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Int. J. Services, Economics and Management, Vol. 2, Nos. 3/4, 2010 267
Copyright © 2010 Inderscience Enterprises Ltd.
Literature review of covering problem
in operations management
C.N. Vijeyamurthy
Department of Management Studies,
Pallavan College of Engineering,
Kancheepuram 631 502, India
Email: vijeyamurthy@gmail.com
R. Panneerselvam*
Department of Management Studies,
School of Management,
Pondicherry University,
Pondicherry 605 014, India
Email: panneer_dms@yahoo.co.in
*Corresponding author
Abstract: This paper discusses a review of literature of covering problem.
Facility location problem is an important area in operations management.
Within the facility location problem, covering problem aims to locate the
minimum number of facilities to cover the given number of customers. This
problem comes under combinatorial category. Hence, researchers have
concentrated in the development of heuristics to get near optimal solution. In
the process of developing heuristics, some researchers have developed
mathematical models for the same problem for the purpose of comparing their
results with optimal solutions. In this paper, a comprehensive review of
literature of the covering problem is presented.
Keywords: covering problem; potential site; facility; customer.
Reference to this paper should be made as follows: Vijeyamurthy, C.N. and
Panneerselvam, R. (2010) ‘Literature review of covering problem in operations
management’, Int. J. Services, Economics and Management, Vol. 2, Nos. 3/4,
pp.267–285.
Biographical notes: C.N. Vijeyamurthy is an Assistant Professor in the
Department of Management, Pallavan College of Engineering, Kancheepuram,
India. He holds MBA from IGNOU. His specialisations include operations
management and telecommunications. He served BSNL for 35 years and
retired as Assistant General Manager.
R. Panneerselvam is serving the Department of Management Studies,
Pondicherry University from 1987. He served in the Industrial Engineering
Department, Anna University for about seven years. He is the former Dean of
School of Management, Pondicherry University. He holds BE in Mechanical
Engineering and ME as well as PhD in Industrial Engineering. His research
areas include facility location and layout, line balancing, production scheduling
and simulation. He published eight leading text books, viz. Production
and Operations Management, Engineering Economics, Operations Research,
268 C.N. Vijeyamurthy and R. Panneerselvam
Database Management System, Research Methodology, Project Management
with P. Senthilkumar, Design and Analysis of Algorithms, and Interest Table
for Engineering Economics, all through PHI. He has published 75 research
articles in leading international and national journals.
1 Introduction
Consider a set of residential regions (customers) in a district which are to be covered by a
set of dealers (facilities) of a particular company. A residential region is said to be
covered if there is a dealer located within (say) 10 km from that residential region. In this
example the objective is to cover all the residential regions with the minimum number of
dealers. This problem of covering all the residential regions with the minimum number
of dealers is called as Total Covering Problem (TCP). In this problem, the objective is
to find the minimum number of sites for locating facilities to cover all the customers. In
general, all the emergency-services/the essential-service related situations are formulated
as TCP. Some more examples are locating schools, banks, post offices, healthcare
centres, emergency grain reserve centres, railway accident relief vans, etc. In continuation
to this TCP, the other types of Covering Problem (CP) are presented below.
If the policy of the management is to provide a limited number of facilities to cover
the customers, then the objective is to cover as many customers as possible with the
given utmost number of facilities. This is called as Partial Covering Problem (PCP). In
the PCP, the objective is to find the minimum number of sites without exceeding the
given utmost number of potential sites for locating facilities, to cover as many customers
as possible. This is also known as Maximal Covering Problem (MCP).
Set Covering Problem (SCP) is a covering problem in which a minimum number of
subsets are selected from the given input set so that the union of the selected subsets
contains all the elements of the input set and minimises the total cost of selection of the
subsets. If the cost co-efficients are unity, then the SCP is same as the TCP.
Anti-Covering Location Problem (ACLP) aims to locate a maximum weighted
number of facilities such that no two facilities are within a specified distance from each
other.
Further the covering problem can be classified as listed below:
1 the covering problem with 0-1 co-efficient
2 the covering problem with non 0-1 co-efficient.
All the emergency-facility location problems and other public-utility location problems
such as the problems related to post-office locations and locations of banks’ branches,
police stations, etc. will have 0-1 covering co-efficient matrix. The problems, such as
warehouse location problems, wholesaler location problems, etc. will have non 0-1
covering co-efficient matrix.
