Designing a Genetic Algorithm to Solve an Integrated Model in Supply Chain Management Using Fuzzy Goal Programming Approach

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1 Designing a Geneti Algorithm to Solve an Integrated Model in Supply Chain Management Using Fuzzy Goal Programming Approah M. Rostami N. K.i *, J. Razmi, and F. Jolai Department of Industrial Engineering, College of Engineering, University of Tehran, Iran {jrazmi,marostamy,fjolai}@ut.a.ir Abstrat. Appliation of fuzzy goal programming and geneti algorithm is onsidered in this paper. We extend a multi-objetive model for integrated inventory-prodution-distribution planning in supply hain (SC). We onsider a supply hain network whih onsists of a manufature, with multiple plants, multiple distribution enters (DCs), multiple retailers and multiple ustomers. The manufaturer produes several items. Deision maker s impreise aspiration levels of goals are inorporated into model using fuzzy goal programming approah. Due to the omplexity of problem in large size and in order to get a satisfatory near optimal solution with great speed, a new geneti algorithm is proposed to solve onstrained problems. To show the effiieny of the used model and fuzzy goal programming approah and geneti algorithm for the ollaborative inventory-prodution-distribution problem, omputational experiments are performed on a hypothetially onstruted ase problem. Keywords: fuzzy goal programming, supply hain management, geneti algorithm. 1 Introdution Industries around the world are now all rushing the territory of globalization and speialization. Cooperating with good strategi partner is the sure way to takle the potential problems arising from ompetition. Companies an ahieve the optimum operating effiieny by working with other ompanies through ommuniation and speialization whih evolve a new type of relationship, the SC relationship, among these ompanies and further faster a new onept in management: the supply hain management (SCM). Prodution and distribution operations are the two most important operational funtions in a SC. To ahieve optimal operational performane in a SC, it is ritial to integrate these two funtions and plan them jointly in a oordinated way [[1],[9]]. Different aspets of integrated problems for different parts of supply hain have been onsidered in the literatures. Nozik and Turnquist [9] tried to integrate inventory, transportation and loation funtions of a supply hain. The proposed model has been onfined to a single period, single ehelon problem with no apaity onstraint. * Corresponding author. Á. Ortiz Bas, R.D. Frano, P. Gómez Gasquet (Eds.): BASYS 2010, IFIP AICT 322, pp , IFIP International Federation for Information Proessing 2010

2 Designing a Geneti Algorithm to Solve an Integrated Model in SCM 169 Some researhers have investigated fuzzy prodution planning problems in SC ([10], [1], [5]). Deision analysis in e-supply hain using fuzzy set approah mainly on the basis of fuzzy reasoning Petry nets has been onsidered by Gao et al [5]. The method proposed by Liang [8] aims at simultaneous minimization of total distribution osts and the total delivery time on the basis of fuzzy available supply, total budget, fuzzy foreast demand, and warehouse spae. Chen et al. [4] suggested an approah to deriving the membership funtion of the fuzzy minimum total ost of the multi-period SC model with fuzzy parameters. The onsidered fuzzy prodution planning models and methods are mainly a single produt type and separate prodution planning model without integration with distribution and inventory problems. Computational intelligene is widely used in eonomis and finane [Chen [3], Sheen [11], Zadro et al. [14]]. Torabi and Hassini [13] onsidered a supply hain master planning model onsisting of multiple suppliers, one manufaturer and multiple distribution enters. They proposed a method to solve multiobjetive linear programming (MOLP) models, named TH. Although many researhers have been devoted to solve prodution distribution problems in SC environment, there is a few researhes that integrates an aggregate inventory-prodution and distribution plan in a ollaborative manner using fuzzy modeling approahes. This paper ontributes to the literature by presenting a multi objetive model with resoure onstraints and lead times and onsidering a manufaturer with states whih is near to real world rather than other models and using FGP approah for handling the ollaborative inventory- prodution distribution planning problems SC and presenting a new geneti algorithm approah for onstrained problems. Evolutionary algorithms suh as geneti algorithm were used to solve ombinatorial problems. A Geneti algorithm (GA) based method has been used to develop an effiient solution algorithm by Tasan[12]. Fuzzy-Geneti approah has been proposed to aggregate prodution and distribution planning in supply hain management by Aliev et al [2]. Many hallenges arise when the algorithm is applied to heavily onstrained problems where feasible regions may be sparse or disonneted. This study proposes an approah to obtain solutions whih are lose to the feasible region. 2 Mathematial Model 2.1 Indies i: index of prodution items (i=1,,i) t: index of time periods(t=1,,t) p: index of plants(p=1,,p) d: index of DC s(d=1,,d) r: index of retailers(r=1,,r) :index of ustomers(=1,,c) 2.2 Problem Assumptions The assumptions to model the problem are as follows: 1. It has been onsidered a manufaturer with p plants, d distributors, r retailers, and ustomers 2. All of the deisions are made within multi-periods.

