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The capacitated clustering problem (CCP) consists of forming a specified number of clusters or groups from a set of elements in such a way that the sum of the weights of the elements in each cluster is within some capacity limits, and the sum of the benefits between the pairs of elements in the same cluster is maximized. This problem—which has been recently tackled with a GRASP/VNS approach—arises in the context of facility planners at mail processing and distribution. We propose a tabu search and several GRASP variants to find high quality solutions to this NP-hard problem. These variants are based on several neighborhoods, including a new one, in which we implement a one-for-two swapping strategy. We also hybridize both methodologies to achieve improved outcomes. The maximally diverse grouping problem (MDGP) is a special case of the CCP in which all the elements have a weight of 1 U. This problem has been recently studied in the academic context when forming student groups, and we adapt the best method reported in the literature, a tabu search with strategic oscillation (TS_SO), to the CCP. On the other hand, the handover minimization in mobility networks is a problem equivalent to the CCP in which we minimize the sum of the benefits (costs) of the edges between different clusters. GRASP with Path Relinking has been recently applied to it. Our empirical study with 133 instances shows the superiority of the new GRASP with tabu search for the CCP with respect to these three previous approaches: the GRASP/VNS, the adapted TS_SO, and the GRASP with Path Relinking.
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Tabu Search and GRASP
for the Capacitated Clustering Problem
ANNA MARTÍNEZ-GAVARA
Departamento de Estadística e Investigación Operativa, Universidad de Valencia, Spain
Gavara@uv.es
VICENTE CAMPOS
Departamento de Estadística e Investigación Operativa, Universidad de Valencia, Spain
Vicente.Campos@uv.es
MICAEL GALLEGO
Departamento de Ciencias de la Computación, Universidad Rey Juan Carlos, Spain.
Micael.Gallego@urjc.es
MANUEL LAGUNA
Leeds School of Business, University of Colorado at Boulder, USA
laguna@colorado.edu
RAFAEL MARTÍ
Departamento de Estadística e Investigación Operativa, Universidad de Valencia, Spain
Rafael.Marti@uv.es
ABSTRACT
The Capacitated Clustering Problem (CCP) consists of forming a specified number of clusters or
groups from a set of elements in such a way that the sum of the weights of the elements in each
cluster is within some capacity limits, and the sum of the benefits between the pairs of elements in
the same cluster is maximized. This problem which has been recently tackled with a GRASP/VNS
approach arises in the context of facility planners at mail processing and distribution. We propose
a tabu search and several GRASP variants to find high quality solutions to this NP-hard problem.
These variants are based on several neighborhoods, including a new one, in which we implement a
one-for-two swapping strategy. We also hybridize both methodologies to achieve improved
outcomes.
The Maximally Diverse Grouping Problem (MDGP) is a special case of the CCP in which all the
elements have a weight of one unit. This problem has been recently studied in the academic context
when forming student groups, and we adapt the best method reported in the literature, a tabu
search with strategic oscillation (TS_SO), to the CCP. Our empirical study with 50 large instances
shows the superiority of the new GRASP with tabu search for the CCP with respect to both the
previous GRASP/VNS approach and the adapted TS_SO.
Keywords: Capacitated Clustering, Diversity problems, tabu search, metaheuristics.
Draft version: July 29, 2013.
Capacitated Clustering Problem 2
1. Introduction
Given a graph   where is a set of nodes and is a set of edges, let   be the
weight of node    and let  be the benefit of edge  . The Capacitated Clustering
Problem (CCP) consists of partition into clusters in such a way that the sum of the weights of the
elements in each cluster is within some integer capacity limits, and , and the sum of the benefits
between the pairs of elements in the same cluster is maximized.
The CCP can be formulated as a quadratic integer program with binary variables  that take the
value of 1 if element is in cluster and 0 otherwise.
  





     
  
     
       
The objective function adds the total benefit of all pairs of elements that belong to the same cluster.
The first set of constraints forces the assignment of each element to a cluster. The second set of
constraints forces the sum of the weights of the pairs of elements in the same cluster to be between
and .
