Study on Fuzzy Game Problem in Icosikaitetragonal Fuzzy Number
S. Sasikumar1 and V. Raju2
1Research Scholar, Department of Mathematics,
Vels Institute of Science, Technology and Advanced Studies (VISTAS), Chennai-600 117, Tamil Nadu, India
2 Assistant Professor, Department of Mathematics School of Basic Sciences
Vels Institute of Science, Technology and Advanced Studies (VISTAS), Chennai-600 117, India
1 [email protected] and 2 [email protected] ABSTRACT
In this article, we bring in the fuzzy game problem by Icosikaitetragonal fuzzy number. We generate the value of Payoff matrix by Icosikaitetragonal fuzzy number. We modify the fuzzy game problem into crisp valued game problem by using ranking to pay off. The crisp valued game problem can be solved by Oddment method and illustrations are exemplified
Keywords
Icosikaitetragonal fuzzy number, fuzzy game problem, fuzzy ranking, pay off matrix.
1.Introduction
Fuzzy set theory was introduced by Zadeh [1].The concept of fuzzy set theory reveals imprecision and vagueness. We make use of the fuzzy set almost in all business and also in our many day to day life activities. We bring together more information from the environment in each day is fuzzy. We are not able to move our daily life without crossing fuzzy connected circumstance. Fuzzy set is efficient in many ongoing world situations. Jain [2] was the first to propose method of ranking fuzzy numbers for decision making in fuzzy related situation. Raju and Jayagopal [3] was the first to introduce the Icosikaitetragonal fuzzy number. Game theory makes it easier to have interaction of decision makers between the scenario of co operation and spirited approach.
Game theory has extensive range of applications in various fields such as business model development, Economics, diplomacy and military strategy. In a game theory each player is to take good decision by selecting various strategies from the all available strategies. When uncertainties occur in game theory, fuzzy set is the best tool to study this kind of game in which pay off matrices is denoted by fuzzy numbers. In this paper, we have taken two person zero sum game, in which imprecise values are Icosikaitetragonal fuzzy numbers. We have made clear it with converting to crisp valued game problem using ranking technique. We have analyzed fuzzy game problem using Icosikaitetragonal fuzzy number with illustrations and solved the game problem with oddment method
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2. PRELIMINARIES
In this section, we give the preliminaries that are required for this study.
Definition 2.1. A fuzzy set 𝐴 is defined by 𝐴 = {(𝑥, 𝜇𝐴(𝑥)): 𝑥 ∈ 𝐴, 𝜇𝐴(𝑥) ∈ [0,1]}. Here 𝑥 is crisp set 𝐴 and 𝜇𝐴(𝑥) is membership function in the interval [0,1].
Definition 2.2.
The fuzzy number 𝐴 is a fuzzy set whose membership function must satisfy the following conditions.
(i) A fuzzy set 𝐴 of the universe of discourse 𝑋 is convex
(ii) A fuzzy set 𝐴 of the universe of discourse 𝑋 is a normal fuzzy set if 𝑥𝑖 ∈ 𝑋 exists (iii) 𝜇𝐴(𝑥) is piecewise continuous
Definition 2.3.
A fuzzy number 𝐴 = (𝑎, 𝑏, 𝑐), where 𝑎 ≤ 𝑏 ≤ 𝑐, is triangular fuzzy number and its membership function is given by
𝜇𝐴(𝑥) =
𝑥−𝑎
𝑏−𝑎, 𝑓𝑜𝑟𝑎 ≤ 𝑥 ≤ 𝑏
𝑐−𝑥
𝑐−𝑏, 𝑓𝑜𝑟𝑏 ≤ 𝑥 ≤ 𝑐 0, 𝑥 > 𝑐
Definition 2.4
A fuzzy number 𝐴 = (𝑎, 𝑏, 𝑐, 𝑑), where 𝑎 ≤ 𝑏 ≤ 𝑐 ≤ 𝑑, is trapezoidal fuzzy number and its membership function is given by
𝜇𝐴(𝑥) =
0, 𝑓𝑜𝑟𝑥 < 𝑎
𝑥−𝑎
𝑏−𝑎, 𝑓𝑜𝑟𝑎 ≤ 𝑥 ≤ 𝑏 1, 𝑓𝑜𝑟𝑏 ≤ 𝑥 ≤ 𝑐
𝑑−𝑥
𝑑−𝑐, 𝑓𝑜𝑟𝑐 ≤ 𝑥 ≤ 𝑑 0, 𝑥 > 𝑑
Definition 2.5
An 𝛼-cut of fuzzy set 𝐴 is classical set defined as A =
xX A
x
Definition 2.6
A fuzzy set 𝐴 is a convex fuzzy set iff each of its 𝛼-cut A is a convex set.
