A naive algorithm to solve pentagonal fuzzy transportation problem

Introduction

The purpose of the fuzzy transportation problem is to decide the transportation schedule that reduces the entire fuzzy transportation cost and time. An FTP is a problem in which all the quantities are fuzzy quantities such as transportation cost, supply, and demand. These quantities are unknown due to many unmanageable factors. To manage exact details in building decisions Zadeh 1978 [10] established the concept of fuzziness. [1] Suggested a method to solve FTP working on PFN. In [2] the best optimality condition has been checked. Also [3] proposed octagonal FTP using a different ranking method. A simple algorithm is given to check the feasibility of FTP by [4] . [5] Solved TP with fuzzy numbers by the ranking method [6,7] . Solved FTP using the Best Candidate method. Balanced and unbalanced FTP by using Octagonal Fuzzy Numbers introduced by [8] . The [9] proposed solution of FTP uses robust rank techniques and an improved method. A new algorithm is suggested to discover an optimum solution for FTP in which PFN is involved. We obtain the least value for the optimum solution by applying the ranking technique and proposed algorithm.

Preliminaries Definition (PFN) [1]

A fuzzy number A is said a PFN, which is named as (s 1, s 2 , s 3, s 4, s 5 ) whose membership function 𝜇𝐴 (x) is given by

Proposed ranking technique

To calculate the interquartile range, we need to follow the following steps: 1. Arrange the numbers in increasing order. 2. Find the Median. 3. Calculate the median of both upper (Q 3 ) and lower (Q 1 ) data. 4. Observe the difference between Q 3 and Q 1 . 5. The inter-quartile range = Q3 -Q1 Proposed ranking method 1. Set up the transportation problem table and examine whether it is balanced or not. If it is not balanced, then add a dummy row/column. 2. We convert the FTP into crisp values using the proposed ranking method. 3. For each row we take the sum of the row maximum and just less than the maximum and divide it by several columns. 4. For each column we take the sum of the column maximum and just less than the maximum and divide it by several rows. 5. We choose the greatest of the resultant value. After that, we choose the least value of the cost and make the allocation. If we get multiple greatest values, we can choose any value. 6. Repeat steps 3 to 5 to complete the allocations and the optimal solution. Now we must find the greatest value from the result and notice the least value of the cost and make the allocation. Now again we must find the greatest value from the result and notice the least value of the cost and make the allocation. The same technique is followed repeatedly until we get the final allocation.

Numerical examples 1. Consider a balanced FTP

Comparison with other techniques

In the below table the analogy of the proposed technique along with the other techniques are given. It is evidence presented that the proposed technique gives the optimum solution. Example 2) Consider an unbalanced FTP. The stated Problem is unbalanced so we add 0 columns to make it balance. Now we must find the greatest value from the result and notice the least value of the cost and make the allocation. Again, we must find the greatest value from the result and notice the least value of the cost and make the allocation. The same technique is followed repeatedly until we get the final allocation.

Conclusion

A new range technique i.e., the interquartile range is proposed to get a crisp value which is very easy, and a new algorithm is applied to solve FTP. We get an optimal solution using this method as compared to other known techniques.

References

1. Sathya Geetha S, Selvaumari K. A New Method for Solving Fuzzy Transformation Problem Using Pentagonal Fuzzy Numbers, Journal of Critical Reviews. 2020;7(9):171-174. ISSN-2394-5125.
2. Dr. Shraddha Mishra. Solving Transportation Problem by Various Methods and Their Comparison, International Journal of Mathematics Trends and Technology (IJMIT). 2017 Apr;44(4):270-275. ISSN:2231-5373
3. Dr. Rajarajeswari P, Menaka G. Octagonal Fuzzy Transportation Problem Using Different Ranking Method, International Journal of Trend in Scientific Research and Development. 2020 Aug;4(5):8-13. EISSN: 2456-6470.
4. Fegade MR, Jadhav VA, Muley AA. Solving Fuzzy Transportation Problem using Zero Suffix and Robust Ranking Technology, IOSR Journal of Engineering (IOSRJEN). 2012 Jul;2(7):36-39. ISSN:2250-3021.
5. Kirtiwant Ghadle P, Priyanka Pathade P. Solving Transportation Problem with Generalized Hexagonal and Generalized Octagonal Fuzzy Numbers by Ranking Method, Global Journal of Pure and Applied Mathematics. 2017;13(9):6367-6376. ISSN 0973-1768.
6. Stephen Dinagar D, Keerthivasan R. Solving Fuzzy Transportation Problem Using Modified Best Candidate Method, Journal of Computer and Mathematical Sciences. 2018 Sep;9(9):1179-1186. ISSN 0976-5727 (Print) ISSN 2319-8133 (Online)
7. Annie Christi MS. Solutions of Fuzzy Transportation Problem Using Best Candidates Method and Different Ranking Techniques, World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences. 2017 Nov 5;11(4):182-7.
8. Priyanka Pathade A, Kirtiwant Ghadle P. Optimal Solution of Balanced and Unbalanced Fuzzy Transportation Problem by Using Octagonal Fuzzy Numbers, International Journal of Pure and Applied Mathematics. 2018;119(4):617-625. ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
9. Jayarama1 P, Jahirhussian R. Fuzzy Optimal Transportation Problems by Improved Zero Suffix Method via Robust Rank Techniques, International Journal of Fuzzy Mathematics and Systems. 2013;3(4):303-311. ISSN 2248-9940.
10. Zadeh LA. Fuzzy set as a basis for a theory of possibility, Fuzzy sets, and systems. 1978;1:3-28.