An Improved Relax-and-Fix Algorithm for the Fixed Charge Network
Design Problem with User-optimal Flow
Pedro Henrique Gonz
´
alez
1,2
, Luidi Gelabert Simonetti
1
, Carlos Alberto de Jesus Martinhon
1
,
Edcarllos Santos
1
and Philippe Yves Paul Michelon
2
1
Institute of Computing, Fluminense Federal University, Niter
´
oi, Brazil
2
Laboratoire d’Informatique d’Avignon, Universit
´
e d’Avignon et des Pays de Vaucluse, Avignon, France
Keywords:
Network Design Problem, Dynamic Programming, Relax-and-Fix, Bi-level Problem.
Abstract:
Due to the constant development of society, increasing quantities of commodities have to be transported in
large urban centers. Therefore, network planning problems arise as tools to support decision-making, aim-
ing to meet the need of finding efficient ways to perform such transportations. This paper review a bi-level
formulation, an one level formulation obtained by applying the complementary slackness theorem, Bellman’s
optimality conditions and presents an improved Relax-and-Fix heuristic, through combining a randomized
constructive algorithm with a Relax-and-Fix heuristic, so high quality solutions could be found. Besides that,
our computational results are compared with the results found by an one-level formulation and other heuristics
found in the literature, showing the efficiency of the proposed method.
1 INTRODUCTION
The Fixed Charge Network Design Problem (FC-
NDP) involves selecting a subset of edges from a
graph, in such a way that a given set of commodities
can be transported from their origins to their destina-
tions. The problem consists in minimizing the sum
of the fixed costs (due to selected edges) and variable
costs (depending on the flow of goods on the edges).
Fixed and variable costs can be represented by lin-
ear functions and arcs are not capacitated. The FC-
NDP belongs to a large class of network design prob-
lems (Magnanti and Wong, 1984). In the literature,
one can find several variations of FCNDP (Boesch,
1976) such as shortest path problem, minimum span-
ning tree problem, vehicle routing problem, travel-
ing salesman problem and network Steiner problem
(Magnanti and Wong, 1984). Moreover, as illus-
trated by several books and papers (Boesch, 1976)
(Boyce and Janson, 1980) (Mandl, 1981), generic
network design problem has numerous applications.
Mathematical formulations for FCNDP not only rep-
resent the FCNDP, but also problems of communi-
cation, transportation, sewage systems and resource
planning. It also appears in other contexts, such as
flexible production systems (Kimemia and Gershwin,
1978) and automated manufacturing systems (Graves
and Lamar, 1983). Finally, network design problems
arise in many vehicle fleet applications that do not in-
volve the construction of physical facilities, but rather
model decision problems such as sending a vehicle
through a road or not (Simpson, 1969); (Magnanti,
1981).
In network planning problems, not only the
simplest versions are NP-Hard (Johnson et al.,
1978);(Wong, 1978), but also the task of finding feasi-
ble solutions (for problems with budget constraint on
the fixed cost) is extremely complex (Wong, 1980).
Due to the natural difficulties of the problem, heuris-
tics methods are presented as a good alternative in the
search for quality solutions.
In the paper, we intend to address a specific
variation of FCNDP. The Fixed-Charge Uncapacited
Network Design Problem with User-optimal Flows
(FCNDP-UOF), which consists of adding multiple
shortest path problems to the original problem. The
FCNDP-UOF can be modeled as a bilevel discrete
linear programming problem. This type of problem
involves two distinct agents acting simultaneously
rather than sequentially when making decisions. On
the upper level, the leader (1
st
agent) is in charge of
choosing a subset of edges to be opened in order to
minimize the sum of fixed and variable costs. In re-
sponse, on the lower level, the follower (2
nd
agent)
must choose a set of shortest paths in the network,
resulting in the paths through which each commod-
100
González P., Simonetti L., de Jesus Martinhon C., Santos E. and Paul Michelon P..
An Improved Relax-and-Fix Algorithm for the Fixed Charge Network Design Problem with User-optimal Flow.
DOI: 10.5220/0004832601000107
In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 100-107
ISBN: 978-989-758-017-8
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
ity will be sent. The effect of an agent on the other
is indirect: the decision of the followers is affected
by the network designed on the upper level, while the
leader’s decision is affected by variable costs imposed
by the routes setted in the lower level.
