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Abstract
Multi-level mathematical programming (MLP) problems are identified as mathematical programming that solves decentralized planning problems with multiple decision-makers (DMs) in a multi-level or hierarchical organization. Bi-level and three-level programming problems are classes of MLP problems in which there are two or three independent decision-makers (DMs), respectively. Each DM attempts to optimize its objectives function and is affected by the actions of the other DMs.
Recently, the balance space approach has been developed based on monotonic set contraction (expansion) that solves multi-objective programming problems in their original form and provides a new instrument, called balance set, which is represented by a collection of equations in the function value space that allow us to choose in advance the quality of a particular optimal solution from a set of the like ones. The balance space approach is based on minimal deviations from optimality of the individual objectives. Considering either equal or different deviations for each objective function, the concept of the balance number, respectively, the balance point are derived.
This thesis presents bi-level multi-objective decision-making (BL-MODM) model and an interactive algorithm for solving such model using the concept of Satisfactoriness and the Balance Space Approach. The algorithm simplifies bi-level multi-objective decision-making problems by transforming it into separate multi-objective decision making problems at each level, thereby avoiding the difficulty associated with non-convex mathematical programming. Also, an extension to this algorithm is introduced to solve the three-level programming problems as a generalization to solve multi-level programming problems.
A new formula is introduced to interconnect the satisfactoriness and the apportioned numbers. Therefore, we have the apportioned numbers including satisfactoriness. Also, we present, according to the solution of the balance space approach, new definitions for the satisfactoriness and the preferred solution in view of singular-level multi-objective decision making problems, and also new definitions for the feasible solution and the preferred solution (point) of bi-level and three-level programming problems are presented. η-optimalη-optimal
The Supply-Demand Interactions in Electronic Commerce can be formulated as BL-MODM problem. A comparison between the presented interactive balance space approach for solving BL-MODM and the fuzzy approach for solving BL-MODM will be studied and applied to the Supply-Demand Interactions in Electronic Commerce.
Illustrative numerical examples are given to demonstrate the algorithms and the comparative study.
Keywords: Multi-level programming problem; Multi-objective Optimization; Nonscalarized Global Optimization Methods; Balance Set; Balance Space Approach ; Interactive methods; Satisfactoriness; Electronic Commerce; Supply-Demand Interaction; Fuzzy approach.