COMPARISON OF METAHEURISTICS FOR WORKFORCE
DISTRIBUTION IN MULTI-SKILL CALL CENTRES
David Millán-Ruiz
Telefonica Research & Development, Madrid, Spain
J. Ignacio Hidalgo
Complutense U. of Madrid, Madrid, Spain
Josefa Díaz
University of Extremadura, Mérida, Spain
Keywords: Memetic algorithms, Variable neighbourhood search, Iterated local search, Simulated annealing.
Abstract: Call centre technology requires the assignment of a large volume of incoming calls to agents with the
required skills to process them. In order to determine the right assignment among incoming calls and agents
for a real production environment, a comparative study of meta-heuristics has been carried out. The aim of
this study is to implement and empirically compare various representative meta-heuristics, which represent
distinct search strategies to reach accurate, feasible solutions, for two different instances of the workforce
distribution problem. This study points out how memetic algorithms can outperform other acknowledged
meta-heuristics for two different problem instances from a real multi-skill call centre from one of the
world's largest telecommunications companies.
1 INTRODUCTION
A Multi-Skill Call Centre (MSCC) is a centralised
office where there are groups of agents who may
have a variable number of skills to handle large
volumes of heterogeneous incoming telephone calls.
Even though MSCCs have been widely studied,
there are still some lacks on optimisation which may
imply huge losses of money every year and client
dissatisfaction due to everlasting delays.
A key feature of an MSCC is the Automatic Call
Distributor (ACD). The ACD is a system which
models incoming calls and automatically distributes
them over different queues from which certain
agents can pull work. The routing strategy is a rule-
based set of operations that guides the ACD to
process a given incoming call within the system.
Typically, once the call has been classified and
assigned to a Call Group (CG) in relation to its
nature; a second algorithm either selects the best
available agent to reply to a given incoming call or
reconfigures the assignments call group-agent.
The basic variant of the workforce distribution
problem in MSCCs requires the assignment of
incoming calls to the agents who have the required
skills to handle them over time, satisfying a given
set of additional constraints and respecting
dependencies among individual incoming calls and
differences in the execution skills of the agents. In
our case, these constraints are associated to
incoming calls, agents, timings and
desired/undesired actions.
Workload distribution in MSCCs has been
generally faced by a Skill-Based Routing algorithm
(SBR). Garnett (2000) defines SBR as a call-
assignment protocol used in CCs to assign an
incoming (customer) call to the most appropriate
agent, instead of merely opting for the next existing
agent. The major handicap of this approach is that
online (ad-hoc) routing heuristics cannot be very
complex in view of the fact that a very short
response time is required. These fast, unplanned
decisions may imply suboptimal task assignments to
existing agents.
Nevertheless, Millán-Ruiz (2010) has recently
demonstrated that Memetic Algorithms (MAs) can
352
Millán-Ruiz D., Hidalgo J. and Díaz J..
COMPARISON OF METAHEURISTICS FOR WORKFORCE DISTRIBUTION IN MULTI-SKILL CALL CENTRES.
DOI: 10.5220/0003083503520357
In Proceedings of the International Conference on Evolutionary Computation (ICEC-2010), pages 352-357
ISBN: 978-989-8425-31-7
Copyright
c
 2010 SCITEPRESS (Science and Technology Publications, Lda.)

outperform other classical call centre techniques
(including SBR) when combining middle-term
predictions with optimisation. But, given this
formulation of the problem, other AI-style
techniques could be also applied to this
environment. The present paper investigates whether
MAs can also outperform, for two different real-
world instances of the problem of workforce
distribution in MSCCs, other Meta-Heuristics (MHs)
such as Iterated Local Search (ILS), Simulated
Annealing (SA) or Variable Neighbourhood Search
(VNS). These acknowledged techniques have been
carefully chosen because they provide different
search strategies to obtain feasible solutions.
The rest of this document is organised as
follows: Section 2 describes the problem definition.
Section 3 presents the encoding and the target
function for the workforce distribution problem.
Section 4 describes the MHs to be compared and
their tuning. Section 5 conducts an analysis of the
results. Finally, Section 6 concludes our work with a
summary of major contributions and points out
prospects for future work.