The inclusion of shortest path problem constraints
in a mixed integer linear programming is not straight-
forward. Difficulties arise both in modeling and de-
signing efficient methods. As far as we know, there
are few works done on FCNDP-UOF in the literature,
and most of them address to a particular variant. This
problem or its variant could be seen on (Billheimer
and Gray, 1973); (Kara and Verter, 2004); (Erkut
et al., 2007); (Mauttone et al., 2008); (Erkut and
Gzara, 2008); (Amaldi et al., 2011); (Gonz
´
alez et al.,
2013) and has been treated as part of larger problems
in some applications on (Holmberg and Yuan, 2004).
The FCNDP-UOF problem appears in the design of
a road network for hazardous materials transportation
(Kara and Verter, 2004); (Erkut et al., 2007); (Erkut
and Gzara, 2008) and (Amaldi et al., 2011). Dur-
ing the solution of this problem the government de-
fines a selection of road segments to be opened/closed
to the transportation of hazardous materials assuming
that hazmat shipments in the resulting network will
be done along shortest paths. There are no costs as-
sociated with the selection of roads to compose the
network but the government wants to minimize the
population exposure in case of an incident during a
dangerous-goods transportation. This is a particular
case of the FCNDP-UOF problem where the fixed
costs are equal to zero.
It is interesting to specify the contributions of
each work cited above. (Billheimer and Gray, 1973)
present and formally define the FCNDP-UOF. (Kara
and Verter, 2004) and (Erkut et al., 2007) works focus
on exact methods, presenting a mathematical formu-
lation and several metrics for the hazardous materials
transportation problem. (Mauttone et al., 2008) not
only presented a different model, but also presented
a Tabu Search for the FCNDP-UOF. Both, (Erkut
and Gzara, 2008) and (Amaldi et al., 2011) presented
heuristic approaches to tackle the hazardous materi-
als transportation problem. At last, (Gonz
´
alez et al.,
2013), presented a extension of the model proposed
by Kara and Verter and also a GRASP.
This text is organized as follows. In Section 2, we
start by describing the problem followed by a bi-level
and an one-level formulation, presented by (Mauttone
et al., 2008). Then in Section 3 we present our solu-
tion approach. Section 4 reports on our computational
results. In Section 5 we will compare our results with
the mathematical formulation and with heuristic re-
sults found in the literature. At last, in Section 6 the
conclusion and future works are presented.
2 GENERAL DESCRIPTION OF
FCNDP-UOF
In this section we describe the problem and present a
bi-level and an one-level formulation for the FCNDP-
UOF proposed respectively by (Colson et al., 2005)
and (Mauttone et al., 2008) for the FCNDP-UOF,
which we address as MLF Model.
Since the structure of the problem can be easily
represented by a graph, the basic structures to create
a network are a set of nodes V that represents the fa-
cilities and a set of uncapacited and undirected edges
E representing the connection between installations.
Furthermore, the set K is the set of commodities to be
transported over the network, and these commodities
may represent physical goods as raw material for
industry, hazardous material or even people. Each
commodity k ∈ K, has a flow to be delivered through
a shortest path between its source o(k) and its
destination d(k). The formulation presented here
works with variants presenting commodities with
multiple origins and destinations, and for treating
such a case, it is sufficient to consider that for each
pair (o(k), d(k)), there is a new commodity resulting
from the dissociation of one into several commodities.
2.1 Mathematical Formulation
This subsection shows a small review of FCNDP-
UOF in order to exemplify the characteristics and
make easier the understanding of it.
The model for FCNDP-UOF has two types of vari-
ables, one for the construction of the network and an-
other related to representing the flow. Let y
i j
be a
binary variable, we have that y
i j
= 1 if the edge (i, j)
is chosen as part of the network and y
i j
= 0 other-
wise. In this case, x
k
i j
denotes the commodity k flow
through the arc (i, j). Although the edges have no di-
rection, they may be referred to as arcs, because each
commodity flow is directed. Treating y = (y
i j
) and
x = (x
k
i j
), respectively, as vectors of adding edge and
flow variables, a mixed integer programming formu-
lation can be elaborated.
AnImprovedRelax-and-FixAlgorithmfortheFixedChargeNetworkDesignProblemwithUser-optimalFlow
101
2.1.1 List of Symbols
V Set of Nodes.
E Set of admissible bi-directed Edges.
K Set of Commodities.
δ
+
i
Set of all arcs leaving node i.
