在存貨系統中，最常使用傳統EOQ模式，但是傳統EOQ模式，未考慮退化性因素，與一般現實狀況不合。關於退化性產品的存貨系統模式之探討，最早是由Ghare和Schrader在1963年提出來的，他們建立了一個退化率與需求率為常數的存貨模式。事實上，對於很多的產品或複雜的系統而言，大部分的產品或系統的壽命都是具有浴缸型退化率函數。此外，在傳統EOQ模式中，我們經常假設需求率為固定已知的常數。然而，在Ouyang et al. (2003)和Teng et al. (2005)中我們可以觀察到需求率通常與存貨水準有關。因此，基於上述理由，本研究考慮在有限的計畫期間內，分別建構產品壽命為Xie等人(2002)、Hjorth (1980)與Mudholkar 和 Srivastava (1993)所定義的浴缸型退化率及隨著存貨水準變動的需求率之存貨模式。

最後，本文將建立三個含有持有成本、退化成本、訂購成本、缺貨成本及產品銷售損失成本的浴缸型退化性之存貨模式。並且利用數值範例來說明求解的程序，以決定最佳訂購策略。

Classical EOQ model is used most frequently in the inventory system, but it is not considered deteriorated factor. It is not conformed with the general realistic state. First, Ghare and Schrader (1963) suggest an inventory model for deteriorating items with fixed deteriorated rate and demand rate. In fact, most products or system life all have bathtub-shaped distributed deterioration for many products or complicated system. In addition, it is generally assumed that the demand rate is constant in the classical EOQ model. However, we could observe that the demand rate usually depends on inventory-level in Ouyang et al. (2003) and Teng et al. (2005). So, in this thesis, we will consider product life with bathtub-shaped distributed deterioration by Xie et al. (2002), Hjorth (1980), Mudholdar and Srivastava (1993) defined and inventory model with stock-dependent demand rate over a fixed planning horizon, respectively.
   Finally, we construct three bathtub-shaped distributed deterioration inventory models with holding cost, deteriorating cost, ordering cost, shortage cost, and sale loss cost. We give some numerical examples to illustrate solution procedure and decide the optimal replenishment policy.

目錄
目錄  ……………………………………………………………….   I
表目錄  ……………………………………………………………. III
圖目錄  …………………………………………………………….  VI

第一章	 緒論  …………………………………………………  1
1-1 研究動機與目的  ………………………………………….   1
1-2 文獻探討  ………………………………………………….   3
1-3 本文結構  ........................................  4


第二章	  具有浴缸型退化率及隨著存貨水準變動的需求率之
.          存貨模式  ………………………………………     5
2-1符號說明與假設  …………………………………………     5
2-2模式建立  ………………………………………………      12
2-2-1具有Xie等人(2002)定義的退化率之存貨模式  ………   12
2-2-2 具有Hjorth (1980)定義的退化率之存貨模式  ………  19
2-2-3 具有Mudholkar和Srivastava(1993)定義的退化率
      之存貨模式  ……………………………………………   24
2-3模式的求解程序  ……………………………………………  30

第三章	存貨模式之數值範例及敏感度分析  ……………    31
3-1有關Xie等人(2002)定義的浴缸型退化率之存貨模式       32
3-1-1數值範例  …………………………………………        32
3-1-2敏感度分析  .………………………………………       36
3-2有關Hjorth (1980)定義的浴缸型退化率之存貨模式  …   39
3-2-1數值範例  ……………………………………………      39
3-2-2敏感度分析  …………………………………………      46
3-3有關Mudholkar和Srivastava(1993)定義的浴缸型
   退化率之存貨模式  ………………………………………    49
3-3-1數值範例  ……………………………………………      49
3-3-2敏感度分析  ………………………………………        59

第四章 結論與未來研究方向  …………………………………  62
4-1 結論  ……………………………………………………     62
4-2 未來研究方向  …………………………………………     65

