《系统工程》课程教学资源（英文文献）Optimal Design of Two Levels Reverse Logistic

Optimal Design of Two Levels Reverse Logistic Supply Chain byConsidering the Uncertain Quantity of Collected Multi -productsAbstract: In the recent years, the reverse logistic supply chain has taken an important part inmanufacturing systems. Due to the pressures from both environmental awareness and businesssustainability, several companies must work in collaboration to solve partially the problem of wasterecovery.The purpose of this work is to help those companies who would like to pursue reverselogistics as axe of profit. In this paper we present a mixed-integer linear programming (MILP) modelwhich includes several costs of collection, treatment and sales income from secondary market. Thegoal of study is to investigate the best material flows between different specialized facilities undercapacities constraints and costs minimization. Themodel is sub divides in two levels: the first levelconcerns the collect process while a size of each product is compared to a standard size to optimizethe storage capacity of collection centers. Then to unify the request of several products or classes ofproducts, in the second level a running time is added instead a processing capacity by product andused to facilitate the study of multi-product case. This proposed optimization model can help todetermine the optimal facility location and the material flows of network for multiproduct in pseudodynamic case.The usefulness ofthe proposed model is a real design of reverse network withconsideringtheuncertainquantity of collectedproduct anda progressive investment.Togivetothestudy an illustration, we present an example to regard the recovery rate using Lingo 12.0 solver.Keywords: design of reverse supplychain-location-allocation optimisation1.INTRODUCTIONWith the development of social economy,resource and environmental situations are becomingmore and more critical. It is an important task for every country to develop the circle economy andachieve sustainabledevelopment ofeconomy byraising theutilization rate of resources andalleviating the pressures from environmental pollution and shortage of resources. Reverse logistics,which can promote the recycling of nature resources, has become a key factor in thedevelopment ofcircle economy.In order to collect products efficient at the end of their life cycle, a simulation modelofareverselogisticsnetworkscanbedevelopedtocalculatethecollectioncostinapredictablemanner in the actual produce collection as well as developing a facility location model for logistics

Optimal Design of Two Levels Reverse Logistic Supply Chain by Considering the Uncertain Quantity of Collected Multi –products Abstract: In the recent years, the reverse logistic supply chain has taken an important part in manufacturing systems. Due to the pressures from both environmental awareness and business sustainability, several companies must work in collaboration to solve partially the problem of waste recovery. The purpose of this work is to help those companies who would like to pursue reverse logistics as axe of profit. In this paper we present a mixed-integer linear programming (MILP) model which includes several costs of collection, treatment and sales income from secondary market. The goal of study is to investigate the best material flows between different specialized facilities under capacities constraints and costs minimization. The model is sub divides in two levels: the first level concerns the collect process while a size of each product is compared to a standard size to optimize the storage capacity of collection centers. Then to unify the request of several products or classes of products, in the second level a running time is added instead a processing capacity by product and used to facilitate the study of multi-product case. This proposed optimization model can help to determine the optimal facility location and the material flows of network for multiproduct in pseudo dynamic case. The usefulness of the proposed model is a real design of reverse network with considering the uncertain quantity of collected product and a progressive investment. To give to the study an illustration, we present an example to regard the recovery rate using Lingo 12.0 solver. Keywords: design of reverse supply chain -location –allocation optimisation. 1. INTRODUCTION With the development of social economy, resource and environmental situations are becoming more and more critical. It is an important task for every country to develop the circle economy and achieve sustainable development of economy by raising the utilization rate of resources and alleviating the pressures from environmental pollution and shortage of resources. Reverse logistics, which can promote the recycling of nature resources, has become a key factor in the development of circle economy. In order to collect products efficient at the end of their life cycle, a simulation model of a reverse logistics networks can be developed to calculate the collection cost in a predictable manner in the actual produce collection as well as developing a facility location model for logistics

