Binary fuzzy goal programming

Goal programming is an important technique for solving many decision/management problems. Fuzzy goal programming involves applying the fuzzy set theory to goal programming, thus allowing the model to take into account the vague aspirations of a decision-maker. Using preference-based membership functions, we can define the fuzzy problem through natural language terms or vague phenomena. In fact, decision-making involves the achievement of fuzzy goals, some of them are met and some not because these goals are subject to the function of environment/resource constraints. Thus, binary fuzzy goal programming is employed where the problem cannot be solved by conventional goal programming approaches. This paper proposes a new idea of how to program the binary fuzzy goal programming model. The binary fuzzy goal programming model can then be solved using the integer programming method. Finally, an illustrative example is included to demonstrate the correctness and usefulness of the proposed model.     

A procedure for one-parametric linear programming

A procedure is proposed for the parametric linear programming problem where all the coefficients are linear or polynomial functions of a scalar parameter. The solution vector and the optimum value are determined explicitly as rational functions of the parameter. In addition to standard linear programming technique, only the determination of eigenvalues is required. The procedure is compared to those by Dinkelbach and Zsigmond, and a numerical example is given.     

Interactive fuzzy programming approach to Bi-level quadratic fractional programming problems

In this paper we propose an interactive fuzzy programming method for obtaining a satisfactory solution to a “bi-level quadratic fractional programming problem” with two decision makers (DMs) interacting with their optimal solutions. After determining the fuzzy goals of the DMs at both levels, a satisfactory solution is efficiently derived by updating the satisfactory level of the DM at the upper level with consideration of overall satisfactory balance between both levels. Optimal solutions to the formulated programming problems are obtained by combined use of some of the proper methods. Theoretical results are illustrated with the help of a numerical example.     

Selecting tolerances in chance-constrained programming: A multiple objective linear programming approach

An importance issue concerning the practical application of chance-constrained programming is the lack of a rational method for choosing risk levels or tolerances on the chance constraints. While there has also been much recent debate on the relationship, equivalence, usefulness, and other characteristics of chance-constrained programming relative to stochastic programming with recourse, this paper focuses on the problem of improving the selection of tolerances within the chance-constrained framework. An approach is presented, based on multiple objective linear programming, which allows the decision maker to be more involved in the tolerance selection process, but does not demand a priori decisions on appropriate tolerances. An example is presented which illustrates the approach.     

Possibility linear programming with trapezoidal fuzzy numbers

Fuzzy linear programming with trapezoidal fuzzy numbers (TrFNs) is considered and a new method is developed to solve it. In this method, TrFNs are used to capture imprecise or uncertain information for the imprecise objective coefficients and/or the imprecise technological coefficients and/or available resources. The auxiliary multi-objective programming is constructed to solve the corresponding possibility linear programming with TrFNs. The auxiliary multi-objective programming involves four objectives: minimizing the left spread, maximizing the right spread, maximizing the left endpoint of the mode and maximizing the middle point of the mode. Three approaches are proposed to solve the constructed auxiliary multi-objective programming, including optimistic approach, pessimistic approach and linear sum approach based on membership function. An investment example and a transportation problem are presented to demonstrate the implementation process of this method. The comparison analysis shows that the fuzzy linear programming with TrFNs developed in this paper generalizes the possibility linear programming with triangular fuzzy numbers.     

A compromise procedure for the multiple objective linear fractional programming problem

A generalization of a well-known multiple objective linear fractional programming (MOLFP) problem, the multiple objective fractional programming (MOFP) problem, is formulated. A concept of multiple objective programming (MOP) problem corresponding to MOFP is introduced and some relations between those problems are examined. Based on these results, a compromise procedure for MOLFP problem is proposed. A numerical example is given to show how the procedure works.     

