AN EXACT ALGORITHM FOR THE CLOSE ENOUGH TRAVELING
SALESMAN PROBLEM WITH ARC COVERING CONSTRAINTS
Minh Hoang Ha
1,3
, Nathalie Bostel
2
, Andr´e Langevin
1
and Louis-Martin Rousseau
1
1
Department of Mathematics and Industrial Engineering and CIRRELT,
´
Ecole Polytechnique de Montr´eal, C.P. 6079,
Succursale Centre-ville, Montr´eal, Qu´e., H3C 3A7, Canada
2
IUT de Saint-Nazaire, IRCCyN, 58 rue Michel Ange, B.P. 420, 44606 Saint-Nazaire Cedex, France
3
´
Ecole des Mines de Nantes, IRCCyN, 4 rue Alfred Kastler, 44307 Nantes Cedex 3, France
Keywords:
Close enough traveling salesman problem, Automated meter reading, Radio frequency identification, Cutting
planes.
Abstract:
In this paper, weconsider a problem still seldom studied in the literature, the Close Enough Traveling Salesman
problem with covering constraints on the arcs. This problem arises in the context of utility companies that use
automated meter reading (AMR) with radio frequency identification (RFID) technology to read meters from
a distance. The contribution of this paper is to introduce a new mathematical formulation and to propose a
first exact algorithm for this problem. Computational results show that our algorithm is capable of solving
to optimality instances of realistic size, such as those introduced in (Golden et al., 2008), with 1000 arcs and
9000 customers in less than 2 hours.
1 INTRODUCTION
Recent technological advances in the combined us-
age of automated meter reading and radio frequency
identification are allowing an increasing number of
utility companies to retrieve customer information re-
motely. With this technology, the exact consumption
of a resource (electricity, gas or water) by customers is
coded in a unique radio frequencysignal and transmit-
ted from metering devices, situated at the customer
location, to portable receivers, which can be mounted
in moving vehicles. The effective radius of automated
meter reading (AMR), also called read range, is nor-
mally between 150 and 300 meters, but may be as
high as 381 meters (see (Golden et al., 2008)). Al-
though AMR is based on various technologies, we
consider the most common version (radio frequency
based) to study a variant of an arc routing problem
dealing with mobile or ”drive-by” meter reading. In
this variant, a reading device is installed in a vehi-
cle, which drives around to collect remotely the data
sent by metering devices. Thus, the reader does not
have to reach each customer to collect the data, but
only needs to pass within its read range. Each vehicle
must traverse a service area and pass close enough to
each meter so that they can all be read.
More formally, let G = (V, A) be a directed graph,
where V is the vertex set, V = {v
0
, ..., v
n−1
}, vertex v
0
is the depot, and A = {(v
i
, v
j
) : v
i
, v
j
∈ V} is the arc
set of size m. Numbering the arcs enables A to be
expressed as {a
1
, a
2
, ..., a
m
} and a cost c
a
to be asso-
ciated with each arc of A. Let W = {w
1
, w
2
, ..., w
l
} be
a set of customers that must be covered. These cus-
tomers can be located everywhere on the area covered
by the network, and havea different vertex. The Close
Enough Traveling Salesman Problem with arc cover-
ing constraints (CETSP) consists in finding a mini-
mum cost tour, which begins and ends at the depot,
such that every customer of W is covered by the tour,
i.e. lies within a distance r from an arc of the tour.
The CETSP is NP-hard as it reduces to a Rural Post-
man Problem (RPP) when r = 0, i.e. every customer
of W coincides with a point on an arc of A, and the
number of arcs containing the customers (called re-
quired arcs) is less than m, the number of arcs in the
graph. Note that, if the number of required arcs is
equal to m, the problem becomes a Chinese Postman
Problem (CPP).
To the best of our knowledge, the only work on
the CETSP can be credited to (Golden et al., 2008).
The authors call the problem a CETSP over a street
network and propose four heuristics to solve 18 in-
stances with an average of about 900 arcs and 9000
customers each. Among these instances, only two of
them are described in more detail: a sparse instance
that includes 3345 customers and 405 arcs and a dense
instance with 10,230 customers and 1099 arcs. Basi-
cally, the heuristics of (Golden et al., 2008) are imple-
233
Hoang Ha M., Bostel N., Langevin A. and Rousseau L..
AN EXACT ALGORITHM FOR THE CLOSE ENOUGH TRAVELING SALESMAN PROBLEM WITH ARC COVERING CONSTRAINTS.
DOI: 10.5220/0003698702330238
In Proceedings of the 1st International Conference on Operations Research and Enterprise Systems (ICORES-2012), pages 233-238
ISBN: 978-989-8425-97-3
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

