An Optimal Algorithm for a Fuzzy Transportation Problem

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西南交通大学学报

第 55 卷第 3 期

2020 年 6 月

JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY

Vol. 55 No. 3 June 2020

ISSN: 0258-2724 DOI：10.35741/issn.0258-2724.55.3.7 Research article

Mathematics

A

N

O

PTIMAL

A

LGORITHM FOR A

F

UZZY

T

RANSPORTATION

P

ROBLEM

一种模糊运输问题的最优算法

Rasha Jalal Mitlif, Mohammed Rasheed, Suha Shihab

Applied Science Department, University of Technology

Baghdad, Iraq, [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]

Received: February 1, 2020 ▪ Review: February 28, 2020 ▪ Accepted: April 20, 2020

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

Abstract

This paper deals with the optimal approximate solution to a special type of optimization problem called a fuzzy transportation problem using pentagonal fuzzy numbers. The values of the cost, supply, and demand for fuzzy transportation problems are taken as pentagonal fuzzy numbers. The pentagonal fuzzy numbers are converted into crisp values using a novel suggested ranking function. By comparing this with the conventional ranking methods, we can achieve better results with the aid of the proposed new ranking method. Vogel’s Approximation Method is then applied to obtain the solution. Several experiments have been conducted in order to investigate the suggested technique.

Keywords:Pentagonal Fuzzy Number, Experiment, Fuzzy Optimization Transportation, Ranking Function

摘要本文针对使用五边形模糊数的一种特殊类型的优化问题（称为模糊运输问题）提出了最佳近似解。模糊运输问题的成本，供给和需求的值被当作五边形模糊数。使用新颖的建议排序函数将五边形模糊数转换为清晰的值。通过将其与常规排名方法进行比较，我们可以借助提出的新排名方法获得更好的结果。然后应用沃格尔的近似方法来获得解。为了研究建议的技术，已经进行了一些实验。 关键词: 五角模糊数，实验，模糊优化运输，排序函数

I. I

NTRODUCTION

Transportation problems are special kinds of problems in optimization. They are associated with real-world activities that are managed with logistics. The problem for transportation includes

transportation with a single manufacturing product in different supplies to a number of different destinations. The aim is to achieve the minimum total transportation cost of items that will satisfy the demands at various destinations. Furthermore, a problem of fuzzy transportation is

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one in which the transportation quantities regarding cost, supply, and demand are all fuzzy. The objective of the present work is to calculate the shipped schdule for fuzzy transportation problem in order to minimize the total cost of fuzzy transportation while maintaining the limit of fuzzy supply and demand. The basic transportation problem was originally developed in [1]. Authors in [2] suggested an improved technique for the approximate solution of the fuzzy optimization problem. The authors in [3], [4], [5], [6], [7], [8], [9] studied fuzzy transportation problems depending on the ranking function to find the optimal solution. In the context of decision-making, ranking fuzzy numbers is very important. There are different ranking methods for fuzzy numbers. Some researchers [10], [11], [12], [13], [14], [15], [16], [17] discussed the solution of fuzzy transportation problems with various fuzzy numbers. Saman et al. [18] introduced an optimal solution of full fuzzy transportation problems using total integral ranking function value and fuzzy numbers. Jahirhussain et al. [19] proposed a new solution of the vector fuzzy transportation problem with interval integer form. Many applications can be used with fuzzy transportation problems, such as [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47].

In the present work, a problem based on fuzzy transportation is considered. The demand, cost, and supply values for a fuzzy transportation problem is taken as pentagonal fuzzy numbers. Pentagonal fuzzy numbers are converted into crisp values using the proposed ranking. The problem is then solved by the usual Vogel’s Approximation Method (VAM) technique in order to obtain the best results.

The structure of this work is arranged in the following steps. In section two, fundamentals of interpretations are given. Section three presents an introduction of the fuzzy transportation problem. In section four, procedure, numerical example for the proposed method, and the comparative study are given, and section five is the conclusion.

II. P

RELIMINARIES

Some basic definitions have been investigated in this section.

A. Fuzzy Number (FN) [48]

Assume that is a standard set and is a function from to the interval [0,1]. A fuzzy set with the membership function is

represented by =

{ ,

B. Pentagonal Fuzzy Number (PFN) [49]

A Pentagon Fuzzy Number

where and are real numbers and with membership function obtained below:

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C. Problems of Fuzzy Transportation [50]

Problems in fuzzy transportation can be written as follows: (2) Subject to (3) (4) where

number of fuzzy units carries over from source to destination.

D. Ranking Function [51]

The fuzzy number is based on ranking function . The set of all fuzzy numbers represents the set of real numbers, which maps each fuzzy number into a real number.

