In this section, we recall some basic definitions and notations which are useful

throughout the paper.

Definition 1. A fuzzy set A ˜ is the collection of ordered pairs (x, µ (x)) defined on

the universal set X and denoted by A ˜ = {(x, µ (x))|x ∈ X and µ (x) ∈ [0, 1]}, where

A ˜ A ˜

µ : X → [0, 1] is the membership function of A ˜It is commonly believed that all

details accessed by the entities are represented in crisp numbers. However, most real-

life systems are engaged when the goals and constraints are often vague or unclear.

Definition 2. An interval valued fuzzy set A ˜ defined on the universal set X is

represented by A ˜ = {(x, (µ (x), µ (x))) |x ∈ X}, where µ : X → [0, λ] and

A ˜U A ˜L A ˜U µ ˜L : X → [0, ω] are the upper and lower membership functions with the maximum

values λ and ω respectively.

Definition 3. A non linear interval valued triangular fuzzy number A ˜ is

nLIV F N

denoted by A ˜ = [{(a , b, c ; λ) , (a, b, c; ω)}; ξ , ξ , ξ , ξ ] where 0 < ω ≤ λ ≤ 1

nLIV F N 1 1 1 2 3 4

and a1 < a < b < c < c1. The upper and lower membership functions are defined as

follows:  ξ

 x−a1  λ ; a1 ≤ x ≤ b  b−a1 λ ; x = b µ ˜U (x) = (1)

A ξ2

 c1−x  λ ; b ≤ x ≤ c  c −b 1 0 ; else

and   x−a  ω ; a ≤ x ≤ b  b−a ω ; x = b µ ˜L (x) = (2)

A ξ4

 c−x  ω ; b ≤ x ≤ c  c−b 0 ; else

representations

Within this section, we have established distinct categories of pentagonal fuzzy num-

bers by leveraging the continuity of the membership function—a dimension that was

notably absent in the framework presented by [6].

Definition 4. A pentagonal fuzzy number with µ ˜(x) as membership function is

represented by A ˜ = (a , a , a , a , a ) and satisfies the following conditions:

1 2 3 4 5

1. µ ˜(x) is continuous in [0, 1].

2. µ ˜(x) is strictly increasing continuous function on [a1, a2] and [a2, a3].

3. µ ˜(x) is strictly decreasing continuous function on [a3, a4] and [a4, a5].

Definition 5. Linear pentagonal fuzzy number with symmetry (LS) is represented by

A ˜ = ⟨a , a , a , a , a ; k⟩. The revised definition by the Figure 1 with a continuous

LS 1 2 3 4 5

membership function is given  as follows:

 x−a1  k ; a1 ≤ x ≤ a2  a2−a1  x−a  k + (1 − k) ; a2 ≤ x ≤ a3  a3−a2 1 ; x = a3 µ ˜ (x) = (3)

ALS a −x

 k + (1 − k) ; a3 ≤ x ≤ a4  a4−a3  a −x  k ; a4 ≤ x ≤ a5  a5−a4 0 ; else Fig. 1 Linear PFN with symmetry (LS).

Definition 6. The revised α-cut of linear pentagonal fuzzy number with symmetry is defined as the collection of all x ∈ X having membership function µ ˜ (x) as greater

A LS

than or equal to α i.e. Aα = {x ∈ X|µ ˜ (x) ≥ α} is given by

ALS

 A1L(α) = a1 + (a2 − a1) ; α ∈ [0, k]  α−k A2L(α) = a2 + (a3 − a2) ; α ∈ [k, 1] Aα = α−k (4)  A2R(α) = a4 − (a4 − a3) ; α ∈ [k, 1]  1−k A1R(α) = a5 − (a5 − a4) ; α ∈ [0, k]

Here A1L(α), A2L(α) are the increasing functions of α and A1R(α), A2R(α) are the

decreasing functions of α.

Definition 7. Non linear pentagonal fuzzy number with symmetry (N S) is represented

by A ˜ = (a , a , a , a , a ; k) . The revised definition by the Figure 2 with

N S 1 2 3 4 5 (ξ ,ξ ;ζ ,ζ )

1 2 1 2

a continuous membership function is given as follows:

 x−a1  k ; a1 ≤ x ≤ a2  a2−a1  ξ2  x−a  k + (1 − k) ; a2 ≤ x ≤ a3  a3−a2 1 ; x = a3 µ ˜ (x) = ζ (5)

AN S 1

 a4−x  k + (1 − k) ; a ≤ x ≤ a  a −a 3 4  ζ2  a5−x  k ; a4 ≤ x ≤ a5  a −a  5 4 0 ; else Fig. 2 Non Linear PFN with Symmetry A ˜

NS.

