Solution of Fuzzy Matrix Game Problem by Method of Oddments using Octadecagonal Fuzzy Numbers

80 https://researchtrendsjournal.com Online at: https://researchtrendsjournal.com ISSN No: 2584-282X Indexed Journal Peer Reviewed Journal INTERNATIONAL JOURNAL OF TRENDS IN EMERGING RESEARCH AND DEVELOPMENT Volume 3; Issue 5; 2025; Page No. 80-85 Received: 14-07-2025 Accepted: 20-08-2025 Published: 06-10-2025 Solution of Fuzzy Matrix Game Problem by Method of Oddments using Octadecagonal Fuzzy Numbers 1Anuj Sharma and 2Dr. Dharmendra Badal 1Research Scholar, Department of Mathematical Sciences and Computer Applications, Bundelkhand University, Jhansi, Uttar Pradesh, India 2Assistant Professor, Department of Mathematical Sciences and Computer Applications, Bundelkhand University, Jhansi, Uttar Pradesh, India DOI: https://doi.org/10.5281/zenodo.17279088 Corresponding Author: Anuj Sharma Abstract In this paper we have considered fuzzy game matrix. The inexact values in the fuzzy game matrix are Octadecagonal fuzzy numbers. We convert the fuzzy valued game problem to a crisp valued game problem by using ranking which we have tried to solve using method of oddments. Keywords: Fuzzy sets, Octadecagonal Fuzzy Numbers (ODFN), Ranking of Fuzzy Numbers, Fuzzy Game Problem Introduction The theory of games started in the 20th century. But the mathematical treatment of games took fire in 1944 when Neumann, J.V. and Morgenstern, O. [15] published their renowned article on “Theory of games and economic behavior”. The Neumann’s approach utilizes the mini-max principle which involves the elementary idea of minimization of maximum loss. All the parameters in the fuzzy game problem are fuzzy numbers. The fuzzy numbers can be triangular, trapezoidal, hexagonal, octagonal, hendecagonal, octadecagonal etc. The ranking method using α-cuts, for ranking of Fuzzy numbers was proposed by Basirzadeh [11] in which he has ranked triangular and trapezoidal fuzzy numbers. For Octadecagonal fuzzy numbers, the arithmetic operations, alpha cut, and ranking technique are introduced by Barya V, Sharma A and Badal D [1]. By using this ranking, the Fuzzy Game problem is converted to a crisp value problem, which can be solved using the method of oddments. Preliminaries The aim of this section is to throw some light on some notations, notions and results which are used further. Fuzzy Set: Let X be a non-empty set. A fuzzy set “A” in X is characterized by its membership function A: X [0, 1] and A(x) is interpreted as the degree of membership of element x in fuzzy A for each x Ɛ X. Complete non-membership is represented by the value zero; complete participation is represented by the value one and intermediate degrees of membership are represented by values in between. The membership function of fuzzy set A is also known as the mapping A. International Journal of Trends in Emerging Research and Development https://researchtrendsjournal.com 81 https://researchtrendsjournal.com Fuzzy Number A fuzzy number “A” is a convex normalized fuzzy set on the real line R, such that: a. There exists at least one x˳ Ɛ R with µᴀ (x)= 1 b. µᴀ (x) is piecewise continuous. Octadecagonal Fuzzy Number A Octadecagoanal fuzzy number is denoted as ᾹODFN = (a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, a16, a17, a18) and its membership function is given by: International Journal of Trends in Emerging Research and Development https://researchtrendsjournal.com 82 https://researchtrendsjournal.com Parametric Form of Octadecagonal Fuzzy Number The parametric form of Octadecagonal fuzzy number is defined as U = (P1(r), Q1(s), R1(t), S1(u), T1(v), U1(w), V1(x), W1(y), P2(r), Q2(s), R2(t), S2(u), T2(v), U2(w), V2(x), W2(y)) for r ϵ [0,1/8], s ϵ [1/8,2/8], t ϵ [2/8,3/8], u ϵ [3/8,4/8], v ϵ [4/8,5/8], w ϵ [5/8,6/8], x ϵ [6/8,7/8], y ϵ [7/8,1] where, a. P1(r), Q1(s), R1(t), S1(u), T1(v), U1(w), V1(x) and W1(y) are bounded left continuous