Fuzzy Multi-objective Linear Programming Problem Research Articles

A novel approach for multi-objective linear programming model under spherical fuzzy environment and its application

Every decision-making process particularly those involving real-life issues is disproportionately plagued by uncertainty. It is also unavoidable and obvious. Since its conception are several ways for representing uncertainty have been proposed by numerous academics to cope with uncertainty. Fuzzy sets and hierarchical such as picture fuzzy sets stand out among them as excellent representation techniques for modeling uncertainty. However, there are several significant drawbacks to the current uncertainty modeling techniques. Due to its vast versatility and benefits we here embrace the idea of the spherical fuzzy set, an extension of the picture fuzzy set. On the other hand amid uncertainty in real life the multi-objective plays a critical role. In this research paper determining a Multi-Objective Linear Programming Problem of Spherical fuzzy sets serves to stimulate nous. The score function corresponding to the degree positive, negative and neutral is the foundation upon which the suggested approach is developed. Additionally we apply the suggested strategy to the solution of the multi-objective linear programming problem to demonstrate its superiority through certain numerical examples. Maximization or Minimizing of the cost is the primary goal of the multi-objective linear programming problem. Using an explicitly defined score function the suggested solution transformed the Spherical Fuzzy Multi-Objective Linear Programming Problem into a Crisp Multi-Objective Linear Programming Problem (CMOLPP). We establish some theorems to show that the efficient solution of CMOLPP is likewise an efficient solution of SFMOLPP. The CMOLPP is then further simplified into a single-objective Linear Programming Problem (LPP) thus we revamp the modified Zimmermann’s approach in the environment of a nonlinear membership function with the aid of the suggested technique. It is possible to simply solve this single-objective LPP using any software or standard LPP algorithm. The suggested approach achieves the fuzzy optimum result without altering the nature of the issue. An application of the suggested approach has been used to illustrate it and its results have been distinguished from those of other preexisting methods found in the literature. To determine the importance of the suggested technique which adjudicate thorough theorem and result analysis is conducted.

Fuzzy Mathematical Approach for Solving Multi-Objective Fuzzy Linear Fractional Programming Problem with Trapezoidal Fuzzy Numbers

Fuzzy Mathematical Approach for Solving Multi-Objective Fuzzy Linear Fractional Programming Problem with Trapezoidal Fuzzy Numbers

A New Method to Solve Fully Intuitionistic Fuzzy Multi-objective Linear Programming Problem Through Conflict and Non-conflict

A New Method to Solve Fully Intuitionistic Fuzzy Multi-objective Linear Programming Problem Through Conflict and Non-conflict

An ε-Constraint Method for Multiobjective Linear Programming in Intuitionistic Fuzzy Environment

Effective decision-making requires well-founded optimization models and algorithms tolerant of real-world uncertainties. In the mid-1980s, intuitionistic fuzzy set theory emerged as another mathematical framework to deal with the uncertainty of subjective judgments and made it possible to represent hesitancy in a decision-making problem. Nowadays, intuitionistic fuzzy multiobjective linear programming (IFMOLP) problems are a topic of extensive research, for which a considerable number of solution approaches are being developed. Among the available solution approaches, ranking function-based approaches stand out for their simplicity to transform these problems into conventional ones. However, these approaches do not always guarantee Pareto optimal solutions. In this study, the concepts of dominance and Pareto optimality are extended to the intuitionistic fuzzy case by using lexicographic criteria for ranking triangular intuitionistic fuzzy numbers (TIFNs). Furthermore, an intuitionistic fuzzy ε-constraint method is proposed to solve IFMOLP problems with TIFNs. The proposed method is illustrated by solving two intuitionistic fuzzy transportation problems addressed in two studies (S. Mahajan and S. K. Gupta’s, “On fully intuitionistic fuzzy multiobjective transportation problems using different membership functions,” Ann Oper Res, vol. 296, no. 1, pp. 211–241, 2021, and Ghosh et al.’s, “Multi-objective fully intuitionistic fuzzy fixed-charge solid transportation problem,” Complex Intell Syst, vol. 7, no. 2, pp. 1009–1023, 2021). Results show that, in contrast with Mahajan and Gupta’s and Ghosh et al.’s methods, the proposed method guarantees Pareto optimality and also makes it possible to obtain multiple solutions to the problems.