mented through a two-stage process. In stage one, the
heuristic identifies a subset of arcs to be traversed, ei-
ther with some simple greedy procedures or through
solving an integer program. In stage two, the prob-
lem becomes the well-knownRural Postman Problem
(RPP) and is solved by a sophisticated heuristic.
There is another problem which can be seen as
a CETSP with the vertex covering constraints. It is
named the covering tour problem by (Gendreau et al.,
1997) and is similar to the CETSP with arc cover-
ing constraints except that a closed tour has to be
determined so that every vertex of W lies within a
distance r from a vertex of the tour. In (Gendreau
et al., 1997), an exact algorithm and a heuristic are
presented. Heuristics have also been proposed for
problems belonging to the family of the covering tour
problem, such as the CETSP in the plane in (Gulczyn-
ski et al., 2006) and (Dong et al., 2007).
In this paper, we address the CETSP, which we
formulate as an integer program and solve through a
cutting plane approach. Computational experiments
show that our approach is efficient, being capable of
solving instances in some cases with up to 1500 arcs
in less than 2 hours.
The remainder of the paper is organized as fol-
lows: Section 2 presents in detail our exact algo-
rithm to solve the CETSP. Computational results are
reported and analysed in Section 3. Finally, Section 4
summarizes our conclusions.
2 EXACT ALGORITHM FOR
CETSP
Given a node subset, S ⊆ V, let δ
+
(S) denote the set
of outgoing arcs of S and δ
−
(S) denote the set of in-
coming arcs of S. If S = {v
k
}, we simply write δ
+
(k)
(or δ
−
(k)) instead of δ
+
({v
k
}) (or δ
−
({v
k
})). E(S)
is the set of arcs with both end-points in S. Let x
a
be
the number of times arc a is traversed, and c
a
the as-
sociated cost (distance or travel time). We define the
binary coefficients λ
lk
equal to 1 if and only if w
l
∈ W
can be covered by a
k
∈ A. Given x ∈ N
|A|
and T ⊂ A,
x(T) denotes
∑
e∈T
x
e
. Then the CETSP can be stated
as:
Minimize
∑
a∈A
c
a
x
a
(1)
subject to x

δ
+
(0)

≥ 1 (2)
x

δ
+
(i)

− x

δ
−
(i)

= 0 ∀i ∈ V (3)
∑
a∈A
x
a
.λ
wa
≥ 1 ∀w ∈ W (4)
Mx

δ
+
(S)

− x

E(S)