In this paper, a comprehensive review of literature of the covering problems is
presented.
Covering problem in operations management 269
1.1 Model for total covering problem
A 0-1 programming model for the TCP is presented in this section. To illustrate the TCP
with 0-1 covering co-efficients, consider the data as given in Table 1.
Table 1 Covering coefficient matrix
Potential site j
1 2 3 J n
1 C11 C12 C13 C1j C1n
2 C21 C22 C23 C2j C2n
3 C31 C32 C33 C3j C3n
i Ci1 Ci2 Ci3 Cij Cin
Customer i
m Cm1 Cm2 Cm3 Cmj C
mn
In Table 1, let,
m be the number of customers.
n be the number of potential sites for locating facilities.
Cij = 1, if the distance of the customer i from the potential site j is less than or equal to a
specified maximum distance;
= 0, otherwise.
A 0-1 programming model to determine the minimum number of sites for locating
facilities to cover all the customers is presented below:
Minimise ij j
Z
CX=
∑
Subject to
1
1 for 1, 2,3,...,
n
ij j
j
CX i m
=
≥=
∑ (1)
where
Xj = 0 or 1, for j = 1, 2, 3, …, n
In the above model,
Xj = 1, if the potential site j is assigned with a facility;
= 0, otherwise.
The objective function minimises the number of sites which are assigned with
facilities to cover all the customers. Each of the constraints ensures that each customer is
covered (served) by at least one potential site which is assigned with a facility.
1.2 Model for partial covering problem
In this section, model for the PCP (Panneerselvam et al., 1996; Panneerselvam, 2005) is
presented. As stated earlier, the objective of this problem is to cover as many customers
as possible with the given utmost number of facilities.
270 C.N. Vijeyamurthy and R. Panneerselvam
Let,
Yj = 1, if the site j is selected for assigning a facility;
= 0, otherwise.
Xi = 1, if the customer i is covered by a facility;
= 0, otherwise.
m be the total number of customers.
n be the total number of potential sites.
k be the maximum number of facilities available.
kk be the actual number of sites to be selected for assigning facilities (kk ≤ k).
Cij be the covering co-efficient as defined in the TCP.
H be a very large value (say, H > m).
The objectives of this PCP are to maximise the number of customers covered and to
minimise the number of potential sites selected for assigning facilities whose maximum
value is given as k.
11
Maximise
mn
ij
ij
Z
HX Y
==
=−
∑
∑
Subject to
1
n
j
j
Yk
=
≤
∑ (2)
()
1
1 1 for 1, 2,3, ...,
n
ij j i
j
CY X i m
=
+− ≥ =
∑ (3)
()
1
1 for 1, 2,3,...,
n
ij j i
j
CYnXni m
=
+− ≤ =
∑ (4)
where
Xi = 0 or 1, for i = 1, 2, 3, …, m
Yj = 0 or 1, for j = 1, 2, 3, …, n
The objective function has two components, viz. maximising the number of customers
covered and minimising the number of potential sites selected for assigning facilities.
The first component is assigned a weight which is more when compared to that of the
second component. The first constraint restricts the total number of potential sites which
are assigned with facilities to utmost k. The second set and the third set of constraints
jointly assure the following:
1 If a customer is covered, then it is served by at least one potential site which is
assigned with a facility.
2 If a customer is not covered, then it is not served by any of the sites which are
assigned with facilities.
Covering problem in operations management 271
The covering problem comes under combinatorial category, which requires exponential
time to obtain its optimal solution. This necessitates the use of heuristics to obtain near
optimal solution. Under heuristic category, there are different types, viz. single pass
heuristic, meta-heuristic, etc. In this paper, a comprehensive review of literature of the
covering problems and the relevant techniques to solve such problems is presented.
2 Classification of techniques for covering problem
This section gives a classification of techniques for the covering problem as shown in
Figure 1.
Figure 1 Classification of techniques for covering problem (see online version for colours)
CP
Ex He M S
p
H
y
The exact procedures (Ex) aim to obtain optimal solution for the SCP. Dynamic
programming, branch and bound techniques and mathematical models fall in this
category. Heuristic (He) aims to obtain near optimal solution. Alternatively, this is called
single pass heuristic, which means that based on set guidelines the procedure will
terminate very quickly if there is no improvement in the objective function. Meta-
heuristics (M) consist of tabu search, simulated annealing, Genetic Algorithm (GA), etc.
