MULTI-ENTERPRISE SUPPLY CHAIN OPTIMIZATION BY MEANS OF ...

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MULTI-ENTERPRISE SUPPLY CHAIN OPTIMIZATION BY MEANS OF ...

  1. 1. nd 2 Mercosur Congress on Chemical Engineering th 4 Mercosur Congress on Process Systems Engineering MULTI-ENTERPRISE SUPPLY CHAIN OPTIMIZATION BY MEANS OF EVOLUTIONARY STRATEGIES C. M. Silva11, E. C. Biscaia Jr.1 1 Programa de Engenharia Química/COPPE - Universidade Federal do Rio de Janeiro Abstract. In this contribution, the multi-objective optimization of a multi-enterprise supply chain network is investigated. A multi-product, multistage and multi-period production and distribution-planning model is addressed. The optimization problem is formulated as a multi-objective mixed-integer nonlinear programming problem (MOMINLP). Objective functions including maximizing profit of each participant enterprise, customer service level and safe inventory level are considered. A multi-objective optimization algorithm based on evolutionary strategies is applied to solve the problem. A new fuzzy penalization strategy is proposed to improve the performance of the algorithm in dealing with the huge number of decision variables and constraints that comprises such systems. The method is based on a hierarchy classification of the constraints, which are penalized according to the level and frequency of violation. The proposed method has successfully attained a compromise solution among all participant enterprises of the supply chain. It has also provided a fair distribution profit with high satisfaction for all objectives. The results of an hypothetical case study confirmed the ability of the proposed method in providing an improved compensatory solution for multi-objective problems in supply chain systems. Keywords: Multi-enterprise supply chain management, multi-objective optimization, evolutionary strategies. 1. Introduction Supply chain networks are complex structures that involve integration and coordination of several enterprises. Suppliers, manufacturers, distribution network and customers are interconnected echelons of these structures. An efficient and flexible supply chain system is demanding for every enterprise in order to face global competition. Traditionally, supply chain optimization was focused on local issues as optimization of plant operations, individual logistic activities of distribution and inventory management. In recent years, however, the increasing market requirements have placed great emphasis on companies towards coordination and integration of all their business operations. In order to achieve a profitable supply chain, it is necessary to consider all interactions and limitations among members, as well as individual behaviors, which also influence the performance of the whole system. In most studies concerning supply chain management, minimization of costs or maximization of profit as a single objective function is adopted. Although these strategies can generate an optimal result for the whole system, they can also increase costs or decrease profit margins for some of the participant enterprises. In supply chain optimization involving multiple enterprises, policies that ensure fair profit distribution between echelons are also required. Other important requirements such as customer service are not taken into account, since it is difficult to quantify them as a monetary amount (Chen et al, 2003). A multi-objective formulation is therefore more appropriate to describe the general production and distribution-planning problem. 1* To whom all correspondence should be addressed. Address: Programa de Engenharia Química, PEQ/COPPE/UFRJ - Centro de Tecnologia, bloco G, sala G-115, Cidade Universitária - 21945-970 Rio de Janeiro – Brazil 1