δ
−
i
Set of all arcs arriving at node i.
c
e
Length of edge e.
o(k) Origin node for commodity k.
d(k) Destination node for commodity k.
g
k
i j
Variable cost of transporting commodity
k through the edge (i, j) ∈ E.
f
i j
Fixed cost of opening the edge (i, j) ∈ E.
y
i j
Indicates if edge (i, j) belongs in the solution.
x
k
i j
Indicates if commodity k passes through
the arc (i, j).
2.1.2 Bi-level Formulation
FCNDP-UOF is a variation of the FCNDP where each
k ∈ K has to be transported through a shortest path
between its origin o(k) and its destination d(k). This
change entails adding new constraints to the general
problem. In FCNDP-UOF, besides selecting a subset
of E whose sum of fixed and variable costs is mini-
mal (leading problem), each commodity k ∈ K must
be transported through the shortest path between o(k)
and d(k) (follower problem). The FCNDP-UOF be-
longs to the class of NP-Hard problems and can be
modeled as a bi-level discrete integer programming
problem (Colson et al., 2005), as follows:
min
∑
(i, j)∈E
f
i j
y
i j
+
∑
k∈K
∑
(i, j)∈E
g
k
i j
x
k
i j
s.t. y
i j
∈ {0, 1}, ∀e = (i, j) ∈ E, (1)
where x
k
i j
is a solution of the problem:
min
∑
k∈K
∑
(i, j)∈E
c
i j
x
k
i j
s.t.
∑
(i, j)∈δ
+
(i)
x
k
i j
−
∑
(i, j)∈δ
−
(i)
x
k
i j
= b
k
i
, ∀i, j ∈ V, ∀k ∈ K,
x
k
i j
+ x
k
ji
≤ y
e
, ∀e = (i, j) ∈ E, ∀k ∈ K,
x
k
i j
≥ 0, ∀e = (i, j) ∈ E, ∀k ∈ K.
(2)
(3)
(4)
where:
b
k
i
=
−1 if i = d(k),
1 if i = o(k),
0 otherwise.
Analyzing the model described by constraints (1) -
(4), we can see that the set of constraints (1) ensures
that y
e
assume only binary values. In (2), we have
flow constraints. Constraints (3) do not allow flow
into arcs whose corresponding edges are closed.
Finally, (4) imposes the non-negativity restriction
of the variables x
k
i j
. An interesting remark is that
solving the follower problem is equivalent to solving
|K| shortest paths problems independently.
2.1.3 One-level Formulation
The FCNDP-UOF can be formulated as an one-level
integer programming problem replacing the objective
function and the constraints defined by (2), (3) and
(4) of the follower problem for its optimality condi-
tions (Mauttone et al., 2008). This could be done
by applying the fundamental theorem of duality and
the complementary slackness theorem (Bazaraa et al.,
2004). However, optimality conditions for the prob-
lem in the lower level are, in fact, the optimality con-
ditions of the shortest path problem and they could be
expressed in a more compact and efficient way if we
consider the Bellman’s optimality conditions for the
shortest path problem (Ahuja et al., 1993) and using a
simple lifting process (Luigi De Giovanni, 2004).
Unfortunately this new formulation loses the interest-
ing feature of being linear. To bypass this problem a
Big-M linearization is applied. After these modifica-
tions, one can write the model as an one-level mixed
integer linear programming problem, as follows:
min
∑
(i, j)∈E
f
i j
y
i j
+
∑
k∈K
∑
(i, j)∈E
g
k
i j
x
k
i j
s.t.
∑
(i, j)∈δ
+
(i)
x
k
i j
−
∑
(i, j)∈δ
−
(i)
x
k
i j
= b
k
i
, ∀i, j ∈ V, ∀k ∈ K,
x
k
i j
+ x
k
ji
≤ y
i j
, ∀e = (i, j) ∈ E, ∀k ∈ K,
π
k
i
− π
k
j
≤ M − y
e)
(M − c
e
) − 2c
e
x
k
ji
, ∀e = (i, j) ∈ E, k ∈ K,
π
k
i
≥ 0, ∀i ∈ V, ∀k ∈ K,
π
k
i
= 0, ∀i = d(k), ∀k ∈ K,
π
k
i
∈ R, ∀i ∈ V, ∀k ∈ K,
x
k
i j
∈ {0, 1}, ∀(i, j) ∈ E, ∀k ∈ K,
y
i j
∈ {0, 1}, ∀(i, j) ∈ E.
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
where:
b
k
i
=
−1 if i = d(k),
1 if i = o(k),
0 otherwise.
