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Nom original: usthb17.pdfTitre: Stochastic Discrete Multiple Objective

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Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

Stochastic Discrete Multiple Objective

30th International Conference of Jangjeon Mathematical Society
Pur and Applied Mathematics

ICJMS

USTHB-ALGERIA 2017

Operations Research Department
Faculty of Mathematics - USTHB - ALGERIA

July 19, 2017
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

Plan
1

Introduction
The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

2

Theoretical Developments

3

Principal Components of the Technique

4

The algorithm

5

Illustration

6

Conclusion
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Plan
1

Introduction
The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

2

Theoretical Developments

3

Principal Components of the Technique

4

The algorithm

5

Illustration

6

Conclusion
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

A N ew A lgorithme for Determinating the whole S et of I nteger E fficient S tochastic S olutioins

I T he problem consists of searching for the solution
Pareto-set in stochastic environment, when, p criteria are
to be optimized simultaneously.
Consider (Ω, Ξ, P) a probability space, with discrete
distribution.
Given a Multiple Objective Integer Stochastic Linear
Programming problem (MOISLP)

“ min ”Fk



s.t.
(PSC )




= ck (ξ)x; k = 1, · · · , p
Ax = b;
T(ξ)x = h(ξ);
x∈N

• A, b matrices: (m × n), (m × 1): decision constraint
• ck , T, h are random matrices : (K × n), (` × n) and (` × 1)
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

A N ew A lgorithme for Determinating the whole S et of I nteger E fficient S tochastic S olutioins

I T he problem consists of searching for the solution
Pareto-set in stochastic environment, when, p criteria are
to be optimized simultaneously.
Consider (Ω, Ξ, P) a probability space, with discrete
distribution.
Given a Multiple Objective Integer Stochastic Linear
Programming problem (MOISLP)

“ min ”Fk



s.t.
(PSC )




= ck (ξ)x; k = 1, · · · , p
Ax = b;
T(ξ)x = h(ξ);
x∈N

• A, b matrices: (m × n), (m × 1): decision constraint
• ck , T, h are random matrices : (K × n), (` × n) and (` × 1)
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

A N ew A lgorithme for Determinating the whole S et of I nteger E fficient S tochastic S olutioins

I T he problem consists of searching for the solution
Pareto-set in stochastic environment, when, p criteria are
to be optimized simultaneously.
Consider (Ω, Ξ, P) a probability space, with discrete
distribution.
Given a Multiple Objective Integer Stochastic Linear
Programming problem (MOISLP)

“ min ”Fk



s.t.
(PSC )




= ck (ξ)x; k = 1, · · · , p
Ax = b;
T(ξ)x = h(ξ);
x∈N

• A, b matrices: (m × n), (m × 1): decision constraint
• ck , T, h are random matrices : (K × n), (` × n) and (` × 1)
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

A N ew A lgorithme for Determinating the whole S et of I nteger E fficient S tochastic S olutioins

I T he problem consists of searching for the solution
Pareto-set in stochastic environment, when, p criteria are
to be optimized simultaneously.
Consider (Ω, Ξ, P) a probability space, with discrete
distribution.
Given a Multiple Objective Integer Stochastic Linear
Programming problem (MOISLP)

“ min ”Fk



s.t.
(PSC )




= ck (ξ)x; k = 1, · · · , p
Ax = b;
T(ξ)x = h(ξ);
x∈N

• A, b matrices: (m × n), (m × 1): decision constraint
• ck , T, h are random matrices : (K × n), (` × n) and (` × 1)
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

A N ew A lgorithme for Determinating the whole S et of I nteger E fficient S tochastic S olutioins

I T he problem consists of searching for the solution
Pareto-set in stochastic environment, when, p criteria are
to be optimized simultaneously.
Consider (Ω, Ξ, P) a probability space, with discrete
distribution.
Given a Multiple Objective Integer Stochastic Linear
Programming problem (MOISLP)

“ min ”Fk



s.t.
(PSC )




= ck (ξ)x; k = 1, · · · , p
Ax = b;
T(ξ)x = h(ξ);
x∈N

• A, b matrices: (m × n), (m × 1): decision constraint
• ck , T, h are random matrices : (K × n), (` × n) and (` × 1)
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

A N ew A lgorithme for Determinating the whole S et of I nteger E fficient S tochastic S olutioins