systems includingreverseflow.Uncertainty is one of the characteristics of reverse logistics networks, which is leading to thecomplex and vibrating operations. Many companies with limited resources outsource their reverselogistics operation demands to third-party providers.Moreover, the reverse logistics network is oftenaffectedbymanyvariousdisruptions,suchasbadweathers,workdispute,fireandtransportationcondition.Therefore, it is importantfor many manufacturers to selectfeasible reverse logistics supplychain in order to mitigate the risk, the contingency, and reduce the logistics cost under perfectcollaboration.For example, when and where the old products returned, the quantity and quality of thereturned products as well as the requirement of the remanufactured products, would have a complexinfluence on a series of problems like location, the number of facilities, processing ability as well asarrangement plans of manufacture and distribution.Therefore, howto deal with the special effect onthe remanufacturing reverse logistics networks by these uncertain factors is the key to build asuccessful locatingmodel.In this paper, we focus on the causes of uncertainty within reverse logistics.The paper proceedsby firstly reviewing some important works that address the dynamic and stochastic cases. Then, wepresent a realistic pseudo dynamic model formulated in mixed integer linear programming to optimizeseveral costs2.STATEOFARTEANDPROBLEMDESCRIPTIONA.StateofthearteIn the past decades, due to dwindling landfill and incineration capacities and growingenvironmental concern to the limited natural resources, waste reduction has become a primarychallenge for industrialized countries. More and more environmental legislations are imposed on theOEM (Old Equipment Manufacturing) to take correspondingresponsibility for the whole life cycle oftheir products [3]. In many European countries, producers are required to take back packaging fromtheir customers. In the U. S., local and state governments are often responsible for the recyclingpackaging. In the EU, the WEEE (waste from electrical and electronic equipment) directive requiresend of life equipment to be collected for recovery, recycling and reuse by end 2003. However,regarding the recovered products face a competition from the new products, the investment on productrecoverybecomesariskyadventurewhereacostofrecoveredproductscanbereducedbyoptimallocationsand allocationsoffacilities.In the literature several studies have discussed the problem of design of dynamic reverse logistic

systems including reverse flow. Uncertainty is one of the characteristics of reverse logistics networks, which is leading to the complex and vibrating operations. Many companies with limited resources outsource their reverse logistics operation demands to third-party providers. Moreover, the reverse logistics network is often affected by many various disruptions, such as bad weathers, work dispute, fire and transportation condition. Therefore, it is important for many manufacturers to select feasible reverse logistics supply chain in order to mitigate the risk, the contingency, and reduce the logistics cost under perfect collaboration. For example, when and where the old products returned, the quantity and quality of the returned products as well as the requirement of the remanufactured products, would have a complex influence on a series of problems like location, the number of facilities, processing ability as well as arrangement plans of manufacture and distribution. Therefore, how to deal with the special effect on the remanufacturing reverse logistics networks by these uncertain factors is the key to build a successful locating model. In this paper, we focus on the causes of uncertainty within reverse logistics. The paper proceeds by firstly reviewing some important works that address the dynamic and stochastic cases. Then, we present a realistic pseudo dynamic model formulated in mixed integer linear programming to optimize several costs. 2. STATE OF ARTE AND PROBLEM DESCRIPTION A. State of the arte In the past decades, due to dwindling landfill and incineration capacities and growing environmental concern to the limited natural resources, waste reduction has become a primary challenge for industrialized countries. More and more environmental legislations are imposed on the OEM (Old Equipment Manufacturing) to take corresponding responsibility for the whole life cycle of their products [3]. In many European countries, producers are required to take back packaging from their customers. In the U. S., local and state governments are often responsible for the recycling packaging. In the EU, the WEEE (waste from electrical and electronic equipment) directive requires end of life equipment to be collected for recovery, recycling and reuse by end 2003. However, regarding the recovered products face a competition from the new products, the investment on product recovery becomes a risky adventure where a cost of recovered products can be reduced by optimal locations and allocations of facilities. In the literature several studies have discussed the problem of design of dynamic reverse logistic

