Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 296 https://internationalpubls.com A Comparison of Proposed Ranking Method for Fuzzy Transportation Problem with the Existing Method E. Fany Helena 1, D. Stephen Dinagar 2 1Assistant Professor in PG and Research Department of Mathematics T.B.M.L. College (Affiliated to Annamalai University)Porayar-609307, Tamil Nadu, India. E-mail: fanyhelena3@gmail.com 2Associate Professor in PG and Research Department of Mathematics T.B.M.L. College (Affiliated to Annamalai University)Porayar-609307, Tamil Nadu, India. E-mail: dsdina@rediffmail.com Article History: Received: 15-05-2024 Revised: 25-06-2024 Accepted: 09-07-2024 Abstract: In this paper, we proposed two ranking methods to solve the transportation problems. Here we introduced new methods for problem solving. In this article, we proposed two ranking functions called ῡ_μ (Value Ranking) and Ᾱ_μ(Ambiguity Ranking) to solve the transportation problems. This work's main aim is to compare the suggested ranking function with the existing ranking method (Roubast Ranking). We have compared it with an example and justified it.. Keywords: Trapezoidal fuzzy numbers, Fuzzy Transportation Problem, Roubast Ranking Method, Value Ranking, Ambiguity Ranking. 1. Introduction Transportation models can be widely utilized in supply chain and logistics to minimize costs when there is accurate knowledge of the demand and supply quantities, as well as the cost coefficients. In these situations, efficient algorithms for solving the transportation problem have been developed. An Optimal Algorithm for a FTP (Fuzzy Transportation Problem) was studied in [1]. Randomness and imprecision are inevitable in the real world due to unanticipated events. They put forth a novel strategy in [2,3] to address the transportation issues with fuzzy numbers. In certain situations, uncontrollable factors may lead to uncertainty in the cost coefficients and demand & supply quantities of a transportation problem. Finding the schedule of shipping which reduces overall fuzzy transportation costs while meeting demand limits & fuzzy supply is the aim of the fuzzy transportation problem. We proposed a similarity measure in [4,5,6] using vector length for a TrIFNs, sign distance, and value ambiguity indices. Various ranking procedures are used to propose the measures. In this work, we addressed more practical issues, such as the fuzzy cost ƈ𝑖𝑗 transportation problem. The objective function is also regarded as a FN since the goal is to maximize total profit or minimize total cost, subject to certain fuzzy constraints. To find the best option, the objective function’s fuzzy objective values are ranked according to value using the ambiguity ranking approach for numbers. The FTP has been transformed using the α cut solution. The concept involves converting a fuzzy parameter problem into an LPP and using the Vogel Approximation Method to solve it. