A LOT-SIZING PROBLEM WITH TIME VARIATION IMPACT IN CLOSED-LOOP SUPPLY CHAINS

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1 A LOT-SIZING PROBLEM WITH TIME VARIATION IMPACT IN CLOSED-LOOP SUPPLY CHAINS Aya Ishigaki*, Tokyo University of Science, Japan Tetsuo Yamada, The University of Electro-Communications, Japan, ABSTRACT In this research, closed-loop supply chains where product returns are integrated with traditional forward supply chains are considered. Customer demands can be satisfied from manufacturing of new products or from remanufacturing of returned products. In the literature, several papers analyze EOQ-type inventory control model and assume same batch sizes for production batch and remanufacturing batch. However, the same batch size does not necessarily become an optimum. Furthermore, generally demand/return data differ for every period. This research addresses the inventory management issue in closed-loop supply chains, and develops deterministic and dynamic model for a two-echelon system under generalized cost structures. Also, various numerical examples are investigated using the developed model. Keywords: Multi-Echelon Closed-Loop Supply Chain, Economic Order Quantity, Dynamic Return Data, Dynamic Programming INTRODUCTION In recent years, the activity which reduces environmental impacts, such as recycling and reuse, is increasing. To design the closed-loop supply chain in consideration of recycling or reuse is needed also in order to construct social responsibility and competitive superiority. However, in order to establish supply chains for sustainability, it is necessary to consider into not only the environment but also economic efficiency. Many research on the closed-loop supply chain in consideration of economic efficiency is performed from various viewpoints [1, 2]. Deterministic models mainly investigate lot-sizing problems for static (EOQ-type [3]) demand/return data. Only a handful of references that deal with stochastic multi-echelon closed-loop supply chains are available in the literature [4, 5]. The problem of EOQ-type lot-sizing in systems with product returns has been addressed from both, repairable items and a remanufacturing products. Schrady (1967) analyzes a system where returns are repaired in several consecutive, equally sized batches [6]. Mitra (2012) discussed about the influence of economic ordering quantity under the economic environment with the ordering cost and inventory holding cost for collected products [7, 8]. They assumed that repair/recovery batch sizes are same, and treats as what does not carry out a time variation. However, it is necessary to consider that the quantity of recoverable products changes with the time variation of the demand or the quality of returned products in practice. Furthermore, the quality of recoverable products changes in collection timing. As a result, the time variation of the quantity of recoverable products should be also considered. The aims of this research are to investigate the time variation of the collected quantity at clarifying influence which it has on the economic order quantity in a closed-loop supply chain. The difference in the time distribution of a collected quantity to what was sold at a certain time considers the influence which it has on the economic order quantity.

2 MODEL FORMULATION In order to understand a basic structure, two-echelon inventory system comprising three stages is considered. Figure 1 shows an inventory-control model for a closed-loop supply chain with a retailer, a manufacturer, a maker and a collection agency. Stage 1 (retailer) faces customer demand. And stage 2 (maker) gets supplies from an outside supplier and provides product to the stage 1. Stage 3 (collection agency), which recovers returns, belongs to stage 1 and supplements the product to stage 1. This model considers deterministic, stationary and uniform demand. Returns are remanufactured, which are interchangeable with new items that are procured from a maker to meet customer demand from the serviceable stock. outside supplier stage 2 delivery stage 1 delivery maker retailer order demand delivery order customer collection return agency stage 3 material flow information flow Figure 1 Two-echelon inventory system comprising three stage WAGNER/WHITIN MODEL EXTENSION This study investigates the influence of economic order quantity for dynamic return data under the economic environment. As an extension of the Wager/Whitin dynamic programming [9] that includes a recovery option, the following assumptions are made. A constant and deterministic demand rate has to be satisfied, and backorders are not permitted. Also, a return fraction of demand is deterministic and dynamic. It is assumed that lead times are ignored. Remanufactured products are as good as new products, and all return parts have the same quality. The following parameters and notation is used in our model formulation: A i setup cost at stage i (i=1, 2, 3) h i inventory holding cost per unit per period at stage i (i=1, 2, 3) μ demand per period r fraction of demand returned at period t

3 δ 1,t binary variable (If a retailer orders to a maker at period t, it is set to 1. Otherwise, it is set to 0.) δ 2,t binary variable (If a maker orders to an outside supplier at period t, it is set to 1. Otherwise, it is set to 0.) δ 3,t binary variable (If a retailer orders to a collection agency at period t, it is set to 1. Otherwise, it is set to 0.) x 1, t order quantity from a retailer to a maker at period t x 2, t order quantity from a maker to an outside supplier at period t x 3, t order quantity from a retailer to a collection agency at period t I 1, t retailer s inventory at period t I 2, t maker s inventory at period t I 3, t collection agency s inventory at period t M Big M T planning horizon J product lifetime a t,j quantity of collected product at period (t+j) in the product sold at period t (It is shown by the rate to demand) When the above notation is used, this problem is formulated as a dynamic programming. Minimize: (A 1 δ 1,t + A 2 δ 2,t + A 3 δ 3,t + h 1 I 1,t + h 2 I 2,t + h 3 I 3,t ) (1) Subject to: I 1,t = I 1,t-1 + x 1,t + x 3,t μ, I 2,t = I 2,t-1 + x 2,t - x 1,t, I 3,t = I 3,t-1 + a t-j,j μ - x 3,t, (2) (3) (4) x i,t Mδ i,t, I 1,0 = I 2,0 = I 3,0 = 0 (5) (6) x 1,t, x 2,t, x 3,t, I 1,t, I 2,t, I 3,t 0, δ 1,t, δ 2,t, δ 3,t {0,1}, (7) (8) Equations (2)-(4) is an inventory preservation formula at period t. Equation (5) expresses that the order quantity of the period which is not ordered is 0. The Big M introduces surplus and artificial variables to convert all inequalities into that form. Equation (6) expresses that an initial inventory is 0. And equation (7) shows nonnegative conditions.

