Cost Analysis of a Remanufacturing System

Transcription

1 Asia Pacific Management Review (2004) 9(4), Cost Analysis of a Remanufacturing System Kenichi Nakashima, Hiroyuki Arimitsu *, Toyokazu Nose * and Sennosuke Kuriyama Abstract This paper deals with a cost management problem of a remanufacturing system with stochastic variability such as demand, remanufacturing rate and discard rate. We model the system with consideration for two types of inventories. One is the actual product inventory in a factory. The other is the virtual inventory that is used by customer. We define the state of the remanufacturing system by the both of the inventory levels. It is assumed that the cost function is composed of various cost factors such as holding, backlog and some kinds of manufacturing costs etc. The expected average cost per period is obtained by numerical computation. We also consider some scenarios under various conditions using design of experiments and discuss the effects of the cost factors on the remanufacturing system. Keywords: Remanufacturing system; Optimal control; Design of experiments 1. Introduction The escalating growth in consumer waste in recent years has started to threaten the environment. Product recovery is mainly driven by the escalating deterioration of the environment and aims to minimize the amount of waste sent to landfills by recovering materials and parts from old or outdated products by means of recycling and remanufacturing. Product recovery includes collection, disassembly, cleaning, sorting, repairing and reconditioning broken components, reassembling and testing [1,5]. Here we focus on product recovery in a remanufacturing system under stochastic variability. This paper deals with an optimal control problem of a remanufacturing system under stochastic variability such as demand. The system is formulated into a Markov Decision Process (MDP)[6,16]. The MDP is a class of stochastic sequential processes in which the reward and transition probability depend only on the current state of the system and the current action. The MDP models have gained recognition in such diverse fields as economics, communications, transportation and so on. Each model consists of Osaka Institute of Technology, Department of Industrial Management, Omiya, Asahi-ku, Osaka, JAPAN, Telephone: , (nakasima, arimitsu, nose)@dim.oit.ac.jp Setsunan University, Faculty of Business Administration and Information, 17-8, Ikedanakamachi, Neyagawa, Osaka, JAPAN 595

2 Kenichi Nakashima et al. states, actions, rewards, and transition probability. In the engineering field, for example, the approach is used for controlling the production system [14,15]. Choosing an action as production quantity in a state generates rewards and/or costs, and determines the state at the next decision epoch through a transition probability function. Then, we can obtain the optimal production policy that minimizes the expected average cost per period in the optimal production control problem. We here consider two types of inventtories. One is the actual product inventory in a factory, the other is the virtual inventory which is used by customer. We define the state of the remanufacturing system by the both of the inventory levels. We can obtain the optimal production policy that minimizes the expected average cost per period. We also consider some scenarios under various conditions using design of experiments. In section 2, we describe a brief review of the literature on the remanufacturing systems. In section 3, we consider a single-item remanufacturing system under stochastic demand. The system is formulated into an undiscounted MDP to determine the optimal control policy that minimizes the expected average cost per period. Finally, we discuss the effects of cost factors on the remanufacturing system under various conditions. 2. Literature Review We present a brief review of the literature in the areas of product recovery modeling of remanufacturing systems with stochastic variability. Gungor and Gupta [3] and Moyer and Gupta [10] reviewed the literature in the area of environmentally conscious manufacturing and product recovery. Minner [9] pointed out that there are the two well-known streams in product recovery research area. One is stochastic inventory control (SIC) and the other is material requirement planning (MRP). In this paper, we restrict ourselves to SIC. As for the periodic review models, Cohen et al. [2] developed the product recovery model in which the collected products are used directly. Inderfurth [7] discussed effect of non-zero leadtimes for orders and recovery in the different model. As for continuous review models, Muckstadt and Isaac [11] dealt with a model for a remanufacturing system with non-zero leadtimes and a control policy with the traditional (Q, r) rule. Van der Laan and Salomon [17] suggested push and pull strategies for the remanufacturing system. Guide and Gupta [4] presented a queueing model to study a remanufacturing system. 596

