MOLP -...
12
http://jnrm.srbiau.ac.ir * MOLP MOLP MOLP * . Corresponding author, Email: [email protected] پژوهشاضیوین در ری های نحقیقاتم و تاحد علومی، وه آزاد اسنشگا دا
Transcript of MOLP -...
http://jnrm.srbiau.ac.ir
*
MOLP
MOLP
MOLP
*. Corresponding author, Email: [email protected]
های نوین در ریاضیپژوهش دانشگاه آزاد اسالمی، واحد علوم و تحقیقات
MOLP
MOLP
MOLP
MOLP
MOLP
خمینت
MOLP
yxy
x
1
t t
,
1 1
t
2 2 2
max
s.t.
max
s.t. ,
x y
y
c x d y
A x B y b
e y
A x B y b
1, nc x2, , nd e y1A
1 1m n1B1 2m n2A
2 1m n2B2 2m n
1
t t
,
1 1
max
s.t.
x yc x d y
A x B y b
t
2 2 2
max
s.t.
ye y
A x B y b
( , )x y
y
x
11 1 . A x B y b
MOLP
MOLP
t
1
t
2
t
min
min
min
s.t.
,
p
c x
c x
c x
Ax b
x 0
, , ,1 2n
pc c cm nA
nxmb
{ | , } X xAx b x 0
MOLP
{ | } Y Cxx X
MOLP
x
Cp n
it
ic
x XMOLP
x X
, Cx Cx Cx Cx
xy Cx
EX
NY
{ | ; , }E X x X x X Cx Cx Cx Cx
{ | }.N E Y Cx x X
x X
x X
Cx Cx
xy Cx
WEX
WNY
WE { | ; } X x X x X Cx Cx
WN WE{ | }. Y Cx x X
E WEX X. N WNY Y
y Y
tmax
s.t.
,
p
1 s
y s Y
s 0
pp1
s
1
2
s
s
s p
s
y Y
max s
s.t. s1
s 0,
p
y Y
s
I I I I(y ,y , ,y )1 2 py
I
N
y min y
s.t.
k k
y Y
1,2, ,k pMOLP
ykky
N N N(y ,y , ,y )1 2
Npy
N
N
y max y
s.t.
k k
y Y
1,2, ,k pMOLP
k
Iy min y
s.t. y
k k
Y
I Ny y y , 1, 2, ,k k k k p
Ny Y
p
Ny
Ny
MOLP
Ny
MOLP
MOLP
t
N
max
s.t.
c y
y Y
Ny Y
Ny Y
0s
M
t t
,
t
max
s.t.
max
s.t.
p
p
M
y s
s
c y 1 s
y Y
1 s
y s Y
s 0
sy
s
y
( , )y s
s
y Y
t
1
N
max
s.t.
z
c y
y Y
2
t
t tmax,
s.t.
max
s.t.
p
Mpz
s
c y 1 sy s
y Y
1 s
y s Y
s 0
NY
NY
*y
*y*Yy
*( , )0y
2
* *t( , ) .z 0y yc
1 2 2
*( , )z z z 0y
2 1z z
* *( , )y s
*0s
* *( , )y s
*0s*Yy* *( , ) 0y s
M
2 2
* ** *( , ) ( , ),z z 0y ys s
* *( , )y s
2 1z z
*( , )0y
*s
*yNY
*y
2 2 1 1
* * *t ( , ) ( ) .z z z z 0y y yc
2 1z z
t
WN
max
s.t.
c y
y Y
WNy Y
WNy Y
M
tmax (s),
s.t.
max s
s
s.t. s1
s 0
p
Ms
c yy
y Y
y Y
s
tc y
MOLPi
N(y )i
ic eiei
i
N t
, , ,
t
,
y max y
s.t.
max
s.t.