Definition 2.7
Game theory provides a mathematical framework for analyzing the decision-making processes and strategies of adversaries (or players) in different types of competitive situations. The simplest type of competitive situations is two-person, zero-sum games. These games involve only two players; they are called zero-sum games because one player wins whatever the other player loses.
Definition 2.8 [3] A fuzzy number A =
a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,...a24
is Icosikaitetragonal fuzzy number and its membership function is given by
24 23
23 24
24 1
23 22
1
22 21
21 22
22 1 2 1
21 20
2
20 19
19 20
20 2 3 2
19 18
3
18 17
17 18
18 3 4 3
17 16
4
16 15
15 16
16 4 5 4
15 14
5
14 13
13 14
14 5 5
13 12
12 11
11 12
11 5
5
11 10
5
10 9
9 10
9 4
5 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
, ,
, )
( ,
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( ,
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1 ( , 1
, )
1 ( ,
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( ,
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( ,
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( ,
, )
( ,
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0
) (
a x a a for
a x k a
a x a for k
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a x k a
k k
a x a for k
a x a a for
a x k a
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a x a for k
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a x k a
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a x a for k
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a x k a
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a x a a for
a a k x
k
a x a for k
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a a k x
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a a k x
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a x for
A x
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3. Mathematical formulation of Fuzzy Game problem:
Consider a two person zero sum game in which all the entries of the payoff matrix are Icosikaitetragonal fuzzy numbers. Let us take the player P has m strategies and player Q has n strategies. We will assume that each player has to choose from the pure strategies.
Player P is always considered to be gainer and the player Q is always loser. Then the payoff matrix m x n is
G=
mn m
m
n n
p p
p
p p
p
p p
p
. . . . .
. .
2 1
2 22
21
1 12
11
3.1 Numerical Examples:
Consider the fuzzy game problem with payoff matrix as Icosikaitetragonal fuzzy numbers. This problem is worked out by taking the values
6 , 5 6 , 4 6 , 3 6 , 2 6 1
5 4
3 2
1 k k k k
k We get the values of 𝜇𝐼𝑐𝑠𝑘𝑡𝑒𝑡𝑟𝑎(𝑎𝑖𝑗)
Fuzzy game problem is reduced to the following payoff matrix A=
2 12.5 1.67 25 14.54 24
Minimum of first row is 2
