On solving the multi-period location-assignment problem under - - PowerPoint PPT Presentation

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On solving the multi-period location-assignment problem under uncertainty

Mar´ ıa Albareda-Sambola2 Antonio Alonso-Ayuso1 Laureano Escudero 1 Elena Fern´ andez2 Celeste Pizarro Romero1

• 1. Departamento de Estad´

ıstica e Investigaci´

• n Operativa

Universidad Rey Juan Carlos, Madrid

• 2. Departamento de Estad´

ıstica e Investigaci´

• n Operativa

Universidad Polit´ ecnica de Catalu˜ na, Barcelona 13th Combinatorial Optimization Workshop Aussois (France), January 12-17, 2009

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

Outline

1

Introduction

2

Problem description

3

Uncertainty

4

Impulse-Step variables based (DEM)

5

Algorithmic framework

6

Computational comparison

7

Conclusions

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

The MISFLP problem

Given a time horizon, a set of customers and a set of facilities (e.g., production plants), Multi-period Incremental Service Facility Location Problem (MISFLP) is concerned with: locating the facilities within a given discrete set of potential sites and assigning the customers to the facilities along given periods in a time horizon.

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

The MISFLP problem

Assumptions Ensuring at each single period t the service of a minimum number of customers, say nt. The allocation of any customer to the servers might change in different periods. Once a customer is served in a time period it must be served at any subsequent period. Once a facility is opened it remains open until the end of the time horizon.

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

Variation of the MISFLP problem

We present a variation of the MISFLP where: Each customer needs to be serviced only in a subset of the periods of the time horizon we assume that this set of periods is known for each customer There is uncertainty in some parameters of the problem

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

The problem

Consider a network including a set of facilities and a set of customers

t = 1

a b c 1 2 3 4 5

n1 = 2, p1 = 2

t = 2

a b c 1 2 3 4 5

n2 = 4, p2 = 1

t = 3

a b c 1 2 3 4 5

n3 = 5, p3 = 0

f

facilities

1

customers

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

The problem

For the costumers:

The allocation of a customer to the facilities might change at different time periods but Once a customer is assigned in a given time period, he must continue to be assigned to one facility. A costumer cannot be assigned to more than one facility at each period All customers must be found to have been assigned at the end of the time horizon

At each single period

exactly pt facilities are opened at least nt new customers are covered

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

The problem

For the costumers:

The allocation of a customer to the facilities might change at different time periods but Once a customer is assigned in a given time period, he must continue to be assigned to one facility. A costumer cannot be assigned to more than one facility at each period All customers must be found to have been assigned at the end of the time horizon

At each single period

exactly pt facilities are opened at least nt new customers are covered

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

The problem

Costs Assigning a customer to a facility at a given period incurs a assignment cost, ct

ij, even if the customer does not have a

need for service in this period. There is a setup depreciation cost, f t

i

for the open facilities. the penalty cost, ρj, for the customers not served in time by the facilities

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

Uncertainty: Modeling via scenario tree

scenario 1 scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7 scenario 8 scenario 9 Two stages scenario 1 scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7 scenario 8 scenario 9 Multi stages

Scenario is an execution of uncertain and deterministic parameters along different stages of the temporal horizon. Scenario group for a given stage is the set of scenarios with the same realization of the uncertain parameters up to the stage. Scenario tree scheme is a technique used to model and interpret the uncertainty.

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Parameters

Uncertainty in the problem

The most important uncertainty that we find in this problem is the number of costumers that will need to be serviced in each time period Parameters dg

j , coefficient that takes the value 1 or 0 depending on

whether or not customer j is available for being serviced at time period t(g) under scenario group g, ∀j ∈ J . ng, minimum number of customers to be serviced in time period t(g) under scenario group g.

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Uncertainty in the problem

Variables

scenario 1 scenario 2 scenario 3 scenario 4 scenario 5 scenario 6 scenario 7 scenario 8 scenario 9

Non-anticipativity principle (Rockafellar and Wets)

If two different scenarios s and s

′ are identical until stage t as to as the

disponible information in that stage, then the decisions (variables) in both scenarios must be the same too until stage t.