4 THE TIME VARIATION OF THE QUANTITY OF COLLECTED PRODUCT Figure 2 shows the image which a collection product reaches to collection agency. For example, the products sold to the 1st period are collected after the 5th period. Generally, collection product of what was sold at the same period is increasing with time. Furthermore, a collection period is limited in order that there is a lifetime in a product. If these assumptions are used, the number of the collection products which can be used at the time of a retailer's order can be estimated. In this research, the time variation of the quantity of collected product is expressed by adding two definitions to a model: R t fraction of demand returned which can be used at period t, Q j j-th order quantity. The following relations are formed in these notations. R t = J a t j, j j = 1 J r = j= 1 a t, j (9) (10) The example of the figure shows that the collection product can be used by the 2nd order. collection agency s inventory t Q 1 Q 2 Figure 2 The image which a collection product reaches to collection agency ECONOMIC IMPACT OF TIME VARIATION FOR ORDER QUANTITY Using the extended Wagner/Whitin model, the influence which the time variation of a return rate has on economic order quantity is investigated. In order to understand easily, it is assumed that return is collected uniformly during a collection period. Thus, the following equations are formed about a quantity of collected product. a t, j = r J

5 If this equation is used, a quantity of collected product increase by an initial state. Then a quantity of collected product becomes constant, i.e., a stationary state. Many previous research has treated the stationary state of a quantity of collected product. However, our interest is in the state in which a quantity of collected product like an initial state performs a time variation. So, a numerical experiment is performed to the initial state. In experiment, each parameter is set as follows: μ= 100, A 1 = 25, A 2 = 100, A 3 = 50, h 1 = 2, h 2 = 1, h 3 = 0.5, r = 0.5, J = 40 and T = 50. Figure 3 and 4 show the result of having calculated economic lot size using the extended Wagner/Whitin model. The vertical line shows the economic lot size, and the horizontal line shows the period. Figure 3 shows the result in the case of assuming the model ordering to a collection agency at the same time a retailer is ordering to a maker. This means consuming return product preferentially from consideration of an environmental factor. On the other hand, figure 4 shows the result of economic lot size on condition that return product is used up within a planning horizon. As a result of numerical experiment, the average total became in the case of Figure 3, and it became in the case of Figure 4. After all, when return product is dynamic data, it became more economical to add return product to the order depending on the situation x1,t x2,t x3,t order quantity period Figure 3 Economic ordering quantity when order to a maker and a collection agency is the same period x1,t x,2,7 x3,4 order quantity period Figure 4 Economic ordering quantity in case there is no constraint to an environmental factor

6 SUMMARY In this research, the time variation of the collected quantity at the influence which it has on the economic ordering quantity in a closed-loop supply chain was investigated. As a result, Although the model which uses up return product preferentially can reduce environmental factors, it turned out that it does not become economical by the time variation of a return rate or life time of used product. This research focuses on the relation between time variation and economics. Future, it is necessary to extend as a multi-objective optimization problem including not only time variation and economics but environmental impact. ACKNOWLEDGMENTS The authors would like to thank Mr. Junya Murakami for having collected a lot of information about this research. This research was partially supported by the Japan Society for the Promotion of Science (JSPS), KAKENHI, Grant-in-Aid for Scientific Research (B), Grant Number from 2014 to REFERENCES [1] Fleischmann, M., Bloemhof-Ruward, J.M, Dekker, R., van der Laan, E., van Nunen, J.A.E.E, van Wassenhove, L.N. Quantitative models for reverse logistics: A review, European Journal of Operational Research, 1997, Vol. 103, pp.1-17 [2] Ilgin, M.A., Gupta, S.M., Environmentally conscious manufacturing and product recovery (ECMPRO): A review of the state of the art, Journal of Environmental Management, 2010, Vol.91, pp [3] Harris, F. W. "What Quantity to Make at Once", Shaw Factory Management series, 1915, Vol. V, Operation and Costs, Chicago: A. W. Shaw Company, pp [4] Kurugan, A., Gupta, S.M., A multi-echelon inventory system with returns, Computers and Industrial Engineering, 1998, Vol.35, pp [5] Minner, S., Strategic safety stocks in reverse logistics supply chains International Journal of Production Economics, 2006, Vol. 51, pp [6] Schrady, D.A., A Deterministic Inventory Model for Reparable Items, Naval Research Logistics Quarterly, 1967, Vol.14, pp [7] Mitra, S., Analysis of a two-echelon inventory system with returns, Omega, 2009,Vol.37, pp [8] Mitra, S., Inventory management in a two-echelon closed-loop supply chain with correlated demands and returns, Computers and Industrial Engineering, 2012, Vol.62, pp [9] Wagner, H.M., Whitin, T.M., Dynamic version of the economic lot size model, Management Science, 1958, Vol.5, pp