The variables π
k
i
, k ∈ K, i ∈ V , are the shortest dis-
tance between vertex i and vertex d(k). Then we de-
fine that π
k
d(k)
will always be equal to zero. Assum-
ing y and x binary and assuming that the inequalities
(7) are satisfied, it is easy to see that constraints (8)
are equivalent to Bellman’s optimality conditions for
a |K| set of pairs (o(k), d(k)).
ICORES2014-InternationalConferenceonOperationsResearchandEnterpriseSystems
102
3 SOLUTION APPROACH
We address this section to present and explain the
Partial Decoupling Heuristic and the Relax and Fix
Heuristics. Before explaining the improved Relax-
and-Fix heuristic, called DPRF, a small review of the
Relax-and-Fix heuristic is presented.
3.1 Partial Decoupling Heuristic
A total decoupling heuristic for the FCNDP-UOF, is
based on the idea of dissociating the problem of build-
ing a network from the shortest path problem. How-
ever, as discussed in (Erkut and Gzara, 2008), the de-
coupling of the original problem can provide worst
results than when addressing both problems simul-
taneously. Therefore, this algorithm proposes what
we call partial decoupling, where certain aspects of
the follower problem are considered when trying to
build a solution to the leading problem. So in or-
der to build the network the following cost is used:
( f
e
×(1−y
e
))+(α×g
k
0
i j
+(1−α)×c
e
), which means
that we consider whether the is edge open or not,
plus a linear combination of the variable cost and the
length of the edge. The α works as a mediator of
the importance of the g
k
0
i j
and c
e
values. In the be-
ginning of the iterations α prioritizes the variable cost
(g
k
0
i j
), while in the end it prioritizes the edge lenght
(c
e
).After building the network, a shortest path algo-
rithm is applied to take every product from its origin
o(k) to its destination d(k), considering c
e
as the edge
cost. It is important to note that g
k
i j
= q
k
β
i j
, where q
k
represents the amount of commodity k and β
i j
repre-
sents the shipping cost through the edge e = (i, j).
The algorithm presented here is a small variation
of the Partial Decoupling Heuristic presented in
(Gonz
´
alez et al., 2013). The procedure is further ex-
plained on Algorithm 1.
The partial decoupling heuristic consists in using
the Dijkstra algorithm for the shortest path problem.
Procedures DijkstraLeader and DijkstraFollower, se-
quentially solve the problem of network construction,
followed by the shortest path problem for each com-
modity k ∈ K, so that in the end of the procedure,
all commodities have been transported from its ori-
gin to its destination. The DLCost and DSCost are re-
spectively DijkstraLeader and DijkstraFollower pro-
cedures costs. The notation s ←< y, x > represents
that the solution s is storing the values of the vari-
ables y and x that were just defined by DijkstraLeader
and DijkstraFollower. The function CloseEdge closes
all the edges that at the end of the DijkstraFollower
procedure are open and do not have flow. The ran-
dom function returns a random element from the set
Algorithm 1: Partial Decoupling Heuristic.
Input: γ
Data: MinCost ← ∞, α ← 1, y ← 0, x ← 0;
begin
ˆ
K ← K;
for numIterDP in 1 . . . MaxIterDP do
while K 6=
/
0 do
K ← CandidateList(K, γ);
k
0
← Random(K);
for each e = (i, j) ∈ E do
DLCost(e, k
0
) ← ( f
e
× (1 − y
e
)) +
(α × g
k
0
i j
+ (1 − α) × c
e
);
y ← Di jkstraLeader(DLCost, k
0
);
K ← K\{k
0
};
for each e = (i, j) ∈ E do
DSCost(e) ← c
e
;
for k ∈
ˆ
K do
x ← Di jkstraFollower(DSCost, k);
s ←< y, x >;
CloseEdge(s);
if Cost(s) < MinCost then
s
best
← s;
MinCost ← Cost(s
best
);
α ← α −
1
MaxIterDP
;
Rearm(K);
return s
best
passed as a parameter. In order to choose the inser-
tion order of |K| commodities, the procedure uses a
candidate list consisting of a subset of products not
yet routed, whose amount is greater than or equal to
γ times the largest amount of commodity not routed.
The function Rearm(K) adds all commodities to set
K and makes all variables return to its initial state.