  2. 2. nd 2 Mercosur Congress on Chemical Engineering th 4 Mercosur Congress on Process Systems Engineering Several attempts have been made to optimize supply chain models by deterministic approaches. Perea-López et al. (2003) describe a model predict control strategy for profit maximization of a multi-product, multi-echelon distribution network with multi-product batch plant. These authors use a rolling horizon MPC approach to include the disturbance influences in the MILP dynamic model. Chen et al (2003) address the problem of a fair distribution for a multi-enterprise supply chain. A two-phase fuzzy decision-making method is proposed to attain a compromise solution that ensures a fair profit distribution among all participant companies. Mokashi and Kokossis (2003) apply customized dispersion algorithms to solve a problem involving production planning and distribution scheduling. These authors present an approach to decompose the overall problem into sub-parts that are modeled using dispersion framework. Stochastic methodologies are also employed to solve supply chain multiobjective optimization problems. Guillén et al. (2005) solve a supply chain design problem as a multiobjective stochastic MILP model by considering profit, customer satisfaction and financial risk. Chan et al. (2005) develop a hybrid genetic algorithm to solve production and distribution problems in multi-factory supply chain models. An analytic hierarchy process is used to calculate fitness values. Zhou and Hua (2000) use two multiobjective decision-making methods – goal programming and analytic hierarchy process – to address sustainable supply chain optimization and scheduling of continuous process industries. Azapagic and Clift (1999) use life cycle assessment in environmental management to solve a multiobjective optimization system in which complete supply chains are considered. In this contribution, a multiobjective optimization algorithm based on evolutionary strategies is applied to determine the best configuration of a multi-enterprise supply chain network. A fuzzy penalization approach is proposed to deal with the large number of constraints characteristic of such systems. The proposed approach allows the formulation of the problem as a multi-objective mixed-integer nonlinear problem (MOMINLP). The fair profit distribution problem is also addressed. 2. Problem Statement The supply chain considered in this work consists of a centralized three-echelon structure – manufacturing, storage and market. The first echelon comprises two retailers or markets, from which products are sold to customers. Two warehouses, where products are stored, constitute the second level. The warehouses are distribution centers that deliver the products from the plant to retailers. The third level is the plant. The distribution channels between plants and retailers consist of a smaller-scale distributor with fast delivery service and a larger-scale distributor with a slower delivery service. The faster service implies higher operating costs. The slower one, on the other hand, uses economies of scale to lower its costs, but has a transportation lead- time of one week. Delayed shipment problem is considered in the distribution system. The plant batch manufactures two different products. Each product is manufactured during a particular period of time and has a fixed manufacturing cost associated. If the production line is idle, a fixed idle cost is added to the total manufacturing cost. The manufacturing can be conducted in regular time or overtime, to satisfy customer demand. The raw material purchasing cost is included in the manufacturing cost. E-mail: cmartins@peq.coppe.ufrj.br 2
  3. 3. nd 2 Mercosur Congress on Chemical Engineering th 4 Mercosur Congress on Process Systems Engineering The planning horizon considered in this problem is of 3 weeks. Each week correspond to one period. The overall problem is stated as follows: Given: a) product sale prices at each retailer, distribution center and plant; b) direct cost parameters such as unit manufacturing, transport, handling and inventory costs; c) manufacturing data, such as production in regular time and overtime; d) transportation data, such as capacity level and lead time; e) inventory data of each node, such as inventory capacity and safe inventory quantity and f) forecasted customer demand for each product on each