3 170 M. Rostami N. K.i, J. Razmi, and F. Jolai 3. Customers assign ost for bakorder as a penalty for retailers. 4. There is transportation apaity onstraint for all ehelons. 5. Delivery due dates have been assigned by distributors. 2.3 Parameters CD : Amount of produt i demanded by ustomer from retailer r in period t. PCP : Unit prodution ost of produt i. SPMD : Unit sales prie for produt i from manufaturer to DCs in period t. This inluded produt ost, the ost of transportation, and ordering ost. SPD : Unit sales prie for produt i from DC d to retailer r in period t. TCPD : Unit transportation ost for produt i from plant p to DC d in period t. DC pay it. HCF : Unit inventory holding ost of final produt i in plant p. HCD : Unit inventory holding ost of final produt i in DC d. FHCP : Final produt i warehouse apaity in plant p in period t. PHCP : Final produt i warehouse apaity in DC d in period t. PHCR : Final produt i warehouse apaity in retailer r in period t. TFP : Unit proessing time for final produt i. TDR : Transportation apaity from DC d to retailer r in period t. DUEM : Delivery due date for produt i from DC d to manufaturer in period t. TPD : The time required to ship the produts from plant p to DC d. TCM : Transportation apaity from plant p to DC d in period t. 2.4 Deision Variables PTD : Amount of produt i to be transported from DC d to retailer r in period t. TRC : Amount of produt i to be transported from retailer r to ustomer in period t. PTPD : Amount of produt i to be transported from plant p to DC d in period t. ID : Inventory of produt i at the DC d in period t. IR : Inventory of final produt i in retailer r in period t. PQ : Prodution quantity of produt i in plant p in period t. BCR : The amount of bakorder of produt i from retailer r to ustomer at the end of period t. TIP : The time in whih final produt i is maintain in plant p in period t. FP : The amount of inventory of produt i in plant p in period t 2.5 Objetive Funtions Manufaturer Profit :( Maximization)=sales revenue-total manufaturing ost - total inventory types holding ost = SPMD PTPD (PCP PQ ) (HCF FP ) (1)

4 Designing a Geneti Algorithm to Solve an Integrated Model in SCM 171 inventory ost-total transportation ost = ( SPD PTD )- ( SPMD PTPD ) - i ( t HCD id ID idt ) - i P ( t TCPD ipdt PTPD ipdt ) (2) Bakorder Level For Retailer r : (Minimization) = i ( t BCR irt ) (3) 2.6 Constraints FP ipt FHCP ipt i, p, t (4) ID idt PHCP idt i, d, t (5) IR irt PHCR irt i, r, t (6) (4), (5), (6) state that inventory level in eah ehelon is restrited by inventory apaity. Sine, distributors set due date for delivering items from manufaturer, thus sum of prodution time in manufaturer and inventory holding time in plants in eah period and delivery time items to distributor, should be less than produt delivery due date. Thus we have: PQ ipt TFP i + TIP ipt + TPD pd PTPD ipdt DUEM ipdt i, p, d, t (7) i PTPD ipdt TCM pdt d, p, t (8) i PTD idrt TDR drt d, r, t (9) (8), (9), mean that the amount of produt i to be transported among ehelons is limited by transportation apaity. ID idt = ID idt 1 + p PTPD ipdt r PTD idrt i, d, t (10) (10) is balane equation for DCs; the amount of produts that enter to DC d must be equal to the amount of produts that leave from and stored at this DC. Inventory of produt i in plant p in period t is equal to inventory of that produt in previous period plus prodution quantity of produt i in plant p in period t minus amount of produt i transported from plant p to DCs in period t. As shown by (11): FP ipt = FP ipt 1 +PQ ipt d PTPD ipdt i, p, t (11) Bakorder level of produt i inurred by retailer r in period t equals to bakorder level of that produt in previous period plus total demand of the produt by retailer r in that period. This is shown by (12).

