59 This work is licensed under a Creative Commons Attribution 4.0 International License. Modified Ranking Function to Compute Fuzzy Matrix Games Rasha Jalal Mitlif Department of Applied Sciences, Branch of Mathematics and Computer Applications, University of Technology, Baghdad, Iraq. rasha.j.mitlif@uotechnology.edu.iq Abstract Game theory problems (GTP) frequently occur in Economy, Business Studies, Sociology, Political Science, Military Activities, and so on are some of the subjects covered. To tackle the uncertainty in Games, the analysis of games in which the payoffs are represented by fuzzy numbers (FN) will benefit from fuzzy set theory (FST). The purpose of this paper is to develop an efficient technique for solving constraint matrix games (MG) with payoff trapezoidal fuzzy numbers (TFN). The description of the new ranking method is introduced for a constrained matrix with TFN and values. Stock market forecasting has been one of the most important research areas for decades. Stock market values are volatile, non-linear, complicated and chaotic. Based on a ranking function (RF), we used a new algorithm to solve the fuzzy game problem (FGP) employing TFN and also to try to get a desirable gain. Centered on the latest proposed ranking algorithm, the Fuzzy decision method is designed to analyze possible stock opportunists. The paper considers a zero-sum game for two persons in which TFN are fuzzy payoffs. A ranking method (RM) is proposed to convert TFN into crisp numbers (CN) and it is used to solve FGP. The fuzzy game (FG) issue with concept strategies pure minimax maximin is presented. This problem is converted into the crisp problem by a new RF and then solved using the arithmetic (oddment) method. With the help of numerical examples, the suggested technique is explained. This paper finalizes the conclusion and includes an outlook for future study in this direction. Keywords: Ranking of Trapezoidal Fuzzy Numbers (RTFN), Game Theory (GTH), Saddle point (SP), Pure Strategy (PS). Ibn Al Haitham Journal for Pure and Applied Sciences Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/35.2.2765 Article history: Received 30, December, 2021, Accepted, 8,Fabruary, 2022, Published in April 2022. https://creativecommons.org/licenses/by/4.0/ mailto:rasha.j.mitlif@uotechnology.edu.iq Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 60 1. Introduction Fuzzy Game (FG) has been used in many areas, including operations analysis, control theory and management sciences, etc. When J.Newmann and O.Morgenstern [1] published the influential essay 'Theory of Games and Economic Performance' in 1944, the mathematical treatment of G TH became available. The important paper ' Economic results and G TH ' by J.Newmann and O.Morgenstern [1] published in 1944 presented the first statistical treatment of Game Theory (G TH). FG is a body of information that discusses decision-making (DM) while two and most experienced and rational competitors are in dispute and competition conditions. G TH has played a significant role in decision-making analysis fields like economics and management. We need to determine the right values of payoffs when we can use G TH to design any specific problems we face in real life. Knowing the actual value with payoffs, on the other hand, is difficult, and we could only know the calculated solution. In such situations, modeling the issues as games with uncertain payoffs is