Definition 8. The revised α-cut of non linear pentagonal fuzzy number with symmetry

is defined as the collection of all x ∈ X with membership function µ ˜ (x) as greater

AN S

than or equal to α i.e. Aα = {x ∈ X|µ ˜ (x) ≥ α} is

AN S

 α k (a2 − a1) ; α ∈ [0, k]  A1L(α) = a1 + 1

 α−k (a − a ) ; α ∈ [k, 1]  A2L(α) = a2 + 1−k 3 2  (6) Aα = α−k  A (α) = a − 1−k (a4 − a3) ; α ∈ [k, 1]  2R 4  α 2  A (α) = a − k (a5 − a4) ; α ∈ [0, k]  1R 5

Here A1L(α), A2L(α) are the increasing functions of α and A1R(α), A2R(α) are the

decreasing functions of α.

Definition 9. The revised definition of linear interval valued pentagonal fuzzy num-

ber with symmetry (LIP S) A ˜ = ⟨(a , a , c, a , a ; k, p) , (b , b , c, b , b ; w, q)⟩ is

LIP S 1 2 4 5 1 2 4 5 defined by the Figure 3 with upper and lower membership functions as follows:

 x−a1  p ; a1 ≤ x ≤ a2  a2−a1  x−a  p + (k − p) ; a2 ≤ x ≤ c  c−a2 k ; x = c µ ˜U (x) = (7)

A a −x

LIP S  4  p + (k − p) ; c ≤ x ≤ a4  a4−c  a −x  p ; a4 ≤ x ≤ a5  a5−a4 0 ; else

and   x−b1  q ; b1 ≤ x ≤ b2  b2−b1  x−b  q + (w − q) ; b2 ≤ x ≤ c  c−b2 w ; x = c µ ˜L (x) = (8)

A b −x

LIP S  4  q + (w − q) ; c ≤ x ≤ b4  b4−c  b −x  q ; b4 ≤ x ≤ b5  b5−b4 0 ; else

Definition 10. The revised α-cut of linear interval valued pentagonal fuzzy number

˜ ˜U ˜L with symmetry is defined as ALIP Sα = ALIP S ⊖ ALIP S for α1 ∈ [0, w];

α α2 α α1 α2 ∈ [0, k].

Where  U α2 (a − a ) ; α ∈ [0, p]  A (α ) = a + p 2 1 2  1L 2 1  U α2−p  A (α ) = a + (c − a2) ; α2 ∈ [p, k] 2L 2 2 k−p (9) ˜U U α2−p A = A (α ) = a − (a4 − c) ; α2 ∈ [p, k]

LIP S 2R 2 4 k−p

α2  α  U 2 (a − a ) ; α ∈ [0, p]  A 1R(α2) = a5 − p 5 4 2

and  L α1 (b − b ) ; α ∈ [0, q]  A (α ) = b + q 2 1 1  1L 1 1  L α1−q  A (α ) = b + (c − b2) ; α1 ∈ [q, w] 2L 1 2 w−q (10) ˜L L α1−q A = A (α ) = b − (b4 − c) ; α1 ∈ [q, w]

LIP S 2R 1 4 w−q

α1  α  L 1 (b − b ) ; α ∈ [0, q]  A 1R(α1) = b5 − q 5 4 1

U U L L

Here A 1L(α2), A 2L(α2), A 1L(α1), A 2L(α1) are the increasing functions of

U U L L

α2 and α1 correspondingly and A 1R(α2), A 2R(α2), A 1R(α1), A 2R(α1) are the

decreasing functions of α2 and α1 correspondingly.

