Post on 04-Jun-2018
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Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
International Doctoral School Algorithmic Decision
Theory: MCDA and MOO
Lecture 2: Multiobjective Linear Programming
Matthias Ehrgott
Department of Engineering Science, The University of Auckland, New Zealand
Laboratoire dInformatique de Nantes Atlantique, CNRS, Universite de Nantes,
France
MCDA and MOO, Han sur Lesse, September 17 21 2007
Matthias Ehrgott MOLP I
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Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Overview
1 Multiobjective Linear ProgrammingFormulation and ExampleSolving MOLPs by Weighted Sums
2 Biobjective LPs and Parametric SimplexThe Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
3 Multiobjective Simplex MethodA Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Matthias Ehrgott MOLP I
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Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
Overview
1 Multiobjective Linear ProgrammingFormulation and ExampleSolving MOLPs by Weighted Sums
2 Biobjective LPs and Parametric SimplexThe Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
3 Multiobjective Simplex MethodA Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Matthias Ehrgott MOLP I
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Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
Variables x Rn
Objective function Cx where C Rpn
Constraints Ax=bwhere A Rmn and b Rm
min {Cx :Ax=b, x 0} (1)
X ={x Rn :Ax=b, x 0}
is thefeasible set in decision space
Y ={Cx :xX}
is thefeasible set in objective space
Matthias Ehrgott MOLP I
M l i bj i Li P i
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Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
Variables x Rn
Objective function Cx where C Rpn
Constraints Ax=bwhere A Rmn and b Rm
min {Cx :Ax=b, x 0} (1)
X ={x Rn :Ax=b, x 0}
is thefeasible set in decision space
Y ={Cx :xX}
is thefeasible set in objective space
Matthias Ehrgott MOLP I
M lti bj ti Li P i
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Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
Variables x Rn
Objective function Cx where C Rpn
Constraints Ax=bwhere A Rmn and b Rm
min {Cx :Ax=b, x 0} (1)
X ={x Rn :Ax=b, x 0}
is thefeasible set in decision space
Y ={Cx :xX}
is thefeasible set in objective space
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
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Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
Variables x Rn
Objective function Cx where C Rpn
Constraints Ax=bwhere A Rmn and b Rm
min {Cx :Ax=b, x 0} (1)
X ={x Rn :Ax=b, x 0}
is thefeasible set in decision space
Y ={Cx :xX}
is thefeasible set in objective space
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
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Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
Example
min
3x1+x2x1 2x2
subject to x2 3
3x1 x2 6
x 0
C =
3 11 2
A=
0 1 1 03 1 0 1
b=
36
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
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Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
Example
min
3x1+x2x1 2x2
subject to x2 3
3x1 x2 6
x 0
C =
3 11 2
A=
0 1 1 03 1 0 1
b=
36
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingF l i d E l
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Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
0 1 2 3 4
0
1
2
3
4
X
x1
x2 x3
x4
Feasible set in decision space
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingF l ti d E l
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j g gBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
Y
Cx1
Cx2
Cx3
Cx4
Feasible set in objective space
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingFormulation and Example
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j g gBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
Definition
Let xXbe a feasible solution of the MOLP (1) and let y=Cx.x is calledweakly efficientif there is no xX such thatCx0 such that for all i and x with cTi x0 such that cTj x>cTj x and
cTi x cTi x
cTj x cTj x
M.
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingFormulation and Example
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Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
Definition
Let xXbe a feasible solution of the MOLP (1) and let y=Cx.x is calledweakly efficientif there is no xX such thatCx0 such that for all i and x with cTi x0 such that cTj x>cTj x and
cTi x cTi x
cTj x cTj x
M.
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBi bj i LP d P i Si l
Formulation and Example
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Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
Definition
Let xXbe a feasible solution of the MOLP (1) and let y=Cx.x is calledweakly efficientif there is no xX such thatCx0 such that for all i and x with cTi x0 such that cTj x>cTj x and
cTi x cTi x
cTj x cTj x
M.
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBi bj ti LP d P t i Si l
Formulation and Example
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Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
Definition
Let xXbe a feasible solution of the MOLP (1) and let y=Cx.x is calledweakly efficientif there is no xX such thatCx0 such that for all i and x with cTi x0 such that cTj x>cTj x and
cTi x cTi x
cTj x cTj x
M.
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjecti e LPs a d Pa a et ic Si le
Formulation and Example
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Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
pSolving MOLPs by Weighted Sums
0 1 2 3 4 5 6 7 8 9 10 11 12 1 3 14
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
Y
Cx1
Cx2
Cx3
Cx4
Nondominated set
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Formulation and Example
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Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
pSolving MOLPs by Weighted Sums
0 1 2 3 4
0
1
2
3
4
X
x1
x2 x3
x4
Efficient set
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Formulation and Example
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Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
Solving MOLPs by Weighted Sums
Overview
1 Multiobjective Linear ProgrammingFormulation and ExampleSolving MOLPs by Weighted Sums
2 Biobjective LPs and Parametric SimplexThe Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
3 Multiobjective Simplex MethodA Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Formulation and ExampleS l i MOLP b W i h d S
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Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
Solving MOLPs by Weighted Sums
Let 1, . . . , p 0 and consider
LP() min
pk=1
kcTk x= min
TCx
subject to Ax = b
x 0
with some vector 0 (Why not = 0 or 0?)
LP() is a linear programme that can be solved by theSimplex method
If >0 then optimal solution ofLP() isproperly efficient
If 0 then optimal solution ofLP() isweakly efficient
Converse also true, because Y convex
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Formulation and ExampleS l i MOLP b W i ht d S
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object e s a d a a et c S p eMultiobjective Simplex Method
Solving MOLPs by Weighted Sums
Let 1, . . . , p 0 and consider
LP() min
pk=1
kcTk x= min
TCx
subject to Ax = b
x 0
with some vector 0 (Why not = 0 or 0?)
LP() is a linear programme that can be solved by theSimplex method
If >0 then optimal solution ofLP() isproperly efficient
If 0 then optimal solution ofLP() isweakly efficient
Converse also true, because Y convex
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Formulation and ExampleSolving MOLPs by Weighted Sums
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j pMultiobjective Simplex Method
Solving MOLPs by Weighted Sums
Let 1, . . . , p 0 and consider
LP() min
pk=1
kcTk x= min
TCx
subject to Ax = b
x 0
with some vector 0 (Why not = 0 or 0?)
LP() is a linear programme that can be solved by theSimplex method
If >0 then optimal solution ofLP() isproperly efficient
If 0 then optimal solution ofLP() isweakly efficient
Converse also true, because Y convex
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Formulation and ExampleSolving MOLPs by Weighted Sums
http://find/8/14/2019 HanLecture2 ME
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Multiobjective Simplex MethodSolving MOLPs by Weighted Sums
Let 1, . . . , p 0 and consider
LP() min
pk=1
kcTk x= min
TCx
subject to Ax = b
x 0
with some vector 0 (Why not = 0 or 0?)
