Efficiency for continuous facility location problems with attraction - - PDF document

Efficiency for continuous facility location problems

• A. Jourani

Efficiency for continuous facility location problems with attraction and repulsion∗ Abderrahim Jourani Universit´ e de Dijon Institut de Math´ ematiques de Bourgogne UMR 5584-CNRS jourani@u-bourgogne.fr Journ´ ees Franco-Chiliennes d’Optimisation Toulon, may 2008

∗ collaboration with C. Michelot and M. Ndiaye ∗ to appear in AOR ∗http ://math.u-bourgogne.fr/IMB/IMB.html

Universit´ e de Dijon France Page 1

Efficiency for continuous facility location problems

• A. Jourani
• (X, · ) normed linear space
• a+

i ∈ X, ω+ i > 0 ∀i = 1, · · · , m

• a−

i ∈ X, ω− i > 0 ∀i = 1, · · · , n

(P +) min

x

• ω+

i x − a+ i

(P +−)min

x

• ω+

i x − a+ i −

• ω−

i x − a− i

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Efficiency for continuous facility location problems

• A. Jourani

Mω set of solutions to (P +) ◮ X inner product space ⇒ Mω ⊂ co{a+

i : i = 1, · · · , p}

◮ dimX = 2 = ⇒ Mω ∩ co{a+

i : i = 1, · · · , p} = ∅

Can we replace co{a+

i : i = 1, · · · , p} ?

What happens in the case of (P +−) ?

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Efficiency for continuous facility location problems

• A. Jourani

Vector Optimization

F : Y → I Rp mapping, D ⊂ Y (P) min

y∈D F(y)

• y0 weakly efficient if

∄y ∈ D; F(y) − F(y0) ∈ intI Rp

−

• y0 strictly efficient if

∄y ∈ D, y = y0; F(y) − F(y0) ∈ I Rp

−

• y0 efficient if

∄y ∈ D; F(y) − F(y0) ∈ I Rp

−,

F(y) = F(y0)

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Efficiency for continuous facility location problems

• A. Jourani

Basic elements

• X euclidean
• Ω ⊂ X convex and closed
• A+ ⊂ X, A− ⊂ X compact, A+ ∩ A− = ∅
• x ∈ X

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Efficiency for continuous facility location problems

• A. Jourani

Weak efficiency : x ∈ WE(A+, A−, Ω) ∀y ∈ Ω,        ∃a+ ∈ A+, a+ − x ≤ a+ − y

• u

∃a− ∈ A−, a− − x ≥ a− − y Strict efficiency : x ∈ SE(A+, A−, Ω) ∀y ∈ Ω, y = x,    ∃a+ ∈ A+, a+ − x < a+ − y

• u ∃a− ∈ A−, a− − x > a− − y

Efficiency : x ∈ E(A+, A−, Ω) ∀y ∈ Ω, y = x,                              ∃a+ ∈ A+, a+ − x < a+ − y

• u

∃a− ∈ A−, a− − x > a− − y

• u

       ∀a+ ∈ A+, a+ − x ≤ a+ − y et ∀a− ∈ A−, a− − x ≥ a− − y

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Efficiency for continuous facility location problems

• A. Jourani

NSOC

◮ (Carrizosa-Plastria) A− = ∅. x weakly efficient ⇐ ⇒ 0 ∈ co[

• a+∈A+

∂( · −a+)(x)] + N(Ω, x) A+ = {a+

1 , · · · , a+ n }, A− = {a− 1 , · · · , a− m}

◮ If x is locally weakly efficient, then ∃λ+ ≥ 0, ∃λ− ≥ 0,

n

• i=1

λ+

i + m

• j=1

λ−

j = 1

[m

j=1 λ− j ∂( · −a− j )(x)]

[n

i=1 λ+ i ∂( · −a+ i )(x) + N(Ω, x)] = ∅

The last one is also sufficient provided that the norm and Ω are locally polyhedral.

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Efficiency for continuous facility location problems

• A. Jourani

Notations.

