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.     

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).     

Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach

In this paper, two new algorithms are presented to solve multi-level multi-objective linear programming (ML-MOLP) problems through the fuzzy goal programming (FGP) approach. The membership functions for the defined fuzzy goals of all objective functions at all levels are developed in the model formulation of the problem; so also are the membership functions for vectors of fuzzy goals of the decision variables, controlled by decision makers at the top levels. Then the fuzzy goal programming approach is used to achieve the highest degree of each of the membership goals by minimizing their deviational variables and thereby obtain the most satisfactory solution for all decision makers.     

Fuzzy goal programming with nested priorities

This paper proposes a new approach to formulating fuzzy priorities in a goal programming problem. The proposed methodology remedies certain shortcomings of the composite membership function approach discussed in previous works [7, 10]. The principal advantage of the proposed method is that it leads to a formulation in which tradeoffs between goals more closely reflect the decision maker's intentions than in other noninteractive approaches [8, 9, 10, 14], in some of which a fixed hierarchy of goals is assumed.     

A note on article “A tolerance approach to the fuzzy goal programming problems with unbalanced triangular membership function”

Kim and Whang use a tolerance approach for solving fuzzy goal programming problems with unbalanced membership functions [J.S. Kim, K. Whang, A tolerance approach to the fuzzy goal programming problems with unbalanced triangular membership function, European Journal of Operational Research 107 (1998) 614–624]. In this note it is shown that some results in that article are incorrect. The necessary corrections are proposed.     

A fuzzy goal programming approach for mid-term assortment planning in supermarkets

We develop a fuzzy mixed integer non-linear goal programming model for the mid-term assortment planning of supermarkets in which three conflicting objectives namely profitability, customer service, and space utilization are incorporated. The items and brands in a supermarket compete to obtain more space and better shelf level. This model offers different service levels to loyal and disloyal customers, applies joint replenishment policy, and accounts for the holding time limitation of perishable items. We propose a fuzzy approach due to the imprecise nature of the goals’ target levels and priorities as well as critical data. A heuristic method inspiring by the problem-specific rules is developed to solve this complex model approximately within a reasonable time. Finally, the proposed approach is validated through several numerical examples and results are reported.     

A fuzzy goal programming approach to multi-objective optimization problem with priorities

In goal programming problem, the general equilibrium and optimization are often two conflicting factors. This paper proposes a generalized varying-domain optimization method for fuzzy goal programming (FGP) incorporating multiple priorities. According to the three possible styles of the objective function, the varying-domain optimization method and its generalization are proposed. This method can generate the results consistent with the decision-maker (DM)’s expectation, that the goal with higher priority may have higher level of satisfaction. Using this new method, it is a simple process to balance between the equilibrium and optimization, and the result is the consequence of a synthetic decision between them. In contrast to the previous method, the proposed method can make that the higher priority achieving the higher satisfactory degree. To get the global solution of the nonlinear nonconvex programming problem resulting from the original problem and the varying-domain optimization method, the co-evolutionary genetic algorithms (GAs), called GENOCOPIII, is used instead of the SQP method. In this way the DM can get the optimum of the optimization problem. We demonstrate the power of this proposed method by illustrative examples.     

Enhanced interactive satisficing method via alternative tolerance for fuzzy goal programming with progressive preference

This paper proposes an enhanced interactive satisficing method via alternative tolerance for fuzzy goal programming with progressive preference. The alternative tolerances of the fuzzy objectives with three types of fuzzy relations are used to model progressive preference of decision maker. In order to improve the dissatisficing objectives, the relaxed satisficing objectives are sacrificed by modifying their tolerant limits. By means of attainable reference point, the auxiliary programming is designed to generate the tolerances of the dissatisficing objectives for ensuring feasibility. Correspondingly, the membership functions are updated or the objective constraints are added. The Max–Min goal programming model (or the revised one) and the test model of the M-Pareto optimality are solved lexicographically. By our method, the dissatisficing objectives are improved iteratively till the preferred result is acquired. Illustrative examples show its power.     

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.     

On fuzzy linear programming

Fuzzy multi-objective and fuzzy Goal Programming are discussed in connection with several membership functions which are used to transform the original problem into three equivalent linear programming problems. Existence and uniqueness theorems are given. Fuzzy duality is presented, and an extension of the initial fuzzy problem arises immediately from it.     

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.     

Fuzzy stochastic goal programming problems

In this paper, we present a model to measure attainment value of fuzzy stochastic goals. Then, the new measure is used to de-randomize and de-fuzzify the fuzzy stochastic goal programming problem and obtain a standard linear program (LP). A numerical example is provided to illustrate the proposed method.     