supply chain by considering several parameters, the model proposed by Pourmohammadi et al, havestudiedtherecovery of industrialaluminumwasteinthecityof LosAnglesinUSA.Intheirwork,they have formulated the model in mixed integer linear representation to minimize several costs. Newparameters related to environmental cost of transportation and disposal were introduced (energy,water consumed and Co2). Tang and Xie, have presented a motivating work that intends to complywith new laws related to the environment protection, where it is possible to integrate a product or apart into a global chain. Their work have focused on optimization of multiple variable costs (related tothe transport, inventory, operational, backorder and penalty) of many products returned from customerfor repair. In addition to sets of customers and factories, the global chain includes the periodic openingorexpendingofcollectioncentersandrepaircentersMoreover, Min et al., have proposed a nonlinear model formulated in mixed subdivided into twostages. To meet the needs of service and under the condition that the number and capacity size ofcollection centers are considered in the choice of the strategic decision. The objective of this work isto search for optimal locations of potential sites (collection centers and Centralized Return Center).Eventually, this is done in parallel to the minimization of several costs where the benefit observed isthe inclusion of time as a variable in the objective function, in order to quantify time of the operationand periods ofcollection.Intermsof serviceand economyanddespiteoftheimportancethatbringstherepairserviceforproducts returned under guarantee (Following a quality defect), a minority of work were interested tointegrate therepair process in logistics of their origin supply chain.However, Min et al.too, haveprovided a reference case of study that takes into account the activity of repair. Working in a dynamicmulti-product environment, they propose a supply chain which contains three traditional entities (plant,distribution center and customer) and repair center as a potential to ensure the recovery process. Theobjective of the proposed work is to find the optimal number and size of distribution and repaircenters to locate with taking into account the coordination between customers and plants. Also, theauthors include in their model the capacities of different sites and the possibility to use the distributioncenter as a repair center. For such size of the chain, the use of genetic algorithms to solve the problemof optimization model developed in mixed integer programming is inevitable.In order to improve the performance ofthe quality of customer service, Liu and Ni, are interestedto studying a closed looped supply chain. Assuming that the proposed approach is dynamicmulti-product with limited capacity of sites (plant, intermediate centers (recovery + warehouse), the

supply chain by considering several parameters, the model proposed by Pourmohammadi et aI, have studied the recovery of industrial aluminum waste in the city of Los Angles in USA. In their work, they have formulated the model in mixed integer linear representation to minimize several costs. New parameters related to environmental cost of transportation and disposal were introduced (energy, water consumed and Co2). Tang and Xie, have presented a motivating work that intends to comply with new laws related to the environment protection, where it is possible to integrate a product or a part into a global chain. Their work have focused on optimization of multiple variable costs (related to the transport, inventory, operational, backorder and penalty) of many products returned from customer for repair. In addition to sets of customers and factories, the global chain includes the periodic opening or expending of collection centers and repair centers. Moreover, Min et al., have proposed a nonlinear model formulated in mixed subdivided into two stages. To meet the needs of service and under the condition that the number and capacity size of collection centers are considered in the choice of the strategic decision. The objective of this work is to search for optimal locations of potential sites (collection centers and Centralized Return Center). Eventually, this is done in parallel to the minimization of several costs where the benefit observed is the inclusion of time as a variable in the objective function, in order to quantify time of the operation and periods of collection. In terms of service and economy and despite of the importance that brings the repair service for products returned under guarantee (Following a quality defect), a minority of work were interested to integrate the repair process in logistics of their origin supply chain. However, Min et al. too, have provided a reference case of study that takes into account the activity of repair. Working in a dynamic multi-product environment, they propose a supply chain which contains three traditional entities (plant, distribution center and customer) and repair center as a potential to ensure the recovery process. The objective of the proposed work is to find the optimal number and size of distribution and repair centers to locate with taking into account the coordination between customers and plants. Also, the authors include in their model the capacities of different sites and the possibility to use the distribution center as a repair center. For such size of the chain, the use of genetic algorithms to solve the problem of optimization model developed in mixed integer programming is inevitable . In order to improve the performance of the quality of customer service, Liu and Ni, are interested to studying a closed looped supply chain. Assuming that the proposed approach is dynamic, multi-product with limited capacity of sites (plant, intermediate centers (recovery + warehouse)), the

