On solving the multi-period
location-assignment problem under
uncertainty
Marı́a Albareda-Sambola2
Elena Fernández2
Antonio Alonso-Ayuso1
Laureano Escudero 1
Celeste Pizarro Romero1
1. Departamento de Estadı́stica e Investigación Operativa
Universidad Rey Juan Carlos, Madrid
2. Departamento de Estadı́stica e Investigación Operativa
Universidad Politécnica de Cataluña, Barcelona
13th Combinatorial Optimization Workshop
Aussois (France), January 12-17, 2009
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Outline
1
Introduction
2
Problem description
3
Uncertainty
4
Impulse-Step variables based (DEM)
5
Algorithmic framework
6
Computational comparison
7
Conclusions
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
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.
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
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.
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
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
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
The problem
t=1
t=2
1
1
a
a
2
3
b
3
b
4
c
3
b
4
c
5
5
1
2
n = 2, p = 2
facilities
2
4
c
f
1
a
2
1
t=3
2
n = 4, p = 1
1
5
3
3
n = 5, p = 0
customers
Consider a network including a set of facilities and a set of customers
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
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
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
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
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
The problem
Costs
Assigning a customer to a facility at a given period incurs a
assignment cost, cijt , even if the customer does not have a
need for service in this period.
There is a setup depreciation cost, fit for the open facilities.
the penalty cost, ρj , for the customers not served in time by
the facilities
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Uncertainty: Modeling via scenario tree
scenario 1
scenario 1
scenario 2
scenario 2
scenario 3
scenario 3
scenario 4
scenario 4
scenario 5
scenario 5
scenario 6
scenario 6
scenario 7
scenario 7
scenario 8
scenario 8
scenario 9
Two stages
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.
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
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
g
dj , 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.
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Uncertainty in the problem
Variables
Non-anticipativity principle (Rockafellar and Wets)
0
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.
scenario 1
scenario 2
scenario 3
scenario 4
scenario 5
scenario 6
scenario 7
scenario 8
scenario 9
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Uncertainty in the problem
Impulse-Step variables based formulation
0–1 variables
g
yi =
8
>
1,
>
<
if facility i is open by time period t(g) under
scenario group g
∀i ∈ I, g ∈ G : t(g) ∈ T ∗
>
>
:
otherwise
0,
and
g
xij =
8
>
1,
>
<
if customer j is assigned to facility i at time
period t(g) under scenario group g
∀i ∈ I, j ∈ J , g ∈ G − .
>
>
:
otherwise
0,
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.
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Impulse-Step variables based formulation (DEM)
Pure 0–1 Model
Objective function
min
X
fi0 yi0 +
g∈G −
i∈I
XX
g∈G j∈J
X
g
w g ρj dj + min
wg
»X“
t(g)
fi
g
γ(g)
(yi −yi
)+
t(g) g
xij
cij
” X
X g ´–
g`
+
ρj dj 1−
xij
=
j∈J
j∈J
i∈I
X»
X
t(g)
X
w g (fi
i∈I g∈G:t(g)∈T ∗
t(g)+1
− fi
g
)yi +
X
g∈G −
i∈I
wg
X
g g
–
c ij xij