2 PROBLEM DESCRIPTION
In an MSCC, there are n customer calls queued in k
CGs and m agents that may have up to l different
skills s
i
(l ≤ k) to process these types of calls. Note
that s
i
represents the skill to process incoming calls
from CG
i
(s
i
~CG
i
). In an MSCC, each agent can
process calls from different CGs and, given a CG, it
may be answered by several agents who have the
required skill. Obviously, this scenario can be
simpler in some special CCs with only one CG in
which agents have a single skill. These CCs can be
modelled with q single queues working in parallel
(all of them for the same CG). In other cases, every
agent has all possible skills; hence all customer calls
are queued in a single queue so that every agent can
take and process any customer call. The system is
noticeably easier to analyse in these two extreme
cases. With all agents having all skills, the system is
also more efficient (shorter waiting times, fewer
abandonment rates) when the service time
distribution for a given call type does not depend on
the agent’s skill set. However, this assumption turns
out to be wrong in practice: agents are usually faster
when they handle a smaller set of call types (even if
their training gives them more skills). Agents with
more skills are also more high-priced as their
salaries depend on their skill sets. Thus, for large
volumes of call types, it makes sense to dedicate
various single-skill agents (specialists) to handle
most of the load. A small number of agents,
proportional to the calls of each type, with two or
more skills can cover potential fluctuations in the
arriving load.
To address the abovementioned fluctuations, the
skills are grouped in skill profiles P
i
. A skill profile
P
i
can be any subset of agent skills, containing up to
l skills (P
i
={s
j
(1), s
j
(2),...,s
j
(l)}) so that we can assign
an agent to specific types of calls during a given
period of time, despite this agent may have
additional skills to process other type of calls. Figure
1 illustrates the relationship among clients’ calls,
queues and agents and Figure 2 shows an example of
a feasible assignment. Millan-Ruiz (2010) provides
further details of the problem of workforce
distribution within MSCCs.
The solution to the problem of the workforce
distribution in MSCCs is defined as the right
assignment for every agent a
i
to the most suitable
skill profile P
j
from his/her real skill profiles for
each v seconds, where v is the size of the time-frame
considered (in the CC studied, this must be done
each 300 seconds). To determine whether (or not) a
given solution is suitable, we need to define a
quality metric to evaluate the rightness of each
feasible solution. There are very significant metrics
to measure the quality of a CC such as the
abandonment and service rates. These metrics
somehow hinge on the (customer) service level
(Koole, 2006) which is defined as the percentage of
customer calls that have to queue shorter than a
specified amount of time. Our work has been
conducted by applying this metric.
Figure 1: Inbound scheme in MSCCs with 5 agents, 9
incoming calls and 4 CGs.
The complexity of this problem is huge because
we are not only dealing with an NP-hard problem
like in the job assignment problem (Chauvet, 2000),
but also considering high dynamism, massive
incoming customer calls and large number of agents
COMPARISON OF METAHEURISTICS FOR WORKFORCE DISTRIBUTION IN MULTI-SKILL CALL CENTRES
353

having multiple skills. Besides, since customer calls
are not planned, this makes the call assignment a
truly laborious task.
3 ENCODING AND TARGET
FUNCTION
In order to present a fair comparison among the
analysed MHs (ILS, SA, VNS and MAs), this
section provides a common problem representation
and a target function to evaluate the rightness of the
candidate solutions for every MH.
3.1 Encoding
The first stage when designing an MH is to define a
problem representation to encode candidate
solutions to the problem in a form that every
computer can interpret. There are multiple forms to
encode candidate solutions which range from binary
strings, arrays of integers or arrays of decimal
numbers to strings of letters. Specifically, our
solution consists of an integer representation. We
just need an array of integers whose indexes
represent the available agents at a given instant and
the array contents refer to the profile, P
j
, assigned to
each agent a
i
. Then, incoming calls are “routed” to
the agents, according to the profiles assigned. Of
course, we can also encode the solution as an array
of integers whose indexes symbolise the task types
and its respective contents represent the number of
agents assigned to each task type. This option is
recommended whether there are too many agents
and hardware capacity is very limited (with respect
the total number of available agents). In contrast, we
are missing the capability of working at agent’s
profile level. As we have not this capacity
constraint, we will employ the first codification
proposed.
