Fuzzy Multi-Objective Mathematical Model for ...

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Fuzzy Multi-Objective Mathematical Model for Supplier Selection and Network Configuration in Reverse Logistics Systems Kamran S. Moghaddam Craig School of Business, California State University, Fresno, Fresno, CA, USA. [email protected] ABSTRACT Evaluating and selecting suppliers are crucial strategic decisions in reverse logistic systems. After identifying the candidate suppliers, allocating optimal orders to them are also considered as important tactical decisions. In this research, we develop a fuzzy multi-objective mathematical model to identify the best suppliers, determine the manufacturing and refurbishing plants, and find the optimal number of unfinished parts and final products in a reverse logistic network configuration. The objective functions in this study are total profit, total defective parts, total late delivered parts, and economic risk factors associated with the candidate suppliers whereas the uncertainty associated with customers’ demand, suppliers’ capacity, and percentage of returned products are treated in a fuzzy environment. The proposed fuzzy model is then solved by an integrated Monte Carlo simulation and fuzzy goal programming. The effectiveness of the mathematical model and the proposed solution method in obtaining trade-off solutions is demonstrated in a numerical example.

I. INTRODUCTION Reverse logistics stands for all operations related to the reuse of products and materials. It is the process of planning, implementing, and controlling the efficient, cost effective flow of raw materials, in-process inventory, finished goods and related information from the point of consumption to the point of origin for the purpose of recapturing value or proper disposal. Remanufacturing and refurbishing activities also may be included in the definition of reverse logistics (Fleischmann et al., 1997; Hawks, 16

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2006). Normally, logistics deal with events that bring the product towards the customer. In the case of reverse logistics, the resource goes at least one step back in the supply chain. One of the important decisions that impact the company’s performance as well as the entire supply chain competitiveness is the supplier selection and order allocation to the selected suppliers. In addition to identifying the candidate suppliers and allocate optimal orders to them, reverse logistic systems consist of reuse, resale, repair, refurbishing, remanufacturing, and recycling operations. In the remanufacturing process, returned products are disassembled in disassembly sites and then usable parts are collected, cleaned, refurbished, and transmitted to part inventory. In the next stage the new products are manufactured from the old and new parts (Kim, Song and Jeong, 2006). In reverse logistics, the new parts are purchased from external suppliers and the used parts are obtained from used or returned products. Not only the cost of purchase is important, but also other criteria of suppliers’ performance play a vital role. For example, late delivery can affect the production schedule and increase the final costs tremendously. As a result, suppliers should be assessed based on several conflicting criteria and purchasing cost is only one of them. Supplier selection problem is a multicriteria decision making problem that includes both quantitative and qualitative factors such as total cost, on-time delivery, quality, customer satisfaction level, etc. In addition to cost related factors in supplier evaluation and selection, when consolidating and reducing the number of suppliers, companies will encounter the risk of not having sufficient raw materials to meet their fluctuating demand. These risks may be caused by natural disasters or man-made actions (Li and Zabinsky, 2011). Environmental criteria are another group of characteristics that should be emphasized in reverse logistics configurations. Recycling, clean technology, pollution reduction capacity, and environmental costs are examples of environmental aspects to be considered. It is noticeable that conservation of environment is one of the designated goals of reverse logistics systems (Amin and Zhang, 2012). In this research, we extend the work of (Amin and Zhang, 2012; Arikan, 2013; 17