Minimum of second row is 1.67 Minimum of third row is 14.54 Maximum of first column is 14.54 Maximum of second column is 25 a11 -13,-12,-11,-10,-9,-8,-7,-6,-4,-3,-2,
0,1,2,4,6,8,10,12,14,16,18,20,22
𝜇𝐼𝑐𝑠𝑘𝑡𝑒𝑡𝑟𝑎( 𝑎11) = 2 a12 1,2,3,4,5,6,7,8,9,10,11,12,13,14,
15,16,17,18,19,20,21,22,23,24
𝜇𝐼𝑐𝑠𝑘𝑡𝑒𝑡𝑟𝑎( 𝑎12) = 12.5 a21 -12,-11,,-10,-9,-8,-7,-6,-5,-4,-3,
-2,-1,0,3,5,6,7,8,10,12,14,15,16,18
𝜇𝐼𝑐𝑠𝑘 𝑡𝑒𝑡𝑟𝑎( 𝑎21) = 1.67 a22 2,4,6,8,10,12,14,16,18,20,22,24,26,
28,30,32,34,36,38,40,42,44,46,48
𝜇𝐼𝑐𝑠𝑘𝑡𝑒𝑡𝑟𝑎( 𝑎22) = 25 a31 0,1,2,3,4,5,6,7,8,9,10,11,13,15,
17,19,21,23,25,27,29,30,31,33
𝜇𝐼𝑐𝑠𝑘𝑡𝑒𝑡𝑟𝑎( 𝑎31) = 14.54 a32 1,3,5,7,9,11,13,15,17,19,21,23,25,
27,29,31,33,35,37,39,41,43,45,47
𝜇𝐼𝑐𝑠𝑘𝑡𝑒𝑡𝑟𝑎( 𝑎32) = 24
Max(min) = 14.54 Min(max) = 14.54 It has a saddle point Value of the game is 14.54
3.2 Diagram of Icosikaitetragonal fuzzy number:
3.3 Ranking of Icosikaitetragonal fuzzy number:
Let I be a normal Icosikaitetragonal fuzzy number. The value 𝑀 (𝐼), called as measure of I is calculated as
1
5 2 1 4
2 1 3
2 1 2
1 2
1 1
2 1
5 4
3
2 2
1 1
2 1 2
) 1 (
k k
k k
k k
k k
k k
dq q q dp p p do o o dn n n dm m m d
I
M
1 0k1k2 k3k4k5 where
) 1 )(
( ) )(
( ) )(
(
) )(
( ) )(
( ) (
4 ) 1 (
5 14 13 12 11 4 5 16 15 10 9 3 4 18 17 8 7
2 3 20 19 6 5 1 2 22 21 4 3 1 24 23 2 1
k a
a a a k k a a a a k k a a a a
k k a a a a k k a a a a k a a a L a
M
where 0k1k2 k3k4 k5 1
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6
, 5 6 , 4 6 , 3 6 , 2 6 1
5 4
3 2
1 k k k k
k for values the
take we
3.4 Numerical Examples:
Let us consider the matrix
18 , 17 , 16 , 15 , 14 , 13 , 12 , 11 , 10 , 9 , 8
, 67 , 5 , 4 , 3 , 2 , 1 , 0 , 1 , 2 , 3 , 4 , 5 39
, 37 , 35 , 34 , 32 , 30 , 28 , 25 , 23 , 22
, 20 , 19 , 17 , 16 , 15 , 13 , 12 , 10 , 9 , 8 , 6 , 3 , 2 , 1 29 , 28 , 27 , 26 , 25 , 24 , 22 , 21 , 19 , 17
, 15 , 14 , 13 , 11 , 10 , 9 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0
30 , 29 , 28 , 27 , 26 , 24 , 22 , 20 , 18 , 16
, 14 , 13 , 12 , 11 , 10 , 9 , 8 , 7 , 6 , 5 , 4 , 2 , 1 , 0 19
, 18 , 17 , 16 , 15 , 14 , 13 , 12 , 11 , 10
, 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0 , 1 , 2 , 3 , 4 18
, 16 , 15 , 14 , 13 , 12 , 11 , 10 , 9 , 8 , 7 , 6
, 5 , 4 , 3 , 2 , 1 , 0 , 1 , 2 , 3 , 4 , 5 , 6
24 , 23 , 22 , 21 , 20 , 19 , 18 , 17 , 16 , 15
, 14 , 13 , 12 , 11 , 10 , 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 15 , 14 , 13 , 12 , 11 , 10 , 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2
, 1 , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 17
, 16 , 15 , 14 , 13 , 12 , 11 , 10 , 9 , 8 , 7 , 6
, 5 , 4 , 3 , 2 , 1 , 0 , 1 , 2 , 3 , 4 , 5 , 6
Step 1:
We obtain the values of 𝜇𝐼𝑐𝑠𝑘𝑜𝑐𝑡( 𝑎𝑖𝑗) of the given fuzzy game problem and convert the fuzzy game problem into crisp valued problem which is shown in the given table.