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 297 https://internationalpubls.com 2. Objectives Definition:2.1 A FN �̃� has to be generalized TrFNs (Trapezoidal Fuzzy Number) having a parameter ≤ 𝑎1 ≤ 𝑎2 ≤ 𝑎3 ≤ 𝑎4 and represented by �̃� = (𝑎1, 𝑎2, 𝑎3, 𝑎4) and its membership functions are followed below. 0 𝑖𝑓 𝑥 < 𝑎1 ( 𝑥−𝑎1 𝑎2−𝑎1 ) 𝑖𝑓 𝑎1 ≤ 𝑥 ≤ 𝑎2 𝜇�̃�(𝑥) = 1 𝑖𝑓 𝑎2 ≤ 𝑥 ≤ 𝑎3 ( 𝑎4−𝑥 𝑎4−𝑎3 ) 𝑖𝑓 𝑎3 ≤ 𝑥 ≤ 𝑎4 0 𝑖𝑓 𝑥 > 𝑎4 Definition: 2.2 Tough ranking algorithm that yields results consistent with human intuition while meeting linearity, additivity, and compensation requirements. If ã then the Robust Ranking which has been defined by R(�̃�) = ∫ (0.5)(𝑎𝛼 𝐿 𝑎𝛼 𝑈)𝑑𝛼 1 0 where (𝑎𝛼 𝐿 𝑎𝛼 𝑈) is the 𝛼 level cut of the FN ã we use this approach for objective values ranking. The R(ã) provides the representative value of FN ã. 2.3 Arithmetic operations on Generalized Trapezoidal “Fuzzy Numbers Let �̃�= (𝑎1, 𝑎2, 𝑎3, 𝑎4) and �̃� = (𝑏1, 𝑏2, 𝑏3, 𝑏4) be 2 generalized TrFNs & 𝜆 be a real number. Then 1. �̃�+�̃� = (𝑎1 + 𝑏1, 𝑎2 + 𝑏2, 𝑎3 + 𝑏3, 𝑎4 + 𝑏4) 2. �̃�-�̃� = (𝑎1 − 𝑏4, 𝑎2 − 𝑏3, 𝑎3 − 𝑏2, 𝑎4 − 𝑏1) 3. 𝛌�̃� = (𝛌𝑎1, 𝛌𝑎2, 𝛌𝑎3, 𝛌𝑎4); if 𝛌 > 0 (𝛌𝑎4, 𝛌𝑎3, 𝛌𝑎2, 𝛌𝑎1) if 𝛌 > 0 4.Let 𝑅(�̃�) = 𝑏1+𝑏2+𝑏3+𝑏4 4 �̃� × �̃�= (𝑎1𝑅(�̃�), 𝑎2𝑅(�̃�), 𝑎3𝑅(�̃�), 𝑎4𝑅(�̃�)) 5. Let 𝑅(�̃�) = 𝑏1+𝑏2+𝑏3+𝑏4 4 �̃� ÷ �̃� = ( 𝑎1 𝑅(�̃�) , 𝑎2 𝑅(�̃�) , 𝑎3 𝑅(�̃�) , 𝑎4 𝑅(�̃�) ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 298 https://internationalpubls.com 3. Methods” (i) Vague ranking function ῡ𝑖𝑗(𝑥) = ∫ (𝐿�̃�𝑖(𝛼) 1 0 + 𝑅�̃�𝑖(𝛼))𝑓(𝛼)𝑑𝛼 ῡ𝑖𝑗(𝑥) = ∫ [𝑎1 + 𝛼(𝑎2 − 𝑎1) + 𝑎4 − 𝛼(𝑎4 − 𝑎3)]𝛼 𝑑𝛼 1 0 After simplification, we get ῡ𝑖𝑗(𝑥) = 𝑎1 + 2𝑎2 + 2𝑎3 + 𝑎4 6 The vague ranking function of TrFN is defined as ῡ𝜇(𝑥) = 𝑎1+2𝑎2+2𝑎3+𝑎4 6 (ii) Ambiguity ranking function Ᾱ𝑖𝑗(𝑥) = ∫ ( 1 0 𝑅�̃�𝑖(𝛼) − 𝐿�̃�𝑖(𝛼))𝑓(𝛼)𝑑𝛼 �̃�𝑖𝑗(𝑥) = ∫ [𝑎4 − 𝛼(𝑎4 − 𝑎3) − 𝑎1 + 𝛼(𝑎2 − 𝑎1)]𝛼 𝑑𝛼 1 0 After simplification, we get Ᾱ𝜇(𝑥) = −𝑎1 − 2𝑎2 + 2𝑎3 + 𝑎4 6 The ambiguity ranking function of TrFN is defined as Ᾱ𝜇(𝑥) = −𝑎1 − 2𝑎2 + 2𝑎3 + 𝑎4 6 4. Results Transportation Problem Using Proposed Ranking Function The company has four sources, denoted as 𝑠1, 𝑠2, 𝑠3, and 𝑠4, and four destinations, also denoted as 𝑠1, 𝑠2, 𝑠3, and 𝑠4. The fuzzy transportation cost for transferring one unit of the product from the ith source to the jth destination is determined by the given formula. Cij =( (1,2,3,4)(1,3,5,6)(9,11,12,14)(5,7,8,11) (0,1,2,4)(−1,0,1,2)(5,6,7,8)(0,1,2,3) (3,5,6,8)(5,8,9,12)(12,15,16,19)(7,9,10,12) ) The availability of the sources are ((1,6,7,12), (0,1,2,3), (5,10,12,17)) and the product’s fuzzy demand at destinations are ((5,7,8,10), (1,5,6,10), (1,3,4,6) (1,2,3,4)) correspondingly. Then the problem becomes as Table:1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 299 https://internationalpubls.com Solution: The following mathematical programming form can be used to formulate the fuzzy transportation