I T he problem consists of searching for the solution
Pareto-set in stochastic environment, when, p criteria are
to be optimized simultaneously.
Consider (Ω, Ξ, P) a probability space, with discrete
distribution.
Given a Multiple Objective Integer Stochastic Linear
Programming problem (MOISLP)

“ min ”Fk



s.t.
(PSC )




= ck (ξ)x; k = 1, · · · , p
Ax = b;
T(ξ)x = h(ξ);
x∈N

• A, b matrices: (m × n), (m × 1): decision constraint
• ck , T, h are random matrices : (K × n), (` × n) and (` × 1)
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

A N ew A lgorithme for Determinating the whole S et of I nteger E fficient S tochastic S olutioins

I T he problem consists of searching for the solution
Pareto-set in stochastic environment, when, p criteria are
to be optimized simultaneously.
Consider (Ω, Ξ, P) a probability space, with discrete
distribution.
Given a Multiple Objective Integer Stochastic Linear
Programming problem (MOISLP)

“ min ”Fk



s.t.
(PSC )




= ck (ξ)x; k = 1, · · · , p
Ax = b;
T(ξ)x = h(ξ);
x∈N

• A, b matrices: (m × n), (m × 1): decision constraint
• ck , T, h are random matrices : (K × n), (` × n) and (` × 1)
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

A N ew A lgorithme for Determinating the whole S et of I nteger E fficient S tochastic S olutioins

I T he problem consists of searching for the solution
Pareto-set in stochastic environment, when, p criteria are
to be optimized simultaneously.
Consider (Ω, Ξ, P) a probability space, with discrete
distribution.
Given a Multiple Objective Integer Stochastic Linear
Programming problem (MOISLP)

“ min ”Fk



s.t.
(PSC )




= ck (ξ)x; k = 1, · · · , p
Ax = b;
T(ξ)x = h(ξ);
x∈N

• A, b matrices: (m × n), (m × 1): decision constraint
• ck , T, h are random matrices : (K × n), (` × n) and (` × 1)
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Plan
1

Introduction
The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

2

Theoretical Developments

3

Principal Components of the Technique

4

The algorithm

5

Illustration

6

Conclusion
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

M otivations : Why optimizing over integer efficient set?

The problem is well known when the parameters are deterministic; unfortunately, it is rare in
stochastic environment/
As far as we know, the only method that solves
such problem is published in (EJOR 2006) .
This does not provide all integer efficient solutions.
Some real world problems have to be modeled
as (PSC ).

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

M otivations : Why optimizing over integer efficient set?

The problem is well known when the parameters are deterministic; unfortunately, it is rare in
stochastic environment/
As far as we know, the only method that solves
such problem is published in (EJOR 2006) .
This does not provide all integer efficient solutions.
Some real world problems have to be modeled
as (PSC ).

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

M otivations : Why optimizing over integer efficient set?

The problem is well known when the parameters are deterministic; unfortunately, it is rare in
stochastic environment/
As far as we know, the only method that solves
such problem is published in (EJOR 2006) .
This does not provide all integer efficient solutions.
Some real world problems have to be modeled
as (PSC ).

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

M otivations : Why optimizing over integer efficient set?

The problem is well known when the parameters are deterministic; unfortunately, it is rare in
stochastic environment/
As far as we know, the only method that solves
such problem is published in (EJOR 2006) .
This does not provide all integer efficient solutions.
Some real world problems have to be modeled
as (PSC ).

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Plan
1

Introduction
The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

2

Theoretical Developments

3

Principal Components of the Technique

4

The algorithm

5

Illustration

6

Conclusion
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Plan
1

Introduction
The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

2

Theoretical Developments

3

Principal Components of the Technique

4

The algorithm

5

Illustration

6

Conclusion
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Deterministic E quivalent Problem
H ypothesis
ä We suppose some probability space (Ω, Ξ, P) that defines
the probabilistic aspect of random parameters in our
problem.
ä The decision on x has to be taken before the realization of
the random variables is known “here and now".
ä We suppose a fixed recourse matrix W and the recourse
costs is taken as linear defined by q0 y

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Deterministic E quivalent Problem
H ypothesis
ä We suppose some probability space (Ω, Ξ, P) that defines
the probabilistic aspect of random parameters in our
problem.
ä The decision on x has to be taken before the realization of
the random variables is known “here and now".
ä We suppose a fixed recourse matrix W and the recourse
costs is taken as linear defined by q0 y