參考文獻  ……………………………………………………     67

表目錄
表3-1-1：Xie等人(2002)定義的退化率( )之存貨模式的
         求解程序  ………………………………………      34
表3-1-2：Xie等人(2002)定義的退化率( )
         之存貨模式的求解程序  ………………………      34
表3-1-3：Xie等人(2002)定義的退化率( )
         之存貨模式.的求解程序  ……………………       35
表3-1-4：Xie等人(2002)定義的退化率( )
         之存貨模式的求解程序  …………………………    35
表3-1-5：Xie等人(2002)定義的浴缸型退化率之存貨模式
         的敏感度分析  ………………………………………  37
表3-2-1：Hjorth(1980)定義的退化率( )
         之存貨模式.的求解程序  …………………………   42
表3-2-2：Hjorth(1980)定義的退化率( )
         之存貨模式.的求解程序  …………………………   42
表3-2-3：Hjorth(1980)定義的退化率( )
         之存貨模式的求解程序  …………………………    43
表3-2-4：Hjorth(1980)定義的退化率( )
         之存貨模式的求解程序  …………………………    43
表3-2-5：Hjorth(1980)定義的退化率( )
         之存貨模式的求解程序  ……………………………  44
表3-2-6：Hjorth(1980)定義的退化率( )
         之存貨模式的求解程序  ……………………………  44
表3-2-7：Hjorth(1980)定義的退化率( )
         之存貨模式的求解程序  …………………………    45
表3-2-8：Hjorth(1980)定義的退化率( )
         之存貨模式的求解程序  ……………………………  45

表3-2-9：Hjorth(1980)定義的浴缸型退化率之存貨模式
         的敏感度分析 ………………………………………   47
表3-3-1：Mudholkar和Srivastava (1993)定義的退化率
         ( )之存貨模式的求解程序  ………………………   53
表3-3-2：Mudholkar和Srivastava (1993)定義的退化率
         ( )之存貨模式的求解程序  ………………………   53
表3-3-3：Mudholkar和Srivastava (1993)定義的退化率
         ( )之存貨模式的求解程序  ………………………   54
表3-3-4：Mudholkar和Srivastava (1993)定義的退化率
         ( )之存貨模式的求解程序  ………………………   54
表3-3-5：Mudholkar和Srivastava (1993)定義的退化率
         ( )之存貨模式的求解程序  ………………………   55
表3-3-6：Mudholkar和Srivastava (1993)定義的退化率
         ( )之存貨模式的求解程序  ………………………   55
表3-3-7：Mudholkar和Srivastava (1993)定義的退化率
         ( )之存貨模式的求解程序  ………………………   56
表3-3-8：Mudholkar和Srivastava (1993)定義的退化率
        ( )之存貨模式的求解程序  ………………………    56
表3-3-9：Mudholkar和Srivastava (1993)定義的退化率
        ( )之存貨模式的求解程序  ………………………    57
表3-3-10：Mudholkar和Srivastava (1993)定義的退化率
        ( )之存貨模式的求解程序  ………………………    57
表3-3-11：Mudholkar和Srivastava (1993)定義的退化率
        ( )之存貨模式的求解程序  ………………………    58
表3-3-12：Mudholkar和Srivastava (1993)定義的浴缸型
          退化率之存貨模式的敏感度分析  ……………     60

表4-1：Xie等人(2002)定義的浴缸型退化率之存貨模式
       參數值變動與最佳訂購策略之關係  ………………    64
表4-2：Hjorth(1980)定義的浴缸型退化率之存貨模式
      參數值變動與最佳訂購策略之關係  …………………   64
表4-3：Mudholkar和Srivastava (1993)定義的浴缸型退化率
       之存貨模式參數值變動與最佳訂購策略之關係  ……  65

圖目錄
圖2-1：Xie等人(2002)定義的退化率  …………………………  9
圖2-2：Hjorth (1980)定義的退化率  ………………………… 10
圖2-3：Mudholkar和Srivastava(1993)定義的退化率  ……   12

參考文獻
中文部分
[1]  許桀瑋，2003，「產品壽命為混合分配且具有可控制的欠撥
     率之存貨模式研究」，淡江大學統計學系應用統計研究所碩
     士論文。
[2]  黃士逢，2001，「產品壽命為混合分配的存貨模式之研究」，
     淡江大學統計學系應用統計研究所碩士論文。
[3]  游俊彥，2002，「需求率與存貨水準有關且考慮部分欠撥之
     退化性產品的生產存貨模型」，淡江大學管理科學研究所碩
     士論文。