LP() is a linear programme that can be solved by theSimplex method
If >0 then optimal solution ofLP() isproperly efficient
If 0 then optimal solution ofLP() isweakly efficient
Converse also true, because Y convex
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Formulation and ExampleSolving MOLPs by Weighted Sums
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23/176
Multiobjective Simplex MethodSolving MOLPs by Weighted Sums
Let 1, . . . , p 0 and consider
LP() min
pk=1
kcTk x= min
TCx
subject to Ax = b
x 0
with some vector 0 (Why not = 0 or 0?)
LP() is a linear programme that can be solved by theSimplex method
If >0 then optimal solution ofLP() isproperly efficient
If 0 then optimal solution ofLP() isweakly efficient
Converse also true, because Y convex
Matthias Ehrgott MOLP I
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http://find/8/14/2019 HanLecture2 ME
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Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Multiobjective Simplex Methodg y g
y Rp satisfyingTy= define a straight line (hyperplane)
Since y=Cx and TCxis minimised, we push the linetowards the origin (left and down)
When the line only touches Ynondominated points are foundNondominated points YNare on the boundary ofY
Yis convex polyhedron and has finite number of facets. YNconsists of finitely many facets ofY. The normal of the facetcan serve as weight vector
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Multiobjective Simplex Method
y Rp satisfyingTy= define a straight line (hyperplane)
Since y=Cx and TCxis minimised, we push the linetowards the origin (left and down)
When the line only touches Ynondominated points are foundNondominated points YNare on the boundary ofY
Yis convex polyhedron and has finite number of facets. YNconsists of finitely many facets ofY. The normal of the facetcan serve as weight vector
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Multiobjective Simplex Method
y Rp satisfyingTy= define a straight line (hyperplane)
Since y=Cx and TCxis minimised, we push the linetowards the origin (left and down)
When the line only touches Ynondominated points are foundNondominated points YNare on the boundary ofY
Yis convex polyhedron and has finite number of facets. YNconsists of finitely many facets ofY. The normal of the facetcan serve as weight vector
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
http://find/http://goback/8/14/2019 HanLecture2 ME
29/176
Multiobjective Simplex Method
y Rp satisfyingTy= define a straight line (hyperplane)
Since y=Cx and TCxis minimised, we push the linetowards the origin (left and down)
When the line only touches Ynondominated points are foundNondominated points YNare on the boundary ofY
Yis convex polyhedron and has finite number of facets. YNconsists of finitely many facets ofY. The normal of the facetcan serve as weight vector
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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30/176
j p
y Rp satisfyingTy= define a straight line (hyperplane)
Since y=Cx and TCxis minimised, we push the linetowards the origin (left and down)
When the line only touches Ynondominated points are foundNondominated points YNare on the boundary ofY
Yis convex polyhedron and has finite number of facets. YNconsists of finitely many facets ofY. The normal of the facetcan serve as weight vector
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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j p
Question: Can all efficient solutions be found using weighted sums?If xX is efficient, does there exist >0 such that x is optimal
solution to min{TCx :Ax=b, x 0}?
Lemma
A feasible solution x0 X is efficient if and only if the linearprogramme
max eTzsubject to Ax = b
Cx+Iz = Cx0
x, z 0,
(2)
where eT = (1, . . . , 1) Rp and I is the p p identity matrix, hasan optimal solution(x, z) with z= 0.
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Question: Can all efficient solutions be found using weighted sums?If xX is efficient, does there exist >0 such that x is optimal
solution to min{TCx :Ax=b, x 0}?
Lemma
A feasible solution x0 X is efficient if and only if the linearprogramme
max eTzsubject to Ax = b
Cx+Iz = Cx0
x, z 0,
(2)
where eT = (1, . . . , 1) Rp and I is the p p identity matrix, hasan optimal solution(x, z) with z= 0.
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Question: Can all efficient solutions be found using weighted sums?If xX is efficient, does there exist >0 such that x is optimal
solution to min{TCx :Ax=b, x 0}?
Lemma
A feasible solution x0 X is efficient if and only if the linearprogramme
max eTzsubject to Ax = b
Cx+Iz = Cx0
x, z 0,
(2)
where eT = (1, . . . , 1) Rp and I is the p p identity matrix, hasan optimal solution(x, z) with z= 0.
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Proof.LP is always feasible with x=x0, z= 0 (and value 0)
Let (x, z) be optimal solution
If z= 0 then z=Cx0 Cx= 0 Cx0 =Cx
There is no xX such that CxCx0
because (x, Cx0
Cx)would be better solution x0 efficient
If x0 efficient there is no xX with CxCx0
there is no z with z=Cx0 Cx0
max eT
z 0 max eT
z= 0
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric SimplexMultiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Proof.
LP is always feasible with x=x0, z= 0 (and value 0)
Let (x, z) be optimal solution
If z= 0 then z=Cx0 Cx= 0 Cx0 =Cx
There is no xX such that CxCx0
because (x, Cx0
Cx)would be better solution x0 efficient
If x0 efficient there is no xX with CxCx0
there is no z with z=Cx0 Cx0
max eT
z 0 max eT
z= 0
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric SimplexMultiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Proof.
LP is always feasible with x=x0, z= 0 (and value 0)
Let (x, z) be optimal solution
If z= 0 then z=Cx0 Cx= 0 Cx0 =Cx
There is no xX such that CxCx0
because (x, Cx0
Cx)would be better solution x0 efficient
If x0 efficient there is no xX with CxCx0
there is no z with z=Cx0 Cx0
max eT
z 0 max eT
z= 0
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric SimplexMultiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Proof.
LP is always feasible with x=x0, z= 0 (and value 0)
Let (x, z) be optimal solution
If z= 0 then z=Cx0 Cx= 0 Cx0 =Cx
There is no xX such that CxCx0
because (x, Cx0
Cx)would be better solution x0 efficient
If x0 efficient there is no xX with CxCx0
there is no z with z=Cx0 Cx0
max eT
z 0 max eT
z= 0
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric SimplexMultiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Proof.
LP is always feasible with x=x0, z= 0 (and value 0)
Let (x, z) be optimal solution
If z= 0 then z=Cx0 Cx= 0 Cx0 =Cx
There is no xX such that CxCx0
because (x, Cx0
Cx)would be better solution x0 efficient
If x0 efficient there is no xX with CxCx0
there is no z with z=Cx0 Cx0
max eT
z 0 max eT
z= 0
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric SimplexMultiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Proof.
LP is always feasible with x=x0, z= 0 (and value 0)
Let (x, z) be optimal solution
If z= 0 then z=Cx0 Cx= 0 Cx0 =Cx
There is no xX such that CxCx0
because (x, Cx0
Cx)would be better solution x0 efficient
If x0 efficient there is no xX with CxCx0
there is no z with z=Cx0 Cx0
max eT
z 0 max eT
z= 0
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric SimplexMultiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Proof.