These algorithms aim to obtain global optimum for the given problem. Hybrid heuristics
(Hy) are constructed by combining two or more of the heuristics/meta-heuristics. The
techniques for the covering problems which are not coming under any of these categories
are grouped under special techniques (Sp).
3 Exact procedures
Curry and Keitch (1969) have developed dynamic programming method to solve TCP.
ReVelle and Swain (1970) studied the TCP and presented a branch and bound method to
obtain the optimal solution. Toregas et al. (1971) have developed a linear programming
model to solve the traditional SCP with equal cost in the objective. Patel (1979) has used
dynamic programming approach for locating rural social service centres for the
Dharmpur taluka in South Gujarat in India. Klastorin (1979) applied assignment method
to solve the conventional SCP.
272 C.N. Vijeyamurthy and R. Panneerselvam
Daskin and Stern (1981) have developed a model for locating emergency medical
service vehicle deployment with the objective of minimising the number of facilities and
maximising the back-up coverage. Daskin (1983) has refined the above model and
provided a heuristic to obtain better results.
Saatcioglu (1982) has developed a mathematical model for airport site selection
based on Turkish data. Moore and ReVelle (1982) have used relaxed linear programming
supplemented by branch and bound technique to solve hierarchical service problem in
which, the number of facilities is minimised at the first level and the covering is
maximised at the second level.
Pirkul and Schilling (1988) have developed a model and solution procedure based on
Lagrangian relaxation procedure for the covering problem applied to siting of emergency
facilities with workload capacities and back-up service. Pirkul and Schilling (1991) have
developed a model and solution procedure based on the MCP with capacities on total
workload. Batta and Mannur (1990) have developed integer programming models for
emergency situations that require multiple response units.
Panneerselvam et al. (1990a) have modelled the warehouse location problem as a
covering problem (p-median problem). Panneerselvam et al. (1990b) have presented a
mathematical model to identify the machine component groupings of group technology
problem using the similarity index matrix. They have formulated this problem as a TCP.
Goldberg and Paz (1991) have developed a model for locating emergency vehicle
bases when service time depends on call location by taking stochastic travel time,
unequal vehicle utilisation, various call types and service times that depend on call
location into account.
Chan and Yano (1992) have developed a branch and bound algorithm with multiplier
adjustment for the traditional SCP.
Williams (2003) has addressed the maximal covering sub-tree problem which has
application in transportation network design and extensive facility location. Finding an
optimal sub-tree may involve two objectives, viz. minimising the total arc cost or
distance of the sub-tree and maximising the sub-tree’s coverage of population or demand
at nodes. The author has developed four new bi-objective zero-one programming models.
These models have ‘integer-friendly’ solution properties and are relatively small in terms
of the number of decision variables and constraints. They offer both increased flexibility
and potential for more diverse array of solutions.
Murry (2005) has developed a model using the spatial structure to address
complementary partial service of areas. The service coverage has been a fundamental
aspect of geographic research. In particular, facility placement and associated coverage
are central concerns in emergency services, transit route design, cartographic simplification,
natural resource management and weather monitoring among others. The author
discusses widely applied SCP, focusing on its use in geographic analysis. Thereby,
problematic aspects of set coverage modelling across space are identified. The geographic
information systems and enhanced spatial information have accentuated spatial
representation issues in need of greater consideration in modelling service coverage. This
model decreases modifiable aerial unit problem impact known to be an issue in the
geographic application of SCP.
Shavandi and Mahlooji (2006, 2007) have presented a fuzzy location allocation
model for congested systems. They utilised the fuzzy theory to develop a queuing
maximal covering location allocation model, which they call fuzzy queuing maximal
covering location allocation model. They considered the fuzzified queuing parameters as
Covering problem in operations management 273
well as fuzzified constraints to develop a new mathematical model which they converted
to a single objective integer programming model. Their model considers one type of
service call, one type of server and includes one constraint on the quality of service in the
form of service time or a queue length constraint. A genetic algorithm is developed to
solve and test the model with networks having 50 nodes.
Younies and Wesolowsky (2007) introduced a new model for the Planar Maximal
Covering Location Problem (PMCLP) under different block norms. The authors have
addressed a problem for locating ‘g’ facilities anywhere on the plane in order to cover the
maximum number of ‘n’ given demand points. They have modelled different block
norms as a maximum clique partition problem on an equivalent multi-interval graph.