,
i i p
p
M
x y x s
x s
1 s
Cx y
Ax b
x 0
1 s
Cx y s
Ax b
x 0 s 0
y
KKT
N t
, , ,
t t
max y
s.t.
i i py M
x y x s1 s
Cx y
Ax b
x 0
Cx y s
Ax b
v C w A 0
t t
t t
t t
t
( ) 0
( ) 0
( ) 0
, ,
p
p
v 1
v C w A x
v 1 s
w b Ax
x 0 s 0 w 0
0K
t t t t, , ( ) 0 v C w A 0 x 0 v C w A x
t t t t, , ( ) 0p p v 1 s 0 v 1 s
t, , ( ) 0 w 0 Ax b w b Ax
t t , ( ),
{0,1}, 1,2, ,
n
j
K K
p j n
0 v C w A p 0 x 1 p
t t , ( ),
{0,1}, 1,2, ,
p p
i
K K
q i p
0 v 1 q 0 s 1 q
, ( ),
{0,1}, 1,2, ,i
K K m
t i m
0 w t 0 b Ax 1 t
s 0
Min -11 -11 -12 -9 - +9x 9xx x x x2 3 4 5 6 7
Min -11 -11 -9 -12 - +99xx x x x x1 3 4 5 76
Min -11 -11 -9 -9 - -12x x x x 12x x1 2 4 5 6 7
s.t. + + + + + =1x x x x x x x1 2 3 4 5 6 7
, , , , , , 0x x x x x x x1 2 3 4 5 6 7
N(0, 0, 0).y
yMax -M( + + )s s s1 2 31
s.t. = -11 -11 -12 -9 - +y 9x 9xx x x x2 3 4 5 6 71
= -11 -11 -9 -12 - +9y 9xx x x x x1 3 4 5 762
= -11 -11 -9 -9 - -12y x x x x 12x x1 2 4 5 6 73
+ + + + + + =1x x x x x x x1 2 3 4 5 6 7
, , , , , , 0x x x x x x x1 2 3 4 5 6 7
Max + +s s s1 2 3
s.t. = -11 -11 -12 -9y s x x x2 3 4 511
- +9x 9xx 6 7
= -11 -11 -9 -12 - +9y s 9xx x x x x1 3 4 5 72 62
= -11 -11 -9 -9 - -12y s x x x x 12x x1 2 4 5 6 733
+ + + + + + =1x x x x x x x1 2 3 4 5 6 7
, , , , , , 0x x x x x x x1 2 3 4 5 6 7
K
yMax 1
s.t. -11 11 12 9y 9x 9xx x x x2 3 4 5 6 71
-11 11 9 12 9y 9xx x x x x1 3 4 5 762
-11 11 9 9 12y x x x x 12x x1 2 4 5 6 73
1x x x x x x x1 2 3 4 5 6 7
, , , , , , 0x x x x x x x1 2 3 4 5 6 7
-11 11 12 9y 9x 9xx x x x2 3 4 5 6 7 11
-11 11 9 12 9y 9xx x x x x1 3 4 5 76 22
-11 11 9 9 12y x x x x 12x x1 2 4 5 6 7 33
1x x x x x x x1 2 3 4 5 6 7
0 11 12
s
s
s
v
1 3 1
0 11 111 3 2
0 11 111 2 3
0 12 9 91 2 3 4
0 9 12 91 2 3 5
0 9 9 121 2 3 6
0 9 9 121 2 3 7
1, 1, 11 2 3
0 (1 )x1 1
0 (1 )x2 2
0 (1 )x3 3
0 (1 )x4 4
0 x5
w K pv
w K pv v
w K pv v
w K pv v v
w K pv v v
w K pv v v
w K pv v v
v v v
K p
K p
K p
K p
(1 )5
0 (1 )x6 6
0 (1 )x7 7
, , , , , , {0,1}.2 3 4 5 6 71
K p
K p
K p
p p p p p p p
MOLP
MOLP
[1] Armand, P., Malivert, C., (1991).
Determination of the efficient set in multiobjective linear programming.
Journal of Optimization Theory and
Applications, 70, 467-489.
[2] Benson, H. P., (1984). Optimization
over the efficient set. Journal of
Mathematical Analysis and Applications,
98, 562-580.
[3] Benson, H. P., Lee, D., McClure, J. P.,
(1998). Global optimization in practice:
An application to interactivemultiple
objective linear programming. Journal of
Global Optimization, 12, 353-372.
[4] Benson, H. P., Sun, E., (2002). A
weight set decomposition algorithm for
finding all efficient extreme points in the
outcome set of a multiple objective linear program. European Journal of
Operational Research, 139, 26-41.
[5] Calvete, H. I., Gal𝑒′, C., Mateo, P. M.,
(2008). A new approach for solving
linear bilevel problems usin ggenetic
algorithms. European Journal of
Operational Research, 188,14-28.