Figure 2 shows a fictitious example (related to
Figure 1) of encoding for 9 customer calls (c
1
-c
9
)
queued in 4 different CGs (cg
1
-cg
4
) depending on
the nature of the calls, 5 agents (a
1
-a
5
) and 7 profiles
(P
1
-P
7
), where P
1
={s
1
}, P
2
={s
1
, s
2
}, P
3
={s
2
}, P
4
={s
2
,
s
3
}, P
5
={s
1
, s
3
}, P
6
={s
3
} and P
7
={s
4
}. Now, suppose
that the agents have the following potential skill
profiles: a
1
~{P
1
,P
2
}, a
2
~{P
1
,P
3
,P
7
}, a
3
~{P
4
,P
5
},
a
4
~{P
6
} and a
5
~{P
2
,P
3
,P
7
}. We have seen the
potential profiles for every agent but only one
profile can be assigned to each agent at a given
instant t; therefore, a feasible solution would be
Figure 2. Note that more than one agent can have
assigned the same profile (e.g. a
1
and a
5
).
Index (agents)  1 2 3 4 5
Content (profiles)  2 7 4 6 2
Figure 2: Example of the problem encoding.
3.2 Target Function
The target function is an evaluating mechanism
which is defined over the encoding to measure the
quality of a given candidate solution. This function
often guides the search and decides which solutions
must be selected for the next iteration. The target
function is inherently linked to the problem.
Frequently, the hardest action when defining an MH
is to identify the right target function. Sporadically,
it is hard (sometimes impossible) to characterise the
target expression. In other cases, long evaluating
times imply that an approximate function is needed.
The target function that we will use is the one
provided by Millán-Ruiz (2010).
4 DEFINITION
OF META-HEURISTICS
AND TUNING
In this section, we briefly explain some of the
previously hinted MHs and their configuration for
our problem. For all of them, we start from a
feasible randomly generated solution.
4.1 Simulated Annealing
SA is an MH of variable environment, which
generalises Monte Carlo’s method (Kirkpatrick,
1983). SA proposes that the current state of a
thermodynamic system is equivalent to the candidate
solution in optimisation, the energy equation for a
thermodynamic system is analogous to a target
function, and ground state corresponds to the global
minimum. This technique has the ability to hinder
getting trapped in local optima since the algorithm
allows for changes that decrease the values returned
by the target function with a given probability
(temperature). The main complexity is to determine
the right value for the temperature.
Concretely, we consider Cauchy’s scheme to
cool off the temperature because it is faster than
the Bolzmann’s criterion and we only have 300
seconds to provide the system with a new
solution. In Cauchy’s scheme, the temperature is
defined as
it) + (1/ T = T
0it
, where it is the
iteration number and the initial temperature is
ICEC 2010 - International Conference on Evolutionary Computation
354

f(S*)*))( log-/( = T
0


where f(S*) is the cost of
the initial solution, Φ is the probability of accepting
a “μ” worse solution than the current one (in our
experiments, Φ= μ =0.3). Finally, the maximum of
neighbour solutions generated each time is L(T)=30
and the probability of accepting a worse solution is
exp(-δ/Tit) given that δ=f(Neighbour_Solution)-
f(Current_Solution) and T
it
is the temperature at
iteration it.
4.2 Iterated Local Search
The basic idea of ILS (Stützle, 2006) is to
concentrate the search on a smaller subspace defined
by the solutions which are locally optimal to the
current one. ILS consists of the iterative application
of a more or less simple LS method. To avoid
getting trapped in local optima, a perturbation is
applied before executing each LS.
In our case, the complete process is repeated
during 300 seconds as this is the size of the time-
frame imposed by the MSCC which is being
analysed. The perturbation applied after each LS
affects to the 3% of agents (the higher this
perturbation is, the more random the search gets).
Finally, we present the pseudo-code of an LS based
on the best neighbour scheme:
Local_Search (candidate_solution)
{
best_solution  candidate_solution;
neighbour  candidate_solution;
For(i=0;i<size(candidate_solution);i++)
{
Agent a = neighbour.getAgent(i);
For(j=0;j<a.number_profiles();j++)
{
neighbour.change_profile(i,j);
if(neighbour.fitness()>best_solutio
n.fitness())
best_solution = neighbour;
}
neighbour = best_solution;
}
candidate_solution = best_solution;}
4.3 Variable Neighbourhood Search
VNS is an MH whose fundamental idea is to cause
systematic changes in the neighbourhood of an LS
procedure (Mladenovic, 1997). VNS escapes from
local optima by changing of environment e,
increasing the size of the neighbourhood nh
e
until a
local minimum better than the current one is
reached.
We consider three different environments e
max
=3:
nnn


332211
nhe;5.0nhe;3.0nhe
Similarly to the previously described techniques,
these steps are repeated during 300 seconds
(stopping condition).