• Ω = X : SE(A+, A−), E(A+, A−), WE(A+, A−)
• SE(A+) = SE(A+, ∅)
• E(A+) = E(A+, ∅)
• WE(A+) = WE(A+, ∅)

Properties ◮ SE(A+) ⊂ SE(A+, A−), E(A+) ⊂ E(A+, A−), WE(A+) ⊂ WE(A+, A−) ◮ A+ ∩ Ω ⊂ SE(A+, A−, Ω) ⊂ E(A+, A−, Ω) ⊂ WE(A+, A−, Ω) ◮ SE(A+, A−) ∩ Ω ⊂ SE(A+, A−, Ω) E(A+, A−) ∩ Ω ⊂ E(A+, A−, Ω) WE(A+, A−) ∩ Ω ⊂ WE(A+, A−, Ω)

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Efficiency for continuous facility location problems

• A. Jourani

Weak efficiency

Theorem 1. x ∈ WE(A+, A−) ⇐ ⇒ co(A+) ∩ co({x} ∪ A−) = ∅. Corollary 1. WE(A+, A−) compact ⇐ ⇒ A− = ∅

• WE(A+, A−) = co(A+)

Proposition 1. co(A+) ∩ co(A−) = ∅ ⇐ ⇒ WE(A+, A−) = X

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Efficiency for continuous facility location problems

• A. Jourani

Strict efficiency

Theorem 2.

• co(A+) ∩ ri[co({x} ∪ A−)] = ∅ =

⇒ x ∈ SE(A+, A−)

• SE(A+, A−) = co(A+) + cl[cone[co(A+) − co(A−)]]

Corollary 2.

• SE(A+, A−) closed and convex
• co(A+) et co(A−) polyhedral :

SE(A+, A−) = co(A+) + cone[co(A+) − co(A−)] Corollary 3.

• co(A+) ∩ co(A−) = ∅ =

⇒ co(A−) ⊂ SE(A+, A−)

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Efficiency for continuous facility location problems

• A. Jourani

Theorem 2. x ∈ ri[SE(A+, A−)] ⇔ ri[co(A+)]∩ri[co({x}∪A−)] = ∅.

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Efficiency for continuous facility location problems

• A. Jourani

Efficiency

Proposition 2. ri[co(A+)] ∩ ri[co(A−)] = ∅ ⇐ ⇒ E(A+, A−) = X Theorem 3.

• A+ et A− not contained in the same hyperplan :

E(A+, A−) = SE(A+, A−)

• A+ et A− contained in the same hyperplan H :

ri[co(A+)] ∩ ri[co(A−)] = ∅ ⇐ ⇒ E(A+, A−) ⊂ H

• ri[co(A+)] ∩ ri[co(A−)] = ∅
• E(A+, A−) = SE(A+, A−) = X

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Efficiency for continuous facility location problems

• A. Jourani

Coincidence

Theorem 5. K = co(A+) + cl[cone[co(A+) − co(A−)]]

• E(A+, A−) = WE(A+, A−)
• u

E(A+, A−) = SE(A+, A−)

• co(A+) ∩ co(A−) = ∅

⇓ SE(A+, A−) = E(A+, A−) = WE(A+, A−) = K Corollary 4. E(A+, A−) et WE(A+, A−) closed and convex

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Efficiency for continuous facility location problems

• A. Jourani

Constrained efficiency

ProjΩWE(A+, A−) ⊂ WE(A+, A−, Ω)

Inner product spaces

(X, · ) linear normed space Theorem 6. dimX ≥ 3. i) X inner product space

• ii) ∀A+, A− ⊂ X with A+ ∩ A− = ∅, cardA+ < +∞,

cardA− < +∞, we have x ∈ WE(A+, A−) ⇐ ⇒ co(A+) ∩ co({x} ∪ A−) = ∅.

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Efficiency for continuous facility location problems

• A. Jourani

Complexity

Theorem 7. A+, A− ⊂ I R2, |A+| = n, |A−| = m, coA+∩ coA− = ∅ ⇓ SE(A+, A−) can be computed in O(nm) + O(n log n) time

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