Then the equivalent graph problem is modelled as an Unconstrained Binary Quadratic
Problem (UQP). The authors have solved the UQP format through a genetic algorithm.
Rajagopalan et al. (2008) developed a multi-period location model for dynamic
redeployment of ambulances. The objective of the model is to determine the minimum
number of ambulances and their locations for each time-cluster in which significant
changes in pattern occur while meeting coverage requirement with a predetermined
reliability. They have further enhanced the model by calculating ambulance-specific busy
probabilities and validated it by a comprehensive simulation model.
Plastria and Vanhaverbeke (2008) have extended the classical maximal covering
model in a competitive environment by including a price decision. They have formulated
a revenue maximisation model and proposed two procedures to solve it. Computational
experiments show promising results for small, medium and large case studies.
Wagner (2008) has proposed improved model formulations for the hub covering
problems. The author has discussed multiple and single allocation problems, including
non-increasing quantity-dependent transport time functions for transport links. The
computational results show that the new solution approaches for problems with quantity-
independent transport times outperform previous works.
4 Heuristic
As the covering problem is combinatorial in nature, development of heuristic is
inevitable. Kuehn and Hamburger (1963) have developed a heuristic program for
locating warehouses by considering it as a covering problem. Shannon and Ignizio (1972)
have developed a heuristic programming algorithm for locating warehouses by considering
the cost of travel between the warehouses and the plants only. This algorithm is primarily
based on add-and-drop technique. In this work, an upper limit for the number of
warehouses to be located is assumed.
Neebe (1988) has presented a procedure for locating emergency-service facilities for
all possible response distances. Boffey (1989a) has discussed the location problems
arising in computer networks. Later, Boffey (1989b) has given direction for locating
software in distributed computing system based on SCP.
Panneerselvam and Balasubramanian (1985) have developed a set covering heuristic
to determine the economical number of manufacturing cells and cell arrangements. The
algorithm addresses the total facilities design problem including machine grouping, cell
layout, cell loading, and estimation of machine requirements and its impact on idle time
and overtime of the machines. Later, Balasubramanian and Panneerselvam (1993) have
presented an improved version of this paper.
274 C.N. Vijeyamurthy and R. Panneerselvam
Panneerselvam (1990) has discussed the application of TCP and presented a heuristic
to determine the minimum number of potential sites for assigning facilities to cover all
the customers. Rajkumar and Panneerselvam (1991) have developed an improved
heuristic for the TCP. Panneerselvam (1996) developed an efficient heuristic for TCP. He
compared its results with that of a mathematical model (Schilling et al., 1993) and also
with the results of the heuristic developed by Rajkumar and Panneerselvam (1991).
Panneerselvam (1998) has presented an algorithm for the TCP with the cost
consideration and probabilistic demand. In this paper, the cost of providing service to the
customers and the probabilistic demand of the customers are considered while
determining the minimum number of sites. The probabilistic aspect of demand causes the
challenges in solving the problem. In this research, the best result is obtained after
carrying out a specified number of iterations. Specifying the number of iteration
beforehand is a limitation of this research.
Karmarkar et al. (1991) presented an interior point algorithm to solve computationally
difficult SCPs. The interior point approach to the 0-1 integer programming feasibility
problem is based on the minimisation of a non-convex potential function. The procedure
generates a sequence of strict interior points of a polytope defined by a set of inequalities,
such that each consecutive point reduces the value of potential function. The algorithm
has in-built module for two routines schemes to show how to proceed when a non-global
local minimum is encountered.
Fontanari (1995) has employed the annealed approximation to study two stochastic
models. The author has studied the constant density model and the Karp model. It gives
lower bounds for the optimal cost for different scaling P and N in an incidence matrix.
El-Darzi and Mitra (1995) reviewed graph theoretic relaxations of the SCP and set
partitioning problem. Greedy algorithm has been developed by them with a matching
relaxation and a graph covering relaxation.
Galvao and ReVelle (1996) developed a Lagrangian heuristic for the Maximal
Covering Location Problem (MCLP) where upper bounds are given by vertex-addition
and vertex-substitution heuristic, and the lower bounds are given by the sub-gradient
optimisation algorithm.