A method for solving fuzzy goal programming problems based on MINMAX approach

Narasimhan incorporated fuzzy set theory within goal programming formulation in 1980. Since then numerous research has been carried out in this field. One of the well-known models for solving fuzzy goal programming problems was proposed by Hannan in 1981. In this paper the conventional MINMAX approach in goal programming is applied to solve fuzzy goal programming problems. It is proved that the proposed model is an extension to Hannan model that deals with unbalanced triangular linear membership functions. In addition, it is shown that the new model is equivalent to a model proposed in 1991 by Yang et al. Moreover, a weighted model of the new approach is introduced and is compared with Kim and Whang’s model presented in 1998. A numerical example is given to demonstrate the validity and strengths of the new models.     

Disjunctive cuts for continuous linear bilevel programming

This work shows how disjunctive cuts can be generated for a bilevel linear programming problem (BLP) with continuous variables. First, a brief summary on disjunctive programming and bilevel programming is presented. Then duality theory is used to reformulate BLP as a disjunctive program and, from there, disjunctive programming results are applied to derive valid cuts. These cuts tighten the domain of the linear relaxation of BLP. An example is given to illustrate this idea, and a discussion follows on how these cuts may be incorporated in an algorithm for solving BLP.     

Revised multi-choice goal programming

Chang [C.-T. Chang, Multi-choice goal programming, Omega, The Inter. J. Manage. Sci. 35 (2007) 389–396] has recently proposed a new method namely multi-choice goal programming (MCGP) for multi-objective decision problems. The multi-choice goal programming allows the decision maker to set multi-choice aspiration levels for each goal to avoid underestimation of the decision. However, to express the multi-choice aspiration levels, multiplicative terms of binary variables are involved in their model. This leads to difficult implementation and it is not easily understood by industrial participants. In this paper, we propose an alternative method to formulate the multi-choice aspiration levels with two contributions: (1) the alternative approach does not involve multiplicative terms of binary variables, this leads to more efficient use of MCGP and is easily understood by industrial participants, and (2) the alternative approach represents a linear form of MCGP which can easily be solved by common linear programming packages, not requiring the use of integer programming packages. In addition, a new concept of constrained MCGP is introduced for constructing the relationships between goals in this paper. Finally, to demonstrate the usefulness of the proposed method, an illustrate example is included.     

Multi-objective integer programming: A general approach for generating all non-dominated solutions

In this paper we develop a general approach to generate all non-dominated solutions of the multi-objective integer programming (MOIP) Problem. Our approach, which is based on the identification of objective efficiency ranges, is an improvement over classical ε-constraint method. Objective efficiency ranges are identified by solving simpler MOIP problems with fewer objectives. We first provide the classical ε-constraint method on the bi-objective integer programming problem for the sake of completeness and comment on its efficiency. Then present our method on tri-objective integer programming problem and then extend it to the general MOIP problem with k objectives. A numerical example considering tri-objective assignment problem is also provided.     

Composite geometric programming

Geometric programming is based on functions called posynomials, the terms of which are log-linear. This class of programs is extended from the composition of an exponential and a linear function to an exponential and a convex function. The resulting duality theory for composite geometric programs retains many of the qualities of geometric programming duality, while at the same time encompassing new areas of application. As an application, composite geometric programming is applied to exponential geometric programming. A pure dual is developed for the first time and used to solve a problem from the literature.This research was supported by the Air Force Office of Scientific Research, Grant No. AFOSR-83-0234.     

Fuzzy goal programming approach to multilevel programming problems

The purpose of this paper is to propose a procedure for solving multilevel programming problems in a large hierarchical decentralized organization through linear fuzzy goal programming approach. Here, the tolerance membership functions for the fuzzily described objectives of all levels as well as the control vectors of the higher level decision makers are defined by determining individual optimal solution of each of the level decision makers. Since the objectives are potentially conflicting in nature, a possible relaxation of the higher level decision is considered for avoiding decision deadlock. Then fuzzy goal programming approach is used for achieving highest degree of each of the membership goals by minimizing negative deviational variables. Sensitivity analysis with variation of tolerance values on decision vectors is performed to present how the solution is sensitive to the change of tolerance values. The efficiency of our concept is ascertained by comparing results with other fuzzy programming approaches.     

