Fuzzy approach to solve interval-valued transportation problem and comparison of the effectiveness of different fuzzy numbers

Abstract

In real world, we come across transportation problems, wherein the associated data involves some sort of uncertainty, which at times can be most appropriately represented in the form of interval numbers. Since ordering of intervals involves complexity, hence, we have used fuzzy concept to solve Interval-Valued Transportation Problems (IVTPs). We have proposed new fuzzification methods for conversion of interval number to trapezoidal, pentagonal, hexagonal and heptagonal fuzzy numbers, thereby converting IVTP to Fuzzy Transportation Problem (FTP). Further, we have proposed new ranking functions for conversion of these fuzzy numbers to crisp numbers, which can also be used in other fields of decision making. The crisp-valued transportation problem is then solved using Vogel’s Approximation Method followed by Modified Distribution method. Numerical illustration for the proposed algorithm is given in a later section. The solutions obtained for these examples are used to approximate the solutions for octagonal, nonagonal and decagonal FTPs corresponding to IVTP, using Newton’s Polynomial. On the basis of these solutions, ordering of effectiveness of various types of fuzzy numbers in solving IVTP is done. Lastly, comparison is made between the optimal solutions obtained by various methods. The proposed method can be applied to industrial transportation problems in which the difference between the actual and the proposed demand and supply is quite significant.

Optimal solution of pentagonal fuzzy transportation problem using a new ranking technique

“…Definition 2 (Pentagonal Fuzzy Number (PFN) [24]). A fuzzy number ÃP = (ϱ 1 , ϱ 2 , ϱ 3 , ϱ 4 , ϱ 5 ; ω 1 , ω 2 ) is said to be a PFN (see Figure 1) if it satisfies the following properties: In realworld problems, the data available is not always in the crisp form due to various reasons, such as inaccuracy in measurement, change in cost with time, weather conditions etc.…”

“…The concept of ranking fuzzy numbers was introduced by Jain [22]. Recently, Bisht and Dangwal [23], [24] proposed new ranking functions and applied them to solve game problem and interval valued transportation problems through fuzzy approach, respectively. Many defuzzification methods have been proposed by various authors and used in different decision making problems.…”