Simulation optimization using the Particle Swarm Optimization with optimal computing budget allocation

Si Zhang, Pan Chen, Loo Hay Lee, Chew Ek Peng, Chun-Hung Chen
2011 Proceedings of the 2011 Winter Simulation Conference (WSC)  
Simulation has been applied in many optimization problems to evaluate their solutions' performance under stochastic environment. For many approaches solving this kind of simulation optimization problems, most of the attention is on the searching mechanism. The computing efficiency problems are seldom considered and computing replications are usually equally allocated to solutions. In this paper, we integrate the notion of optimal computing budget allocation (OCBA) into a simulation optimization
more » ... approach, Particle Swarm Optimization (PSO), to improve the efficiency of PSO. The computing budget allocation models for two versions of PSO are built and two allocation rules PSOs_OCBA and PSObw_OCBA are derived by some approximations. The numerical result shows the computational efficiency of PSO can be improved by applying these two allocation rules. 1 Zhang, Chen, Lee, Chew, and Chen The origin of particle swarm optimization (PSO) is from the computer animation requirement of forming what appeared to be a fuzzy object. Kennedy and Eberhart (1995) develop the basic model for PSO and lead to the application of PSO in finding the optimal solution of mathematical functions. In PSO, each time certain number of solutions in the search space will be selected as particles to form a swarm. Each particle in the swarm will move through a search space according to its velocity value based on the location information of both the best solution that it has found individually (personal best) and the best solution that is found by any of the particles that this particle can communicate with (global best). To avoid the case of rapid convergence to local optimal and the case of finding the global optimal but with very slow convergence rate, Shi and Eberhart (1998) incorporate an inertia weight into the velocity update equation to improve the basic PSO model. Based on the same consideration, Clerc and Kennedy (2002) introduce another factor, named as constriction factor, into the velocity update equation to build a generalized PSO model. By incorporating the idea of cluster analysis, Kennedy (2000) modifies the original PSO and develops a new version of PSO algorithm by using cluster centers as personal best of particles. For different specific simulation optimization problems, a lot of other versions of PSO have been developed in recent years. Banks, Anyakoha (2007, 2008) give a comprehensive review of these developments. Bratton and Kennedy (2007) take these recent developments into account and define a standard for PSO. In all of these PSO versions, a comparison among all candidate solutions within each iteration is required to update particles' locations. And the computing budget is usually equally allocated to these candidate solutions under stochastic environment. Because the number of particles at a swarm is limited, some approaches in ranking and selection procedures can be applied into the comparison process to efficiently allocate computing replications to these competing candidate solutions. Among these approaches, the optimal computing budget allocation (OCBA) procedures developed by Chen et al. (2000) aims at maximizing the probability of correctly selecting the best design(s) from finite number of designs under limited computing budget constraint. It has shown great potential in improving simulation efficiency for tackling simulation optimization problems. Chen et al. (2008) show numerical examples about the performance of the algorithm combining OCBA-m with Cross-Entropy (CE). The theoretical part about the integration of OCBA with CE is then further analyzed in He et al. (2010) . Chew et al. (2009) integrate MOCBA with Nested Partition (NP) to handle multi-objective inventory policies problems and Lee, Wong, and Jaruphongsa (2009) integrate MOCBA with GA to solve an aircraft spare part allocation problem. In all these papers, the numerical results demonstrate the significant improvements gained by integrating OCBA into these simulation optimization approaches. The application of OCBA into PSO is considered in Pan, Wang, and Liu (2006) , where they do not analyze the PSO from the OCBA perspective but just directly apply OCBA allocation rule from Chen et al. (2000) to select the best particle at a swarm. In this paper, we integrate OCBA into two versions of PSO and model the computing budget allocation problems of PSO by maximizing the convergence rate of the probability of incorrect selection. The conditions for the asymptotical optimal allocation rules for the standard PSO and PSObw are derived. Under some assumptions, we get the optimal allocation rules, named as PSOs_OCBA and PSObw_OCBA, which are closed-form and easy to implement. Numerical testing indicates that the resulting integrated procedure can lead to computational efficiency gains for both the standard PSO and PSObw. We reiterate that our objective is not to find the best PSO algorithm or compare the standard PSO with PSObw, but rather to demonstrate that an intelligent control of simulation budget allocation can improve the computational efficiency of PSO. The framework in this paper can also be flexibly applied to other versions of PSO or other simulation optimization approaches to seek the computational efficiency improvement. The rest of this paper is organized as follows. In section 2, we introduce the simulation optimization problem setting and build computing budget allocation models for both the standard PSO and the PSObw from a large deviation perspective. Section 3 derives the asymptotically optimal simulation allocation rules to minimize the probability of incorrect selection. In section 4, we show two numerical experiments to compare the performance of PSOs_OCBA and PSObw_OCBA with the equal allocation rule PSOs_EA and PSObw_EA. Section 5 concludes the whole paper.
doi:10.1109/wsc.2011.6148117 dblp:conf/wsc/ZhangCLCC11 fatcat:hphziyvmy5atxnhr4dmsvlhhw4