retailer over a time horizon. Determine: a) production schedule, including production rates of each product for all time intervals; b) transportation of products between plants, distribution centers and retailers; c) sale quantity of each retailer, distribution center and plant; d) costs and revenue of each enterprise and e) inventory level of each enterprise. The objective is to determine the configuration of the supply chain that maximizes the profit of each enterprise, the customer service level and safe inventory level, taking into account a fair distribution of these targets among all the participants. 3. Multiobjective Model Formulation The mathematical formulation of the supply chain model, as proposed by Chen et al. (2003), is adopted. All parameters related to system information are presented in the abovementioned reference. 3.1. Mathematical Relations of Retailer Inventory balance constraints. The inventory level of product i at retailer r corresponds to the difference between the amounts received from all distribution centers, considering the delayed shipment caused by the transportation lead time, and the amounts sold to costumers during period t, plus the amount stored during the previous period: I irt = I irt ,t −1 + ∑ S idr ,t −TLTdr − S irt (1) d All inventories should be greater than the safe inventory quantities on the last period: I irT ≥ SIQir (2) Backlog level constraints. The backlog level of product i at retailer r is the amount at the previous period plus the difference between the forecasted customer demand and the amounts sold to costumers during period t: Birt = Birt ,t −1 + FCDirt − S irt (3) The backlog at last period should be zero to fulfill customer demand. All physical variables should be positive: BirT = 0 I irt ≥ 0 Birt ≥ 0 S irt ≥ 0 (4) Maximum inventory capacity constraints. The total amount of products at each retailer during period t is limited: ∑ I irt ≤ MIC r (5) i 3
  4. 4. nd 2 Mercosur Congress on Chemical Engineering th 4 Mercosur Congress on Process Systems Engineering Safe inventory constraints. The shortage in safe inventory of product i at retailer r during period t cannot exceed the safe inventory quantity. In the last period, there should be no shortage in safe inventory for all retailers: SIQir − I irt ≤ Dirt ≤ SIQir (6) DirT = 0 Dirt ≥ 0 (7) Purchase cost. The total purchase cost corresponds to the sum of the unit sales revenue multiplied by the quantity of products sold by the distribution centers to retailer, during period t: TPC rt = ∑∑ USRidr S idrt (8) d i Inventory cost. The total inventory cost is equal to the sum of the unit inventory cost of each product multiplied by the product inventory level at the end of the period t: TIC rt = ∑ UICir I irt (9) i Handling cost. The total handling cost is equal to the sum of the unit handling cost of each product multiplied by the total amount of each product handled during period t: THC rt = ∑ UHCir (∑ S idr ,t −TLTdr + S idrt ) (10) i d Product sales revenue. The revenue is the sum of the unit sales revenue of each product multiplied by the quantity of product sold by retailer to customer, during period t: PSRrt = ∑ USRir S irt (11) i Customer service level. The customer service level is defined as the average percentafe ratio of actual sales quantity of each product from retailer to customer during period t, to the total demand quantity, which comprises the sum of the forecasted demand and the backlog level at the end of the previous period: 100 S CSLrt (%) = ∑ FCD irt B (12) I i irt + ir ,t −1 Safe inventory level. The safe inventory level corresponds to the average percentage of 1 minus the ratio of the shortage in safe inventory to the safe inventory level for each product: 100  Dirt  SILrt (%) = ∑ 1 − SIQ  I i    (13) ir  Objective functions. Overall profit max ∑ Z rt = PSRrt − TPC rt − TIC rt − THC rt (14) t 1 Average customer service level max ∑ CSLrt T t (15) 4