≥ 0 ∀S ⊂ V − {v
0
}
and 2 ≤ |S| ≤ n− 2 (5)
x
a
∈ Z
+
∀a ∈ A (6)
where M is a large number. The objective (1) is
to minimize the total travel cost. Constraint (2) en-
sures that the depot belongs to the tour, while (3)
are the flow conservation constraints. Constraints (4)
enforce that every customer of W is covered by the
tour and constraints (5) are the disjoint subtour elim-
ination constraints. (In arc routing, an optimal tour
can contain a cycle. The disjoint ”subtour elimination
constraints” eliminate subtours disconnected from the
tour containing the depot). These constraints force
the presence of at least one outgoing arc of any set S,
for every possible subset S of V containing an arc be-
longing to the tour. Constraints (6) define the variable
domains.
We propose to solve the CETSP optimally through
a cutting plane approach. The main strategy of this al-
gorithm is to solve iteratively an integer program in-
cluding the constraints (2), (3), (4), and (6). At each
iteration, the disjoint subtour elimination constraints
(5) violated by the optimal solution are added to the
model. A disjoint subtour can be identified through
a depth-first search which, starting from a given ver-
tex, traverses all the solution arcs to reach all other
vertices. The encountered vertices marked are placed
on a stack in the order in which they are visited. Af-
ter an arc is traversed, it is removed from the current
graph. A disjoint subtour is created when, starting
from some vertex, it is not possible to mark all the
vertices present in the solution. Once a disjoint sub-
tour is identified, the process can be repeated starting
from any unmarked vertex of the solution.
Note that the constraints (5) can only be applied
to the disjoint subtours that do not contain the depot.
This is because, for the subtours S
0
containing the de-
pot, the disjoint subtour elimination can be obtained
by the use of arcs not already used in the solution but
with head points belonging to S
0
, the set of nodes of
the subtour containing the depot. In order to improve
the performance of the algorithm, we first check the
covering of such subtours. If we can not cover all the
customers with the arcs of S
0
(used and unused), the
constraint (5) is then still applied; otherwise, we add
the following constraint for S
0
:
Mx

δ
∗
(S
0
)

− x

E
′
(S
0
)

≥ 0 (7)
where E
′
(S
0
) denotes the set of arcs used in S
0
while δ
∗
(S
0
) denotes the set of arcs that are not used
and whose heads are in S
0
.
ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems
234