useful. All parameters are FN in a FGP. Standard or irregular triangular, trapezoidal, or octagonal numbers can be fuzzy. The vague FGP is transformed to a crisp estimation problem using the RF and then solved using the traditional approach. Some researchers extend FGP applications to crisp games and suggest appropriate solution methods and principles to solve FGP with ranking features [2-12]. Recently, the theory of fuzzy sets (FS) has been established and is the most suitable to date theory to deal with uncertainties. Many important G TH applications have been extended by embedding the FS ideas [13-18]. Several authors [19 -23] investigated two-person zero sum games with fuzzy payoffs and fuzzy aims G TH. We have taken two people's zero-sum games in this article, in which Trapezoidal fuzzy numbers (TFN) are imprecise values. We have explained it using the ranking technique to convert it to a crisply valued game problem. Using TFN, we have exposed a fuzzy game problem. The paper is structured as follows. Section 2 presents the concept of certain basic definitions and preliminaries regarding the crisp matrix games (CMG). In section 3, we present the mathematical formulation of the FGP. Section 4 introduces the proposed ranking technique. Section 5 presents the procedure for solving FGP using the matrix oddment method. In section 6, a numerical illustration is given to demonstrate the proposed method's efficacy. Finally, Section 7 presents the conclusions of this work. 2. Basic concepts: Definition (2.1): Fuzzy set (FS) [24] If the membership function ŊǪ ∶ Ʈ → [0,1] of a FS Đ̃ defined on the set of points Ʈ has the following properties, it is said to be a FN. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 61 (i) Đ̃ is an upper semi- continuous membership function; (ii) Đ̃ is convex, i. e.,ŊǪĘ(𝒽 𝜗 + (1 − 𝒽))𝑛𝑖𝑚 {ŊǪ(𝜗), ŊǪ(𝓋) },𝒽 ∈ [0,1] , For all 𝜗,𝓋 ∈ Ʈ. (iii) Đ̃ is normal, i. e., ∃ 𝜗0 ∈ Ʈ for which ŊǪ(𝜗0) = 1. (iv) 𝑆𝑢𝑝𝑝 (Đ̃) = {𝜗 ∈ Ʈ ∶ ŊǪ(𝜗) > 0} is the support of the Đ̃ , and its closure 𝑐𝑙 (𝑆𝑢𝑝𝑝 (Đ̃)) is a compact set. Definition (2.2): Pure strategy (PS) [25] If a player knows precisely what the other player is going to do is called pure strategy. Definition (2.3): Saddle point (SP) [26] A saddle point has been said to occur when the maximum value equals the minimum value, and optimal strategies are known as the complementary strategies that give the saddle point. The crisp game value (CGV) of the game matrix (GM) is the number of payoffs at an equilibrium point. Definition (2.4): Trapezoidal Fuzzy Numbers (TFN) [27]: A fuzzy number Đ̃ = (𝓉𝓅, 𝓉ç, ę, ᶁ) is said to be a TFN if its membership function is defined as follows: ŊǪĘ = { 𝜗−(𝓉𝓅−ę) ę , 𝓉𝓅 − ę ≤ 𝜗 ≤ 𝓉𝓅 1 , 𝓉𝓅 ≤ 𝜗 ≤ 𝓉ç (𝓉ç+𝛽)−𝜗 ᶁ , 𝓉ç ≤ 𝜗 ≤ 𝓉ç + ᶁ 0 , 𝑜𝑡ℎ𝑒𝑟 𝑤𝑖𝑠𝑒 Definition (2.5): Ranking function (RF) [28]: The function: Ʈ: Ƒ(Ʈ) → Ʈ that maps each FN into the real line is a RF where a natural order exists. If �̌�, ᶌ̌ ∈ Ƒ(Ʈ) , then a) �̌� ≥ ᶌ̌ if and only if Ʈ(�̌� ) ≥ Ʈ( ᶌ̌); b) �̌� > ᶌ̌ if and only if Ʈ(�̌� ) > Ʈ( ᶌ̌); c) �̌� = ᶌ̌ if and only if Ʈ(�̌� ) = Ʈ( ᶌ̌); d) �̌� ≤ ᶌ̌ if and only if Ʈ(�̌� ) ≤ Ʈ( ᶌ̌). 3. Mathematical Formulation of FGP: Consider a two-player zero-sum FG in which the payoffs matrix contains only TFN. Let us suggest each player has ɱ strategies and ɲ strategies. It is presumed that each player must choose a pure strategy from the available options. The gainer is always assumed to be the player 𝛿 , and the loser is always assumed to be player Ƴ. ɱ × ɲ is the payoff matrix. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 62 Ƕ = ( ǥ11̃ ǥ12̌ ǥ13̃ ǥ21̃ ǥ22̃ ǥ23̃ ǥ31̃ ǥ32̃ ǥ33̃ … ǥ1�̃� … ǥ2�̃� … ǥ3�̃� … … … … … … ǥ𝑚1̃ ǥ𝑚2̃ ǥ𝑚3̃ … … … … … ǥ𝑚�̃�) 4. Proposed Ranking Technique: In this section, we propose a new ranking function that depends on the idea of the Maleki ranking function with a new weight ( 7 8 ) . Assume ǥ̃ = ( 𝓉𝓅, 𝓉ç,ę,ᶁ ) is a trapezoidal fuzzy number, using the following formula to find the ranking function of �̃�. Ʈ(Đ̃) = 7 8 ∫ [ 𝑖𝑛𝑓 1 0 �̃�ȿ + 𝑠𝑢𝑝 �̃�ȿ ] 𝑑ȿ Ʈ(Đ) = 7 8 ∫ (( 𝓉𝓅 − 1 0 ęȿ) 𝑑𝛾 + ( 𝓉ç + ᶁȿ ) ) 𝑑ȿ Ʈ(Đ̃) = 7 8 [ 𝓉𝓅ȿ − 1 2 ę ȿ2 + 𝓉ç ȿ + 1 2 ᶁ ȿ2 | 0 1 Ʈ(Đ̃) = 7 8 (𝓉𝓅 − 1 2 ę + 𝓉ç + 1 2 ᶁ ) Ʈ(Đ̃) = 7 8 (𝓉𝓅 + 𝓉ç + 1 2 (ᶁ − ę ) ) Ʈ(Đ̃) = 7 8 (𝓉𝓅 + 𝓉ç) + 7 16 (ᶁ − ę ) . 5. Procedures for solving FGP using the matrix oddment method: The following steps are used to implement a solution procedure for solving the two-person zero-sum matrix game problems (MGP) in this section: Step1. In the FGP, describe the TFN in its parametric form. Step2. Translate the fuzzy game theory (FGTH) into a CV problem by applying the new RF. Step3. Check and see if the problem has a SP. The solution can be obtained directly if it exists. Proceed to the next level if the SP does not exist. Testing this fuzzy matrix game has SP exists or does not exist, we will continue in the following cases: (𝑖) Take a SP to test. (𝑖𝑖) If there is no a SP, find equalizing strategies to solve the problem. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 63 For players, the best-mixed strategies are 𝛿 = (ơ1 ,ơ2) in addition to the player Ƴ = (ᶉ1 , ᶉ2) Where ơ1 = ǥ22 −ǥ12 (ǥ11+ǥ22)−(ǥ12+ǥ21) , ơ2 = 1- ơ1 ᶉ1 = ǥ22 −ǥ21 (ǥ11+ǥ22)−(ǥ12+ǥ21) , ᶉ2 = 1- ᶉ1 and Value of the game ℂ = ǥ11∗ǥ22−ǥ12∗ǥ21 (ǥ11+ǥ22)−(ǥ12+ǥ21) Step4. To solve problems in G TH, use the arithmetic (oddment) approach to find the best strategies and the corresponding matrix game value for players. 6. Numerical examples: Example 1: Consider the payoff problem of the following fuzzy game as TFN. Ƴ 𝛿 ( 30 12 7 9 19 10 28 21 32 ) The minimum in the first row = 7 The minimum in the second row= 9 The minimum in the third row = 21 The maximum in the first column = 30 The maximum in the second column = 21 The maximum in the third column =32 Maximum (minimum) = 21 also Minimum (maximum) =21 It has saddle point. Value of Game = 21 Consider the following fuzzy game problem (FGP): Ƴ 𝛿 ( (28,32,0.3,1.7) (10,14,0.3,1.7) (5,9,0.3,1.7) (7,11,0.4,1.6) (17,21,0.4,1.6) (8,12,0.4,1.6) (26,30,0.2,1.8) (19,23,0.2,1.8) (30,34,0.2,1.8) ) Step1. We obtain the values of Ʈ (Đ) for the given FGP and transform it to a CV problem, as shown in the table below. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 64 Table 1. The TFN problem is transformed CV in example1. Step2. The given FGP is reduced in the following payoff matrix: Ƴ 𝛿 ( 53.1125 21.6125 12.8625 16.2750 33.7750 18.0250 49.7000 37.4500 56.7000 ) The minimum in the first row =12.8625 The minimum in the second row = 16.2750 The minimum in the third row = 37.4500 The maximum in the first column = 53.1125 The maximum in the second column = 37.4500 The maximum in the third column = 56.7000 Maximum (minimum) = 37.4500 also Minimum (maximum) = 37.4500 It has saddle point. Value of Game = 37.4500 Example 2: Consider the payoff problem of the following FG as TFN. Ƴ 𝛿 ( 3 13 5 7 14 8 12 16 6 ) ǥ11 = (28,32,0.3,1.7) Ʈ (ǥ11) = 53.1125 ǥ12 = (10,14,0.3,1.7) Ʈ (ǥ12 ) = 21.6125 ǥ13 = (5,9,0.3,1.7) Ʈ (ǥ13) = 12.8625 ǥ21 = (7,11,0.4,1.6) Ʈ (ǥ21) = 16.2750 ǥ22 = ((17,21,0.4,1.6) Ʈ (ǥ22) = 33.7750 ǥ23 = ( 8,12,0.4,1.6) Ʈ (ǥ23) = 18.0250 ǥ31 = (26,30,0.2,1.8) Ʈ (ǥ31) = 49.7000 ǥ32 = ((19,23,0.2,1.8) Ʈ (ǥ32) = 37.4500 ǥ33 = (30,34,0.2,1.8) Ʈ (ǥ33) = 56.7000 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 65 The minimum in the first row = 3 The minimum in the second row = 7 The minimum in the third row = 6 The maximum in the first column = 12 The maximum in the second column = 16 The maximum in the third column = 8 Maximum (minimum) = 7 but Minimum (maximum) = 8 Here Maximum (minimum) ≠ Minimum (maximum). It has no SP. The dominance approach is used. The second and third rows clearly dominate the first row, as all of the elements in the first row are lower than those in the second and third rows. As a result of removing the first row, we get Ƴ 𝛿 ( 7 14 8 12 16 6 ) Since all of the elements in the second column are greater than those in the first and third columns, the second column is once again dominated by the first and third columns. As a consequence of removing the second column, we get Ƴ 𝛿 ( 7 8 12 6 ) To determine the best mixed strategy and game value: Now, ǥ11 = 7 ,ǥ12 = 8 ,ǥ21 = 12 ,ǥ22 = 6 . ơ1 = ǥ22 − ǥ12 (ǥ11 + ǥ22) − (ǥ12 + ǥ21) = 6 − 8 (7 + 6) − (8 + 12) = −2 −7 = 0.285. ơ2 = 1 − ơ1 = 1 − 0.285 = 0.715. 𝑞1 = ǥ22 − ǥ21 (ǥ11 + ǥ22) − (ǥ12 + ǥ21) = 6 − 12 (7 + 6) − (8 + 12) = −6 −7 = 0.857. ᶉ2 = 1 − ᶉ1 = 1 − 0.857 = 0.143. Strategy for player 𝛿 = (ơ1,ơ2) = (0.285 ,0.715) Strategy for player Ƴ = (ᶉ1, ᶉ2) = (0.857 ,0.143) Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 66 Value of the game ℂ = ǥ11∗ǥ22−ǥ12∗ǥ21 (ǥ11+ǥ22)−(ǥ12+𝑎ǥ21) = (7∗6)−(8∗12) (7+6)−(8+12) = − 54 − 7 = 7.714. Consider the following FGP: Ƴ 𝛿 ( (1,5,0.7,1.3) (11,15,0.7,1.3) (3,7,0.7,1.3) (5,9,0.6,1.4) (12,16,0.6,1.4) (6,10,0.6,1.4) (10,14,0.8,1.2) (14,18,0.8,1.2) (4,8,0.8,1.2) ) Step1. We obtain the values of Ʈ (Đ) for the given FGP and transform it to a CV problem, as shown in the table below. Table 2. The TFN problem is transformed CV in the example. ǥ11 = (1,5,0.7,1.3) Ʈ (ǥ11) = 5.5125 ǥ12 = (11,15,0.7,1.3) Ʈ (ǥ12 ) = 23.0125 ǥ13 = (3,7,0.7,1.3) Ʈ (ǥ13) = 9.0125 ǥ21 = (5,9,0.6,1.4) Ʈ (ǥ21) = 12.6000 ǥ22 = (12,16,0.6,1.4) Ʈ (ǥ22) = 24.8500 ǥ23 = (6,10,0.6,1.4) Ʈ (ǥ23) = 14.3500 ǥ31 = (10,14,0.8,1.2) Ʈ (ǥ31) = 21.1750 ǥ32 = (14,18,0.8,1.2) Ʈ (ǥ32) = 28.1750 ǥ33 = (4,8,0.8,1.2) Ʈ (ǥ33) = 10.6750 Step 2. The given FGP is reduced in the following payoff matrix: Ƴ 𝛿 ( 5.5125 23.0125 9.0125 12.6000 24.8500 14.3500 21.1750 28.1750 10.6750 ) The minimum in the first row = 5.5125 The minimum in the second row = 12.6000 The minimum in the third row = 10.6750 The maximum in the first column = 21.1750 The maximum in the second column = 28.1750 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 67 The maximum in the third column = 14.3500 Maximum (minimum) = 12.6000 but Minimum (maximum) = 14.3500 Here Maximum (minimum) ≠ Minimum (maximum). It has no SP. The dominance approach is used. The second and third rows clearly dominate the first row, as all of the elements in the first row are lower than those in the second and third rows. As a result of removing the first row, we get Ƴ 𝛿 ( 12.6000 24.8500 14.3500 21.1750 28.1750 10.6750 ) Since all of the elements in the second column are greater than those in