ISSN 1816-6075 (Print), 1818-0523 (Online) Journal of System and Management Sienes Vol. 2 (2012) No. 3, pp. 9-17 An integrated optimization model of a Closed- Loop Supply Chain under unertainty Xiaoxia Zhu 1, Xiuquan Xu 2 1 Shool of Sienes, Hebei University of Siene and Tehnology, Shijiazhuang Hebei 050018, China 2 Shools of Eonomis and Management, Beijing Jiaotong University Beijing100044, China Abstrat: This paper studies a losed-loop supply hain, mainly onsisting of positive selling, repairing, and remanufaturing under unertainty. Then, we build a model for the losed-loop supply hain integration and optimization under unertainty, in whih a omprehensive utility funtion is used to larify the random parameters, so that we an transform the model into a mixed integer programming model and do further analysis by hybrid geneti algorithm. The feasibility and validity of the model are verified by illustrative examples. This paper provides an effetive deisionmaking tool for the losed-loop supply hain design. It is useful for the integration of forward and reverse supply hains. Keywords: Closed-loop supply hain, Produts repair, Modular reutilization, Unertainty environment, Hybrid geneti algorithm 1. Introdution Early supply hains were mostly profit-oriented and aimed at reduing osts and enhaning ompetitiveness. They had one-way linear struture whih laked the awareness of sustainable development. However, through the growth of the global market, more and more ompanies realized that ustomers satisfation is extremely important for them to survive. The development of the buyer s 9
market leaded to fierer ompetition by inreasing produt alternatives and shortening produts life yles. The more and more important role of the market, espeially the role of onsumers hoie, has fored ompanies to extend produts liability to a longer period, sometimes even lifelong. Therefore, the supply hain requires innovation urgently. A series of operating proedures and assoiated networks should be added to the traditional supply hains, suh as testing, repairing, reproessing, and reyling. A new supply hain will thus be onstruted by integrating the traditional one with various reverse ativities. In this new hain, all materials of the produts in the irular flow should be managed effetively in their life yles, thus the negative impats on environment would be redued. Suh a supply hain is named as Closed-Loop Supply Chain (CLSC) (Harold, K., et al., 2001, Van Wassenhove, L. N., et al., 2003). The losed-loop supply hain onsists of both the positive supply hain and the reverse supply hain in the produt life yle. There are a number of methods that we an use to onstrut supply hains. The losed-loop supply hain networks are mostly under unertainties. Most studies established optimized model of losed-loop supply hain in a supposed approximation of ertainty; but it is against the existene of the unertain information suh as vagueness or randomness, suh as Mao H.J., et al., (2006); Liu, Q., et al., (2007); El-Sayed, et al.,(2008), and so on. In this paper, by taking the random environmental fators into onsideration, we will establish the model, whih it is based on the unertainties of produts repair, reyling, sales, and the time value of money. Afterwards, we will give numerial examples to present the effets of parameters on the losed-loop supply hain model under unertainty, the results indiate that the model is not only aommodates the existing unertainty deision-making methods, but also suessfully inorporates the deision preferene into the optimization proess. 2. Modeling design 2.1. Deision variables and parameters desription Consider varieties of produts and the fat of modular prodution. Every fatory an make and repair goods at the same time, but the unit osts of prodution and 10
repairing are different; during every period, demands of onsumer markets for various produts are treated as random numbers. Demands among onsumer markets are independent; the demand of onsumers for eah produt an be supplied partially or exessively. Both situations will indue penalty ost; the number of tested produts and reyled produts is proportional to the demands of onsumers. Only useful produts will be reyled; all the failities will be built at known loations, and restritions are set on apaity onstraints; all failities should be onstruted or extended in the orresponding periods, aording to the demands of onsumer markets. But the extension sale is limited, and the time spent is negligible; for eah faility, the handling osts, transportation osts and penalty osts are known; seleted failities annot be removed but an only be extended. In this setion, symbols for parameter variables and deision variables in the model are as follows: I Number of fatories loations,iε{1,2,, I}, J Number of disassembling enters loations; jε{1,2,, J} M Number of ustomers loations, mε{1,2,, M}; N Number of testing enter loations, nε{1,2,, N}; T Number of time periods, tε{1,2,, T} ; L Number of produt varieties, d lε{1,2,, L}; Q ablt Number of produt l that transports from a to b during the t period ; X at If a new faility is built in loation a during the t period, then X at = 0; Y at If the faility is extended in loation a during the t period, then Y at = 1; otherwise Y at = 0; Z at The sale of the faility extended in loation a during the t th period; FC at The ost of building a new faility in loation a during the t th period; KC at The ost of extending one unit sale of faility in loation a during the t th period; VC pm ilt the ost of produing one unit of produt l in fatory i during the t th period; VC pr ilt The ost of repairing one unit of produt l in fatory i during the t th period; VC alt The ost of disposing one unit of produt l by faility in loation a during the t th d period; SC ablt The ost of transporting one unit of produt l from a to b during the t th period; A a The maximal apaity of faility in loation; B a The inreased apaity of extending one unit of sale of faility in loation; C a The maximal sale of extending faility in loation; D The maximal number of new faility that an be built; U il pm The apaity ost of produing one unit of produt l in 11
