A Possibilistic Approach for Multiobjective Programming Problems. Efficiency of Solutions

In this paper, we present a multi-objective fractional probabilistic programming problem where the parameters in the objective functions and right-hand side constraints are multi-choice in nature. Multi-choice multi-objective objective fractional probabilistic programming problem is formulated. The multi-choice parameters in the right-hand side of constraints are random variables following Cauchy distribution. Multi-choice multi-objective fractional probabilistic programming problem is transformed to deterministic equivalent multi-choice multi-objective fractional programming problem using chance-constrained programming problem. Lagrange interpolating polynomial approach is proposed to deal with multi-choice parameters. The multi-choice multi-objective fractional programming problem is transformed to non-linear mixed integer multi-objective programming problem. An \(\epsilon -\) constrained method is applied to find the compromise solutions of non-linear mixed integer multi-objective programming problem using different values of \(\epsilon \). LINGO software is applied to find the optimal solution of single objective non-linear mixed integer programming problem. Finally, a numerical example is given to illustrate the proposed programming problem.KeywordsMulti-choice programming problemMulti-objective programming problemProbabilistic programming problemFractional programming problem\(\epsilon \)-constraint method

On computational methods for solutions of multiobjective linear production programming games

A bipolar approach in fuzzy multi-objective linear programming

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.

This paper presents a fuzzy goal programming (FGP) approach for modeling and solving multiobjective fractional programming problems (MOFPPs) with fuzzy numbers parameter sets. In the proposed approach, first the notion of α-cut in fuzzy sets (FSs) is used to transform a problem into conventional MOFPP by using the tolerance membership functions in FSs. In model formulation, membership functions are converted into fuzzy goals for measuring the degree of satisfaction of decision maker (DM) with the solution for achievement of fuzzily described objectives of the problem. In the solution process, an iterative parametric method is addressed within the framework of minsum FGP model to reach the highest membership value (unity) to the extent possible in the decision making environment. The efficiency of the proposed approach is illustrated by a numerical example. The model solution is also compared with the solution obtained by using other approaches studied previously.

Two new powerful mathematical languages, fuzzy set theory and possibility theory, have led to two optimization types that explicitly incorporate data whose values are not real-valued nor probabilistic: 1) flexible optimization and 2) optimization under generalized uncertainty. Our aim is to make clear what these two types are, make distinctions, and show how they can be applied. Flexible optimization arises when it is necessary to relax the meaning of the mathematical relation of belonging to a set (a constraint set in the context of optimization). The mathematical language of relaxed set belonging is fuzzy set theory. Optimization under generalized uncertainty arises when it is necessary to represent parameters of a model whose values are only known partially or incompletely. A natural mathematical language for the representation of partial or incomplete information about the value of a parameter is possibility theory. Flexible optimization, as delineated here, includes much of what has been called fuzzy optimization whereas optimization under generalized uncertainty includes what has been called possibilistic optimization. We explore why flexible optimization and optimization under generalized uncertainty are distinct and important types of optimization problems. Possibility theory in the context of optimization leads to two distinct types of optimization under generalized uncertainty, single distribution and dual distribution optimization. Dual (possibility/necessity pairs) distribution optimization is new. Mathematical subject classification: 90C70, 65G40.

A multi-objective particle swarm optimization (MOPSO) approach for multi-objective economic load dispatch problem in power system is presented in this paper. The economic load dispatch problem is a non-linear constrained multi-objective optimization problem. The proposed MOPSO approach handles the problem as a multi-objective problem with competing and non-commensurable fuel cost, emission and system loss objectives and has a diversity-preserving mechanism using an external memory (call “repository”) and a geographically-based approach to find widely different Pareto-optimal solutions. In addition, fuzzy set theory is employed to extract the best compromise solution. Several optimization runs of the proposed MOPSO approach were carried out on the standard IEEE 30-bus test system. The results revealed the capabilities of the proposed MOPSO approach to generate well-distributed Pareto-optimal non-dominated solutions of multi-objective economic load dispatch. Comparison with Multi-objective Evolutionary Algorithm (MOEA) showed the superiority of the proposed MOPSO approach and confirmed its potential for solving multi-objective economic load dispatch.

Convergence of exponential penalty function method for multiobjective fractional programming problems

A neural network approach to multiobjective and multilevel programming problems

Since L. Zadeh proposed the concept of a fuzzy set in 1965 the relationships between probability theory and possibility theory have been discussed. Both theories seem to be similar in the sense that they both are concerned with some type of uncertainty and both use the [0,1] interval for their measures as the range of their respective functions. (At least as long as one considers normalized fuzzy sets only!) Other uncertainty measures, which were already mentioned in chapter 4, also focus on uncertainty and could therefore be included in such a discussion. The comparison between probability theory and fuzzy set theory is difficult primarily for two reasons: 1. The comparison could be made on very different levels, that is, mathematically, semantically, linguistically, and so on. 2. Fuzzy set theory is not or is no longer a uniquely defined mathematical structure, such as Boolean algebra or dual logic. It is rather a very general family of theories (consider, for instance, all the possible operations which were already defined in chapter 3 or of different types of membership functions). In this respect, fuzzy set theory could rather be compared with the different existing theories of multivalued logic.

