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A linear bilevel programming algorithm: A comment
Comput. Opns Res.
Vol.15,No.3,
pp. 297-298,
0305-0548/88 $3.00+0.00 Copyright0 1988 Pergamon Press plc
1988
Printedin Great Britain. All rights reserved
NOTE A LINEAR
BILEVEL
PROGRAMMING WILFRED
ALGORITHM:
A COMMENT*
CANDLER?
World Bank, 1818 H Street N.W., Washington, DC 20433, U.S.A.:
(Received May 1987)
INTRODUCTION
Theorem 2, due to Bard [l, 21, used in Gulseven Unlu’s recent paper [3], seems to be in error. The theorem : Theorem: 31’~(0, l] such that the corresponding solution programming problem (1) is the optimum solution to (2).
(x0, y”) of the parametric
linear
max &IX + by) + (1 - 1)dy subject to
Ax + By d u
(1)
x,y30. Where problem (2) can be written: maxax+by, X
where y solves max cx + dy
subject ti
(2)
Ax + By 6 u x,y>o.
COUNTER
EXAMPLE
Augmenting the numerical example given in [3] by the addition of a vector y,, and a restraint, we have: Counter example:
max Z, = 3x + yr + y, - 5y,,
where y,, y, and y, solve
x
max Z, = - 5y1 - Yz +
Y3
Yl.YZ.Yl
subject to
x+
y1 +y,+y3<3
2x+2yr
+y,-y3<4
2x - 2y,
6 l/2 y, < 0.04
X,Y,,Y,,Y,~O. *Editor’s Note: Dr Bard has had an opportunity to review this note and has done so. He feels that it is correct, and is now working to strengthen the proof in r21. tWilfred Candler is a Senior Ag&&ural Economist in the Operations Evaluation Department of the World Bank in Washinaton. DC. He holds a B.Aar.Sc. and M.Aar.Sc. from Massey University. New Zealand, and a Ph.D. from Iowa State University. Current research interests include Aghculture Policy and Linear Programming and hence Bilevel Programming. $The World Bank does not accept responsibility for the views expressed herein which are those of the author and should not be attributed to the World Bank or its affrhated organizations.
297
298
Note
The parametric linear program yields the following solutions: K
Anin
1max
I 2 3 4 5
0
0 0.5 0.55 0.67 1
0.5 0.55 0.67
x
0.67 0.25 0.25 0.625 1.125
Yl
Y2
0 0 0 0.375 0.875
0 0 2.75 2 0
Y3 0.04 0.04 0 0 0
22
Z,
0 0.55 3.5 4.438 4.688
0.04 0.04 2.75 -3.875 -4.375
With x fixed at these parametric values, the solutions to the inner problems are: K 1 2,3 4 5
x 0 0.25 0.625 1.125
4’1
YZ
0.008 0.375 0.875
0 0 0 0
0
Y3 0.04 0.04 0.04 0.04
Z2 0 0.55 2.237 4.487
Z, 0.04 0.04 - 1.835 -4.335
For given values of the policy variables, the solution to the inner problem may, or may not, lie on the efficiency frontier traced out by parametric programming. Parametric programming maximizes the value of one objective for a gioen level of the other objective function; linear bilevel programming maximizes the value of one objective given the setting of the policy variables. Which is quite a different thing! Dr Unlu is in good company in making her proposal; which is particularly attractive since the twolevel solutions may lie on the parametric efficiency frontier. REFERENCES 1. J. F. Bard, Optimality conditions for the bilevel programming problem. Nav. Res. Logist. Q. 31, 13-26 (1984). 2. J. F. Bard, An efficient point algorithm for a linear two-stage optimization problem. Opns Res. 31,670-684 (1983). 3. G. Unlu, A linear bilevel programming algorithm based on bicriteria programming. Comput. Opns Res. 14,173-179 (1987).
Vol.15,No.3,
pp. 297-298,
0305-0548/88 $3.00+0.00 Copyright0 1988 Pergamon Press plc
1988
Printedin Great Britain. All rights reserved
NOTE A LINEAR
BILEVEL
PROGRAMMING WILFRED
ALGORITHM:
A COMMENT*
CANDLER?
World Bank, 1818 H Street N.W., Washington, DC 20433, U.S.A.:
(Received May 1987)
INTRODUCTION
Theorem 2, due to Bard [l, 21, used in Gulseven Unlu’s recent paper [3], seems to be in error. The theorem : Theorem: 31’~(0, l] such that the corresponding solution programming problem (1) is the optimum solution to (2).
(x0, y”) of the parametric
linear
max &IX + by) + (1 - 1)dy subject to
Ax + By d u
(1)
x,y30. Where problem (2) can be written: maxax+by, X
where y solves max cx + dy
subject ti
(2)
Ax + By 6 u x,y>o.
COUNTER
EXAMPLE
Augmenting the numerical example given in [3] by the addition of a vector y,, and a restraint, we have: Counter example:
max Z, = 3x + yr + y, - 5y,,
where y,, y, and y, solve
x
max Z, = - 5y1 - Yz +
Y3
Yl.YZ.Yl
subject to
x+
y1 +y,+y3<3
2x+2yr
+y,-y3<4
2x - 2y,
6 l/2 y, < 0.04
X,Y,,Y,,Y,~O. *Editor’s Note: Dr Bard has had an opportunity to review this note and has done so. He feels that it is correct, and is now working to strengthen the proof in r21. tWilfred Candler is a Senior Ag&&ural Economist in the Operations Evaluation Department of the World Bank in Washinaton. DC. He holds a B.Aar.Sc. and M.Aar.Sc. from Massey University. New Zealand, and a Ph.D. from Iowa State University. Current research interests include Aghculture Policy and Linear Programming and hence Bilevel Programming. $The World Bank does not accept responsibility for the views expressed herein which are those of the author and should not be attributed to the World Bank or its affrhated organizations.
297
298
Note
The parametric linear program yields the following solutions: K
Anin
1max
I 2 3 4 5
0
0 0.5 0.55 0.67 1
0.5 0.55 0.67
x
0.67 0.25 0.25 0.625 1.125
Yl
Y2
0 0 0 0.375 0.875
0 0 2.75 2 0
Y3 0.04 0.04 0 0 0
22
Z,
0 0.55 3.5 4.438 4.688
0.04 0.04 2.75 -3.875 -4.375
With x fixed at these parametric values, the solutions to the inner problems are: K 1 2,3 4 5
x 0 0.25 0.625 1.125
4’1
YZ
0.008 0.375 0.875
0 0 0 0
0
Y3 0.04 0.04 0.04 0.04
Z2 0 0.55 2.237 4.487
Z, 0.04 0.04 - 1.835 -4.335
For given values of the policy variables, the solution to the inner problem may, or may not, lie on the efficiency frontier traced out by parametric programming. Parametric programming maximizes the value of one objective for a gioen level of the other objective function; linear bilevel programming maximizes the value of one objective given the setting of the policy variables. Which is quite a different thing! Dr Unlu is in good company in making her proposal; which is particularly attractive since the twolevel solutions may lie on the parametric efficiency frontier. REFERENCES 1. J. F. Bard, Optimality conditions for the bilevel programming problem. Nav. Res. Logist. Q. 31, 13-26 (1984). 2. J. F. Bard, An efficient point algorithm for a linear two-stage optimization problem. Opns Res. 31,670-684 (1983). 3. G. Unlu, A linear bilevel programming algorithm based on bicriteria programming. Comput. Opns Res. 14,173-179 (1987).