The literature on the CCP and related problems is vast and a recent paper (Deng and Bard, 2011)
summarizes previous heuristics and formulations for this problem. We therefore do not duplicate
the work here and refer the reader to Deng and Bard’s excellent survey. These authors proposed a
GRASP (Greedy Randomized Adaptive Search Procedure) with VNS (Variable Neighborhood Search)
that, according to their computational study, outperforms previous approaches. An interesting
application of the problem, as mentioned by Deng and Bard, arises in the context of facility planners
at mail processing and distribution centers within the US Postal Service. In particular, the design of
the zones to help rationalize the bulk movement of mail can be framed as a CCP.
An interesting characteristic of the GRASP in Deng and Bard (2011) is that it constructs the
customary restricted candidate list (RCL) using both nodes and edges, which to the best of our
knowledge is an unconventional strategy (Festa and Resende, 2009). This design triggered our
interest on this problem, to compare this unconventional GRASP implementation with a standard
one in which the RCL is formed with either nodes or edges but not both simultaneously. Deng and
Bard (2011) computational study does not consider general CCP instances and limits itself to the
case in which the node weights, , are equal to 1. This means that although they proposed a
procedure for the CPP, their tests were performed on the special case of the MDGP. We extend their
testing to include general CCP instances to compare several GRASP designs.
Capacitated Clustering Problem 3
When reviewing the recent clustering literature, we found an interesting connection not previously
reported. In particular, the maximally diverse grouping problem (MDGP) consists of grouping a set of
elements into mutually disjoint groups in such a way that the diversity among the elements in each
group is maximized. The diversity among the elements in a group is calculated as the sum of the
individual distance between each pair of elements. The objective of the problem is to maximize the
overall diversity, i.e., the sum of the diversity of all groups, when the size of each group is within a
specified range. Clearly, the MDGP is a special case of the CCP for which   for all node , and
the distance between each pair of nodes  is the benefit . Therefore, from the CCP formulation
above, the MDGP can be formulated as:
  





     
  
     
       
The MDGP is called the -partition problem in Feo et al. (1992) and the equitable partition problem
in O’Brien and Mingers (1995). It belongs to the family of diversity problems (Gallego et al. 2009,
Martí and Sandoya 2013). One of the most popular MDGP applications appears in the academic
context of forming student groups (Weitz and Jelassi, 1992). Gallego et al. (2013) proposed a tabu
search with strategic oscillation, TS_SO, for the MDGP that outperforms the previous approaches for
this problem. Recognizing the connection between MDGP and CCP, as part of our work, we also
extend the TS_SO to tackle the general CCP.
The main contributions of our current development are:
A new (simplified) GRASP for the CCP and a thorough
comparison with Deng’s and Bard’s GRASP
An adaptation of TS_SO to handle instances of the CCP
A new TS for the CCP to be hybridized with GRASP
Testing of all the methods on a new set of CCP instances
The next section summarizes the Deng and Bard GRASP for the CCP and the Gallego et al. TS_SO for
the MDGP. Section 3 describes the proposed new procedures. This is followed by the description of
our extensive experimentation that includes 50 new CCP instances with   , and .
Statistical analysis shows the merit of our approach when compared to existing methods (including
the adaptation of TS_SO to the CCP).
Capacitated Clustering Problem 4
2. Existing CCP and Adapted MDGP Approaches
Deng and Bard (2011) proposed a GRASP for the CCP, which as customary in GRASP
implementations, it alternates between two phases: construction and improvement. In the
construction phase, the clusters are first seeded with the heaviest weight edges algorithm (HWE),
and then completed with a greedy randomized procedure. Specifically, the HWE identifies the
nodes with the largest weights and assigns them, separately, to the clusters. The heaviest edges
incident to these nodes are then identified, and their endpoints are assigned to the corresponding
clusters. An alternative constructive method, labeled CMC, proposed by Deng and Bard was shown
to be inferior to HWE and therefore we do not consider it here.
