Fuzzy Multi-objective Transportation Problem Research Articles

Improved fuzzy multi-objective transportation problem with Triangular fuzzy numbers

The present study investigates a Multi-Objective Transportation Problem within a fuzzy environment. The cost of transportation, supply, and demand data are assumed to be inaccurate due to market variations. As a result, the parameters are imprecise or fuzzy data. We offer a multi-objective, balanced transportation problem during this work, where all the parameters are fuzzy numbers. Following a mathematical formulation, fuzzy arithmetic will be used to divide the Fuzzy MOTP into three levels MOTP (lower, medium, upper). After reducing the problem to a crisp MOTP and applying a harmonic mean to each objective function, a suggested solution procedure is presented. Determining the optimal solutions for the FMOTP under unknown situations is, thus, the most important objective of this research.

Dual hesitant fuzzy set in multi-objective transportation problems in time sequence frame work

Decision makers may have different information in different situations when making decisions about the same attributes, depending on the circumstances. A time-sequential framework is necessary for navigating these kinds of circumstances. We established the notion of a Time-sequential dual hesitant fuzzy set (TS-DHFS) to illustrate the significance of time sequence in decision-making with the aid of a dual hesitant fuzzy set (DHFS), one of the important tools to handle the uncertainty embedded in the real world. The proposed set describes the hesitant situations more accurately by inducing a time sequence framework. We defined the fundamental operations, two score function for ranking and a series distance measures for the proposed set. For the pragmatic application of our proposed set, we devised the mathematical model in three different types of Time-sequential dual hesitant fuzzy multi-objective transportation problems (TS-DHF-MOTPs). The proposed three models are classified based on the nature of the parameters such as transportation cost (TC), transportation time (TT), availability, and demand. The goal of the proposed transportation systems is TC and TT simultaneously in a single framework. We established an approach utilizing one of the ranking functions that have been proposed, followed by the fuzzy programming (FP) and weighted sum technique (WST), to optimize the constructed models. Additionally, numerical computations are also carried out to emphasize the efficiency and benefits of MOTP under the TS-DHF environment.

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.

A New Approach to Solve Fully Fuzzy Multi-Objective Transportation Problem

The transportation problem is the problem of transferring goods from several sources or producers to multiple destinations or consumers in a cost-effective way, which is one of the most important problems in the supply chain management problems. The application of this problem in addition to the distribution of goods in the location and production planning problems is also important. Many real-life transportation problems encounter multiple, conflicting, and incommensurable objective functions. In addition, in real applications, due to lack of information, it is not possible to accurately estimate the parameters of this problem. Therefore, the main goal of this paper is to find the Pareto optimal solutions of fully fuzzy multi-objective transportation problem under the conditions of uncertainty. In accordingly, a new approach based on nearest interval approximation is proposed to solve the problem. Numerical examples are provided to illustrate the proposed approach and results.

Fuzzy harmonic mean technique for solving fully fuzzy multi-objective transportation problem

In this research article, an algorithm is developed using a fuzzy harmonic mean technique to solve a fully fuzzy multi-objective transportation problem (FFMOTP). First, the FFMOTP will be formalized mathematically, and then by using fuzzy arithmetic, it will be decomposed into three-level (lower, middle, & upper) multi-objective linear programming problems. After then, by using the fuzzy harmonic mean as a tool, the three-level multi-objective linear programming problems are converted into three-level single objective linear programming problems. Further, these single objective linear programming problems for all the three levels are solved to give the Combine fuzzy optimal solution. In the end, two numerical examples are solved to elaborate our proposed algorithm completely and compare the result obtained by our proposed algorithm with the results obtained using the other methods.

From fuzzy transportation problem to non-linear intuitionistic fuzzy multi-objective transportation problem: A literature review

ABSTRACT To deal with the unpredictability of certain parameters of the transportation problem like fuel charges, climatic conditions, supply and demand of the product, etc., the concept of fuzzy transportation problem came into the limelight. It has been proved to be an efficient approach and extended in literature by various scholars. In the given paper, the existing methods to obtain the solution of different types of fuzzy transportation problem and its extensions are critically reviewed. Intuitionistic fuzzy transportation problems having single and multiple objectives have been focused upon. A review of the past ten years has been done to know the recent advancement. Tabular and graphical comparisons of different approaches are incorporated to understand their benefits and drawbacks.