[6] Deb, K., Chaudhuri, S., Miettinen, K.,
(2006). Towards estimating nadir
objective vector using evolutionary
approaches. In: Keijzer, M. et al. (Eds.),
2006 Genetic and Evolutionary
Computation Conference (GECCO2006),
Seattle,Washington, USA, 1, 643-650.
[7] Dempe, S., (2002). Foundations of
Bilevel Programming. Kluwer Academic
Publishers.
[8] Dessouky, M. I., Ghiassi, M., Davis, W.
J., (1986). Estimates of the minimum
nondominated criterion values in
multiple-criteria decision-making.
Engineering Costs and Production
Economics, 10, 95-104.
[9] Dyer, J.S., Fishburn, P. C., Steuer, R. E.,
Wallenius, J., Zionts, S., (1992). Multiple
criterion decision making, multi attribute
utility theory: The next ten years.
Management Science, 38, 645-654.
[10] Ecker, J. G., Hegner, N. S., Kouada, I.
A., (1980). Generating all maximal
efficient faces for multiple objective linear programs. Journal of Optimization
Theory and Applications, 30, 353-381.
[11] Ehrgott, M., (2005). Multicriteria
optimization. Springer, Second Edithion,
Berlin, Germany.
[12] Ehrgott, M., Tenfelde-Podehl, D.,
(2003). Computation of ideal and nadir
values and implications for their use in
MCDM methods. European Journal of
Operational Research, 151,119-139.
[13] Horst, R., Thoai, N. V., Yamamoto, Y.,
Zenke, D., (2007). On optimization over
the efficient set in linear multi criteria programming. Journal of Optimization
Theory and Applications, 134,433-443.
[14] Isermann, H., (1977). The
enumeration of the set of all efficient solutions for a linear multiple objective
program. Operational Research
Quarterly, 28,711-725.
[15] Isermann, H., Steuer, R. E., (1987).
Computational experience concerning payoff tables and minimum criterion
values over the efficient set. European
Journal of Operational Research, 33, 91-
97.
[16] Jorge, J. M., (2005). A bilinear
algorithm for optimizing a linear function over the efficient set of a multiple
objective linear programming problem.
Journal of Global Optimization, 31, 1-6.
[17] Korhonen, P., Salo, S., Steuer, R. E.,
(1997). A heuristic method for estimating
nadir criterion values in multiple objective linear programming.
Operations Research, 45, 751-757.
[18] Leschine, T. M., Wallenius, H.,
Verdini, W. A. (1992). Interactive
multiobjective analysis and assimilative
capacity based ocean disposal decision.
European Journal of Operational
Research, 56, 278-289.
[19] Lv, Y., Hu, T., Wang, G., Wan, Z.,
(2007). A penalty function method based
on Kuhn-Tucker condition for solving
linear bilevel programming. Applied
Mathematics and Computation, 188, 808-
813.
[20] Metev, B., Vassilev, V., (2003). A
Method for Nadir Point Estimation in MOLP Problems. Cyberneticsand
Information Technologies, 3, 15-24.
[21] Pourkarimi, L., Zarepisheh, M.,
(2007). A dual-based algorithm for
solving lexicographic multiple objective programs. European Journal of
Operational Research, 176, 1348-1356.
[22] Reeves, G. R., Reid, R. C., (1988).
Minimum values over the efficient set in multiple objective decision making.
European Journal of Operational
Research, 36, 334-338.
[23] Shi, C., Lu, J., Zhang, G., (2005). An
extended Kuhn-Tucker approach for linear bilevel programming. Applied
Mathematics and Computation, 162, 51-
63.
[24] Shi, C., Lu, J., Zhang, G., Zhou, H.,
(2006). An extended branch and bound
algorithm for linear bilevel programming.
Applied Mathematics and Computation,
180, 529-537.
[25] Shi, C., Zhou, H., Lu, J., Zhang, G.,
Zhang, Z., (2007). The Kth-best approach
for linear bilevel multi follower programming with partial shared
variables among followers. Applied
Mathematics and Computation, 188,
1686-1698.
[26] Tu, T. V., (2000). Optimization over
the efficient set of a parametric multiple
objective linear programming problem.
European Journal of Operational
Research, 122, 570-583.
[27] Yamamoto, Y., (2002). Optimization
over the efficient set: Overview. Journal
of Global Optimization, 22, 285-317.
[28] Yan, H., Wei, J., (2005). Constructing
efficient solutions structure of
multiobjective linear programming.
Journal of Mathematical Analysis and
Applications, 307, 504-523.