3 Asia Pacific Management Review (2004) 9(4), They, however, considered that demand and procurement were independent in the inventory systems. Nakashima et al. [12] dealt with a product recovery system with a single class product life cycle. The quantity of the virtual inventory the customers have in the system depended on the demand. They proposed new analytical approach to evaluating the system using a discrete time Markov chain and gave the numerical examples in various conditions. Furthermore, they optimized the system with constant cost parameters under minimum cost criterion [13]. In this paper, we optimize the remanufacturing system using the MDP model and discuss the effect of the cost factors and demand variance on the optimal production policy. 3. The Remanufacturing Model Let us consider a single process that produces single item product. The finished products are stocked in the factory and faced according to the customer demand. Traditional inventory management focuses on only the inventory in the factory. In the remanufacturing system, however, we should focus on the outdated products that are collected from the customers. That is, the remanufacturing producers have to consider the products in use as the part of the future inventory. We here take the products used by customers as the virtual inventory. It is important managing the virtual inventory until products are collected and used in the remanufacturing as well as controlling the inventory on hand. Figure 1 shows a remanufacturing production system. Remanufacturing preserves the product's or the part's identity, and performs the required disassembly and refurbishing operations to bring the product to a desired level of quality with remanufacturing cost. On the other hand, we define normal manufacturing as producing the products using new resources. The number of products by normal manufacturing at period t, P(t) is chosen as an action, k, that is k=p(t). Products are produced by normal manufacturing and/or remanufacturing using the parts taken back from the customers. All production begins at the start of the period and all products are completed by the end of the period. All the products bought by customers are new. We assume that the number of finished products and that of the products bought by customers are I(t) and J(t), respectively. If the backlog occurs, I(t) is negative value. Demand in successive periods, D(t) is independent random variables with known identical distribution. When sold, products are remanufactured at the remanufacturing rate, λ with remanufacturing cost, and products in use are discarded at the discarded rate, μ with out-of-date cost. It is supposed that λ + μ

4 Kenichi Nakashima et al. P(t) Inven tory on hand I(t) Factory λ o oo D(t) Custom er Virtual Inven tory J(t ) μ Figure 1 A Remanufacturing System The state of the system is denoted by s(t) = ( I(t), J(t) ). (1) The transition of the each inventory is given by I( t + 1) = I( t) + k + λ J ( t) D( t) (2) and J(t + 1) = J(t) λj(t) μj(t) + min{[i(t)] + + P(t)+ λj(t), D(t) + [I(t)] + } (3) where [x] + =min{0, x}. The action space in s(t), K(s(t)) is defined by K(s(t)) = { 0,, max{0, Imax-I(t)-λ J(t)} }. (4) The expected cost per period in state s(t) under k, r(s(t),k) is given by r(s(t), k) = C N k + C R λj(t) + C H [I(t)] + + C B [ I(t)] + + C O μj(t) (5) where the parameters are as follows: C N : the normal manufacturing cost of a new product C R : the remanufacturing cost of a product 598

5 Asia Pacific Management Review (2004) 9(4), C H : the holding cost per unit C B : the backlog cost per unit C O : the out-of-date cost per unit We can also define the same transition probability as in Nakashima et al. [13]. Denote by S and S a set of all possible states and the total number of the states, respectively. Let number the state s n by s (= 1 S ). An undiscounted MDP that minimizes the expected average cost per period, g is formulated as the following optimality equation: g + v s = min r(s, k) + P ss' (k) v s' ( s S). (6) k K(s) s' S where v s denotes the relative value when the production system starts from state s [6]. An optimal production policy is determined as a set of k that minimizes the right-hand side of equation (6) for each state s using policy iteration method [6, 16]. 4. Computational Results In this section, we show the numerical results using design of experiments[8] and investigate the effects of the some factors on the system by analysis of variance. The distribution of the demand is given by Pr{D n = D 1 2 Q + j} = Q j 1 Q 2,(0 j Q), where D=2 and Q is an even number and variance(σ 2 ) is Q/4. This is the shifted binomial distribution. It helps toward investigating the effect of the demand variance under the constant mean. The maximum number of actual inventory and that of virtual inventory are 5 and 10, respectively. The maximum number of backlog demand is set as -5. It is assumed that the remanufacturing rate is 0.6 and the discarded rate is 0.2. We use L 8 (2 7 ) orthogonal array and take the five cost parameters defined in eq.(5) as the factors as well as the variance of demand. Each factor has two levels as below: Table 1 Factors and Levels A (C N ): A 1 =1, A 2 =5 D(C O ): D 1 =10, D 2 =30 B (C H ): B 1 =1, B 2 =2 F(C R ): F 1 =2, F 2 =8 C(C B ): C 1 =10, C 2 =30 G(σ 2 ) : G 1 =0.5, G 2 =

