International Journal of Trend in Scientific Research and Development (IJTSRD) Volume 5 Issue 6, September-October 2021 Available Online: www.ijtsrd.com e-ISSN: 2456 – 6470 @ IJTSRD | Unique Paper ID – IJTSRD47553 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1194 Icosagonal Fuzzy Number in Decision Making Problem V. Raju 1 , S. Ramachandran 2 1Department of Mathematics, School of Basic Sciences, Vels Institute of Science, Technology and Advanced Studies (VISTAS), Chennai, Tamil Nadu, India 2Department of Mathematics, St. Thomas College of Arts and Science, Chennai, Tamil Nadu, India ABSTRACT This article introduces Icosagonal fuzzy number in the fuzzy decision making problem by. The value of Payoff matrix is specified by Icosagonal fuzzy number. We adapt the fuzzy decision making problem into crisp valued decision making problem by way of ranking to pay off. The crisp valued decision making problem can be efficiently gave out with Savage mini max regret criterion. KEYWORDS: Icosagonal fuzzy number, fuzzy decision making, fuzzy ranking and Savage mini max How to cite this paper: V. Raju | S. Ramachandran "Icosagonal Fuzzy Number in Decision Making Problem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-6, October 2021, pp.1194-1199, URL: www.ijtsrd.com/papers/ijtsrd47553.pdf Copyright © 2021 by author (s) and International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0) (http://creativecommons.org/licenses/by/4.0) 1. INTRODUCTION Fuzzy set theory was initiated by Zadeh [1].The concept of fuzzy set theory gives out ambiguity and indistinctness. This is the reason of uncertainty in decision making problem. Uncertainty arises for having the lack of understanding about insertion or omission. We get a lot information from our environment day by day is fuzzy. We are unable to have our daily life without crossing fuzzy connected circumstances. Fuzzy set is resourceful in many existent world condition .Jain [2] was the first to propose method of ranking fuzzy numbers for decision making in fuzzy related condition. Raju and Jayagopal [3] was the first to introduce the Icosagonal fuzzy number. Some decisions are being in use based on the compilation of data. Decision makers are applying Icosagonal fuzzy numbers rather than real numbers to express their judgments. Solving problems and making decisions are necessary skills for business and life. Problem solving shows the importance of decision making. Decision making is very significant and essential for management and leadership and individual. In today’s situation there are many process and techniques to get better the decision making and the quality of decision making In this paper, we have taken decision making problem in which imprecise values are Icosagonal fuzzy numbers. We have completed it with converting to crisp valued decision making problem using ranking technique. We have described fuzzy decision making problem using Icosagonal fuzzy number with illustrations. The crisp valued decision making problem can be efficiently explained with Savage mini max regret criterion 2. PRELIMINARIES In this section, we give the preliminaries that are required for this study. Definition 2.1. A fuzzy set is defined by . Here is crisp set and is membership function in the interval . Definition 2.2. The fuzzy number is a fuzzy set whose membership function must satisfy the following conditions. 1. A fuzzy set of the universe of discourse is convex IJTSRD47553 International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47553 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1195 2. A fuzzy set of the universe of discourse is a normal fuzzy set if exists 3. is piecewise continuous Definition 2.3. A fuzzy number , where , is triangular fuzzy number and its membership function is given by Definition 2.4 A fuzzy number , where , is trapezoidal fuzzy number and its membership function is given by Definition 2.5 An -cut of fuzzy set is classical set defined as Definition 2.6 A fuzzy set is a convex fuzzy set iff each of its - cut A α is a convex set. Definition 2.7 Decision theory concerns about decisions.Decisions are taken based on the available datas.If available datas are false datas that decisions awill be false. We will have the loss. So we search for good datas, itit is received the decisions will be good. Definition 2.8 [3] A fuzzy number is called as Icosagonal fuzzy number and is denoted by A = ( ) ,........ , , , , , , , , , 20 10 9 8 7 6 5 4 3 2 1 r r r r r r r r r r r and its membership function is given by 1 0 4 3 2 1 ≤ ≤ ≤ ≤ ≤ k k k k where                                                    > ≤ ≤       − − ≤ ≤ ≤ ≤       − − − + ≤ ≤ ≤ ≤       − − − + ≤ ≤ ≤ ≤       − − − + ≤ ≤ ≤ ≤       − − − + ≤ ≤ ≤ ≤       − − − + ≤ ≤ ≤ ≤       − − − + ≤ ≤ ≤ ≤       − − − + ≤ ≤ ≤ ≤       − − − + ≤ ≤ ≤ ≤       − − = 20 20 19 19 20 20 1 19 18 1 18 17 17 18 18 1 2 1 17 16 2 16 15 15 16 16 2 3 2 15 14 3 14 13 13 14 14 3 4 3 13 12 4 12 11 11 12 12 4 4 11 10 10 9 9 10 9 4 4 9 8 4 8 7 7 8 7 3 4 3 7 6 3 6 5 5 6 5 2 3 2 5 4 2 4 3 3 4 3 1 2 1 3 2 1 2 1 1 2 1 1 1 , 0 , , , ) ( , , ) ( , , ) ( , , ) 1 ( ,1 , ) 1 ( , , ) ( , , ) ( , , ) ( , , , 0 ) ( r x for r x r for r r x r k r x r for k r x r for r r x r k k k r x r for k r x r for r r x r k k k r x r for k r x r for r r x r k k k r x r for k r x r for r r x r k k r x r for r x r for r r r x k k r x r for k r x r for r r r x k k k r x r k r x r for r r r x k k k r x r for k r x r for r r r x k k k r x r for k r x r for r r r x k r x for x A p µ International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47553 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1196 2.9 Graphical representation of Icosagonal fuzzy number: 3. Mathematical formulation of Fuzzy Decision making problem: Consider a fuzzy decision making problem in which all the entries of the payoff matrix are Icosagonal fuzzy numbers. Let us take the problem P has m strategies and problem Q has n strategies. Then the payoff matrix m x n is A=               mn m m n n p p p p p p p p p . . . . . . . 2 1 2 22 21 1 12 11 3.1. Procedures for solving Savage Minimax regret criterion: Step 1: Construct a regret (opportunity loss) table of each alternative for every state of nature from the given pay off matrix Step 2: Pick out the maximum pay off in each column and subtract all the elements in that column from this maximum value Step 3: For each decision alternative ( row ) , pick out the maximum row value and enter this in the last decision column Step 4: Choose the decision alternative with the smallest value in the decision column 3.2. Numerical Examples: Consider the fuzzy decision making problem with payoff matrix as Icosikaioctagonal fuzzy numbers. This problem is worked out by taking the values k1=0.2, k2=0.4, k3=0.6, k4=0.8 We get the values of ( ) Step 1: Fuzzy decision making problem is reduced to the following payoff profit matrix Alternatives Expected level of sale ( Rupees ) I II A 3 10.5 B 3.9 11.3 C 21 20 a11 10,-9,-8,-7,-6,-4,-3,-2,-1,0,1,2,4,6,8,10,12,14,16,18,20 ( ) = 3 a12 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 ( ) = 10.5 a21 -3,-7,-6,-5,-4,-3,-2,-1,0,3,5,6,7,8,10,12,14,15,16,18 ( ) = 3.9 a22 0,1,2,3,4,5,6,7,8,9,10,11,13,15,17,19,21,23,25,27 ( ) = 11.3 a31 2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40 ( ) = 21 a32 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39 ( ) = 20 International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47553 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1197 Step 2: The opportunity loss table for each alternative with the states of nature is depicted below Alternatives Expected level of sale ( Rupees ) I II A 18 9.5 B 17.1 8.7 C 0 0 Column Maximum 21 20 Step 3: The opportunity loss table and the maximum loss in each row is entered and shown in the below table Alternatives Expected level of sale ( Rupees ) Decision Column ( Maximum Loss) I II A 18 9.5 18 B 17.1 8.7 17.1 C 0 0 0 Result: Since the minimum of maximum loss is in alternative C = 0 rupee, this alternative should be selected. 