3.2 Relax and Fix Heuristic
Given a mixed integer programming formulation:
min c
1
z
1
+ c
2
z
2
;
s.t. A
1
z
1
+ A
2
z
2
= b;
z
1
∈ Z
n
1
+
, z
2
∈ Z
n
2
+
;
(14)
(15)
without loss of generality, let’s suppose that the
variables z
1
j
for j ∈ N
1
are more important than the
variables z
2
j
for j ∈ N
2
, with n
i
= |N
i
| for i = 1, 2.
The idea of the Relax and Fix, consists in solving two
(or more) easier LPs or MIPs. The first one allows
us to fix (i.e. z
i
j
= w, w ∈ Z
n
i
+
) or limit the range of
more important variables, while the second allows us
to choose good values for other variables z
2
.
In order to do so, first it is necessary to solve a
relaxation like:
AnImprovedRelax-and-FixAlgorithmfortheFixedChargeNetworkDesignProblemwithUser-optimalFlow
103
min c
1
z
1
+ c
2
z
2
;
s.t. A
1
z
1
+ A
2
z
2
= b;
z
1
∈ Z
n
1
+
, z
2
∈ R
n
2
+
;
(16)
(17)
in which the integrality of z
2
variables is dropped. Let
(¯z
1
, ¯z
2
) be the corresponding solution. Secondly fix
the important variables, according to criterias based
on the problem peculiarity, and solve the new prob-
lem. After that, (¯z
1
, ¯z
2
) becomes the corresponding
solution if the solution of the relaxed model is feasi-
ble. At last, the algorithm returns z
H
= (¯z
1
,
¯
¯z
2
).
In terms of algorithm, the Relax and Fix procedure
can be seen as:
Algorithm 2: Relax and Fix Heuristic.
Input: n
1
, n
2
, N
1
, N
2
Data: MinCost ← ∞
begin
for i = 1 . . . 2 do
for j ∈ N
2
do
z
j
i
∈ {0, 1};
s ← SolveLR(N
1
, N
2
);
for j ∈ N
1
do
if z
j
i
= w then
z
j
i
= w;
if Cost(s) < MinCost and
Feas(s) = T RUE then
s
best
← s ;
MinCost ← Cost(s
best
) ;
return s
best
The function SolveLR(N
1
, N
2
) solves the linear relax-
ation of the Generalized Model for the sets N
1
and
N
2
. The function Feas(s) returns true if the solution s
passed as parameter is a feasible solution to the prob-
lem and returns false otherwise.
3.3 DPRF
In order to adapt the Relax and Fix for the FCNDP-
UOF, we separate the set of variables x
k
i j
, (i, j) ∈ E,
k ∈ K, in |K| disjoint sets, where |K| is the number of
commodities on the model, so that the heuristic per-
forms |K| iterations. At each iteration k, the variables
x
k
i j
∈ Q
k
are defined as binary. After solving the re-
laxed model, if it returns a feasible solution, we fix
the variables y
e
, that are both zero and attend to the
reduced cost criterion for variable fixing, as zero.
The impact of the ordering of the commodities in the
fixing procedure was not the focus of this work, so
we followed the order in which the commodities ap-
peared in the instance.
The function SolveLR(V, E, K, MinCost) solves the
linear relaxation of the MLF Model for the sets V ,
E and K, taking into consideration the primal bound
MinCost. The RCV F(y
e
) function returns TRUE if
the Linear Relaxation cost plus the Reduced Cost of
y
e
is lower than the current Relax and Fix solution.
Since y
e
and x
k
e
are decision variables in the integer
programming model. The function Feas(s) returns
true if the solution s passed as parameter is a feasible
solution to the problem and returns false otherwise.
Algorithm 3: DPRF.
Data: MinCost ← ∞
begin
s ← PartialDecoupling(γ);
MinCost ← Cost(s) ;
for k ∈ K do
for e ∈ E do
x
k
e
∈ {0, 1};
s ← SolveLR(V, E, K, MinCost);
for e ∈ E do
if y
e
= 0 and RCV F(y
e
) = T RUE
then
y
e
= 0;
if Cost(s) < MinCost and
Feas(s) = T RUE then
s
best
← s ;
MinCost ← Cost(s
best
) ;
return s
best
Since the Partial Decoupling Heuristic provides a fea-
sible solution, no recovery strategy was developed in
case the current fixing of the variables turns out to be
at infeasible.
4 COMPUTATIONAL RESULTS
In this section we present computational results for
the one-level model and for the Relax-and-Fix pre-
sented in the previous section.