non-decreasing function over [0,1/8], [1/8,2/8], [2/8,3/8], [3/8,4/8], [4/8,5/8], [5/8,6/8], [6/8,7/8], and [7/8,1]. b. P2(r), Q2(s), R2(t), S2(u), T2(v), U2(w), V2(x) and W2(y) are bounded left continuous non-increasing function over [0,1/8], [1/8,2/8], [2/8,3/8], [3/8,4/8], [4/8,5/8], [5/8,6/8], [6/8,7/8], and [7/8,1]. Fig 1: Graphical Representation of Octadecagonal Fuzzy number Ranking of Octadecagonal Fuzzy Number The ranking function r:F(R) ⟶ R where F(R) is a set of fuzzy numbers defined on set of real numbers, which maps each fuzzy number into the real line, where the natural order exists (Yager 1986), i.e. a. A > B iff r(A) > r(B) b. A < B iff r(A) < r(B) c. A = B iff r(A) = r(B) Let A = (a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15, a16, a17, a18) and B = (b1, b2, b3, b4, b5, b6, b7, b8, b9, b10, b11, b12, b13, b14, b15, b16, b17, b18) be two Octadecagonal fuzzy numbers then 𝑟(𝐴) = 𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 + 𝑎5 + 𝑎6 + 𝑎7 + 𝑎8 + 𝑎9 + 𝑎10 + 𝑎11 + 𝑎12 + 𝑎13 + 𝑎14 + 𝑎15 + 𝑎16 + 𝑎17 + 𝑎18 18 And 𝑟(𝐵)=𝑏1 + 𝑏2 + 𝑏3 + 𝑏4 + 𝑏5 + 𝑏6 + 𝑏7 + 𝑏8 + 𝑏9 + 𝑏10 + 𝑏11 + 𝑏12 + 𝑏13 + 𝑏14 + 𝑏15 + 𝑏16 + 𝑏17 + 𝑏18 18 Solution of Fuzzy Matrix Game by Method of Oddments Consider the general (3x3) game matrix, whose elements are Octadecagonal fuzzy numbers. Algorithm ▪ First we shall convert the given fuzzy game problem into a crisp value problem ▪ Check for the saddle point in the payoff matrix. ▪ If there is no saddle point nor it is reducible by dominance, solve using method of oddments. International Journal of Trends in Emerging Research and Development https://researchtrendsjournal.com 83 https://researchtrendsjournal.com Step: 1 Subtract each row from the row above i.e. subtract 2nd row from 1st and 3rd row from the 2nd and write the differences in the form of two successive rows beneath the matrix's rows. Step: 2 Now Subtract each column form the column to its left i.e. subtract 2nd column from 1st and 3rd column from 2nd and write the differences in the form of two successive columns to the right of the matrix. Step: 3 Now calculate the oddments for the A’s 1, 2, 3 strategies and B’s I, II, III strategies. Step: 4 Write these oddments, neglecting the signs. Step: 5 Now check if the sum of oddments of both the players is same then both the players uses their all pure strategies and hence game is conformable for matrix method. But if the sum of oddments of both the players is different, then both the players do not uses their all pure strategies. Step: 6 Now divide the oddments by the sum of the oddments to get the optimal strategies of both the players. Step: 7 Finally calculate the value of game by using the formula given below, Or Example: Consider the following fuzzy game problem, Solution: First of all we shall use ranking of Octadecagonal fuzzy numbers to convert the given fuzzy game problem to a crisp value problem, a11 = (-14, -12, -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20) a12 = (-16, -14, -12, -10, -8, -6, -4, -2, 2, 3, 4, 5, 6, 7, 8, 9, 20, 26) a13 = (-16, -14, -12, -10, -8, -6, -4, -2, 2, 3, 4, 5, 6, 7, 8, 9, 20, 26) a21 = (-16, -14, -12, -10, -8, -6, -4, -2, 2, 3, 4, 5, 6, 7, 8, 9, 20, 26) a22 = (-16, -14, -12, -10, -8, -6, -4, -2, 2, 3, 4, 5, 6, 7, 8, 9, 20, 26) a23 = (-16, -14, -12, -10, -8, -6, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24) a31 = (-16, -14, -12, -10, -8, -6, -4, -2, 2, 3, 4, 5, 6, 7, 8, 9, 20, 26) a32 = (-9, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13) a33 = (-16, -14, -12, -10, -8, -6, -4, -2, 2, 3, 4, 5, 6, 7, 8, 9, 20, 26) International Journal of Trends in Emerging Research and Development https://researchtrendsjournal.com 84 https://researchtrendsjournal.com Reduced Crisp value problem is given by the following matrix, Now Subtracting 2nd row form the 1st row and 3rd row from the 2nd row and write the differences in the form of two successive rows below the matrix, Player