M/M/S機器修理系統含有同步休假及多個門檻之最佳管理

Abstract

工業工程與管理學系 計畫名稱：M/M/S 機器修理系統含有同步休假及多個門檻策略之最佳管理 研究者：彭文理 經費來源：行政院國家科學委員會 第一年 計畫名稱：探討M/M/S 機器修理系統含有同步休假及多個門檻策略之操作特性 分析 考慮一機器修理系統，其服務者會有群體休假或暫時離開系統的情況， 即每次休假的服務者數量並不相等，且並不限定為一個。我們對於M/M/S 機 器修理系統含有同步休假及多個門檻策略，使用矩陣技巧推導出其穩態機率 向量間的遞迴關係式並整理為明確而簡易的求解法。於求得系統穩態機率解 之後，我們推導出重要的系統特徵之顯示公式，如系統期望機器數；閒置、 忙碌、休假的服務者期望數等。並用我們所獲得的結果與過去學者的研究成 果做比較，以驗證結果之正確性與一般性。 第二年 計畫名稱：探討M/M/S 機器修理系統含有同步休假及多個門檻策略與服務者 會故障之操作特性分析 考量到現實情況中，服務者除了有群體 (同步) 休假之行為外，還常有故障 或失效的突發情況發生，必需維修或休養一段時間之後才能回到系統繼續提供服 務。對於這種生產或製造系統，一般管理者會備有備用機台，以應付生產機具當 機的情況；在此同樣考量備用機台也有故障的可能性。我們利用隨機過程之技 巧分析含有多門檻同步休假與服務者會故障之 M/M/S 機器修理系統。嘗試 利用矩陣手法處理穩態方程式、得到各穩態機率向量間的遞迴關係式，並提 供一矩陣的遞迴演算法用於系統穩態機率向量解。再以穩態點的機率表示重 要的系統特徵之顯示公式，如等候線期望機器數；閒置、忙碌、休假、故障 的服務者期望數等。同時和第一年的計畫成果與過去學者的研究成果做比 較，以驗證結果之正確性與一般性。 第三年 計畫名稱：探討M/M/S 機器修理系統含有同步休假及多個門檻策略與服務者 會故障之最佳控制 機器修理系統與一般排隊系統之假設不同，它假設顧客到達之來源為一有限 集合。儘管為有限顧客來源，其服務品質與系統使用率亦不能太差，所以平均等 候線長度與機台使用率等都可能是管理者所在意的重要系統特徵。如何利用排隊 理論的方法來研究含有多門檻同步休假策略與服務者會故障M/M/S 機器修理系 統之最佳管理問題，建立成本模式來決定最低成本時之最佳系統參數 (含服務者 個數、休假門檻、與休假服務者數等) 設定。我們將撰寫不同階段之Maple 電腦 程式，第一階段針對離散型變數最佳化問題，使用直接搜尋法解出其最佳解，並 於最佳設定後針對各服務者之服務率進行第二階段的最佳化。第二階段面對連續 型變數最佳化問題將使用Quasi-Newton 方法搜尋其最佳解。

本計畫預定執行三年，英文摘要分別概述如下： 第一年：Operating Characteristic Analysis of the M/M/S Machine Repair Problem with Multi-threshold Synchronous Vacation Policy 英文摘要 Considering a machine repair system, for some reason, partial servers may leave the system synchronously. That is, the number of vacation servers doesn’t be limited to one. We consider a multi-server machine repair system with multi-threshold synchronous vacation policy. The useful matrix-analytic approach is employed for obtaining the steady-state probabilities. The expressions for some system performance measures are derived such as the expected number of failed machines in the system and the expected number of idle, busy, vacation servers are evaluated by using the steady-state probability. Finally, some remarks and probability interpretations are presented. 第二年：Operating Characteristic Analysis of the M/M/S Machine Repair Problem with Multi-threshold Synchronous Vacation Policy and Server subject to Breakdown 英文摘要 In real situation, the servers may take a synchronous vacation; in addition, they may be unreliable and fail anytime. When a server breakdown, it is sent to the maintain department and is repaired immediately. Some stochastic process techniques are employed to analysis the M/M/S machine repair system with multi-threshold synchronous vacation policy and server breakdown. We deal with the steady-state equations by matrix-analytic method. A matrix-analytic recursive algorithm is provided to obtain the stationary probabilities. The explicit expressions of some important system performance measures are developed such as the expected length of waiting line, and the idle, busy, vacation, and breakdown servers. Finally, some remarks and probability interpretations are presented. 第三年：Optimal Control of the M/M/S Machine Repair Problem with Multi-threshold Synchronous Vacation Policy and Server subject to Breakdown 英文摘要 Machine repair system assumes that the source of arriving is finite, therefore, the capacity of system is assumed finite. Although the arriving is finite, the manger also cares about the length of the waiting line, the service quality, the system availability, and some important system performance measures. How to decide the optimal system parameter settings (including the number of servers and synchronous vacation policy) to achieve the minimum total expected cost per unit time is the problem that the decision maker should solve. In this project, we utilize queueing theory to study the optimization control investigation of a multi-server machine repair system with multi-threshold synchronous vacation policy and server breakdown. A cost model is derived for this machine repair system. The direct search approach and Quasi-Newton method are implemented to deal with optimization problem of the discrete and continuous variables, respectively. We use an efficient Maple computer program to calculate the optimal system parameters that correspond minimum cost.