The algorithms were coded in Xpress Mosel using
FICO Xpress Optimization Suite, on an Intel (R) Core
TM 2 CPU 6400@2.13GHz computer with 2GB of
RAM. Computing times are reported in seconds. In
order to test not only the performance of the one-
level model, but also the performance of the presented
heuristic, we used networks data obtained through
private communication with one of the authors of
ICORES2014-InternationalConferenceonOperationsResearchandEnterpriseSystems
104
Table 1: Computational results for Tabu Search and GRASP approach.
Exact Tabu Search MLF GRASP
Opt Best Sol Best Time GAP Avg Sol Avg Time Dev Sol Dev Time Best Sol Best Time GAP
30-0.8-30-001 4830 4927 1110 0.020 4871 332.144 0 9.227 4871 330.908 0.008
30-0.8-30-002 6989 7322 93 0.048 7122.2 328.295 182.39 4.115 6989 325.357 0.000
30-0.8-30-003 7746 8142 565 0.051 8124 337.191 16.43 33.634 8112 321.838 0.047
30-0.8-30-004 8384 8828 1287 0.053 8384 318.062 0 26.091 8384 338.249 0.000
30-0.8-30-005 7428 7502 794 0.010 7442.8 321.434 33.09 17.889 7428 344.367 0.000
Avg 769.8 0.04 327.42 0.01
Table 2: Computational results for Tabu Search and DPRF approach.
Exact Tabu Search MLF DPRF
Opt Best Sol Best Time GAP Avg Sol Avg Time Dev Time Best Sol Best Time GAP
30-0.8-30-001 4830 4927 1110 0.020 4830 8.88 0.04 4830 8.8 0
30-0.8-30-002 6989 7322 93 0.048 7322 33.52 0.02 7322 33.49 0.048
30-0.8-30-003 7746 8142 565 0.051 8112 35.64 0.03 8112 35.61 0.047
30-0.8-30-004 8384 8828 1287 0.053 8828 60.45 0.19 8828 60.29 0.053
30-0.8-30-005 7428 7502 794 0.010 7585 14.5 0.04 7585 14.46 0.021
Avg 769.8 0.04 30.6 0.03
Table 3: Computational results for GRASP and DPRF approach.
Exact GRASP DPRF
Opt Avg Sol Avg Time Dev Sol Dev Time Best Sol Best Time GAP Avg Sol Avg Time Dev Time Best Sol Best Time GAP
20-0.3-10-001 5978 6513.58 15.65 136.48 0.34 6411 15.50 0.07 5978.00 0.16 0.0424 5978 0.14 0.00
20-0.3-10-002 10469 10813.30 16.57 185.69 0.58 10664 16.38 0.02 10724.00 0.63 0.0011 10724 0.63 0.02
20-0.3-10-003 7020 7286.40 15.99 132.14 0.34 7200 15.67 0.03 7020.00 0.90 0.0199 7020 0.86 0.00
20-0.3-10-004 5484 5754.74 15.84 116.73 0.33 5598 15.71 0.02 5543.00 1.73 0.0143 5543 1.72 0.01
20-0.3-10-005 7932 8322.00 16.04 0.00 0.40 8322 16.01 0.05 8070.00 0.38 0.0011 8070 0.38 0.02
20-0.3-20-001 9488 9488.00 32.10 0.00 1.36 9488 31.84 0.00 9488.00 0.45 0.0004 9488 0.45 0.00
20-0.3-20-002 11521 11699.86 31.64 201.31 0.91 11607 30.94 0.01 11522.00 1.06 0.0031 11522 1.05 0.00
20-0.3-20-003 8270 8670.82 32.57 222.90 0.72 8568 32.44 0.04 8270.00 1.39 0.0044 8270 1.39 0.00
20-0.3-20-004 11901 12320.58 31.94 300.06 1.07 11985 31.62 0.01 12400.00 1.53 0.0024 12400 1.53 0.04
20-0.3-20-005 9656 10379.38 32.12 178.59 0.46 10297 31.93 0.07 9656.00 1.21 0.0008 9656 1.21 0.00
20-0.3-30-001 12510 13244.00 49.28 0.00 0.76 13244 48.69 0.06 12510.00 1.03 0.0016 12510 1.03 0.00
20-0.3-30-002 14216 14854.90 49.81 364.81 1.76 14737 49.41 0.04 14216.00 1.22 0.0045 14216 1.22 0.00
20-0.3-30-003 13393 14687.52 48.18 577.28 1.41 14629 47.79 0.09 13393.00 3.19 0.0036 13393 3.18 0.00
20-0.3-30-004 14452 15420.97 48.62 327.77 0.63 15329 48.32 0.06 14452.00 1.96 0.0034 14452 1.96 0.00