HWE produces clusters containing two nodes. At this point, a candidate list  of elements is built to
continue the construction process according to the GRASP methodology. In particular,  is formed
with the nodes and edges (pairs of nodes) that can be inserted into a solution cluster without
exceeding the upper capacity limit . For each node   , let  be the increase in the
objective function if node is added to cluster . In other words, the sum of the benefits  for all
the nodes already in cluster . Similarly, let  be the increase in the objective function when
edge is added to cluster (i.e., when its two endpoint nodes are in the cluster). The restricted
candidate list  is formed with the best elements in  according to this evaluation. This set of
top candidates is computed as a fraction of the number of elements in , where   . Deng
and Bard implemented a so-called reactive GRASP in which effective values for this search
parameter are self-adjusted as the iterations progress.
An important characteristic of this method is that the  consists of both, nodes and edges. That is,
the next element to be included in the solution may be either a single node or a pair of nodes (if an
edge is selected). This is somewhat unconventional in GRASP, if one considers the implementations
in the annotated bibliography by Festa and Resende (2009). We will compare this design with a
more conventional design in which the  consists of either nodes or edges, but not both.
In the second phase of their GRASP, Deng and Bard proposed three different neighborhoods to
improve a constructed solution: , , and . Let  be the set of nodes in cluster
of this solution, and let be the sum of the weights of the nodes in (i.e., ); then,
must be within the capacity limits:     for    .
is the result of extended insertion moves, in which a node is moved from a cluster to a
cluster . The node is only removed from if this cluster remains feasible after the move (i.e., if
   ), but if there is not enough capacity in cluster for node (i.e., if  ),
instead of discarding the move, a node in is moved from cluster to a different one, say cluster
, with enough capacity for node . The move is performed if the final solution is feasible, in other
words, if    ,     , and  .
consists of edge insertions. Given an edge   if both nodes are in the same cluster
(i.e.,    ), an edge insertion considers moving the nodes to another cluster , as long as the
resulting solution remains feasible. In mathematical terms, we move  from cluster to cluster
if
   and   
Capacitated Clustering Problem 5
This neighborhood also contains moves of edges with endpoints in two different clusters. In
particular, given an edge   with    and   , it examines moving to cluster , moving
to cluster , or moving both nodes to another cluster . In any case, only capacity-feasible moves are
considered.
implements a classical swap move, that is, one in which a node is moved from a cluster to
a cluster , and simultaneously a node is moved from the cluster to the cluster . As in the other
neighborhoods, the move is performed if the resulting solution is feasible. In mathematical terms:
     and    .
Deng and Bard proposed two improvement methods based on these three neighborhoods. The first
one, called CNS, examines the neighborhoods in a sequential/cyclical fashion, terminating when no
improvement is possible. The second one implements the so-called variable neighborhood descent,
VND, in which the method switches from the current neighborhood to the next one,, when
cannot produce an improvement of the current solution. Moreover, if a better solution is
obtained with with   , the method switches back to . The VND terminates when is
applied and no further improvement is possible. Finally, the authors implemented a Randomized
VND, called RNVD, in which the neighborhood to be searched in the next iteration is probabilistically
selected, where the probability of selection is linked to the merit of each neighborhood as
determined by the quality of the solutions found during the search.
In the computational testing, Deng and Bard (2011) compared their designs and concluded that the
combination of HWE with RVND resulted in the best overall performance. We use this variant for the
purpose of comparison and call it Prev_GRASP. As a side note, Deng and Bard tested a Path Relinking
post-processing strategy and concluded that its inclusion did not result in a significant improvement
and therefore we have left it out of our experiments.
Gallego et al. (2013) proposed a tabu search with strategic oscillation, TS_SO, for the maximally
diverse grouping problem (MDGP). TS_SO is the state-of-the-art solution method for MDGPs.
Empirical evidence shows that TS_SO outperforms the improvement method LCW (Weitz and
Lakshminarayanan, 1998) and the hybrid genetic algorithm LSGA (Fan et al., 2011). As described in
the previous section, the MDGP is a special case of the CCP. We briefly describe TS_SO, which is
based on three steps: 1) construction of the initial solution, 2) neighborhood search and 3) strategic
oscillation, and our adaptation of TS_SO to the CCP.
The construction step (GC, for greedy construction) for the MDGP starts by randomly selecting
nodes and assigning each of these elements to a separate group (cluster in our case). Therefore, at
the end of the first step, each group has one node assigned to it. Then, the procedure performs
iterations to assign the remaining unassigned elements to groups. In order to generate a feasible
solution, the iterations are divided into two phases. In the first phase, the elements are assigned to
groups that currently contain fewer elements than the desired minimum number of elements. In the
second phase, the remaining elements are assigned to groups with a number of elements that is
smaller than the desired maximum number of elements.
