 I 48 11 32 Io 122 1121 Io I 3 3 60 154 1 17 1750 ~24 1724 I 3 Is 1259 I I 22 I 19 I 35 I 75 I (After Crossover) Iss Chromosome B' I 20 Chromosome A' 1132 I0 l6o ls4 117 1750 136 148 1122 1121 I3 1724 I3 1259 I I0 Fig. 7. Schcm<ttic gr<tph of the two-point crossover operiltion. [Before Mutation I Chromosome I 58 (After Mutation) Chromosome' I 58 lm I lz59 I ~ mutation point lm I 0 l6o 1121 I 0 I 3 I 3 lm I l2s9 I !132 I0 1122 11 21 0 Fig. 8. Schemiltic graph of the simple point mutiltion method.
  7. 7. Y.-H Chong / Expert SysrC"ms with Applications 37 (20 10) 691 9-6930 T.able 2 Sum mary of study co nslraints. Number Consrr.aim s The quantsty ordered by a retasler must m.nch the qu.antity purch.ased by • distribution center The quantity purchased by the distribution center must march the quantity manufactured The raw material needed by the manuf.tcturer must match the quantity it purchases from the supplirr The stor~e constrainls of disuibutioo center must be s.atisfied The productiOn C.Jpaciry of manufacture r must be S.ltssfied The supply constro~ints of ro~w mo~rerio~l supplier mu st be satisfied d. Repeat operations (a)-( c). Once the fitness values of a crite· rion are close and exceed the threshold value. then use all criteria to evaluate the chromosome until the near optimum solution is found. If the population size is 100, the number of criteria is 6. Table 2 shows the criteria. After the system creates the initial popu lation and evaluates this population using the coevolutionary mode (Fi g. 9 ), the first step is to calculate how many criteria all chromosomes match. When chromosomes matching criterirt 1- 6 are 20, 30, 70, 45, 40. and 0, respectively, the fitness value of each criterion is 80( - 100 - 20), 70. 30. 55. 60, and 100. After retrieving the fitness value of each criterion. the roulette wheel method is applied to pick the evaluation criteria for this generation as criteria I. 6. 1. 2, 6, and 1. Thus. if all chromosomes are evaluated by criteria 1, 2, and 6. and is this circle in each generation is retained until the fitness values of all criteria are >90, then all criteria are used to evaluate the chromosomes unti l the system converge. (5) Constraint-satisfaction mode When the initial chromosome is created, one must determin e whether it matches all constraints. If one of initial chromosomes violates the constraint, the chromosome should be created again. The creation method is as follows. First. distribute the demand of last levels to next leve l. For example, distribution center D1 demands 240 units thar shou ld be distributed ro th ree manufacturers. The first two manufacturers are distributed randomly by the sysrem. If manufacturer f 1 has 100 units, manufacturer F2 has 60 units and the remain- a. b. c. cnfcriun cri t<"nl>ll nit..:rion 1-fit n c~ · !W 2 - titnot~><· 7!) 3-fi t n.:' ~: 30 55 60 ~·nt.:ntiO .J.-fi tnc" : nitcrion !- rl tnes< chmmn,;omr c hrnmosom<.· ~·ntcnon o-i1 t nc'~ Ch ll•OUhOOI<' 100 ]!)() critcnon I. 2. o chrum"sonlC Chll llllOS(lffit: 1- f1tnc~~. 10~000 2 · filti<.'S' 204!:!0 cn rcn.•n I cnten.:• n 2 criteriu n6 (<. ah:u lale f•t n c":.; , hycnr.-na t.:::.M n f <: h rt'>m.>• 0m<:~ Fig.. 9. Co-evoluuon ~ ry e v o~ luarion d. (6) mode. 6925 ing 80 (240 - 100 - 60 ) units are distributed to manufacturer f 3 . After initial population was created, the chromosome that already meets the requirement must be verified by production capacity and the upper bound of storage. If any chromosome does not match requirements. the chromosome must be created again until all chromosomes match the constr.1ints. This method ensures that the initial chromosome must fall into the optimum solution region. However. even if the initial population matches constraints, after crossover and mutation, the chromosome can still be outside the feasible region. Thus. the ch romosome that does not match constraints should be adj usted such chat the search space falls inside the feasible solution space. This method narrows down the search space to find the near optimum solution quickly. The adjustments to invalid chromosome are as follows. When the distribution result cannot meet the demands of the level s. pick