Fuzzy Goal Programming Problem Research Articles

Maritime Supply Chain Optimization by Using Fuzzy Goal Programming

Fuzzy goal programming has important applications in many areas of supply chain, logistics, transportation and shipping business. Business management has complications, and there exist many interactions between the factors of its components. The locomotive of world trade is maritime transport and approximately 90% of the products in the world are transported by sea. Optimization of maritime operations is a challenge in order to provide technical, operational and financial benefits. Fuzzy goal programming models attract interests of many scholars, therefore the objective of this paper is to investigate the problem of minimization of total cost and minimization of loss or damage of containers returned from destination port. There are various types of fuzzy goal programming problems based on models and solution methods. This paper employs fuzzy goal programming with triangular fuzzy numbers, membership functions, constraints, assumptions as well as the variables and parameters for optimizing the solution of the model problem. The proposed model presents the mathematical algorithm, and reveals the optimal solution according to satisfaction rank from 0 to 1. Providing a theoretical background, this study offers novel ideas to researchers, decision makers and authorities.

An Extended Multi-Objective Capacitated Transportation Problem with Mixed Constraints in Fuzzy Environment

In this paper, we study a multi-objective capacitated transportation problem (MOCTP) with mixed constraints. This paper is comprised of the modelling and optimisation of an MOCTP in a fuzzy environment in which some goals are fractional and some are linear. In real life application of the fuzzy goal programming (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 or uncertain. Also, we developed the concept of linearisation of fractional goal for solving the MOCTP. In this paper imprecision of the parameter is handled by the concept of fuzzy set theory by considering these parameters as a trapezoidal fuzzy number. α-cut approach is used to get the crisp value of the parameters. Numerical examples are used to illustrate the method for solving MOCTP.

Goal programming approach for solving multi-objective fractional transportation problem with fuzzy parameters

In this paper, authors studied a goal programming approach for solving multi-objective fractional transportation problem by representing the parameters (γ, δ) in terms of interval valued fuzzy numbers. Fuzzy goal programming problem with multiple objectives is difficult for the decision makers to determine the goal valued of each objective precisely. The proposed model presents a special type of non-linear (hyperbolic) membership functions to solve multi-objective fractional transportation problem with fuzzy parameters. To illustrate the proposed method numerical examples are solved.

A joint maxmin-lexicographic maximisation approach in fuzzy goal programming using dominance possibility and necessity criteria

In this paper, a new approach for solving fuzzy goal programming problems is introduced. The coefficients and the aspiration level of each fuzzy goal are considered either trapezoidal or triangular fuzzy numbers. Four dominance criteria (dominance possibility, strict dominance possibility, dominance necessity, and strict dominance necessity) are utilised for comparing the fuzzy numbers. The proposed approach is based on merging the maxmin approach and the lexicographic approach in a two-phase process. The first phase applies the maxmin technique by maximising the minimum achievement degree of the fuzzy goals. The second phase lexicographically maximises the achievement degrees of the fuzzy goals according to their preemptive priorities. This methodology provides the decision maker with the advantage of improving the results of his preemptive priority structure model by initially maximising the lowest achievement of the fuzzy goals, and hence guarantee that the ultimate achievement of any fuzzy goal will never be lower than a specific percentage of the achieved maxmin value. The suggested approach is illustrated by a numerical example.

A joint maxmin-lexicographic maximisation approach in fuzzy goal programming using dominance possibility and necessity criteria

In this paper, a new approach for solving fuzzy goal programming problems is introduced. The coefficients and the aspiration level of each fuzzy goal are considered either trapezoidal or triangular fuzzy numbers. Four dominance criteria (dominance possibility, strict dominance possibility, dominance necessity, and strict dominance necessity) are utilised for comparing the fuzzy numbers. The proposed approach is based on merging the maxmin approach and the lexicographic approach in a two-phase process. The first phase applies the maxmin technique by maximising the minimum achievement degree of the fuzzy goals. The second phase lexicographically maximises the achievement degrees of the fuzzy goals according to their preemptive priorities. This methodology provides the decision maker with the advantage of improving the results of his preemptive priority structure model by initially maximising the lowest achievement of the fuzzy goals, and hence guarantee that the ultimate achievement of any fuzzy goal will never be lower than a specific percentage of the achieved maxmin value. The suggested approach is illustrated by a numerical example.