20-0.3-30-005 11419 12599.00 51.32 0.00 1.08 12599 51.02 0.10 11419.00 1.01 0.0018 11419 1.01 0.00
20-0.5-10-001 4784 4784.00 21.56 0.00 0.83 4784 21.43 0.00 4932.00 0.98 0.0038 4932 0.97 0.03
20-0.5-10-002 7689 7689.00 21.86 0.00 0.57 7689 21.73 0.00 7689.00 0.62 0.0025 7689 0.62 0.00
20-0.5-10-003 6184 6184.00 22.68 0.00 0.47 6184 22.45 0.00 6237.00 0.47 0.0005 6237 0.47 0.01
20-0.5-10-004 5189 5532.91 22.41 95.20 0.29 5489 22.19 0.06 5444.00 0.20 0.0004 5444 0.20 0.05
20-0.5-10-005 6051 6233.72 22.78 80.47 0.59 6172 22.74 0.02 6051.00 1.51 0.0077 6051 1.50 0.00
20-0.5-20-001 8816 9964.00 46.50 0.00 0.95 9964 45.85 0.13 8816.00 1.18 0.0015 8816 1.18 0.00
20-0.5-20-002 8584 8721.34 47.45 150.45 1.83 8584 46.89 0.00 8584.00 0.81 0.0005 8584 0.81 0.00
20-0.5-20-003 7560 8354.83 45.72 214.84 0.92 8305 44.65 0.10 7560.00 1.78 0.0057 7560 1.78 0.00
20-0.5-20-004 7634 7750.74 45.28 100.06 0.84 7674 44.92 0.01 7634.00 0.72 0.0008 7634 0.72 0.00
20-0.5-20-005 8270 8636.00 44.86 0.00 1.12 8636 44.77 0.04 8270.00 1.98 0.0042 8270 1.97 0.00
20-0.5-30-001 10156 12600.00 67.99 0.00 2.34 12600 67.99 0.24 10156.00 1.35 0.0005 10156 1.35 0.00
20-0.5-30-002 11403 12932.00 68.66 0.00 1.91 12932 68.66 0.13 11403.00 3.15 0.0026 11403 3.15 0.00
20-0.5-30-003 11600 13021.40 73.29 334.74 1.35 12867 71.57 0.11 11671.00 8.47 0.0328 11671 8.45 0.01
20-0.5-30-004 11785 12333.56 70.88 317.15 1.32 12260 68.82 0.04 11978.00 2.68 0.0015 11978 2.68 0.02
20-0.5-30-005 9559 10989.00 69.47 0.00 1.82 10989 69.33 0.15 9559.00 2.90 0.0029 9559 2.90 0.00
20-0.8-10-001 3947 4120.80 34.32 105.35 0.90 4040 34.32 0.02 3947.00 0.27 0.0004 3947 0.27 0.00
20-0.8-10-002 3743 3915.00 34.51 0.00 1.13 3915 34.02 0.05 3809.00 1.45 0.0150 3809 1.44 0.02
20-0.8-10-003 3412 3480.24 34.81 74.75 0.58 3412 34.39 0.00 3412.00 0.05 0.0004 3412 0.05 0.00
20-0.8-10-004 4086 4209.00 35.27 0.00 0.80 4209 34.99 0.03 4086.00 0.79 0.0008 4086 0.79 0.00
20-0.8-10-005 4498 4542.98 35.64 97.51 0.77 4498 35.28 0.00 4574.00 0.73 0.0008 4574 0.73 0.02
20-0.8-20-001 5796 6909.00 70.88 0.00 1.73 6909 69.22 0.19 5992.00 1.69 0.0099 5992 1.69 0.03
20-0.8-20-002 7037 7635.54 71.48 187.03 1.02 7590 70.34 0.08 7321.00 15.45 0.0134 7321 15.44 0.04
20-0.8-20-003 4596 6251.89 69.00 89.48 1.84 5422 68.18 0.18 4596.00 3.51 0.0008 4596 3.51 0.00
20-0.8-20-004 4851 5187.00 70.26 69.01 2.45 5250 69.98 0.08 4851.00 1.11 0.0019 4851 1.11 0.00
20-0.8-20-005 6086 6855.53 72.13 86.23 1.93 6267 71.42 0.03 6086.00 4.08 0.0110 6086 4.08 0.00
20-0.8-30-001 7769 9425.00 105.01 0.00 2.17 9425 101.23 0.21 7769.00 2.21 0.0037 7769 2.21 0.00
20-0.8-30-002 7681 8735.33 110.77 126.42 1.98 8666 109.89 0.13 7681.00 4.80 0.0131 7681 4.79 0.00
20-0.8-30-003 5144 5947.89 107.30 201.43 2.67 5889 106.24 0.14 5709.00 3.07 0.0268 5709 3.04 0.11
20-0.8-30-004 7188 8768.08 104.77 177.53 3.74 8630 104.56 0.20 7387.00 17.34 0.0154 7387 17.33 0.03
20-0.8-30-005 7374 8175.16 108.08 127.82 1.46 7942 108.08 0.08 7374.00 3.94 0.0050 7374 3.93 0.00
AVG 49.32 0.07 2.38 0.01
AnImprovedRelax-and-FixAlgorithmfortheFixedChargeNetworkDesignProblemwithUser-optimalFlow
105
(Mauttone et al., 2008).