forward supply chain contains the three classical entities. For the reverse flow a recovery centerensures the link between the consumer and the plan. Given the complexity of the problem (multiproduct, multi period) and for high quality customer service, the location of recovery centers andwarehouses, as much old as new, is very strategic and requires a powerful computational too. Thus,the authors have developed a new meta-heuristic called "Immune Genetic Algorithm", which isinspired from biology and faster than genetic algorithms.Also the work developed by Lee and Dong, in several stages, presents one of more distinctiveworks which concerns the design of global logistics network in dynamic and stochastic approachesTo provide more flexibility to the chain, the authors have implemented in the design a new entitycalled Hybrid Processing center. Depending on need and timing of demand, the role of HPC is totransfer from a distribution centerto a collection center or the opposite.Tomake decisions related tothechoiceofnetworkconfiguration,thecapacityofthethreeintermediatecenters(distribution,collection and HPF)must be adjusted shrewdly for optimal response to the command. Moreover, inthe stochastic case, the use of vector representation of the model combined with a probability functionbecomes inevitable; thus definding the possible scenarios depending on uncertain parameters. In bothcases (dynamic or stochastic), the approach of the resolution used is SAA (Sample AverageApproximation) hybridized with SA (simulated annealing: used to make the coding domain of thefeasibilityregionoflocalization)Considering relatively some uncertain parameters (time, place and quantity) which intervening inthe recovery process of one type of product, Mao et al, have proposed a model of processing singletype of product in several periods in a closed loop supply chain. The proposed chain structure is builtfrom return centers, disassembly centers, processing plants (remanufacturing)and distribution centersUnder the criteria of the robustness of the network structure to be located, the aim is to minimize thetotal investment costs (storage, transportation, infrastructure, openness) respecting the stability ofproduction in several envelopes of variable duration time. To help solve the optimization problem oflocalization, the authors have developed hybridized intelligent algorithms where the tools of solvingapplied are Matlab and Ralcsoftwares . As the system of e-waste recovery is complex, Fleischmannproposed a method by using mathematics to help the decision-making for e-waste reverse logistics. Inordertoanalyzetheinteractionbetweene-wasteacquisition,disassemblyandtreatment,ThomasSpengler applied mixed integer linear programming model to solve the integrated planning problemfordifferent stages

forward supply chain contains the three classical entities. For the reverse flow a recovery center ensures the link between the consumer and the plan. Given the complexity of the problem (multi product, multi period) and for high quality customer service, the location of recovery centers and warehouses, as much old as new, is very strategic and requires a powerful computational too. Thus, the authors have developed a new meta-heuristic called "Immune Genetic Algorithm", which is inspired from biology and faster than genetic algorithms. Also the work developed by Lee and Dong, in several stages, presents one of more distinctive works which concerns the design of global logistics network in dynamic and stochastic approaches. To provide more flexibility to the chain, the authors have implemented in the design a new entity called Hybrid Processing center. Depending on need and timing of demand, the role of HPC is to transfer from a distribution center to a collection center or the opposite. To make decisions related to the choice of network configuration, the capacity of the three intermediate centers (distribution, collection and HPF) must be adjusted shrewdly for optimal response to the command. Moreover, in the stochastic case, the use of vector representation of the model combined with a probability function becomes inevitable; thus definding the possible scenarios depending on uncertain parameters. In both cases (dynamic or stochastic), the approach of the resolution used is SAA (Sample Average Approximation) hybridized with SA (simulated annealing: used to make the coding domain of the feasibility region of localization). Considering relatively some uncertain parameters (time, place and quantity) which intervening in the recovery process of one type of product, Mao et aI, have proposed a model of processing single type of product in several periods in a closed loop supply chain. The proposed chain structure is built from return centers, disassembly centers, processing plants (remanufacturing) and distribution centers. Under the criteria of the robustness of the network structure to be located, the aim is to minimize the total investment costs (storage, transportation, infrastructure, openness) respecting the stability of production in several envelopes of variable duration time. To help solve the optimization problem of localization, the authors have developed hybridized intelligent algorithms where the tools of solving applied are Matlab and Ralcsoftwares . As the system of e-waste recovery is complex, Fleischmann proposed a method by using mathematics to help the decision-making for e-waste reverse logistics. In order to analyze the interaction between e-waste acquisition, disassembly and treatment, Thomas Spengler applied mixed integer linear programming model to solve the integrated planning problem for different stages

Grit Walther built a linear, activity-based model for German ewaste recycling network.Themodel optimized the allocation of discarded products, disassembly activities and disassemblyfractions to facilities of the treatment system. A variety of products are covered, and material flowsbetween disassembly facilities are possible.This model is more close to the actual recycling network.But it is used to determine material flows other than the facility location problem of e-waste recoveryFrom analysis of these studies we can conclude that the structure of a global supply chain isconsisted with more than three entities and then the study of design and control problem is morecomplicated. According to the objectives set by the authors and in the goal to attain maximum profitsmany design proposals have been described. For the cited papers, the studies have been conducted onimaginary cases. Firstly, this reflects the willingness of researchers to develop several structures toserve these axes of research which are relatively new. Secondly,the lack of true collaboration at thelevel of supply chain partners implies absence of parameters that indicate good prognostics. Despitethis, several tools for modeling and optimization have been applied in order to respond to differentmechanisms of design at different levels (strategic and tactical) and in different environments(deterministic, dynamic, and stochastic, with or without consideration of capacity, based on two levelsandmultilevels.)We note the complexity arising in the mathematical formulation of their models, especially whenit comes to treating a problem of several objectives and constraints. Some authors, and within thelimits of their possibility,have decomposed their models to sub-models. Others have validated theirmodels on a very complex case but with a small system size by justifying that the tool of calculationused is limitedandthedecompositionof themodel isdifficult.Todevelopacomprehensiveoptimization model with more flexibility and applicability which concerns the optimal design offundamental third partners reverse supply chain. In this study, the first level concerns the collect ofseveral classes of products from collect points toward collection centers during a time interval. At theend of this time interval, a new flow of these classes of products is opened from collection centerslocated toward remanufacturing centers or disposal centers during the same time interval. Undercapacity constraints and request, the supply order is manifested in order of priority from the collectioncenter to: remanufacturing centers, and then to the disposal center. The purpose of the introduction ofan alternating time interval between the first level and the second level is introduced to eliminate theuncertainty about the quantity to be treated. Also the need to address several products or productcategories is applied to reduce the transportation distances, the number of remanufacturing centers and

