本研究針對遇缺補貨型存貨問題發展馬可夫期望成本模式與求解方法，其中考慮再訂購點與固定訂購數量存貨政策，即如當期期末存貨量小於或等於再訂購點時，即訂購固定數量，並於固定前置時間後到達，反之則不訂購，當存貨量小於零時，即發生缺貨，須待補貨後才能優先補足，此模式考慮離散時間，故與古典連續時間之再訂購點與訂購數量存貨模式不同。本研究建立之期望成本模式，考慮訂購、存貨，以及遇缺補貨之缺貨成本，並發展有效率之求解方法，可快速求得最佳或近似最佳再訂購點與訂購數量。
在本論文所建立之馬可夫存貨模式中，考慮隨機需求與不同補貨前置時間，首先根據馬可夫性質建立存貨轉移機率矩陣與極限機率分配，再利用極限機率發展單位時間期望總成本模式。而後針對成本模式進行分析，當在期末訂貨，可於下一期期初到貨時，發現在兩存貨政策中如訂購數量固定，只有再訂購點變動時，兩政策之極限機率俱對等關係，且當訂購數量固定時，其單位時間期望總成本為再訂購點之凸函數。為確認模式之精確性，並以模擬實驗進行分析與比較。
本研究發展兩個適用於求解不同前置時間存貨模式之啟發式演算法，對於期末訂貨，可於下一期期初到貨之問題，主要依據所發現之理論性質首先搜尋再訂購點，再以黃金分割法求得最佳訂購數量，直到期望成本不再改善為止。當在期末訂貨，於下一期之後才能到貨之問題中，則皆以黃金分割法為基礎，進行再訂購點與訂購數量之求解。在總共192組演算實驗中，可求得最佳決策者約佔81.7%，與局部窮舉法比較，其整體平均成本偏差小於0.7%，在求解效率上，兩演算法之平均搜尋次數約為2至4次，顯示演算法之求解品質良好且俱效率。

A markovian (s, Q) inventory problem with backorder is investigated in this thesis, where demand is a random variable over the infinite periods. When the inventory position falls below the reorder point at any period, a fixed quantity of Q is ordered and the replenishment arrives after a constant leadtime. Shortage is backordered and is filled upon restocking. The reorder point and fixed quantity are to be determined to minimize the expected total relevant cost per unit time, which consists of ordering cost, inventory carrying cost and shortage cost.
The (s, Q) inventory problem is formulated as a markovian model. For the basic model where an order placed at the end of one period will arrive at the beginning of next period, an exact analysis is performed. Two properties have been established, and are used to develop an efficient heuristic for finding the optimal (s, Q). An approximate analysis is then performed for the extended model with leadtime greater than one. A 2-D search procedure is developed to find the near optimal (s, Q).
Computational results with 192 instances have shown that the probability of finding optimum policy is approximately 81.7%, and the average deviation of expected cost from those in the local enumeration is less than 0.7%. The average running times for solving an instance are approximately 1 and 28 seconds for the basic and extended models, respectively. It also indicates that the heuristics is robust and efficient.

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