LP is always feasible with x=x0, z= 0 (and value 0)
Let (x, z) be optimal solution
If z= 0 then z=Cx0 Cx= 0 Cx0 =Cx
There is no xX such that CxCx0
because (x, Cx0
Cx)would be better solution x0 efficient
If x0 efficient there is no xX with CxCx0
there is no z with z=Cx0 Cx0
max eT
z 0 max eT
z= 0
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric SimplexMultiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Lemma
A feasible solution x0 X is efficient if and only if the linearprogramme
min uTb+wTCx0
subject to uTA+wTC 0w e
u Rm(3)
has an optimal solution(u, w) with uTb+ wTCx0 = 0.
Proof.
The LP (3) is the dual of the LP (2)
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric SimplexMultiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Lemma
A feasible solution x0 X is efficient if and only if the linearprogramme
min uTb+wTCx0
subject to uTA+wTC 0w e
u Rm(3)
has an optimal solution(u, w) with uTb+ wTCx0 = 0.
Proof.
The LP (3) is the dual of the LP (2)
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric SimplexMultiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Theorem
A feasible solution x0 X is an efficient solution of the MOLP (1)if and only if there exists a Rp> such that
TCx0 TCx (4)
for all xX .
Note: We already know that optimal solutions of weighted sum
problems are efficient
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric SimplexMultiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Theorem
A feasible solution x0 X is an efficient solution of the MOLP (1)if and only if there exists a Rp> such that
TCx0 TCx (4)
for all xX .
Note: We already know that optimal solutions of weighted sum
problems are efficient
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric SimplexMultiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Proof.
Let x0 XE
By Lemma4LP (3) has an optimal solution (u,w) such that
uTb=wTCx0 (5)
u is also an optimal solution of the LP
min
uTb:uTA wTC
, (6)
which is (3) with w= w fixed
There is an optimal solution of the dual of (6)
max
wTCx :Ax=b, x 0
(7)
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric SimplexMultiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Proof.
Let x0 XE
By Lemma4LP (3) has an optimal solution (u,w) such that
uTb=wTCx0 (5)
u is also an optimal solution of the LP
min
uTb:uTA wTC
, (6)
which is (3) with w= w fixed
There is an optimal solution of the dual of (6)
max
wTCx :Ax=b, x 0
(7)
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Proof.
Let x0 XE
By Lemma4LP (3) has an optimal solution (u,w) such that
uTb=wTCx0 (5)
u is also an optimal solution of the LP
min
uTb:uTA wTC
, (6)
which is (3) with w= w fixed
There is an optimal solution of the dual of (6)
max
wTCx :Ax=b, x 0
(7)
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
Formulation and ExampleSolving MOLPs by Weighted Sums
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Proof.
Let x0 XE
By Lemma4LP (3) has an optimal solution (u,w) such that
uTb=wTCx0 (5)
u is also an optimal solution of the LP
min
uTb:uTA wTC
, (6)
which is (3) with w= w fixed
There is an optimal solution of the dual of (6)
max
wTCx :Ax=b, x 0
(7)
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex MethodFormulation and ExampleSolving MOLPs by Weighted Sums
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Proof.
By weak duality uTb wTCx for all feasible solutions uof(6) and for all feasible solutions x of (7)
We already know that uTb=wTCx0 from (5)
x0 is an optimal solution of (7)
Note that (7) is equivalent to
min
wTCx :Ax=b, x 0
with w e>0 from the constraints in (3)
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex MethodFormulation and ExampleSolving MOLPs by Weighted Sums
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Proof.
By weak duality uTb wTCx for all feasible solutions uof(6) and for all feasible solutions x of (7)
We already know that uTb=wTCx0 from (5)
x0 is an optimal solution of (7)
Note that (7) is equivalent to
min
wTCx :Ax=b, x 0
with w e>0 from the constraints in (3)
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex MethodFormulation and ExampleSolving MOLPs by Weighted Sums
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Proof.
By weak duality uTb wTCx for all feasible solutions uof(6) and for all feasible solutions x of (7)
We already know that uTb=wTCx0 from (5)
x0 is an optimal solution of (7)
Note that (7) is equivalent to
min
wTCx :Ax=b, x 0
with w e>0 from the constraints in (3)
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex MethodFormulation and ExampleSolving MOLPs by Weighted Sums
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Proof.
By weak duality uTb wTCx for all feasible solutions uof(6) and for all feasible solutions x of (7)
We already know that uTb=wTCx0 from (5)
x0 is an optimal solution of (7)
Note that (7) is equivalent to
min
wTCx :Ax=b, x 0
with w e>0 from the constraints in (3)
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex MethodThe Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
Overview
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Overview
1 Multiobjective Linear ProgrammingFormulation and ExampleSolving MOLPs by Weighted Sums
2 Biobjective LPs and Parametric SimplexThe Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
3 Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex MethodThe Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
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Modification of theSimplex algorithmfor LPs with twoobjectives
min
(c1)Tx, (c2)Tx
subject to Ax = b
x
0
(8)
We can find all efficient solutions by solving the parametric LP
min
1(c1)Tx+2(c
2)Tx :Ax=b, x 0
for all = (1, 2)> 0
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex MethodThe Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
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Modification of theSimplex algorithmfor LPs with twoobjectives
min
(c1)Tx, (c2)Tx
subject to Ax = b
x
0
(8)
We can find all efficient solutions by solving the parametric LP
min
1(c1)Tx+2(c
2)Tx :Ax=b, x 0
for all = (1, 2)> 0
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
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We can divide the objective by 1+2 without changing theoptima, i.e. 1=1/(1+2),
2=2/(1+2 and
1+2= 1 or
2= 1 1
LPs with one parameter 0 1 and parametric objective
c() :=c1 + (1 )c2
min
c()Tx :Ax=b, x 0
(9)
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
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We can divide the objective by 1+2 without changing theoptima, i.e. 1=1/(1+2),
2=2/(1+2 and
1+2= 1 or
2= 1 1
LPs with one parameter 0 1 and parametric objective
c() :=c1 + (1 )c2
min
c()Tx :Ax=b, x 0
(9)
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
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LetBbe a feasible basis
Recall reduced cost cN =cN cTBB
1N
Reduced cost for the parametric LP
c() =c1 + (1 )c2 (10)
Suppose Bis an optimal basis of (9) for some
c() 0
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
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LetBbe a feasible basis
Recall reduced cost cN =cN cTBB
1N
Reduced cost for the parametric LP
c() =c1 + (1 )c2 (10)
Suppose Bis an optimal basis of (9) for some
c() 0
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
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LetBbe a feasible basis
Recall reduced cost cN =cN cTBB
1N
Reduced cost for the parametric LP
c() =c1 + (1 )c2 (10)
Suppose Bis an optimal basis of (9) for some
c() 0
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
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LetBbe a feasible basis
Recall reduced cost cN =cN cTBB
1N
Reduced cost for the parametric LP
c() =c1 + (1 )c2 (10)
Suppose Bis an optimal basis of (9) for some
c() 0
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
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LetBbe a feasible basis
Recall reduced cost cN =cN cTBB
1N
Reduced cost for the parametric LP
c() =c1 + (1 )c2 (10)
Suppose Bis an optimal basis of (9) for some
c() 0
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
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Case 1: c2 0
From (10) c() 0 for all
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Case 1: c2 0
From (10) c() 0 for all
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Case 1: c2 0
From (10) c() 0 for all
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Case 1: c2 0
From (10) c() 0 for all
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Case 1: c2 0
From (10) c() 0 for all
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Case 1: c2 0
From (10) c() 0 for all
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Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
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Case 1: c2 0
From (10) c() 0 for all
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Case 1: c2 0
From (10) c() 0 for all
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I={i N : c2i
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I={i N : c2i
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I={i N : c2i
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I={i N : c2i
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I={i N : c2i
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Input: Data A, b, C for a biobjective LP.
