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Feb 3, 2012 ... On multi-objective linear programming approach for solving fuzzy matrix games. Diâna Dance. Department of Mathematics, University of Latvia.
On multi-ob jective linear programming approach for solving fuzzy matrix games

Diâna Dance

Department of Mathematics, University of Latvia

Eleventh international conference on fuzzy set theory and applications Liptovsky Jan, Slovak Republic January 30 - February 3, 2012

Diâna Dance

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

In this talk we deal with non cooperative two-person games with fuzzy pay-offs. Namely, we consider matrix games where each component of the pay-off matrix is a fuzzy number. We describe the formal definition of the value of a fuzzy pay-off matrix game and develop a fuzzy programming method to find it by solving the corresponding bilevel linear programming problem. To realize fuzzy programming on two levels we apply specially designed aggregation of objectives.

Diâna Dance

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Matrix games

Matrix games are zero - sum two - person games. th

A is called the pay-off matrix: if Player I chooses i strategy and Player II chooses j

th

(j = 1, n)

(i = 1, m)

strategy then aij is

amount paid by Player II to Player I:

 A

a11

a12

 a21 =  ...

a22

am1 Lower value of the game: v

... am2

=

max

min

=1,m

j

i

Upper value of the game: w

=

min j

If v

=w

Diâna Dance

... ... ... ...

=1,n



a2n

  

... amn

=1,n

max i

a1n

=1,m

aij . aij .

then it is called value of the game.

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Expected pay-off function

( , )=

EA x y

m n X X

aij xi yj

,

=1 j=1

i

when Player I chooses mixed strategy x and Player II - y:

m

S

= {x = (x1 , x2 , . . . , xm ) :

m X

xi

= 1,

xi

∈ [0, 1]},

yj

= 1,

yj

∈ [0, 1]}.

=1

i

n

S

= {y = (y1 , y2 , . . . , yn ) :

n X

=1

j

The function EA has a seddle point

(x∗ , y∗ )



and x , y



are

optimal strategies for Player I and Player II.

( ∗ , y∗ ).

Value of the game EA x Diâna Dance

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Linear programming and matrix game equivalence

Player II:

Player I: v

→ max, m X

aij xi

w

>v

n X

(j = 1, n),

=1 xi

n X

= 1,

=1

(i = 1, m),

yj

= 1,

j

> 0 (i = 1, m).

yj



Solutions of the problems: x , y

Diâna Dance

6w

=1

i

xi

aij yj

=1

j

i

m X

→ min,



>0

and v

(j = 1, n).

∗ , w∗ .

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Fuzzy matrix games

Pay-off matrix:



˜11 a  a˜21 ˜ = A  ... am1 ˜ ˜

where aij (i

Diâna Dance

= 1, m; j = 1, n)

˜12 a ˜22 a ... am2 ˜

... ... ... ...

 ˜1n a ˜2n  a  ...  amn ˜

are fuzzy numbers.

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Fuzzy numbers (FN)

˜ : R → [0, 1]

Fuzzy number is a function aij

M

there exists the unique point a

ij

α

˜ |α

- cuts aij

are closed for all

˜Lij = a˜ij |]−∞;a

We denote: a

α

˜

- cut for aij will be

Diâna Dance

M ij

L

],

if:

˜ (

such that aij a

M ij

) = 1;

α ∈ [0, 1].

˜U a = a˜ij |[a ij

M ij

;∞[ .

U

[aij |α ; aij |α ]. Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Triangular fuzzy numbers (TFN) Triangular fuzzy number:

L M U aij = (aij , aij , aij ), ˜

L

where aij

U < aM ij < aij .

Membership function:

 0,    L    aij − aij , M − aL aij (aij ) = ˜ a ij ij    aij − aU  ij   M U, a ij − aij Diâna Dance

if aij

6 aLij , aij > aU ij ,

L

6 aij 6 aM ij ,

M

6 aij 6 aU ij .

if aij

if aij

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Multi-objective programming approach: Li's model

Player 1

Player 2

(vL , vM , vU ) → max

(wL , wM , wU ) → min

m X i=1 m X i=1 m X i=1

L

aij xi

M

aij xi

U

aij xi

m X i=1 Diâna Dance

xi

> vL ,

(j = 1, .., n)

> vM ,

(j = 1, .., n)

> vU ,

(j = 1, .., n)

n X j=1 n X j=1

= 1, xi > 0

n X j=1

L

6 wL ,

(i = 1, .., m)

M

6 wM ,

(i = 1, .., m)

U

6 wU ,

(i = 1, .., m)

aij yj

aij yj

aij yj

n X

yj

= 1, yj > 0

j=1 Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Reasonable solution of fuzzy matrix games Ordering of TFN's

˜t = (tL , tM , tU ), τ˜ = (τ L , τ M , τ U ) L L M t 6 τ , t 6 τ M and tU 6 τ U .

