A Genetic Algorithm for Finding Good Balanced Sequences in a Customer Assignment Problem with no State Information

In this paper, we study the control problem of optimal assignment of tasks to servers in a multi-server queue with inhomogeneous servers. In order to improve the performance of the system, we use a periodic deterministic sequence of job assignments to servers called a billiard sequence. We then use a genetic algorithm (GA) for computing a near-optimal billiard sequence. By means of a recent result obtained in the area of ordinal optimization, we show that the solution found by the GA belongs to the top 1% of possible choices for such a billiard sequence. As illustrated by numerical examples, not only is the performance under a billiard sequence better than that of the corresponding randomized policy, the optimal billiard sequence even outperforms the billiard implementation of the optimal randomized policy. The framework we introduce in this paper is suitable for general optimization problems over (periodic) deterministic decision sequences. Given the significant performance improvement that a switch from randomized policies to billiard sequences yields, this framework is of importance in practical applications. Finally, we show that constrained or multi-objective optimization can be dealt with in our framework as well.

[1]  Ger Koole,et al.  Analysis of a Customer Assignment Model with No State Information , 1994, Probability in the Engineering and Informational Sciences.

[2]  Yu-Chi Ho,et al.  Ordinal Optimization: Soft Computing for Hard Problems (International Series on Discrete Event Dynamic Systems) , 2007 .

[3]  Xin Guo,et al.  Optimal probabilistic routing in distributed parallel queues , 2004, PERV.

[4]  G. A. Hedlund,et al.  Symbolic Dynamics II. Sturmian Trajectories , 1940 .

[5]  Eitan Altman,et al.  Multimodularity, Convexity, and Optimization Properties , 2000, Math. Oper. Res..

[6]  Qing-Shan Jia,et al.  Quantifying Heuristics in the Ordinal Optimization Framework , 2010, Discret. Event Dyn. Syst..

[7]  Pierre Arnoux,et al.  Complexity of sequences defined by billiard in the cube , 1994 .

[8]  Dorothea Heiss-Czedik,et al.  An Introduction to Genetic Algorithms. , 1997, Artificial Life.

[9]  C. Hicks,et al.  An ordinal optimization based evolution strategy to schedule complex make-to-order products , 2006 .

[10]  E. Altman Constrained Markov Decision Processes , 1999 .

[11]  Eugene A. Feinberg,et al.  Handbook of Markov Decision Processes , 2002 .

[12]  A. E. Eiben,et al.  Introduction to Evolutionary Computing , 2003, Natural Computing Series.

[13]  Gideon Weiss,et al.  The Stochastic Optimality of SEPT in Parallel Machine Scheduling , 1994, Probability in the Engineering and Informational Sciences.

[14]  Eitan Altman,et al.  Time-Sharing Policies for Controlled Markov Chains , 1993, Oper. Res..

[15]  Eitan Altman,et al.  Balanced sequences and optimal routing , 2000, JACM.

[16]  Naoto Miyoshi,et al.  m-Balanced words: A generalization of balanced words , 2004, Theor. Comput. Sci..

[17]  Bruce E. Hajek,et al.  Extremal Splittings of Point Processes , 1985, Math. Oper. Res..

[18]  Kenli Li,et al.  A DAG scheduling scheme on heterogeneous computing systems using double molecular structure-based chemical reaction optimization , 2013, J. Parallel Distributed Comput..

[19]  Arie Hordijk,et al.  On the Average Waiting Time for Regular Routing to Deterministic Queues , 2005, Math. Oper. Res..

[20]  Kenli Li,et al.  A genetic algorithm for task scheduling on heterogeneous computing systems using multiple priority queues , 2014, Inf. Sci..

[21]  Eitan Altman,et al.  Regular ordering and applications in control policies , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[22]  Ger Koole,et al.  On the Assignment of Customers to Parallel Queues , 1992, Probability in the Engineering and Informational Sciences.

[23]  D. E. Goldberg,et al.  Genetic Algorithms in Search , 1989 .

[24]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .

[25]  Eitan Altman,et al.  Discrete-Event Control of Stochastic Networks - Multimodularity and Regularity , 2004, Lecture notes in mathematics.

[26]  Goldberg,et al.  Genetic algorithms , 1993, Robust Control Systems with Genetic Algorithms.

[27]  Martin L. Puterman,et al.  Markov Decision Processes: Discrete Stochastic Dynamic Programming , 1994 .

[28]  Arie Hordijk,et al.  Periodic routing to parallel queues and billiard sequences , 2004, Math. Methods Oper. Res..

[29]  Lawrence. Davis,et al.  Handbook Of Genetic Algorithms , 1990 .

[30]  Sandjai Bhulai,et al.  Optimal balanced control for call centers , 2012, Ann. Oper. Res..