  5. 5. nd 2 Mercosur Congress on Chemical Engineering th 4 Mercosur Congress on Process Systems Engineering 1 Average safe inventory level max ∑ SILrt T t (16) 3.2. Mathematical Relations of Distribution Center Similar mathematical relations are used to describe the distribution center model. Inventory balance constraints I idt = I idt ,t −1 + ∑ S ipd ,t −TLTpd − ∑ S idrt (17) p r I idT ≥ SIQid I idt ≥ 0 S idrt ≥ 0 (18) Maximum inventories constraints ∑ I idt ≤ MIC d (19) i Shortage in safe inventories constraints SIQid − I idt ≤ Didt ≤ SIQid (20) DidT = 0 Didt ≥ 0 (21) Output transportation constraints. The sum of all products sold by the distribution center to retailer during period tthe is equal to total quantity of products shippped to retailer: ∑ Qkdrt = ∑ S idrt (22) k i TCLk −1,dr Ykdrt ≤ Qkdrt ≤ TCLkdr Ykdrt (23) The products are shipped by only one mode of transportation: ∑ Ykdrt ≤1 (24) k ∑∑ TCLkdrYkdrt ≤MOTC d (25) r k Input transportation constraints ∑ Qk ' pdt = ∑ S ipdt (26) k' i TCL k '−1, pd Yk ' pdt ≤ Qk ' pdt ≤ TCL k ' pd Yk ' pdt (27) ∑ Yk ' pdt ≤ 1 (28) k' ∑∑ TCLk ' pd Yk ' pdt ≤MITC d (29) p k' Purchase cost TPC dt = ∑∑ USRipd S ipdt (30) p i Inventory cost TIC dt = ∑ UICid I idt (31) i Handling cost THC dt = ∑ UHCid (∑ S ipd ,t −TLTpd + S idrt ) (32) i d Transportation cost. The transportation cost is the sum of the input and output transportation costs: 5
  6. 6. nd 2 Mercosur Congress on Chemical Engineering th 4 Mercosur Congress on Process Systems Engineering TTC dt = ∑ ∑ ( FTC kdr Ykdrt + UTC kdr Qkdrt ) + ∑ ∑ ( FTC k ' pd Yk ' pdt + UTC k ' pd Qk ' pdt ) (33) k r k' p Product sales revenue PSRdt = ∑∑ USRidr S idrt (34) r i 100  Didt  Safe inventory level SILdt (%) = ∑ 1 − SIQ  I i    (35) id  Objective functions Overall profit max ∑ Z dt = PSRdt − TPC dt − TIC dt − THC dt − TTC dt (36) t 1 Average safe inventory level max ∑ SILdt T t (37) 3.3. Mathematical Relations of Plant Similar mathematical relations are used to describe the plant model. Inventory balance constraints I ipt = I ipt ,t −1 + FMQip α ip ,t −1 + OMQip ο ip ,t −1 − ∑ S ipdt (38) d I ipT ≥ SIQip I ipt ≥ 0 S ipdt ≥ 0 (39) Maximum inventory constraints ∑ I ipt ≤ MIC p (40) i Shortage in safe inventory constraints SIQip − I ipt ≤ Dipt ≤ SIQip (41) DipT = 0 Dipt ≥ 0 (42) Manufacturing constraints. The plant manufactures just one product at time. The plant has to be set up to manufacture the product in regular time.The plant can change over to manufacture a product. The overtime production takes place when regular production is not sufficient. Total number of overtime periods is limited. The number of continuous overtime period is limited. ∑ βipt =1 α ipt ≤ β ipt (43) i γ ipt ≥ β ipt − β ip ,t −1 ο ipt ≤ α ipt (44) ∑∑ οipt ≤MTO p (45) i t ∑ ∑ ο ip ,t −n+1 ≤N − 1 (46) i n Manufacturing cost TMC pt = −∑ [ FMCip γ ipt + FICip (β ipt − α ipt ) +UMCip FMQip α ipt + OMCip OMQip ο ipt (47) i Inventory cost TIC pt = ∑ UICip I ipt (48) i 6
  7. 7. nd 2 Mercosur Congress on Chemical Engineering th 4 Mercosur Congress on Process Systems Engineering Handling cost THC pt = ∑ UHCip ( FMQip α ip ,t −1 + OMQip ο ip ,t −1 + ∑ S ipdt ) (49) i d Product sales revenue PSR pt = ∑∑ USRipd S ipdt (50) d i 100  Dipt  Safe inventory level SIL pt (%) = ∑ 1 − SIQ  I i    (51) ip  Objective functions Overall profit max ∑ Z pt = PSR pt − TMC pt − TIC pt − THC pt (52) t 1 Average safe inventory level max ∑ SIL pt T t (53) 4. Multi-objective Optimization Multi-objective optimization consists of a search for non-dominated solutions, which are superior to the others on all objectives. The process is guided by the fitness evaluation and continues until no improvement is obtained in any objective without deteriorating at least one of the other objectives. The optimal solution constitutes a family of points, called Pareto optimal front, that equally satisfy the set of objective functions. Any of these points is an acceptable solution of the problem, thus, its selection requires additional knowledge of the problem. Evolutionary algorithms are robust stochastic methods for global and parallel optimization. These methods are founded on the principles of natural genetics, in which the fittest species survive and propagate