On generating fuzzy Pareto solutions in fully fuzzy multiobjective linear programmingviaa compromise method

In the present paper, it is unified and extended recent contributions on fully fuzzy multiobjective linear programming, and it is proposed a new method for obtaining fuzzy Pareto solutions of a fully fuzzy multiobjective linear programming problem. For its formulation, triangular fuzzy numbers and variables are combined with fuzzy partial orders and fuzzy arithmetic, and no ranking functions are required. By means of solving related crisp multiobjective linear problems, it is provided algorithms to generate fuzzy Pareto solutions; in particular, to generate compromise fuzzy Pareto solutions, what is a novelty in this field.

An Application of Fully Intuitionistic Fuzzy Multi-objective Linear Fractional Programming Problem in E-education System

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

Fuzzy Multi-Objective Linear Programming Problem Using DM's Perspective

In this paper, a two-stage method has been proposed for solving Fuzzy Multi-objective Linear Programming Problem (FMOLPP) with Interval Type-2 Triangular Fuzzy Numbers (IT2TFNs) as its coefficients. In the first stage of problem solving, the imprecise nature of the problem has been handled. All technological coefficients given by IT2TFNs are first converted to a closed interval and then the objectives are made crisp by reducing a closed interval into a crisp number and constraints are made crisp by using the concept of acceptability index. The amount by which a specific constraint can be relaxed is decided by the decision maker and thus the problem reduces to a crisp multi-objective linear programming problem (MOLPP). In the second stage of problem solving, the multi-objective nature of the problem is handled by using fuzzy mathematical programming approach. In order to explain the methodology, two numerical examples of the proposed methodology in Production planning and Diet planning problems have also been worked out in this paper.

Solving Fuzzy Multi-Objective Linear Programming Problem by Applying Statistical Method

In this paper, the statistical averaging method and the new statistical averaging methods have been used to solve the fuzzy multi-objective linear programming problems. These methods have been applied to form a single objective function from the fuzzy multi-objective linear programming problems. At first, a numerical example of solving fuzzy multi-objective linear programming problem has been provided to validate the maximum risk reduction by the proposed method. The proposed method has been applied to assess the risk of damage due to natural calamities like flood, cyclone, sidor, and storms at the coastal areas in Bangladesh. The proposed method of solving the fuzzy multi-objective linear programming problems by the statistical method has been compared with the Chandra Sen’s method. The numerical results show that the proposed method maximizes the risk reduction capacity better than Chandra Sen’s method.

Finding compromise solutions for fully fuzzy multi-objective linear programming problems by using game theory approach

Solving multi-objective linear programming (MOLP) problems and fully fuzzy multi-objective linear programming (FFMOLP) problems involves the trade-off process among several objectives. A new algorithm extended where FFMOLP problems are solved using a 2-player zero-sum game approach to deal with this case. Firstly, The FFMOLP problem is separated into a certain number of fully fuzzy linear programming (FFLP) problems and each is solved by applying any method. After forming a ratio matrix, a game theory approach is applied for finding the weights of objective functions and a weighted LP problem is constructed by these weights. Solving the weighted LP problem, a fuzzy compromise solution of the FFMOLP problem is found. Constructing different ratio matrices, it is also possible to obtain more than one compromise solution to be offered to the decision-maker(s). Some examples are given to show the applicability of the algorithm.

Fuzzy solution of fully fuzzy multi-objective linear fractional programming problems

In this study, we present a novel method for solving fully fuzzy multi-objective linear fractional programming problems without transforming to equivalent crisp problems. First, we calculate the fuzzy optimal value for each fractional objective function and then we convert the fully fuzzy multi-objective linear fractional programming problem to a single objective fuzzy linear fractional programming problem and find its fuzzy optimal solution which inturn yields a fuzzy Pareto optimal solution for the given fully fuzzy multi-objective linear fractional programming problem. To demonstrate the proposed strategy, a numerical example is provided.