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Shaw, Shankara, Yadava and Thakurb, 2012) and develop a general reverse logistics network that includes suppliers, manufacturing, disassembly, and refurbishing facilities. The manufacturer uses new parts purchased from suppliers and refurbished parts from returned products to manufacture new final products that are in demand by the customers. The main decisions of this system are the optimal set of candidate suppliers and refurbishing strategies along with the optimal number of products and parts to be positioned in each stage of the network. Since some important parameters of this decision making problem are associated with uncertainty the problem is formulated as a fuzzy multi-objective mathematical model in which the objective functions are total profit to be maximized, total number of defective parts purchased from suppliers to be minimized, total number of late delivered parts to be minimized, and finally economic risk factors of the candidate suppliers to be minimized. The proposed model can be used by decision makers for supplier and refurbishing sites selection (strategic decisions) and also for determining the number of products and parts in each part of the network (tactical decisions). We propose a Monte Carlo simulation model integrated with fuzzy goal programming to simultaneously find Pareto-optimal solutions (aka non-dominated solutions) of the proposed model. This paper is organized as follows: Section 2 reviews the current literature on network configuration in reverse logistics. Section 3 gives the problem formulation as a fuzzy multi-objective optimization model. Section 4 develops the solution methodology in which Monte Carlo simulation is combined with fuzzy goal programming in order to find the non-dominated solutions of the proposed multi-objective model. Section 5 presents computational results for a sample problem and Section 6 summarizes the paper, points out the importance of inclusion of the multiple objectives into modeling, and the advantages of using hybrid method to determine the supplier selection decisions. II. LITERATURE REVIEW Fleischmann et al. (1997) reviewed and categorized reverse logistic networks and

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closed-loop supply chain systems literature into three main categories; distribution planning, inventory planning, and production planning. More recently, Melo, Nickel and Saldanha-da-Gama (2009) examined the application of facility location models in supply chain management and segmented the literature of reverse logistics to closedloop supply chains, and recovery networks. Network configuration is one of the main research venues in reverse logistic systems and the majority of researchers apply facility location models to formulate such networks. However, there are few studies in which supplier selection techniques have been utilized during reverse logistics configuration. Kim, Song and Jeong (2006) developed a mathematical model to determine the quantity of parts and products processed in manufacturing and refurbishing facilities and the number of parts to be purchased from a single supplier while maximizing manufacturing cost savings. A mixed integer nonlinear programming model was formulated by Ko and Evans (2007) in order to model multi-period, two-echelon, multicommodity, and capacitated network design problem with simultaneous forward and reverse flows of parts and final products. Pati, Vrat and Kumar (2008) proposed a goal programming model to determine the facility location, route and flow of different types of recyclable wastepaper in a multi-product, multi-echelon and multi-facility decision making context. Lee, Gen and Rhee (2009) formulated a mathematical model for a general closed-loop supply chain network and used genetic algorithm to solve the model. A mathematical model to maximize the profit of a remanufacturing system was presented in (Shi, Zhang and Sha, 2011). Amin and Zhang (2012) examined general closed-loop supply chain network containing manufacturer, disassembly, refurbishing, and disposal sites and proposed a two-phase integrated model. In an another study, Amin and Zhang (2013) expanded their previous work by developing three-stage model including evaluation, network configuration, and selection and order allocation by simultaneously considering the closed-loop supply chain system and selection process under uncertain demand and in an uncertain decision-making environment. A multi objective linear programming problem was developed by Arikan (2013) to model and solve multiple sourcing supplier selection problem. In order to solve the problem, a 19

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fuzzy mathematical model was proposed to satisfy the decision maker’s aspirations for fuzzy goals. III. PROBLEM FORMULATION 3.1. Reverse Logistics System Specifications In this research, a reverse logistic system with manufacturing /refurbishing, disassembly, and disposal facilities is assumed in which the manufacturer produces final products according to the demand in which the flow of parts, final products, and information in the network is controlled by manufacturer. After using the products by customers, some of them are to be returned back to the supply chain system. The returned products are collected and disassembled at a disassembly facility and the parts are categorized into reusable and unusable parts from which the unusable parts will be transported to the disposal facilities outside of the network. The reusable parts are then sent to production/refurbishing facilities to be tested and refurbished. These parts are ultimately added to part inventory as new parts. According to the demand and refurbished parts, the manufacturer purchases new parts from external suppliers to be able to manufacture demanded products. Not only the purchase price and other associated costs such as transportation and inventory costs of new parts are important to manufacturer, but also other criteria such as quality, delivery, and risk factors of economic environment associated with suppliers should be considered. The manufacturer encounters two types of decisions. First, suppliers’ evaluation and selection have to be considered as strategic decisions. Second, the optimal number of final products and parts at each stage of the network should to be determined as tactical decisions. We will use the following notation for sets, indices, decision variables, and parameters to develop and express the fuzzy multi-objective mathematical model. Table 1. List of Notations Sets I J

