Research article · 2022 · Full text
International Journal of Statistics and Applied MathematicsOptimal solution for fuzzy transportation problem using median ranking
Volume 7(4) · Pages 01–06 · DOI: 10.22271/maths.2022.v7.i3b.838 ↗
Research areasOptimization and Mathematical Programming
Abstract
Author-writtenIn this paper, a new ranking method known as median ranking is used to convert fuzzy transportation problems to crisp valued transportation problems and then solve them by using the Max-Min technique. We can apply this ranking to both types of fuzzy numbers such as odd numbers of fuzzy numbers and even numbers of fuzzy numbers. Using the method, we can reach an optimal solution of balanced as well as unbalanced FTP.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- Transportation problems are a type of linear programming used to minimize cost functions when allocating resources optimally. View evidence ↗
- Fuzzy numbers are employed in fields like Artificial Intelligence and experimental sciences to handle uncertainty. View evidence ↗
- A fuzzy transportation problem (FTP) is one where supply, demand, and transportation costs are expressed as fuzzy numbers. View evidence ↗
The question
- The study aims to use a new 'median ranking' method to convert fuzzy transportation problems into crisp-valued problems, which are then solved using the Max-Min technique. View evidence ↗
What they did
- The proposed ranking technique treats the elements of fuzzy numbers as ungrouped data and applies the median formula based on whether the number of elements is odd or even to obtain a crisp value. View evidence ↗
- The median ranking method is designed to be applicable to both balanced (BFTP) and unbalanced (UBFTP) fuzzy transportation problems. View evidence ↗
- The algorithm involves constructing the table, balancing it with dummy rows/columns if necessary, converting fuzzy values to crisp values via median ranking, and solving for the optimal solution using the Max-Min method. View evidence ↗
What they found
- For a balanced FTP using dodecagonal fuzzy numbers, the median ranking method yielded an optimal solution of 121. View evidence ↗
- In the dodecagonal BFTP example, median ranking (121) produced a lower cost than the sub-interval average technique (125.95) and the range technique (541). View evidence ↗
- For an unbalanced FTP using dodecagonal fuzzy numbers, the median ranking method resulted in an optimal solution of 62.75. View evidence ↗
- In the dodecagonal UBFTP example, median ranking (62.75) outperformed the sub-interval average technique (63.5) and the range technique (341). View evidence ↗
- For a balanced FTP using hendecagonal fuzzy numbers, the median ranking method produced an optimal solution of 350. View evidence ↗
- In the hendecagonal BFTP example, median ranking (350) was more optimal than the sub-interval average technique (353.24) and the range technique (834). View evidence ↗
- For an unbalanced FTP using hendecagonal fuzzy numbers, the median ranking method yielded an optimal solution of 147. View evidence ↗
- In the hendecagonal UBFTP example, median ranking (147) provided a lower cost than the sub-interval average technique (148) and the range technique (850). View evidence ↗
What it means
- The authors conclude that the new median ranking technique successfully obtains optimal solutions for FTPs involving hendecagonal and dodecagonal fuzzy numbers. View evidence ↗
- The authors state that the proposed ranking is versatile, applicable to any type of fuzzy number (even or odd) and both balanced and unbalanced FTPs. View evidence ↗
Visual summary
3 visualsIllustrations explain the paper; charts are replotted from the paper's own table values.
FTP Solution Workflow
AI-generated conceptual illustration (A mathematical transportation network diagram). It explains the idea and does not show study data.
View source evidence ↗
Optimal Solution: Range Technique vs Sub interval Average... vs Median Ranking
Replotted from a table (page 3) of the original paper. Every bar is the paper's own table value.
View source evidence ↗
Optimal Solution: Range Technique vs Sub interval Average... vs Median Ranking
Replotted from a table (page 4) of the original paper. Every bar is the paper's own table value.
