Fuzzy Sets and Interactive Multiobjective Optimization
The main characteristics of the real-world decision-making problems facing humans today are multidimensional and have multiple objectives including eco nomic, environmental, social, and technical ones. Hence, it seems natural that the consideration of many objectives in the actual decision-making process re quires multiobjective approaches rather than single-objective. One ofthe major systems-analytic multiobjective approaches to decision-making under constraints is multiobjective optimization as a generalization of traditional single-objective optimization. Although multiobjective optimization problems differ from single objective optimization problems only in the plurality of objective functions, it is significant to realize that multiple objectives are often noncom mensurable and conflict with each other in multiobjective optimization problems. With this ob servation, in multiobjective optimization, the notion of Pareto optimality or effi ciency has been introduced instead of the optimality concept for single-objective optimization. However, decisions with Pareto optimality or efficiency are not uniquely determined; the final decision must be selected from among the set of Pareto optimal or efficient solutions. Therefore, the question is, how does one find the preferred point as a compromise or satisficing solution with rational pro cedure? This is the starting point of multiobjective optimization. To be more specific, the aim is to determine how one derives a compromise or satisficing so lution of a decision maker (DM), which well represents the subjective judgments, from a Pareto optimal or an efficient solution set.
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FUNDAMENTALS OF FUZZY SET THEORY
FUZZY LINEAR PROGRAMMING
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a-level set a-MONLP a-Pareto admissible set asking the DM assumed augmented minimax problem Bellman and Zadeh bisection method chapter constraint problem convex convex sets decision maker DM defined degree denoted DM's Example exists extension principle extreme points fi(x Figure following theorem formulated fuzzy equal fuzzy goals fuzzy linear regression fuzzy multiobjective linear fuzzy numbers fuzzy parameters fuzzy set goal programming hyperbolic inverse hyperplane interactive fuzzy multiobjective Interactive multiobjective isoquant Lagrange multipliers linear membership function linear programming problem M-a-Pareto maximize minimize minimum and maximum MOLP MONLP multiobjective linear programming multiobjective nonlinear programming multiobjective optimization nonlinear programming problem objective functions obtained optimal solution set Pareto optimal concept Pareto optimal solution problems with fuzzy programming method reference levels reference membership levels reference membership values reference point Sakawa and Yano satisficing solution simplex method solve the corresponding solving the following Step strict inequality holding trade-off information updating the reference vector Weak Pareto optimal zi(x