A priority based fuzzy programming approach for multiobjective probabilistic linear fractional programming

This paper presents a fuzzy goal programming approach for modelling and solving multi-objective decision making problem having fuzzy random variables and fuzzy numbers associated with the system constraints. In the model formulation process, the problem is converted into an equivalent fuzzy programming problem by using chance constrained programming technique. The problem is then decomposed into sub problems by considering the tolerance limits of fuzzy numbers relating to the system constraints. The individual optimal solution of each objective is found to construct the membership goals. A two-phase fuzzy goal programming model is developed to achieve the highest degree of each of the defined membership goals of the objectives to the extent possible by minimising under deviational variables and thereby obtaining most satisfactory solution in the decision making environment. A numerical example is solved to illustrate the proposed approach and the achieved solutions are compared with other existing methodologies.

In this paper, we propose an alternate modified solution procedure based on fuzzy goal programming (FGP) approach for solving bi-level multiobjective linear fractional programming problems (BL-MOLFPPs) with some major modifications in the algorithm suggested by Abo-Sinna and Baky (2010). The proposed modified technique is simpler and requires less computational works than that of earlier technique. In the formulation of FGP models, each objective function (at both the levels - leader's level and follower's level) as well as control variables (of leader's level) are transformed into fuzzy goals with modified membership functions. Then FGP approach is used for achieving the highest degree of each of membership goals. To avoid decision deadlock situations and difficulties associated with bi-level programming problems, solution preferences by decision makers (at leader's and follower's levels) are not taken into account in proposed algorithm. A comparative analysis is also carried out with numerical example in order to show efficiency of proposed algorithm over earlier technique.

A multiobjective linear programming (MOLP) problem with fuzzy parameters in the constraints and objective functions in considered. Assuming fuzzy aspiration levels for the particular objectives and involving the comparison of fuzzy numbers, the problem is transformed into a multiobjective linear fractional program. For its solution an interactive method involving a linear programming procedure is proposed. This method has been developed for an application to long-term development planning of a water supply system. It has been first described in Slowinski (1986) and in this paper it is outlined and shown in the context of an updated review of recent proposals for multiobjective fuzzy linear programming.

This chapter presents two methodologies for solving quadratic programming problems with multiple objectives under fuzzy stochastic environments. The right side parameters of the chance constraints of both the models are chosen as fuzzy random variables (FRVs) following different probability distributions. Like the previous chapters, chance constrained programming (CCP) methodology is employed to the fuzzy chance constraints to develop fuzzy programming model. In the first model, cut of fuzzy sets and fuzzy partial order relations are incorporated to the fuzzy programming model to develop an equivalent deterministic model. For the second model, defuzzification method of fuzzy numbers (FNs), which are presented in Chapter 2, are taken into consideration to generate equivalent quadratic programming model in a crisp environment. As the objective functions are quadratic in nature, it is easy to understand that the membership functions obtained through methodological development process are also quadratic in nature. To linearize the quadratic membership functions, linearization techniques are employed in this chapter. Finally, for achieving the maximum degree of each of the membership goals of the objectives, a fuzzy goal programming (FGP) approach is developed for the linearized membership goals and solved by minimizing under-deviational variables and satisfying modified system constraints in fuzzy stochastic decision-making environments. To illustrate the acceptability of the developed methodology presented in this chapter, some numerical examples are included.

This paper proposes an interactive fuzzy programming method for seeking a satisfactory solution for multiobjective two‐level linear fractional programming problems in which the decision makers in the upper and lower levels have several objectives, by first setting up the fuzzy goals for several objectives of each decision maker and seeking a satisfactory solution for the degree of satisfaction of two decision makers in a cooperative manner. In the proposed method, the decision maker at the upper level sets the minimal satisfactory levels for each fuzzy goal and the decision maker at the lower level determines the aspiration levels. The minimal satisfactory levels are treated as a constraint and the solution closest to the aspiration levels of the decision maker at the lower level is computed. The ratio of the aggregative degrees of satisfaction of the decision makers in the upper and lower levels with the obtained solution is evaluated by using partial information on the preference of the decision makers. If the ratio of the degrees of satisfaction satisfies the given condition and the decision maker at the upper level is satisfied with this solution, then the interaction is complete. Otherwise, a satisfactory solution is sought by updating the minimal satisfactory and aspiration levels. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 86(9): 54–62, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10047

ABSTRACTIn this paper, we consider a Fuzzy Stochastic Linear Fractional Programming problem (FSLFP). In this problem, the coefficients and scalars in the objective function are the triangular fuzzy number and technological coefficients and the quantities on the right side of the constraints are fuzzy random variables with the specific distribution. Here we change an FSLFP problem to an equivalent deterministic Multi-objective Linear Fractional Programming (MOLFP) problem. Then by using Fuzzy Mathematical programming approach transformed MOLFP problem is reduced single objective Linear programming (LP) problem. A numerical example is presented to demonstrate the effectiveness of the proposed method.