View source evidence ↗Figures & tables
From the original paper
Figure 1
Figure 1
Figure 2Introduction
A transportation problem represents a particular type of linear programming problem used for optimally allocating resources. It works in a way of minimizing a cost function. Because of their uncertainty, fuzzy numbers and values are used in various fields such as experimental sciences, Artificial Intelligence, etc. L.A. Zadeh introduced the fuzzy set theory [1] , which is very useful in real-life situations. A fuzzy transportation problem is a transportation problem with the quantities like supply, demand, and transportation cost in fuzzy numerals. There are different methods for ranking fuzzy numbers. S. Sathya Geetha, and K. Selvkumari [2] proposed a new method for solving FTP using pentagonal fuzzy numbers using the range technique for ranking in 2020. The Sub-Interval Average Method for Ranking of Linear Fuzzy Numbers was proposed by S. Kamalnath and Rameshan Natarajan [3] , Dr. A. Sahya Sudha, S. Karunam [4] proposed a new ranking method using heptagonal Fuzzy number. New ranking function of the Nanogonal Fuzzy Number was proposed by K. Deepika, S. Rekha [5] . Kirtiwant P. Ghadle and Priyanka A. Pathade [6] solved FTP with generalized Hexagonal and generalized Octagonal Fuzzy Number by ranking method. A suitable defuzzification method to find minimum transportation cost using decagonal fuzzy number was developed by S. Nagadevi and G.M. Rosario [7] . Edithstine Rani Mathew [8] , Sunny Joseph Kalayathankal proposed a new ranking method using dodecagonal fuzzy numbers to solve FTP. Sanjivani Ingle and Kirtiwant Ghadle [9] derived two formulas related to odd and even numbers of fuzzy numbers for the optimal solution to Fuzzy Assignment Problem. In this paper, we introduced a new ranking technique considering elements fuzzy numbers as ungrouped data, and according to the odd or even number of elements we apply the formula of median and convert the fuzzy number into a crisp value. Then using Max-Min Method [2] we will get an optimal solution. This ranking can be applicable for balanced and unbalanced fuzzy transportation problems. The first two examples are BFTP and UBFTP using dodecagonal Fuzzy Numbers. The next Two are BFTP and UBFTP using hendecagonal Fuzzy Numbers.
Preliminaries Fuzzy Number [9] : A fuzzy number is a convex normalized fuzzy set on the real line R such that 1. There exists at least one y ∈ R with µA˜(y) = 1. 2. µ A˜( y) is piecewise continuous. [8] The membership function of dodecagonal fuzzy number 𝐴̃ = (𝑎 1 , 𝑎 2 , 𝑎 3 , 𝑎 4 , 𝑎 5 , 𝑎 6 , 𝑎 7 , 𝑎 8 , 𝑎 9 , 𝑎 10 , 𝑎 11 , 𝑎 12 )
Dodecagonal fuzzy number
where 𝑎 1 , 𝑎 2 , 𝑎 3 , 𝑎 4 , 𝑎 5 , 𝑎 6 , 𝑎 7 , 𝑎 8 , 𝑎 9 , 𝑎 10 , 𝑎 11 , 𝑎 12 are real numbers, are given by, Where 0 < 𝑘 1 < 𝑘 2 < 1 Hendecagonal Fuzzy Number [10] A Hendecagonal fuzzy number HeD is denoted as (a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 , a 9 , a 10 , a 11 ), and the membership function is defined as 0, otherwise
Algorithm
Step 1: Construct the transportation table and check whether it is BTP or UBTP.
Step 2: If the problem is unbalanced then make it balanced by adding a dummy row or column as per the requirement of supply and demand, otherwise go to step 3.
Step 3: Convert the fuzzy transportation table to a crisp transportation table using the proposed ranking method.
Step 4: Use the Max-Min method [2] to solve the problem's optimal solution.
Numerical Examples: A. Consider the balanced fuzzy transportation problem using Dodecagonal Fuzzy Numbers
The crisp transportation table using the proposed ranking is as follows:
1.5 0.5 6.5 1.5 1.5 F3 5.5 8.5 15.5 9.5 11 Demand 7.5 5.5 3.5 2.5 We got the optimal solution as: (5.5) (7.5) + (3.5) (5.5) + (6.5) (1.5) + (11.5) (1) + (9.5) (2.5) + (15.5) (1) = 121 (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
Sr.NO Ranking Technique
F4 (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) (
Figures and tables from the paper
References
2 in the original paper- Sathya S, Geetha Selvakumari K. A new method for solving the fuzzy trans problem using Pentagonal fuzzy numbers. Journal of critical reviews, 2020, 7.
- Kamalnath S, Ramesh Natarajan. Subinterval average method or ranking of linear fuzzy numbers. Article in international journal of pure and applied mathematics, 2017, Jan.