B 3 1 1 Player A 1 1 5 1 4 1 A1 - A2 2 0 -4 A2 - A3 0 -3 4 Similarly, subtracting 2nd column form the 1st column and 3rd column from the 2nd column and write the differences in the form of two successive columns to the right of the matrix, Player B B1 - B2 B2 - B3 3 1 1 2 0 Player A 1 1 5 0 -4 1 4 1 -3 3 A1 - A2 2 0 -4 A2 - A3 0 -3 4 Now we shall calculate the oddments for A’s 1,2,3 strategies and B’s I, II, III strategies, Oddment for A’s 1st Strategy = det |0 −4 −3 3 | = -12 Oddment for A’s 2nd Strategy = det |2 0 −3 3| = 6 Oddment for A’s 3rd Strategy = det |2 0 0 −4| = -8 Oddment for B’s Ist Strategy = det |0 −4 −3 4 | = -12 Oddment for B’s IInd Strategy = det |2 −4 0 4 | = 8 Oddment for B’s IIIrdStrategy = det |2 0 0 −3| = -6 Now we shall write these oddments, neglecting the signs as shown in the table below, Now check the sum of oddments of both the players A and B which is same i.e. 26 as shown in the table above. So we can say that both the players A and B uses their all pure strategies and hence the game is solvable by matrix method but if the sum of oddments of both the players is not equal that means both the players do not use their all pure strategies and hence the game is not solvable by matrix method. International Journal of Trends in Emerging Research and Development https://researchtrendsjournal.com 85 https://researchtrendsjournal.com Now we shall calculate the optimal strategies of players A and B, by dividing the oddments by the sum of oddments, therefore Conclusion ▪ In this paper, Method of Oddments is used for solving the fuzzy matrix game problem. ▪ In the above example we have considered a fuzzy matrix game problem whose elements are Octadecagonal fuzzy numbers and solved using method of oddments. References 1. Barya V, Sharma A, Badal D. A new octadecagonal fuzzy number and its fuzzy arithmetic operations. Vidyawarta. 2023;45:47-49. 2. Sharma A, Badal D. Solution of fuzzy game problem using Tridecagonal fuzzy number. Review of Business and Technology Research. 2019;16:129-134. 3. Arockiaraj JJ, Sivasankari N. Solution of fuzzy game problem using hexagonal fuzzy numbers. International Journal of Mathematics and its Applications. 2016;4:381-387. 4. Nayak PK, Pal M. Solution of rectangular interval games using graphical method. Tamsui Oxford Journal of Mathematical Sciences. 2016;221:95-115. 5. Christi Anni MS, Malini D. Solving transportation problems with hexagonal fuzzy numbers using Best Candidate Method and different ranking techniques. International Journal of Engineering Research and Applications. 2016;624:76-81. 6. Selvakumari K, Lavanya S. An approach for solving fuzzy game problem. Indian Journal of Science and Technology. 2015;8(15):1-5. 7. Selvakumari K, Lavanya S. On solving fuzzy game problem using octagonal fuzzy numbers. Annals of Pure and Applied Mathematics. 2014;82:211-217. 8. Kamble AJ, Venkatesh T. Some results on fuzzy numbers. Annals of Pure and Applied Mathematics. 2014;72:90-97. 9. Rajarajeshwari P, Sudha AS. Ordering generalized hexagonal fuzzy numbers using rank mode divergence and spread. IOSR Journal of Mathematics. 2014;1032:15-22. 10. Rajarajeshwari P, Sudha AS, Karthika R. A new operation on hexagonal fuzzy numbers. International Journal of Fuzzy Logic Systems. 2013;33:15-26. 11. Basirzadeh H, Abbasi R. A new approach for ranking fuzzy numbers based on P-cuts. Journal of Applied Mathematics and Informatics. 2008;26:767-778. 12. Lee KH. First course on fuzzy theory and application. Springer; 2005:137-145. 13. Murthy PR. Operations research. New Age International Publishers; c2005. 14. Nishizaki I, Sakawa M. Equilibrium solution for multi-objective bimatrix games in corporation fuzzy goals. Journal of Optimization Theory and Applications. 1995;86:433-457. 15. Neumann JV, Morgenstern O. Theory of games and economic behavior. Princeton University Press; c1947. 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