3 COMPUTATIONAL
EXPERIMENTS
In this section, we describe the CETSP instances
and the computational evaluation of the proposed ap-
proach. Our algorithm is coded in C/C++ using Cplex
12.1 with Callable Library and was run on a 2.4 GHz
CPU with 2GB RAM. The running time for each in-
stance was limited to 2 hours. We have tested differ-
ent values of M (5000, 10,000 and 20,000) and ob-
served that its impact on the performance of the algo-
rithm is negligible. We decided to predetermine M at
a value of 10,000 in the implementation.
3.1 Data Instances
To build the CETSP instances, we randomly generate
graphs that imitate real street networks by the follow-
ing procedure:
• The coordinates of n vertices are randomly gener-
ated in a unitary square. Then a heuristic is used to
find the shortest tour passing through all the nodes
exactly once. This tour is a Hamiltonian cycle
and is used as a framework to construct the full
graph. The resulting graph is therefore strongly
connected.
• In order to imitate real networks, random arcs are
added to the current tour to reach a total number of
arcs m = nd, where n denotes the number of ver-
tices and d the ratio between the number of arcs
and the number of nodes, in such a way that: (i)
the arcs are not too long, and (ii) there is no inter-
section between any two arcs.
In our tests, we use the graphs with the num-
ber of vertices n ∈ {300, 400, 500} and the ratio be-
tween the number of arcs and the number of nodes
d ∈ {1.5, 2, 2.5, 3}. For each couple of n and d, we
have generated 10 different graphs. Arc costs are de-
fined as c
ij
kilometres, where c
ij
is the Euclidean dis-
tance between v
i
and v
j
multiplied by 5 to obtain an
average length of arcs close to reality (from about 0.2
to 0.4 kilometres).
Once the graphs are created, the CETSP instances
are generated by randomly positioning q = mt cus-
tomer nodes on the square containing the graph,
where m denotes the number of arcs and t the ratio
between the number of customers and the number of
arcs, t ∈ {0.5, 1, 5, 10}. Thus, for each graph, four
CETSP instances are created. The effective RFID ra-
dius r is set at a value of 150 meters. In order to en-
sure the existence of a solution, we delete all the cus-
tomers that can not be covered by any arc. We also
examine the impact of increasing the radius parame-
ter from 150 meters to 200 meters. To do this, we still
use the graph created with r = 150 meters but change
the read range to 200 meters. In other words, the arcs
and coordinates of the customers and vertices are kept
constant while r is increased.
For each value of r, we thus generate 480 CETSP
instances named ce-n-k-t, where n is the number of
nodes, k indicates the number of arcs and t is the ra-
tio between the number of customers and the number
of arcs. For example, ce-300-450-10 stands for an in-
stance with 300 nodes, 450 arcs, and t =10.
In order to analyse further the impact of customer
number on our algorithm, we use a customer reduc-
tion procedure. Given w
l
∈ W, let Z(w
l
) be the set
of arcs that can cover w
l
. Consider each pair of cus-
tomers w
i
and w
j
, if Z(w
i
) ⊆ Z(w
j
) then customer w
j
can be eliminated. This is because when we service
w
i
, w
j
is covered at the same time. Note that, the
number of remaining customers is also the maximum
number of arcs that have to be activated for covering
purpose.
3.2 Computational Results
Tables 1 and 2 present the characteristics of the in-
stances and the computational results obtained for
two different values of r (radius of AMR). They in-
clude the name of the instance and the average num-
ber of remaining customers after the reduction pro-
cedure (in columns 1 and 2). #ofOpt indicates the
number of optimal solutions obtained for each set and
OptVal the average optimal value (in km) for these
solutions. #ofIPIter presents the average, minimum
and maximum number of integer programming itera-
tions. Time shows the average, minimum and maxi-
mum running time (in seconds).
The results presented on Table 1 and 2 indicate
that the ratio between the number of remaining cus-
tomers and the number of arcs in graph is often be-
tween 0.2 and 0.5. Therefore, the number of arcs that
must be activated for covering purpose is smaller than
the total number of arcs and our instances are thus far
from a CPP.
As can be observed in Table 1, for t = {5,10} our
algorithm was able to solve all butone of the instances
with 1250 arcs. It was also capable of solving some
instances with up to 1500 arcs. But when t is smaller,
the instances become more difficult. This counterin-
tuitive behaviour can be explained as follows. When
the number of customers decreases, the number of
arcs that must be activated for covering purpose also
decreases. Thus, there are more potential combina-
tions to connect these ”covering” arcs. More MIP it-
AN EXACT ALGORITHM FOR THE CLOSE ENOUGH TRAVELING SALESMAN PROBLEM WITH ARC
COVERING CONSTRAINTS
235

