Fuzzy Transportation Problem Using New Ranking 
Techniques to Order Pentagonal Fuzzy Numbers with 
Error Using Lagrange’s Interpolation Formula

Ashok Mhaske; Ambadas Deshmukh

doi:https://doi.org/10.14445/22315373/IJMTT-V70I8P101

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 70 | Issue 8 | Year 2024 | Article Id. IJMTT-V70I8P101 | DOI : https://doi.org/10.14445/22315373/IJMTT-V70I8P101

Fuzzy Transportation Problem Using New Ranking Techniques to Order Pentagonal Fuzzy Numbers with Error Using Lagrange’s Interpolation Formula

Ashok Mhaske, Ambadas Deshmukh

Received	Revised	Accepted	Published
19 Jun 2024	28 Jul 2024	13 Aug 2024	31 Aug 2024

Abstract

In this research article, we transform the conventional transportation problem into a fuzzy transportation problem employing symmetric pentagonal fuzzy numbers (a1-2d, a1-d, a1, a1+d, a2+2d). The ordering of fuzzy pentagonal numbers is accomplished through the alpha-cut method. To quantify the discrepancy between the crisp and fuzzy transportation problems, we examine the error for varying values of d (1, 2, and 3). The data obtained is subjected to Lagrange's polynomial fit to model the error term. Subsequently, we conduct a comparative analysis of the errors derived from the fuzzy transportation method and those obtained through Lagrange's polynomial for different d values, specifically d=4.

Keywords

Fuzzy, Ranking, α-cut, Pentagonal, Transportation, Python, Lagrange’s, Error.