4.4 Memetic Algorithms
MAs represent a growing research area in
evolutionary computation. MAs are a variety of
population-based techniques for heuristic search in
optimization problems. This technique is much
faster than traditional Evolutionary Algorithms
(EAs) for many problem domains. Fundamentally,
these combine EAs’ operators with Local Search
(LS) heuristics to refine candidate solutions. To
design our MA, we have considered a steady-state
genetic algorithm combined with the basic local
search described for the ILS in Section 4.2.
Now, we briefly comment the final configuration
of the evolutionary operators of our MA as the
reasoning of this choice is a very significant topic
and deserves to be presented in a separate study.
The configuration of the evolutionary operators is
the following one:
Population: The population contains 20 different
individuals.
Selection: Since the population needs to be bred
each successive generation, we have chosen a binary
tournament selection (Prügel-Bennett, 2000).
Crossover: The following step is to produce a new
generation from selected individuals. Concretely, we
consider that children will inherit the common points
in their parents and randomly receive the rest of
genes from them.
Mutation: This operator causes tiny changes in the
genes of the chromosome to explicitly maintain
diversity (actually there are much more
mechanisms). In this study, we apply a perturbation
of a 3% over the chromosome.
Replacement policy: Finally, we decide which
individuals are incorporated (or maybe reinserted)
into the population. In this study, we consider
elitism (Chakraborty, 2003) with a probability of
0.93 to replace the worst individuals of the
population for next generation. And, with a
probability of 0.07, a worse individual may be
captured.
Subpopulation for LS: The LS is applied over the
best 25% of individuals.
LS frequency: The LS is applied over the selected
individuals each 10 generations.
Stopping condition: All steps are carried out until
our 300-second termination criterion is met.
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355

5 EVALUATION OF RESULTS
In this section, we analyse the results achieved by all
the MHs for two different problem instances
(medium and high difficulty, respectively). These
two instances are real data taken from our MSCC’s
production environment during two different days at
the same hour (from 12:40 to 12:45, 300 seconds): a
one-day campaign and a normal day. The size of the
time-frame to execute all the MHs is 300 seconds (5
minutes) because we need to provide the system
with a solution each 300 seconds (continuous re-
planning of the ACD). We have selected this time
interval because this hour (between 12:30 and
13:00) is very representative as this is precisely the
most critical hour of the day (highest load of the
day: n/m). Note that around 800 incoming calls (n)
simultaneously arrive during a normal day in such a
time interval, whereas up to 2450 simultaneous
incoming calls may arrive during this interval during
a commercial campaign. The number of agents (m),
for each time interval, oscillates between 700 and
2100, having 16 different skills for each agent on
average (minimum=1 and maximum=108), grouped
in profiles of 7 skills on average. The total number
of CGs considered for this study is 167. Therefore,
when the workload (n/m) is really high, finding the
right assignment among agents and incoming calls
becomes fundamental. In this way, we have run
every MH under two double-core processors of a
Sun Fire E4900 server (one processor for the
interfaces and data pre-processing, and the other one
for each MH).
Once the magnitude of our MSCC has been
presented, each MH is compared alongside the
others. Table 1 summarises the results obtained by
each MH in 50 executions, starting from 50 different
randomly generated initial solutions.
In our comparative study, we present dissimilar
MHs which cover diverse strategies. Theoretically,
due to the local character of the basic LS, it is
complicated to reach a high-quality solution because
the algorithm usually gets trapped in a
neighbourhood when a local minimum is found.
This occurs because the engine is always looking for
better solutions which probably do not actually exist
in the neighbourhood. For this reason, sometimes, it
is more appropriate to allow deterioration
movements in order to switch to other regions of the
search space. This is precisely the shrewd policy of
SA whose temperature allows for many oscillations
(the probability of accepting a worse solution
decreases according to the time) at the beginning of
the process and only few ones at the end (fewer
chances to select a worse solution as the algorithm is
supposed to be refining the solution at this point).
Specifically, we have chosen Cauchy’s criterion
because the convergence is faster than Boltzmann’s
and we only have 30
0 seconds to run the complete
process. Besides, this scheme avoids decreasing the
distance between two solutions when the process
converges (jumps in the neighbourhood). Therefore,
the temperature must be high enough at the
beginning to better explore the search space (its
neighbourhood) and low enough at the end to
intensify the search as well (exploitation of
promising areas). The value for speed is, therefore,
the stopping condition which must agree with the
number of neighbours generated.