Genetic Algorithm Based Solution of Fuzzy Multi-Objective Transportation Problem

Optimizing problems in the modern era, the single objective optimization problems are insufficient to hold the full data of the problem. Therefore, multi-objective optimization problems come to the rescue. Similarly, in daily life problems, the parameters used in the optimization problem are not always fixed but there may be some uncertainty and it can characterize by fuzzy number. This work underlines the genetic algorithm (GA) based solution of fuzzy transportation problem with more than one objective. With a view to providing the multifaceted choices to decision-maker (DM), the exponential membership function is used with the decision-makers desired number of cases which consisted of shape parameter and aspiration level. Here, we consider the objective functions which are non-commensurable and conflict with each other. To interpret, evaluate and exhibit the usefulness of the proposed method, a numerical example is given.

Solving the fully fuzzy multi-objective transportation problem based on the common set of weights in DEA

A transportation problem basically deals with the problem which aims to minimize the total transportation cost or maximize the total transportation profit of distributing a product from a number of sources or origins to a number of destinations. While, in general, most of the real life applications are modeled as a transportation problem (TP) with the multiple, conflicting and incommensurate objective functions. On the other hand, for some reason such as shortage of information, insufficient data or lack of evidence, the data of the mentioned problem are not always exact but can be fuzzy. This type of problem is called fuzzy multi-objective transportation problem (FMOTP). There are a few approaches to solve the FMOTPs. In this paper, a new fuzzy DEA based approach is developed to solve the Fully Fuzzy MOTPs (FFMOTPs) in which, in addition to parameters of the MOTPs, all of the variables are considered fuzzy. This approach considers each arc in a FFMOTP as a decision making unit which produces multiple fuzzy outputs using the multiple fuzzy inputs. Then, by using the concept of the common set of weights (CSW) in DEA, a unique fuzzy relative efficiency is defined for each arc. In the following, the unique fuzzy relative efficiency is considered as the only attribute for the arcs. In this way, a single objective fully fuzzy TP (FFTP) is obtained that can be solved using the existing standard algorithms for solving this kind of TPs. A numerical example is provided to illustrate the developed approach.

Fuzzy arithmetic DEA approach for fuzzy multi-objective transportation problem

In this paper, a transportation problem (TP) with fuzzy costs in the presence of multiple and conflicting objectives is investigated. In fact, a fuzzy data envelopment analysis (DEA) approach is proposed to solve the fuzzy multi-objective TP (FMOTP). To this end, each arc in FMOTP will be considered as a decision-making unit (DMU). Next, those objective functions that needs to be maximized will be used to define the outputs of DMU and those that needs to be minimized will be used to define the inputs of DMU. Consequently, two different fuzzy efficiency scores will be derived for each arc by solving fuzzy DEA models. So, a unique fuzzy attribute will be defined for each arc by combining the resulting fuzzy efficiency scores. Therefore, the FMOTP will be converted into a single objective fuzzy TP that can be solved using the standard algorithms. Finally, using a numerical example the proposed approach has been illustrated.

Solution of Multi Objective Transportation Problem Under Fuzzy Environment

The crisp transportation problem is one of the special case of linear programming problems. It is explained with the hypothesis that the cost parameters are indicated inaccurate way. But in the present world, the transportation parameters may be uncertain due to many uncontrolled factors; this gives rise to fuzzy environment. Here, we proposed a method for the solution of fully fuzzy multi objective transportation problems involving triangular fuzzy numbers. We solve the fully fuzzy multi objective transportation problems without converting it into crisp form. A numerical example is discussed to demonstrate the theory formulated in this article.

Correction to: On fully intuitionistic fuzzy multiobjective transportation problems using different membership functions

This erratum is published because vendor overlooked corrections related to Theorems 3 to 5.

On fully intuitionistic fuzzy multiobjective transportation problems using different membership functions

Intuitionistic fuzzy set theory is a generalized version of classical fuzzy set theory, where membership degree of acceptance and membership degree of rejection are both measured. In the present study, we formulate a balanced transportation problem with multiple objectives having all parameters and variables as intuitionistic fuzzy numbers. The problem is first reduced to a crisp multiobjective transportation problem using accuracy function on each objective function and then an algorithm is proposed to solve the problem. In the solution procedure, linear, exponential and hyperbolic membership functions are used to tackle intuitionistic fuzzy constraints related with each objective. To show the relations among the intuitionistic fuzzy transportation problem, its equivalent crisp formulation and the problems obtained by applying different membership functions, various theorems are established. Finally, two numerical examples are illustrated to clarify the steps involved in the proposed algorithm and to draw the comparison among linear, exponential and hyperbolic membership functions.