Fuzzy Goal Programming Problem Based on Minmax Approach for Optimal System Design

Every system in nature evolved in order to carry on their existence and reach their targets with minimal losses. The fundamental condition of a system’s success lies on making the correct decision by evaluating multiple, complicated, and conflicting goals based on the present constraints. Many mathematical programming problems are make up of objective functions combined by the decision maker based on the constrains. This study investigates how an optimal design can be reached based on Minmax approach. Goal Programming and a Fuzzy Goal Programming known as MA approach are used in this study. The solution of a problem organized as a Multiple De novo programming in order to determine the resource amounts for a business in handcrafts is carried out based on these two approaches. Budget constrain is organized as a goal to solve the problem based on MA approach, and a solution is proposed accordingly. The acquired results suggest that the solution results of Minmax Goal Programming and MA approach are the same.

An Improvement for Fuzzy Stochastic Goal Programming Problems

We examined the solution process for linear programming problems under a fuzzy and random environment to transform fuzzy stochastic goal programming problems into standard linear programming problems. A previous paper that revised the solution process with the lower-side attainment index motivated our work. In this paper, we worked on a revision for both-side attainment index to amend its definition and theorems. Two previous examples were used to examine and demonstrate our improvement over previous results. Our findings not only improve the previous paper with both-side attainment index, but also provide a theoretical extension from lower-side attainment index to the both-side attainment index.

Hedging an option portfolio with minimum transaction lots: A fuzzy goal programming problem

Options are designed to hedge against risks to their underlying assets such as stocks. One method of forming option-hedging portfolios is using stochastic programming models. Stochastic programming models depend heavily on scenario generation, a challenging task. Another method is neutralizing the Greek risks derived from the Black–Scholes formula for pricing options. The formula expresses the option price as a function of the stock price, strike price, volatility, risk-free interest rate, and time to maturity. Greek risks are the derivatives of the option price with respect to these variables. Hedging Greek risks requires no human intervention for generating scenarios. Linear programming models have been proposed for constructing option portfolios with neutralized risks and maximized investment profit. However, problems with these models exist. First, feasible solutions that can perfectly neutralize the Greek risks might not exist. Second, models that involve multiple assets and their derivatives were incorrectly formulated. Finally, these models lack practicability because they consider no minimum transaction lots. Considering minimum transaction lots can exacerbate the infeasibility problem. These problems must be resolved before option hedging models can be applied further. This study presents a revised linear programming model for option portfolios with multiple underlying assets, and extends the model by incorporating it with a fuzzy goal programming method for considering minimum transaction lots. Numerical examples show that current models failed to obtain feasible solutions when minimum transaction lots were considered. By contrast, while the proposed model solved the problems efficiently.

A hybrid probabilistic fuzzy goal programming approach for agricultural decision-making

In this paper, we present a probabilistic fuzzy goal programming model to capture different uncertainties in an agricultural decision-making environment. First, we construct the goals of the model as probabilistic fuzzy goals. Next, we convert the probabilistic fuzzy goal programming problem to a probabilistic constrained programming problem. While a deterministic solution to this problem cannot be derived, we use a hybrid approach comprising Monte-Carlo simulation and a real-coded genetic algorithm. The application of the proposed model and the solution technique is demonstrated by considering the agricultural planning of the Danton-II community development block of Paschim Medinipur District, West Bengal, India.

Probabilistic Fuzzy Goal Programming Problems Involving Pareto Distribution: Some Additive Approaches

In many real-life decision making problems, probabilistic fuzzy goal programming problems are used where some of the input parameters of the problem are considered as random variables with fuzzy aspiration levels. In the present paper, a linearly constrained probabilistic fuzzy goal programming programming problem is presented where the right hand side parameters in some constraints follows Pareto distribution with known mean and variance. Also the aspiration levels are considered as fuzzy. Further, simple, weighted, and preemptive additive approaches are discussed for probabilistic fuzzy goal programming model. These additive approaches are employed to aggregating the membership values and form crisp equivalent deterministic models. The resulting models are then solved by using standard linear mathematical programming techniques. The developed methodology and solution procedures are illustrated with a numerical example.