fatory i; U il pr The apaity ost of repairing one unit of produt l in fatory i; U al The apaity ost of disposing one unit of produt l by faility in loation a; D mlt The demand of ustom market m for produt l during the t th + ; λ mlt The penalty ost one unit of produt l when the supply of it is greater than the demand of ustom market m during the t th period; λ mlt The penalty ost one unit of produt l when the supply of it is less than the demand of ustom market m during the t th period; r mlt The failure (return) rate of produt l in ustrom market m during the t th period; g mlt The reyling rate of produt l in ustom market m during the t th period; e mlt The failure (return) rate of produt l whih need to be repaired in ustom market m during the t th period; θ mlt The average value of one unit of produt l reyled in ustom market m during the t th period; IR t The Annual Perentage Rate (APR) of the bank during the t th period. The subsripts a and b in deision variables and parameters; i, j, m and n symbolize the loation of the new built fatory, the new built disassembling enter; the onsumer market loation and the testing enter loation, respetively; The supersript and d in deision variables and parameters; p, w, z and r symbolize fatories, disassembling enter, onsumer markets and testing enters respetively. 2.2. Network optimization model The objetive funtions of the model: min TC = FixC + OperC + TranC + PuniC ModC (1) (2) (3) 12
(4) (5) Formula (2) expresses the fixed ost of building and extending fatories, disassembling entres and testing entres. Formula (3) expresses the operating ost of all the failities. Formula (4) expresses the transportation ost among all the failities Formula (5) expresses the penalty ost when the supply is either greater or less than the demand. Formula (6) expresses the re-use value of the reyled produts. Constraints: (6) (7) (8) (9) 13
(10) (11) (12) (13) (14) (15) (16) (17) (18) (19) 14
(20) Constraints (7) and (8) desribe the limited apaity of eah faility; Constraints (9) assure the amount of input and output logistis should onserve. Constraint (10) presents the number of produt need to be repaired. Constraints (11) assure that a faility has input and output logistis only if the faility has already been set up. M is an infinite number. Constraint (12) assures the total number of produts ustomer returns is equal to the number of market expeted value of repairing. Constraint (13) assures the total number of produts disassembled is equal to the number of market expeted value of reovery. Constraint (14) presents that eah faility is only built one. Constraint (15) desribes that the number of failities is limited. Constraint (16) assures that eah faility would be extended only if it already exits. Constraint (17) assures the extending sale of the failities is a positive integer only if they are to be extended. Constraint (18) assures the extending sale annot exeed the maximal sale. Constraints (19) to (20) present the ranges of eah deision variables. 3. Numerial results An equipment enterprise onsiders building a losed -loop supply hain network whih will inlude both distribution hannels and the reverse logistis network of produt returns and reyling. We onsider three periods, three andidates of fatories, five andidates of testing enters, two andidates of dismantling enters, and 10 onsumer markets in our design. Parameters of ustomer demands are assumed to have normal distribution the data of other Parameters is omission. In order to solve the model, we design and prepare the optimized program with the geneti algorithm optimal-toolbox of MATLAB, Parameters of the designed GA are: rnax_generation 100, poi_size 80, ross_rate 0.5, mutation_rate 0.1, and the generation gap 90%. In this paper, the random data is larified by the third random effets omprehensive funtion. 15
Then, we use this algorithm to solve the above-mentioned model. Results after alulation are shown in the Table 1. Table 1The summary of the solution in the model The total ost of SCLSC model is $1.6992E +06, in whih the fixed apital ost is $1.6879E +06, the extension ost is $1.0376E +04, penalty ost is $783.00, transportation ost is $3.2731E +03; the utilizing value generated by the reovery and demolition is $735.14. In short, in the results from multiperiod optimization model, failities are relatively small and entralized. Therefore, it is good for the operation and management of user-friendly losedloop supply hain. 4. Conlusions and future work This paper mainly fouses on the optimized design of losed-loop supply hain in unertain environment. It offers dynami model of loation seletion based on different fators. It takes the dynami state of loation, faility extension, and apaity improvement into onsideration. The feasibility and effiieny has been analyzed and verified through ases. Numerial results show that the algorithm based on optimal perturbation analysis is feasible and effetive. We also analyzed some parameters of the model. Aknowledgements This paper is supported by the projet from National Key Tehnologies R & D Program of China (2012BAH20B03) and Supported by Foundation of Hebei University of Siene and Tehnology (XL200761). Referenes 16
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