3.3. Ranking of Icosagonal fuzzy number : Let I be a normal Icosagonal fuzzy number. The value , called as measure of I is calculated as ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∫ ∫ ∫ + + + + + + + + + = 1 4 3 2 2 1 1 4 2 1 3 2 1 2 1 2 1 1 2 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 1 ) ( ) ( 2 1 ) ( k k k k k k k k k dv v E v E du u D u D dt j C j C ds a B a B dr r A r A I M 1 0 4 3 2 1 ≤ ≤ ≤ ≤ ≤ k k k k where       − + + + + − + + + + − + + + + − + + + + + + + = ) 1 )( ( ) )( ( ) )( ( ) )( ( ) ( 4 1 ) ( 4 12 11 10 9 3 4 13 14 8 7 2 3 16 15 6 5 1 2 18 17 4 3 1 20 19 2 1 k r r r r k k r r r r k k r r r r k k r r r r k r r r r I M 1 0 4 3 2 1 ≤ ≤ ≤ ≤ ≤ k k k k where 3.4. Numerical Example: Let us consider the matrix                                                                         ,16 1,12,13,14 6,8,9,10,1 3,4,5, ,-1,0,1,2, 5,-4,-3,-2 - ,28,30 2,23,25,27 19,20,21,2 4,16,17, 10,12,13,1 2,3,6,8,9, ,25 8,20,22,23 14,15,16,1 ,11,13, 6,7,8,9,10 0,1,3,4,5, ,20,22 5,16,18,19 10,12,13,1 ,7,9, ,0,2,3,5,6 4,-3,-2,-1 - 25,26 ,21,23,24, 16,17,1819 13,15, 8,9,10,11, 1,2,3,5,6, ,20 4,16,17,19 10,11,13,1 ,6,7,9, ,0,1,2,3,4 4,-3,-2,-1 - 3,14 10,11,12,1 5,6,7,8,9, 3,4, ,-1,0,1,2, 5,-4,-3,-2 - ,22,23 8,19,20,21 15,16,17,1 ,13,14, 9,10,11,12 4,5,6,7,8, 1,12 7,8,9,10,1 2,3,4,5,6, 0,1, ,-3,-2,-1, 7,-6,-5,-4 - Step 1: We obtain the values of ( ) of the given fuzzy decision making problem and convert the fuzzy decision making problem into crisp valued decision making problem which is shown in the given table. a11 -7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12 ( ) = 2.5 a12 4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 ( ) = 13.5 a13 -5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14 ( ) = 4.5 a21 -4,-3,-2,-1,0,1,2,3,4,6,7,9,10,11,13,14,16,17,19,20 ( ) = 7.1 a22 1,2,3,5,6,8,9,10,11,13,15,16,17,1819,21,23,24,25,26 ( ) = 13.6 a23 -4,-3,-2,-1,0,2,3,5,6,7,9,10,12,13,15,16,18,19,20,22 ( ) = 8.35 a31 0,1,3,4,5,6,7,8,9,10,11,13,14,15,16,18,20,22,23,25 ( ) =11.5 a32 2,3,6,8,9,10,12,13,14,16,17,19,20,21,22,23,25,27,28,30 ( ) = 16.25 a33 -5,-4,-3,-2,-1,0,1,2,3,4,5,6,8,9,10,11,12,13,14,16 ( ) = 4.95 International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47553 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 1198 Step 2: The given fuzzy decision making problem is reduced to the following payoff profit matrix Alternatives Expected level of Sale ( in Rupees ) I II III A 2.5 13.5 4.5 B 7.1 13.6 8.35 C 11.5 16.25 4.95 Step 3: The opportunity loss table for each alternative with the states of nature is depicted below Alternatives Expected level of Sale ( in Rupees ) I II III A 9 2.75 3.85 B 4.4 2.65 0 C 0 0 3.4 Column Maximum 11.5 16.25 8.35 Step 4: The opportunity loss table and the maximum loss in each row is entered and shown in the below table Alternatives Expected level of Sale ( in Rupees ) Decision Column ( Maximum Loss) I II III A 9 2.75 3.85 9 B 4.4 2.65 0 4.4 C 0 0 3.4 3.4 Result: Since the minimum of maximum loss is in alternative C =3.4 rupees, this alternative should be selected. Conclusion: In this paper we have explained and solved fuzzy decision making problem and its pay off matrix whose elements are Icosagonal fuzzy number. We have illustrated the alternative selection of the fuzzy valued decision making problem converting to crisp valued decision making problem using ranking techniques. The Crisp valued decision making problem is worked out by savage minimax regret criterion. References: [1] L. A. Zadeh, , Fuzzy sets, Information and Control, 8(3) , 1965, 338-353. [2] R. E. Bellman and L. A. Zadeh, Decision making in fuzzy environment, Management Science, 17, 1970, 141- 164. [3] V. Raju and R. Jayagopal “An Arithmetic Operations of Icosagonal fuzzy number using Alpha cut” International Journal of Pure and Applied Mathematics. Volume 120, No. 8, 2018, 137-145 [4] S. Sasikumar and V. Raju ‘’ Study on Fuzzy game problem in Icosikaitetragonal Fuzzy number’’ Annals of Romanian Society for Cell biology , Volume 25, Issue 6, 2021, Page No : 10500-10508 [5] V. 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