An iterative approach to solve multiobjective linear fractional programming problems

This paper suggests an iterative parametric approach for solving multiobjective linear fractional programming (MOLFP) problems which only uses linear programming to obtain efficient solutions and always converges to an efficient solution. A numerical example shows that this approach performs better than some existing algorithms. Randomly generated MOLFP problems are also solved to demonstrate the performance of new introduced algorithm.     

A procedure for parameterizing a constraint in linear programming

A post-optimal procedure for parameterizing a constraint in linear programming is proposed. In the derivation of the procedure, the technique of pivotal operations (Jordan eliminations) is applied. The procedure is compared to another by Orchard-Hays [2], and a numerical example of the procedure is provided.     

Fuzzy multi-choice goal programming

Goal programming as a well known technique has been widely used for solving multi objective decision making problems. However, in some practical cases, there may exist situations where the decision maker is interested in setting multi aspiration levels for objectives that may not be expressed in a precise manner. In this paper, a novel formulation of fuzzy multi-choice goal programming (FMCGP) is presented. The proposed approach not only improves the applicability of goal programming in real world situations but also provides useful insight about the solution of a new class of problems. To illustrate and clarify the proposed approach, a numerical example is presented.     

Posynomial geometric programming as a special case of semi-infinite linear programming

This paper develops a wholly linear formulation of the posynomial geometric programming problem. It is shown that the primal geometric programming problem is equivalent to a semi-infinite linear program, and the dual problem is equivalent to a generalized linear program. Furthermore, the duality results that are available for the traditionally defined primal-dual pair are readily obtained from the duality theory for semi-infinite linear programs. It is also shown that two efficient algorithms (one primal based and the other dual based) for geometric programming actually operate on the semi-infinite linear program and its dual.     

A note on chance constrained programming with fuzzy coefficients

This paper deals with nonlinear chance constrained programming as well as multiobjective case and goal programming with fuzzy coefficients occurring in not only constraints but also objectives. We also present a fuzzy simulation technique for handling fuzzy objective constraints and fuzzy goal constraints. Finally, a fuzzy simulation based genetic algorithm is employed to solve a numerical example.     

Constrained optimization using multiple objective programming

In practical applications of mathematical programming it is frequently observed that the decision maker prefers apparently suboptimal solutions. A natural explanation for this phenomenon is that the applied mathematical model was not sufficiently realistic and did not fully represent all the decision makers criteria and constraints. Since multicriteria optimization approaches are specifically designed to incorporate such complex preference structures, they gain more and more importance in application areas as, for example, engineering design and capital budgeting. The aim of this paper is to analyze optimization problems both from a constrained programming and a multicriteria programming perspective. It is shown that both formulations share important properties, and that many classical solution approaches have correspondences in the respective models. The analysis naturally leads to a discussion of the applicability of some recent approximation techniques for multicriteria programming problems for the approximation of optimal solutions and of Lagrange multipliers in convex constrained programming. Convergence results are proven for convex and nonconvex problems.     

Interactive fuzzy goal programming approach for bilevel programming problem

This paper presents an interactive fuzzy goal programming (FGP) approach for bilevel programming problems with the characteristics of dynamic programming (DP).     

Comment to “interactive fuzzy goal programming approach for bilevel programming problem” by S.R. Arora and R. Gupta

The fuzzy bilevel programming problem is solved in the paper [1] using an approach of fuzzy goal programming. Using a simple example we show that this will not lead to a satisfactory solution. Especially the fuzzy constraint that the upper level variable is near some desired one has no influence on the computed solution in the example. This constraint is used to model the hierarchy in the approach in [1]. At the end of the paper we suggest one straightforward possibility for modeling the fuzzy bilevel programming problem and converting it into a crisp substitute.