Lee et al. (1996) have applied a stochastic algorithm to solve the graph covering
problem in which a set of patterns that fully covers a graph with a minimal cost is sought.
It is a problem-specific encoding scheme to reduce the size of the search space which
yields good solution. Technology mapping problem from the VLSI synthesis area is
solved by this algorithm.
Haddadi (1997) proposed a simple Lagrangian heuristic for the set cover problem
from simple and classical ideas. It produces an efficient solution only for the low-density
set cover problems, which is its limitation.
Paschos (1997) did a comprehensive survey on approximation minimum set covering,
the minimum vertex covering, the maximum set packing and the maximum independent
set problems. The author discussed their approximation performance and their
complexities.
Plastria and Carrizosa (1999) have viewed the covering problem differently, for
minimal covering objectives to locate undesirable facility. They used two criteria, viz. a
radius of influence to be maximised, indicating within which distance from the facility
population disturbance is taken into consideration and the total covered population lying
within influence radius from the facilities which is to be minimised. The objective of this
research is to find the largest circle which is not covering more than a given total size of
Covering problem in operations management 275
the population with respect to the undesirable facility. It turns out to be a special type of
problem where the decision maker needs to be sensitive to the threshold value of radius
when an undesirable facility location is selected and that does not cause any damage to
the health of the population who live.
Chakravarty and Shekhawat (1999) analysed four serial heuristics and four parallel
heuristics for the minimum set cover problem. These algorithms perform a trade-off
between run time and quality of solution. The parallel heuristics are derived from the
serial heuristics where it is demonstrated that an increase in number of processors does
not degrade the quality of the solution.
Flores et al. (1999) studied test set compaction problem, which is a fundamental
problem in digital system testing. This paper studied the application of set covering
models to the compaction of test sets, which can be used with any heuristic test
compaction procedure. For this purpose, effective set covering algorithms were used.
They found that the size of the computed test sets can often be reduced by using the set
covering algorithms. They concluded that it is preferable not to use fault simulation when
the final objective is test set compaction.
Chuzhoy and Naor (2002) have considered the classical vertex cover and set cover
problems with the addition of hard capacity constraints. It means that the set can cover
only a limited number of its elements and the number of available copies of each set is
bounded. This is a natural generalisation of a classical problem that captures the
resource limitation in reality. They developed a 3-approximation algorithm which is
based on randomised rounding with algorithms for unweighted vertex covering problem
with hard capacities. They proved that the weighted version is at least as hard as set cover
problem.
Tsuyoshi and Toshihiro (2002) developed a modified greedy algorithm for set cover
problem with two weights. They have developed two sets of weights, viz. subsets
weights restricted to one and a constant weight ‘d’. The modified greedy algorithm
produces same approximation bound.
Berman and Huang (2002) considered the location of new facilities which serve only
a certain proportion of demand. They minimised the total weighted distances of the
served demand. They presented a location for one facility on a plane and on a network of
m facilities. A heuristic algorithm is proposed for the median problem in a plane. The
authors assumed that all the demands need not be served.
Moore et al. (2003) have compared two heuristics methods, simple greedy algorithm
and progressive rarity algorithm with optimal solution and concluded that heuristics often
provide good optimal solution for the area selection of land cover to identify the
minimum number of areas required to represent all species over some geographic region.
This progressive rarity algorithm is efficient, producing near optimal set and does not
make most efficient selection at every step, and likely to perform poorly if the number of
areas selected is very few relative to the number required to represent all species.
Beraldi and Ruszczynski (2002) have analysed the structure of the set of
probabilistically efficient points of binary random vectors and proposed a branch and
bound algorithm for probabilistic SCP. They have developed a heuristic method for
generating upper bound for the initial solution. The dual heuristic aims at finding the
Probabilistically Efficient Points (PEP), with lower dual, producing good results. The
PEP is a vector that can be substituted for the right hand side of SCP to obtain a solution
of the probabilistic problem.
276 C.N. Vijeyamurthy and R. Panneerselvam
Haouari and Chaouachi (2002) developed probabilistic greedy search method for
SCP yielding a simple robust and quite-fast heuristic, resulting consistently near optimal
solution. The proposed approach is an alternative for the user not to invest the
significant amount of time and efforts, although the special purpose methods are
available for the set cover problem.