Definition 11. The membership functions of linear interval val-

ued pentagonal fuzzy number with asymmetry (LIP AS) A ˜ =

LIP AS

⟨(a1, a2, c, a4, a5; k, p, r) , (b1, b2, c, b4, b5; w, q, s)⟩ are defined by the Figure 4 with upper

and lower membership functions as follows:

; a1 ≤ x ≤ a2  a2−a1  x−a1  p x−a2 ; a ≤ x ≤ c  p + (k − p) c−a 2

 k ; x = c (11)

 a4−x ; c ≤ x ≤ a4 r + (k − r) a4−c µ ˜U (x) = a −x A 5 ; a ≤ x ≤ a LIP AS r a −a 4 5  5 4

 0 ; else

and  x−b ; b1 ≤ x ≤ b2  b2−b1 1  x−b2  q + (w − q) ; b2 ≤ x ≤ c  c−b2  w ; x = c (12)

b4−x ; c ≤ x ≤ b4 µ ˜L (x) = s + (w − s) b4−c LIP AS   b5−x ; b ≤ x ≤ b  s b −b 4 5  5 4  0 ; else

Definition 12. The revised α-cut of linear interval valued pentagonal fuzzy number

˜U ˜L with asymmetry is defined as ALIP ASα = ALIP AS ⊖ ALIP AS for α1 ∈

α α2 α α1

[0, w]; α2 ∈ [0, k].

Where 

U α2

A (α ) = a + (a2 − a1) ; α2 ∈ [0, p]  1L 2 1 p  α −p  U 2 A (α ) = a + (c − a2) ; α2 ∈ [p, k] U 2L 2 2 k−p (13) A ˜ = U α −r LIP AS 2 (a − c) ; α ∈ [r, k] A 2R(α2) = a4 − k−r 4 2 α2   U α  2 (a − a ) ; α ∈ [0, r] A 1R(α2) = a5 − r 5 4 2

and  L α1 (b − b ) ; α ∈ [0, q] A (α ) = b + q 2 1 1  1L 1 1  L α1−q  A (α ) = b + (c − b2) ; α1 ∈ [q, w] 2L 1 2 w−q (14) ˜L α −s A = L 1 (b − c) ; α ∈ [s, w] LIP AS A 2R(α1) = b4 − w−s 4 1 α1   L α1  A (α ) = b − (b5 − b4) ; α1 ∈ [0, s]

1R 1 5 s U U L L

Here A 1L(α2), A 2L(α2), A 1L(α1), A 2L(α1) are the increasing functions of

U U L L

α2 and α1 correspondingly and A 1R(α2), A 2R(α2), A 1R(α1), A 2R(α1) are the

decreasing functions of α2 and α1 correspondingly.

Definition 13. The membership functions of non linear interval val-

ued pentagonal fuzzy number with symmetry (N IP S) A ˜ = D E N IP S

(a1, a2, c, a4, a5; k, p) , (b1, b2, c, b4, b5; w, q) are defined by the Figures 5 to 8 for different (ξ1,ξ2;ζparameters 1,ζ2) ξ1, ξ2, ζ1, ζ2 with (ξ1,ξrevised 2;ζ1,ζ2) continuous upper and

lower membership functions as follows:

NIPS 1 2 1 2

 x−a1 ξ1  ; a1 ≤ x ≤ a2  a2−a1

x−a2 ; a ≤ x ≤ c  p + (k − p) c−a 2 k ; x = c (15) µ ˜U (x) = ζ1 N IP S   a4−x ; c ≤ x ≤ a4  p + (k − p) a4−c

 a5−x ; a ≤ x ≤ a  p a −a 4 5

 0 ; else

and ξ1 ; b1 ≤ x ≤ b2  bx 2− −bb1 1  q ξ x−b2 ; b ≤ x ≤ c  q + (w − q) c−b 2  w ; x = c (16)  ζ1 b4−x ; c ≤ x ≤ b4 µ ˜L (x) = q + (w − q) b4−c N IP S  2 b5−x ; b ≤ x ≤ b  q b −b 4 5  5 4  0 ; else

Definition 14. The revised α-cut of non linear interval valued pentagonal fuzzy num-

˜U ˜L ber with symmetry is defined as follow: AN IP Sα = AN IP S ⊖ AN IP S for

α α2 α α1

α1 ∈ [0, w]; α2 ∈ [0, k].

