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標題: 以粒子群優化演算法計算具有阻礙,放棄及服務壓力係數之暖備機器修理問題的最低成本
Apply particle swarm optimization for the warm-standby M/M/R machine repair problem with balking, reneging, and service pressure coefficient
作者: 郭瓊穗
關鍵字: Particle Swarm Optimization;粒子群優化演算法;balking;cost;reneging;service pressure coefficient;machine repair problem;阻礙;成本;放棄;服務壓力係數;機器修理問題
出版社: 應用數學系所
引用: [1]. 商紓瑋.具有阻礙,放棄及服務壓力係數之M/M/R暖備機器修理問題.國立中興大學統計學研究所碩士學位論文(2010)。 [2]. 藍俊雄和郭美貝.考慮服務壓力係數下M/M/s/k等候模式之建構與推導.管理科學研究 第一屆管理與決策2005年學術研討會特刊:151-159。 [3]. 紀梓民.粒子群演算法之改善及探討.國立中興大學機械工程研究所碩士論文(2008) [4]. F. Benson and D.R. Cox. The productivity of machine requiring attention at random intervals. Journal of the Royal Statistical Society 13 (1951): 65-82。 [5]. Jr. C.J. Ancker and A.V. Gafarian. Some queueing problems with balking and reneging: II. Operations Research 11 (1963): 928-937。 [6]. B.D. Sivazlian and K.-H. Wang. Economic analysis of the M/M/R machine repair problem with warm standbys. Microelectronics and Reliability 29 (1989): 25-35。 [7]. M.O. Abou-El-Ata and A.I. Shawky. The single-server Markovian overflow queue with balking, reneging and an additional server for longer queues. Microelectronics and Reliability 32 (1992): 1389-1394。 [8]. M.O. Abou-El-Ata and A.M.A. Hariri. The M/M/c/N queue with balking and reneging. Computers and Operations Research 19 (1992): 713-716。 [9]. J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” Proc. IEEE Int. Conf. Neural Networks, pp. 1942-1948, 1995。 [10]. A.I. Shawky. The single-server machine interference model with balking, reneging, and an additional server for longer queues. Microelectronics and Reliability 37 (1997): 355-357。 [11]. J.Kennedy and R.C.Eberhart. A discreet binary version of the particle swarm algorithm,1997。 [12]. Clerc M. The swarm and the queen: Towards a determininistic and adaptive particle swarm optimization. In Congress on Evolutionary Computation(CEC99),pages 1951-1957,1999。 [13]. J.-C. Ke and K.-H. Wang. Cost analysis of the M/M/R machine repair problem with balking, reneging, and server breakdowns. Journal of the Operational Research Society 50 (1999): 275-282。 [14]. F.S. Hiller and G.J. Lieberman. Introduction to Operations Research, 6rd edn. McGraw Hill Higher Education (2001)。 [15]. J.Kennedy and R.Eberhart. Swarm Intelligence, Morgan Kaufmann Publishers,Inc.,San Francisco,CA,2001. [16]. R. C. Eberhart and Y. Shi, “Particle swarm optimization:development, applications and resources,” Evolutionary Computation,2001. Proceedings of the 2001 Conqress on Vol. 1, pp. 81-86, 2001。 [17]. K.-H. Wang and Y.-Ch. Chang. Cost analysis of a finite M/M/R queueing system with balking, reneging, and server breakdowns. Mathematical Methods of Operations Research 56 (2002): 169-180。 [18]. K.-H. Wang and J.-C. Ke. The reliability analysis of balking and reneging in a repairable system. Quality and Reliability Engineering International 18 (2002): 467-478。 [19]. V. Gerhard and S. S. Jaroslaw, “Particle swarm optimization,” Source:Collection of Technical Papers-AIAA/ASME/ASCE/AHS/ASCStructures ,Structural Dynamics and Materials Conference, Vol. 1,.282-290, 2002 [20]. K.-H. Wang and J.-C. Ke. Probabilistic analysis of a repairable system with warm standbys plus balking and reneging. Applied Mathematical Modelling 27 (2003): 327-336。 [21]. S. Drekic and D.G. Woolford. A preemptive priority queue with balking. European Journal of Operational Research 164 (2005): 387–401。 [22]. D. Yue, W. Yue, and Y. Sun. Performance analysis of an M/M/c/N queueing system with balking, reneging and synchronous vacations of partial servers. International Symposium on OR and Iits Applications (2006): 128–143。 [23]. J.-C. Ke. Operating characteristic analysis on the M[x]/G/1 system with a variant vacation policy and balking. Applied Mathematical Modelling 31 (2007): 1321–1337。 [24]. K.-H. Wang, J.-B. Ke, and J.-C. Ke. Profit analysis of the M/M/R machine repair problem with balking, reneging, and standby switching failures. Computers and Operations Research 34 (2007): 835-847。 [25]. M. Lozano and P. Moreno. A discrete time single-server queue with balking: economic applications. Applied Economics 40 (2008): 735–748。
排隊是現代文明社會的表徵,越是文明的社會這種現象就越普遍;接受服務的顧客不喜歡在隊伍中等待,提供服務的管理者也不喜歡因排隊所增加的成本。排隊理論於焉誕生。我們考慮具有阻礙、放棄和服務壓力係數的M/M/R暖備機器修理問題,先建立穩態方程式,再利用生死過程(birth-and-death process)計算穩態機率;並撰寫Maple程式計算系統效能測度。針對成本函數 F(R,S,μ,r),我們利用直接搜尋法,先固定μ和r 的值,尋找最佳修理人員個數(R)和暖備機器個數(S)的最佳值(R*,S*),最後再利用準牛頓法,以固定的(R*,S*),尋找成本之最佳值 F(R*,S*,μ*,r*)。
近年來新興的人工智慧優化演算法-粒子群優化演算法(Particle Swarm Optimization, PSO),是以亂數族群搜尋為基礎的隨機搜尋演算法。本文中,我們以Maple實作PSO演算法,透過同時多點搜尋以迭代方式尋找排隊系統中的最低成本。以數個例子比較PSO和準牛頓法的最小成本,發現PSO可以找到比準牛頓法更低的成本,且改善率高達30%以上。

Queuing is one of the features of modern civilized society. The more civilized the society, the more common the phenomenon is. Generally, guests do not like to wait, neither do the service providers like the resulting increase cost. Therefore, the Queueing Theorem has come to existence. We consider M/M/R warm standby machine repair problem with balking, reneging, and service pressure coefficient. First, we establish the steady-state equation, and then apply the birth-and-death process to calculate the steady-state probability, following by coding Maple programs to execute the system performance measurement. Focusing on the cost function F(R,S,μ,r), we use the direct search method to find the optimum value(R*,S*), with the optimum number of repairmen R and standby machines S , having μ and r fixed. Next, we use the Quasi-Newton method to calculate the optimum cost F(R*,S*,μ*,r*) with fixed(R*,S*).
An artificial intelligence method of optimization that rises in recent years, the Particle Swarm Optimization, PSO, is a random search algorithm with a base of random population. In this paper, we use Maple to operate the PSO optimization, finding the minimal cost in the queueing system, with the aid of simultaneous multiple particle searching and the iterative method. We apply several examples to compare the minimal cost by the PSO and the Quasi-Newton method respectively, suggesting that the PSO can offer lower cost than the Quasi-Newton method does, with the rate of improvement as high as more than 30%.
其他識別: U0005-3006201217560500
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