A 2-approximation algorithm for capacitated partial vertex cover with demands have
been proposed by Takatoshi and Toshihiro (2003). The vertex cover problem is the
minimum size problem. This problem is shown as 2 approximiable. This paper has
considered various forms of its extensions. Each vertex can cover only a prescribed
number of incident edges, forming a fraction of all demands covered. This paper attempts
to unify all extensions, under a general version of vertex cover problem.
Xiao et al. (2004) studied the issue of the PCP such that a part of mobile nodes may
get covered. Then they proposed an algorithm with polynomial time complexity and
applied under resource and topology constraints. This algorithm produced an
improvement over 3-approximation algorithm for the k-centre problem. Covering all
given mobile nodes within limited number of mobile servers in the communication system
enables cost effectiveness under resource and topology constraints.
Hwang (2004) studied SCP for both ameliorating and deteriorating item to determine
minimum number of storage facility among a discrete set of location sites, so that the
probability of each customer being covered is not less than a critical value. They have
developed a computer program for the stochastic SCP and applied for the fish culture and
the storage and supply centres. Further the impacts of ameliorating and deteriorating rates
on location problems are shown.
Ablanedo and Rego (2005) have studied surrogate constraint normalisation for the
SCP. They introduced a number of normalisation rules and demonstrated their
superiority. The study motivates the development of normalisation rules for generating
good surrogate constraints for SCP and other combinatorial problems.
Hassin and Levin (2005) developed an algorithm, which is better than greedy
approximation algorithm. It gives an improvement on the approximation ratio for all
constant values. They demonstrated that greedy algorithm is not the best possible
approximation algorithm for the weighted set cover problem.
Bautista and Pereria (2005) studied the reverse logistics problem arising in municipal
waste management in Union of European countries. They established relationship
between SCP and MAX-SAT problem and then they used genetic algorithm and GRASP
heuristic, respectively to solve the problem.
Sherali et al. (2006) have considered a situation in which the group of facilities needs
to be constructed to serve a set of customers by maximising the total service reliability of
the system, subject to budgetary constraints. They have formulated a probabilistic PCP
and studied the effect of problem parameters on solution difficulty in understanding the
behaviour of the parameters of the system.
Zhang et al. (2006) developed an effective learning algorithm for the minimum set
cover problem. It is capable of adjusting the balance between constraint term and energy
function, so that the local minimum that the network falls into vanishes and the network
can continue updating in a gradient descent direction of energy. It is a very useful
algorithm in mobile phones, where the user is able to understand or caution the status of
battery current availability, so that the user could inform the other-end user for the
expected disconnection of speech.
Covering problem in operations management 277
Dom et al. (2006) initiated the study of MMSC, the Minimum Membership Set Cover
Problem, in the context of interference-reduction in the cellular networks. It is more
generally known as Red Blue Hitting Set (RBHS), where c red = c blue. This paper
investigates the generalisation of the problem behaviour in terms of Consecutive Ones
Property (COP). They concluded that the Red-Blue Hitting Set obeys the COP. This
paper provides polynomial time solvability, NP completeness and approximability for
various cases.
Monaci and Toth (2006) developed an algorithm for set covering based heuristic
approach for solving two types of problems – constraint bin-packing problem and basic
bin-packing problem, where the objects cannot be rotated.
Berman and Huang (2008) provided a heuristic for a minimum weighted covering
location problem with distance constraints to minimise the total demand covered,
subjected to the condition that no two facilities are allowed to be closer than a pre-
specified distance. They have proved that, there exists a Dominating Location Set. This
paper provides a comparison of the mathematical formulations to solve the problem.
5 Meta-heuristic
A new heuristic by Jacobs and Brusco (1995) provides a local search heuristic for large
SCP. This heuristic is based on the simulated annealing algorithm and uses an
improvement routine designed to provide low-cost solutions within a reasonable amount
of CPU time.
Huang et al. (1994) incorporated new penalty function and mutation operator in their
genetic algorithm approach for the SCP to handle the constraints. While approaching the
optima, the mutation operator approaches on either side of the feasible/infeasible borders.
This in fact reduces the search time to half. They utilised the new penalty function for
handling the constraints.
Beasley and Chu (1994) developed a genetic algorithm based heuristic for non-
unicost SCP by modifying the basic genetic procedure including a new fitness-based
crossover operator (fusion), a variable mutation rate and a heuristic feasibility operator
tailored specifically for the set cover problem. The heuristic is suitable for generating
optimal solution for small size problems, which is its limitation.
