Optimal Computing Cost Budget Allocation For Solving Buffer Allocation Problem Mahmoud H. Alrefaei * and Ahmad A. Abubaker † * Department of Mathematics and physics Qatar University P.O. Box 2713, Doha, QATAR E-mail: malrefaei@qu.edu.qa † College of Computer and Information Science Al-Imam Muhammad Ibn Saud Islamic University P.O.Box 84880, Alriyadh 11681, Saudia Arabia E-mail: a abubaker2000@yahoo.com In this paper, we study the problem of selecting a sys- tem that has the best performance or selecting one of the best systems when the objective function is the ex- pected performance of a complex stochastic system. The focus here is on a huge size but finite feasible so- lution set, that is not necessarily well structured. We consider the use of the ordinal optimization technique that concerns with the order of the designs in the state space rather than estimating each design alternative accurately. We also assume that the computing cost is limited. We discuss how to allocate the available computational resources on the alternative designs to maximize the probability of making a correct selection (selecting the best or one of the best systems). Keywords: Simulation Optimization, Optimal Comput- ing Budget Allocation, Simulated Annealing, Ordinal Op- timization. 1. Introduction We study the discrete stochastic optimization problems where the objective function is the expected performance of a complex stochastic system. The problem can be ex- pressed as follows: min θ ∈Θ J (θ ) ≡ E [L(θ , ξ θ )], . . . . . . . . . (1) where L is a function of θ and ξ θ , ξ θ is a random variable depends on the design θ , and Θ is the set of all potential solution candidates or the search space, is a huge finite set and not necessarily well structured. Θ is said to be well structured if for any θ ∈ Θ, 5 a set N(θ ), contains all neighbors of θ and all alternatives can be connected through a sequence of neighbors. This type of problems can be found in many real life complex stochastic sys- tems such as communication, manufacturing, and other systems. When the sample space Θ is small, then numeral meth- ods can be used, where the objective function values in (1) are estimated using simulation. Random samples of stochastic process ξ 1 θ , ξ 2 θ ,..., ξ N θ of the random variable ξ θ are generated, and E [L(θ , ξ )] is estimated as follows: J (θ )= E [L(θ , ξ θ )] ≈ 1 N N ∑ i=1 L(θ , ξ i θ ) . . . . (2) where ξ i θ represents the ith sample of design θ , and N is the number of replications (i.e., the number of simula- tion runs). Then use any deterministic optimization tech- nique to solve the alternative problem by replacing J (θ ) by J (θ ). This method requires large computational time to do simulation runs, especially when the set of feasible solutions is large. Therefore, this method is considered infeasible. Several techniques are proposed to tackle this problem. In this paper, we focus on one of these tech- niques, particularly the ordinal optimization technique and compare its performance with the simulated anneal- ing algorithm in a simple practical example. The rest of this paper is organized as follows; in Section 2, we preview a simulated annealing algorithm, in Sec- tion 3, we review the ordinal optimization approach and in Section 4, we present the optimal computing cost bud- get allocation technique. In Section 5, we present numeri- cal results obtained from applying the proposed approach to a buffer allocation problem for homogeneous asymp- totically reliable serial production lines. We compare the results and computational time to results obtained by ap- plying the most recent simulated annealing algorithms to the same problem. Finally, in Section 6, we end by some concluding remarks. 2. Simulated annealing with the standard clock tech- nique Some researchers have studied the use of simulated an- nealing to solve discrete simulation optimization prob- lems. The simulated annealing algorithm has originally been proposed by Kirkpatrick et.al. [9] who use the idea of simulated annealing to solve deterministic optimiza- tion problems. Gelfand and Mitter [6] and Gutjahr and Pflug [7] have proposed and analyzed the simulated an- nealing for solving stochastic optimization problems. Al-