Rule sets based bilevel decision model and algorithm Zheng Zheng a,b , Jie Lu a, * , Guangquan Zhang a , Qing He c a Faculty of Information Technology, University of Technology, P.O. Box 123, Broadway, Sydney, NSW 2007, Australia b Beijing University of Aeronautics and Astronautics, Beijing, China c Key Laboratory of Intelligent Information Processing, Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China Abstract Bilevel decision addresses the problem in which two levels of decision makers, each tries to optimize their individual objectives under certain constraints, act and react in an uncooperative, sequential manner. As bilevel decision making often involves many uncertain fac- tors in real world problems, it is hard to formulate the objective functions and constraints of the leader and the follower in modelling a real bilevel decision problem. This study explores a new approach that uses rule sets to formulate a bilevel decision problem. It first devel- ops related theories to prove the feasibility to model a bilevel decision problem by rule sets. It then proposes an algorithm to describe the modelling process. A case study is discussed to illustrate the functions and effectiveness of the proposed rule sets based bilevel decision modelling algorithm. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Rule sets; Bilevel programming; Uncertainty bilevel decision making; Decision model; Rough sets 1. Introduction Organizational decision making often involves two lev- els of decision makers. In general, the decision maker at the upper level known as the leader, will influence or induce the behaviour of the decision maker at the lower level, called follower, but not completely control his/her action. In addition, the follower gains his/her objective under a given region, although his/her decision is in a subordinate position. In such a bilevel decision situation, decision maker at each level has individual payoff function, and the upper level the decision maker is at, the more important and global his/her decision is. Therefore, a bilevel decision making model intends to reach certain goals, which reflect the leader’s objective and also consider the reaction of the follower on the final decision. Bilevel decision problems were introduced by Von Stac- kelberg in the context of unbalanced economic markets (Stackelberg, 1952). After that moment this field has obtained a rapid development and intensive investigation in both theories and applications (Bard, 1998; Bard & Falk, 1982; Bard & Moore, 1992; Bialas & Karwan, 1982; Candler & Norton, 1977; Chen & Gruz, 1972; Dempe, 2002; Shi, Zhang, & Lu, 2005; Lu, Shi, & Zhang, 2006). Much efforts have been done on the development of both linear or nonlinear modeling and solution methods, such as the Kth-Best approach (Bialas & Karwan, 1984; Candler & Townsley, 1982) and Kuhn–Tucker approach (Bard & Falk, 1982; Bialas & Karwan, 1982; Hansen, Jaumard, & Savard, 1992; Shi, Lu, & Zhang, 2005) for solving linear bilevel programming problems, and Penalty function approach (Aiyoshi & Shimizu, 1982; White & Anandalingam, 1993) or stability based approach (Liang & Sheng, 1992) for solving nonlinear bilevel programming problems. Contributions to this field have been delivered by mathematicians, economists, engineers and many other researchers and developers. However, we have found that this interest often stems from the inherent complexity and consequent challenge of the underlying mathematics, as well as the uncertainty existed in a bilevel decision problem which causes the difficulties to build a bilevel decision model and the applicability of the bilevel decision model to a real-world situation. 0957-4174/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2007.09.011 * Corresponding author. Tel.: +61 2 95141838; fax: +61 2 95144535. E-mail address: jielu@it.uts.edu.au (J. Lu). www.elsevier.com/locate/eswa Available online at www.sciencedirect.com Expert Systems with Applications 36 (2009) 18–26 Expert Systems with Applications