Solving fuzzy multiobjective linear bilevel programming problems based on the extension principle

Fuzzy multiobjective linear bilevel programming (FMOLBP) problems are studied in this paper. The existing methods replace one or some deterministic model(s) instead of the problem and solve the model(s). Doing this work, we lose much information about the compromise decision, and it does not make sense for the uncertain conditions. To overcome the difficulties, Zadeh’s extension principle is applied to solve the FMOLBP problems. Two crisp multiobjective linear three-level programming problems are proposed to find the lower and upper bound of its objective values in different levels. The problems are reduced to some linear optimization problems using one of the scalarization approaches, called the weighting method, the dual theory, and the vertex enumeration method. The lower and upper bounds are estimated by the resolution of the corresponding linear optimization problems. Hence, the membership functions of compromise objective values are produced, which is the main contribution of this paper. This technique is applied for the problem for the first time. This method applies all information of a fuzzy number and does not estimate it by a crisp number. Hence, the compromise decision resulted from the proposed method is consistent with reality. This point can minimize the gap between theory and practice. The results are compared with the results of existing approaches. It shows the efficiency of the proposed approach.

Solving intuitionistic fuzzy multiobjective linear programming problem under neutrosophic environment

The existence of neutral /indeterminacy degrees reflects the more practical aspects of decision-making scenarios. Thus, this paper has studied the intuitionistic fuzzy multiobjective linear programming problems (IFMOLPPs) under neutrosophic uncertainty. To highlight the degrees of neutrality in IFMOLPPs, we have investigated the neutrosophic optimization techniques with intuitionistic fuzzy parameters. The marginal evaluation of each objective is determined by three different membership functions, such as truth, indeterminacy, and falsity membership degrees under the neutrosophic environment. The marginal evaluation of each objective function is elicited by various sorts of membership functions such as linear, exponential, and hyperbolic types of membership functions, which signifies an opportunity for decision-makers to select the desired membership functions. The developed neutrosophic optimization technique is implemented on existing numerical problems that reveal the validity and applicability of the proposed methods. A comparative study is also presented with other approaches. At last, conclusions and future research directions are addressed based on the proposed work.

An Interactive Method for Solving Fuzzy Multi-objective Linear Programming Problems

An Interactive Method for Solving Fuzzy Multi-objective Linear Programming Problems

A solving approach for fuzzy multi-objective linear fractional programming and application to an agricultural planting structure optimization problem

In this paper, an algorithm is presented to solve fuzzy multi-objective linear fractional programming (FMOLFP) problems through an approach based on superiority and inferiority measures method (SIMM). In the model for the proposed approach, each of fuzzy goals defined for the fractional objectives and some of constraints have fuzzy numbers. To achieve the highest membership value, SIMM is adopted to deal with fuzzy number in constraints, then a linear goal programming methodology is introduced to solve the problem in which the fractional objectives is fuzzy goals. A case of agricultural planting structures optimization problem is solved to illustrate the application of the algorithm. The results show that winter wheat and summer corn acreage should be 38,386.4 ha, and cotton acreage should be 20,669.6 ha. Because of high risk in cotton cultivation at present, the ratio of grain planted area to cotton planted area is unreasonable. An improved support in policy is necessary for the government to enhance the enthusiasm of farmers to plant cotton and sustain the development of cotton market in the long term.

Solving fully fuzzy multi-objective linear programming problem with fuzzy dominant degrees

In this paper, we investigate all relative relationships between two fuzzy numbers. Then, we introduce new relative measures to compare two fuzzy numbers instead of using absolute value to represent the fuzzy number. These measures address the dominant level that one fuzzy number is better than the other in terms of its position and shape. The so-called absolute fuzzy dominant degree and relative fuzzy dominant degree are developed to measure the differences between two fuzzy numbers applying for different types of constraint. These measures could capture all the shape’s characteristics and relative positions of fuzzy numbers. Finally, the fully fuzzy multi-objective decision making (FFMODM) problem is solved by using these fuzzy dominant degrees. For validation, we compare our approach to the fuzzy ranking method of the linear ranking function. Our obtained results show better performance.