Grit Walther built a linear, activity-based model for German ewaste recycling network. The model optimized the allocation of discarded products, disassembly activities and disassembly fractions to facilities of the treatment system. A variety of products are covered, and material flows between disassembly facilities are possible. This model is more close to the actual recycling network. But it is used to determine material flows other than the facility location problem of e-waste recovery . From analysis of these studies we can conclude that the structure of a global supply chain is consisted with more than three entities and then the study of design and control problem is more complicated. According to the objectives set by the authors and in the goal to attain maximum profits, many design proposals have been described. For the cited papers, the studies have been conducted on imaginary cases. Firstly, this reflects the willingness of researchers to develop several structures to serve these axes of research which are relatively new. Secondly, the lack of true collaboration at the level of supply chain partners implies absence of parameters that indicate good prognostics. Despite this, several tools for modeling and optimization have been applied in order to respond to different mechanisms of design at different levels (strategic and tactical) and in different environments (deterministic, dynamic, and stochastic, with or without consideration of capacity, based on two levels and multilevels.) We note the complexity arising in the mathematical formulation of their models, especially when it comes to treating a problem of several objectives and constraints. Some authors, and within the limits of their possibility, have decomposed their models to sub-models. Others have validated their models on a very complex case but with a small system size by justifying that the tool of calculation used is limited and the decomposition of the model is difficult. To develop a comprehensive optimization model with more flexibility and applicability which concerns the optimal design of fundamental third partners reverse supply chain. In this study, the first level concerns the collect of several classes of products from collect points toward collection centers during a time interval. At the end of this time interval, a new flow of these classes of products is opened from collection centers located toward remanufacturing centers or disposal centers during the same time interval. Under capacity constraints and request, the supply order is manifested in order of priority from the collection center to: remanufacturing centers, and then to the disposal center. The purpose of the introduction of an alternating time interval between the first level and the second level is introduced to eliminate the uncertainty about the quantity to be treated. Also the need to address several products or product categories is applied to reduce the transportation distances, the number of remanufacturing centers and

to ensure continuous functioning.Two new ideas are proposed: the size of each product is comparedto a standard size to optimize the storage capacity of collection centers, and, to unify the request ofseveral products or classes of product, a running time is considered instead of processing capacity byproduct. This is used to facilitate the study of multi-product case, where the operating time means thetime required from processing product until its storage.B.Problem descriptionTo describe the reverse logistic supply chain in a realistic environment, it has been assumed somehypothesis established from reality.Since the quantity of thereturned products is unknown, and toensure a permanent working of remanufacturing centers, it is more beneficial to recover severalproducts or class of products able to be treated at these centers. This assumption is taken to minimizethe number of collect centers and to bring them nearer to remanufacturing centers and others potentialfactories, in order to reduce the rate of transport pollution. This is assumed because the reverselogisticstransportationtakesan importantpartofeconomicalbalance.Atoperationallevel,andtoensure a continuous procurement of products that should be treated, it had assumed a time lag betweencollection and processing, i.e. the quantity of goods or class of products collected in time T must betreated at remanufacturing centers in period T+ 1. The advantage of this assumption promotes tomake the best decisions concerning the choice of assignments of types of products by consideringcriteria related to cost and recovery rate.About the design of the reverse logistic supply chain, the critical question that remainsunanswered is who should be responsiblefor this supply chain? Themanufacturer, theconsumer orthe authorities! At level of cooperation these three are all responsible, but at the level of investment,the negotiations between the manufacturer and the authorities remain in conflict although theconsumer pays. The manufacturer sees his company in a competitive situation in the world B & Bwhile the authorities stipulate laws to reduce pollution. It is clear that this conflict can be raised onlygradually and in the logic of ERR (Extended Recourses Responsibility). The Costs of plants involvedin the design of the chain become theresponsibility of all firms that produce the same goods or classesof products and so huge investment is divided. On the other hand, the authorities will provide morebenefit to those enterprises in consumer market (of course this logic is valid in the case of acompetitivemarket).Moreover, this vision can shout a collaborative environment between firms, which form the chainof recovery, and the authorities. Once this reconciliation is solidified, information and interaction can

