Research On Strategies And Application Of Ill-Posed Bilevel Programming Problem By The Satisfactory Degree

Bilevel programming problem is a hierarchical mathematical problem which involves two optimization problems:upper level programming problem and lower level programming problem. The upper level decision maker (also called the leader) makes a decision first, then, the lower level decision maker (also called the follower) chooses his strategy according to the leader’s action. Each decision maker independently seeks its own interest, but is affected by the action of the other decision maker. As the foundation of multi-level programming problem, bilevel programming plays exceedingly important role in different application fields, such as power market, transportation, engineering, economics, ecology, principal-agent and others. So, it has been developed and researched by many authors.This paper mainly discusses ill-posed bilevel programming problem whose solution of the lower level problem is not unique, it is difficult for the leader to predict which point the follower will choose. Then, the leader is difficult to decide its solution But, the leader can acquire its best or worst solution by the assumptions of absolutely cooperation or non-cooperation from the follower to the leader. Based on above, we have optimistic model and pessimistic model. Since these two models are the two extreme situations, it would be not appropriate and could not express the decision’s truly intentions if the leader considers only optimistic model or pessimistic model in the factual decision making. Then, the intermediate problem between the optimistic and pessimistic model becomes one of the hottest and most difficult subjects of the ill-posed bilevel programming problem. Moreover, the optimistic solution is always the best decision to the upper level but is not always be the best choice for the lower level which is not fit for the situation in application because the lower level decision is also a rational decision who also wants to achieve his own maximum profits. Based on Simon’s "satisfaction criterion", many authors present the definition of satisfying solution to make both the upper level decision and the lower level decision be willing to accept. Based on all above discussion, the main contributions of the dissertation are organized as followes:(1) This paper summaries the recent research and development of ill-posed bilevel programming problem.(2)Based on D.Cao’s partial cooperation model, this paper transferres the coopera-tion degree to a profit allocation proportion for the leader to motivate the follower to maximize its intrests for ill-posed linear bilevel programming problem.(3) A.Aboussoror’s has introduced a partial cooperation model by using a cooperation index to describe the degree of follower’s cooperation to the leader. But, in his model, the cooperation degree is a constant indicating the leader’s expectation coefficient for the follower’s action, but it is not the follower’s willingness. To solve this situation, we develops A.Aboussoror’s model by using the follower’s satisfactory degree as the cooperation degree. Because this new cooperation degree is no longer a constant but a function which is dependent on the leader’s choice and decided by the follower’s satisfactory degree, we can not only get an optimal value between the optimistic value and pessimistic value but also get a more satisfactory solution than A. Aboussoror’s model. Finally, we give an example to demonstrate the feasibility of these theorems.(4) To acquire a more satisfactory solution than the optimistic solution, we not only considers the leader’s satisfactory degree but also considers the follower’s. We construct a minimization problem of the two objective function by weighted summation and transfer this two level problem to a single level problem by using the duality gap of the lower level problem as a penalty function. We can acquire even a more satisfactory solution than the optimistic solution from the efficient solutions to such minimization problem by different weight.(5) The principal-agent problem with asymmetric information is considered. Using the method in chapter4for its optimistic formulation, we acquire a satisfactory contract which is derived for both principal and agent so that winners.At last, the main works of this dissertation is summarized and the future research is pointed out.