Fractional Goal Programming for Fuzzy Solid Transportation Problem with Interval Cost

In this paper, we study a solid transportation problem with interval cost using fractional goal programming approach (FGP). In real life applications of the FGP problem with multiple objectives, it is difficult for the decision-maker(s) to determine the goal value of each objective precisely as the goal values are imprecise, vague, or uncertain. Therefore, a fuzzy goal programming model is developed for this purpose. The proposed model presents an application of fuzzy goal programming to the solid transportation problem. Also, we use a special type of non-linear (hyperbolic) membership functions to solve multi-objective transportation problem. It gives an optimal compromise solution. The proposed model is illustrated by using an example.     

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.     

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).     

Linear programming with multiple fuzzy goals

This paper describes the use of fuzzy set theory in goal programming (GP) problems. In particular, it is demonstrated how fuzzy or imprecise aspirations of the decision maker (DM) can be quantified through the use of piecewise linear and continuous functions. Models are then presented for the use of fuzzy goal programming with preemptive priorities, with Archimedean weights, and with the maximization of the membership function corresponding to the minimum goal. Examples are also provided.     

Derivation of intuitionistic fuzzy weights based on intuitionistic fuzzy preference relations

This paper proposes linear goal programming models for deriving intuitionistic fuzzy weights from intuitionistic fuzzy preference relations. Novel definitions are put forward to define additive consistency and weak transitivity for intuitionistic fuzzy preference relations, followed by a study of their corresponding properties. For any given normalized intuitionistic fuzzy weight vector, a transformation formula is furnished to convert the weights into a consistent intuitionistic fuzzy preference relation. For any intuitionistic fuzzy preference relation, a linear goal programming model is developed to obtain its intuitionistic fuzzy weights by minimizing its deviation from the converted consistent intuitionistic fuzzy preference relation. This approach is then extended to group decision-making situations. Three numerical examples are provided to illustrate the validity and applicability of the proposed models.     

Three fuzzy goal programming models for index portfolios

Studies show that most actively managed mutual funds struggle to beat the market, driving an increase in the popularity of index investing. Index investing instruments, including index funds and Exchange-traded Funds, aim to track market performance. This study pursues both tracking error minimization and excess return maximization, two conflicting objectives, to construct an index portfolio. In the real-world financial environment, the desires and expectations of decision makers are generally imprecise. This study applies fuzzy theory to deal with imprecise objectives. This study represents minimizing tracking error and maximizing excess return as ‘fuzzy goals’ to improve traditional goal programming, which is suitable for handling multiple conflicting objectives, but subject to establishing crisp goals. Three fuzzy goal programming (FGP) models that track indexes are compared and discussed, and the results show that through certain membership functions and tracking models, an index tracking portfolio with a tracking error lower than the 0050 index fund, and a similar excess return to 0050 index fund can be constructed using additive type FGP. max-min type FGP underperforms the additive type FGP in index fund construction.     

A weighted max–min model for fuzzy goal programming

After Narasimhan's pioneering study of applying fuzzy set theory to goal programming in 1980, many achievements in the field have been recorded. Most of them followed the max–min approach. However, when objectives have different levels of importance, only the weighted additive model of Tiwari et al. seems to be applicable. However, the shortcoming of the additive model is that the summation of quasiconcave functions may not be quasiconcave. This study proposes a novel weighted max–min model for fuzzy goal programming (FGP) and for fuzzy multiple objective decision-making. The proposed model adapts well to even the most complicated membership functions. Numerical examples demonstrate that the proposed model can be effectively incorporated with other approaches to FGP and is superior to the weighted additive approach.     

Note on “Some models for deriving the priority weights from interval fuzzy preference relations”

Deriving accurate interval weights from interval fuzzy preference relations is key to successfully solving decision making problems. Xu and Chen (2008) proposed a number of linear programming models to derive interval weights, but the definitions for the additive consistent interval fuzzy preference relation and the linear programming model still need to be improved. In this paper, a numerical example is given to show how these definitions and models can be improved to increase accuracy. A new additive consistency definition for interval fuzzy preference relations is proposed and novel linear programming models are established to demonstrate the generation of interval weights from an interval fuzzy preference relation.     

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.     