an item randomly and add the insufficient quantity directly to the distribution quantity. For example, rete~il er R, demand s 250 unitsP, , and distribution center D, di stributes 140 units. distribution center D2 distributes 100 units. However, the retailer is short 10 units (250- 140- 100). Thu s, the system selects a discribution center randomly and adds 10 units to the distribution quantity. If distribution center D2 is selected. the distribution quantity of D2 must be increased to 110 units ( 100 + 10). When the distribution amoum exceeds the demand of a level, then select an item random ly and use the distribution quantity to deduct the quantity exceeding that needed . If the distribution quantity is still excessive, the next distribution quantity of the next item shou ld be deducted until the distribution quantity matches demand. For instance. retailer R, orders 250 units P, . distribu tion center D1 distributes 260 units and distribution center 0 2 distributes 100 units. However, the retailer receives an additional 11 0 units (260 + I 00 - 250 ). Thus. the system selects a distribution center randomly. If it se!ectsD 2 • the distribution quantity of D2 is reduced by 100 units and the others should be deducted 10 from next item D, 's distribution quantity. Thus, the demand equals the distribution quantity. The distribution quantity of 0 1 is 250 units and the distribution quantity of D2 is 0 un its. When the total distribution resu lt exceeds the upper bound, then select an item random ly and use the distribution quantity to deduct the excessive quantity. If the rota I distribution quantity is sriH excessive. the disrriburion quamity of t he next item should be deducted unt il the total distribution quantiry equals the upper bound. For example, distribution center D, of retailer R1 orders 150 unitsP 1 , and the distribution center 0 1 of retailer R, order 100 unitsP 2 • However. the upper bound of distribution center 0 1 is 100 units. Thus, the system then selects an item randomly. !fit se lects retailer D, of retailer R1 order P2 , the distribution quantity should be deducted 1 DO units, insufficient part should be deducted 50 from next item 0 1 's distribution quantity of retailer R, order P1 • Thus. the distribution quantity is 100(150 - 50 ), and the total distribution quantity (0 + 100) equals the upper quamity. Repeat operations (a}-(c) until total distribution quantity equals demand and is less than or equal to the upper bound quantity. Co-evolutionary constraint mode This mode combines the fearures of the co-evo lutionary mode and constraint-satisfaction mode. The constraint-satisfaction mode was introduced in item 5. Because the adjustment mechanism of the constraint-satisfaction mode can
  8. 8. Y.-H. Cha ug j txpm Sys ttms wirll Applications 37 (1010) 6919-6930 6926 Tabl~ 1 Summary constramts for co-evolutiOnary constraint genetiC algorit hm. Number Constraints The The The The The quantity of products transported from a distribution center to .1 ret.liler multiplied by th~ unit tr.lnsporution cost should be mimmized distribution qu;mtity of a disrnbution center's products multiplied by the unit distribution cost should be minimized quantity of products rransport~d from manufacturers ro a distribution center multiplit'd by the unit mmsportation cost should bf' minimiud quantity of m,mufactured prod ucts multiplied by the unit manufacturing cost should be minimized raw materi.tl quantity of a suppli~r's r.lw llldteridl multiplied by the unit purchase cost should be minimized the search space for a feasib le solution that reduces unnecessary search cost. Then, the co-evolutionary mode and const rai nt are combined as the tensile force. As the fitness value of a constraint increases, the power of evolution force increases. and the constraint has more needed to dominant its evolvement. This concept is opposite to chromosome. The system ca n use the interactive evaluation between constraint and chromosome to find the near optimum solution. Fig. 10 shows the concept co-evolutionary constraint mode. 4. Experiments and verification Fig. 10. Conceptual graph of the co-evolurion.Jry constraint genetic <~lgorithm . modify chromosomes tha t do not match const raims. if adopt constrain evaluation method on co-evolutionary geneti c algorithm. the degree of chromosome that matches constrc:lints canno t be distinguished. Thus. const raint-s<Hisfaction mode can be adjusted as follows. a. When the demands of all levels and resource constraints are mer. the minimum cost is the basic rule (Table 3). b. Take the chromosome with highest cost as the fitness value of th e constraint. For instance. if the purchase cost of one chromosome is highest, the fitness value of constraint 5 is purchase cost plus 1. Fitness of the constraint equals the sum of all chromosomes with the highest cost. it shows that higher the cost is. more need of evo lvement to the constraint. c. Take the fitness value of constra ints as the probability va lue of roulette wheel method. d. According to the selected constraint s, evaluate all chromo-somes and calc ulate their fitness values. The fitness of a chromosome can be calcu lated as described in Section 3.2(2). Only calcula te the fitness values of related costs that are re levant to this constraint select ion. e. Re peat operations (a)-{d) until the near optimum solution is found. A beneficial feature of the co-evolutionary constraint genetic algorithm is that it uses constraint-satisfaction mode to narrow down This srudy uses the above mathematical model to simulate a supply chain network de sign problem. To make the experiment closely resemble practical supply chain operat ion, the Taiwanese textile export industry is adopted as the experimental object. Th is study adopts proportionaliry information acquired from interviews with manufacturers to set parameter values for the model. This is a real supply chain mechanism ro a large extent. Because the supply chain network scales in the textile industry differ, this study divided the test case into 10 large and small-scales (supply chai n network nodes below 1000 is small-scale. the rest are large-scale ) ro ensure the model can be applied ro various scales. Table 4 shows the test case. With mathematical programming. the simple genetic algorithm, constraint-satisfaction genetic algorithm. co-evolutionary genetic algorithm. and co-evolutionary constraint genetic algorithm a re used to find the solution that verifies solution search efficiency and accu racy. Table 5 presenrs parameters of the text ile industry. In experimen ts. population size is 100, crossover rate is 1, and mutation rate is 0.01. The terminati on condi tion when fi tness values of chromosomes in a population are the same. When the number of generations exceeds 100,000, searchi ng will stop automatically. This experiment is written using the Delphi development tool, and is executed on a Pent ium rv 2.8G with lG RAM to continuously test 10 times as a benchmark for comparison. The mathematical programming method, however, uses LINGO 9.0 software for comparison. and it is commonly used for non -linear integer programming with the same computer Table 6 presents comparison results. Fig. 11 compares the time costs in obtaining a solution. Table 4 Tes1 cases .lnd parameter table. C.Jse R~tailer Distribution center M<~nufilcturers Supplier Product MCJterial Smaii-SC.llt'S Combination number of n~rwork .ucs 12 96 324 768 t..uge-sules 10 10 12 14 16 18 20 10 10 12 14 16 18 20 10 12 14 16 18 20 10 12 14 16 18 20 !SOO 10 2592 4116 6144 8748 12.000 '
  9. 9. I Y.- H. Ch ang/ Expert Sys rems wuh Applicatio ns 37 (20 10) 6919-6930 6927 Table 5 P01rameter romdom range of the textile industry. Code Parameter Range Unit RetAiler dem<~nd Textile r.aw m.ueri o~l Unit purc.hase cost of textile raw materi.al by a rexrile supplier Unit manuf<~cturing cost of .a product 100- 10,000 0.09-0.3 60-120 5-12 Yard kg Unit distribution cost of a disrribution center 0.9-1.6 dollil r/kg doll.lr/ V.1rd 0.6-7.5 dollar/ Y.Jrd dollar/ Unit tra nsport.alion cost of .1 product uansported from a distribution center to a re u iler 1-8 v.ud dollar/ Troxtile supplier Tdw ffidteri.ll up~ r bound of supply Manufacturer upper bound of productivity Distribution upper bound of .a distribution center 1.5-2 times of m.anufdctur~r's production cap.lCity 2- 3 times of distribution cente(s capacity 1.5-2.5 rimes of retailer's total demand quantity Unit tr.Jnsport.ltion cost of a product tro~uspone<l from manuf.acturers to distribution cenrers Y~rd 10 T~ble Y.m! 6 Perform~nce Case of UNGO •nd rh~ genetic 10 Case ~lgonthms. Simple CA UNGO Time Cost 937.546 3,710.342 9.868,523 22.526.384 43.206.351 63,557.214 135.063.824 175.395. 