Capacitated Clustering Problem 6
In our adaptation of GC to the CCP, the first phase adds a randomly selected node to the cluster
that maximizes and for which  . When all the clusters have a sum of weights of their
nodes larger than or equal to , the second phase proceeds in a similar way, by adding randomly
selected elements to the cluster that maximizes , without exceeding the upper capacity limit
.
Once a solution has been constructed, the neighborhood search (phase 2 of the TS_SO method) is
applied. The search neighborhood in TS_SO consists of all node insertions and swaps. Note that
insertions and swaps are considered simultaneously instead of sequentially, as done by Deng and
Bard (2011). Moreover, the TS_SO implements simple insertions, involving only two groups, instead
of the extended version (involving three groups) implemented in . We denote the simple
insertion neighborhood as . In mathematical terms, at each step the method selects the best
feasible move in . TS_SO includes a short-term tabu memory (Glover and Laguna,
1997) that enables the method to search beyond the first local optimal point. Specifically, when the
local search reaches a point where no improving moves are available, the best non-improving move
is selected and executed. At this point, elements that are moved from their current group to another
are not allowed to move again for    iterations. The process terminates when
   consecutive iterations have been performed without improving the best solution
found during the search. TS_SO can be straightforwardly adapted to the CCP by simply checking the
feasibility of moves in terms of the sum of the weights of the nodes in each cluster instead of the
cluster cardinality.
So far, we have assumed that feasibility is maintained through the search. However, the third phase
of TS_SO the strategic oscillation (SO) phase explores solutions for which the group cardinality
restrictions may be violated. In particular, the method applies the neighborhood search described
above but the number of the elements in a group is allowed to be outside the specified limits by a
certain amount  that ranges between 0 and . This means that when    the search is
allowed to visit infeasible solutions. To create the oscillation pattern, the value of  is reset to one
after every successful application of the improvement method, otherwise  is increased by one unit
until it reaches .
In our adaptation of the SO phase to the CCP, we consider that the sum of the weights of the
elements in a cluster has to be within the following limits:
      
Note that this method does not guarantee that the final solution is feasible. When this happens, we
apply a repair mechanism that consists of removing elements from clusters for which   and
adding elements to clusters for  . The elements are selected at random and the process
continues until the weight constraints of the clusters are satisfied. We refer to the entire adaptation
of TS_SO to the CPP as CCP_TS_SO.
3. A Simplified GRASP for the CCP
In our view, the GRASP implementation of Deng and Bard (2011) is overly complicated and a
simplified version should be able to obtain solutions of similar quality. The best way to describe their
Capacitated Clustering Problem 7
method is that it is the result of hybridizing Reactive GRASP with VND with three different
neighborhoods that are combined in a probabilistic fashion, where the neighborhoods consist of a
set of compound moves that may change the node assignment of up to three clusters. In addition,
the candidate list in the construction process includes both nodes and edges. In contrast, we
propose a simplified GRASP implementation in which only nodes are candidates in the construction
process and two simple neighborhoods are combined into a deterministic VND design.
Our GRASP starts by seeding the clusters with randomly selected nodes. Then, we
explore the clusters in lexicographical order assigning elements until all of them satisfy the lower
bound constraint (i.e., until the sum of the weights of the nodes already assigned to the cluster is
larger than or equal to ). To do so, the candidate set  is formed with all the unassigned nodes
and the value  is calculated for all pairs  of nodes and clusters. that is, the
restricted candidate list of nodes for cluster is formed with all nodes for which  is within a
percentage    of the maximum value  in :
       where   
 
Our constructive method randomly selects an element in , and performs the corresponding
assignment. The method proceeds in this way, starting with cluster   and assigning elements to
it until the lower bound constraint is satisfied (   where ). Then, the process
moves to cluster 2 and proceeds in this way until the sum of the weights of the elements assigned to
each cluster is larger than or equal to .