˜ 6 τ˜

are TFN's. Then t

if

Definition

˜ = (vL , vM , vU ) and w ˜ = (wL , wM , wU ) be TFN's. Then (˜ v, w ˜ ) is called a reasonable solution of the fuzzy matrix game m n there exist x ∈ S , y ∈ S such that

Let v

( , )>v ˜

E˜ x y A

( , )6w ˜

E˜ x y A

If

(˜ v, w ˜)

for all y for all x

if

∈ Sn ; ∈ Sm . ˜

is a reasonable solution of fuzzy game then v

˜

(respectively w) is called the reasonable value of Player I (Player II). Diâna Dance

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Solutions of fuzzy matrix games

( respectively W) is v ˜ ( respectively w ˜ ) for V

the set of all reasonable values Player I (Player II).

Definition An element

L M U (˜ v∗ = (v∗ , v∗ , v∗ ), w ˜ ∗ = (w∗L , w∗M , w∗U )) ∈ V × W

is

called a solution of the fuzzy game if

˜ = (vL , vM , vU ) ∈ V , v∗ );

there does not exist any v L

M

U

L

M

(v , v , v ) > (v∗ , v∗

such that

U

˜ = (wL , wM , wU ) ∈ W (wL , wM , wU ) 6 (w∗L , w∗M , w∗U ). there does not exist any w

Diâna Dance

such that

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Step 1 v

M

→ max x

∈D

M

Solution: v∗

and x



Step 2 L , vU → max m X L ∗ L aij xi > v ,

v

(j = 1, .., n)

=1

i

m X

U



aij xi

> vU ,

(j = 1, .., n)

=1

i

Solution of the problem is:

Diâna Dance

(v∗L , v∗M , v∗U ). Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Numerical example 1

˜ =

A



˜ ˜ 10

20

˜ 5 ˜ 20

 ,

˜ = (10, 20, 30), ˜ ˜ = (9, 10, 15), 5 = (1, 5, 20), 10 ˜ = (2, 20, 30) are TFN's. 20 where 20

If we solve individual problems of

(vL , vM , vU ) → max ∈D1

x

then

M

x∗

Diâna Dance

= (0.4, 0.6), xL∗ = (0, 1), xU ∗ = (0.6, 0.4) Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Bi-level linear programming

Player 2

Player 1

1 P1 1 P2

2

M

P1

: v → max : vL → max U v → max m X

aLij xi > vL

2 P2

(j = 1, n),

M aM ij xi > v

(j = 1, n),

U aU ij xi > v

(j = 1, n),

i=1

Diâna Dance

(i = 1, m),

n X

M aM ij yj 6 w

(i = 1, m),

n X

U aU ij yj 6 w

(i = 1, m),

j=1

i=1 m X

aLij yj 6 wL

j=1

i=1 m X

n X j=1

i=1 m X

: wM → min : wL → min U w → min

xi = 1, xi > 0.

n X

yj = 1, yj > 0.

j=1

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Individual solutions

v

M

→ max D1

Solutions: M

vmax v

L

= vM (xM ),

M

where x

M = (xM 1 , ..., xm )

→ max D1

Solutions: L

vmax v

U

= vL (xL ),

L

where x

= (xL1 , ..., xLm )

→ max D1

Solutions: U

vmax

Diâna Dance

= vU (xU ),

U

where x

U = (xU 1 , ..., xm )

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Membership function

µM (vM ) =

M

where v0

Diâna Dance

  0,    vM − vM 0

M M  vmax − v  0   1,

if v

,

M

6 v0M ,

M

M 6 vM 6 vmax ,

M

M > vmax .

if v0 if v

= min(vM (xL ), vM (xU )).

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

To maximize membership functions:

µM (vM ), µL (vL ), µU (vU ) −→ max, ∈D1

x

we maximize the smallest extreme degree of achievement among all functions:

{µM (vM ), µL (vL ), µU (vU )} −→ max

min

x

We denote

∈D1

σ : hµM (vM ) > σ, µL (vL ) > σ, µU (vU ) > σi

and write

problem in this form:

σ −→ max x,σ  M M   µ (v ) > σ   µL (vL ) > σ  µU (vU ) > σ    x∈D 1 Diâna Dance

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Diâna Dance

membership function

µL (vL )

membership function

µM (v M )

membership function

µU (vU )

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

σ=

max

x

Diâna Dance

:µM (vM )=α

{µL (vL ), µU (vU )}

min

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

This graph is obtained by solving the following problems:

σ −→ max x



  µ M (v M ) = α     µL (vL ) > σ  µU ( v U ) > σ    x∈D 1

Diâna Dance

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Numerical example 2

Numerical example (Li and Yang, Campos)

˜ = A



˜ ˜ 156 180

˜ ˜ 180 90

 ,

˜ = (175, 180, 190), 156 ˜ = (150, 156, 158), ˜ 90 = (80, 90, 100) are TFN's.

where 180

Diâna Dance

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Individual solutions: v

M

→ max D1 M

= 161.052,

x

L

= 155.208,

x

U

= 166.393,

x

Solutions: vmax v

L

M

= (0.789, 0.210)

→ max D1

Solutions: vmax v

U

L

= (0.791, 0.208)

U

= (0.737, 0.262)

→ max D1

Solutions: vmax

M

v0

Diâna Dance

= 156.393, v0L = 150.081, v0U = 164.666

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Membership functions

 0,   

M v

− 156.393 µM (vM ) = ,    161.052 − 156.393 1,   0, 

L v

− 150.081 µ (v ) = ,  155 . 208 − 150.081   1,  0,    U − 164.666 v µU (vU ) = ,    166.393 − 164.666 1, L

Diâna Dance

L

if v

M

6 156.393, .

if 156 393 if v

if v

M

L

> 161.052.

6 150.081, .

if 150 081 if v

L

if v

U

6 164.666, .

U

6 vL 6 155.208,

> 155.208.

if 164 666 if v

6 vM 6 161.052,

6 vU 6 166.393,

> 166.393.

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

σ −→ max x



 M v − 156.393    >σ   − 156.393  161.052  L  v − 150.081  >σ 155.208 − 150.081  U  v − 164.666   >σ     166.393 − 164.666  x ∈ D1 Solution:

σ ∗ = 0.50000001796808

Diâna Dance

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

σ −→ max ,σ

x

 M v − 156.393    =α   − 156.393  161.052  L  v − 150.081  >σ 155.208 − 150.081  U  v − 164.666   >σ   166.393 − 164.666   x∈D 1

Diâna Dance

α

σ

v

M

0

0.500000017968089905

156.3934

0.1

0.500000017968088351

156.8593

0.2

0.500000017968085908

157.3252

0.3

0.500000017968091681

157.7911

0.4

0.500000017968088573

158.2571

0.5

0.500000017968092902

158.7230

0.6

0.424383778061159233

159.1889

0.7

0.328447736942724334

159.6548

0.8

0.232511695824289433

160.1207

0.9

0.136575654705872296

160.5867

1

0.0406396135874516064

161.0526

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

δ = 0.8 then = (0.78, 0.22) M L U µM (v∗∗ ) = 0.8, µL (v∗∗ ) = 0.232511701, µU (v∗∗ ) = 0.232511687.

If we choose level x∗∗

Diâna Dance

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Numerical example 1

˜ =

A



˜ ˜ 10

20

˜ 5 ˜ 20

 ,

˜ = (10, 20, 30), ˜ ˜ = (9, 10, 15), 5 = (1, 5, 20), 10 ˜ = (2, 20, 30) are TFN's. 20 where 20

Diâna Dance

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

σ ∗ = 0.50000000017500 If we choose level δ = 0.8 then x∗∗ = (0.32, 0.68) M L U µM (v∗∗ ) = 0.8, µL (v∗∗ ) = 0.46666667, µU (v∗∗ ) = 0.46666667. Diâna Dance

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Multi-level approach

TFN's

FN's

=⇒

(Elements of the pay-off matrix

are fuzzy numbers)

are triangular fuzzy numbers)

BLLP (Bi-level linear programming)

Diâna Dance

(Elements of the pay-off matrix

MLLP

=⇒

(Multi-level linear programming)

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Multi-level linear programming m X

M aM ij xi > v

(j = 1, n),

i=1 m X

aLij |α1 xi > vL |α1

(j = 1, n),

i=1 m X

1

P1

1 P2

... 1

Pk

: vM → max : vL |α1 → max U v |α1 → max : vL |αk → max U v |αk → max

U aU ij |α1 xi > v |α1

(j = 1, n),

i=1

... m X

aLij |αk xi > vL |α1

(j = 1, n),

i=1 m X

U aU ij |αk xi > v |α1

(j = 1, n),

i=1 m X

xi = 1, xi > 0.

i=1 Diâna Dance

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games

Thank you for your attention!

Diâna Dance

Department of Mathematics, University of Latvia

On multi-objective linear programming approach for solving fuzzy matrix games