128 356.951,353 395.298,246 Avg. cost 268 1632 23,141 37,682 50.924 93,951 173.952 313,825 937,546 3,7 10.342 9,868.523 22.526.384 43,206.351 67.072.073 142.577.049 187.904.091 378.041 .78 1 426.351,982 Co-evoluriotury GA Avg. time 937.546 3.710.342 9,868.523 22.526,384 43.206,351 66.569.586 141.297,516 185.910.201 377.086.922 4 21.965,2 15 Avg. gene. Avg. cost 68 937,546 3.710.342 9,868.523 22,526.384 43.206.351 66,826.246 142.038.588 187,333,036 377.879.078 425.841.874 Ill ! 58 75 1 654 1351 2133 3649 4997 6824 Constraint-satisfaction GA Avg. cost 10 kg Yard 15.049 24.973 32.089 67.842 82.684 127.682 179.72 1 184,932 Avg. time Avg. gene 129 602 594 11 98 2033 3249 4970 6592 43 78 12,492 20,439 24.03 1 52.968 76,843 119.192 168.518 176.821 Co-evolutionary Constraint GA Avg. time Avg. gene. Avg. cost Avg. time Avg. gene. 4 153 501 583 1105 1900 2594 4733 6413 28 70 10,726 14.392 20,195 44.395 73.824 105.835 156.418 166.492 937.546 3,710,342 9,868.523 22.526,384 43.206,351 66.153,035 141.11 5.368 185,476.283 366.865.767 409,754.413 84 342 540 983 1503 2347 4008 6348 20 69 9038 10,832 19.824 37,521 70,952 96,382 148.254 159,824 Nare : the unit of the target value is dollars and time is in seconds. is the abbrevi<ltion of "average·. "gene." is the ~bbrev1ation of "generaliOil ~. "avg.~ This study analyzes the time and solution search cost fo r large and small-scale cases. Fig. 11 and Table 6 shows the solution search rime fo r the small -scale case. Cases 1 and 2 are smallest. With both LINGO and the genetic algori thms . solution time is <6 s. However. as the number of leve ls increases in cases 3 and 4, time cost for LINGO increases from several seconds to roughly 30 min ( 1632 s); the simple generic algorithm only takes 12 min (751 s): and the co-evolutionary const raint genetic algorithm takes 6 min (342 s). Thus. genetic algori thm is fas ter than LINGO in for the small-scale cases. For the small -sca le cost (Fig. 12 ). the genetic algorithm and LINGO solution search qualities are almost the same. For the large-scale cases, as the case parameter increases. complexity increases. The time cost for LIN GO increases dramatically (Fig. 13). For case 5, LINGO needs over 6 h; for case 7. LINGO needs over 14 h. Although ti me cost for the genetic algorith ms also increases (e.g.• the simp le generic algo rithm needs 35 min and coevo luti onary constraint genetic algori thm only needs 25 min ). but rema ins consi derably less than that by LINGO. For case 8. time cost fo r LINGO inc reases by days. For case 10, LI NGO needs over 3 days to search the solution. Thus. LINGO is very ineffi cie nt when supply chain management needs every second. However. all genetic algorithms reta ined high performance levels. Time cost was within 1.8 h for all genetic algorithms. Thus, th e average tim e for so lution search for all genetic algorithm s was better than that with LINGO, especially the co-evolutionary constra int genetic algorithms. which performed best in complex problems. Fig. 14 li sts the cost for the large-sca le case. Although case complexity increases and the solution searc h quality of the ge ne ric algorithms and LINGO are almost the same. the difference is 8%. In the most complicated case, case 10. the co-evolutionary constraint genetic algorithm only has a 3.6% difference from LINGO. Therefore, the co-evolutionary constraint generic algorithm has some efficiency for large-scale solution cost The comparison of so lution search efficiency and quali ty for la rge and small cases indicates that although costs by the generic algori thms are sli ght ly more than those for LINGO, time cost for the genetic algorithms is much less than for LINGO. Norably, the co-evolutionary constraint ge neti c algorithm us es the
