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In the real-world optimization problems, coefficients of the objective function are not known precisely and can be interpreted as fuzzy numbers. In this paper we define the concepts of optimality for linear programming problems with fuzzy parameters based on those for multiobjective linear programming problems. Then by using the concept of comparison of fuzzy numbers, we transform a linear programming problem with fuzzy parameters to a multiobjective linear programming problem. To this end, we propose several theorems which are used to obtain optimal solutions of linear programming problems with fuzzy parameters. Finally some examples are given for illustrating the proposed method of solving linear programming problem with fuzzy parameters.

Bus network design is an important problem in public transportation. In practice, some parameters of this problem are uncertain. We propose two models for the bus terminal location problem with fuzzy parameters. In the first formulation, the number of passengers corresponding to each node is a fuzzy number. In the second formulation, an additional assumption of fuzzy neighborhood is considered. These problems being NP-hard, we use a genetic algorithm (GA) and a simulated annealing (SA) algorithm for solving them. We also propose an idea to hybridize these algorithms. In our hybrid algorithm, SA is applied as a neighborhood search procedure of GA on the best individual of the population, which is the best available approximation of the optimal solution, with a varying probability that is gradually increased with the increase in the number of iterations in GA. We then implement GA, SA, our hybrid algorithm, and a recently proposed hybrid algorithm making use of a constant probability for application of SA on all the individuals of the population of GA, and use a nonparametric statistical test to compare their performances on a collection of randomly generated medium to large-scale test problems. Results of computational experiments demonstrating the efficiency and practicability of our proposed algorithm are reported.

World is full of uncertainties. In decision-making problems, uncertainty occurs in many forms such as fuzzy, rough, interval, soft and researchers use data in one of these forms for presenting uncontrollable factors. This paper aims to develop an effective method for solving bi-matrix game problem with interval payoffs assuming that the players know the upper and lower bounds on payoffs. First, bi-matrix game model with interval payoffs is considered. This model is then transformed to another model with fuzzy payoffs. Finally, ranking approach is used to convert the model to crisp-valued bi-matrix game model. Further, due to correspondence between games and programming problem, the Nash equilibrium of interval bimatrix game is obtained by solving a deterministic nonlinear programming problem with nonlinear objective and linear constraints. Finally, a real-life problem of marketing management is solved to validate, approve and illustrate the effectiveness of the proposed model and its solution method. The results derived are compared with some previously defined methods and conclusions are drawn further.

Game theory is of substantial significance in diverse domains, acting as a potent instrument to comprehend and assess strategic engagements among rational decision-makers. It formulates mathematical models to represent strategic interactions among rational decision-makers in the competitive world. Due to ambiguity in the real-world problems, acquiring the precise payoff values of a matrix game proves challenging. However, in numerous scenarios, these payoffs fluctuate within specific ranges, making them suitable for consideration as interval numbers. This leads to the formulation of a special form of game problem known as the interval valued game problem (IVGP). Some methodologies exist in the literature to find the optimal strategies as well as the value of game for IVGP, but most of them possess some limitations, resulting in the need for proposing a new methodology to find the optimal strategies and value of game. Thus, in this paper, a new solution method for game problems with payoffs represented as interval numbers is presented, utilizing the fuzzy concept. The process begins by transforming the interval payoffs into fuzzy numbers using a ranking function. Subsequently, these fuzzy payoffs are converted into crisp values, leading to the formulation of the crisp matrix game. The resulting crisp matrix game is then solved using linear programming approach. Additionally, MATLAB code for the proposed method is developed and proposed to streamline the computation process, enhancing comparison and decision-making efficiency, particularly when dealing with large payoff matrices. Furthermore, three numerical examples are provided to illustrate the validity of the proposed approach as well as its MATLAB code. A real-life example of IVGP in the realm of tourism industry is also provided. Finally, a comparative analysis is conducted, comparing the proposed method with some existing methods.

To the extent of our knowledge, there is no method in fuzzy environment to solving the fully LR-intuitionistic fuzzy transportation problems (LR-IFTPs) in which all the parameters are represented by LR-intuitionistic fuzzy numbers (LR-IFNs). In this paper, a novel ranking function is proposed to finding an optimal solution of fully LR-intuitionistic fuzzy transportation problem by using the distance minimizer of two LR-IFNs. It is shown that the proposed ranking method for LR-intuitionistic fuzzy numbers satisfies the general axioms of ranking functions. Further, we have applied ranking approach to solve an LR-intuitionistic fuzzy transportation problem in which all the parameters (supply, cost and demand) are transformed into LR-intuitionistic fuzzy numbers. The proposed method is illustrated with a numerical example to show the solution procedure and to demonstrate the efficiency of the proposed method by comparison with some existing ranking methods available in the literature.

In this paper, we introduce a new method to solve Interval-Valued Transportation Problem (IVTP) to deal with those problems of transportation wherein the information available is imprecise. First, a newly proposed fuzzification method is used to convert the IVTP to octagonal fuzzy transportation problem and then with the help of ranking function proposed in this paper, the fuzzy transportation problem is converted into crisp transportation problem. Lastly, Initial Basic Feasible Solution (IBFS) of this problem is obtained using Vogel’s Approximation Method and the solution is improved using Modified Distribution (MODI) method. A numerical example with interval data is solved using the proposed algorithm to make comparison of the solution with some other methods. Also, a numerical example with parameters in the form of octagonal fuzzy numbers is illustrated to compare the effectiveness of the proposed ranking technique. The proposed fuzzification and ranking technique can be used in the other fields of decision making dealing with the data in the same form as considered in this paper.

In real world, we come across transportation problems, wherein the associated data involves some sort of uncertainty, which at times can be most appropriately represented in the form of interval numbers. Since ordering of intervals involves complexity, hence, we have used fuzzy concept to solve Interval-Valued Transportation Problems (IVTPs). We have proposed new fuzzification methods for conversion of interval number to trapezoidal, pentagonal, hexagonal and heptagonal fuzzy numbers, thereby converting IVTP to Fuzzy Transportation Problem (FTP). Further, we have proposed new ranking functions for conversion of these fuzzy numbers to crisp numbers, which can also be used in other fields of decision making. The crisp-valued transportation problem is then solved using Vogel’s Approximation Method followed by Modified Distribution method. Numerical illustration for the proposed algorithm is given in a later section. The solutions obtained for these examples are used to approximate the solutions for octagonal, nonagonal and decagonal FTPs corresponding to IVTP, using Newton’s Polynomial. On the basis of these solutions, ordering of effectiveness of various types of fuzzy numbers in solving IVTP is done. Lastly, comparison is made between the optimal solutions obtained by various methods. The proposed method can be applied to industrial transportation problems in which the difference between the actual and the proposed demand and supply is quite significant.

This research paper develops an algorithm to solve a neutrosophic linear fractional optimization problem, where the cost of the objective functions, the technology coefficients, and the resources are neutrosophic numbers (triangular and trapezoidal numbers). Two defuzzification methods are used here to convert the neutrosophic linear fractional optimization problem into a multi-objective linear fractional optimization problem (MOLFOP). The fractional objective function is defuzzified using the component-wise optimization method, and the linear constraints are defuzzified using the aggregation ranking function strategy. Hence, the MOLFOP is then solved using the fuzzy goal programming method. In order to further explain our suggested algorithm, two numerical examples are solved at the end, and the outcomes obtained by our method are compared with the outcomes obtained using the other algorithm.