5 172 M. Rostami N. K.i, J. Razmi, and F. Jolai BCR irt = BCR ir(t 1) + CD irt TRC irt i, r, t (12) And finally, the bakorder level at the last period should be zero for fulfilling the ustomer demand: All of variables are ontinuous and positive. BCR irt = 0 i, r (13) 3 Methodology 3.1 The Proposed Geneti Algorithm In this problem we are dealing with a multiple objetive possibilisti linear programming model. To solve this problem, we apply a two-step approah. In the first step, the original problem is onverted into an equivalent auxiliary risp multiple objetive linear model. In the seond step, a fuzzy goal programming approah whih is named TH is used. In this method the risp multi-objetive model onvert to a single objetive model onsidering deision maker s impreise aspiration levels for goals. To solve the problem, we propose a geneti algorithm (PGA) onsidering onstrained model and ompare the results with algorithm whih is proposed by Haupt and Haupt[7] (HGA). We use ontinuous geneti algorithm versus binary geneti algorithm, beause variables are ontinuous. For dealing with our onstrained model when model s size is large, we propose a geneti algorithm whih its solutions are seleted from feasible region as far as possible beause it is impossible to selet all members of population from feasible region. Then we ompare obtained results through this method with method whih is proposed by Haupt and Haupt [7] named HGA. In HGA method, onstraints are added to objetive funtion by using penalty oeffiients. A onstrained model an be stated as follows: Min f(x) s.t ( ) 0 = 1,, (x)=0 i=1,,i In majority of algorithms, for generating initial population random numbers are used for generating initial population. While, in a problem whih inludes onstraints in addition to objetive funtions, using this random numbers annot imply that initial population has been seleted from a feasible region or has minimum violation. Also, initial population is the base for onstruting next populations. In geneti algorithm whih is presented here, we onsider this problem. Also initial solution is onstruted with minimization of violation. We do it, using PSO algorithm whih is an evolutionary algorithm similar to

6 Designing a Geneti Algorithm to Solve an Integrated Model in SCM 173 geneti algorithm but it have operators less than geneti algorithm. Thus, for generating initial population, we minimize below fitness funtion: Violation= max (0, ) + Another differene between our proposed geneti algorithm and other algorithms is that in this algorithm fitness funtion is the same as objetive funtion. The method for mating whih has been proposed by Haupt and Haupt[7] is as follows: The blending method finds ways to ombine variable values from the two parents into new variable values in the offspring. A single offspring variable value,, omes from a ombination of the two orresponding offspring variable values = +(1- ) Where: = random number on the interval [0, 1], = nth variable in the mother hromosome, = nth variable in the father hromosome. In our proposed geneti algorithm for mating, hange while the violation for new offspring is less than the average violation in initial population. Also for mutation and seleting the member whih should be affeted by mutation,we ontinue generating random numbers while interested violation will be ourred, then replae the amount of seleted member with the amount of same member in hromosome whih has minimum violation. 3.2 Analysis of Variane In addition to ahievement level, another harateristi whih has been onsidered to ompare performane of HGA and PGA is violation. It has been done by analysis of variane (ANOVA). An independent t-test has been onduted to test the differene between mean of violations. It is assumed that: M1: violation whih was resulted from HGA M2: violation whih was resulted from PGA Analysis of variane was done using Minitab. Thus following 2 tests of hypothesis were performed. : = : Amount of obtained p-value for test of hypothesis 1 is Considering, test of hypothesis is performed with 0.95 for onfidene level, and p-value for test of hypothesis is less than 0.05, thus, null hypothesis is rejeted. It shows that mean of violation in PGA is less than HGA (P-value<0.05). 4 Numerial Example To demonstrate the pratiality of the proposed model, we onsidered an assumed SC. In our hypothetially onstruted SC, there is a manufaturer ompany onsists of multiple plants, multiple DCs, retailers and ustomers. First, a problem with a small size has been designed and then results have been

