在市場競爭日益激烈的環境下，企業為了維護其市場，必須不斷的尋求以及維持自身的競爭優勢。藉由生產管理的運用，企業能夠有效的降低其成本並提高競爭力。然而，由於現今市場環境的快速變遷下，存貨管理對於企業的營運績效也越來越顯現出其重要性。因此，如何有效的控制存貨水準已經變成企業管理者一個所必須要面對的議題。
本論文主要是探討前置時間為變動且當需求大於供給時允許欠撥的存貨模式。在本研究中，前置時間可以分割為幾個組成成分，每個組成成分有各自的成本函數。我們的目標是同時考慮訂購量以及前置時間的情況下，找出最適訂購量，並且使總成本為最小。在過去多數的研究中，都假設年需求為固定已知且欠撥率為常數或是跟價格折扣成線性關係。然而，在實務上存在著許多無法量化的不確定因素。有鑒於此，在本論文中藉由模糊的概念，來將一些不確定因素模糊化，並且建立求解的演算法。對每一個模式，我們利用signed distance (一個利用排序求解模糊數的方法) 來解糊模化，然後找出所對應的最佳解。最後舉出數值範例來加以說明。

關鍵字：存貨管理、模糊理論、前置時間、欠撥率

In today’s competitive marketplace, companies have to earn and maintain the advantages themselves. Through the production management, companies can effectively reduce their cost and raise competition. However, the inventory problem plays a more important role in production management then before. Therefore, how to efficiently control the inventory level becomes a very important issue.
This paper investigates the backorder rate inventory problem with variable lead time. In this paper, the variable lead time can be decomposed into several components, each having a crashing cost function for the respective reduced lead time. The objective is to find the optimal order quantity and the lead time simultaneously, and then minimizing the total inventory cost. In the past, most of the published data assumes that the annual demand is deterministic and the backorder rate is in proportion to the price discount offered by the supplier. However, there are many uncertain factors in practical situations. Therefore, they can be described by fuzzy sense. Here, we use the concept of fuzziness to joint the mixture inventory system, and construct the solution procedure to find the optimal order quantity and lead time. For each model, we utilize the signed distance, a ranking method for fuzzy numbers, to find the estimate of annual demand and backorder rate in the fuzzy sense, and then derive the corresponding optimal solution. Numerical examples are included to illustrate the procedures of the solution.

Keywords: Inventory; Fuzzy set; Lead time; Signed distance; Backorder rate

[1] Ben-Daya, M. and Raouf, A., “Inventory models involving lead time as a decision variable,” Journal of Operational Research Society, Vol. 45, 1994, pp.579-584.
[2] Chang, H.C., “An application of fuzzy sets theory to the EOQ model with imperfect quality items,” Computers and Operations Research, Vol. 31, 2004, pp.2079-2092.
[3] Chang, S.C., Yao, J.S. and Lee, H.M., “Economic reorder point for fuzzy backorder quantity,” European Journal of Operational Research, Vol. 109, 1998, pp.183-202.
[4] Chen, S.H. and Wang, C.C., “Backorder fuzzy inventory model under functional principle,” Information Sciences, Vol. 95, 1996, pp.71-79.
[5] Gallego, G. and Moon, I., “The distribution free newsboy problem: Review and extensions,” Journal of Operational Research Society, Vol. 44, 1993, pp.825-834.
[6] Kaufmann, A. and Gupta, M.M., Introduction to fuzzy arithmetic: theory and applications, New York: Van Nostrand Reinhold, 1991.
[7] Krajewski, L.J. and Ritzman, L.P., Operations Management: Strategy and Analysis, Addison-Wesley, Reading, MA, 1996.
[8] Liao, C.J. and Shyu, C.H., “Analytical determination of lead time with normal demand,” International Journal of Operations and Production Management, Vol. 11, 1991, pp.72-78.
[9] Moon, I. and Yun, W., “The distribution free job control problem,” Computers and Industrial Engineering, Vol. 32, 1997, pp.109-113.
[10] Naddor, E., Inventory Systems, New York: John Wiley, 1966.
[11] Ouyang, L.Y. and Wu, K.S., “A minimax distribution free procedure for mix inventory model with variable lead time,” International Journal of Production Economics, Vol. 56, 1998, pp.511-516.
[12] Ouyang, L.Y., Yen, N.C. and Wu, K.S., “Mixture inventory model with backorders and lost sales for variable lead time,” Journal of the Operational Research Society, Vol. 47, 1996, pp.829-832.
[13] Pan, J.C. and Hsiao, Y.C., “Inventory models with backorder discounts and variable lead time,” International Journal of Systems Science, Vol. 32, 2001, pp.925-929.
[14] Pan, J.C., Hsiao, Y.C. and Lee, C.J., “Inventory models with fixed and variable lead time,” Journal of the Operational Research Society, Vol. 53, 2002, pp.1048-1053.
[15] Pan, J.C., Lo, M.C. and Hsiao, Y.C., “Optimal reorder point inventory models with variable lead time and backorder discount considerations,” European Journal of Operational Research, Vol. 158, 2004, pp.488-505.
[16] Park, K.S., “Fuzzy-set theoretic interpretation of economic order quantity,” IEEE Transactions on Systems, Man, and Cybernetics, Vol. 17, 1987, pp.1082-1084.
[17] Pu, P.M. and Liu, Y.M., “Fuzzy topology 1 neighborhood structure of a fuzzy point and Moor-Smith convergence,” Journal of Mathematical Analysis and Applications, Vol. 76, 1980, pp.571-599.
[18] Ravindran, A., Phillips, D.T. and Solberg, J.J., Operations Research: Principle and Practices, New York: John Wiley, 1987.
[19] Tersine, R.J., Principles of Inventory and Material Management, New York: North Holland, 1982.
[20] Vojosevic, M., Petrovic, D. and Petrovic, R., “EOQ formula when inventory cost is fuzzy,” International Journal of Production Economics, Vol. 45, 1996, pp.499-504.
[21] Vollmann, T.E., Berry, W.L. and Whybark, D.C., Manufacturing Planning and Control Systems, third edition, Chicago: Irwin, 1992.
[22] Yao, J.S. and Lee, H.M., “Fuzzy inventory with backorder for fuzzy order,” Information Sciences, Vol. 93, 1996, pp.283-319.
[23] Yao, J.S. and Wu, K., “Ranking fuzzy numbers based on decomposition principle and signed distance,” Fuzzy sets and Systems, Vol. 116, 2000, pp.275-288.
[24] Zimmermann, H.J., Fuzzy set theory and its application, third edition, Dordrecht: Kluwer Academic Publishers, 1996.