Ranking function of a pentagonal fuzzy number [23]:

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E. New Ranking Function

The of

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(6) (7) (8) (9) (10) and (11) (12) (13) (14) so (15) (16)

By using the following ranking function (17) With and (18) (19)

III. P

ROCEDURE

First, convert the cost, supply and demand values of the fuzzy transportation problem, which are all pentagonal fuzzy numbers, into crisp values by using proposed ranking.

Check the balance condition to ensure total demand is equal to total supply.

Acquire the optimal solution by the VAM method.

Subtract the cell cost in the column (row) that has the lowest value from the next cell cost in the same row (column) in order to calculate the penalty cost for all rows and columns.

Choose the largest penalty cost in the column (row).

Utilize cell with the lowest transportation cost in the row or column with the highest penalty cost.

Determine the total cost of transportation of feasible allocations.

Determine the total minimum cost of the product.

IV. N

UMERICAL

E

XAMPLE

A. Example 1

The first problem is the pentagonal fuzzy transportation of the three sources , together with the destinations . The cost values of transporting from source to destination , which are pentagonal fuzzy numbers, are listed in Table 1.

Table 1.

Pentagonal fuzzy transportation table

Destinations Sources Supply (1,3,4,5,7) (0,2,3,4,6) (2,4,5,6,8) (4,6,7,8,10) (3,5,6,7,9) (5,7,8,9,11) (4,6,7,8,10) (6,8,9,10,12) (1,3,4,5,7) (2,4,5,6,8) (3,5,6,7,9) (10,12,13,14,16) Demand (3,5,6,7,9) (5,7,8,9,11) (12,14,15,16,18)

Here, the sum of supply = [20, 26, 29, 32, 38] and the sum of demand = [20, 26, 29, 32, 38]. It is clear that the condition balance is satisfied.

Applying to all values with the new ranking function yields the following table:

Table 2.

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Destinations Sources Supply 4 3 5 7 6 8 7 9 4 5 6 13 Demand 6 8 15

The solution after VAM application is given in Table 3.

Table 3.

Reduced table of VAM method

Destinations Sources Supply 4 3 [7] 5 7 6 8 7 [9] 9 4 [6] 5 [1] 6 [6] 13 Demand 6 8 15

The total cost of transportation is given below:

(20) Then, apply the suggested ranking function to compute the total cost of transportation.

Table 4.

Reduced table of exist ranking function

Table 5.

Destinations Sources Supply 12 9 [21] 15 21 18 24 21 [27] 27 12 [18] 15[3] 18 [18] 39 Demand 18 24 45

The following total cost of transportation is obtained:

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B. Example 2

The first problem is the pentagonal fuzzy transportation of the three sources , together with destinations . The cost values of transporting from source i to destination j, which are pentagonal fuzzy numbers, are listed in Table 6.

Table 6.

Pentagonal fuzzy transportation table

Destinations Sources Supply (7, 9, 10, 11, 13) (9, 11, 12, 13, 15) (13, 15, 16, 17, 19) (22, 24, 25, 26, 28) (8, 10, 11, 12, 14) (12, 14, 15, 16, 18) (10, 12, 13, 14, 16) (27, 29, 30, 31, 33) (11, 13, 14, 15, 17) (15, 17, 18, 19, 21) (9, 11, 12, 13, 15) (37, 39, 40, 41, 43) Demand (32, 34, 35, 36, 38) (42, 44, 45, 46, 48) (12, 14, 15, 16, 18)

In this case, the sum of supply = [86, 92, 95, 98, 104] and the sum of demand = [86, 92, 95, 98, 104]. It is clear that the condition balance is satisfied.

Applying to all values with the new ranking function yields the following table:

Table 7.

Reduced table of new ranking function

Table 8.

Destinations Sources Supply 10 12 [25] 16 25 11 [30] 15 13 30 14 [5] 18 [20] 12 [15] 40 Demand 35 45 15

The total cost of transportation is given below:

(22) Then, apply the suggested ranking function to compute the total cost of transportation.

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Reduced table of exist ranking function Destinations Sources Supply 30 36 48 75 33 45 39 90 42 54 36 120 Demand 105 135 45

The solution obtained via VAM is given below:

Table 10.

Destinations Sources Supply 30 36 [75] 48 75 33 [90] 45 39 90 42 [15] 54 [60] 36 [45] 120 Demand 105 135 45

The total cost of transportation is

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Table 11. Comparison table

Examples New ranking function Existing ranking function Example 1 149 1341 Example 2 1120 11160

V. C

ONCLUSION

This study considers some examples of fuzzy transportation with cost values obtained as pentagonal fuzzy numbers. These are converted into crisp values using a proposed ranking function. The optimal solution is obtained via conventional VAM. The proposed ranking function proves better able than rival formulas to obtain the optimal solution for the minimum total cost of transportation. Data and results are presented in tables for comparison.

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