Trapezoidal Intuitionistic Fuzzy Fractional Transportation Problem

This paper is devoted to a modem approach for solving type-2 duality fuzzy fractional transportation problem by using interval-valued trapezoidal intuitionistic fuzzy numbers using goal programming technique. The proposed method solves three types of models in which variables are taken to be type-2 fuzzy variables which have given rise to a fuzzy fractional transportation problem. In these models, duality is applied and a special type of non-linear (hyperbolic) membership functions are used to solve duality fractional transportation problem with fuzzy parameters. The computational process is carried out by program solving software called Tingo. A numerical example for inspecting the execution of the proposed model is inclined here.

In conventional transportation problem (TP), supplies, demands and costs are always certain. In this paper, the author tried to categories the TP under the mixture of certain and uncertain environment and formulates the problem and utilizes the crisp numbers, triangular fuzzy numbers (TFNs) and trapezoidal fuzzy numbers (TrFNs) to solve the TP. The existing ranking procedure of Liou and Wang is used to transform the type-1 and type-3 fuzzy transportation problem (FTP) into a crisp one so that the conventional method may be applied to solve the TP. The solution procedure differs from TP to type-1 and type-3 FTP in allocation step only. Therefore, the new method called PSK method and new multiplication operation on TrFN is proposed to find the mixed optimal solution in terms of crisp numbers, TFNs and TrFNs. The main advantage of this method is computationally very simple, easy to understand and also the optimum objective value obtained by our method is physically meaningful. The effectiveness of the proposed method is illustrated by means of a numerical example.

The cost of goods per unit transported from the source to the destination is considered to be fixed regardless of the number of units transported. But, in reality, the cost is often not fixed. Quantity discount is often allowed for large shipments. Furthermore, the transportation cost and the price break quantities are not deterministic. In this study, we introduce the concept of Value- and Ambiguity-based approach for solving the intuitionistic fuzzy transportation problem with total quantity discounts and incremental quantity discounts. Here, the cost and quantity price breakpoints are represented by trapezoidal intuitionistic fuzzy numbers. The Values and Ambiguities are defined as the degree of acceptance and rejection for trapezoidal intuitionistic fuzzy numbers. The trapezoidal intuitionistic fuzzy transportation problem is converted to a parametric transportation problem based on their Value indices and Ambiguity indices. Then, for different Values of the parameter, the transformed problem is reduced to the linear programming problem. Then, the linear programming problem is solved by using the classical methods. The proposed method is demonstrated with a numerical example. In conclusion, the intuitionistic fuzzy transportation problem with total quantity discounts is compared with the intuitionistic fuzzy transportation problem with incremental quantity discounts.

In classical fractional transportation programming problem, we want to optimize the objective function in the form of one or several ratios subject to some linear constraints. If in multi-objective transportation problem, objective function is in ration of two linear function under some linear restrictions, then the problem is called multi-objective linear fractional transportation problem (MOLFTP). In this paper we propose a new method to solve multi-objective linear fractional transportation problem which is extension of Nomani et al. (Int J Manag Sci Eng Manag (2016) [9]). Two numerical problems are presented to validate the proposed algorithm.

<abstract><p>The concept of an interval-valued Fermatean fuzzy set (IVFFS), an extension of Fermatean fuzzy sets, is a more resilient and reliable tool for dealing with uncertain and incomplete data in practical applications. The purpose of this paper is to define a triangular interval-valued Fermatean fuzzy number (TIVFFN) and its arithmetic operations. Fractional transportation problems (FTPs) have important implications for cost reduction and service improvement in logistics and supply management. However, in practical problems, the parameters in the model are not precise due to some unpredictable factors, including diesel prices, road conditions, weather conditions and traffic conditions. Therefore, decision makers encounter uncertainty when estimating transportation costs and profits. To address these challenges, we consider a FTP with TIVFFN as its parameter and call it an interval-valued Fermatean fuzzy fractional transportation problem (IVFFFTP). A new method for solving this IVFFFTP is proposed without re-transforming the original problem into an equivalent crisp problem. Illustrative examples are discussed to evaluate the precision and accuracy of the proposed method. Finally, the results of the proposed method are compared with those of existing methods.</p></abstract>

This research uses trapezoidal fuzzy numbers to solve a fuzzy transportation problem in a fuzzy environment. Applied mathematics has extensively researched the transportation problem, which involves solving a set of linear equations with various scientific applications. Simultaneous mathematical models play a crucial role in operational research. This research paper's primary goal is to determine the lowest possible transportation cost for certain supplies via a capacitated network, where the demand and availability notes are previously represented by fuzzy trapezoidal values. The membership function is demonstrated by using trapezoidal fuzzy numbers. This approach is demonstrated by solving a fuzzy transportation problem with trapezoidal fuzzy numbers and the ranking strategy. A numerical example is provided to illustrate that it is the best conceivable solution.