NIPS 1 2 1 2

NIPS 1 2 1 2 NIPS 1 2 1 2

Where

 U α2  A (α ) = a + p (a2 − a1) ; α2 ∈ [0, p]  1L 2 1  ξ2  U α −p  2 (c − a ) ; α ∈ [p, k]  A 2L(α2) = a2 + k−p 2 2 (17) A = ζ1

N IP S U α2−p

α  A (α ) = a − k−p (a4 − c) ; α2 ∈ [p, k] 2  2R 2 4

 U α2  A (α ) = a − p (a5 − a4) ; α2 ∈ [0, p]  1R 2 5

and 1  L α1 (b2 − b1) ; α1 ∈ [0, q]  A (α ) = b + q  1L 1 1 1  α −q  L 1 (c − b ) ; α ∈ [q, w]  A 2L(α1) = b2 + w−q 2 1 (18) ˜L ζ1 A = L α1−q N IP S A (α ) = b − w−q (b4 − c) ; α1 ∈ [q, w] α1  2R 1 4  ζ2  L α  1 (b − b ) ; α ∈ [0, q]  A 1R(α1) = b5 − q 5 4 1

U U L L

Here A 1L(α2), A 2L(α2), A 1L(α1), A 2L(α1) are the increasing functions of

U U L L

α2 and α1 correspondingly and A 1R(α2), A 2R(α2), A 1R(α1), A 2R(α1) are the

decreasing functions of α2 and α1 correspondingly.

Definition 15. The membership functions of non linear interval val-

ued pentagonal fuzzy number with asymmetry (N IP AS) A ˜ =

N IP AS (a1, a2, c, a4, a5; k, p, s) , (b1, b2, c, b4, b5; w, q, t) are defined by

(ξ1,ξ2;ζ1,ζ2) (ξ1,ξ2;ζ1,ζ2)

the Figures 9 to 12 for different parameters ξ1, ξ2, ζ1, ζ2 having continuous upper and

lower membership functions as follows:

 x−a1 ξ1  ; a1 ≤ x ≤ a2  a2−a1

x−a2 ; a ≤ x ≤ c  p + (k − p) c−a 2 k ; x = c (19) µ ˜U (x) = ζ1 N IP AS   a4−x ; c ≤ x ≤ a4  s + (k − s) a4−c

a5−x ; a ≤ x ≤ a  s a −a 4 5  5 4  0 ; else

and ξ1 ; b1 ≤ x ≤ b2  bx 2− −bb1 1  q ξ2  x−b2 ; b ≤ x ≤ c  q + (w − q) c−b 2  w ; x = c (20)

b4−x ; c ≤ x ≤ b4 µ ˜L (x) = t + (w − t) b4−c N IP AS  2  b −x  5 ; b ≤ x ≤ b  t b −b 4 5  5 4  0 ; else

NIPAS 1 2 1 2 NIPAS 1 2 1 2

Definition 16. The revised α-cut of non linear interval valued pentagonal fuzzy num-

˜U ˜L ber with asymmetry defined as AN IP ASα = AN IP AS ⊖ AN IP AS for

α α2 α α1

α1 ∈ [0, w]; α2 ∈ [0, k].

Where

 α p (a2 − a1) ; α2 ∈ [0, p]  U 2  A 1L(α2) = a1 + 1  U α2−p  A (α ) = a + k−p (c − a2) ; α2 ∈ [p, k] (21)  2L 2 2 ˜U ζ A = 1 N IP AS U α2−s (a − c) ; α ∈ [s, k] α  A (α ) = a − k−s 4 2 2  2R 2 4  U α2 2  A (α ) = a − s (a5 − a4) ; α2 ∈ [0, s]  1R 2 5

and 1

 L α1 (b − b ) ; α ∈ [0, q]  A (α ) = b + q 2 1 1  1L 1 1 1  ξ2  L α1−q  A (α ) = b + w−q (c − b2) ; α1 ∈ [q, w] (22)

2L 1 2

A ˜ =

N IP AS ζ1

α  L α1−t (b − c) ; α ∈ [t, w] 1  A (α ) = b − w−t 4 1  2R 1 4  L α1 2  A (α ) = b − t (b5 − b4) ; α1 ∈ [0, t]  1R 1 5 NIPAS 1 2 1 2

NIPAS 1 2 1 2 U U L L

Here A 1L(α2), A 2L(α2), A 1L(α1), A 2L(α1)”are the increasing functions of

U U L L

α2 and α1 correspondingly and A 1R(α2), A 2R(α2), A 1R(α1), A 2R(α1) are the

decreasing functions of α2 and α1 correspondingly.