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Deterministic E quivalent Problem
H ypothesis
ä We suppose some probability space (Ω, Ξ, P) that defines
the probabilistic aspect of random parameters in our
problem.
ä The decision on x has to be taken before the realization of
the random variables is known “here and now".
ä We suppose a fixed recourse matrix W and the recourse
costs is taken as linear defined by q0 y

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Deterministic E quivalent Problem

ä Given the penalties qr = q(ξ r ) to the constraint violations
are given
ek (x) = E(Fk (x)) =
ä F

R
X

pr Fk,r (x) =

r=1

ä A recourse function

Q(x, ξ r )

R
X

pr ck (ξ r )x

r=1

is added to each criterion Fkr ,

ä the corresponding penalty is given by
Q(x, ξ r ) = min {(qr )t f |W(ξ r )f = h(ξ r ) − T(ξ r )x; f ≥ 0}
f

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Deterministic E quivalent Problem

ä Given the penalties qr = q(ξ r ) to the constraint violations
are given
ek (x) = E(Fk (x)) =
ä F

R
X

pr Fk,r (x) =

r=1

ä A recourse function

Q(x, ξ r )

R
X

pr ck (ξ r )x

r=1

is added to each criterion Fkr ,

ä the corresponding penalty is given by
Q(x, ξ r ) = min {(qr )t f |W(ξ r )f = h(ξ r ) − T(ξ r )x; f ≥ 0}
f

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Deterministic E quivalent Problem

ä Given the penalties qr = q(ξ r ) to the constraint violations
are given
ek (x) = E(Fk (x)) =
ä F

R
X

pr Fk,r (x) =

r=1

ä A recourse function

Q(x, ξ r )

R
X

pr ck (ξ r )x

r=1

is added to each criterion Fkr ,

ä the corresponding penalty is given by
Q(x, ξ r ) = min {(qr )t f |W(ξ r )f = h(ξ r ) − T(ξ r )x; f ≥ 0}
f

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Deterministic E quivalent Problem

ä Given the penalties qr = q(ξ r ) to the constraint violations
are given
ek (x) = E(Fk (x)) =
ä F

R
X

pr Fk,r (x) =

r=1

ä A recourse function

Q(x, ξ r )

R
X

pr ck (ξ r )x

r=1

is added to each criterion Fkr ,

ä the corresponding penalty is given by
Q(x, ξ r ) = min {(qr )t f |W(ξ r )f = h(ξ r ) − T(ξ r )x; f ≥ 0}
f

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Deterministic E quivalent Problem

ä The deterministic multiple objective integer linear
programming problem


R
X


e


min

F
=
E
(F
)
=
pr crk x

k
k
e
r=1
P

s.t.
Ax = b



x ∈ Nn

k = 1, · · · , p

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Deterministic E quivalent Problem

• The parametric problem is given by




p
 min (E(F)) = X λ E(F )

k
k
λ

P
k=1


s.t. x ∈ D = {x ∈ Rn |Ax = b, x ∈ N}

λ = (λ)k=1,··· ,p : is an integer parameter vector.

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Deterministic E quivalent Problem

• The parametric problem is given by




p
 min (E(F)) = X λ E(F )

k
k
λ

P
k=1


s.t. x ∈ D = {x ∈ Rn |Ax = b, x ∈ N}

λ = (λ)k=1,··· ,p : is an integer parameter vector.

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Deterministic E quivalent Problem

ä The parametric deterministic problem including penalty
variable

p
 min (E(F)) = X λ E(F ) + θ

k
k
λ

P
k=1


s.t.
x∈D

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

Plan
1

Introduction
The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

2

Theoretical Developments

3

Principal Components of the Technique

4

The algorithm

5

Illustration

6

Conclusion
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

ä First Stage
Feasibility and optimality tests as was introduced in
First Set
L-shapped technique
ä Second Stage
Basic definitions and results in multiple objective integer linear programming theory
Second Set

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

ä First Stage
Feasibility and optimality tests as was introduced in
First Set
L-shapped technique
ä Second Stage
Basic definitions and results in multiple objective integer linear programming theory
Second Set

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

A dmissibility T est

• To check out for feasibility of the second stage-problems,
we solve the following problem :
For all possible realizations of ξ


0 h(ξ r ) − T(ξ r )x0
max
u



s.t. ut W ≤ 0
(FDual )
kuk1 ≤ 1



u ∈ R`

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

A dmissibility T est



• In case where utr h (ξ r ) − T (ξ r ) x0 > 0 for some
ξ r ; r ∈ {1, · · · , R} and ur is an optimal solution of the
previous problem ;