Phase I: Solve the auxiliary LP for Phase I using the Simplex algorithm. If the optimal value is positive, STOP, X =. Otherwise letB be an optimalbasis.
Phase II: Solve the LP (9) for= 1 starting from basisB found in Phase Iyielding an optimal basisB. ComputeA andb.
Phase III: WhileI={i N : c2
i 0
o.
LetB:= (B \ {r}) {s} and updateA andb.
End while.
Output: Sequence of -values and sequence of optimal BFSs.
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
Overview
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1 Multiobjective Linear ProgrammingFormulation and ExampleSolving MOLPs by Weighted Sums
2
Biobjective LPs and Parametric SimplexThe Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
3 Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
Example
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Example
min
3x1+x2x1 2x2
subject to x2 3
3x1 x2 6x 0
LP()
min (4 1)x1 + (3 2)x2subject to x2 + x3 = 3
3x1 x2 + x4 = 6x 0.
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
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Use Simplex tableaus showing reduced cost vectors c1 and c2
Optimal basis for = 1 is B={3, 4}, optimal basic feasible
solution x= (0, 0, 3, 6)Start with Phase 3
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
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Use Simplex tableaus showing reduced cost vectors c1 and c2
Optimal basis for = 1 is B={3, 4}, optimal basic feasible
solution x= (0, 0, 3, 6)Start with Phase 3
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
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Use Simplex tableaus showing reduced cost vectors c1 and c2
Optimal basis for = 1 is B={3, 4}, optimal basic feasible
solution x= (0, 0, 3, 6)Start with Phase 3
Matthias Ehrgott MOLP I
Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
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Iteration 1:
c1 3 1 0 0 0
c2 -1 -2 0 0 0
x3 0 1 1 0 3
x4 3 -1 0 1 6
= 1, c() = (3, 1, 0, 0),B1 ={3, 4}, x1 = (0, 0, 3, 6)
I={1, 2}, = max
13+1
, 21+2
= 2
3
s= 2, r= 3
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
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Iteration 2
c1 3 0 -1 0 -3
c2 -1 0 2 0 6
x2 0 1 1 0 3
x4 3 0 1 1 9
= 2/3, c() = (5/3, 0, 0, 0),B2 ={2, 4}, x2 = (0, 3, 0, 9)
I={1}, = max
13+1
= 1
4
s= 1, r= 4
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
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Iteration 3
c1 0 0 -2 -1 -12
c2 0 0 7/3 1/3 9
x2 0 1 1 0 3
x1 1 0 1/3 1/3 3
= 1/4, c() = (0, 0, 5/4, 0),B3 ={1, 2}, x3 = (3, 3, 0, 0)I=
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
Weight values 1 = 1 2 = 2/3 3 = 1/4 4 = 0
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Weight values = 1, = 2/3, = 1/4, = 0
Basic feasible solutions x1
, x2
, x3
In each iteration c() can be calculated with the previous andcurrent c1 and c2.
BasisB1 = (3, 4) and BFS x1 = (0, 0, 3, 6) are optimal for [2/3, 1].
BasisB2 = (2, 4) and BFS x2 = (0, 3, 0, 9) are optimal for [1/4, 2/3], and
BasisB3 = (1, 2) and BFS x3 = (3, 3, 0, 0) are optimal for [0, 1/4].
Objective vectors for basic feasible solutions: Cx1 = (0, 0),Cx2 = (3, 6), and Cx3 = (12, 9)
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
Weight values 1 = 1, 2 = 2/3, 3 = 1/4, 4 = 0
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Weight values 1, 2/3, 1/4, 0
Basic feasible solutions x1
, x2
, x3
In each iteration c() can be calculated with the previous andcurrent c1 and c2.
BasisB1 = (3, 4) and BFS x1 = (0, 0, 3, 6) are optimal for [2/3, 1].
BasisB2 = (2, 4) and BFS x2 = (0, 3, 0, 9) are optimal for [1/4, 2/3], and
BasisB3 = (1, 2) and BFS x3 = (3, 3, 0, 0) are optimal for [0, 1/4].
Objective vectors for basic feasible solutions: Cx1 = (0, 0),Cx2 = (3, 6), and Cx3 = (12, 9)
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
Weight values 1 = 1, 2 = 2/3, 3 = 1/4, 4 = 0
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Weight values 1, 2/3, 1/4, 0
Basic feasible solutions x1
, x2
, x3
In each iteration c() can be calculated with the previous andcurrent c1 and c2.
BasisB1 = (3, 4) and BFS x1 = (0, 0, 3, 6) are optimal for [2/3, 1].
BasisB2 = (2, 4) and BFS x2 = (0, 3, 0, 9) are optimal for [1/4, 2/3], and
BasisB3 = (1, 2) and BFS x3 = (3, 3, 0, 0) are optimal for [0, 1/4].
Objective vectors for basic feasible solutions: Cx1 = (0, 0),Cx2 = (3, 6), and Cx3 = (12, 9)
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
Weight values 1 = 1, 2 = 2/3, 3 = 1/4, 4 = 0
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g , / , / ,
Basic feasible solutions x1
, x2
, x3
In each iteration c() can be calculated with the previous andcurrent c1 and c2.
BasisB1 = (3, 4) and BFS x1 = (0, 0, 3, 6) are optimal for [2/3, 1].
BasisB2 = (2, 4) and BFS x2 = (0, 3, 0, 9) are optimal for [1/4, 2/3], and
BasisB3 = (1, 2) and BFS x3 = (3, 3, 0, 0) are optimal for [0, 1/4].
Objective vectors for basic feasible solutions: Cx1 = (0, 0),Cx2 = (3, 6), and Cx3 = (12, 9)
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
Weight values 1 = 1, 2 = 2/3, 3 = 1/4, 4 = 0
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g , / , / ,
Basic feasible solutions x
1
, x
2
, x
3
In each iteration c() can be calculated with the previous andcurrent c1 and c2.
BasisB1 = (3, 4) and BFS x1 = (0, 0, 3, 6) are optimal for [2/3, 1].
BasisB2 = (2, 4) and BFS x2 = (0, 3, 0, 9) are optimal for [1/4, 2/3], and
BasisB3 = (1, 2) and BFS x3 = (3, 3, 0, 0) are optimal for [0, 1/4].