while the less successful tend to disappear. The evolution strategy consists of performing the population with genetic operators to generate the next population. The basic genetic operators simulate the processes of selection, crossover and mutation. Selection is based on the survival potential, expressed by the fitness function. Crossover involves random exchange of characters between pairs of individuals, in order to produce new ones. Mutation is an occasional change in individual’s characters randomly chosen. It introduces diversity to a model population. These processes happen according to pre-established probabilities. Evolutionary methods are able to cope with ill-behaved problem domains, such as the ones presenting multimodality, discontinuity, time-variance, randomness and noise. 4.1 The Proposed Strategy A multi-objective optimization algorithm based on evolutionary strategies has been proposed to generate optimal configurations for the supply chain problem. In this algorithm, the concept of population ranking initially suggested by Goldberg (1989) has been extended to treat multidimensional problems. The probability of reproduction has been determined by a fitness function, which takes into account the rank level and the populational density on the rank. In the ranking procedure, points are categorized into groups according to their non-dominance levels. The closer to the Pareto set a group of points is, the higher its probability to propagate to the next generation. 7
  8. 8. nd 2 Mercosur Congress on Chemical Engineering th 4 Mercosur Congress on Process Systems Engineering Some operators have been introduced to enhance the algorithm performance: (a) a niche operator, which prevents genetic drift and maintains a uniformly distributed population along the optimal set; (b) a Pareto-set filter, which avoids missing optimal points during the evolution process; (c) an elitism operator, which insures the propagation of the best result of each individual objective function. Real codification, which is computational fast and stable in converging to global optima, is also adopted. The original algorithm is presented in a previous work (Silva and Biscaia, 2002). In this contribution, a new penalty function method based on fuzzy logic theory has been proposed to deal with the huge number of constraints. In this approach, the constraints are classified according to a hierarchy methodology. First-stage constraints correspond to extremely rigid ones, whose violation results in losing the physical meaning of variables. Second stage involves intermediary constraints, which must be satisfied during the optimization process. Third stage includes equality constraints that directly influence the objective function values. These constraints restrict the optimization process, as their satisfaction represents one less degree of freedom in the process. Hence, third-stage constraints are just satisfied in the end of the optimization process. The fuzzy approach consists of different levels of penalization, which are incorporated into the fitness function. The levels are graduated regarding the stage, being stronger in the first stage, intermediate in the second and weak in the third stage. The penalization method also involves two different strategies to improve the performance of the algorithm, based on the intensity and frequency of violation. The optimization is conducted for each class of constraints following the strategies described, until the feasible region is reached. High mutation probabilities are also used to increase the exploitation ability, improving the robustness of the algorithm and the convergence of the optimization process. The original algorithm has been adapted to operate simultaneously in continuous and discrete variable space, in order to treat the mixed-integer nonlinear problem. The operators were also reformulated to deal with a variable number of decision variables. 