Multi-Objective Fuzzy Chance Constrained Fuzzy Goal Programming for Capacitated Transportation Problem

paper presents a multi-objective fuzzy chance constrained capacitated transportation problem based on fuzzy goal programming problem with capacity restrictions on commodities which are shipped from different sources to different destinations. The capacity of each origin and the demand of each destination are considered random in nature with fuzzy normal stochastic parameters following normal distribution with fuzzy mean and fuzzy variance respectively. These inequality constraints are also considered as fuzzy probabilistic in nature assuming to be triangular fuzzy numbers. Further, the supply and demand constraints are converted into equivalent deterministic forms. Then, we define the fuzzy goal tolerance limit of each of the objective functions which are then characterized by the associated membership functions. In the solution process, the fuzzy parameters are defuzzied by applying graded mean integration method which provided a satisfactory result. Further, an illustrative example is solved to demonstrate the effectiveness of the proposed model. Keywordsgoal programming, fuzzy chance constrained programming, capacitated transportation problem, multi objective decision making, graded mean integration method.

Fuzzy Multi-fractional Programming for Land Use Planning in Agricultural Production System

In this paper, we present a multi-objective linear fractional programming (MOLFP) approach for multi-objective linear fuzzy goal programming (MOLFGP) problem. Here, we consider a problem in which a set of pair of goals are optimized in ratio rather than optimizing them individually. In particular, we consider the optimization of profit to cash expenditure and crop production in various seasons to land utilization as a fractional objectives and used remaining goals in its original form. Further, the goals set in agricultural production planning are conflicting in nature; thus we use the concept of conflict and nonconflict between goals for computation of appropriate aspiration level. The method is illustrated on a problem of agricultural production system for comparison with Biswas and Pal [1] method to show its suitability.

Interval Goal Programming Approach to Multiobjective Fuzzy Goal Programming Problem with Interval Weights

Abstract This article presents interval goal programming (IGP) approach for solving linear multiobjective fuzzy goal programming problem with interval weights. In the proposed approach, interval weights for achievement of fuzzy goals to their aspired levels on the basis of their relative importance are considered in an uncertain environment. In the model formulation of the problem, the membership functions for each of the fuzzy goals are defined first. Then, the membership functions are transformed into membership goals by assigning the highest membership value (unity) and introducing under-and over-deviational variables to each of them. In the solution process, the interval weights (derived from pairwise interval judgment matrix) associated with the unwanted deviational variables is introduced in the goal achievement function for minimizing them to reach the aspired goal levels of the problem. To illustrate the proposed approach, a numerical example is solved.

Multi-Objective Chance Constrained Capacitated Transportation Problem based on Fuzzy Goal Programming

This paper presents chance constrained multi-objective capacitated transportation problem based on fuzzy goal programming problem. Generally, in transportation problem the capacity of each origin and the demand of each destination are random in nature. The inequality constraints representing supplies and demands are probabilistically described. In many real situations, there are capacity restrictions on units of commodities which are shipped from different sources to different destinations. In the model formulation, supply and demand constraints are converted into equivalent deterministic forms. Then, we define the fuzzy goal levels of the objective functions. The fuzzy objective goals are then characterized by the associated membership functions. In the solution process, two fuzzy goal programming models are considered by minimizing negative deviational variables to obtain compromise solution. Distance function is used in order to obtain the most compromise optimal solution. In order to demonstrate the effectiveness of the proposed approach, an illustrative example of chance constrained multi- objective capacitated transportation problem is solved.

Two Weighted Fuzzy Goal Programming Methods to Solve Multiobjective Goal Programming Problem

We propose two new methods to find the solution of fuzzy goal programming (FGP) problem by weighting method. Here, the relative weights represent the relative importance of the objective functions. The proposed methods involve one additional goal constraint by introducing only underdeviation variables to the fuzzy operator λ (resp., 1‐λ), which is more efficient than some well‐known existing methods such as those proposed by Zimmermann, Hannan, Tiwari, and Mohamed. Mohamed proposed that every fuzzy linear program has an equivalent weighted linear goal program where the weights are restricted as the reciprocals of the admissible violation constants. But the above proposition of Mohamed is not always true. Furthermore, the proposed methods are easy to apply in real‐life situations which give better solution in the sense that the objective values are sufficiently closer to their aspiration levels. Finally, for illustration, two real examples are used to demonstrate the correctness and usefulness of the proposed methods.