Description set of suppliers set of manufacturing/refurbishing plants 20

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M N K Indices i j m n k Decision variables xijm yjn rn om dm refjm si bdn Parameters selln costjn priceim shipij invj setdisn disam dispm refcostjm qualityim deliveryim econriski dem n reqmn supmax im

supminim reusem return n

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set of parts set of final products set of objective functions Description index of suppliers, i = 1,…, I index of manufacturing/refurbishing plants, j = 1,…, J index of parts, m = 1,…, M index of final products, n = 1,…, N index of objective functions, k = 1,…, K Description number of part m purchased from supplier i by plant j, i  I, j  J, and m  M number of product n produced at plant j, j  J, and n  N number of returned product n to be disassembled at disassembly site, n  N number of part m obtained at disassembly site, m  M number of part m to be disposed from disassembly site, m  M number of part m to be refurbished at plant j, j  J, and m  M binary variable, has value of 1 if supplier i is chosen, 0 otherwise, i  I binary variable for setup of disassembly site for product n, n  N Description unit selling price of product n, n  N unit production cost of product n at plant j, j  J, and n  N unit price of part m purchased from supplier i, i  I, and m  M transportation cost from supplier i to plant j, i  I, and j  J inventory holding cost at plant j, j  J setup cost of disassembly site for product n, n  N unit disassembly cost for part m, m  M unit disposing cost for part m, m  M unit refurbishing cost for part m at plant j, j  J, and m  M fraction of poor quality parts of type m purchased from supplier i, i  I, and m  M fraction of late delivered parts of type m purchased from supplier i, i  I, and m  M risk factors of economic environment associated with supplier i, i  I fuzzy demand for product n, n  N unit requirements for part m to produce one unit of product n, m  M, and n  N fuzzy maximum capacity available of part m provided of supplier i, i  I, and m  M minimum purchase quantity of part m from supplier i, i  I, and m  M maximum percent of reusable part m, m  M fuzzy percentage of returned product n, n  N

3.2. Fuzzy Multi-Objective Optimization Model Max f 1    ( sell n  cost jn ) y jn   jJ nN

  setdis n bd n  nN

 ( price

im

 ship ij  inv j ) xijm

iI jJ mM

 (disa

mM

m

o m  disp m d m )  

 refcost

jm

ref jm

(1)

jJ mM

Min f 2    qualityim xijm

(2)

iI jJ mM

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Min f 3    deliveryim xijm

(3)

Min f 4   econrisk i xijm

(4)

iI jJ mM

iI

Subject to:

y

 dem n

jn

n  N

(5)

jJ

 req

mn

nN

 ref

y jn   xijm  ref jm

j  J , m  M

(6)

iI

jm

 d m  om

m  M

(7)

jJ

om   req mn rn

m  M

(8)

nN

x

ijm

~  supmax im s i

i  I , m  M

(9)

x

ijm

 supmin im si

i  I , m  M

(10)

 reuse m om

m  M

(11)

d m  (1  reuse m )o m

m  M

(12)

jJ

jJ

 ref

jm

jJ

 return

n

~ y jn  rn

n  N

(13)

jJ

(14)

rn  B. bd n n  N

si , bd n  0,1

i  I , n  N

xijm , y jn , rn , om , d m , ref jm  0

i  I , j  J , m  M , n  N

(15)