to ensure continuous functioning. Two new ideas are proposed: the size of each product is compared to a standard size to optimize the storage capacity of collection centers, and, to unify the request of several products or classes of product, a running time is considered instead of processing capacity by product. This is used to facilitate the study of multi-product case, where the operating time means the time required from processing product until its storage. B. Problem description To describe the reverse logistic supply chain in a realistic environment, it has been assumed some hypothesis established from reality. Since the quantity of the returned products is unknown, and to ensure a permanent working of remanufacturing centers, it is more beneficial to recover several products or class of products able to be treated at these centers. This assumption is taken to minimize the number of collect centers and to bring them nearer to remanufacturing centers and others potential factories, in order to reduce the rate of transport pollution. This is assumed because the reverse logistics transportation takes an important part of economical balance. At operational level, and to ensure a continuous procurement of products that should be treated, it had assumed a time lag between collection and processing, i.e. the quantity of goods or class of products collected in time T must be treated at remanufacturing centers in period T + 1. The advantage of this assumption promotes to make the best decisions concerning the choice of assignments of types of products by considering criteria related to cost and recovery rate. About the design of the reverse logistic supply chain, the critical question that remains unanswered is who should be responsible for this supply chain? The manufacturer, the consumer or the authorities! At level of cooperation these three are all responsible, but at the level of investment, the negotiations between the manufacturer and the authorities remain in conflict although the consumer pays. The manufacturer sees his company in a competitive situation in the world B & B while the authorities stipulate laws to reduce pollution. It is clear that this conflict can be raised only gradually and in the logic of ERR (Extended Recourses Responsibility). The Costs of plants involved in the design of the chain become the responsibility of all firms that produce the same goods or classes of products and so huge investment is divided. On the other hand, the authorities will provide more benefit to those enterprises in consumer market (of course this logic is valid in the case of a competitive market). Moreover, this vision can shout a collaborative environment between firms, which form the chain of recovery, and the authorities. Once this reconciliation is solidified, information and interaction can

occur automatically between the three main partners. About the design of the reverse logistic supplychain, the critical question that remains unanswered is who should be responsible for this supply chain?The manufacturer,the consumer or the authorities! At level of cooperation thesethree are allresponsible, but at the level of investment, the negotiations between the manufacturer and theauthorities remain in conflict although the consumer pays. The manufacturer sees his company in acompetitive situation in the worldB &Bwhile the authorities stipulate laws to reduce pollution.It isclear that this conflict can be raised only gradually and in the logic of ERR (Extended RecoursesResponsibility).The Costs of plants involved in the design of the chain become the responsibility ofall firms that produce the same goods or classes of products and so huge investment is divided. On theother hand, the authorities will provide more benefit to those enterprises in consumer market (ofcoursethislogicisvalidinthecaseofacompetitivemarket)Moreover, this vision can shout a collaborative environment between firms, which form the chainof recovery,and the authorities.Once this reconciliation is solidified, information and interaction canoccur automatically between the three main partners. In this paper we are interested in designing areverse logistics chain at two levels. The first step is the collection stage between the collection pointsand the collection center.The second step includes different entities of recovery.To get close toreality, several types of plant are, at this level, candidate to participate: the secondary market, thesubcontractor, the reprocessing center and ultimately the landfill. After an inspection of the quality ofproducts in the collection center, the order of shipment occurs from collect center respectively to:secondary market, then to remanufacturing center, next to subcontractor and in thelast to disposalfacility (see figure 1). In the formulated model is assumed that, the quantity of treated products atopened remanufacturing and subcontractor centers can't surpass the total quantity of all productscollected.Good productsSecondarymarketSupply orderCollect pointCollect centerRemanufacturingcenterubcontractoDisposal fucilityFigurel.Reverselogisticsupplychainconsidered3.MODELFORMULATION