The aim of this paper is twofold. Firstly, to define a solution concept of Pareto-optimality for a multi-objective flexible linear programming (MOFLP) problem (or multi-objective fuzzy linear programming problem) and design a method to extract a Pareto-optimal solution of MOFLP problem from the set of optimal solutions of equivalent optimization problem formulated by Dubey and Mehra (2013). Secondly, to extend this study to multi-objective linear programming problem involving hard and flexible constraints with interval uncertainty. A flexible constraint with interval uncertainty generalizes a flexible constraint by allowing preferences to be expressed in the form of intervals. An optimistic-pessimistic approach is proposed to solve multi-objective flexible linear programming with interval uncertainty (MOFLPIU) using an interval-valued fuzzy set representation and the Hurwicz optimism-pessimism criterion.

Particle swarm optimization (PSO) was proposed by Kennedy et al. as a general approximate solution method for nonlinear programming problems. Its efficiency has been shown, but there have been left some shortcomings of the method. Thus, the authors proposed a revised PSO (rPSO) method incorporating the homomorphous mapping and the multiple stretching in order to cope with these shortcomings. In this paper, we construct an interactive fuzzy satisficing method for multiobjective nonlinear programming problems based on the rPSO. Furthermore, in order to obtain better solutions in consideration of the property of multiobjective programming problems, we incorporate the direction to nondominated solutions into the rPSO. Finally, we show the efficiency of the proposed method by applying it to numerical examples

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.

The goal of a new area of Computing with Words (CWW) is solving computationally tasks formulated in a natural language (NL). The extreme uncertainty of NL is the major challenge to meet this ambitious goal requiring computational approaches to handle NL uncertainties. Attempts to solve various CWW tasks lead to the methodological questions about rigorous and heuristic formulations and solutions of the CWW tasks. These attempts immediately reincarnated the long-time discussion about different methodologies to model uncertainty, namely: Probability Theory, Multi-valued logic, Fuzzy Sets, Fuzzy Logic, and the Possibility theory. The main forum of the recent discussion was an on-line Berkeley Initiative on Soft Computing group in 2014. Zadeh claims that some computing with words tasks are in the province of the fuzzy logic and possibility theory, and probabilistic solutions are not appropriate for these tasks. In this work we propose a useful constructive probabilistic approach for CWW based on sets of specialized K-Measure (Kolmogorov’s measure) probability spaces that differs from the random sets. This work clarifies the relationships between probability and possibility theories in the context of CWW.

This report summarizes techniques that use possibility theory to estimate the risk of terrorist acts. These techniques were developed under the sponsorship of the Department of Homeland Security (DHS) as part of the National Infrastructure Simulation Analysis Center (NISAC) project. The techniques have been used to estimate the risk of various terrorist scenarios to support NISAC analyses during 2004. The techniques are based on the Logic Evolved Decision (LED) methodology developed over the past few years by Terry Bott and Steve Eisenhawer at LANL. [LED] The LED methodology involves the use of fuzzy sets, possibility theory, and approximate reasoning. LED captures the uncertainty due to vagueness and imprecision that is inherent in the fidelity of the information available for terrorist acts; probability theory cannot capture these uncertainties. This report does not address the philosophy supporting the development of nonprobabilistic approaches, and it does not discuss possibility theory in detail. The references provide a detailed discussion of these subjects. [Shafer] [Klir and Yuan] [Dubois and Prade] Suffice to say that these approaches were developed to address types of uncertainty that cannot be addressed by a probability measure. An earlier report discussed in detail the problems with using a probability measure to evaluate terrorist risk. [Darby Methodology]. Two related techniques are discussed in this report: (1) a numerical technique, and (2) a linguistic technique. The numerical technique uses traditional possibility theory applied to crisp sets, while the linguistic technique applies possibility theory to fuzzy sets. Both of these techniques as applied to terrorist risk for NISAC applications are implemented in software called PossibleRisk. The techniques implemented in PossibleRisk were developed specifically for use in estimating terrorist risk for the NISAC program. The LEDTools code can be used to perform the same linguistic evaluation as performed in PossibleRisk. [LEDTools] LEDTools is a general purpose linguistic evaluation tool and allows user defined universes of discourse and approximate reasoning rules, whereas PossibleRisk uses predefined universes of discourse (risk, attack, success, loss, and consequence) and rules. Also LEDTools has the capability to model a large number of threat scenarios with a graph and to integrate the scenarios (paths from the graph) into the linguistic evaluation. Example uses of PossibleRisk and LEDTools for the possibilistic evaluation of terrorist risk are provided in this report.