MCDAandMOO,HansurLesse,September17–212007 Lecture2:MultiobjectiveLinearProgrammingMatthiasEhrgott InternationalDoctoralSchoolAlgorithmicDecisionTheory:MCDAandMOO

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 d’Informatique de Nantes Atlantique, CNRS, Universit´e de Nantes,

France

MCDA and MOO, Han sur Lesse, September 17 – 21 2007

Overview

1 Multiobjective Linear Programming Formulation and Example

Solving MOLPs by Weighted Sums

2 Biobjective LPs and Parametric Simplex The Parametric Simplex Algorithm Biobjective Linear Programmes: Example

3 Multiobjective Simplex Method A Multiobjective Simplex Algorithm Multiobjective Simplex: Examples

Overview

1 Multiobjective Linear Programming Formulation and Example

Solving MOLPs by Weighted Sums

2 Biobjective LPs and Parametric Simplex The Parametric Simplex Algorithm Biobjective Linear Programmes: Example

3 Multiobjective Simplex Method A Multiobjective Simplex Algorithm Multiobjective Simplex: Examples

Variables x∈Rⁿ

Objective function Cx whereC ∈R^p×n

Constraints Ax =b whereA∈R^m×n andb ∈R^m

min{Cx :Ax =b,x=0} (1)

X ={x ∈Rⁿ:Ax =b,x =0}

is thefeasible set in decision space

Y ={Cx :x ∈X} is thefeasible set in objective space

Variables x∈Rⁿ

Objective function Cx whereC ∈R^p×n

Constraints Ax =b whereA∈R^m×n andb ∈R^m

min{Cx :Ax =b,x=0} (1)

X ={x ∈Rⁿ:Ax =b,x =0}

is thefeasible set in decision space

Y ={Cx :x ∈X} is thefeasible set in objective space

Variables x∈Rⁿ

Objective function Cx whereC ∈R^p×n

Constraints Ax =b whereA∈R^m×n andb ∈R^m

min{Cx :Ax =b,x=0} (1)

X ={x ∈Rⁿ:Ax =b,x =0}

is thefeasible set in decision space

Y ={Cx :x ∈X} is thefeasible set in objective space

Variables x∈Rⁿ

Objective function Cx whereC ∈R^p×n

Constraints Ax =b whereA∈R^m×n andb ∈R^m

min{Cx :Ax =b,x=0} (1)

X ={x ∈Rⁿ:Ax =b,x =0}

is thefeasible set in decision space

Y ={Cx :x ∈X} is thefeasible set in objective space

Example

min

3x1+x2

−x₁−2x₂

subject tox2 5 3 3x1−x2 5 6

x = 0

C =

3 1

−1 −2

A=

0 1 1 0

3 −1 0 1

b =

3

6

Example

min

3x1+x2

−x₁−2x₂

subject tox2 5 3 3x1−x2 5 6

x = 0

C =

3 1

−1 −2

A=

0 1 1 0 3 −1 0 1

b =

3 6

0 1 2 3 4 0

1 2 3 4

X x ¹

x ² x ³

x ⁴

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 Cx

Cx

Cx

Cx

Definition

Let ˆx∈X be a feasible solution of the MOLP (1) and let ˆy =Cˆx.

ˆ

x is called weakly efficientif there is nox ∈X such that Cx <Cˆx; ˆy =Cˆx is called weakly nondominated.

ˆ

x is called efficientif there is nox ∈X such that Cx ≤Cx;ˆ ˆ

y =Cˆx is called nondominated.

ˆ

x is calledproperly efficientif it is efficient and if there exists a real numberM >0 such that for alli andx with c_i^Tx <c_i^Tˆx there is an index j andM >0 such thatc_j^Tx >c_j^Tˆx and

c_i^Txˆ−c_i^Tx c_j^Tx−c_j^Txˆ ≤M.

Definition

Let ˆx∈X be a feasible solution of the MOLP (1) and let ˆy =Cˆx.

ˆ

x is called weakly efficientif there is nox ∈X such that Cx <Cˆx; ˆy =Cˆx is called weakly nondominated.

ˆ

x is called efficientif there is nox ∈X such that Cx ≤Cx;ˆ ˆ

y =Cˆx is called nondominated.

ˆ

x is calledproperly efficientif it is efficient and if there exists a real numberM >0 such that for alli andx with c_i^Tx <c_i^Tˆx there is an index j andM >0 such thatc_j^Tx >c_j^Tˆx and

c_i^Txˆ−c_i^Tx c_j^Tx−c_j^Txˆ ≤M.

Definition

Let ˆx∈X be a feasible solution of the MOLP (1) and let ˆy =Cˆx.