The defuzzification method converts the fuzzy inference result into an appropriate

exact value. It plays a critical role in a fuzzy environment. For fuzzy logic, several

defuzzification techniques are available, such as the center of the area (centroid), great-

est of maxima, smallest of maxima, mean of maxima, the bisector of area, graded

mean integral value, and so on. We investigate three methods to defuzzify the linear

pentagonal fuzzy number with symmetry (LS) based on different approaches. Our

results are compared with [6].

Example 1. Consider a fuzzy matrix game for two different players A and B whose

pay-offs are linear pentagonal fuzzy numbers with symmetry as follows:

Now using the mean of α- cut method as defuzzification defined by Eq. (32)

for linear pentagonal fuzzy number to convert the given fuzzy pay-off matrix game as Table (1) into the following crisp matrix game as Table 3, we have

The above crisp matrix game defined by Table 3 has no saddle point. Therefore according to the dominance property in matrix games row X1 can be dominated by row X3. Hence we

have a modified matrix game as Table 4.

Again we have that all the elements of column Y1 are greater than or equal to corresponding elements of column Y3. Therefore column Y1 dominates column Y3 and so we can delete the

column Y1 for new improved matrix game as Table 5.

Now each element of average of strategies Y3 and Y4 of player B are less than or

equal to the corresponding elements of strategy Y2 of player B. Therefore on deleting

the column of strategy Y2 by dominance property, we obtain a new modified matrix

game as Table (6).

Again each element of strategy X2 of player A is less than or equal to the corresponding

elements of the average of strategies X3 and X4 of player A. Therefore on deleting the

row of strategy X2 by dominance property, we obtain a new improved matrix game

as Table (7).

2 1 2 1 8

Which are A 0 is, 0 a,2× , 2 matrix and B game 0, 0, without , respectively and the value of game is saddle point, so the strategy of both.the players

3 3 3 3 3 Example 2. Consider a fuzzy matrix game for two different players A and B whose pay-offs are linear pentagonal fuzzy numbers with symmetry as follows:

Now using the mean of α-cut method as defuzzification defined by equation (32) for the linear pentagonal fuzzy number to convert the given fuzzy pay-off matrix game

as Table (8) into the crisp matrix game defined by the Table (9), as follows:

Since the above crisp matrix game defined by Table (8) has no saddle point, there-

fore according to the dominance property in matrix games, strategy X1 of player A

can be dominated by strategy X3 of player A. Hence we have a modified matrix game

defined by Table (9).

Again we have that all the elements of strategy Y1 are greater than or equal to the

corresponding elements of strategy Y3 for player B. Therefore strategy Y1 dominates

strategy Y3 and so we can delete the strategy column Y1 for new improved matrix

game as Table (10).

Table 10 Second step of dominance property.

Strategy Y2 Y3 Y4

X2 5 3 5 X3 3 5 0 X4 4 0 9

Further the matrix game as Table (10) can not be reduced by dominance rule. There-

fore by applying the linear programming approach for matrix game, we have

For Player A 

 max v  such that  5x1 + 3x2 + 4x3 ≥ v 3x1 + 5x2 + 0x3 ≥ v  5x1 + 0x2 + 9x3 ≥ v  x + x + x = 1  1 2 3 and x1, x2, x3, v ≥ 0.

and

For Player B  min w

 such that  5y + 3y + 5y ≤ w  1 2 3 3y1 + 5y2 + 0y3 ≤ w  4y + 0y + 9y ≤ w  1 2 3  y1 + y2 + y3 = 1 and y , y , y , w ≥ 0.

1 2 3

Where v, w are the minimum expected gain and maximum expected loss for player A

and B respectively. The above linear programming problems for player A and player B as x y can be written in the following manner by assuming = Ri, = Sj for i, j = 1, 2, 3 For Player A  v w  min R1 + R2 + R3

 such that

 3R1 + 5R2 + 0R3 ≥ 1  5R1 + 0R2 + 9R3 ≥ 1

and R1, R2, R3 ≥ 0.

and

For Player B  max S1 + S2 + S3

 such that  5S + 3S + 5S ≤ 1

1 2 3

 4S + 0S + 9S ≤ 1  1 2 3 and S1, S2, S3 ≥ 0.