the first and third columns, the second column is once again dominated by the first and third columns. As a consequence of removing the second column, we get Ƴ 𝛿 ( 12.6000 14.3500 21.1750 10.6750 ) To determine the best-mixed strategy and game value: Now, ǥ11 = 12.6000 ,ǥ12 = 14.3500 ,ǥ21 = 21.1750 ,ǥ22 = 10.6750 . ơ1 = ǥ22 − ǥ12 (ǥ11 + ǥ22) − (ǥ12 + ǥ21) = 10.6750 − 14.3500 (12.6000 + 10.6750) − (14.3500 + 21.1750) = −3.6750 −12.2500 = 0.3000. ơ2 = 1 − ơ1 = 1 − 0.3000 = 0.7000. ᶉ1 = ǥ22 − ǥ21 (ǥ11 + ǥ22) − (ǥ12 + ǥ21) = 10.6750 − 21.1750 (12.6000 + 10.6750) − (14.3500 + 21.1750) = −10.5000 −12.2500 = 0.8571. ᶉ2 = 1 − ᶉ1 = 1 − 0.8571 = 0.1429. Strategy for player 𝛿 = (ơ1,ơ2) = (0.3000 ,0.7000) Strategy for player Ƴ = (ᶉ1, ᶉ2) = (0.8571 ,0.1429) Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 68 Value of the game ℂ = ǥ11∗ǥ22−ǥ12∗ǥ21 (ǥ11+ǥ22)−(ǥ12+ǥ21) = (12.6000∗10.6750)−(14.3500∗21.1750) (12.6000+10.6750)−(14.3500+21.1750) = −169.3562 −12.2500 = 13.8250. 7. Conclusion This research evaluates the effectiveness of trapezoidal fuzzy numbers in stock market prediction using a proposed ranking technique based on ranking functions. Our model can achieve a more suitable partition of the universe by using trapezoidal fuzzy numbers, significantly boosting forecasting results. Furthermore, we struggle with games of fuzzy payoffs issues when stock market data is uncertain. We use the trapezoidal membership function to get the best gains, making the data fuzzier. The currently proposed ranking function algorithm uses trapezoidal fuzzy numbers for the decision-maker. We suggest an arithmetic oddment strategy to solve players (fuzzy game problem strategies, all trapezoidal fuzzy numbers) are implemented using the ranking of fuzzy numbers (FN). References 1. Newmann , J.; Morgenstern, O. Theory of Games and Economic Behaviour, Princeton University Press, Princeton, New Jersey (1947) . 2. Raju , V.; vathana, M. P. Discourse on Fuzzy Game Problem in Icosagonal Fuzzy Number, International Journal of Scientific Research and Review. 2019,8, 1384 – 1390. 3. Selvakumari, K.; Lavanya, S. On Solving Fuzzy Game Problem using Octagonal Fuzzy Numbers, Annals of Pure and Applied Mathematics. 2014, 8(2), 211-217. 4. Hussein, I H. ; Mitlif, R J. Ranking Function to Solve a Fuzzy Multiple Objective Function, Baghdad Science Journal, 2021,18, 1, 144-148. 5. Namarta, U.; Gupta, Ch.; Thakur, N.I. Solving Game Problems involving Heptagonal and Hendecagonal Fuzzy Payoffs, International Journal of Innovative Technology and Exploring Engineering (IJITEE). 2019, 8, 2114-2120. 6. Namarta, U.; Gupta, Ch.; Thakur, N. Applications of Game Problems in Fuzzy Enviornment, International Journal for Research in Engineering Application & Management (IJREAM), 2019, 4, 228-232. 7. Mitlif ,R J., Computation the Optimal Solution of Octagonal Fuzzy Numbers, Journal of Al- Qadisiyah for Computer Science and Mathematics, 2020, 12, 4, 71–78. 8. Krishnaveni, G.; Ganesan, K. A new approach for the solution of fuzzy games, National Conference on Mathematical Techniques and its Applications , 2018,1-6. 9. Monisha, P.; Sangeetha, K. To Solve Fuzzy Game Problem Using Pentagonal Fuzzy Numbers, International Journal for Modern Trends in Science and Technology, 2017, 03( 09), , 152-154. 10. Mitlif, R J.; Fatema Ahmad Sadiq, Finding the Critical Path Method for Fuzzy Network with Development Ranking Function, Journal of Al-Qadisiyah for Computer Science and Mathematics. 