problem. (i) Vague ranking function ῡ𝑖𝑗(𝑥) = 𝑎1 + 2𝑎2 + 2𝑎3 + 𝑎4 6 By substituting the values, we get ῡ11(𝑥) = 2.5; ῡ12(𝑥) = 3.5; ῡ13(𝑥) = 11.5; ῡ14(𝑥) = 7.7 ῡ21(𝑥) = 1.7; ῡ22(𝑥) = 0.5; ῡ23(𝑥) = 6.5; ῡ24(𝑥) = 1.5 ῡ31(𝑥) = 5.5; ῡ32(𝑥) = 8.5; ῡ33(𝑥) = 15.5; ῡ34(𝑥) = 9.5 Supply ῡ(𝑓𝑆1 ) = 6.5; ῡ(𝑓𝑆2 ) = 1.5; ῡ(𝑓𝑆3 ) = 11 Demand ῡ(𝑓𝐷1 ) = 7.5; ῡ(𝑓𝐷2 ) = 5.5; ῡ(𝑓𝐷3 ) = 3.5; ῡ(𝑓𝐷4 ) = 2.5 After using the value ranking function the table is Table:2 By using the Vogel approximation method we get Table:3 The transportation cost is = (2.5)(1.0)+(3.5)(5.5)+(1.5)(1.5)+(5.5)(6.5)+(15.5.5)(3.5)+(9.5)(1.0) = 123.5 (ii) Ambiguity ranking function Ᾱ𝜇(𝑥) = −𝑎1 − 2𝑎2 + 2𝑎3 + 𝑎4 6 By the ambiguity ranking function also we get the same table value as in the value ranking function. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 300 https://internationalpubls.com After using VAM we get Table:4 The transportation cost is = 123.5. 5. Discussion and Conclusion In [7] they used the “Roubast Ranking” to solve the transportation. Then we proposed two ranking functions and we compared them in the below table Table:5 In this article, we proposed two new ranking functions called “ Value Ranking and Ambiguity Ranking” in an imprecise, vague area. Then we implemented this in the same transportation problem and we got the same cost as in Robust Ranking. To overcome the drawbacks of the existing method we newly defined the ranking function and we justified it by comparing them. References [1]. Rasha Jalal Mitlif, Mohammed Rasheed, Suha Shihab, An Optimal Algorithm for a Fuzzy Transportation Problem, Journal of Southwest Jiaotong University, 55:3(2020), 1-11. [2]. Ali Ebrahimnejad & Jose Luis Verdegay, A new approach for solving fully intuitionistic fuzzy transportation problems, Fuzzy optimization and decision making17(2018), 447 – 474. [3]. Laxminarayan Sahoo,A new score function based Fermatean fuzzy transportation problem, Results in Control and Optimization 4 (2021), 10040. [4]. D. Stephen Dinagar, E. Fany Helena, Similarity Measure using Sign Distance, Advances in Mathematics: Scientific Journal 10:1 (2021), 193-197. [5]. D. Stephen Dinagar, E. Fany Helena, Similarity Measures of Intuitionistic Trapezoidal Fuzzy Number using Centroids of Horizontal & Vertical Axes and Value & Ambiguity Indices, Advances and Applications in Mathematical Sciences 20 : 5 (2021), 875-885. [6]. D. Stephen Dinagar, E. Fany Helena, Similarity Measures with Vector – Length under Fuzzy Environment, Advances and Applications in Mathematical Sciences 20:8 (2021),1425- 1432. https://link.springer.com/article/10.1007/s10700-017-9280-1#auth-Ali-Ebrahimnejad-Aff1 https://link.springer.com/article/10.1007/s10700-017-9280-1#auth-Jose_Luis-Verdegay-Aff2 https://www.sciencedirect.com/journal/results-in-control-and-optimization https://www.sciencedirect.com/journal/results-in-control-and-optimization