Step 2 : The given fuzzy game problem is reduced to the following payoff matrix B
a11
17 , 16 , 15 , 14 , 13 , 12 , 11 , 10 , 9 , 8 , 7 , 6
, 5 , 4 , 3 , 2 , 1 , 0 , 1 , 2 , 3 , 4 , 5 ,
6
𝜇𝑅( 𝑎11) = 5.5
a12
15 , 14 , 13 , 12 , 11 , 10 , 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2
, 1 , 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 ,
8
𝜇𝑅( 𝑎12) = 3.5
a13
24 , 23 , 22 , 21 , 20 , 19 , 18 , 17 , 16 , 15
, 14 , 13 , 12 , 11 , 10 , 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 ,
1 𝜇𝑅( 𝑎13) = 12.5
a21
18 , 16 , 15 , 14 , 13 , 12 , 11 , 10 , 9 , 8 , 7 , 6
, 5 , 4 , 3 , 2 , 1 , 0 , 1 , 2 , 3 , 4 , 5 ,
6
𝜇𝑅( 𝑎21) = 5.54
a22
19 , 18 , 17 , 16 , 15 , 14 , 13 , 12 , 11 , 10
, 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0 , 1 , 2 , 3 ,
4
𝜇𝑅( 𝑎22) = 7.5
a23
30 , 29 , 28 , 27 , 26 , 24 , 22 , 20 , 18 , 16
, 14 , 13 , 12 , 11 , 10 , 9 , 8 , 7 , 6 , 5 , 4 , 2 , 1 ,
0 𝜇𝑅( 𝑎23) = 14.3
a31
29 , 28 , 27 , 26 , 25 , 24 , 22 , 21 , 19 , 17
, 15 , 14 , 13 , 11 , 10 , 9 , 7 , 6 , 5 , 4 , 3 , 2 , 1 ,
0 𝜇𝑅( 𝑎31) =14.25
a32
39 , 37 , 35 , 34 , 32 , 30 , 28 , 25 , 23 , 22 , 20
, 19 , 17 , 16 , 15 , 13 , 12 , 10 , 9 , 8 , 6 , 3 , 2 ,
1 𝜇𝑅( 𝑎32) = 19
a33
18 , 17 , 16 , 15 , 14 , 13 , 12 , 11 , 10 , 9 , 8 , 7
6 , 5 , 4 , 3 , 2 , 1 , 0 , 1 , 2 , 3 , 4 ,
5
𝜇𝑅( 𝑎33) = 6.5
A
5 . 6
3 . 14
5 . 12
19 5 . 7
5 . 3
25 . 14
54 . 5
5 . 5
Minimum of first row = 3.5 Minimum of second row = 5.54 Minimum of third row = 6.5 Maximum of first column = 14.25 Maximum of second column = 19 Maximum of third column = 14.3 Max (min) = 6.5
Min (max) = 14.25 Max (min) ≠ Min (max) 6.5 ≠14.5
There is no saddle point
Step 3: To solve the value of the problem, we apply dominance method. Clearly first row is dominated by second row as all the elements of first row are less than second row. Hence eliminate first row, we get
B
A
5 . 6
3 . 14 19
5 . 7 25 . 14
54 . 5
Again first column is dominated by second column. So eliminate first column.
B A
5 . 6 19
3 . 14 5 . 7
Now we obtain 2x2 payoff matrix. To solve the reduced matrix, which won‟t have any saddle point. So we apply oddment method. Hence the augmented payoff matrix is
B A
5 . 6 19
3 . 14 5 . 7
(7.5P1) + 14.3(1- P1) = 19 P1 + 6.5(1- P1) P1= 0.4041
1- P1 = 0.5959
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7.5 Q1 + 19 (1- Q1) = 14.3 Q1 + 6.5(1- Q1) Q1 = 0.6477
1- Q1 = 0.3523
Strategy for B = ( 0.4041 ,0, 0.5959 ) Strategy for A = (0.6477, 0, 0.3523 ) Value of the game = (7.5 P1) + 14.3(1- P1)
= (7.5 x 0.4041) + (14.3 x 0.5959) = 11.55212
Conclusion: In this article, we have examined and solved 3 × 3 fuzzy pay off matrix whose elements are Icosikaitetragonal fuzzy number. We have illustrated the optimal solution of the fuzzy valued game problem converting to crisp valued game problem using ranking techniques.
The Crisp valued game problem is solved by oddment method 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.
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3631-3640
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