In the following steps, the candidate set  is formed with the pairs  with unassigned nodes
and those clusters such that the sum of the weights of the elements already assigned to the cluster
plus the weight of is lower than or equal to .
        where 
The method proceeds to evaluate  for all  in  build  with the  pairs with an
evaluation within a percentage    of the maximum value in , and select one pair at
random. It stops when all the nodes have been assigned to clusters.
Once a solution is obtained, we apply our improvement method, which consists of a deterministic
VND (Variable Neighborhood Descent) based on two neighborhoods, and . The method
determines first a best neighbor  of in . If  is better than , then is replaced with  and
the method searches now for the best neighbor in , thus performing a local search in while
it improves the current solution. When the current solution cannot be improved in , then the
method resorts to and determines the best neighbor  of in . If  is better than , then
the method comes back to search in ; otherwise the VND finishes. In short, the algorithm
performs a local search for the best solution in and only resorts to searching when the process
is trapped in a local optimum found in . The improvement method considers only feasible moves.
Capacitated Clustering Problem 8
4. Tabu Search for the CCP
As an alternative to the methods described in the previous sections, we engaged in the development
of a tabu search specifically designed for the CCP. In other words, we studied the characteristics of
the CCP and instead of adapting an existing procedure, we designed an original one. The main
characteristic of this procedure consists of a new neighborhood structure based on 2-1 exchanges
that we refer to as .
 may be considered a variant of  in the sense that it explores exchanges of two nodes,
say and , belonging to the same cluster for a node that belongs to a cluster . In this
neighborhood, unlike in , nodes and may or may not be connected. That is,  does not
require that   . The neighborhood includes moves that are contained neither in  nor in
. For instance, consider the following situation.
Cluster A contains nodes 1 and 2 with weights    and has remaining capacity of 2 units. A
Cluster B contains node 3 with   and remaining capacity of 1. Under , the insertions of
node 1 to cluster B, node 2 to cluster B and node 3 to cluster A are considered. However, none of
these moves are feasible. Likewise, the swaps of nodes 1 with 3 and 2 with 3 under  result in
infeasible solutions. However, the 2-1 exchange that moves nodes 1 and 2 from cluster A to B and
node 3 from cluster B to A is feasible.
Our short-term memory tabu search based on  operates as follows. All 2-1 exchanges are
evaluated and the best move (according to the objective function value) is selected. The three nodes
participating in the exchange are made tabu-active. In subsequent iterations, a move is classified
tabu if it contains one or more tabu-active nodes. Nodes remain tabu-active for  iterations.
The tabu status of a move is waved if the exchange leads to a solution that improves upon the
incumbent.
5. Computational Experiments
This section describes the computational experiments that we performed to test the effectiveness
and efficiency of the procedures discussed above. The GRASP by Deng and Bard (2011), Prev_GRASP,
and our adaptation to the CCP of the TS_SO originally proposed for the MDGP by Gallego et al.
(2013), CCP_TS_SO, were implemented in Java SE 6. Our GRASP (Section 3), simply called GRASP,
and the TS (Section 4) for the CCP were implemented in C. All experiments were conducted on an
Intel Core 2 Quad CPU Q 8300 with 6 GB of RAM.
We employed 50 instances in our experimentation. This benchmark set of instances, referred to as
CCPLIB, is available at http://www.optsicom.es/ccp. The set is divided into two subsets:
1. RanReal This set, originally proposed by Gallego et al. (2013) for the MDGP, consists
of 40 matrices in which the benefit values  are real numbers generated using a
Uniform distribution . We have adapted these instances to the CCP by
generating the node weights with a Uniform distribution . There are 20
instances with   ,   ,   , and   . The remaining 20 instances
have   ,   ,   , and   .
Capacitated Clustering Problem 9
2. DB This set is based on the problem with    and    introduced by Deng and
Bard (2011) in the context of mail delivery. We keep the benefit values as they appear in
that original instances. However, instead of considering all the node weights equal to 1,
as in Deng’s and Bard’s article, we randomly generate the node weights with a Uniform
distribution  and thus changing the instances from MDGP to CCP. The set
consists of 10 instances with   ,   ,   , and   .
We have selected 15 representative instances with different characteristics to perform a
preliminary experimentation to identify effective values for critical search parameters and to
compare different designs. Specifically, we select 6 RanReal instances with , 6 RanReal
instances with   , and 3 DB instances.