Objective vectors for basic feasible solutions: Cx1 = (0, 0),Cx2 = (3, 6), and Cx3 = (12, 9)
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
Weight values 1 = 1, 2 = 2/3, 3 = 1/4, 4 = 0
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g , / , / ,
Basic feasible solutions x
1
, x
2
, x
3
In each iteration c() can be calculated with the previous andcurrent c1 and c2.
BasisB1 = (3, 4) and BFS x1 = (0, 0, 3, 6) are optimal for [2/3, 1].
BasisB2 = (2, 4) and BFS x2 = (0, 3, 0, 9) are optimal for [1/4, 2/3], and
BasisB3 = (1, 2) and BFS x3 = (3, 3, 0, 0) are optimal for [0, 1/4].
Objective vectors for basic feasible solutions: Cx1 = (0, 0),Cx2 = (3, 6), and Cx3 = (12, 9)
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
Weight values 1 = 1, 2 = 2/3, 3 = 1/4, 4 = 0
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/ /
Basic feasible solutions x
1
, x
2
, x
3
In each iteration c() can be calculated with the previous andcurrent c1 and c2.
BasisB1 = (3, 4) and BFS x1 = (0, 0, 3, 6) are optimal for [2/3, 1].
BasisB2 = (2, 4) and BFS x2 = (0, 3, 0, 9) are optimal for [1/4, 2/3], and
BasisB3 = (1, 2) and BFS x3 = (3, 3, 0, 0) are optimal for [0, 1/4].
Objective vectors for basic feasible solutions: Cx1 = (0, 0),Cx2 = (3, 6), and Cx3 = (12, 9)
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
Values = 2/3 and = 1/4 correspond to weight vectors(2/3, 1/3) and (1/4, 3/4)
C
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Contour lines for weighted sum objectives in decision are
parallel to efficient edges
2
3(3x1+x2) +
1
3(x1 2x2) =
5
3x1
1
4 (3x
1+x
2) +
3
4 (x
1 2x
2) =
5
4x
2
0 1 2 3 4
0
1
2
3
4
X
x1
x2
x3
x4
Feasible set in decision space andefficient set
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
Values = 2/3 and = 1/4 correspond to weight vectors(2/3, 1/3) and (1/4, 3/4)
C li f i h d bj i i d i i
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Contour lines for weighted sum objectives in decision are
parallel to efficient edges
2
3(3x1+x2) +
1
3(x1 2x2) =
5
3x1
1
4 (3x
1+x
2) +
3
4 (x
1 2x
2) =
5
4x
2
0 1 2 3 4
0
1
2
3
4
X
x1
x2
x3
x4
Feasible set in decision space andefficient set
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric SimplexMultiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
Values = 2/3 and = 1/4 correspond to weight vectors(2/3, 1/3) and (1/4, 3/4)
C li f i h d bj i i d i i
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Contour lines for weighted sum objectives in decision are
parallel to efficient edges
2
3(3x1+x2) +
1
3(x1 2x2) =
5
3x1
1
4 (3x
1+x
2) +
3
4 (x
1 2x
2) =
5
4x
2
0 1 2 3 4
0
1
2
3
4
X
x1
x2
x3
x4
Feasible set in decision space andefficient set
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric Simplex
Multiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
Weight vectors (2/3, 1/3) and (1/4, 3/4) are normal tonondominated edges
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0 1 2 3 4 5 6 7 8 9 10 1 1 1 2 1 3 1 4
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
Y
Cx1
Cx2
Cx3
Cx4
Objective space and nondominated set
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric Simplex
Multiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
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Algorithm findsall nondominated extreme pointsin objectivespace andone efficient bfsfor each of those
Algorithmdoes not find all efficient solutionsjust as Simplexalgorithm does not find all optimal solutions of an LP
Example
min (x1, x2)T
subject to 0 xi 1 i= 1, 2, 3
Efficient set: {x R3 :x1=x2= 0, 0 x3 1}
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric Simplex
Multiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
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Algorithm findsall nondominated extreme pointsin objectivespace andone efficient bfsfor each of those
Algorithmdoes not find all efficient solutionsjust as Simplexalgorithm does not find all optimal solutions of an LP
Example
min (x1, x2)T
subject to 0 xi 1 i= 1, 2, 3
Efficient set: {x R3 :x1=x2= 0, 0 x3 1}
Matthias Ehrgott MOLP I Multiobjective Linear Programming
Biobjective LPs and Parametric Simplex
Multiobjective Simplex Method
The Parametric Simplex Algorithm
Biobjective Linear Programmes: Example
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Algorithm findsall nondominated extreme pointsin objectivespace andone efficient bfsfor each of those
Algorithmdoes not find all efficient solutionsjust as Simplexalgorithm does not find all optimal solutions of an LP
Example
min (x1, x2)T
subject to 0 xi 1 i= 1, 2, 3
Efficient set: {x R3 :x1=x2= 0, 0 x3 1}
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex Algorithm
Multiobjective Simplex: Examples
Overview
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1 Multiobjective Linear ProgrammingFormulation and ExampleSolving MOLPs by Weighted Sums
2 Biobjective LPs and Parametric SimplexThe Parametric Simplex AlgorithmBiobjective Linear Programmes: Example
3 Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex Algorithm
Multiobjective Simplex: Examples
min{Cx :Ax=b, x 0}
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{ }
LetBbe a basis and C =C CBA1B Aand R= CN
How to calculate critical ifp>2?
AtB1: CN =
3 11 2
, = 2/3, = (2/3, 1/3)T and
T
CN = (5/3, 0)T
AtB2: CN =
3 11 2
, = 1/4, = (1/4, 3/4)T and
TCN = (0, 5/4)T
Find Rp, >0 such that TR 0 (optimality)andTrj = 0 (alternative optimum)for some column rj ofR
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex Algorithm
Multiobjective Simplex: Examples
min{Cx :Ax=b, x 0}
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LetBbe a basis and C =C CBA1B Aand R= CN
How to calculate critical ifp>2?
AtB1: CN =
3 11 2
, = 2/3, = (2/3, 1/3)T and
T
CN = (5/3, 0)T
AtB2: CN =
3 11 2
, = 1/4, = (1/4, 3/4)T and
TCN = (0, 5/4)T
Find Rp, >0 such that TR 0 (optimality)andTrj = 0 (alternative optimum)for some column rj ofR
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex Algorithm
Multiobjective Simplex: Examples
min{Cx :Ax=b, x 0}
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LetBbe a basis and C =C CBA1B Aand R= CN
How to calculate critical ifp>2?
AtB1: CN =
3 11 2
, = 2/3, = (2/3, 1/3)T and
T
CN = (5/3, 0)T
AtB2: CN =
3 11 2
, = 1/4, = (1/4, 3/4)T and
TCN = (0, 5/4)T
Find Rp, >0 such that TR 0 (optimality)andTrj = 0 (alternative optimum)for some column rj ofR
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex Algorithm
Multiobjective Simplex: Examples
min{Cx :Ax=b, x 0}
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LetBbe a basis and C =C CBA1B Aand R= CN
How to calculate critical ifp>2?