  10. 10. Y. ~H. Chang { Ex~rt 6928 Systems wir/1 Applicarions 37 (2010) 6919- 6930 :~I 1,4)) f 1,300 ! ''i ! 1,200 1.100 I,OOJ j: // ' .., F100 1 /i EIL----------------~·~'_:.~J 2::~:=·-~-~-·~-·_;_':_·/_._:_:_ Fig. 11. Se.-.rching solution time cost for sm.-.IJ .sc.-.Je uses. l,OOO 1,000 1,600 11,400 § 1,200 ~ I,OOl .s "" 600 ~ QA - - -· ~OA --~oomryccmlrMt CA Fig. 12. Search soluuon cost for small-scale cases. co-evo lutionary mode to accelerate evo lution. and the constraintsati sfaction mode can narrow down the search space to obtain the opti mu m solution. Co-evolutionary co nstraint generic algorithm is more efficient in searching the near optimum solution than the simp le genetic algorithm, co-evolutionary genetic algorithm and constraint-satisfaction generic algorithm. The cost for co-evolu tionary constraint genetic algorithm is also less than that of othe r generic algorithms. Thus, the co-evolutionary constraint genetic algorithm performs best for the supp ly chain network design problem. 5. Conclusions and future research Market competition has increased and thi s challenge has caused ente rprises to change their operational modes and enhance the efficiency of supply chain management to enhance their competitiveness. One must consider multiple products, levels and resource constraints to establish the best integration mode that matches supply chain manage ment needs. The co-evolutionary constraint genetic algorithm can solve the sup ply chain nerwork design problem. Ma th ematical programming. the simple genetic algorithm, coevo lutionary genetic algorithm, constraint-satisfaction generic algorithm were compared in terms of solution search quality and efficiency. Th e co-evolutionary cons traint genetic algorithm narrows down the search space with the constraint-satisfaction mode. Additionally, the co-evolutionary mode can adjust evaluation constraints dynamically to match a complex reality, has excellent solution search quality and speed when searching for a solution to supp ly chain management production and sale mode problems. and performs better than other meth ods. Analysis of real textile
  11. 11. 6929 Y.-H. Chang / Expert Sysrems wirh Applications 37 (2010) 6919-6930 ,----- mmm ---- i!SQOOO mm~ -: .-·, ! 1 ~ .;! 100.000 !0.(00 '·"" """ 'if4,oo:J "' 3,00) ~ ""2,(ro 1,001 l""""""""'s=;~~~:::::::::~~~:;;;;;:::;::;;;,..--· Fig. 13. Time cost for solution searching in the .J.rge-sc.J.le case. ~ OA --co-e~~ 'I'C<)(IS(r mt C-A Fig. 14. Cost for solution searching in the large-sco.le co.se. industry cases indicates that this mode reduces supply chain management cost and increases the con nectio n among enterprises. Thus, the experiment result can be a decision-making reference for Taiwan's manufacturing industry when developing global plans. and remind decision-makers that planning should consider production planning and not neglect resource integration of all entities in their supply chain. The integrated power of supply chain management is the bes t channel for enterprises to acquire profit and competence. This study takes the textile industry as its expe rimemal case. Because of the scale of supply chain networks, member structures of various industries, products. raw materials and resource constraints differ. rn future research. the supply chain netwo rk struc- tures of different industries can be compared to verify solution search efficiency of various constraint-satisfaction genetic algorithms. This study referred ro the model structure in Altiparmak et a. (2007 ). Because the demands of supply chain modes vary, modes can be adjusted based on supply chain demands such that the mode is close ro actual supply chain operation. For instance. the model can increase the choices in different periods, storage control. batch discounts. and emergency purchase processing. In a supply chain network, different enterprises have different targers. This study only adopted total minimum cost as the measure of supply chain operations. Multiple constraints will be added as the decision-making standard for evaluation in the future. Moreover, genetic algorithms are nexible and have many parameters
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