A satisficing method for fuzzy goal programming problems with different importance and priorities

For treating multiple objectives decision making problems with fuzzy goals and different importance, various kinds of fuzzy goal programming (FGP) models have been developed in the past three decades. Among them, two most widely used methods are: (1) weighted FGP, where the importance of the objectives is represented by weights (2) preemptive priority (also known as “Lexicographic”) FGP, where the priority levels of the goals are set in advance, and the trade-offs among goals placed at different priority levels are implemented strictly. This article presents a satisficing method based on FGP model, which holds that a more important objective has to be achieved as much as possible. The relaxed preemptive priority requirement in the proposed model provides a more efficient, flexible and practicable decision support compared to the weighted and the lexicographic models. In addition, the trade-off between optimization and importance requirement can be realized by the regulation parameter in the presented method. The performance of this method is evaluated by comparing its result with those of the six existing models in the literature.

Fuzzy multi-objective linear programming applying to crop area planning

Abstract Crop area planning plays significant role in agricultural water management. During the planning, because of ambiguous or uncertain information caused by the vagueness of decision makers’ subjective preference or the uncertainty of objective information, conventional multi-objective linear programming (MOLP) model is not suitable for such decision-making in such fuzzy environment. In this study, we proposed the fuzzy multi-objective linear programming (FMOLP) model with triangular fuzzy numbers and transformed the FMOLP model and its corresponding fuzzy goal programming (FGP) problem to crisp ones which can be solved by the conventional programming methods. The FMOLP model was applied to crop area planning of Liang Zhou region, Gansu province of northwest China, and then the optimal cropping patterns under different water-saving levels and satisfaction grades for water resources availability of the decision makers (DM) were obtained. Compared to the MOLP model, the FMOLP model itself expresses the fuzzy information effectively, and its solutions can represent the DMs satisfactory degree of the subjective preference and propose alternative solutions for better decision support when applied in the crop area planning.

Multi-item inventory models with price dependent demand under flexibility and reliability consideration and imprecise space constraint: A geometric programming approach

In this paper, multi-item economic production quantity (EPQ) models with selling price dependent demand, infinite production rate, stock dependent unit production and holding costs are considered. Flexibility and reliability consideration are introduced in the production process. The models are developed under two fuzzy environments–one with fuzzy goal and fuzzy restrictions on storage area and the other with unit cost as fuzzy and possibility–necessity restrictions on storage space. The objective goal and constraint goal are defined by membership functions and the presence of fuzzy parameters in the objective function is dealt with fuzzy possibility/necessity measures. The models are formed as maximization problems. The first one—the fuzzy goal programming problem is solved using Fuzzy Additive Goal Programming (FAGP) and Modified Geometric Programming (MGP) methods. The second model with fuzzy possibility/necessity measures is solved by Geometric Programming (GP) method. The models are illustrated through numerical examples. The sensitivity analyses of the profit function due to different measures of possibility and necessity are performed and presented graphically.

WEIGHTED ADDITIVE MODELS FOR SOLVING FUZZY GOAL PROGRAMMING PROBLEMS

Weighted additive models are well known for dealing with multiple criteria decision making problems. Fuzzy goal programming is a branch of multiple criteria decision making which has been applied to solve real life problems. Several weighted additive models are introduced to handle fuzzy goal programming problems. These models are based on two approaches in fuzzy goal programming namely goal programming and fuzzy programming techniques. However, some of these models are not able to solve all kinds of fuzzy goal programming problems and some of them that appear in current literature suffer from a lack of precision in their formulations. This paper focuses on weighed additive models for fuzzy goal programming. It explains the oversights within some of them and proposes the necessary corrections. A new improved weighted additive model for solving fuzzy goal programming problems is introduced. The relationships between the new model and some of the existing models are discussed and proved. A numerical example is given to demonstrate the validity and strengths of the new model.

A fuzzy goal programming approach with priority for channel allocation problem in steel industry

A model is presented to address a steel supplier's channel allocation problem that includes decisions of channel mix and capacity allocation for each distribution channel. The problem has been formulated as a fuzzy mixed integer multiple goal programming problem that includes business competitive advantages such as maximizing net profits, minimizing the rate of end user claims, and minimizing the rate of late lading, and is subject to constraints regarding manufacturing capacity, customer's demand, channel capacity, channel quota flexibility, budget limitations, and so on. Realistic data from Taiwan's largest steel company is implemented for the effectiveness of the model. The proposed model can also be applied to the allocation problem in other industries with preemptive priority and desired achievement level in a fuzzy environment.