Table 1: Computational results with r = 150 meters.
# of # of Opt Val # of IP Iter Time (sec)
Data Cus Opt (km) Aver. Min Max Aver. Min Max
ce300-450-0.5 102 10 94.5 6.4 2 24 0.52 0.05 2.5
ce300-450-1 142 10 118.8 2.8 1 8 0.08 0.02 0.31
ce300-450-5
202 10 175.6 1.6 1 3 0.08 0.05 0.16
ce300-450-10 212 10 195.4 1.6 1 3 0.15 0.09 0.36
ce300-600-0.5 126 10 100.6 11.5 2 31 3.98 0.17 14.61
ce300-600-1 166 10 123.5 8.5 2 33 3.01 0.14 17.78
ce300-600-5
224 10 182.1 4.9 2 27 1.07 0.16 8.88
ce300-600-10 236 10 195.6 2.1 1 4 0.27 0.16 0.45
ce300-750-0.5 147 9 95.6 36.2 5 112 308.52 1.98 1807.64
ce300-750-1
193 10 120.4 8.2 4 12 5.40 1.59 10.17
ce300-750-5 248 10 172.1 3.3 2 6 0.70 0.25 2.30
ce300-750-10
264 10 186.3 2.9 2 5 0.60 0.30 1.08
ce300-900-0.5 166 7 99.1 12.6 4 32 194.09 1.77 1034.83
ce300-900-1
213 10 122.0 10.9 2 26 45.03 0.56 258.08
ce300-900-5 276 10 163.5 3.9 2 9 3.24 0.56 11.52
ce300-900-10
292 10 175.0 3.1 1 5 2.18 0.61 3.78
ce400-600-0.5 144 10 100.3 11.2 2 49 1.94 0.05 12.45
ce400-600-1 194 10 129.4 3.3 2 7 0.16 0.06 0.41
ce400-600-5
260 10 185.1 1.8 1 3 0.13 0.08 0.22
ce400-600-10 274 10 207.1 1.5 1 3 0.19 0.14 0.30
ce400-800-0.5 174 9 102.1 35.7 4 88 88.25 1.76 356.92
ce400-800-1 226 10 126.5 17.2 3 42 11.49 0.75 33.45
ce400-800-5
296 10 178.8 3.8 2 12 1.09 0.22 7.03
ce400-800-10 309 10 199.0 2.6 2 5 0.52 0.30 1.64
ce400-1000-0.5 198 3 103.5 11.0 10 13 745.15 40.94 2113.89
ce400-1000-1 252 9 122.1 13.1 5 25 492.43 4.63 1992.3
ce400-1000-5
313 10 169.2 4.9 2 13 4.39 0.72 18.20
ce400-1000-10 327 10 183.5 4.2 2 9 4.94 0.59 24.59
ce400-1200-0.5 221 0
ce400-1200-1 279 4 123.0 8.0 4 12 73.65 40.03 135.44
ce400-1200-5 357 10 157.1 5.7 4 9 29.07 6.43 129.79
ce400-1200-10
373 10 167.0 6.9 3 23 26.83 3.21 91.48
ce500-750-0.5 185 10 113.5 39.0 3 323 674.82 0.13 6731.81
ce500-750-1
249 10 141.3 10.7 2 70 6.71 0.11 63.13
ce500-750-5 319 10 201.3 2.6 2 4 0.21 0.17 0.28
ce500-750-10
328 10 227.1 2.2 1 4 0.30 0.23 0.43
ce500-1000-0.5 224 6 126.2 58.0 8 211 1271.11 7.09 6161.19
ce500-1000-1 285 10 131.0 14.3 3 65 30.72 0.95 102.42
ce500-1000-5
360 10 181.8 4.4 3 9 1.80 0.57 7.25
ce500-1000-10 375 10 203.0 4.1 2 6 1.47 0.58 3.00
ce500-1250-0.5 262 0
ce500-1250-1 330 4 122.5 13.8 5 30 1385.46 235.72 2496.89
ce500-1250-5
408 9 172.0 9.0 3 31 62.24 2.17 201.83
ce500-1250-10 428 10 182.2 5.5 3 13 38.53 2.48 124.22
ce500-1500-0.5 284 0
ce500-1500-1 354 0
ce500-1500-5 458 5 163.9 8.0 6 12 402.75 32.19 1008.89
ce500-1500-10
477 8 172.0 5.8 4 8 353.63 15.22 1994.36
ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems
236