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Uncertainty in the problem

Impulse-Step variables based formulation

0–1 variables

yg

i =

8 > > < > > : 1, if facility i is open by time period t(g) under scenario group g 0,

• therwise

∀i ∈ I, g ∈ G : t(g) ∈ T ∗ and xg

ij =

8 > > < > > : 1, if customer j is assigned to facility i at time period t(g) under scenario group g 0,

• therwise

∀i ∈ I, j ∈ J , g ∈ G−. where T ∗ = {t ∈ T : t ≤ |T | − τ} and G− ≡ G \ {0}

Note

The x–variables still are impulse variables, but the y–variables are step variables.

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Impulse-Step variables based formulation (DEM)

Pure 0–1 Model

Objective function

min X

i∈I

f 0

i y0 i +

X

g∈G−

wg »X

i∈I

“ f t(g)

i

(yg

i −yγ(g) i

)+ X

j∈J

ct(g)

ij

xg

ij

” + X

j∈J

ρjdg

j

` 1− X

i∈I

xg

ij

´– = X

g∈G

X

j∈J

wgρjdg

j + min

X

i∈I

» X

g∈G:t(g)∈T ∗

wg(f t(g)

i

− f t(g)+1

i

)yg

i +

X

g∈G−

wg X

j∈J

cg

ij xg ij

– (1)

Note

where cg

ij = ct(g) ij

− ρjdg

j

γ(g), inmediate scenario group to group g

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Impulse-Step variables based formulation (DEM)

Pure 0–1 Model

Objective function

min X

i∈I

f 0

i y0 i +

X

g∈G−

wg »X

i∈I

“ f t(g)

i

(yg

i −yγ(g) i

)+ X

j∈J

ct(g)

ij

xg

ij

” + X

j∈J

ρjdg

j

` 1− X

i∈I

xg

ij

´– = X

g∈G

X

j∈J

wgρjdg

j + min

X

i∈I

» X

g∈G:t(g)∈T ∗

wg(f t(g)

i

− f t(g)+1

i

)yg

i +

X

g∈G−

wg X

j∈J

cg

ij xg ij

– (1)

Note

where cg

ij = ct(g) ij

− ρjdg

j

γ(g), inmediate scenario group to group g

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Impulse-Step variables based formulation (DEM)

Pure 0–1 Model

Constraints

X

i∈I

X

j∈J

xg

ij ≥ ng

∀ g ∈ G− : t(g) < |T | (2) X

i∈I

xg

ij ≤ 1

∀ j ∈ J , g ∈ G− : t(g) < |T | (3) X

i∈I

xg

ij = 1

∀ j ∈ J , g ∈ G|T | (4) X

i∈I

xγ(g)

ij

≤ X

i∈I

xg

ij

∀ j ∈ J , g ∈ G : t(g) > 1 (5) xg

ij ≤ yγk (g) i

∀ i ∈ I, j ∈ J , g ∈ G−, where k = min{t(g), τ} (6) X

i∈I

(yg

i − yγ(g) i

) = pt ∀ g ∈ G− : t(g) ∈ T ∗ (7) X

i∈I

y0

i

= p0 (8) yγ(g)

i

≤ yg

i

∀ i ∈ I, g ∈ G− : t(g) ∈ T ∗ (9) xg

ij ∈ {0, 1}

∀ i ∈ I, j ∈ J , g ∈ G− (10) ytg

i

∈ {0, 1} ∀ i ∈ I, g ∈ G : t(g) ∈ T ∗ (11)

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

Splitting variables representation

For a general problem:

min X

ω∈Ω

pω` cT xω+qωT yω´

• s. t.