The instances are grouped according to the num-
ber of nodes in the graph (10, 20, 30),followed by the
graph density (0.3, 0.5, 0.8) and finally the amount
of different commodities to be transported. For the
presented tables, we report the optimum value found
by exact model (Opt), the best solution (Best Sol) and
best time (Best Time) reached by selected approach,
and the gap value between exact and heuristic (GAP).
We also reported the average values for time (Avg
Time) and for solutions (Avg Sol). Finally, reported
standard deviation values for time(Dev Time) and so-
lution(Dev Sol). In both tables the results in bold rep-
resent the best solution found, while the underlined
ones represent that the optimum has been found.
In Table 1 and 2, we present the results reached
for the instances generated by (Mauttone et al., 2008).
For these five instances, three heuristics were com-
pared: the Tabu Search heuristic proposed by (Maut-
tone et al., 2008), the GRASP heuristic of (Gonz
´
alez
et al., 2013) and the DPRF algorithm. For the Tabu
Search, the average time was high and no optimum
solution was found. When observing the gap value,
the table shows that the GRASP heuristic obtained
best solutions in general, however the computational
time is very high in comparison with the DPRF
heuristic. Moreover, the standard deviation obtained
by GRASP presented high values suggesting the al-
gorithm has a irregular behavior and for the DPRF
algorithm all standard deviation values for solutions
were 0. Although for those instances GRASP outper-
form the DPRF in solution quality (3 out of 5), table
2 shows that DPRF outperform the Tabu Search pre-
sented by (Mauttone et al., 2008).
In Table 3 were used another 45 instances gener-
ated by Mautonne, Labb and Figueiredo, whose re-
sults were not published by them. For this group of
instances, the computational results suggest the effi-
ciency of DPRF heuristic. On average, the DPRF was
20 times faster than GRASP. Also, DPRF found 29
optimal solutions, while GRASP found only 7 opti-
mal solutions. Besides that, the DPRF also improved
or equaled GRASP results for 40 (36 improvements)
out of 45 instances.
5 CONCLUSIONS AND FUTURE
WORKS
We proposed a new algorithm for a variant of the
fixed-charge uncapacitated network design problem
where multiple shortest path problems are added to
the original problem. In the first phase of the algo-
rithm, the Partial Decoupling heuristic is used to build
a initial solution. In the second phase, a Relax and Fix
heuristic is applied to improve the solution cost.
The proposed approach was tested on a set of in-
stances grouped by graph density, number of nodes
and commodities. Our results have shown the effi-
ciency of DPRF in comparison with a GRASP and
Tabu Search heuristic, once that the proposed algo-
rithm presented best average time for all instances,
often reaching optimum solutions. In a few cases,
GRASP reached best solution values, however the
computational time spend was not good when com-
pared with DPRF.
As future work, we intend to work on exact ap-
proaches as Benders decomposition and Lagrangian
relaxation since both are very effective for similar
problems, as could be seen in (Bektas et al., 2007)
and (Costa et al., 2007).
ACKNOWLEDGEMENTS
This work was supported by CAPES (Process
Number: BEX 9877/13-4) and by Laboratoire
d’Informatique d’Avignon, Universit d’Avignon et
des Pays de Vaucluse, Avignon, France.
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