occur automatically between the three main partners. About the design of the reverse logistic supply chain, the critical question that remains unanswered is who should be responsible for this supply chain? The manufacturer, the consumer or the authorities! At level of cooperation these three are all responsible, but at the level of investment, the negotiations between the manufacturer and the authorities remain in conflict although the consumer pays. The manufacturer sees his company in a competitive situation in the world B & B while the authorities stipulate laws to reduce pollution. It is clear that this conflict can be raised only gradually and in the logic of ERR (Extended Recourses Responsibility). The Costs of plants involved in the design of the chain become the responsibility of all firms that produce the same goods or classes of products and so huge investment is divided. On the other hand, the authorities will provide more benefit to those enterprises in consumer market (of course this logic is valid in the case of a competitive market). Moreover, this vision can shout a collaborative environment between firms, which form the chain of recovery, and the authorities. Once this reconciliation is solidified, information and interaction can occur automatically between the three main partners. In this paper we are interested in designing a reverse logistics chain at two levels. The first step is the collection stage between the collection points and the collection center. The second step includes different entities of recovery. To get close to reality, several types of plant are, at this level, candidate to participate: the secondary market, the subcontractor, the reprocessing center and ultimately the landfill. After an inspection of the quality of products in the collection center, the order of shipment occurs from collect center respectively to: secondary market, then to remanufacturing center, next to subcontractor and in the last to disposal facility (see figure 1). In the formulated model is assumed that, the quantity of treated products at opened remanufacturing and subcontractor centers can't surpass the total quantity of all products collected. Figure l. Reverse logistic supply chain considered 3. MODEL FORMULATION

Mathematical formulation of the proposed model consists in developing the CFCFL model(Capacitated Fixed Charge Facility) for the case of multi products with unifled unities of workingThe model is formulated in MIP, which has cost minimization as its objective from the overallperspective of reverse logistics. The factors considered in the model include the cost of collection,treatment,investmentandmaterialflowsbetweendifferentspecializedfacilitiesundercapacitiesconstraints.Index:p:index of class of product,i: index of collect pointj: index of collect center,k: index of secondary market,I: index of remanufacturing center,m: index of subcontractor,n: index of disposal centerParameters:a: thetransportation cost perunit of distance per unit ofweight,Sp: the average size of product p,Wp: the average weight of product p,Cap,: global capacity of collect center j,Feosl/ fixed cost for opening collect center j,Opp/ operational cost of product p at collect centerj (handlingand test),OPpa: operational cost of product p treated at entity a,Feosl/: flxed cost for opening remanufacturing center I,Disl ap: distance between entity a and entity j3,Assuming that the operation of collection occurs from collect point to collect center during a timeinterval t. After this operation ends, the others operations of shipment will occur from located collectcenter to the different entity during the same interval, designed as t+ 1. The below variablesrepresents:Qpi : real quantity of product p collected from collect point I during I,Qmaxpi: maximal quantity of product p collected from collect point i during I, j3p/ percentage of

Mathematical formulation of the proposed model consists in developing the CFCFL model (Capacitated Fixed Charge Facility) for the case of multi products with unifIed unities of working. The model is formulated in MIP, which has cost minimization as its objective from the overall perspective of reverse logistics. The factors considered in the model include the cost of collection, treatment, investment and material flows between different specialized facilities under capacities constraints. Index: p:index of class of product, i: index of collect point, j: index of collect center, k: index of secondary market, I: index of remanufacturing center, m: index of subcontractor, n: index of disposal center Parameters: a: the transportation cost per unit of distance per unit of weight, Sp: the average size of product p, Wp: the average weight of product p, Cap;: global capacity of collect center j, Feosl/ fIxed cost for opening collect center j, 0pp/ operational cost of product p at collect center j (handling and test), OPpa: operational cost of product p treated at entity a, Feosl/: fIxed cost for opening remanufacturing center I, Disl ap: distance between entity a and entity j3 , Assuming that the operation of collection occurs from collect point to collect center during a time interval t. After this operation ends, the others operations of shipment will occur from located collect center to the different entity during the same interval, designed as t+ 1. The below variables represents: Qpi : real quantity of product p collected from collect point I during I, Qmaxpi: maximal quantity of product p collected from collect point i during I, j3p/ percentage of