+


Admissubility cut

utr [h (ξ r ) − T (ξ r ) x] ≤ 0

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

A dmissibility T est



• In case where utr h (ξ r ) − T (ξ r ) x0 > 0 for some
ξ r ; r ∈ {1, · · · , R} and ur is an optimal solution of the
previous problem ;

+


Admissubility cut

utr [h (ξ r ) − T (ξ r ) x] ≤ 0

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

A dmissibility T est

• To test the optimality of a given solution, we solve the
following problem :
For all possible realizations of ξ

 max π 0 (h(ξ r ) − T(ξ r )x)
s.t. π t W ≤ (qr )0
(Ftest )

π ∈ R`

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

A dmissibility T est

• Q(x0 ) =

R
X

p(ξ r )(πr )t [h(ξ r ) − T(ξ r )x0 ]

r=1

• if Q(x0 ) ≤ θ then x0 is optimal solution
• else

+
θ≥

optimality cut

R
X

pi πit (hi − Ti x)

i=1
Return
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

A dmissibility T est

• Q(x0 ) =

R
X

p(ξ r )(πr )t [h(ξ r ) − T(ξ r )x0 ]

r=1

• if Q(x0 ) ≤ θ then x0 is optimal solution
• else

+
θ≥

optimality cut

R
X

pi πit (hi − Ti x)

i=1
Return
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

A dmissibility T est

• Q(x0 ) =

R
X

p(ξ r )(πr )t [h(ξ r ) − T(ξ r )x0 ]

r=1

• if Q(x0 ) ≤ θ then x0 is optimal solution
• else

+
θ≥

optimality cut

R
X

pi πit (hi − Ti x)

i=1
Return
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

Reduction of the Domain
ä At iteration `, using Sylva and Crema’s idea (EJOR 2007)
ä The feasible set D is reduced by
eliminating all dominated solutions by Cˆx`

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

Reduction of the Domain
ä At iteration `, using Sylva and Crema’s idea (EJOR 2007)
ä The feasible set D is reduced by
eliminating all dominated solutions by Cˆx`

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

Reduction of the Domain
ä At iteration `, using Sylva and Crema’s idea (EJOR 2007)
ä The feasible set D is reduced by
eliminating all dominated solutions by Cˆx`

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

let x1 , x2 , ..., x` be efficient
solutions to problem (P) and
∆k = x ∈ Fn | ecx ≥ ecxk (k ∈ 1, · · · , `).
if x? is an optimal solution to
(
!)
`
G
`
0
e−
(Pλ ) = min λ ecx | x ∈ D
∆k
k=1

for some λ ∈ Rp ,λ > 0, then x? is an efficient solution to
problem (PSc )

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

Plan
1

Introduction
The Problem
Motivations
Passage to deterministic equivalent problem of MOSILP

2

Theoretical Developments

3

Principal Components of the Technique

4

The algorithm

5

Illustration

6

Conclusion
Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

T he main Techniques

eλ )
ä Resolution of the parametric problem (P
ä Two Tests for the obtained solution x∗
• Feasibility

• Optimality

ä Updating the list of efficient solutions
ä Reducing the admissible region

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

T he main Techniques

eλ )
ä Resolution of the parametric problem (P
ä Two Tests for the obtained solution x∗
• Feasibility

• Optimality

ä Updating the list of efficient solutions
ä Reducing the admissible region

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

T he main Techniques

eλ )
ä Resolution of the parametric problem (P
ä Two Tests for the obtained solution x∗
• Feasibility

• Optimality

ä Updating the list of efficient solutions
ä Reducing the admissible region

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

T he main Techniques

eλ )
ä Resolution of the parametric problem (P
ä Two Tests for the obtained solution x∗
• Feasibility

• Optimality

ä Updating the list of efficient solutions
ä Reducing the admissible region

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

T he main Techniques

eλ )
ä Resolution of the parametric problem (P
ä Two Tests for the obtained solution x∗
• Feasibility

• Optimality

ä Updating the list of efficient solutions
ä Reducing the admissible region

Djamal CHAABANE & Fatma MEBREK

Introduction
Theoretical Developments
Principal Components of the Technique
The algorithm
Illustration
Conclusion

T he main Techniques

eλ )
ä Resolution of the parametric problem (P
ä Two Tests for the obtained solution x∗
• Feasibility

• Optimality

ä Updating the list of efficient solutions
ä Reducing the admissible region

Djamal CHAABANE & Fatma MEBREK


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