A fuzzy transportation problem (FTP) includes cost, supply and demand of transportation problems. Its numbers are fuzzy numbers. Fuzzy transportation problem works to reduce transportation cost of some commodities through a capacitate network. Present research paper points out a technique with an alpha cut, optimal solution for solving transportation problem. We suggest a technique to find the fuzzy optimal solution on scales of transportation problem and propose a new hexagonal representation of fuzzy numbers. In general, the comparison of balanced fuzzy transportation problem (BFTP) and unbalanced fuzzy transportation problem (UFTP) shows that the optimal transportation cost of UFTP is less than BFTP.

In dealing with real world transportation problems, several issues are frequently encountered as a result of uncontrollable factors. To deal this uncertainties, many authors have suggested transportation problems with intuitionistic fuzzy parameters. In this article, fully intuitionistic fuzzy transportation problem (InFTP) is considered in which the parameters are triangular intuitionistic fuzzy numbers. To solve this, initially intuitionistic fuzzy branch and bound technique are applied to get the initial basic feasible solution and then intuitionistic fuzzy modified distribution (InFMODI) method is applied to acquire the optimal solution of the fully InFTP. Also, a new ordering is developed here in which some properties of compensation, linearity, additive and partial order relations are satisfied. The optimal solution obtained from the proposed method satisfies the condition of optimality and feasibility. Finally, two numerical examples are provided to show the effectiveness of the proposed method.

In this paper, we discuss the solution of transportation problem in intuitionistic fuzzy nature. Optimal solution for the intuitionistic fuzzy transportation problem without converting the intuitionistic fuzzy number into crisp number is obtained. The procedure is illustrated by numerical example.

In any decision-making process, imprecision is a significant issue.To deal with the ambiguous environment of collective decision-making, various tools and approaches have been created.Fuzzy set theory is one of the most recent approaches for coping with imprecision.The Fuzzy Transportation Problem (FTP) is a well-known network planned linear programming problem which exists in a variety of situations and has received a lot of attention recently.Many authors defined and solved the fuzzy transportation problem with frequently utilized fuzzy numbers such as triangular fuzzy numbers or trapezoidal fuzzy numbers.On the other hand, real-world problems usually involve more than four variables.To tackle these concerns, the pentagonal fuzzy number is applied to the problems.This article proposes an approach to solving transportation problems whose parameters are pentagonal fuzzy numbers without requiring an initial feasible solution.An algorithm based on the core and spread method and an extended MODI method is developed to determine the optimal solution to the problem.The proposed process is based on the approximation method and gives a more efficient result.An illustrated example is used to validate the model.As a result, the proposed methodology is both simpler and more computationally efficient than the existing approaches.

In this research work, a soft computing optimization operating approach is developed for a multi-objective aspirational level fractional transportation problem. In the proposed technique, a mathematical model is formulated for the multi-objective aspirational level fractional transportation problem (MOFTP) based on the highest value of one and all objectives of the model. We also used the symmetry concept over our model to identify the best optimum solution based on symmetrical data. We constructed the membership grades for the set of fetched parameters having symmetry. In this work, we also used the concept of ranking function in our mathematical model to obtain the optimum solution of the fuzzy multi-objective fractional transportation. In this proposed algorithm, the aspiration levels are also associated with the objective function of MOFTP. We are also proposing a new approach for the optimization of fractional problems in which the objectives are being optimized by using the numerator function and denominator function simultaneously. Further, a methodology is also developed to find the average cost for each fractional objective of the model. After that, we will find the ranking function for each parameter by using the defuzzification method. By this methodology, we will be able to convert the MOFTP into a bi-objective transportation problem. The provided technique is elaborated with the help of numerical computations to prove the beauty and power of the proposed technique.

A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers

In this chapter, a new method (named as Mehar method-II) is proposed to check the existence of at least one more-for-less solution as well as to find all more-for-less solutions (if exists) of symmetric triangular fuzzy transportation problems with mixed constraints (transportation problems with mixed constraints in which each parameter and each variable is represented by a symmetric triangular fuzzy number). Also, to illustrate the proposed Mehar method-II, an existing crisp transportation problem with mixed constraints and an existing symmetric triangular fuzzy transportation problem with mixed constraints are solved by the proposed Mehar method-II. Furthermore, the superiority of the proposed Mehar method-II over some existing methods is discussed.

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 this paper a new algorithm is proposed to find the optimal solution of the multi objective dual hesitant fuzzy fractional transportation problem. The proposed algorithm is very simple and easy to understand. This algorithm gives the better solution in both crisp environment and fuzzy environment. The numerical example is solved to explain the algorithm. The proposed algorithm gives the better solution than the existing one.