Axω = b Tωxω +Wyω= hω ∀ω ∈ Ω xω − xω+1 = 0 ∀ω ∈ Ω xω, yω∈ {0, 1} ∀ω ∈ Ω

A T1 W I −I A T2 W I −I A T3 W I −I A T4 W I −I A T5 W I −I A T6 W I −I

= = = = = = = = = = = = = = = = = = b h1 b h2 b h3 b h4 b h5 b h6 x1 y1 x2 y2 x3 y3 x4 y4 x5 y5 x6 y6

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

Branch-and-Fix Coordination

Non-anticipativity constraints are relaxed:

• sc. 1
• sc. 2
• sc. 3
• sc. 4

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

Branch-and-Fix Coordination

Non-anticipativity constraints are relaxed:

• sc. 1
• sc. 2
• sc. 3
• sc. 4

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Branch-and-Fix Coordination

Non-anticipativity constraints are relaxed:

• sc. 1
• sc. 2
• sc. 3
• sc. 4

ts Each problem is solved by a Branch and Fix, coordinatting the decisions

Independent problems

• 2

3

• 10

9 11 12

• 7

8 5 6 3 4 1 2

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Branch-and-Fix Coordination

Fix-and-Relax Coordination

Fix-and-Relax Introduced by Dillebenger et al. in 1994. See also Escudero and Salmer´

• n in 2005.

It is an heuristic approach to solve multi-level linear integer problems: Initially only variables of the first level are 0-1 defined. In successive stages, previous level variables are fixed, and current level variables are declared integer. Fix-and-Relax Coordination (FRC) Combine Fix-and-Relax with Branch-and-Fix Coordination: Non-anticipativity conditions are relaxed. Each scenario group is solved by Fix and Relax. Decisions are coordinated via Branch-and-Fix Coordination approach.

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Branch-and-Fix Coordination

Fix-and-Relax Coordination

Fix-and-Relax Introduced by Dillebenger et al. in 1994. See also Escudero and Salmer´

• n in 2005.

It is an heuristic approach to solve multi-level linear integer problems: Initially only variables of the first level are 0-1 defined. In successive stages, previous level variables are fixed, and current level variables are declared integer. Fix-and-Relax Coordination (FRC) Combine Fix-and-Relax with Branch-and-Fix Coordination: Non-anticipativity conditions are relaxed. Each scenario group is solved by Fix and Relax. Decisions are coordinated via Branch-and-Fix Coordination approach.

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational Branch-and-Fix Coordination

FRC scheme

Fix-and-Relax submodel Given V1 . . . VK a partition of K elements of the set of the variables V, problem IP : min

x∈X cx

• s. t. xj ∈ {0, 1}

∀j ∈ Vk, k = 1, . . . , K, This problem can be approximated by the model IPk : min

x∈X cx

• s. t.

xj = xj ∀j ∈ V′

k, k′ < k,

xj ∈ {0, 1} ∀j ∈ Vk, xj ∈ [0, 1] ∀j ∈ V′

k, k < k′,

It is the so-called FR level k

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational FRC scheme

FR levels

t = 1 t = 2 t = 3 t = 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

(a) Scenario tree

t = 1 t = 2 t = 3 t = 4 1 1 1 1 2 2 3 3 4 5 6 7 8 9 10 11 12 13 14 15

(b) Cluster structure Cluster 1 Cluster 2 Cluster 3 Cluster 4

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational FRC scheme

t = 1 t = 2 t = 3 t = 4 1 1 1 1 2 2 3 3 4 5 6 7 8 9 10 11 12 13 14 15

Level 1

t = 1 t = 2 t = 3 t = 4 1 1 1 1 2 2 3 3 4 5 6 7 8 9 10 11 12 13 14 15

Level 2 Level 3

t = 1 t = 2 t = 3 t = 4 1 1 1 1 2 2 3 3 4 5 6 7 8 9 10 11 12 13 14 15

Level 4 Level 5 Level 6 Level 7

t = 1 t = 2 t = 3 t = 4 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 4 4 5 5 6 6 7 7 8 9 10 11 12 13 14 15

Level 8 Level 9 Level 10 Level 11 Level 12 Level 13 Level 14 Level 15 Nodes whose non-anticipativity constraints are not relaxed

n

Nodes whose variables have been 0-1 fixed

n

Nodes whose variables have been 0-1 integer defined

n

Nodes whose variables have been 0-1 continuous defined

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational FRC scheme

Branching strategy We have chosen the depth first strategy for the TNF branching selection The criterion for branching consists of choosing the candidate TNF with the smallest Lagrangean Substitution value among the two sons of the last branched TNF. We have chosen two largest small deterioration strategies for the selection of the branching variable.