Lorena and Lopes (1997) have developed a genetic algorithm and applied it to
computationally difficult SCPs. This genetic algorithm implementation reaches high-
quality computational results for difficult SCPs, arising in computing the 1-width of
incidence matrix of Steiner triple systems.
Parallel randomised heuristic for the SCP proposed by Catalano and Malucelli (2001)
is an iterative and an embedded constructive heuristic within a randomised procedure.
First group of heuristics is obtained by randomising the choices made at each step when
the solution is constructed. Second group of heuristics is obtained by introducing random
perturbation of the cost of the problem instance. They discussed different parallel
implementation of the heuristics.
Yoichi et al. (2001) considered the covering problem applied to geographic feature
analysis. They compared the genetic crossover operator based geographic feature
analysis of the search space of SCP with standard genetic crossover operator, by applying
to Genetic Local Search. They concluded that the crossover-based geographic feature
analysis is the more powerful one.
278 C.N. Vijeyamurthy and R. Panneerselvam
Ermis et al. (2002) have developed a meta-heuristic vibrational genetic algorithm for
solving continuous covering location problem in continuous space in which the demand
centres are independently served from independent supply centres. It is more of a
covering problem rather than set covering location problem. The algorithm uses
vibrational mutation which introduces a wave with random amplitude into population
periodically beginning with the initial step of the genetic process. The assumption of
transportation cost and installation cost is not reflecting the actuals. For instance, the
labour cost in the city is much higher than the labour cost in the rural areas and hence
the assumption of installation charges is misleading.
Aickelin (2002) developed an indirect genetic algorithm for SCP, which has three
separate modular components. Once a suitable order-based encoding is found, only the
decoding and hill climbing criteria need to be modified rather than redesigning
everything. This is a great advantage in implementing this algorithm. The algorithm
outperformed the problem-specific heuristic only and it needs to be evaluated
statistically.
Saydam and Aytug (2003) have addressed the problems of over or underestimation of
the coverage by a significant margin. The accurate estimation of expected coverage
model becomes an important issue. They have presented a genetic algorithm that
combines the expected coverage approach with the hypercube model to solve maximum
expected coverage location problem with increased accuracy.
A Greedy Randomised Adaptive Search Procedure (GRASP) is used to solve the
unicost SCP. Bautista and Pereria (2007) have developed the heuristic to solve unicost
SCP. It is a special case in the SCP.
6 Hybrid heuristics
Caprara et al. (1999) presented a Lagrangian-based heuristic for large-scale SCP and the
same has been refined to give the best solution. The algorithm uses sub-gradient
optimisation coupled with pricing techniques for cutting the computing time. It is a
substitute algorithm for the procedure GREEDY. The variable fixing is a critical step in
the algorithm. Further, the pre-processing is time consuming, which is an undesirable
quality.
Downs and Camm (1998) developed an exact algorithm for the MCP using dual
based solution methods and greedy heuristics in branch and bound techniques. The
hybrid approach developed in this work appears to be effective for the wide range of
MCP’s model parameters. They have compared the results achieved through the new
method with other existing exact methods, and then arrived a conclusion that the hybrid
heuristic solution on MCP is an efficient one.
Genetic algorithm for time-satisfaction based SCP is an integer programming for
minimising the total fixed cost, rather than set covering location problem. This is
designed using mixed genetic algorithm strategies proposed by Yun-Feng et al. (2005)
Vasko et al. (2005) have surveyed several hybrid algorithms for the assessment of
computational performance, using genetic algorithm for the SCP. GRASP, the Greedy
Randomised Adaptive Search Procedure, is designed using genetic algorithm and local
neighbourhood search approach. It is true that hybridisation algorithm generally gives
optimal results and this approach is an addition to the literature.
Covering problem in operations management 279
Laifenfeld et al. (2006) considered the problem of finding the minimum identifying
code in a graph, a designated set of vertices where neighbourhood uniquely overlap at
any vertex on the graph in order to show that this is computationally equivalent to SCP.
They have presented an approximation algorithm, based on greedy approach for SCP.