j∈J
(1)
Note
g
t(g)
where c ij = cij
g
− ρj dj
γ(g), inmediate scenario group to group g
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Impulse-Step variables based formulation (DEM)
Pure 0–1 Model
Objective function
min
X
fi0 yi0 +
g∈G −
i∈I
XX
g∈G j∈J
X
g
w g ρj dj + min
wg
»X“
t(g)
fi
g
γ(g)
(yi −yi
)+
t(g) g
xij
cij
” X
X g ´–
g`
+
ρj dj 1−
xij
=
j∈J
j∈J
i∈I
X»
X
t(g)
X
w g (fi
i∈I g∈G:t(g)∈T ∗
t(g)+1
− fi
g
)yi +
X
g∈G −
i∈I
wg
X
g g
–
c ij xij
j∈J
(1)
Note
g
t(g)
where c ij = cij
g
− ρj dj
γ(g), inmediate scenario group to group g
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Impulse-Step variables based formulation (DEM)
Pure 0–1 Model
Constraints
X X g
g
xij ≥ n
∀g ∈ G
−
: t(g) < |T |
(2)
i∈I j∈J
X g
xij ≤ 1
∀j ∈ J,g ∈ G
−
: t(g) < |T |
(3)
i∈I
X g
xij = 1
∀ j ∈ J , g ∈ G|T |
(4)
∀ j ∈ J , g ∈ G : t(g) > 1
(5)
i∈I
X γ(g)
X g
xij
≤
xij
i∈I
i∈I
γ k (g)
g
xij ≤ yi
X
g
(yi
−
γ(g)
yi
)
=p
t
∀ i ∈ I, j ∈ J , g ∈ G
∀g ∈ G
−
: t(g) ∈ T
−
, where k = min{t(g), τ }
∗
(6)
(7)
i∈I
X 0
0
yi = p
(8)
i∈I
γ(g)
yi
g
≤ yi
g
xij ∈ {0, 1}
tg
yi
∈ {0, 1}
∀ i ∈ I, g ∈ G
−
: t(g) ∈ T
∀ i ∈ I, j ∈ J , g ∈ G
−
∀ i ∈ I, g ∈ G : t(g) ∈ T
∗
(9)
(10)
∗
(11)
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Splitting variables representation
For a general problem:
6
5
4
3
2
1
x1 y x2 y x3 y x4 y x5 y x6 y
min
X
`
´
pω cT xω +qω T yω
ω∈Ω
s. t.
I
Axω
=b
A
= h1
T1 W
=b
=0
−I
=b
A
= h2
T2 W
Tω xω
xω − xω+1
+Wyω =
hω
=0
∀ω ∈ Ω
I
=0
−I
=b
A
T
∀ω ∈ Ω
3
I
= h3
W
=0
−I
=b
A
xω ,
yω ∈
{0, 1}
∀ω ∈ Ω
T
4
I
= h4
W
=0
−I
=b
A
T
5
I
= h5
W
=0
−I
=b
A
T
−I
6
I
W
= h6
=0
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Branch-and-Fix Coordination
Non-anticipativity constraints are relaxed:
sc. 1
sc. 2
sc. 3
sc. 4
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Branch-and-Fix Coordination
Non-anticipativity constraints are relaxed:
sc. 1
sc. 2
sc. 3
sc. 4
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Branch-and-Fix Coordination
Non-anticipativity constraints are relaxed:
Independent
problems
nts
Each problem is solved by a
Branch and Fix, coordinatting
the decisions
sc. 1
2
3
sc. 2
sc. 3
sc. 4
1
2
3
4
5
6
7
8
9
10
11
12
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Branch-and-Fix Coordination
Fix-and-Relax Coordination
Fix-and-Relax
Introduced by Dillebenger et al. in 1994. See also Escudero and
Salmerón in 2005.
It is an heuristic approach to solve multi-level linear integer
problems:
X Initially only variables of the first level are 0-1 defined.
X 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:
X Non-anticipativity conditions are relaxed.
X Each scenario group is solved by Fix and Relax.
X Decisions are coordinated via Branch-and-Fix Coordination
approach.
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Branch-and-Fix Coordination
Fix-and-Relax Coordination
Fix-and-Relax
Introduced by Dillebenger et al. in 1994. See also Escudero and
Salmerón in 2005.
It is an heuristic approach to solve multi-level linear integer
problems:
X Initially only variables of the first level are 0-1 defined.
X 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:
X Non-anticipativity conditions are relaxed.
X Each scenario group is solved by Fix and Relax.
X Decisions are coordinated via Branch-and-Fix Coordination
approach.
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
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 cx
x∈X
s. t. xj ∈ {0, 1}
∀j ∈ Vk , k = 1, . . . , K ,
This problem can be approximated by the model
IP k : min cx
x∈X
s. t.