3.3. Objective Functions The objective function (1) maximizes the total profit which is the net profit gained from selling of final products minus parts purchasing costs from the external suppliers, transportation cost, inventory holding cost of parts, setup cost of products at disassembly sites, disassembly cost of obtained parts, disposal cost of unusable parts, and refurbishing cost of parts at manufacturing/refurbishing facilities. The objective function (2) minimizes the total number of defective parts which is also equivalent to 22

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maximizing the total quality of parts purchased from the suppliers. The objective function (3) minimizes the total number of late delivered parts purchased from the suppliers, and finally the objective function (4) minimizes the total risk factors of economic environment associated with each supplier. 3.4. Functional and Operational Constraints Fuzzy constraint (5) ensures the total number of manufactured products from each type is equal to fuzzy demand for each type of products. Constraint (6) express the number of parts used to manufacture of final products at each plant to be equal to the total number of refurbished parts from refurbishing sites and purchased parts from the suppliers. Constraint (7) shows that the number of disassembled parts is equal to the number of refurbished parts and disposed parts whereas constraint (8) demonstrates the relationship between the number of parts that can be obtained from returned products. Fuzzy constraint (9) represents the fuzzy maximum supplier capacity not to be exceeded and constraint (10) ensures the minimum purchase quantity requirement of each part from suppliers. Constraints (11) and (12) limit the percentage of reusable and disposed parts that can be obtained from returned products at the disassembly facility. Constraint (13) shows the fuzzy percentage of returned products from total manufactured products. Constraint (14) is for set-up initiation of disassembly sites and constraint (15) limits the decision variables of supplier selection and setup of disassembly sites for each type of products to be binary variables and all the other decision variables to be non-negative. 3.5. Transformation of Fuzzy Constraints Fuzzy sets theory can help the decision maker to overcome uncertainty as the fuzzy number is illustrated by a membership function that is a number between 0 and 1. Since the demand constraint (5) is an equality constraint we use triangular fuzzy number dem n  (demnL , demn , demnU ) and the membership function  n (dem n ) to represent the fuzzy constraint (5). Where demnL , demn , demnU represent the imprecise lower bound,

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most likely and upper bound of the fuzzy number dem n respectively. For the supplier maximum capacity inequality constraint we use linear membership function

 im ( supmax im ) to convert the fuzzy constraint (9) into its equivalent deterministic constraint. Here the fuzzy number supmax im is bounded to a lower bound and an upper U bound supmaximL and supmax im . For the parentage of returned products inequality

constraint (13) we also use linear membership function  n (return n ) to convert the fuzzy constraint to its equivalent deterministic constraint. Here again the fuzzy number return n is bounded to a lower bound and an upper bound return nL and return nU enabling

us to develop the membership function. IV. SOLUTION METHOD Because the developed model in this research has multiple objectives along with fuzzy constraints, Fuzzy Goal Programming (FGP) is considered as the solution method. The major drawback of the standard goal programming methods is that they can obtain only one non-dominated solution which is highly dependent to the decision maker’s choice of the goals and the weights of deviation from the predefined goals. To rectify this dependability and in order to obtain the true Pareto-optimal front, the following hybrid Monte Carlo simulation model is proposed in which randomly generated weights for membership functions are used in the fuzzy goal programming submodel in each simulation replication. In a multi-objective optimization problem, all objectives might not be achieved simultaneously under the system constraints so the decision maker may define a tolerance limit such as membership function  k ( f k ) for the kth fuzzy objectives. It is also useful to mention that the fuzzy goal programming uses the nadir solutions as undesired targets and maximizes the lower bounds on the membership functions. V. COMPUTATIONAL RESULTS In this study 10 suppliers are identified as the potential candidate suppliers from

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Table 2. Hybrid Monte Carlo Simulation and Fuzzy Goal Programming Begin Step 1: Calculate the best and the worst solutions (also known as ideal and nadir solutions) of the objective function k, f kmin and f kmax , by incorporating only one of the objectives while ignoring all other objectives subject of the set of functional constraints; Step 2: Current replication = 1; While (current replication