5. Results and Discussion According to the problem description, three-levels of enterprises are integrated in a multi-objective optimization problem consisting of 12 objective functions, 172 constraints and 122 decision variables. A population size of 30 individuals, crossover probability of 80% and mutation probability of 50% are used to solve the problem. Table 1 shows the resulting Pareto optimal set. High values for all objective functions are obtained, which indicates that the proposed strategy leads to an unbiased search process. A balanced exploration process is mandatory to obtain a good compromise solution to all objectives, and, thus, address the fair distribution problem. Previous attempts to solve this problem using other methods resulted in an unbalanced solution, and required additional optimization (Chen et al, 2003). It should be noticed that the best and worse results obtained for the most relevant objective functions – f1, f4, f6, f9 and f11 - which represent each enterprise profit, do not differ in order of magnitude. Hence, any of the solutions would be satisfactory to all participant enterprises. The highest possible value for the average customer service level and the average safe inventory level, 100%, is obtained for more than half of the objective functions. 8
  9. 9. nd 2 Mercosur Congress on Chemical Engineering th 4 Mercosur Congress on Process Systems Engineering Table 1. Pareto optimal set f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 5.16E+05 0.727 0.6075 6.04E+05 0.8229 0.8275 1.67E+04 0.565 1.00E+06 0.81 2.63E+05 0.711 5.25E+05 0.923 0.6075 6.04E+05 0.822 0.8275 1.67E+04 0.565 1.00E+06 0.81 2.63E+05 0.711 7.05E+05 0.873 0.6475 5.72E+05 0.570 1.00 1.67E+04 1.00 1.03E+06 0.5 1.99E+05 1.00 7.05E+05 0.873 0.6475 5.72E+05 0.570 1.00 1.67E+04 1.0 1.03E+06 0.5 1.99E+05 1. 00 7.89E+05 0.770 0.5 9.52E+05 0.907 0.8275 1.67E+04 0.515 1.03E+06 0.81 1.99E+05 1. 00 7.91E+05 0.808 0.7975 6.04E+05 0.822 0.8275 1.67E+04 0.565 1.01E+06 0.81 1.99E+05 0.987 7.91E+05 0.808 0.735 6.04E+05 0.822 0.8275 6.72E+03 0.565 1.03E+06 0.94 1.99E+05 1. 00 7.91E+05 0.808 0.735 6.04E+05 0.822 0.827 6.72E+03 0.565 1.03E+06 0.94 1.99E+05 1. 00 7.91E+05 0.808 0.622 6.04E+05 0.822 0.827 1.67E+04 0.565 1.03E+06 0.81 1.99E+05 1. 00 7.91E+05 0.808 0.695 6.04E+05 0.822 0.827 1.67E+04 1.00 1.03E+06 0.81 1.99E+05 1. 00 7.91E+05 0.808 0.695 6.04E+05 0.822 0.827 1.67E+04 1.00 1.03E+06 0.81 1.99E+05 1.00 7.91E+05 0.808 0.797 6.04E+05 0.822 0.827 1.67E+04 0.565 1.01E+06 0.81 1.99E+05 0.987 7.91E+05 0.808 1.00 6.04E+05 0.822 0.827 1.67E+04 0.565 1.01E+06 0.81 1.99E+05 0.987 7.91E+05 0.808 0.695 6.04E+05 0.822 0.827 1.67E+04 1.00 1.03E+06 0.81 1.99E+05 1.00 7.91E+05 0.808 0.622 6.04E+05 0.822 0.827 1.67E+04 0.565 1.03E+06 0.81 1.99E+05 1.00 7.91E+05 0.808 0.735 6.04E+05 0.822 0.827 6.72E+03 0.565 1.03E+06 0.94 1.99E+05 1.00 7.91E+05 0.808 0.797 6.04E+05 0.822 0.827 1.67E+04 0.565 1.01E+06 0.81 1.99E+05 1.00 9.98E+05 0.803 0.777 9.52E+05 0.907 0.827 6.72E+03 0.515 1.02E+06 0.74 1.99E+05 1.00 9.98E+05 0.803 0.602 1.09E+06 0.907 0.827 6.72E+03 0.515 1.02E+06 0.74 1.99E+05 1.00 6. Conclusions In this contribution, a multi-product, multistage and multi-period production and distribution-planning model is formulated as a multi-objective mixed-integer nonlinear programming problem (MOMINLP). The model is optimized by using a multi-objective optimization algorithm based on evolutionary strategies. The targets of the optimization problem includes maximizing profit of each participant enterprise, customer service level and safe inventory level. A new fuzzy penalization strategy is proposed to deal with the huge number of constraints, characteristic of supply chain systems. This strategy prevents reducing the degree of freedom of the optimization process, which improves the performance of the algorithm in converging to the optimal solution set. A compromise solution among all participant enterprises of the supply chain is also achieved, ensuring a fair distribution profit. The results confirm the efficiency of the proposed optimization approach to solve supply chain problems involving a significant number of decision variables, constraints and objective functions. In case of huge