Solving a multi-objective multi-skilled manpower scheduling model by a fuzzy goal programming approach

Manpower scheduling is an intricate problem in production and service environments with the purpose of generating fair schedules that consider employers’ objectives and employees’ preferences as much as possible. However, sometimes, vagueness of information related to employers’ objectives and employees’ preferences leads to the fuzzy nature of the problem. This paper presents a multi-objective manpower scheduling model regarding the lack of clarity on the target values of employers’ objectives and employees’ preferences. Hence, a fuzzy goal programming model is developed for the presented model. Afterwards, two fuzzy solution approaches are used to convert the fuzzy goal programming model to two single-objective models. Finally, the results obtained by both single-objective models are compared with each other to select the solution that has the greatest degree of the satisfaction level of employers’ objectives and employees’ preferences.     

Linear programming with fuzzy parameters: An interactive method resolution

This paper proposes a method for solving linear programming problems where all the coefficients are, in general, fuzzy numbers. We use a fuzzy ranking method to rank the fuzzy objective values and to deal with the inequality relation on constraints. It allows us to work with the concept of feasibility degree. The bigger the feasibility degree is, the worst the objective value will be. We offer the decision-maker (DM) the optimal solution for several different degrees of feasibility. With this information the DM is able to establish a fuzzy goal. We build a fuzzy subset in the decision space whose membership function represents the balance between feasibility degree of constraints and satisfaction degree of the goal. A reasonable solution is the one that has the biggest membership degree to this fuzzy subset. Finally, to illustrate our method, we solve a numerical example.     

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.     

An efficient method for solving linear goal programming problems

This note proposes a solution algorithm for linear goal programming problems. The proposed method simplifies the traditional solution methods. Also, the proposed method is computationally efficient.     

Fuzzy analytic hierarchy process: A logarithmic fuzzy preference programming methodology

Fuzzy analytic hierarchy process (AHP) proves to be a very useful methodology for multiple criteria decision-making in fuzzy environments, which has found substantial applications in recent years. The vast majority of the applications use a crisp point estimate method such as the extent analysis or the fuzzy preference programming (FPP) based nonlinear method for fuzzy AHP priority derivation. The extent analysis has been revealed to be invalid and the weights derived by this method do not represent the relative importance of decision criteria or alternatives. The FPP-based nonlinear priority method also turns out to be subject to significant drawbacks, one of which is that it may produce multiple, even conflict priority vectors for a fuzzy pairwise comparison matrix, leading to entirely different conclusions. To address these drawbacks and provide a valid yet practical priority method for fuzzy AHP, this paper proposes a logarithmic fuzzy preference programming (LFPP) based methodology for fuzzy AHP priority derivation, which formulates the priorities of a fuzzy pairwise comparison matrix as a logarithmic nonlinear programming and derives crisp priorities from fuzzy pairwise comparison matrices. Numerical examples are tested to show the advantages of the proposed methodology and its potential applications in fuzzy AHP decision-making.     

Satisfying optimization method based on goal programming for fuzzy multiple objective optimization problem

This paper proposes a satisfying optimization method based on goal programming for fuzzy multiple objective optimization problem. The aim of this presented approach is to make the more important objective achieving the higher desirable satisfying degree. For different fuzzy relations and fuzzy importance, the reformulated optimization models based on goal programming is proposed. Not only the satisfying results of all the objectives can be acquired, but also the fuzzy importance requirement can be simultaneously actualized. The balance between optimization and relative importance is realized. We demonstrate the efficiency, flexibility and sensitivity of the proposed method by numerical examples.     

A new approach to fuzzy programming

A fuzzy program is defined in the usual way as a sequence of statements (instruction) which are considered as functions (possibly fuzzy functions) and fuzzy predicates defined on the given input domain. The essential difference in the approach presented in this paper is the new interpretation of the execution of fuzzy programs, and a new method of evaluating fuzzy predicates. The result of the fuzzy program execution is an appropriate fuzzy subset in the output domain.     

Solving fuzzy queueing decision problems via a parametric mixed integer nonlinear programming method

This paper proposes a mathematical programming method to construct the membership functions of the fuzzy objective value of the cost-based queueing decision problem with the cost coefficients and the arrival rate being fuzzy numbers. On the basis of Zadeh’s extension principle, three pairs of mixed integer nonlinear programs (MINLP) parameterized by the possibility level α are formulated to calculate the lower and upper bounds of the minimal expected total cost per unit time at α, through which the membership function of the minimal expected total cost per unit time of the fuzzy objective value is constructed. To provide a suitable optimal service rate for designing queueing systems, the Yager’s ranking index method is adopted. Two numerical examples are solved successfully to demonstrate the validity of the proposed method. Since the objective value is completely expressed by a membership function rather than by a crisp value, it conserves the fuzziness of the input information, thus more information is provided for designing queueing systems. The successful extension of queueing decision models to fuzzy environments permits queueing decision models to have wider applications in practice.