Table 1 gathers the results obtained by each MH
in 50 different executions for two different problem
instances with the purpose of providing a fair
comparison. The first three columns are the best,
worst and mean fitness values, respectively. Then,
we have the standard deviation and the effectiveness
(best fitted solution represents the 100%).
We perceive from Table 1 that SA worse
behaves than the other MHs except for the easiest
instance of the problem. This may occur because we
are not plenty of time in our environment and the
power of SA relies on a progressive cooling. If we
cool off the temperature too fast, we are missing the
effectiveness of accepting worse solutions in some
cases. Instead, if we cool off the temperature too
slowly, we may be accepting worse solutions
systematically without converging. We have applied
a trade-off between exploration and exploitation but
the time seems to be limited to apply SA to our
environment (perhaps, things might change when
having more time).
Another option to increase the diversity in the
solutions is to enlarge the environment, as VNS
does. This philosophy consists of making a
systematic change upon the environment when the
LS is used, increasing the environment when the
process becomes stagnated. In the VNS, the search
is not restricted to only one environment as in the
basic LS; instead, the neighbourhood changes as the
algorithm progresses. Albeit we only consider three
distinct neighbourhoods, the improvement of the
VNS compared to basic LS is noteworthy.
Consequently, the remarkable factor becomes the
change in the number of neighbourhoods and their
sizes as well as to consider how the algorithm reacts
in response.
Table 1 also shows how VNS only slightly
outperforms SA for the hardest instance of the
problem.
Another strategy is to start from different initial
solutions as ILS accomplishes. ILS generates a
ICEC 2010 - International Conference on Evolutionary Computation
356

random initial solution and afterwards applies a
basic LS. Subsequently, this solution is
systematically mutated and thus refined. ILS obtains
solutions which vaguely improve those given by SA
and VNS for the hardest problem instance, although
it performs worse for the simplest problem instance
as Table 1 corroborates.
Another way to find a solution involves using
methods based on populations, such as MAs. If the
diversity of the solution is low, then the MA
converges to the closest neighbour. Nevertheless,
when the selective pressure is high, individuals may
be alike or even identical. To speed-up the
convergence, MAs apply an LS upon a set of
chromosomes (candidate solutions) that are refined
every certain number of generations. Incorporating a
hybridisation mechanism to the GA is valuable as
the algorithm is improved in all respects. This fact is
pointed up in Table 1 as the MA not only
outperforms all the strategies for both instances but
also remains more unwavering (less differences
among best, worst and mean fitness values).
Finally, it is important to remark that differences
among techniques are not huge after reaching a
fitness of 0.8 since the complexity increases
exponentially in our environment. Therefore, minor
improvements on the fitness value after that point
are hard to obtain but very valuable to accomplish a
fair workforce distribution.
6 CONCLUSIONS AND FUTURE
WORK
We have seen the difficulty of assigning incoming
calls to the right agents within a real-world MSCC.
We have then presented several MHs to carry out
this task and how MAs can outperform them in such
an environment. Afterwards, we have done an
exhaustive comparative study of MHs to empirically
evaluate diverse strategies to reach accurate, feasible
solutions, capturing real data from an MSCC within
a large multinational telephone operator during a
peak of load (12:40-12:45). This study has
illustrated how these MHs perform for two different
problem instances. We can conclude that MAs not
only outperform all the strategies but also remain
more unwavering as systematically provide better
results than the rest of techniques. As future work,
we propose to do a similar study considering parallel
MHs and different time-frame sizes.
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Table 1: Results obtained by the MHs in 50 executions starting from random initial solutions for two problem instances:
medium and hard (larger number of incoming calls and high variability). Values refer to the fitness obtained by all MHs.
Algorithm
Best solution Worst Solution Average Standard deviation Effectiveness
Medium Hard Medium Hard Medium Hard Medium Hard Medium Hard
MA 0.796 0.758 0.785 0.751 0.796 0.754 0.001 0.001 100 100
ILS 0.768 0.728 0.755 0.722 0.763 0.725 0.002 0.003 95.85 96.15
VNS 0.790 0.727 0.766 0.723 0.775 0.724 0.005 0.001 97.36 96.02
SA 0.782 0.721 0.773 0.709 0.779 0.716 0.001 0.003 97.86 94.96
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