Optimal Diet Modeling for Diabetics in Fuzzy Environment Using Multi-Objective Fuzzy Linear Programming

Our purpose in this paper is to present a proper diet model for diabetics in the fuzzy environment. In previous studies many efforts were made regarding preparation of a diet model appropriate for different age groups .but in most studies the daily nutrient intake decisions were made based on crisp data. By prescribing a diet based on crisp data, some of realities are neglected. For the same reason, we dealt with human diet problem through fuzzy approach. Also due to the importance of nutrition for individuals, especially patients with diabetic diseases, we have provided nutrition model as multi-objective fuzzy linear programming problem in which minimization of saturated fats, Sugar, and maximization of carbohydrate and total fiber are considered as our objectives. Results indicated uncertainty about amount of nutrients and their intake affects diet quality making it more realistic.

Optimizing Diet Problem for Cardiovascular Disease in Fuzzy Environment by Using Multi-Objective Fuzzy Linear Programming

Our purpose in this paper is to present a proper diet model for cardiovascular patients in the fuzzy environment. In previous studies many efforts were made regarding preparation of a diet model appropriate for different age groups. But in most studies the daily nutrient intake decisions were made based on crisp data. By prescribing a diet based on crisp data, some of realities are neglected. For the same reason, we dealt with human diet problem through fuzzy approach. Also due to the importance of nutrition for individuals, especially patients with cardiovascular diseases, we have provided nutrition model as multi-objective fuzzy linear programming problem in which minimization of saturated fats, salt, monosaccharide and amount of received calery are considered as our objectives. First we review literature of fuzzy mathematics and then we will discuss nutrition model of cardiovascular patients in the fuzzy environment. Results indicated uncertainty about amount of nutrients and their intake affects diet quality making it more realistic.

A new method for solving fuzzy multi-objective linear programming problems

The purpose of this paper is to develop a new two-stage method for fuzzy multi-objective linear program and apply to engineering project portfolio selection. In the fuzzy multi-objective linear program, all the objective coefficients, technological coefficients and resources are trapezoidal fuzzy numbers (TrFNs). An order relationship for TrFNs is introduced by using the interval expectation of TrFNs. In the first stage, the fuzzy multi-objective linear program with TrFNs is transformed into an interval multi-objective linear program according to the order relationship of TrFNs. Combining the ranking order relation between intervals with the satisfactory crisp equivalent forms of interval inequality relations, the interval multi-objective linear program is further transformed into a crisp multi-objective linear program. In the second stage, the positive and negative ideal solutions are calculated as well as the closeness degrees from the positive ideal solution to all objectives on the basis of the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution). Then, using the closeness degrees, we convert the crisp multi-objective linear program into mono-objective program to solve. The proposed method is not only mathematically rigorous, but also can adequately consider the acceptance degree of decision maker that the fuzzy constraints may be violated. The other possible cases of the fuzzy multi-objective linear program are also discussed. The proposed method is illustrated by means of a project portfolio selection problem.

Q-Taylor series method for solving fuzzy multiobjective linear fractional programming problem

In this work, we present a q -Taylor series method for fuzzy multiobjective linear fractional programming problems (FMOLFPPs). In q -calculus, q -Taylor series is a q -series expansion of a function with respect to q -derivatives. In the proposed approach, membership functions are defined to be piecewise linear. Membership functions associated with each objective of fuzy multiobjective linear fractional programming problem transformed by using q -Taylor series are unified. Thus, the problem is reduced to a single objective. To show the efficiency of the q -Taylor series method, we applied the method to some problems.

An Approach for Solving Fuzzy Multi-Objective Linear Fractional Programming Problems

In this paper, an attempt has been taken to develop a method for solving fuzzy multi-objective linear fractional programming (FMOLFP) problem. Here, at first the FMOLFP problem is converted into (crisp) multi-objective linear fractional programming (MOLFP) problem using the graded mean integration representation (GMIR) method proposed by Chen and Hsieh. That is, all the fuzzy parameters of FMOLFP problem are converted into crisp values. Then the MOLFP problem is transformed into a single objective linear programming (LP) problem using a proposal given by Nuran Guzel. Finally the single objective LP problem is solved by regular simplex method which yields an efficient solution of the original FMOLFP problem. To show the efficiency of our proposed method, three numerical examples are illustrated and also for each example, a comparison is drawn between our proposed method and the respected existing method.