6 Kenichi Nakashima et al. Table 2 shows the assignment of the factors and levels. We can obtain the optimal production policy with the expected minimum total cost in each experiment. For example, we assume C N =1, C H =1, C O =10, C B =10, C R =2 and σ 2 =0.5 in #1 experiment, and obtain the optimal production policy which gives us the expected minimum total cost as under the condition. Table 3 shows the result of performing analysis of variance on the total cost data in Table 2. We can find the effects of the normal manufacturing cost, holding cost, remanufacturing cost, out of date cost and the demand of variance. This means that the difference between levels of each factor except the backlog cost makes the total cost changed, respectively. Table 2 L 8 (2 7 ) Orthogonal Array and Numerical Results No. A B D C F G e Total cost Table 3 Analysis of Variance Factor S φ V F 0 A (C N ): * B (C H ): * C(C B ): D(C O ): ** F(C R ): ** G(σ 2 ) : * e Sum

7 Asia Pacific Management Review (2004) 9(4), Conclusion This paper dealt with a cost management problem of the remanufacturing system with consideration for two types of inventories under stochastic variability. We modeled the system as the Markov decision process to obtain the optimal production policy that minimized the expected average cost per period. We also considered some scenarios under various conditions using design of experiments and discussed the effects of the cost factors on the remanufacturing system. The numerical investigation showed the important factors of the remanufacturing management. For future research, we suggest to develop the extended model including MRP mechanism to analyze the remanufacturing systems. Acknowledgment The authors would like to express their thanks to anonymous referees for their valuable comments and helpful suggestions. The work was partially supported by MEXT.KAKENHI( ). References [1] Brennan, L., S.M. Gupta, K.N. Taleb Operations planning issues in an assembly/disassembly environment. International Journal of Operations and Production Management [2] Cohen, M.A., S. Nahmias, W.P. Pierskalla A dynamic inventory system with recycling. Naval Research Logistics Quarterly [3] Gungor, A., S.M. Gupta Issues in environmentally conscious manufacturing and product recovery: A survey. Computers and Industrial Engineering [4] Guide, Jr.V.D.R., S.M. Gupta (Mar. 1-3). A queueing network model for remanufacturing production systems. Proceedings of the Second International Seminar on Reuse, Eindhoven, The Netherlands, [5] Gupta, S.M., K.N. Taleb Scheduling disassembly. International Journal of Production Research [6] Howard, R.A Dynamic Programming and Markov Processes. Cambridge: The M. I. T. Press. [7] Inderfurth, K., Simple optimal replenishment and disposal policies for a product recovery system with lead-times. OR Spektrum [8] Macclave, J.T., P.G. Benson, T. Sincich Statistics for Business 601

8 Kenichi Nakashima et al. and Economics. NJ: Prentice-Hall. [9] Minner, S Strategic safety stocks in reverse logistics supply chains. International Journal of Production economics [10] Moyer, L., S.M. Gupta Environmental concerns and recycling/ disassembly efforts in the electronics industry. Journal of Electronics Manufacturing [11] Muckstadt, J.A., M.H. Isaac An analysis of single item inventory systems with returns. Naval Research Logistics Quarterly [12] Nakashima, K., H. Arimitsu, T. Nose, S. Kuriyama Analysis of a product recovery system. International Journal of Production Research [13], H. Arimitsu, T. Nose, S. Kuriyama Optimal control of a remanufacturing system. Proceedings of the 17 th International conference of Production Research (CD-ROM). [14] Ohno, K., K. Ichiki Computing optimal policies for controlled tandem queueing systems. Operations Research [15], K. Nakashima Optimality of a Just-in-Time production system. Singapore (World Scientific): Proceedings of APORS'94 (Selected Paper) [16] Puterman, M.L Markov Decision Processes. New York (John Wiley & Sons): Discrete Stochastic Dynamic Programming. [17] Van Der Laan, E.A., M. Salomon Production planning and inventory control with remanufacturing and disposal. European Journal of Operational Research