References

[1] Saied Abbasbandy, and Babak Asady, “Ranking of Fuzzy Numbers by Sign Distance,” Information Sciences, vol. 176, no. 16, pp. 2405- 2416, 2006.
[CrossRef] [Google Scholar] [Publisher Link]
[2] James F. Baldwin, and N.C.F. Guild, “Comparison of Fuzzy Sets on the Same Decision Space,” Fuzzy Sets and Systems, vol. 2, no. 3, pp. 213-231, 1979.
[CrossRef] [Google Scholar] [Publisher Link]
[3] Adrian I. Ban, and Lucian Coroianu, “Existence, Uniqueness and Continuity of Trapezoidal Approximations of Fuzzy Numbers Under a General Condition,” Fuzzy Sets and Systems, vol. 257, pp. 3-22, 2014.
[CrossRef] [Google Scholar] [Publisher Link]
[4] Sanhita Banerjee, and Tapan Kumar Roy, “Arithmetic Operations on Generalized Trapezoidal Fuzzy Number and its Applications” Turkish Journal of Fuzzy Systems, vol. 3, no. 1, pp. 16-44, 2012.
[Google Scholar]
[5] Fred Choobineh, and Huishen Li, “An Index for Ordering Fuzzy Numbers,” Fuzzy Sets and Systems, vol. 54, no. 3, pp. 287-294, 1993.
[CrossRef] [Google Scholar] [Publisher Link]
[6] Feng-Tse Lin, and Tzong-Ru Tsai, “A Two-Stage Genetic Algorithm for Solving the Transportation Problem with Fuzzy Demands and Fuzzy Supplies”, Chinese Culture University, 2009.
[Google Scholar] [Publisher Link]
[7] P. Pandian, and G. Natrajan, “A New Algorithm for Finding a Fuzzy Optimal Solution for Fuzzy Transportation Problem,” Applied Mathematical Sciences, vol. 4, no. 2, pp. 79-90, 2010.
[Google Scholar] [Publisher Link]
[8] Stefan Chanas, and Dorota Kuchta, “A Concept of the Optimal Solution of the Transportation Problem with Fuzzy Cost Coefficients, Fuzzy Sets and Systems, vol. 82, no. 3, pp. 299-305, 1996.
[CrossRef] [Google Scholar] [Publisher Link]
[9] Ying-Ming Wang, and Zhi-Ping Fan, “Fuzzy Preference Relations: Aggregation and Weight Determination,” Computers & Industrial Engineering, vol, 53, no. 1, pp. 163-172, 2007.
[CrossRef] [Google Scholar] [Publisher Link]
[10] Kirankumar Laxmanrao Bondar, and Ashok Mhaske, “Fuzzy Transportation Problem with Error by Using Lagrange’s Polynomial” The Journal of Fuzzy Mathematics, vol. 24, no. 4, pp. 825-832, 2016.
[Google Scholar]
[11] Ashok Sahebrao Mhaske, and Kirankumar Laxmanrao Bondar, “Fuzzy Database and Fuzzy Logic for Fetal Growth Condition,” Asian Journal of Fuzzy and Applied Mathematics, vol. 3, no. 3, pp. 95-104, 2015.
[Google Scholar] [Publisher Link]
[12] Ashok Sahebrao Mhaske, and Kirankumar Laxmanrao Bondar, “Fuzzy Transportation by Using Monte Carlo Method,” Advances in Fuzzy Mathematics, vol. 12, no. 1, pp. 111-127, 2017.
[Google Scholar] [Publisher Link]
[13] Ambadas Deshmukh et al., “Fuzzy Transportation Problem by Using Fuzzy Random Number” International Review of Fuzzy Mathematics, vol. 12, no. 1, pp. 81-94, 2017.
[Google Scholar]
[14] Ambadas Deshmukh et al., “Fuzzy Transportation Problem by Using Trapezoidal Fuzzy Numbers,” International Journal of Research and Analytical Reviews, vol. 5, no. 3, pp. 261-265, 2018.
[Google Scholar]
[15] Ashok Sahebrao Mhaske, and Kirankumar Laxmanrao Bondar. “Fuzzy Transportation Problem by Using Triangular, Pentagonal and Heptagonal Fuzzy Numbers With Lagrange’s Polynomial to Approximate Fuzzy Cost for Nonagon and Hendecagon,” International Journal of Fuzzy System Applications, vol. 9, no. 1, pp. 112-129, 2020.
[CrossRef] [Google Scholar] [Publisher Link]
[16] Ashok S. Mhaske “Ranking Triangular Fuzzy Numbers Using Area of Rectangle at Different Level of α-Cut for Fuzzy Transportation Problem” Journal of Emerging Technologies and Innovative Research, vol. 8, no. 3, pp. 2202-2209, 2021.
[Google Scholar] [Publisher Link]
[17] Ashok S. Mhaske “Difference Between Fuzzy and Crisp Transportation Problem Using Pentagonal Fuzzy Numbers with Ranking by αcut Method,” Journal of Emerging Technologies and Innovative Research, vol. 8, no. 3, pp. 2143-2150, 2021.
[Publisher Link]
[18] Ambadas Deshmukh et al., “Fuzzy Transportation Problem by Using Triangular Fuzzy Numbers with Ranking Using Area of Trapezium, Rectangle and Centroid at Different Level of α-Cut,” Turkish Journal of Computer and Mathematics Education, vol. 12, no.12, pp. 3941- 3951, 2021.
[Google Scholar] [Publisher Link]
[19] Sagar Waghmare et al., “New Group Structure of Compatible Systems of First Order Partial Differential Equations,” International Journal of Mathematics Trends and Technology, vol. 67, no. 9, pp. 114-117, 2021.
[CrossRef] [Google Scholar] [Publisher Link]
[20] Ambadas Deshmukh et al., “Optimum Solution to Fuzzy Transportation Problem Using Different Ranking Techniques to Order Triangular Fuzzy Numbers” Stochastic Modelling & Applications, vol. 26, no. 3, pp. 35-40, 2022.
[Google Scholar] [Publisher Link]
[21] Kirankumar Laxmanrao Bondar, and Ashok Mhaske, “Fuzzy Unbalanced Transportation Problem by Using Monte Carlo Method” Aayushi International Interdisciplinary Research Journal (AIIRJ), 2018.
[Google Scholar]
[22] Amit Nalvade et al., “Solving Fuzzy Game Theory Problem Using Pentagonal Fuzzy Numbers and Hexagonal Fuzzy Number,” International Journal of Mathematics Trends and Technology, vol. 69, no. 2, pp. 74-79, 2023.
[CrossRef] [Google Scholar] [Publisher Link]
[23] Ambadas Deshmukh et al., “Fuzzy Database and Fuzzy Logic Using Triangular and Trapezoidal Fuzzy Number for Coronavirus Disease - 2019 Diagnosis,” Mathematical Statistician and Engineering Applications, vol. 71, no. 4, pp. 8196–8207, 2022.
[Google Scholar] [Publisher Link]
[24] Ambadas Deshmukh et al “Computer Integrated Manufacturing Systems” Computer Integrated Manufacturing Systems, vol. 18, no. 11, pp. 257-264, 2022.
[25] Sagar Waghmare et al., “New Ring and Vector Space Structure of Compatible Systems of First Order Partial Differential Equations,” International Journal of Mathematics Trends and Technology, vol. 68, no. 9, pp. 60-65, 2022.
[CrossRef] [Google Scholar] [Publisher Link]
[26] Ashok Mhaske et al., “Optimum Solution to Fuzzy Game Theory Problem Using Triangular Fuzzy Numbers and Trapezoidal Fuzzy Number,” Journal of Information and Computational Science, vol. 12, no. 3, pp. 199-211, 2022.
[Google Scholar] [Publisher Link]

Citation :

Ashok Mhaske, Ambadas Deshmukh, "Fuzzy Transportation Problem Using New Ranking Techniques to Order Pentagonal Fuzzy Numbers with Error Using Lagrange’s Interpolation Formula," International Journal of Mathematics Trends and Technology (IJMTT), vol. 70, no. 8, pp. 1-10, 2024. Crossref, https://doi.org/10.14445/22315373/IJMTT-V70I8P101