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

On candidate TNF bounding

The bounding of a given candidate TNF, Jf, f ∈ F, can be

• btained by using Lagrangian Decomposition (LD):

ZD(µ) = min

• j∈Jf

wj(cjxj + ajyj) +

• j∈Jf

µj(xj − xj+1)

• s. t. Axj + Byj = bj

∀j ∈ Jf 0 ≤ xj ≤ 1, yj ≥ 0 ∀j ∈ Jf, Our aim is to obtain the bound ZD(µ∗), where µ∗ = argmax{ZD(µ)}.

Note

The number of Lagrange multipliers depends on the number of non yet branched on/fixed common variables in vector xj and the number of nodes, |Jf|, in the family.

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

On candidate TNF bounding

The bounding of a given candidate TNF, Jf, f ∈ F, can be

• btained by using Lagrangian Decomposition (LD):

ZD(µ) = min

• j∈Jf

wj(cjxj + ajyj) +

• j∈Jf

µj(xj − xj+1)

• s. t. Axj + Byj = bj

∀j ∈ Jf 0 ≤ xj ≤ 1, yj ≥ 0 ∀j ∈ Jf, Our aim is to obtain the bound ZD(µ∗), where µ∗ = argmax{ZD(µ)}.

Note

The number of Lagrange multipliers depends on the number of non yet branched on/fixed common variables in vector xj and the number of nodes, |Jf|, in the family.

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational BFC algorithm

• ∀→
• ν
• ≥
• →
• ≥

≥ ≥ ≥

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

Why we use this modelization for the problem? A study of the deterministic problem has been done We have done a computational comparison among:

1

Impulse variables based model

2

Step variables based model

3

Impulse-step variables based model

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

Computational Experience Deterministic Version

Implemented in experimental C++ code. CPLEX v11 for solving each instance. Pentium IV, 1.8Ghz, 512 RAM Microsoft Visual Studio C++ compiler v6.0.

|J | |I| |T | 50 30 4 50 30 8 50 30 12 100 30 4 100 30 8 100 30 12 10 instances for each row: Total 60 instances

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

Models’ dimensions

Impulse model Impulse-step model Step model Instance nv nr nel dens (%) nv nr nel dens (%) nv nr nel dens (%) E4-P30-C50 6120 6388 42240 0.108 6120 6448 33390 0.085 6120 12048 30290 0.041 E8-P30-C50 12240 12796 111480 0.071 12240 12976 69870 0.044 12240 24176 60770 0.021 E12-P30-C50 18360 19204 204720 0.058 18360 19504 106350 0.030 18360 36304 91250 0.014 E4-P30-C100 12120 12738 84240 0.055 12120 12798 66390 0.043 12120 23998 60190 0.021 E8-P30-C100 24240 25546 222480 0.036 24240 25726 138870 0.022 24240 48126 120670 0.010 E12-P30-C100 36360 38354 408720 0.029 36360 38654 211350 0.015 36360 72254 181150 0.006

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

Computational experience

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

Computational experience

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

Graphical comparison of the results obtained with the different models

Model M1 with respect to Model M2

• 15

15 30 45 60 75 90 105 120 135 150

• 2
• 1

1 2 3 % o.f. dev. % time dev.

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

Graphical comparison of the results obtained with the different models

Model M3 with respect to Model M2

• 50

50 100 150 200 10 20 30 40 50 % o.f. dev. % time dev.

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Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computational

Conclusions and Future work

A variation of the multi-period incremental service facility location problem has been presented. A variation with uncertainty in the demand and the set of periods when each customer requieres service is presented by using Stochastic Programming. Three 0–1 equivalent formulations are proposed for the deterministic models, based on the impulse and step variables approaches. An intensive computational experimentation has been performed for the deterministic models. Impulse-Step variables based formulation shows better results. Work-in-progress: Computational experience for the stochastic model