AtB1: CN =
3 11 2
, = 2/3, = (2/3, 1/3)T and
T
CN = (5/3, 0)T
AtB2: CN =
3 11 2
, = 1/4, = (1/4, 3/4)T and
TCN = (0, 5/4)T
Find Rp, >0 such that TR 0 (optimality)andTrj = 0 (alternative optimum)for some column rj ofR
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex Algorithm
Multiobjective Simplex: Examples
min{Cx :Ax=b, x 0}
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LetBbe a basis and C =C CBA1B Aand R= CN
How to calculate critical ifp>2?
AtB1: CN =
3 11 2
, = 2/3, = (2/3, 1/3)T and
TCN = (5/3, 0)
T
AtB2: CN =
3 11 2
, = 1/4, = (1/4, 3/4)T and
TCN = (0, 5/4)T
Find Rp, >0 such that TR 0 (optimality)andTrj = 0 (alternative optimum)for some column rj ofR
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex Algorithm
Multiobjective Simplex: Examples
min{Cx :Ax=b, x 0}
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LetBbe a basis and C =C CBA1B Aand R= CN
How to calculate critical ifp>2?
AtB1: CN =
3 11 2
, = 2/3, = (2/3, 1/3)T and
TCN = (5/3, 0)
T
AtB2: CN =
3 11 2
, = 1/4, = (1/4, 3/4)T and
TCN = (0, 5/4)T
Find Rp, >0 such that TR 0 (optimality)andTrj = 0 (alternative optimum)for some column rj ofR
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex Algorithm
Multiobjective Simplex: Examples
Lemma
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Lemma
IfXE= then X has an efficient basic feasible solution.
Proof.
There is some >0 such that minxXTCxhas an optimal
solution
Thus LP() has an optimal basic feasible solution solution,which is an efficient solution of the MOLP
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Lemma
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Lemma
IfXE= then X has an efficient basic feasible solution.
Proof.
There is some >0 such that minxXTCxhas an optimal
solution
Thus LP() has an optimal basic feasible solution solution,which is an efficient solution of the MOLP
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Lemma
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Lemma
IfXE= then X has an efficient basic feasible solution.
Proof.
There is some >0 such that minxXTCxhas an optimal
solution
Thus LP() has an optimal basic feasible solution solution,which is an efficient solution of the MOLP
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Definition
1 A feasible basis B is calledefficient basisifBis an optimal
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basis of LP() for some Rp>.
2 Two basesBand Bare calledadjacentif one can be obtainedfrom the other by a single pivot step.
3 LetBbe an efficient basis. Variable xj,j N is called
efficient nonbasic variableatB if there exists a Rp
> suchthat TR 0 and Trj = 0, where rj is the column ofRcorresponding to variable xj.
4 LetBbe an efficient basis and let xjbe an efficient nonbasicvariable. Then a feasible pivot from Bwith xjentering the
basis (even with negative pivot element) is called an efficientpivotwith respect to Band xj
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Definition
1 A feasible basis B is calledefficient basisifBis an optimal
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basis of LP() for some Rp>.
2 Two basesBand Bare calledadjacentif one can be obtainedfrom the other by a single pivot step.
3 LetBbe an efficient basis. Variable xj,j N is called
efficient nonbasic variableatB if there exists a Rp
> suchthat TR 0 and Trj = 0, where rj is the column ofRcorresponding to variable xj.
4 LetBbe an efficient basis and let xjbe an efficient nonbasicvariable. Then a feasible pivot from Bwith xjentering the
basis (even with negative pivot element) is called an efficientpivotwith respect to Band xj
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Definition
1 A feasible basis B is calledefficient basisifBis an optimalp
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basis of LP() for some Rp>.
2 Two basesBand Bare calledadjacentif one can be obtainedfrom the other by a single pivot step.
3 LetBbe an efficient basis. Variable xj,j N is called
efficient nonbasic variableatB if there exists a Rp
> suchthat TR 0 and Trj = 0, where rj is the column ofRcorresponding to variable xj.
4 LetBbe an efficient basis and let xjbe an efficient nonbasicvariable. Then a feasible pivot from Bwith xjentering the
basis (even with negative pivot element) is called an efficientpivotwith respect to Band xj
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Definition
1 A feasible basis B is calledefficient basisifBis an optimalp
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basis of LP() for some Rp>.
2 Two basesBand Bare calledadjacentif one can be obtainedfrom the other by a single pivot step.
3 LetBbe an efficient basis. Variable xj,j N is called
efficient nonbasic variableatB if there exists a Rp
> suchthat TR 0 and Trj = 0, where rj is the column ofRcorresponding to variable xj.
4 LetBbe an efficient basis and let xjbe an efficient nonbasicvariable. Then a feasible pivot from Bwith xjentering the
basis (even with negative pivot element) is called an efficientpivotwith respect to Band xj
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
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No efficient basis is optimal for all pobjectives at the sametime
ThereforeRalways contains positive and negative entries
Proposition
LetBbe an efficient basis. There exists an efficient nonbasicvariable atB.
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
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No efficient basis is optimal for all pobjectives at the sametime
ThereforeRalways contains positive and negative entries
Proposition
LetBbe an efficient basis. There exists an efficient nonbasicvariable atB.
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
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No efficient basis is optimal for all pobjectives at the sametime
ThereforeRalways contains positive and negative entries
Proposition
LetBbe an efficient basis. There exists an efficient nonbasicvariable atB.
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
It is not possible to define efficient nonbasic variables by the
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p yexistence of a column in Rwith positive and negative entries
Example
R= 3 2
2 1
Tr2 = 0 requires 2= 21
Tr1 0 requires1 0, an impossibility for >0
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Lemma
LetBbe an efficient basis and xjbe an efficient nonbasic variable.Then any efficient pivot fromBleads to an adjacent efficient basis
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B.
Proof.
xjefficient entering variable at basis B
there is Rp
> with
T
R 0 and
T
r
j
= 0 xjis nonbasic variable with reduced cost 0 in LP()
Reduced costs of LP() do not change after a pivot with xjentering
Let Bbe the resulting basis with feasible pivot and xj entering
Because TR 0 and Trj = 0 at B, Bis an optimal basisfor LP() and therefore an adjacent efficient basis
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Lemma
LetBbe an efficient basis and xjbe an efficient nonbasic variable.Then any efficient pivot fromBleads to an adjacent efficient basis
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B.
Proof.
xjefficient entering variable at basis B
there is Rp
> with
T
R 0 and
T
r
j
= 0 xjis nonbasic variable with reduced cost 0 in LP()
Reduced costs of LP() do not change after a pivot with xjentering
Let Bbe the resulting basis with feasible pivot and xj entering
Because TR 0 and Trj = 0 at B, Bis an optimal basisfor LP() and therefore an adjacent efficient basis
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Lemma
LetBbe an efficient basis and xjbe an efficient nonbasic variable.Then any efficient pivot fromBleads to an adjacent efficient basis
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B.