Fuzzy multi objective transportation problem – evolutionary algorithm approach

This paper deals with fuzzy multi objective transportation problem. An fuzzy optimal compromise solution is obtained by using Fuzzy Genetic Algorithm. A numerical example is provided to illustrate the methodology.

New approach for solving intuitionistic fuzzy multi-objective transportation problem

Multi-objective transportation problem (MOTP) under intuitionistic fuzzy (IF) environment is analysed in this paper. Due to the fluctuation of market scenario, we assume that the transportation cost, the supply and the demand parameters are not always precise. Hence, the parameters are imprecise, i.e., they are IF numbers. Considering the specific cut interval, the IF transportation cost matrix is converted to interval cost matrix in our proposed problem. Again, using the same concept, the IF supply and the IF demand of the MOTP are reduced to the interval form. Then the proposed MOTP is changed into the deterministic MOTP, which includes interval form of the objective functions. Two approaches, namely intuitionistic fuzzy programming and goal programming, are used to derive the optimal solutions of our proposed problem, and then the optimal solutions are compared. A numerical example is included to illustrate the feasibility and the applicability of the proposed problem. Finally, we present the conclusions with the future scopes of our study.

Fuzzy multi objective transportation model based on new ranking index on generalized LR fuzzy numbers

In this paper, we present a new method for analyzing a fuzzy multi objective transportation problem using a linear programming model based on a new method for ranking generalized LR fuzzy numbers. First, we present a new method for ranking generalized LR fuzzy numbers from its λ-cut.To the best of our knowledge till now there is no method in the literature to find the ranking order of the generalized LR fuzzy numbers to various linear and non-linear functions of LR fuzzy numbers from its  -cut based on area, mode and spread. Our method can efficiently rank various LR fuzzy numbers, their images and crisp numbers which are consider to be a special case of fuzzy numbers (normal/nonnormal, triangular/trapezoidal, and general), which could not be ranked by the existing ranking methods. Our ranking method also satisfies the linearity property condition. For the corroboration, we used some comparative examples of different existing methods to illustrate the advantages of the proposed method.The reference functions of LR fuzzy numbers of fuzzy multi objective transportation problem are considered to be linear and non-linear functions. This paper develops a procedure to derive the fuzzy objective value of the fuzzy multi objective transportation problem, in that the fuzzy cost coefficients and the fuzzy time are LR fuzzy numbers. The method is illustrated with an example by various cases.

Mehar’s method to find exact fuzzy optimal solution of unbalanced fully fuzzy multi-objective transportation problems

To the best of our knowledge, till now there is no method described in literature to find exact fuzzy optimal solution of balanced as well as unbalanced fully fuzzy multi-objective transportation problems. In this paper, a new method named as Mehar’s method, is proposed to find the exact fuzzy optimal solution of fully fuzzy multi-objective transportation problems (FFMOTP). The advantages of the Mehar’s method over existing methods are also discussed. To show the advantages of the proposed method over existing methods, some FFMOTP, which cannot be solved by using any of the existing methods, are solved by using the proposed method and the results obtained are discussed. To illustrate the applicability of the Mehar’s method, a real life problem is solved.

Distribution planning decisions using interactive fuzzy multi-objective linear programming

This work develops an interactive fuzzy multi-objective linear programming (i-FMOLP) method for solving the fuzzy multi-objective transportation problems with piecewise linear membership function. The proposed i-FMOLP method aims to simultaneously minimize the total distribution costs and the total delivery time with reference to fuzzy available supply and total budget at each source, and fuzzy forecast demand and maximum warehouse space at each destination. Additionally, the proposed method provides a systematic framework that facilitates the fuzzy decision-making process, enabling a decision maker (DM) to interactively modify the fuzzy data and related parameters until a set of satisfactory solutions is obtained. An industrial case is presented to demonstrate the feasibility of applying the proposed method to real transportation problems. Consequently, the proposed method yields an efficient solution and overall degree of DM satisfaction with the determined objective values. Especially, several significant management implications regarding the practical application of the proposed i-FMOLP method are presented.