In our first preliminary experiment we test four different variants of the simplified GRASP
method for the CCP described in Section 3. In particular, we consider four different values of the
parameter α, namely, 0.2, 0.4, 0.6, and 0.8. We generated 100 different solutions to each
instance with each GRASP variant. Table 1 shows, for each variant, the average objective
function value, the average percent deviation from the best solutions obtained within this
experiment, and the number of instances in which the method is able to match the best
solutions obtained within this experiment (out of 15 instances). These measures are “local” in
the sense that the best solutions are those found within the experiment. They are used because
they allow us to discriminate among the procedures being tested and identify the better
alternatives. However, as a point of reference, Table 1 also includes the average deviation
achieved by each variant against the best-know solutions. This is a “global” measure in the sense
that the best-known solutions are those found across all experiments in this paper, which as far
as we know represents the best-known published solutions.
Additionally, Table 1 includes the Score statistic described in Ribeiro et al. (2002). For each
problem instance in an experiment, the Score of a method is the number of methods that found
a solution that is strictly better than the one that the method being scored found. For a set of
instances, the Score is the sum of the individual scores. Therefore, while the overall best score in
any experiment is zero, the worst score depends on the number of alternative procedures and
the size of the test set. Since there are 15 instances in the test set for this experiment and 4
different procedures (one for each value of alpha) then the worst possible score is 45 (i.e., 15
instances times 3 competing procedures).
Metric
α
0.2
0.4
0.6
0.8
Average objective function value
233397.5
233203.9
233392.9
233226.3
Number of best solutions
3
3
5
4
Local average deviation
0.36%
0.44%
0.36%
0.43%
Global average deviation
14.95%
14.91%
14.95%
14.96%
Score (worst = 45)
24
23
19
24
Table 1. GRASP with different values of α.
Table 1 shows the slightly better performance (in terms of both number of best solutions and
deviation) achieved by GRASP with the parameter α set to 0.6. This table also shows that, in
Capacitated Clustering Problem 10
general, the differences observed among these statistics related to the four variants tested are
relatively small. This is particularly true in the deviations with respect to the overall best known
solutions, which confirms that GRASP constructions on their own are not able to produce high
quality solutions. The local measures, including the Score all favor the parameter setting with
  
An important question in the context of multi-start procedures in general, and GRASP in
particular, relates to the origin of the high-quality solutions obtained after applying the
improvement method. Specifically, we are interested in measuring the correlation between the
objective function value of the constructed solutions and the value of the improved solutions
(i.e., the local optima found after applying the improvement method). We would like determine
whether high-quality local optima are related to high-quality constructions. If so, this could lead
us to implementing filter mechanisms to save the computational time invested on attempting to
improve somewhat poor constructions. To this end, we generated 100 solutions for each
problem in our set of 15 representative instances (testing set) and computed the correlation
coefficient between both the initial solution and the local optimum. The resulting correlation
coefficient of 0.01 indicates that good local optima can come from initial solutions of any quality.
Figure 1 shows this lack of correlation for 100 solutions of a representative instance in our test
set. We observe that the points are scattered in the plane without any discernible pattern.
Figure 1. Constructed and Improved values of 100 solutions
In our second preliminary experiment we compare GRASP (with α set to 0.6) with the
Prev_GRASP on the test set of 15 instances. We run both methods for 60 seconds on each
instance and report the results in Table 2.
Prev_GRASP
GRASP(0.6)
194109.8
233841.7
6
9
10.99%
2.84%
20.68%
14.38%
9
6
Table 2. Comparison of GRASP methods.
Capacitated Clustering Problem 11
Table 2 shows that the proposed simplified GRASP seems to perform slightly better than the one
in the literature. GRASP(0.6) is able to match 9 out of 15 best solutions and exhibits a 2.84%
deviation from the best solutions found within this experiment. The global average deviation
(i.e., the one against the overall best-known solutions) of 14.38% compares well with 20.68%
achieved by Prev_GRASP. Note that in a comparison of two methods, the Score provides the
same information obtained by the “number of best solutions found.” Although not shown in
Table 2, it is interesting to point out that Prev_GRASP performs better than GRASP(0.6) on the
medium size RanReal instances (  ), while GRASP(0.6) outperforms Prev_GRASP on the
large RanReal instances (  ). We apply a statistical, non-parametric, test to confirm that
both methods perform similarly. We employ the Wilcoxon test to make a pairwise comparison of
Prev_GRASP and GRASP(0.6). The results of this test (with a -value of 0.233) determined that
the solutions obtained by the two methods could come from the same population.