Proof.
xjefficient entering variable at basis B
there is Rp
> with
T
R 0 and
T
r
j
= 0 xjis nonbasic variable with reduced cost 0 in LP()
Reduced costs of LP() do not change after a pivot with xjentering
Let Bbe the resulting basis with feasible pivot and
xj entering
Because TR 0 and Trj = 0 at B, Bis an optimal basisfor LP() and therefore an adjacent efficient basis
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Lemma
LetBbe an efficient basis and xjbe an efficient nonbasic variable.Then any efficient pivot fromBleads to an adjacent efficient basis
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B.
Proof.
xjefficient entering variable at basis B
there is Rp
> with
T
R
0 and
T
r
j
= 0 xjis nonbasic variable with reduced cost 0 in LP()
Reduced costs of LP() do not change after a pivot with xjentering
Let
Bbe the resulting basis with feasible pivot andxj entering
Because TR 0 and Trj = 0 at B, Bis an optimal basisfor LP() and therefore an adjacent efficient basis
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Lemma
LetBbe an efficient basis and xjbe an efficient nonbasic variable.Then any efficient pivot fromBleads to an adjacent efficient basis
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B.
Proof.
xjefficient entering variable at basis B
there is Rp
> with
T
R
0 and
T
r
j
= 0 xjis nonbasic variable with reduced cost 0 in LP()
Reduced costs of LP() do not change after a pivot with xjentering
Let Bbe the resulting basis with feasible pivot and xj
entering
Because TR 0 and Trj = 0 at B, Bis an optimal basisfor LP() and therefore an adjacent efficient basis
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Lemma
LetBbe an efficient basis and xjbe an efficient nonbasic variable.Then any efficient pivot fromBleads to an adjacent efficient basis
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B.
Proof.
xjefficient entering variable at basis B
there is Rp
> with
TR0 and
Trj
= 0 xjis nonbasic variable with reduced cost 0 in LP()
Reduced costs of LP() do not change after a pivot with xjentering
Let Bbe the resulting basis with feasible pivot and xj
entering
Because TR 0 and Trj = 0 at B, Bis an optimal basisfor LP() and therefore an adjacent efficient basis
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Lemma
LetBbe an efficient basis and xjbe an efficient nonbasic variable.Then any efficient pivot fromBleads to an adjacent efficient basis
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B.
Proof.
xjefficient entering variable at basis B
there is Rp
> with
TR0 and
Trj
= 0 xjis nonbasic variable with reduced cost 0 in LP()
Reduced costs of LP() do not change after a pivot with xjentering
Let Bbe the resulting basis with feasible pivot and xj
entering
Because TR 0 and Trj = 0 at B, Bis an optimal basisfor LP() and therefore an adjacent efficient basis
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
How to identify efficient nonbasic variables?
Th
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TheoremLetBbe an efficient basis and let xj be a nonbasic variable.Variable xj is an efficient nonbasic variable if and only if the LP
max etvsubject to Rz rj+Iv = 0
z, , v 0(12)
has an optimal value of 0.
(12) is always feasible with (z, , v) = 0
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
How to identify efficient nonbasic variables?
Th
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TheoremLetBbe an efficient basis and let xj be a nonbasic variable.Variable xj is an efficient nonbasic variable if and only if the LP
max etv
subject to Rz rj+Iv = 0z, , v 0
(12)
has an optimal value of 0.
(12) is always feasible with (z, , v) = 0
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Proof.
By definition xjis an efficient nonbasic variable if the LP
min 0T = 0T
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subject to RT 0(rj)T = 0
I e
(13)
has an optimal objective value of 0, i.e. if it is feasible(13) is equivalent to
min 0T = 0subject to RT 0
(rj
)T
0I e
(14)
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Proof.
By definition xjis an efficient nonbasic variable if the LP
min 0T = 0T
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subject to RT 0(rj)T = 0
I e
(13)
has an optimal objective value of 0, i.e. if it is feasible(13) is equivalent to
min 0T = 0subject to RT 0
(rj
)T
0I e
(14)
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Proof
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Proof.The dual of (14) is
max eTvsubject to Rz rj+Iv = 0
z, , v 0.
(15)
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Need to show: ALL efficient bases can be reached by efficient
pivots
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pivots
Definition
Two efficient bases Band Bare calledconnectedif one can beobtained from the other by performing only efficient pivots.
Theorem
All efficient bases are connected.
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Need to show: ALL efficient bases can be reached by efficient
pivots
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pivots
Definition
Two efficient bases Band Bare calledconnectedif one can beobtained from the other by performing only efficient pivots.
Theorem
All efficient bases are connected.
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Need to show: ALL efficient bases can be reached by efficient
pivots
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pivots
Definition
Two efficient bases Band Bare calledconnectedif one can beobtained from the other by performing only efficient pivots.
Theorem
All efficient bases are connected.
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Proof.
Band Btwo efficient bases
, Rp> such that Band Bare optimal bases for LP() and
LP()
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LP()Parametric LP ( [0, 1]) with objective function
c() = TC+ (1 )TC (16)
Assume Bis first basis (for = 1)
After several pivots get an optimal basis Bfor LP()
Since = + (1 ) Rp> for all [0, 1] all bases areoptimal for LP() for some Rp>, i.e. efficient
IfB=B, doneOtherwise obtain B from Bby efficient pivots: they arealternative optima for LP()
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Proof.
Band Btwo efficient bases
, Rp> such that Band Bare optimal bases for LP() and
LP()
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LP()Parametric LP ( [0, 1]) with objective function
c() = TC+ (1 )TC (16)
Assume Bis first basis (for = 1)
After several pivots get an optimal basis Bfor LP()
Since = + (1 ) Rp> for all [0, 1] all bases areoptimal for LP() for some Rp>, i.e. efficient
IfB=B, doneOtherwise obtain B from Bby efficient pivots: they arealternative optima for LP()
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Proof.
Band Btwo efficient bases
, Rp> such that Band Bare optimal bases for LP() and
LP()
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LP()Parametric LP ( [0, 1]) with objective function
c() = TC+ (1 )TC (16)
Assume Bis first basis (for = 1)
After several pivots get an optimal basis Bfor LP()
Since = + (1 ) Rp> for all [0, 1] all bases areoptimal for LP() for some Rp>, i.e. efficient
IfB=B, doneOtherwise obtain B from Bby efficient pivots: they arealternative optima for LP()
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Proof.
Band Btwo efficient bases
, Rp> such that Band Bare optimal bases for LP() and
LP()
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LP()Parametric LP ( [0, 1]) with objective function
c() = TC+ (1 )TC (16)
Assume Bis first basis (for = 1)
After several pivots get an optimal basis Bfor LP()
Since = + (1 ) Rp> for all [0, 1] all bases areoptimal for LP() for some Rp>, i.e. efficient
IfB=B, doneOtherwise obtain B from Bby efficient pivots: they arealternative optima for LP()
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Proof.
