The Karush–Kuhn–Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions

2009 ◽  
Vol 196 (1) ◽  
pp. 49-60 ◽  
Author(s):  
Hsien-Chung Wu
Keyword(s):  
Optimality Conditions ◽  
Multiobjective Programming ◽  
Objective Functions ◽  
Multiobjective Programming Problems ◽  
Interval Valued

Multiobjective Programming

2014 ◽  
pp. 1585-1604
Author(s):  
Minghe Sun
Keyword(s):  
Programming Problem ◽  
Multiobjective Programming ◽  
Optimization Problems ◽  
Optimal Solution ◽  
Objective Functions ◽  
Solution Quality ◽  
Solution Methods ◽  
Multiobjective Programming Problem ◽  
Feasible Solutions ◽  
Multiobjective Programming Problems

Optimization problems with multiple criteria measuring solution quality can be modeled as multiobjective programming problems. Because the objective functions are usually in conflict, there is not a single feasible solution that can optimize all objective functions simultaneously. An optimal solution is one that is most preferred by the decision maker (DM) among all feasible solutions. An optimal solution must be nondominated but a multiobjective programming problem may have, possibly infinitely, many nondominated solutions. Therefore, tradeoffs must be made in searching for an optimal solution. Hence, the DM's preference information is elicited and used when a multiobjective programming problem is solved. The model, concepts and definitions of multiobjective programming are presented and solution methods are briefly discussed. Examples are used to demonstrate the concepts and solution methods. Graphics are used in these examples to facilitate understanding.

Some Approaches for Fuzzy Multiobjective Programming Problems

2021 ◽  
Vol 6 (1) ◽  
Author(s):  
Yves Mangongo Tinda ◽  
◽  
Justin Dupar Kampempe Busili ◽  
Keyword(s):  
Optimization Problem ◽  
Multiobjective Programming ◽  
Fuzzy Numbers ◽  
Fuzzy Optimization ◽  
Objective Functions ◽  
Multiobjective Programming Problem ◽  
Effectiveness And Efficiency ◽  
Nearest Interval Approximation ◽  
Multiobjective Programming Problems ◽  
Interval Approximation

In this paper we discuss two approaches to bring a balance between effectiveness and efficiency while solving a multiobjective programming problem with fuzzy objective functions. To convert the original fuzzy optimization problem into deterministic terms, the first approach makes use of the Nearest Interval Approximation Operator (Approximation approach) for fuzzy numbers and the second one takes advantage of an Embedding Theorem for fuzzy numbers (Equivalence approach). The resulting optimization problem related to the first approach is handled via Karush- Kuhn-Tucker like conditions for Pareto Optimality obtained for the resulting interval optimization problem. A Galerkin like scheme is used to tackle the deterministic counterpart associated to the second approach. Our approaches enable both faithful representation of reality and computational tractability. They are thus in sharp contrast with many existing methods that are either effective or efficient but not both. Numerical examples are also supplemented for the sake of illustration.

Multiobjective Two-Level Fuzzy Random Programming Problems with Simple Recourses and Estimated Pareto Stackelberg Solutions

2018 ◽  
Vol 22 (3) ◽  
pp. 359-368 ◽  
Author(s):  
Hitoshi Yano ◽  
Keyword(s):  
Multiobjective Programming ◽  
Objective Functions ◽  
Possibility Measure ◽  
Weighting Method ◽  
Complementarity Conditions ◽  
Stackelberg Solutions ◽  
Weighting Vector ◽  
Fuzzy Random Programming ◽  
Multiobjective Programming Problems ◽  
Fuzzy Random

In this paper, we focus on multiobjective two-level fuzzy random programming problems with simple recourses, in which multiple objective functions are involved in each level, shortages and excesses resulting from the violation of the constraints with fuzzy random variables are penalized, and sums of the objective functions and expectation of the amount of the penalties are minimized. To deal with such problems, the concept of estimated Pareto Stackelberg solutions of the leader is introduced under the assumption that the leader can estimate the preference of the follower as a weighting vector of the weighting problems. Employing the possibility measure for fuzzy numbers, weighting method for multiobjective programming problems, and Kuhn-Tucker approach for two-level programming problems, a nonlinear optimization problem under complementarity conditions is formulated to obtain the estimated Pareto Stackelberg solutions for the leader. A numerical example illustrates the proposed method for a multiobjective two-level fuzzy random programming problem with simple recourses. Several types of estimated Pareto Stackelberg solutions are derived corresponding to the weighting vectors and permissible possibility levels specified by the leader.

An Interval Approximation Approach for a Multiobjective Programming Model with Fuzzy Objective Functions

2015 ◽  
Vol 23 (06) ◽  
pp. 845-860 ◽  
Author(s):  
M. K. Luhandjula
Keyword(s):  
Multiobjective Programming ◽  
Programming Model ◽  
Optimal Solution ◽  
Optimization Under Uncertainty ◽  
Objective Functions ◽  
New Approach ◽  
Different Shapes ◽  
General Goal ◽  
Multiobjective Programming Problems ◽  
Interval Approximation

Research in optimization under uncertainty is alive. It assumes different shapes and forms, all concurring to the general goal of designing effective and efficient tools for handling imprecision in an Optimization setting. In this paper we present a new approach for dealing with multiobjective programming problems with fuzzy objective functions. Similar to many approaches in the literature, our approach relies on the deffuzification of involved fuzzy quantities. Our improvement stem from the choice of a deffuzification operator that captures essential features of fuzzy parameters at hand rather than those that yield single values, leading to a loss of many useful information. Two oracles play a pivotal role in the proposed method. The first one returns a near interval approximation to a given fuzzy number. The other one delivers a Pareto Optimal solution of the resulting multiobjective program with interval coefficient. A numerical example is also provided for the sake of illustration.

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