To finish the analysis of GRASP methods, we compare three different designs for the
construction phase: 1) the one used in Prev_GRASP in which the candidate list  consists of
both, nodes and edges (N&E_GRASP), 2) one in which the  consists of nodes (N_GRASP), and
3) one in which the  consists of edges (E_GRASP). To this end, we generated 100 solutions for
each problem in our set of 15 representative instances with each of these three constructive
methods and report, in Table 3, the results. They clearly shows the superiority of the
conventional GRASP construction, N_GRASP, in which the next element to be included in the
solution is a single node, instead of an edge, or either a node or an edge.
Metric
N_GRASP
E_GRASP
N&E_GRASP
Average objective function value
195495.8
189815.8
181338.4
Number of best solutions
15
0
0
Local average deviation
0.0%
5.4%
8.7%
Global average deviation
28.1%
32.3%
34.6%
Score (worst = 30)
0
16
29
Table 3. Comparison of GRASP constructive variants.
In our fourth preliminary experiment, we test the tabu search method, TS, described in Section
4, with several  values. We set this parameter to 5, 10, 15, and . Table 4 shows the
results (i.e., average values of the five statistics) on the 15 test instances. These results show
that the best outcomes are obtained with    since this variant is able to achieve 8 best
solutions and exhibits a local average percent deviation of 0.29%.
Metric

5
10
15
Average objective function value
282305.3
282739.3
281972.7
281532.8
Number of best solutions
8
4
2
1
Local average deviation
0.29%
0.88%
2.05%
1.03%
Global average deviation
2.40%
2.47%
3.61%
2.88%
Score (worst = 45)
14
23
26
21
Table 4. TS with different values of tenure.
Capacitated Clustering Problem 12
In the final experiment of this block, we add to the comparison set our adaptation to the CCP of
the tabu search developed by Gallego et al. (2013) for the MDGP. As described in Section 2, this
method has the ability to search in the space of infeasible solutions by way of a strategic
oscillation component. To test the merit of this feature, we have considered two variants of this
adaptation: AdTS, without the SO, and AdTS_SO, with the oscillation. We also compare both
methods with our TS with   . As in the previous preliminary experiments, we run each
method for 60 seconds on each instance. The results associated with this experiment are in
Table 5.
Metric
AdTS
AdTS_SO
TS
Average objective function value
256406.2
269549.5
281203.4
Number of best solutions
1
2
12
Local average deviation
6.81%
2.89%
1.06%
Global average deviation
8.02%
4.16%
2.37%
Score (worst = 30)
26
14
6
Table 5. Tabu search methods
The results in Table 5 indicate that, as expected, our tabu search method specifically designed for
the CCP obtains better solutions than the two versions adapted from the MDGP to the CCP. This
table also shows that the strategic oscillation component (AdTS_SO) is effective when compared to
the version without it. TS obtains 12 best solutions out of the 15 instances, while AdTS and AdTS-SO
obtain 1 and 2, respectively. Additionally, both the local and global average deviations as well as the
score favor TS.
We applied the Friedman test for paired samples to the data used to generate Table 5. The resulting
-value of 0.001 obtained in this experiment indicates that there are statistically significant
differences among the three methods tested (we are using the typical significance level of   .
as the threshold between rejecting or not the null hypothesis). A typical post-test analysis consists of
ranking the methods under consideration by their average rank values. The result is that the TS
method is the best with an average rank of 2.60, followed by AdTS_SO (2.10), while AdTS ranks third
(1.30). Finally, we compare the best two methods in this experiment, TS and AdTS_SO, with the
pairwise Wilcoxon test. The resulting -value of 0.002 obtained confirms that the two methods are
significantly different.