Band Btwo efficient bases
, Rp> such that Band Bare optimal bases for LP() and
LP()
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LP()Parametric LP ( [0, 1]) with objective function
c() = TC+ (1 )TC (16)
Assume Bis first basis (for = 1)
After several pivots get an optimal basis Bfor LP()
Since = + (1 ) Rp> for all [0, 1] all bases areoptimal for LP() for some Rp>, i.e. efficient
IfB=B, doneOtherwise obtain B from Bby efficient pivots: they arealternative optima for LP()
Matthias Ehrgott MOLP I
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Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Proof.
Band Btwo efficient bases
, Rp> such that Band Bare optimal bases for LP() and
LP()
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( )Parametric LP ( [0, 1]) with objective function
c() = TC+ (1 )TC (16)
Assume Bis first basis (for = 1)After several pivots get an optimal basis Bfor LP()
Since = + (1 ) Rp> for all [0, 1] all bases areoptimal for LP() for some Rp>, i.e. efficient
IfB=B, doneOtherwise obtain B from Bby efficient pivots: they arealternative optima for LP()
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Proof.
Band Btwo efficient bases
, Rp> such that Band Bare optimal bases for LP() and
LP()
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( )Parametric LP ( [0, 1]) with objective function
c() = TC+ (1 )TC (16)
Assume Bis first basis (for = 1)After several pivots get an optimal basis Bfor LP()
Since = + (1 ) Rp> for all [0, 1] all bases areoptimal for LP() for some Rp>, i.e. efficient
IfB=B, doneOtherwise obtain B from Bby efficient pivots: they arealternative optima for LP()
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
3 casesX =, infeasibilityX = butXE=, no efficient solutionsX =, XE=
result in three phase multiobjective Simplex algorithmS { T A I b }
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p j p gPhase I: Solve min{eTz :Ax+Iz=b, x 0, z 0}If optimal value is nonzero, X =Otherwise find bfs ofAx=b, x 0 from optimal solution
Phase II: Find efficient bfs by solving appropriate LP()Note: LP() can be unbounded even ifXE=Solve min{uTb+wTCx0 :uTA+wTC 0, w e}If unbounded then XE =Otherwise find optimal wand solve
min{w Cx :Ax=b, x 0}Optimal bfs x1 exists and is efficient bfs for MOLPPhase III: Starting from x1 find all efficient bfs by efficientpivots, even with negative pivot elements
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
3 casesX =, infeasibilityX = butXE=, no efficient solutionsX =, XE=
result in three phase multiobjective Simplex algorithmPh I S l i { T A I b 0 0}
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p j p gPhase I: Solve min{eTz :Ax+Iz=b, x 0, z 0}If optimal value is nonzero, X =Otherwise find bfs ofAx=b, x 0 from optimal solution
Phase II: Find efficient bfs by solving appropriate LP()
Note: LP() can be unbounded even ifXE=Solve min{uTb+wTCx0 :uTA+wTC 0, w e}If unbounded then XE =Otherwise find optimal wand solve
min{w Cx :Ax=b, x 0}Optimal bfs x1 exists and is efficient bfs for MOLPPhase III: Starting from x1 find all efficient bfs by efficientpivots, even with negative pivot elements
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
3 casesX =, infeasibilityX = butXE=, no efficient solutionsX =, XE=
result in three phase multiobjective Simplex algorithmPh I S l i { T A + I b 0 0}
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Phase I: Solve min{eTz :Ax+Iz=b, x 0, z 0}If optimal value is nonzero, X =Otherwise find bfs ofAx=b, x 0 from optimal solution
Phase II: Find efficient bfs by solving appropriate LP()
Note: LP() can be unbounded even ifXE=Solve min{uTb+wTCx0 :uTA+wTC 0, w e}If unbounded then XE =Otherwise find optimal wand solve
min{w Cx :Ax=b, x 0}Optimal bfs x1 exists and is efficient bfs for MOLPPhase III: Starting from x1 find all efficient bfs by efficientpivots, even with negative pivot elements
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
3 casesX =, infeasibilityX = butXE=, no efficient solutionsX =, XE=
result in three phase multiobjective Simplex algorithmPh I S l i { T A + I b 0 0}
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Phase I: Solve min{eTz :Ax+Iz=b, x 0, z 0}If optimal value is nonzero, X =Otherwise find bfs ofAx=b, x 0 from optimal solution
Phase II: Find efficient bfs by solving appropriate LP()
Note: LP() can be unbounded even ifXE=Solve min{uTb+wTCx0 :uTA+wTC 0, w e}If unbounded then XE =Otherwise find optimal wand solve
min{w Cx :Ax=b, x 0}Optimal bfs x1 exists and is efficient bfs for MOLPPhase III: Starting from x1 find all efficient bfs by efficientpivots, even with negative pivot elements
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Algorithm (Multicriteria Simplex Algorithm.)
Input: Data A, b, C of an MOLP.
Initialization: SetL1:=, L2:=.
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Phase I: Solve the LPmin{eTz :Ax+Iz=b, x, z 0}.If the optimal value of this LP is nonzero, STOP,X =.Otherwise let x0 be a basic feasible solution of the MOLP.
Phase II: Solve the LPmin{uTb+wTCx0 :uTA+wTC 0, w e}.If the problem is infeasible, STOP,XE =.Otherwise let(u, w) be an optimal solution.Find an optimal basisBof the LPmin{wTCx :Ax=b, x 0}.
L1:={B},L2:=.
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Algorithm
Phase III:
WhileL1 =
ChooseB inL1, setL1 :=L1\ {B}, L2 :=L2 {B}.ComputeA, b, and R according toB.
N N
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EN :=N.For all j N.
Solve the LPmax{eTv :Ry rj+Iv= 0; y, , v 0}.If this LP is unboundedEN :=EN \ {j}.
End forFor all j EN.
For all i B.
IfB = (B \ {i}) {j} is feasible andB L1 L2then L1 :=L1 B
.
End for.End for.
End while.
Output: L2.
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Example
There can be exponentially many efficient bfs
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min xi i= 1, . . . , nmin xi i= 1, . . . , n
subject to xi 1 i= 1, . . . , nxi 1 i= 1, . . . , n.
n variables, m= 2n constraints, p= 2n objective functions
all 2n extreme points of the feasible set are efficient
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Examples
Example
There can be exponentially many efficient bfs
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min xi i= 1, . . . , nmin xi i= 1, . . . , n
subject to xi 1 i= 1, . . . , nxi 1 i= 1, . . . , n.
n variables, m= 2n constraints, p= 2n objective functions
all 2n extreme points of the feasible set are efficient
Matthias Ehrgott MOLP I
Multiobjective Linear ProgrammingBiobjective LPs and Parametric Simplex
Multiobjective Simplex Method
A Multiobjective Simplex AlgorithmMultiobjective Simplex: Example