英文部分
[1]  Abad, P. L.（1996）, “Optimal pricing and lot-sizing 
     under conditions of perishability and partial 
     backordering”, Management Sciences, 42, 1093-1104.
[2]  Abad, P. L.（2000）, “Optimal lot size for a 
     perishable good under conditions of finite production 
     and partial backordering and lot sale”, Computers & 
     Industrial Engineering, 38, 457-465.
[3]  Abad, P. L.（2001）, “Optimal price and order size 
     for a reseller under partial backordering”, 
     Computers & Operations Research, 28, 53-65.
[4]  Aggarwal, S. P. and Goel, V. P.（1982）, “Order 
     level inventory systems with demand pattern for 
     deteriorating items”, Econ. Comput. Econ Cyb. Stud. 
     Res. , 17, 57-69.
[5]  Baker, R. C. and Urban, L. A.（1988）, 
    “Deterministic inventory systems with an inventory-
     level-dependent demand rate”, Journal of  the 
     Operational Research Society, 39, 823-831.
[6]  Benkherouf, L.（1998）, “Note on a deterministic lot-
     size inventory model for deteriorating items with 
     shortages and a declining market”, Computers & 
     Operations Research, 25, 63-65.
[7]  Bhunia, A. K. and Maiti, M.（1999）, “An inventory 
     model of deteriorating items with lot-size dependent 
     replenishment cost and a linear trend in demand”, 
     Applied Mathematical Modelling, 23, 301-308.
[8]  Chang, C. T.（2004）, “Inventory Models with stock-
     dependent demand and nonlinear holding costs for   
     deteriorating items”, Journal of Operational  
     Research, 21, 435-446.
[9]  Chang, H. J. and Dye, C. Y.（1999a）, “An EOQ Model 
     for Deteriorating Items with Exponential Time-Varying 
     Demand and Partial Backlogging”, International 
     Journal of  Information and Management Sciences, 10, 
     1-11.
[10] Chang, H. J. and Dye, C. Y.（1999b）, “An EOQ model  
     for deteriorating items with time-varying demand and 
     partial backlogging”, Journal of  the Operational 
     Research Society, 50, 1176-1182.
[11] Compaq Visual Fortran Professional Edition version 
     6.6 Intel Version and IMSL (2000) , Compaq Computer 
     Corporation.
[12] Covert, R. P. and Philip, G. C.（1973）, “An EOQ 
     model for items with Weibull distribution 
     deterioration”, AIIE Transactions, 5, 323-326.
[13] Datta, T. K. and Pal, A. K.（1990）, “A note on an 
     inventory model with inventory-level-dependent demand 
     rate”, Journal of the Operational Research Society, 
     41, 971-975.
[14] Datta, T. K. and Pal, A. K.（1992）, “A Note on a 
     Replenishment Policy for an Inventory Model with 
     Linear Trend in Demand and Shortages”, Journal of 
     the Operational Research Society, 43, 993-1001.
[15] Deb, M., and Chaudhuri, K. S.（1986）, “An EOQ model 
     for items with finite rate of production and variable 
     rate of deterioration”, Journal of the Operational 
     Research Society of India, 23, 175-181.
[16] Donaldson, W. A.（1977）, “Inventory Replenishment 
     Policy for a Linear Trend in Demand－An Analysis 
     Solution”, Operational Research Quarterly, 28, 663- 
     670.
[17] Ghare,  P. M. and  Schrder,  G. H.（1963）, “A    
     model for exponentially decaying inventory system”, 
     International Journal of Industrial Engineering,14
     (5),238-243.
[18] Giri, B. C., Goswami, A. and Chaudhuri, K. S.
    （1996）, “An EOQ model for deteriorating items with 
     time varying demand and costs”, Journal of the 
      Operational Research Society, 47, 1398-1405.
[19] Goel, V. P. and Aggarwal, S. P.（1981）, “Order 
     level inventory system with power demand pattern for 
     deteriorating items”, Proceedings all India Seminar 
     on Operational Research and Decision making, 
     University of Delhi, Delhi-110007.
[20] Goswami, A., and Chaudhuri, K. S.（1991）, “An EOQ 
     model for deteriorating items with shortages and 
     linear trend in demand”, Journal of the Operational 
     Research Society, 42, 1105-1110.
[21] Goswami, A., and Chaudhuri, K. S.（1993）, “An 
     deterministic inventory model for deteriorating items 
     with stock-dependent demand rate”, International 
     Journal of Production Economics
     , 32, 291-299.
[22] Gupta, R. and Vrat, P.（1986）, “Inventory models 