2021 ,13, 3 , 98–106. 11. Raju, V.; Vathana, M. P. Discourse on Fuzzy Game Problem in Icosagonal Fuzzy Number, International Journal of Scientific Research and Review, 2019, 8, 1384-1390. Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 69 12. Namarta, U. Ch. Gupta, N. I. Thakur, A New -Cut Approach to Solve Fuzzy Game Problem, International Journal of Management, Technology And Engineering.2018,8,2761-2766. 13. J.C. Liu, Y.F. Zhu, W. J.Zhao, Quadratic Programming Method for Cooperative Games with Coalition Values Expressed by Triangular Fuzzy Numbers and Its Application in the Profit Distribution of Logistics Coalition, Hindawi Mathematical Problems in Engineering, 2019, 1- 10. 14. Mitlif, R J, An Efficient Algorithm for Fuzzy Linear Fractional Programming Problems via Ranking Function, Baghdad Science Journal, 2022, 19, 1, 71-76. 15. Ch. Cui; Feng, Z.; Ch Tan, Credibilistic Loss Aversion Nash Equilibrium for Bimatrix Games with Triangular Fuzzy Payoffs, Hindawi Complexity, 2018,1-16. 16. Mitlif R J, Ranking Function Application for Optimal Solution of Fractional Programming Problem, Al-Qadisiyah Journal Of Pure Science, 2020, 25 , 1,27-35. 17. Narmatha, S.; Glory Bebina, E. ; Vishnu Priyaa, R. SOLVING GAME PROBLEM USING ICOSIKAITETRAGONAL FUZZY NUMBERS, The International journal of analytical and experimental modal analysis2020, 12, 55-60. 18. Jahir Hussain, R.; Priya, A. SOLVING FUZZY GAME PROBLEM USING HEXAGONAL FUZZY NUMBER, Journal of Computer . 2016, JoC, l, 53-59. 19. Hamiden Abdelwahed Khalifa , An approach for solving two- person zero- sum matrix games in neutrosophic environment, Journal of Industrial and Systems Engineering, 2019,12(2),186-198. 20. Salman, Israa M., and Eman A. Abdul-Razzaq. "Solving Nonlinear Second Order Delay Eigenvalue Problems by Least Square Method." Ibn AL-Haitham Journal For Pure and Applied Sciences, 2020, 33(4)59-64. 21. Kalaf, Bayda Atiya Bakar RA, Soon LL, Monsi MB, Bakheet AJ, Abbas IT. A modified fuzzy multi-objective linear programming to solve aggregate production planning. International Journal of Pure and Applied Mathematics, 2015,104(3) .339-352 22. Thirucheran, M. ; Meena kumari, E.R. ; Lavanya, S. A NEW APPROACH FORSOLVING FUZZY GAME PROBLEM, International Journal of Pure and Applied Mathematics, 2017,114(6), 67 – 75. 23. Krishnaveni, G. ; Ganesan, K. New approach for the solution of two person zero sum fuzzy games, International Journal of Pure and Applied Mathematics, 2018, 119 ( 9 ), 405-414. 24. Hamiden bdelwahed Khalifa , An approach for solving two- person zero- sum matrix games in neutrosophic environment , Journal of Industrial and Systems Engineering , 2017,12(2),186- 198. 25. Tharani R., S. Rekha , A New Approach of Solving Fuzzy Game Problem of Order 3 X 3 Using Dodecagonal Fuzzy Numbers , International Journal of Scientific Research in Science, Engineering and Technology , 2017, 3 ( 6 ) , 131- 134. 26. Selvakumari, K. ; Lavanya, S. On Solving Fuzzy Game Problem using Octagonal Fuzzy Numbers , Annals of Pure and Applied Mathematics . 2014, 8 (2) , 211-217. 27. Savitha, M T.; Mary, G. New Methods for Ranking of Trapezoidal Fuzzy Numbers, Advances in Fuzzy Mathematics, 2017, 12(5), 1159-1170. 28. Zhong, Y.; Jia, Y.;Chen, D.; Yang, Y. Interior Point Method for Solving Fuzzy Number Linear Programming Problems Using Linear Ranking Function, Journal of Applied Mathematics, 2013, 1-9.