7 174 M. Rostami N. K.i, J. Razmi, and F. Jolai ompared with Lingo s results. Both geneti algorithms run 30 times and satisfation degree for eah goal is alulated. Then we set: (Weight of manufaturer)=0.3, (Weight of DCs)=0.3, (Weight of retailers)=0.4. Also, we set =0.5( is oeffiient in TH method) i=3, t=4, p=3, d=2, r=3, =3. We generate all of required parameters randomly. For reeiving further information about these random numbers, please ontat the orresponding author. To survey the results obtained from Lingo8 and geneti algorithm, a problem with small size has been solved and results have been presented in table1. It is onluded that the average differene between PGA and HGA and lingo is 7% and 10% respetively. Obviously, differene between results obtained from PGA and HGA are not large; thus, methods an be used in problems with large size. Now, onsider a large model and set: i=6, t=8, p=5, d=2, r=3, =3. The satisfation degree of eah goal is shown in table 2; as an be seen, the satisfation degree for goals in PGA algorithm is more than HGA, and it reveals the effiieny of proposed geneti algorithm. Table 1. Ahievement levels obtained by Lingo, HGA, PGA in small size Ahievement Lingo HGA PGA Level(%) μprofitm μdc1profit μdc2profit μret1blg µret2blg µret3blg Table 2. Ahievement levels obtained by HGA, PGA in large size Ahievement HGA PGA Level(%) μprofitm μdc1profit μdc2profit μret1blg µret2blg µret3blg Conlusions Managing operations in today s ompetitive market plae poses signifiant hallenges. Based on the traditional thought there were so many onflits in the multiple demands on the operational funtions that trade-offs were made in ahieving exellene in one or more of these dimensions.

8 Designing a Geneti Algorithm to Solve an Integrated Model in SCM 175 This paper takled with integrated inventory-prodution-distribution planning problem in SC system. A multi-objetive linear programming model has been developed in this onern. Deision maker s impreise aspiration levels for the goals have been inorporated into the model using fuzzy goal programming approah. Computational experiments have been provided from a ase problem. A geneti algorithm has been proposed to solve onstrained problem and results have been ompared to geneti algorithm suggested by Haupt and Haupt(2004). While, in solving problems with onstraints there is no guarantee that initial population is feasible or has minimum violation. Proposed Geneti Algorithm, using PSO algorithm, tries to onstrut initial solution with minimum violation. Also a new mating method has been proposed in our geneti algorithm. In addition we don t use penalty funtion as fitness funtion. Numerial results obtained, showed the effiieny of proposed algorithm rather than algorithm used by Haupt and Haupt[7]. The paper ontributes to literature in designing a multiple objetive linear model for integrated inventory-prodution-distribution planning whih is lose to real world supply hain and using fuzzy goal programming, also proposing a new geneti algorithm. Analysis of variane has been used to ompare mean of violation in HGA and PGA. It is resulted that violation in PGA is less than HGA. Thus, PGA is loser than HGA to feasible solution. Referenes 1. Abbasbandy, S., Asady, B.: Ranking of fuzzy numbers by sign distane. Inform. Si. 176, (2006) 2. Aliev, R.A., Fazlollahi, B., Guirimov, B.G., Aliev, R.R.: Fuzzy-geneti approah to aggregate prodution distribution planning in supply hain management. Information Sienes 177, (2007) 3. Chen, S.-H.: Computational intelligene in eonomis and finane: arrying on the legay of Herbert Simon. Inform. Si. 170(1), (2005) 4. Chen, S.-P., Chang, P.C.: A mathematial programming approah to supply hain models with fuzzy parameters. Eng., Optimiz. 38(6), (2006) 5. Gao, M., Zhou, M.C., Tang, Y.: Intelligent deision making in disassembly proess based on fuzzy reasoning Petri nets. IEEE Trans. Syst. Man. Cyb. B: Cyb. 34(5), (2004) 6. Gottwald, S.: Mathematial fuzzy logi as a tool for the treatment of vague information. Inform. Si. 172(1-2), (2005) 7. Haupt, R.L., Haupt, S.E.: Pratial Geneti Algorithm, 2nd edn. John Willey and Sons, In., Hoboken (2004) 8. Liang, T.-F.: Distribution planning deisions using interative fuzzy multi-objetive linear programming. Fuzzy Sets Syst. 157(10), (2006) 9. Nozik, L.K., urnquist, M.A.: Inventory, Transportation, Servie quality and the loation of distribution enters. European Journal of Operational Researh 129, (2001) 10. Selim, H., Araz, C., Ozkarahan, I.: Collaborative prodution distribution planning in supply hain: A fuzzy goal programming approah. Transportation Researh Part E (2007) 11. Sheen, J.N.: Fuzzy finanial profitability analyses of demand side management alternatives from partiipant perspetive. Inform. Si. 169(3-4), (2005)

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