xj = xj
∀j ∈ Vk0 , k 0 < k ,
xj ∈ {0, 1}
∀j ∈ Vk ,
xj ∈ [0, 1]
∀j ∈ Vk0 , k < k 0 ,
It is the so-called FR level k
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
FRC scheme
FR levels
t=1
t=1
t=2
t=3
t=2
t=3
t=4
t=4
8
4
8
1
2
C lu ster 1
4
9
9
2
10
5
10
1
2
C lu ster 2
5
11
11
1
12
6
12
1
3
C lu ster 3
6
13
13
3
14
7
14
1
15
(a) Scenario tree
3
C lu ster 4
7
15
(b ) C lu ster stru ctu re
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
FRC scheme
t=1
t=2
t=3
t=4
t=1
8
Lev el 1
1
1
2
2
t=2
t=3
1
2
9
10
10
1
2
1
12
1
3
14
1
t=3
2
5
3
11
Lev el 6
12
1
14
15
t=2
9
6
13
7
t=4
4
10
1
13
7
t=1
t=3
Lev el 5
5
Lev el 3
6
3
2
11
12
3
t=2
8
1
9
5
t=1
Lev el 4
4
11
1
t=4
8
Lev el 2
4
3
6
13
Lev el 7
14
1
3
15
7
15
t=4
Lev el 8
1
2
4
Lev el 9
1
2
4
9
Lev el 1 0
1
2
5
10
Lev el 1 1
1
2
5
11
Lev el 1 2
1
3
6
12
Lev el 1 3
1
3
6
13
Lev el 1 4
1
3
7
14
Lev el 1 5
1
3
7
15
8
N o des w h o se n o n -a n tic ip a tiv ity c o n stra in ts a re n o t rela x ed
n
N o des w h o se v a ria b les h a v e b een 0 -1 fi x ed
n
N o des w h o se v a ria b les h a v e b een 0 -1 in teg er defi n ed
n
N o des w h o se v a ria b les h a v e b een 0 -1 c o n tin u o u s defi n ed
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
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.
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
On candidate TNF bounding
The bounding of a given candidate TNF, Jf , f ∈ F, can be
obtained by using Lagrangian Decomposition (LD):
X
X
ZD (µ) = min
w j (cj xj + aj yj ) +
µj (xj − xj+1 )
j∈Jf
j∈Jf
s. t. Axj + Byj = bj
j
j
0 ≤ x ≤ 1, y ≥ 0
∀j ∈ Jf
∀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 x j and the number of nodes,
|Jf |, in the family.
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
On candidate TNF bounding
The bounding of a given candidate TNF, Jf , f ∈ F, can be
obtained by using Lagrangian Decomposition (LD):
X
X
ZD (µ) = min
w j (cj xj + aj yj ) +
µj (xj − xj+1 )
j∈Jf
j∈Jf
s. t. Axj + Byj = bj
j
j
0 ≤ x ≤ 1, y ≥ 0
∀j ∈ Jf
∀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 x j and the number of nodes,
|Jf |, in the family.
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
BFC algorithm
∀→
ν
!"# #$
&
%
→
≥
!$
!"
#
• '
&()!"*
• ≥
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
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
2
3
Impulse variables based model
Step variables based model
Impulse-step variables based model
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
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 |
50
50
50
100
100
100
|I|
30
30
30
30
30
30
|T |
4
8
12
4
8
12
10 instances for each row:
Total 60 instances
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Models’ dimensions
Impulse model
Instance
E4-P30-C50
E8-P30-C50
E12-P30-C50
E4-P30-C100
E8-P30-C100
E12-P30-C100
nv
nr
nel
6120
12240
18360
12120
24240
36360
6388
12796
19204
12738
25546
38354
42240
111480
204720
84240
222480
408720
Impulse-step model
dens
(%)
0.108
0.071
0.058
0.055
0.036
0.029
dens
(%)
6120 6448 33390 0.085
12240 12976 69870 0.044
18360 19504 106350 0.030
12120 12798 66390 0.043
24240 25726 138870 0.022
36360 38654 211350 0.015
nv
nr
nel
Step model
nv
6120
12240
18360
12120
24240
36360
dens
nr
nel
(%)
12048 30290 0.041
24176 60770 0.021
36304 91250 0.014
23998 60190 0.021
48126 120670 0.010
72254 181150 0.006
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Computational experience
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Computational experience
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Graphical comparison of the results obtained with
the different models
Model M1 with respect to Model M2
150
135
120
105
% time dev.
90
75
60
45
30
15
-2
-1
0
-15 0
1
% o.f. dev.
2
3
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
Graphical comparison of the results obtained with
the different models
Model M3 with respect to Model M2
200
% time dev.
150
100
50
0
0
10
20
30
-50
% o.f. dev.
40
50
Introduction Problem description Uncertainty Impulse-Step variables based (DEM) Algorithmic framework Computatio
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