A simple method for the solution of bottleneck-cost transportation problem under fuzzy environment

In this paper, we develop a new algorithm for the initial fuzzy basic feasible solution to a transportation problem with imprecise parameters. All the decision variables are assumed to be triangular fuzzy numbers or trapezoidal fuzzy numbers or real numbers. By applying the efficient ranking function and arithmetic operations the solution is obtained without changing it into its crisp equivalent form. Using the parametric form of fuzzy numbers such as left fuzziness index, right fuzziness index, modal value and by using a proposed algorithm we obtained the initial basic fuzzy feasible solution to the problem. We further discuss the optimality by applying the modified distribution method. A numerical example is solved to show the effectiveness of the proposed algorithm.

In this paper we develop a new algorithm for the initial fuzzy basic feasible solution to a solid transportation problem with imprecise parameters. All the decision variables are assumed to be triangular fuzzy numbers. Using parametric form of triangular fuzzy numbers such as left fuzziness index, right fuzziness index, modal value and by using proposed algorithm we obtained the initial basic fuzzy feasible solution to the problem without changing to its crisp equivalent form. We further discuss the optimality by applying the modified distribution method. A numerical example is solved to show the effectiveness of the proposed algorithm.

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.

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

Naturally decision makers try to maximize ratio of return/risk, return/cost, and time/cost of transportation problem under uncertain environment. In such cases fractional transportation problem (FTP) play perfect role in instead of ordinary transportation problem. Some limited methods for the solution of fractional transportation problem using exact or fuzzy parameters are available in literature, and existing methods cannot deal uncertain FTP properly having hesitation factors due to inexact information. Here, I introduce the concept of intuitionistic fuzzy expectation of trapezoidal intuitionistic fuzzy numbers, and some related theorems are presented. Further, I present the concept of trapezoidal intuitionistic fuzzy fractional transportation (TIFFTP) problem, and a methodology for the solution of TIFFTP problem based on expectation of trapezoidal intuitionistic fuzzy numbers. Finally, I illustrate the presented TIFFTP problem by using an 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.

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.

Multi-criteria decision making method is used in such cases having multiple targets. Multi-objective linear programming problems, which deal with uncertain measurements for both objectives and constraints, are solved by fuzzy multi-objective linear programming methods. Analytic hierarchy process (AHP), one of the multi-criteria decision-making methods based on weighting of objectives, is designed to solve complex problems. In this study, multiple criteria decision making problems and decision makers’ opinions on these problems as well as their solution processes are discussed. Firstly, a fuzzy multi objective linear programming problem is solved using the approaches of Zimmerman and Hybrid. In the literature, these methods are applied using triangular fuzzy numbers based on decision makers’ opinions only in cases where the objective function is minimization. In this study, as an alternative to the triangular fuzzy numbers, trapezoidal fuzzy numbers are defined, and both minimization and maximization of objective functions are examined. Finally, a solution algorithm for a fuzzy multi-objective linear programming problem is proposed under some given conditions. The algorithm is applied to a sample problem that involves modeling of a congress organizing which aims to place attendees to accommodations under specific objectives and constraints. In the evaluation phase of the algorithm, targets are weighted by using triangular and trapezoidal fuzzy numbers. At the end, the results of the proposed algorithm are compared with the results of other methods in the literature.

This paper aims to develop and introduce a fuzzy Interval based Multi-criteria homogeneous Group Decision making technique (IMGD) to make appropriate decision under fuzzy environment. In fuzzy multi-criteria group decision making process, a group of decision makers often considers several subjective criteria for ranking a set of alternatives. Due to vague and imprecise information, decision makers generally utilize linguistic variables which are mandatorily converted into triangular or trapezoidal fuzzy numbers. The total process then becomes very complex and time consuming. The current investigation advocates fuzzy intervals instead of triangular or trapezoidal fuzzy numbers for simplification of the complex situation and ease of calculation. In this method, fuzzy intervals of performance ratings and weights assessed by homogeneous group decision makers under subjective criteria are converted into first mean fuzzy intervals then into normalized crisp numbers. The normalized crisp performance ratings and normalized crisp weights are combined together to determine initially individual contribution and then into aggregate contribution to each alternative for final ranking and selection of the alternative. The new model is demonstrated with an application to airports ranking and selection problem for better clarification and verification. The outcome of the proposed is validated with the results obtained by well-known existing MCDM techniques. The analysis shows that the proposed method is applicable, useful and effective for appropriate decision making under fuzzy MCDM environment.

