This paper introduces a simplified presentation of a new computing procedure for solving the fuzzy Pythagorean transportation problem. To design the algorithm, we have described the Pythagorean fuzzy arithmetic and numerical conditions in three different models in Pythagorean fuzzy environment. To achieve our aim, we have first extended the initial basic feasible solution. Then an existing optimality method is used to obtain the cost of transportation. To justify the proposed method, few numerical experiments are given to show the effectiveness of the new model. Finally, some conclusion and future work are discussed.Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
An elongation of the single-valued neutrosophic set is an interval-valued neutrosophic set. It has been demonstrated to deal indeterminacy in a decision-making problem. Real-world problems have some kind of uncertainty in nature and among them; one of the influential problems is solving the shortest path problem (SPP) in interconnections. In this contribution, we consider SPP through Bellman's algorithm for a network using interval-valued neutrosophic numbers (IVNNs). We proposed a novel algorithm to obtain the neutrosophic shortest path between each pair of nodes. Length of all the edges is accredited an IVNN. Moreover, for the validation of the proposed algorithm, a numerical example has been offered. Also, a comparative analysis has been done with the existing methods which exhibit the advantages of the new algorithm.Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Shortest path problem (SPP) is an important and well-known combinatorial optimization problem in graph theory. Uncertainty exists almost in every real-life application of SPP. The neutrosophic set is one of the popular tools to represent and handle uncertainty in information due to imprecise, incomplete, inconsistent, and indeterminate circumstances. This chapter introduces a mathematical model of SPP in neutrosophic environment. This problem is called as neutrosophic shortest path problem (NSPP). The utility of neutrosophic set as arc lengths and its real-life applications are described in this chapter. Further, the chapter also includes the different operators to handle multi-criteria decision-making problem. This chapter describes three different approaches for solving the neutrosophic shortest path problem. Finally, the numerical examples are illustrated to understand the above discussed algorithms.
The fuzzy transportation problem is a very popular, well-known optimization problem in the area of fuzzy set and system. In most of the cases, researchers use type 1 fuzzy set as the cost of the transportation problem. Type 1 fuzzy number is unable to handle the uncertainty due to the description of human perception. Interval type 2 fuzzy set is an extended version of type 1 fuzzy set which can handle this ambiguity. In this paper, the interval type 2 fuzzy set is used in a fuzzy transportation problem to represent the transportation cost, demand, and supply. We define this transportation problem as interval type 2 fuzzy transportation problems. The utility of this type of fuzzy set as costs in transportation problem and its application in different real-world scenarios are described in this paper. Here, we have modified the classical Vogel’s approximation method for solved this fuzzy transportation problem. To the best of our information, there exists no algorithm based on Vogel’s approximation method in the literature for fuzzy transportation problem with interval type 2 fuzzy set as transportation cost, demand, and supply. We have used two Numerical examples to describe the efficiency of the proposed algorithm.
This paper proposes a generalized formulation for multilevel redundancy allocation problems that can handle redundancies for each unit in a hierarchical reliability system, with structures containing multiple layers of subsystems and components. Multilevel redundancy allocation is an especially powerful approach for improving the system reliability of such hierarchical configurations, and system optimization problems that take advantage of this approach are termed multilevel redundancy allocation optimization problems (MRAOP). Despite the growing interest in MRAOP, a survey of the literature indicates that most redundancy allocation schemes are mainly confined to a single level, and few problem-specific MRAOP have been proposed or solved. The design variables in MRAOP are hierarchically structured. This paper proposes a new variable coding method in which these hierarchical design variables are represented by two types of hierarchical genotype, termed ordinal node, and terminal node. These genotypes preserve the logical linkage among the hierarchical variables, and allow every possible combination of redundancy during the optimization process. Furthermore, this paper developed a hierarchical genetic algorithm (HGA) that uses special genetic operators to handle the hierarchical genotype representation of hierarchical design variables. For comparison, the customized HGA, and a conventional genetic algorithm (GA) in which design variables are coded in vector forms, are applied to solve MRAOP for series systems having two different configurations. The solutions obtained when using HGA are shown to be superior to the conventional GA solutions, indicating that the HGA here is especially suitable for solving MRAOP for series systems.
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