This paper develops a fuzzy programming technique for solving multiobjective stochastic programming problems having right side parameters associated with the system constraints follow exponential distribution. In the model formulation process the coefficients of the objectives as well as the system constraints are taken as fuzzy numbers. The variables are also considered as fuzzy variables. In the proposed solution process the probabilistic problem is first converted into an equivalent fuzzy programming model applying chance constrained programming methodology. Then using the concept of ranking function the problem is transferred into an equivalent deterministic model. The individual optimal value of each objective is found in isolation to construct the membership goals of the objectives. Finally fuzzy goal programming approach is used for achieving the best compromise solution to the extent possible in the decision making context. An illustrative numerical example is given to demonstrate the efficiency of the proposed methodology.

The difficulty in solving either multi-objective linear programming (MOLP) problems or fuzzy multi-objective linear programming (FMOLP) problems is the trade-off among objectives. To deal with this difficulty, we proposed a new algorithm for solving FMOLP problems by using zero-sum game. First, FMOLP problem given is converted to a crisp MOLP problem by using ranking function, and then a payoff matrix is constructed to find the weights of each objective function of the MOLP problem. After that, each weight is multiplied with the corresponding objective function, a single-objective LP problem is obtained and thus, FMOLP problem is solved. The proposed algorithm is illustrated by numerical examples for the FMLOP problems having fuzzy constraints or not.

PurposeThe purpose of this paper is to propose a method for solving multi-objective linear programming (MOLP) with interval coefficients using positioned programming and interactive fuzzy programming approaches.Design/methodology/approachIn the proposed algorithm, first, lower and upper bounds of each objective function in its feasible region will be determined. Afterwards using fuzzy approach, considering a membership function for each objective function and finally using grey linear programming, the solution for this problem will be obtained.FindingsAccording to the presented example, in this paper, the proposed method is both simple in use and suitable for solving different problems. In the numerical example mentioned in this paper, the proposed method provides an acceptable solution for such problems.Practical implicationsAs in most real-world situations, the coefficients of decision models are not known and exact. In this paper, the authors consider the model of MOLP with interval data, since one of the solutions to cover uncertainty is using interval theory.Originality/valueBased on using grey theory and interactive fuzzy programming approaches, an appropriate method has been presented for solving MOLP problems with interval coefficients. The proposed method, against the complex methods, has less effort and offers acceptable solutions.

The multi-objective linear fractional programming is an interesting topic with many applications in different fields. Until now, various algorithms have been proposed in order to solve the multi-objective linear fractional programming (MOLFP) problem. An important point in most of them is the use of non-linear programming with a high computational complexity or the use of linear programming with preferences of the objective functions which are assigned by the decision maker. The current paper, through combining goal programming and data envelopment analysis (DEA), proposes an iterative method to solve MOLFP problems using only linear programming. Moreover, the proposed method provides an efficient solution which fairly optimizes each objective function when the decision maker has no information about the preferences of the objective functions. In fact, along with normalization of the objective functions, their relative preferences are fairly determined using the DEA. The implementation of the proposed method is demonstrated using numerical examples.

A grey fuzzy multiobjective programming approach for the optimal planning of a reservoir watershed. Part B: Application

We attempt to find solutions to the fully fuzzified linear program where all the parameters and variables are fuzzy numbers. We first change the problem of maximizing a fuzzy number, the value of the objective function, into a multiobjective fuzzy linear programming problem. We prove that fuzzy flexible programming can be used to explore the whole undominated set to the multiobjective fuzzy linear program. An evolutionary algorithm is designed to solve the fuzzy flexible program. The program was applied to two applications with good solutions.

Study on multi-objective linear fractional programming problem involving pentagonal intuitionistic fuzzy number

The purpose of the paper is to present some approaches for multiobjective fractional programming with set coefficients in the objective functions. Among other cases, we discuss for linear fractional multiobjective programming some conservative and nonconservative approaches related to Soyster’s mathematical programming with set-inclusive constraints and Dantzig’s generalized linear programming, respectively. The vectorial max-min optimization will be also used as a conservative approach for inexact multiobjective programming. In particular, the conservative approach will be applied to mathematical programming with nondecreasing objectives with respect to the inexact coefficients. Linear-fractional objective functions will be studied from both points of view (conservative and nonconservative). Also, some applications to the fuzzy linear-fractional multiobjective programming will be suggested.

Solving the multiobjective possibilistic linear programming problem