Bharti and Singh ( International Journal of Fuzzy Systems 20 (2018), 1511-1522) proposed a method for solving a special type of interval-valued intuitionistic fuzzy transportation problems (IVIF-TPs) (transportation problems (TPs) in which the quantity of the product to be supplied is represented as a real number, whereas, all the other parameters are represented as interval-valued triangular intuitionistic fuzzy numbers (IVTIFNs)). In this note, an interval-valued intuitionistic fuzzy transportation problem (IVIF-TP) is solved by Bharti and Singh’s method and shown that more than one IVTIFNs, representing the optimal interval-valued intuitionistic fuzzy (IVIF) transportation cost is obtained, which is mathematically incorrect as the obtained distinct IVTIFNs has different physical meanings. Also, it is pointed out that to resolve this flaw of Bharti and Singh’s method may be considered as a challenging open research problem.

Long term supplier selection using a combined fuzzy MCDM approach: A case study for a telecommunication company

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.

The urge to remanufacture and address environmental concerns in various industrial processes has drawn the attention of academics as well as practitioners towards Closed-loop Supply Chain Networks (CLSC). Although everchanging and complex external factors including social and economic ones, adversely impact the sustainable development of closed-loop supply chain networks. The basic aim of the research is to optimize the functioning of CLSC networks. For the above-said, two objective functions are made. The first objective is to minimize the cost of production and assembly expenses of the forward and reverse logistics. Secondly, an endeavour has been made to reduce the fixed costs associated with plants and retailers. For the sake of achieving two objective functions, two methods are employed: triangular fuzzy numbers and triangular intuitionistic fuzzy numbers. Among the two methods, triangular intuitionistic fuzzy numbers achieved the said objectives with greater optimization substantiated by statistics. This method can deal with uncertain external factors without undermining the optimization of the CLSC networks.

Disassembly is one of the important activities in treating with the product at the End of Life time (EOL). Disassembly is defined as a systematic technique in dividing the products into its constituent elements, segments, sub-assemblies, and other groups. We concern with a Fuzzy Disassembly Line Balancing Problem (FDLBP) with multiple objectives in this article that it needs to allocation of disassembly tasks to the ordered group of disassembly Work Stations. Tasks-processing times are fuzzy numbers with triangular membership functions. Four objectives are acquired that include: (1) Minimization of number of disassembly work stations; (2) Minimization of sum of idle time periods from all work stations by ensuring from similar idle time at any work- station; (3) Maximization of preference in removal the hazardous parts at the shortest possible time; and (4) Maximization of preference in removal the high-demand parts before low-demand parts. This suggested model was initially solved by GAMS software and then using Genetic Algorithm (GA) in MATLAB software. This model has been utilized to balance automotive engine disassembly line in fuzzy environment. The fuzzy results derived from two software programs have been compared by ranking technique using mean and fuzzy dispersion with each other. The result of this comparison shows that genetic algorithm and solving it by MATLAB may be assumed as an efficient solution and effective algorithm to solve FDLBP in terms of quality of solution and determination of optimal sequence.

The overall purpose of this work is to calculate the project completion time using the fuzzy critical path method. In this problem all uncertain parameters or activities are triangular fuzzy numbers. We proposed a methodology based on fuzziness index and location index, and is very useful in determining the project’s overall completion period. To demonstrate the validity of the proposed procedure, a numerical illustration is given.