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
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... ... ... ...
=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
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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
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= 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
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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∗
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= (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
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(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
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= 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
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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
,σ
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
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= 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
,σ
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)
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(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