The transportation problem (TP) is popular in operation research due to its versatile applications in real life. Uncertainty exists in most of the real-life problems, which cause it laborious to find the cost (supply/demand) exactly. The fuzzy set is the well-known field for handling the uncertainty but has some limitations. For that reason, in this chapter introduces another set of values called neutrosophic set. It is a generalization of crisp sets, fuzzy set, and intuitionistic fuzzy set, which is handle the uncertain, unpredictable, and insufficient information in real-life problem. Here consider some neutrosophic sets of values for supply, demand, and cell cost. In this chapter, extension of linear programming principle, extension of north west principle, extension of Vogel's approximation method (VAM) principle, and extended principle of MODI method are used for solving the TP with neutrosophic environment called neutrosophic transportation problem (NTP), and these methods are compared using neutrosophic sets of value as well as a combination of neutrosophic and crisp value for analyzing the every real-life uncertain situation.
Top1. Introduction
Uncertainties exist almost in real life problems, which cannot be solved by classical mathematics. Fuzzy set is one of the most popular and well known methods for handling the uncertainty, which is generalization of crisp set. Prof. Zadeh (1965) first introduced the fuzzy set in 1965, which is representing the membership values, but many real life scenarios result and decision are not enough to the level of accuracy. Fuzzy set and it extended version is used in shortest path problem [Dey, Pal, & Pal, 2016; Kumar et al., 2017; 2018; 2019; 2019a; 2019b; 2020; Dey et al., 2018; Broumi et al., 2019c], Spanning tree problem (Dey et al., 2018; Broumi et al., 2018), graph problem (Dey et al., 2015; Dey et al., 2018; Broumi et al., 2017; 2017a) and so on.
Later, Atanassov (1986; 1999) introduced intuitionistic fuzzy set to recover the limitation of accuracy to solve the problem with imprecision information and characterized by its membership and non-membership values (Atanassov 1999). Hence both fuzzy set and intuitionistic fuzzy set are used for handling the real life uncertainty but it have a limitation for solve the problem with indeterminate or inconsistence information.
To overcome this type of limitation Smarandache (1998) introduced neutrosophic set which is extension of classical set, fuzzy set and intuitionistic fuzzy sets. Neutrosophic set is basically used to represent the truth-membership degree, falsity-membership degree and indeterminacy-membership degree of an object and which is considered as an appropriate approach for representing the indeterminacy and inconsistent information. Wang et al. (2010) have introduced the idea of single valued neutrosophic set and it is more effective for solving the real life problem.
Many researchers have worked on fuzzy transportation problem. Some researchers (Guo et al., 2015; Yu et al., 2015; kumar et al. 2019d) have introduced some algorithmic approaches to find the transportation cost in uncertain environment. In 1941 Hitchcock (1941) has introduced the classical transportation problem. In real life, cost, supply and demand of a transportation problem are uncertain due to several reasons.
In 1978 Zimmerman (1978) has introduced linear programming model in fuzzy environment. It is used to solve TP with fuzzy. Chanas et al. (1984) have proposed fuzzy linear programming model with fuzzy supply and demand but transportation cost is a crisp value. Dinagar & Palanivel (2009) have described the TP with fuzzy supply, demand and Transportation cost. Kaur and Kumar (2012) have introduced an algorithm approach for solving the TP with trapezoidal fuzzy number.