Mathematical Problems in Engineering

Volume 2017, Article ID 2139791, 9 pages

https://doi.org/10.1155/2017/2139791

## Modified Approach for Optimization of Real Life Transportation Problem in Neutrosophic Environment

^{1}School of Mathematics, Thapar University, Patiala, Punjab, India^{2}Department of Supply and Chain Management, University of Manitoba, Winnipeg, MB, Canada

Correspondence should be addressed to S. S. Appadoo; ac.abotinamu.cc@oodappa

Received 11 April 2017; Revised 13 July 2017; Accepted 16 July 2017; Published 22 August 2017

Academic Editor: Erik Cuevas

Copyright © 2017 Akanksha Singh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

To the best of our knowledge, there is only one approach for solving neutrosophic cost minimization transportation problems. Since neutrosophic transportation problems are a new area of research, other researchers may be attracted to extend this approach for solving other types of neutrosophic transportation problems like neutrosophic solid transportation problems, neutrosophic time minimization transportation problems, neutrosophic transshipment problems, and so on. However, after a deep study of the existing approach, it is noticed that a mathematical incorrect assumption has been used in these existing approaches; therefore there is a need to modify these existing approaches. Keeping the same in mind, in this paper, the existing approach is modified. Furthermore, the exact results of some existing transportation problems are obtained by the modified approach.

#### 1. Introduction

In daily life problems several times there is a need to transport the product from various sources to different destinations. To find a way to transport the product in such a manner so that the total transportation cost is minimum is called the optimal way and the problem is called cost minimization transportation problems [1]. Different methods have been proposed in the literature to find the optimal way of such cost minimization transportation problems in which cost for transporting unit quantity of the product, availability of the product at the sources, and demand of the product at the destinations are represented as real numbers.

However, to assume these parameters as real numbers is not always valid according to real life situations; for example, the transportation cost depends upon the circumstances like price of petrol/diesel, weather, travel time, traffic jam, and so on. Similarly, availability of crops varies according to the monsoon, fertilizers, chemicals, and so on; demands of the various clothes depend on the season, fashion trends, discount offers, and so on. Furthermore, the opinions of the experts about these parameters indicate that they cannot always be represented as real numbers; for example, generally experts provide their opinion about these parameters in terms of linguistic variables like high, very high, low, very low, and so on.

Being one of the widely adopted ways in the literature, to deal with such situations is to represent these parameters as fuzzy numbers [2] and its extensions [3]. Thamaraiselvi and Santhi [4] pointed out that neutrosophic set [5], one of the extensions of fuzzy set, is used in different research areas. However, till now no one has used the neutrosophic set in transportation problems, while several researchers have used fuzzy numbers for representing various parameters of transportation problems [6–14]. Therefore, Thamaraiselvi and Santhi [4] proposed the approaches for solving neutrosophic transportation problem of Type I (transportation problem in which cost for transporting unit quantity of the product is represented as trapezoidal neutrosophic number, whereas availability and demands are represented as real numbers) and neutrosophic transportation problem of Type II (transportation problem in which cost for transporting unit quantity of the product, availability of a product, and demand of the product are represented as trapezoidal neutrosophic numbers).

Since neutrosophic transportation problems are new area of research, others may be attracted to extend these approaches for solving other types of neutrosophic transportation problems like neutrosophic solid transportation problems, neutrosophic time minimization transportation problems, neutrosophic transshipment problems, and so on. However, after a deep study of these existing approaches, it is noticed that a mathematical incorrect assumption has been used in these existing approaches; therefore there is a need to modify these existing approaches. Keeping the same in mind, in this paper, these existing approaches are modified. Furthermore, the exact results of some existing transportation problems are obtained by the modified approaches.

#### 2. A Brief Review of Thamaraiselvi and Santhi Approaches

To point out the mathematical incorrect assumptions in the approaches, proposed by Thamaraiselvi and Santhi [4], there is a need to describe these approaches. Therefore, in this section, a brief review of the approaches, proposed by the Thamaraiselvi and Santhi [4], for solving both types of neutrosophic transportation problems is discussed in a brief manner.

##### 2.1. Thamaraiselvi and Santhi Approach for Solving Neutrosophic Transportation Problem of Type I

Using the approach, proposed by Thamaraiselvi and Santhi [4], the optimal solution of a neutrosophic transportation problem of Type I can be obtained as follows.

*Step 1. *Formulate the neutrosophic transportation problem as a neutrosophic linear programming problem .where is the number of units of the product transported from th source to th destination, is the neutrosophic cost of one unit quantity transported from source to destination, is the total availability of the product at the source , is the total demand of the product at the destination .

*Step 2. *Transform the neutrosophic linear programming problem into its equivalent crisp linear programming problem .where

*Step 3. *Transform the crisp linear programming problem into its equivalent crisp linear programming problem .

*Step 4. *Represent the crisp linear programming problem into tabular form shown in Table 1.