In our final experiment we compare the best methods on the entire set of 50 instances. Specifically,
we compare:
Prev_GRASP. The GRASP method by Deng and Bard (2011) described in Section 2
AdTS_SO. Our adaptation of the tabu search by Gallego et al. (2013) described in Section 2.
GRASP. Our GRASP method described in Section 3.
TS. Our tabu search method described in Section 4.
GRASP+TS. A combination of the two methods that we have developed.
Capacitated Clustering Problem 13
The GRASP+TS hybrid consists of running GRASP(0.6) for half of the total running time in the
experiment, and then, for the rest of the running time, applying TS starting from the best solution
found by GRASP.
Tables 6, 7, and 8, show respectively the results of the 20 RanReal instances with , 20
RanReal instances with   , and the 10 DB instances. We run each method for 60 seconds on
each instance.
Metric
Prev_GRASP
AdTS_SO
TS
GRASP
GRASP+TS
Global average deviation
9.43%
2.79%
0.15%
14.55%
0.61%
Number of best solutions
0
0
16
0
4
Score (worst = 200)
79
40
4
100
16
Table 6. Comparison of best methods on the 20 RanReal instances with   
Metric
Prev_GRASP
AdTS_SO
TS
GRASP
GRASP+TS
Global average deviation
41.78%
7.54%
1.99%
19.85%
2.07%
Number of best solutions
0
0
10
0
10
Score (worst = 200)
100
37
10
80
13
Table 7. Comparison of best methods on the 20 RanReal instances with   
Metric
Prev_GRASP
AdTS_SO
TS
GRASP
GRASP+TS
Global average deviation
0.33%
0.02%
4.52%
0.08%
0.13%
Number of best solutions
0
6
0
2
2
Score (worst = 200)
41
11
50
20
21
Table 8. Comparison of best methods on the 10 DB instances
The results in Tables 6, 7, and 8 indicate that the performance of the competing methods varies with
the data sets. For instance, Prev_GRASP has difficulties finding solutions of reasonable quality for
RanReal problems with   , producing a large average deviation of 41.78%. However, the same
method performs much better on the DB set, where it obtains an average deviation of 0.33%. On the
other hand, TS shows better performance than the other procedures in the RanReal sets but comes
out last in the DB set.
We apply statistical tests to the data obtained from this last experiment in order to detect
differences among the Prev_GRASP, AdTS_SO and GRASP+TS. From our three designs, GRASP, TS
and GRASP+TS, we chose GRASP+TS because of its better average overall performance. We first
applied the Friedman test to test the hypothesis that there are differences among the procedures.
The hypothesis that they are the same is rejected with a -value lower than 0.001. The resulting
ranking is GRASP+TS (2.74), AdTS_SO (2.19), and Prev_GRASP (1.07). We then applied the pairwise
Wilcoxon test to GRASP+TS and AdTS_SO, with the outcome indicating a better performance of
GRASP+TS. This conclusion is reached with a -value lower than 0.001.
In an attempt to understand how GRASP and TS interact within GRASP+TS, we recorded the
objective function value of the best solution found during the search when applying GRASP+TS to
Capacitated Clustering Problem 14
the first RanReal instance with   . The profile of this run is depicted in Figure 2. The plot
shows that TS is able to provide a boost after the GRASP search stalls.
Figure 2. Search profile of GRASP+TS on the RanReal240_01 instance
6. Conclusions
The CCP is a difficult combinatorial optimization problem that is closely related to the MDGP,
although this connection had not been previously discussed. Of particular interest in our work has
been testing the effects of a variety of search strategies such as those that allow the search to
move outside the feasible region and neighborhood structures within the GRASP framework. We
also explored a tabu search design with a 2-1 exchange neighborhood that proved effective. Our
work is the first one in which the experiments are performed on general instances of the CCP, that is,
those for which the node weights are different from one. These instances and associated tests
establish a benchmark that currently is not in the literature.
Acknowledgments
This research has been partially supported by the Ministerio de Educación Cultura y Deporte of Spain
(Grant Ref. PRX12/00016), the Ministerio de Economía y Competitividad of Spain (Grant Ref.
TIN2012-35632) and the Generalitat Valenciana (Prometeo 2013/049).
Capacitated Clustering Problem 15
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