Abstract
This paper proposes a new methodology for solving the interval bilevel linear programming problem in which all coefficients of both objective functions and constraints are considered as interval numbers. In order to keep as much uncertainty of the original constraint region as possible, the original problem is first converted into an interval bilevel programming problem with interval coefficients in both objective functions only through normal variation of interval number and chance-constrained programming. With the consideration of different preferences of different decision makers, the concept of the preference level that the interval objective function is preferred to a target interval is defined based on the preference-based index. Then a preference-based deterministic bilevel programming problem is constructed in terms of the preference level and the order relation preceq_{mw}. Furthermore, the concept of a preference δ-optimal solution is given. Subsequently, the constructed deterministic nonlinear bilevel problem is solved with the help of estimation of distribution algorithm. Finally, several numerical examples are provided to demonstrate the effectiveness of the proposed approach.
Highlights
The bilevel programming problem is a hierarchical optimization problem involving decision processes with two decision makers, the so-called leader or upper level decision maker and the so-called follower or lower level decision maker
From a point of decision makers’ preferences, our objective is to propose an alternative way based on preference-based index to deal with the interval bilevel linear programming problem in which all coefficients in both objective functions and constraints are intervals
4 Solution methodology we develop a novel approach on the basis of preference-based index to convert and cope with the interval bilevel linear programming problem ( )
Summary
The bilevel programming problem is a hierarchical optimization problem involving decision processes with two decision makers, the so-called leader or upper level decision maker and the so-called follower or lower level decision maker. In such a hierarchical decision framework, the leader first specifies a strategy, and the follower chooses a strategy in view of the leader’s decision. In the past few decades, the bilevel programming problem has widely applied in numerous areas including transport network design [ , ], price control [ ], principal-agent problems [ , ], supply chain management [ , ], engineering design [ , ], electricity markets [ ]. Many researchers have worked on this topic, like Dempe [ ], Colson et al [ ], Kalashnikov et al [ ], Bard [ ], Dempe [ ], Dempe et al [ ] and Zhang et al [ ]
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