supply chain problems, however, hybrid algorithms seem to be more appropriate, as the large number of function evaluation required may become prohibitive. Notation Indices p products r retailers t periods d distribution centers k transportation capacity level from distribution centers to retailers p plants k’ transportation capacity level from plants to distribution centers Parameters USR{i, pd, dr, r} unit sale revenue of i, sold from p to d, from d to r or from r to customer UICi{i, p, d, r} unit inventory cost of i for p, d, r UHC{i, p, d, r} unit handling cost of i for p, d, r UTC{k, dr} kth-level unit transportation cost , from d to r or from p to d 9
  10. 10. nd 2 Mercosur Congress on Chemical Engineering th 4 Mercosur Congress on Process Systems Engineering FTC{k, dr} kth-level fixed transportation cost, from d to r FTC{k’, pd} k’th-level fixed transportation cost, from p to d UMC{i, p} unit manufacturing cost of i OMC{i, p} overtime unit manufacturing cost of i FMC{i, p} fixed manufacturing cost for changing plant to make i FIC{i, p} fixed idle cost to keep plant idle FCD{i, r, t} forecasted customer demand for i TLT{pd, dr} transportation lead time, from p to d or from d to p SIQ{i, p, d, r} safe inventory quantity in p, d , r MIC{i, p, d, r} maximum inventory capacity of p, d, r TCL{k, dr} kth transportation capacity level, from d to r or from p to d MITC{d} maximum input transportation capacity of d MOTC{d} maximum output transportation capacity of d FMQ{i, p} fixed manufacturing quantity of i OMQ{i, p} overtime fixed manufacturing quantity of i MTO{p} maximum total overtime in manufacturing period Binary Variables α{i, p, t} manufacture with regular-time workforce β{i, p, t} set up plant to manufacture i γ{i, p, t} change plant over to manufacture i ο{i, p, t} manufacture with overtime workforce Y{k, dr, t} kth transportation capacity level from d to r, or from p to d Continuous variables S{i, pd, dr, r, t} sales quantity of i, from p to d, from d to r or from r to customer Q{k, dr, t} kth-level transportation quantity from d to r, or from p to d Q{pd, dr, t} total transportation quantity from d to r or from p to d I{i, p, d, r, t} inventory level of i in p, d, r B{i, r, t} backlog level of i in r at end of t D{i, p, d, r, t} shortage in safe inventory level in p, d, r TMC{p, t} total manufacturing cost of p TPC{d, r, t} total purchase cost of d, r TIC{p, d, r, t} total inventory cost of p, d, r THC{p, d, r, t} total handling cost of p, d, r TTC{d, t} total transportation cost of d, from p to d or from d to r PSR{p, d, r, t} product sales revenue of p, d, r SIL{ p, d, r,, t} safe inventory level of p, d, r CSL{r, t}customer service level of r Z{p, d, r, t} net profit of p, d, r References Azapagic. A., Clift. R. (1999). The Application of Life Cycle Assessment to Process Optimization. Comp. Chem. Eng., 10, 1509. Chan. F.S, Chung. S.H, Wadhwa. S. (2005). A Hybrid Genetic Algorithm for Production and Distribution. Omega, 33, 345. Chen. C. L., Wang. B. W., Lee, W. C. (2003). Multiobjective optimization for a Multienterprise Chain Network. Ind. Eng. Chem. Res., 42, 1879. Goldberg, D.E., (1989). Genetic Algorithms in Search. Optimization and Machine Learning. Addison-Wesley. MA. Guillén, G., Mele. F. D., Bagajewicz. M. J., Espuna. A., Puigjaner. L. (2005). Multiobjective Supply Chain Design under Uncertainty. Chemical Engineering Science, 60, 1535. Mokashi. S. D., Kokossis. A. C. (2003). Application of Dispersion Algorithms to Supply Chain Optimization. Comp. Chem. Eng., 27, 927. Perea-Lopez. E., Ydstie. B. E., Grossmann. I. E. (2003). A Model Predictive Control Strategy for Supply Chain Optimization. Comput. Chem. Eng., 27, 1201. Silva. C. M.. Biscaia Jr. E.C. (2003). Genetic Algorithm Development for Muli-Objective Optimization of Batch Free- Radical Polymerization Reactors. Comp. Chem. Eng., 27, 1329. Zhou. Z., Cheng. S., Hua. B. (2000). Supply Chain Optimization of Continuous Process Industries with Sustainability Considerations. Comp. Chem. Eng., 24, 1151. 10

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