On solving the above linear programming problems by the classical simplex method, 0.28. So the optimal strategy of both the players A and B are A (0, 0.1 71429, 0.28571 1, 0) we obtain R1 = 0.2; R2 = 0.08; R3 = 0; S1 = 0; S2 = 0.2; S3 = 0.08; = 0.28 and = and B (0, 0, 0.71429, 0.28571) respectively and the value of game is 3.57143. In [6] the

9 5 9 5

optimal strategies of the players A 0, 0, , and B 0, 0, , and the value of

14 14 14 14

45 were evaluated.

game 14

Example 3. Consider a fuzzy matrix game for two different players A and B whose pay-offs are linear pentagonal fuzzy numbers with symmetry as follows:

Now using the mean of α- cut method as defuzzification defined by Eq. (32) for the linear pentagonal fuzzy number to convert the given fuzzy pay-off matrix game

as Table (11) into the crisp matrix game defined by the Table (13), as follows: Since the above crisp matrix game defined by Table (11) has no saddle point,

therefore according to the dominance property in matrix games, strategy X1 can be

dominated by strategy X3 of player A. Hence we have a modified matrix game defined

by Table (13).

Again we have that all the elements of strategy Y1 are equal to the corresponding

elements of strategy Y3 for player B. Therefore either strategy Y1 or strategy Y3 for

player B can be dominated by each other. Here we delete the strategy Y1 for the new,

improved matrix game as Table (14).

Now each element of the average of strategies Y3 and Y4 of player B are less than or

equal to the corresponding elements of strategy Y2 of player B. Therefore on deleting

the column strategy Y2 by dominance property, we obtain a newly modified matrix

game defined by the following Table (15) as Further the matrix game as Table (14) can not be reduced by dominance rule.

Therefore by applying the graphical approach for m × 2 matrix game according as

Table (15) given by Figure 17.

Hence it is clear that the lowest boundary point P on the bounded region will give

the smallest expected loss to the minimizing player (playerB). So the best strategy

for player A are those which pass through the point P. Therefore we can write the

matrix game as given by the Table (15) in a 2 × 2 matrix game defined by the Table

(16) as follows:

The above 2 × 2 matrix game is without saddle point, so the strategy of both the

6 1 6 1 36

players are A 0, , , 0 and B 0, 0, , respectively and the value of game is . In

7 7 7 7 7 5 3 5 3

[6] the optimal strategies of the players A 0, 0, , , B 0, 0, , and the value of

8 8 8 8

the game were evaluated.

Remark 1. If we delete the strategy Y3 for new improved matrix game as second step

of dominance property and further on going in a similar manner, we obtain the best

1 6 1

6 , , 0 and B , 0, 0, respectively and the strategy for player A and B as A 0, 7 7 7 7 value of game is .

This research paper has systematically explored and elucidated a range of mathemat-

ical concepts central to the pentagonal fuzzy numbers. The investigation developed

various concepts, including distinct representations, intrinsic properties, α-cuts, and defuzzification methodologies. The application of both linear and non-linear member-ship functions in a symmetric fashion has been rigorously examined, contributing to a comprehensive understanding of the subject. The incorporation of numerical exam-ples drawn from matrix game problems, as originally provided by [6], has added depth and context to the discourse. The introduction and analysis of novel defuzzification techniques for pentagonal fuzzy numbers have provided valuable insights into optimal strategies for players, as well as the determination of game values—parameters of sub-stantial significance within real-world gaming scenarios. This paper offers a stepping stone for further scholarly inquiries into the field of pentagonal fuzzy numbers across diverse real-life problem domains. By shedding light on the representation, defuzzifica-tion, and their applications within the context of game theory, this research develops the way for continued exploration and innovative utilization of pentagonal fuzzy num-bers in addressing complex phenomena. Researchers are encouraged to build upon this foundation, unraveling the potential of pentagonal fuzzy numbers as a powerful tool to handle the real-world challenges.

The authors would like to express their appreciation to all individuals whose valuable comments and suggestions contributed to the improvement of this research.

The authors declare that no financial support, grant, or sponsorship was received for conducting this study.

The data used and analyzed during the current study are available from the corre-sponding author upon reasonable request.