Table 2: Computational results with r = 200 meters.
# of # of Opt Val # of IP Iter Time (sec)
Data Cus Opt (km) Aver. Min Max Aver. Min Max
ce300-450-0.5 101 10 74.2 20.9 2 93 7.76 0.05 52.38
ce300-450-1 143 10 89.3 11.3 2 70 1.31 0.05 10.98
ce300-450-5
197 10 115.7 1.5 3 3 0.07 0.05 0.13
ce300-450-10 205 10 129.5 1.3 2 3 0.11 0.08 0.18
ce300-600-0.5 126 9 74.1 53.3 10 201 844.57 4.42 6959.88
ce300-600-1 168 10 83.5 23.2 4 65 16.74 0.72 60.38
ce300-600-5
217 10 115.1 6.4 2 25 1.91 0.13 8.88
ce300-600-10 220 10 125.7 5.6 1 24 1.72 0.22 11.94
ce300-750-0.5 149 4 71.4 29.3 13 61 2981.45 208.53 7088.2
ce300-750-1
192 7 87.5 29.0 8 109 1749.44 14.36 7198.27
ce300-750-5 238 10 113.9 11.2 5 37 31.05 2.41 145.95
ce300-750-10
240 10 122.9 7.0 4 12 7.83 1.88 22.86
ce300-900-0.5 169 1 81.4 87.0 87 87 4352.8 4352.8 4352.8
ce300-900-1
212 3 90.0 35.7 9 52 2895.48 127.5 7070.88
ce300-900-5 264 9 110.4 15.4 4 55 761.51 6.25 2542.66
ce300-900-10
269 10 117.2 8.1 3 25 148.26 2.80 1225.56
ce400-600-0.5 146 10 79.2 13.4 2 50 4.37 0.08 28.67
ce400-600-1 202 9 88.3 22.4 1 157 24.17 0.06 209.91
ce400-600-5
274 10 114.3 6.4 2 21 0.90 0.13 6.06
ce400-600-10 276 10 123.6 4.9 1 25 0.52 0.20 2.66
ce400-800-0.5 180 7 77.6 20.0 7 43 89.64 6.02 264.20
ce400-800-1 239 8 85.2 11.3 6 25 18.18 2.77 62.84
ce400-800-5
314 9 111.6 8.8 4 22 6.61 1.44 16.95
ce400-800-10 321 10 120.3 24.5 4 162 180.89 1.22 1723.02
ce400-1000-0.5 208 0
ce400-1000-1 270 1 95.9 24.0 24 24 7171.13 7171.13 7171.13
ce400-1000-5
331 0
ce400-1000-10 339 6 116.0 16.0 6 39 1874.72 18.91 7186.65
ce500-750-0.5 192 10 78.0 51.8 3 326 384.42 0.36 3534.50
ce500-750-1
263 10 96.3 28.1 3 181 54.15 0.25 505.33
ce500-750-5 363 10 118.6 5.0 2 24 1.44 0.19 11.27
ce500-750-10
377 10 128.8 3.0 2 4 0.54 0.34 0.78
ce500-1000-0.5 242 1 85.1 22.0 22 22 842.83 842.83 842.83
ce500-1000-1
321 5 90.8 19.8 9 52 436.34 13.56 814.22
ce500-1000-5 411 9 109.0 28.2 6 78 1025.59 6.23 5406.33
ce500-1000-10
429 8 117.8 17.3 5 28 483.14 2.81 3248.66
erations are thus needed to solve the problem.
We also observe that, for a given number of ver-
tices, the greater the vertex degree is, the harder the
instance is.
In Table 2, we see the results for the case where
r = 200 meters. The performance of our algorithm
degrades as it can only solve the instances with up
to 1000 arcs. When the read range increases, each
customer can be covered by more arcs, so the solu-
tion space increases and the optimal solution becomes
more difficult to find. Obviously, increasing the read
range leads to a considerable decrease in the travel-
ing distance and cost. The column OptVal confirms
this remark. Developing new technology to allow an
increased read range could therefore be an effective
means to reduce the distance driven to collect cus-
tomer information.
4 CONCLUSIONS
In this paper, we have formulated and solved the
CETSP with arc covering constraints. An integer lin-
ear programming formulation has been proposed and
solved through a cutting plane algorithm. Computa-
tional results on a set of 960 instances have been re-
AN EXACT ALGORITHM FOR THE CLOSE ENOUGH TRAVELING SALESMAN PROBLEM WITH ARC
COVERING CONSTRAINTS
237

ported and analysed. These results show that our al-
gorithm is capable of solving to optimality instances
of realistic size and works better when there are many
customers to be covered. As we notice that RFID
technology is rather used when the customer density
is important, our algorithm is quite suitable to solve
real problems of utility companies.
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