product p collected in collect centerj valid to sell in secondary market during 1+ 1,Dempk: demand of product p at secondary market k during 1+ 1.Ueoslpk: unitary cost to purchase a product p at secondary market during t+ 1,Ueoslp/: unitary proflt cost of product p at remanufacturing center I during t+ 1 ,Timepa:the mean time requisite to treat a product p at entity a (operational time of product p ),Tol limea: total time of functioning at entity a during 1+ 1.Ad jime / : additional time of functioning at remanufacturing center I during t+ 1,Ad Opp/: additional operational cost of product p treated at remanufacturing center I during t+ 1 ,Variables:VpaP :volume of product p shipped from entity a to entity j3 during a I or 1+ 1 ,Ad_lime / : additional time of functioning at remanufacturing center I during I+ 1,Ad Vpj/ : : additional volume of product p shipped from collect center j to remanufacturing center Iduring 1+1,n/: number of remanufacturing center that should be opened.X= (1 if entity is locatedin sitea a0 otherwisey= (I if customer pointi is aftXted to colllect centerjij0otherwiseThe objective function: min ZI + min Z2 (I)Z,=Fcost,X, +αw, Dist, Y,Qmax p(step of collectandduringaperiodt)ZSawp it/i-osplpFostZ,=2pik:-UcostDist,+op0Pplpilpi(αwp Dist , + Ad_opp, -U cost ,AdV6pilpil(awp Dstjm+0pm)pim%(awpDistin pjm21pimpin(Step of treatment of products during aperiod t+1)

product p collected in collect center j valid to sell in secondary market during 1+ 1 , Dempk: demand of product p at secondary market k during 1+ 1. Ueoslpk: unitary cost to purchase a product p at secondary market during t+ 1, Ueoslp/: unitary profIt cost of product p at remanufacturing center I during t+ 1 , Timepa:the mean time requisite to treat a product p at entity a (operational time of product p ), Tol limea: total time of functioning at entity a during 1+ 1. Ad jime / : additional time of functioning at remanufacturing center I during t+ 1, Ad 0pp/: additional operational cost of product p treated at remanufacturing center I during t+ 1 , Variables: VpaP :volume of product p shipped from entity a to entity j3 during a I or 1+ 1 , Ad_lime / : additional time of functioning at remanufacturing center I during 1+ 1, Ad Vpj/ : : additional volume of product p shipped from collect center j to remanufacturing center I during 1+ 1 , n/: number of remanufacturing center that should be opened. X = {1 if entity is locatedin sitea a 0 otherwise y = {I if customer pointi is aftXted to colllect center j ij 0 otherwise The objective function: min Z\ + min Z2 (1)

Subject to:(2)Y, sx,viel&vjeJZmaxprS,y,≤cp,X,j(3)pi2'+2(Vpi+Ad_Vm)+(4)2m+2n=ppeP&jej(5)'n tiep= Tou_time Xpi(6)ZAd_Vpn timepi=Ad_timep X,Vlepi(7)pm timem= Ta_inemVmeMpj(8)_plEpusEBpupePvjeJ(9)(10)EVpikSDempk,peP,keKEp=min(EDem pZβp）VpeP(11)ikIX,efo1,vjeJ(12)X, e (o,1,Vle L(13)ZX,=n(14)Y, efo,1,viel& VjeJ(15)ZY,=, viel(16)iThe objective function (1) mmnDlzes total logistics costs including opening, operating and profitThe constraint (2) assigns the demand from node i to node j, just in case a facility is located at node j.The constraint (3) forces each demand node i to be served by node j under capacity. The constraint (4)ensurestheequilibriumbalanceof shipmentflow.Theconstraints(5-6-7-8)ensuretheequilibriumbalance of working time. The constraints (9-10-11) organize the order of shipment between collectcenter and secondary market.Constraintsset(12-13-15)checksforbinaryvariables and constraint(14)meansthenumberofremanufacturing center to be opened. The last constraint ensures that only one collect center is

The objective function (1) mmnDlzes total logistics costs including opening, operating and profit. The constraint (2) assigns the demand from node i to node j, just in case a facility is located at node j. The constraint (3) forces each demand node i to be served by node j under capacity. The constraint (4) ensures the equilibrium balance of shipment flow. The constraints (5-6-7-8) ensure the equilibrium balance of working time. The constraints (9-10-11) organize the order of shipment between collect center and secondary market. Constraints set (12-13-15) checks for binary variables and constraint (14) means the number of remanufacturing center to be opened. The last constraint ensures that only one collect center is