Stackelberg solutions for fuzzy random two-level linear programming through level sets and fractile criterion optimization

This paper considers Stackelberg solutions for two-level linear programming problems under fuzzy random environments. To deal with the formulated fuzzy random two-level linear programming problem, an α-stochastic two-level linear programming problem is defined through the introduction of α-level sets of fuzzy random variables. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced and the α-stochastic two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through fractile criterion optimization in stochastic programming, the transformed stochastic two-level programming problem can be reduced to a deterministic two-level programming problem. An extended concept of Stackelberg solution is introduced and a numerical example is provided to illustrate the proposed method.     

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.     

An interactive fuzzy goal programming approach for multi-period multi-product production planning problem

This study addresses an interactive multiple fuzzy goal programming (FGP) approach to the multi-period multi-product (MPMP) production planning problem in an imprecise environment. The proposed model attempts to simultaneously minimize total production costs, rates of changes in labor levels, and maximizing machine utilization, while considering individual production routes of parts, inventory levels, labor levels, machine capacity, warehouse space, and the time value of money. Piecewise linear membership functions are utilized to represent decision maker’s (DM’s) overall satisfaction levels. A numerical example demonstrates the feasibility of applying the proposed model to the MPMP problem. Furthermore, the proposed interactive approach facilitates the DM with a systematic framework of decision making process which enables DM to modify the search direction to reach the most satisfactory results during solving process.     

Restoration of efficiency in a goal programming problem with linear fractional criteria

The problem resulting from a goal programming problem with linear fractional criteria is not easy to solve due to the non-linear constraints inherent in its formulation. This paper introduces a simple and reliable test to establish whether a linear fractional goal programming problem has solutions that verify all goals and, if so, how to find them by solving a linear programming problem. This paper also outlines a new technique for restoring efficiency based on a minimax philosophy. An example is presented.     

区间AHP权重计算的目标规划法

本文对区间 AHP的权重计算问题进行了研究 ,给出了几种新的权重计算方法 .每种方法求解一个线性目标规划问题得到各方案的区间权重 .文章最后给出了一个算例 .     

Priority based fuzzy goal programming technique to fractional fuzzy goals using dynamic programming

This paper describes the use of preemptive priority based fuzzy goal programming method to fuzzy multiobjective fractional decision making problems under the framework of multistage dynamic programming. In the proposed approach, the membership functions for the defined objective goals with fuzzy aspiration levels are determined first without linearizing the fractional objectives which may have linear or nonlinear forms. Then the problem is solved recursively for achievement of the highest membership value (unity) by using priority based goal programming methodology at each decision stages and thereby identifying the optimal decision in the present decision making arena. A numerical example is solved to represent potentiality of the proposed approach.     

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.     

一类系数为梯形模糊数的两层线性规划

讨论了一类系数为梯形模糊数的两层线性规划问题,首先是利用模糊结构元理论将梯形模糊数去模糊化,将其转化成常规的两层线性问题,并验证其去模糊化后的常规的两层线性规划的最优解与系数为梯形模糊数的两层线性规划问题的最优解一致,并给出具体的算法,数例进行验证.     

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.     

A linear programming priority method for a fuzzy transportation problem with non-linear constraints

Demand and supply pattern for most products varies during their life cycle in the markets. In this paper, the author presents a transportation problem with non-linear constraints in which supply and demand are symmetric trapezoidal fuzzy value. In order to reflect a more realistic pattern, the unit of transportation cost is assumed to be stochastic. Then, the non-linear constraints are linearized by adding auxiliary constraints. Finally, the optimal solution of the problem is found by solving the linear programming problem with fuzzy and crisp constraints and by applying fuzzy programming technique. A new method proposed to solve this problem, and is illustrated through numerical examples. Multi-objective goal programming methodology is applied to solve this problem. The results of this research were developed and used as one of the Decision Support System models in the Logistics Department of Kayson Co.     

An Algorithm for Solving the Linear Goal Programming Problem by Solving its Dual

Goal programming, and in particular lexicographic goal programming (i.e. goal programming within a so-called ‘pre-emptive priority’ structure or having non-Archimedean weights), has become one of the most widely used of the approaches for multi-objective mathematical programming. While also applicable to non-linear or integer models, most of the literature has considered the lexicographic linear goal-programming model and its solution via primal simplex-based methods. However, in many cases, enhanced efficiency (and significant additional flexibility) may be gained via an investigation of the dual of this problem. In this paper we consider an algorithm for solving such a dual and also indicate how it may be implemented on conventional (i.e. single objective) simplex software.