Fuzzy Transportation Problem by Using Triangular, Pentagonal and Heptagonal Fuzzy Numbers With Lagrange's Polynomial to Approximate Fuzzy Cost for Nonagon and Hendecagon

Fuzzy Transportation Problem by Using Triangular, Pentagonal and Heptagonal Fuzzy Numbers With Lagrange's Polynomial to Approximate Fuzzy Cost for Nonagon and Hendecagon

Ashok Sahebrao Mhaske (Dada Patil Mahavidyalaya, Karjat, India) and Kirankumar Laxmanrao Bondar (NES Science College, Nanded, India)
Copyright: © 2020 |Pages: 18
DOI: 10.4018/IJFSA.2020010105

Abstract

The transportation problem is a main branch of operational research and its main objective is to transport a single uniform good which are initially stored at several origins to different destinations in such a way that the total transportation cost is minimum. In real life applications, available supply and forecast demand, are often fuzzy because some information is incomplete or unavailable. In this article, the authors have converted the crisp transportation problem into the fuzzy transportation problem by using various types of fuzzy numbers such as triangular, pentagonal, and heptagonal fuzzy numbers. This article compares the minimum fuzzy transportation cost obtained from the different method and in the last section, the authors introduce the Lagrange's polynomial to determine the approximate fuzzy transportation cost for the nanogon (n = 9) and hendecagon (n = 11) fuzzy numbers.
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2. Basic Definitions

2.1. Triangular Fuzzy Number

A triangular fuzzy number IJFSA.2020010105.m01 or simply triangular number represented with three points as follows (a1, a2, a3) holds the following conditions:

  • 1.

    a1 to a2 is increasing function

  • 2.

    a2 to a3 is decreasing function

  • 3.

    a1a2a3

Its membership function is defined as follows:

IJFSA.2020010105.m02
Figure 1.

Triangular fuzzy number [a1, a2, a3]

IJFSA.2020010105.f01

2.1.1. Α- Cut for Triangular Fuzzy Number

For any α ∈ [0, 1] from:

IJFSA.2020010105.m03 = α, IJFSA.2020010105.m04 = αIJFSA.2020010105.m05 = (a2a1) α + a1, IJFSA.2020010105.m06 = - (a3 - a2) α + a3

Thus:

IJFSA.2020010105.m07 = IJFSA.2020010105.m08 = [(a2a1) α + a1, - (a3 - a2) α + a3].

2.1.2. Operations on Triangular Fuzzy Number

Addition, Subtraction and Multiplication of any two triangular fuzzy numbers are also triangular fuzzy number. Suppose triangular fuzzy numbers IJFSA.2020010105.m09 and IJFSA.2020010105.m10 are defined as:

IJFSA.2020010105.m11 = (a1, a2, a3) and IJFSA.2020010105.m12 = (b1, b2, b3)
  • 1.

    Addition IJFSA.2020010105.m13 (+) IJFSA.2020010105.m14 = (a1, a2, a3) + (b1, b2, b3) = (a1 + b1, a2 + b2, a3 + b3)

  • 2.

    Subtraction IJFSA.2020010105.m15 (-) IJFSA.2020010105.m16 = (a1, a2, a3) - (b1, b2, b3) = (a1 - b3, a2 - b2, a3 - b1)

  • 3.

    Symmetric image: (-IJFSA.2020010105.m17) = (-a3, -a2, -a1)

  • 4.

    Multiplication: IJFSA.2020010105.m18 × IJFSA.2020010105.m19 = (min (a1b1, a1b3, a3b1, a3b3), a2b2, max (a1b1, a1b3, a3b1, a3b3)).

2.1.3. Ordering Two Triangular Fuzzy Numbers

Suppose IJFSA.2020010105.m20 = (x1, x2, x3) and IJFSA.2020010105.m21= (y1, y2, y3) be given any two triangular fuzzy number then we find here α- cut say IJFSA.2020010105.m22α = (IJFSA.2020010105.m23,IJFSA.2020010105.m24) and IJFSA.2020010105.m25α = (m1, m2). If IJFSA.2020010105.m26IJFSA.2020010105.m27 for any α > 0.5 then we can say that fuzzy number IJFSA.2020010105.m28IJFSA.2020010105.m29.

2.1.4. Defuzzification

If IJFSA.2020010105.m30 = (a, b, c) be any given triangular fuzzy number then we use mean and center method to Defuzzify i.e. x = IJFSA.2020010105.m31 and method x = b respectively.

If (-a, 0, a) is given fuzzy triangular number then its crisp value is zero.

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