ˆ

x is called weakly efficientif there is nox ∈X such that Cx <Cˆx; ˆy =Cˆx is called weakly nondominated.

ˆ

x is called efficientif there is nox ∈X such that Cx ≤Cx;ˆ ˆ

y =Cˆx is called nondominated.

ˆ

x is calledproperly efficientif it is efficient and if there exists a real numberM >0 such that for alli andx with c_i^Tx <c_i^Tˆx there is an index j andM >0 such thatc_j^Tx >c_j^Tˆx and

c_i^Txˆ−c_i^Tx c_j^Tx−c_j^Txˆ ≤M.

Definition

Let ˆx∈X be a feasible solution of the MOLP (1) and let ˆy =Cˆx.

ˆ

x is called weakly efficientif there is nox ∈X such that Cx <Cˆx; ˆy =Cˆx is called weakly nondominated.

ˆ

x is called efficientif there is nox ∈X such that Cx ≤Cx;ˆ ˆ

y =Cˆx is called nondominated.

ˆ

x is calledproperly efficientif it is efficient and if there exists a real numberM >0 such that for alli andx with c_i^Tx <c_i^Tˆx there is an index j andM >0 such thatc_j^Tx >c_j^Tˆx and

c_i^Txˆ−c_i^Tx c_j^Tx−c_j^Txˆ ≤M.

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 Cx

Cx

Cx

Cx

0 1 2 3 4 0

1 2 3 4

X x ¹

x ² x ³

x ⁴

Overview

1 Multiobjective Linear Programming Formulation and Example

Solving MOLPs by Weighted Sums

2 Biobjective LPs and Parametric Simplex The Parametric Simplex Algorithm Biobjective Linear Programmes: Example

3 Multiobjective Simplex Method A Multiobjective Simplex Algorithm Multiobjective Simplex: Examples

Let λ₁, . . . , λ_p =0 and consider

LP(λ) min

p

X

k=1

λ_kc_k^Tx = minλ^TCx

subject toAx = b

x = 0

with some vector λ≥0 (Why not λ= 0 or λ≤0?) LP(λ) is a linear programme that can be solved by the Simplex method

Ifλ >0then optimal solution of LP(λ) is properly efficient Ifλ≥0then optimal solution of LP(λ) is weakly efficient Converse also true, becauseY convex

Let λ₁, . . . , λ_p =0 and consider

LP(λ) min

p

X

k=1

λ_kc_k^Tx = minλ^TCx

subject toAx = b

x = 0

with some vector λ≥0 (Why not λ= 0 or λ≤0?) LP(λ) is a linear programme that can be solved by the Simplex method

Ifλ >0then optimal solution of LP(λ) is properly efficient Ifλ≥0then optimal solution of LP(λ) is weakly efficient Converse also true, becauseY convex

Let λ₁, . . . , λ_p =0 and consider

LP(λ) min

p

X

k=1

λ_kc_k^Tx = minλ^TCx

subject toAx = b

x = 0

with some vector λ≥0 (Why not λ= 0 or λ≤0?) LP(λ) is a linear programme that can be solved by the Simplex method

Ifλ >0then optimal solution of LP(λ) is properly efficient Ifλ≥0then optimal solution of LP(λ) is weakly efficient Converse also true, becauseY convex

Let λ₁, . . . , λ_p =0 and consider

LP(λ) min

p

X

k=1

λ_kc_k^Tx = minλ^TCx

subject toAx = b

x = 0

with some vector λ≥0 (Why not λ= 0 or λ≤0?) LP(λ) is a linear programme that can be solved by the Simplex method

Ifλ >0then optimal solution of LP(λ) is properly efficient Ifλ≥0then optimal solution of LP(λ) is weakly efficient Converse also true, becauseY convex

Let λ₁, . . . , λ_p =0 and consider

LP(λ) min

p

X

k=1

λ_kc_k^Tx = minλ^TCx

subject toAx = b

x = 0

with some vector λ≥0 (Why not λ= 0 or λ≤0?) LP(λ) is a linear programme that can be solved by the Simplex method

Ifλ >0then optimal solution of LP(λ) is properly efficient Ifλ≥0then optimal solution of LP(λ) is weakly efficient Converse also true, becauseY convex

Illustration in objective space

2 4 6 8 10 12 14

−10

−8

−6

−4

−2 0 2

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

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

Y=CX

•

•

•

• Cx¹

Cx²

Cx³ Cx⁴

y1

y2

{(λ¹)^TCx=α¹}

{(λ²)^TCx=α²}

{(λ³)^TCx=α³}

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λ¹ = (2,1), λ² = (1,3), λ³ = (1,1)