     for stock-dependent consumption rate”, Opsearch, 23, 
     19-24.
[23] Hariga, M. A.（1995）, “An EOQ model for  
     deteriorating items with shortages and time-varying 
     demand”, Journal of the Operational Research 
     Society, 46, 398-404.
[24] Hjorth, U.（1980）, “ A reliability distribution 
     with increasing decreasing ,constant and bathtub-
     shaped failure rates”,Technometrics, 22, 99-107.
[25] Kishan, H. and Mishra, P. N.（1995）, “An inventory 
     model with exponential demand and constant 
     deterioration with shortages”,Indian Journal of 
     Mathematics, 37, 275-279.
[26] Levin, R. I., Mclaughlin, C. P., Lamone R. P. and   
     Kothas, J. F.（1972）, “Production/Operations 
     Management: Contemporary Policy for Managing 
     Operating Systems”, McGraw-Hill, New York.
[27] Liao, H. C. and Su, C. T. （1998）, “Inventory 
     Policies for Weibull Distribution Items When a Decay 
     in Payments is Permissible”, Proceedings of 
     Quantitative Management Techniques and Applications 
     in Taiwan Conference,80-85.
[28] Mandal , B. and Pal, A. K.（1998）, “Order level 
     inventory system with ramp type demand rate for 
     deteriorating items”, Journal of Interdisciplinary 
     Mathematics, 1, 49-66.
[29] Mandal , B. N. and Phaujdar, S.（1989）, “An 
     inventory model for deteriorating items and stock-
     dependent consumption rate”, Journal of the 
     Operational Research Society, 40, 483-488.
[30] Mudholkar, G. S. and Srivastava, D. K.（1993）, 
    “Exponentiated Weibull family for analyzing bathtub 
     failure-rate data”,IEEE Transactions on Reliability 
     ,42, 299-302.
[31] Ouyang, L. Y., Hsieh, T. P., Dye C. Y. and Chang, H. 
     C.（2003）, “An inventory model for deteriorating 
     items with stock-dependent demand under the 
     conditions of inflation and time-value of money”, 
     The Engineering Economist, 48, 52-68.
[32] Padmanabhan, G. and Vrat, P.（1990）, “Inventory 
     model with a mixture of back orders and lot sales”, 
     International Journal of Systems Science, 21, 1721-
     1726.
[33] Padmanabhan, G. and Vrat, P.（1995）, “EOQ models 
     for perishable items under stock dependent selling 
     rate”, European Journal of Operational Research, 86, 
     281-292.
[34] Philip, G. C.（1974）, ”A generalized EOQ model for 
     items with Weibull distribution deterioration”, AIIE 
     Transactions, 6, 159-162.
[35] Raffat, F.（1988）, “An inventory model with a 
     monotonically increasing deterioration rate
     function”, American Institute for Decision Sciences 
     Seventeenth Annual Meeting－Western Regional 
     Conference－Proceedings and Abstracts, 495-505.
[36] Silver, E. A. and Peterson, R.（1982）, Decision 
     Systems for Inventory Management and Production 
     Planning, Wiley,New York.
[37] Tadikamalla, P. R.（1978）, “An EOQ Inventory Model 
     for Items with Gamma Distributed Deterioration”, 
     AIIE Transactions, 10,100-103.	
[38] Teng, J. T., Ouyang, L. Y., and Cheng, M. C.（2005）, 
    “An EOQ Model for Deteriorating Items with Power-Form 
     Stock-Dependent Demand”, Information and Management 
     Sciences, 16, 1-16.
[39] Wang, F. K.（2000）, “A new model with bathtub-
     shaped failure rate using an additive Burr XII 
     distribution”, Reliability   Engineering and System 
     Safety, 70, 305-312.
[40] Wu, K. S.（1998）, “An ordering Policy for Items 
     with Weibull Distribution Deterioration under 
     Permissible Delay in Payments”, Tamsui Oxford 
     Journal of Management Science, 14, 39-54.
[41] Wu, K. S.（2002）, “EOQ inventory model for items 
     with Weibull distribution deterioration, time-varying 
     demand and partial backlogging”, International 
     Journal of Systems Science, 33, 323-329.
[42] Xie, M. and Lai, C. D.（1996）, “Reliability 
     analysis using an additive Weibull model with bathtub-
     shaped failure rate function”, Reliability 
     Engineering and System Safety, 52, 87-93.
[43] Xie, M., Tang, Y. and Goh, T. N.（2002）, “A 
     modified Weibull extension with bathtub-shaped 
     failure rate function”, Reliability Engineering and 
     System Safety, 76, 279-285.