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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, Kirankumar Laxmanrao Bondar

Source Title: International Journal of Fuzzy System Applications (IJFSA) 9(1)

DOI: 10.4018/IJFSA.2020010105

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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 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.
a₁ ≤ a₂ ≤ a₃

Its membership function is defined as follows:

Figure 1.

Triangular fuzzy number [a₁, a₂, a₃]

2.1.1. Α- Cut for Triangular Fuzzy Number

For any α ∈ [0, 1] from:

= α,

= α

= (a₂ – a₁) α + a₁,

= - (a₃ - a₂) α + a₃

Thus:

= [(a₂ – a₁) 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 and are defined as:

= (a₁, a₂, a₃) and

= (b₁, b₂, b₃)

1.
Addition (+) = (a₁, a₂, a₃) + (b₁, b₂, b₃) = (a₁ + b₁, a₂ + b₂, a₃ + b₃)
2.
Subtraction (-) = (a₁, a₂, a₃) - (b₁, b₂, b₃) = (a₁ - b₃, a₂ - b₂, a₃ - b₁)
3.
Symmetric image: (-) = (-a₃, -a₂, -a₁)
4.
Multiplication: × = (min (a₁b₁, a₁b₃, a₃b₁, a₃b₃), a₂b₂, max (a₁b₁, a₁b₃, a₃b₁, a₃b₃)).

2.1.3. Ordering Two Triangular Fuzzy Numbers

Suppose = (x₁, x₂, x₃) and = (y₁, y_2, y₃) be given any two triangular fuzzy number then we find here α- cut say _α = (,) and _α = (m₁, m₂). If ≤ for any α > 0.5 then we can say that fuzzy number ≤ .