Identifying code problem is a special case of the test problem and therefore, test-case
approximation can be applied to produce ‘good-identifying’ codes. Application of the
greedy approximation to graphs of size n generates identifying code with the same
performance ratio. The entropy-based approximation applied to graphs gives different
edge probabilities. The source identifying problem is to find the smallest set, where the
sensors which distinguish between any two vertices in a directed graph. Time constrained
version of the source identification problem, where the sensors which distinguish
between two given vertices, does not progress much and no approximation algorithm is
known in literature, which is its weakness. Further the algorithm has slow convergence
ratio.
Chaudhry (2006) has developed genetic algorithm for the ACLP to locate maximum
weighted number of facilities, such that no two facilities are within a specified distance
from each other. It is also a covering problem in the sense that the facilities do not fall
within stipulated distance. This approach enables to cover all the people of the region but
keeping an undesirable facility at maximum distance and its influence is minimal to the
covering population.
Hybrid evolutionary algorithm on minimal vertex cover for random graphs by
Kalapala et al. (2007) analyses the hierarchical Bayesian Optimisation Algorithm
(hBOA) on minimum vertex cover for standard class of random graphs and transformed
SAT instances. The hBOA is compared with genetic algorithm, parallel simulated
annealing algorithm, and branch and bound algorithm. The branch and bound algorithm
outperforms the other algorithms. Only a marginal difference is noticed between hBOA
and other evolutionary algorithms.
Seda (2007) developed an approach to genetic algorithm based heuristic for solving
SCP to find a minimal logical function in the design of logical circuits. Its results have
been compared with that of Quine-McCluskey method. The proposed algorithm is used
in setting parameter for minimising the Boolean functions.
Umetani and Yagiura (2007) derived a number of heuristic algorithms to obtain good
lower and upper bounds for SCP including the linear programming, the sub-gradient
method, construction algorithm, meta-heuristic and their combinations, focusing on the
contributions of the mathematical programming techniques that provide good lower
bound and also help to obtain good upper bound, when incorporated with heuristic
algorithm. On analysis, they concluded that the hybridisation of meta-heuristic and
mathematical programming approaches will be helpful to handle large-scale instances of
other combinatorial optimisation problems such as 0-1 programming problems. Study of
literature reveals that recent applications of the SCP are found in probe selection in
hybridisation experiments in DNA sequencing and feature selection and pattern
construction in logical analysis of data. In fact, the researcher used the hybridisation
with meta-heuristic approach to solve the TCP, which is a variety of the covering
problem.
Gouwanda and Ponnambalam (2008) developed an evolutionary search technique to
solve the SCP by using three techniques, viz. mathematical model using LINGO, GA
tool box from MATLAB and ACO programmed in MATLAB. The authors have
concluded that LINGO performed well as an optimisation tool in solving SCP but GA
280 C.N. Vijeyamurthy and R. Panneerselvam
tool does not perform well despite its flexibility. The solution obtained through ACO
based on past experience, the pheromone tool and heuristic information is near optimal to
all the benchmark problems, though it takes more computational time. The combination
of all the three techniques for solving SCP takes more time to solve the problem.
7 Special techniques
Espejo et al. (2001) presented a two-level hierarchical extension of the MCLP. They have
developed an effective method for its solution. A combined Lagrangian-surrogate (L-S)
relaxation is defined which reduces the MCP to a 0-1 knapsack problem. Tests were
carried out using a surrogate-based heuristic incorporating the L-S relaxation. These were
compared with exact results obtained by using CPLEX. It was found that the computing
times were reasonable.
Solar et al. (2002) presented a Parallel Genetic Algorithm (PGA) model to solve the
SCP. Experimental results obtained with a binary representation of the SCP show that in
terms of the number of generations (computational time) needed to achieve solutions of
an acceptable quality, PGA only evaluates a pth part of the global population, instead of
the process followed in sequential genetic algorithm.
Stern et al. (2003) proposed a use of genetic algorithm with new selection approach
called the queen genetic algorithm. This approach is not to select both parents from the
entire population but to create a group of better solutions (a queen) and to use one of its
members in each performed crowd. The author demonstrates the use of the queen genetic
algorithm for the problem of moving dynamic observers about the polygonal area in
order to maximise visual area coverage for a given time horizon.
Taejin and Ryu (2004) introduced a genetic algorithm incorporating unexpressed
genes to solve large-scaled MCPs efficiently. The authors employed a new crossover and
mutation operators specially designed to work for the chromosomes of set-oriented
representation. The unexpressed genes are the genes which are not reflected in the
evaluation of the individuals. These genes play the role of preserving information
susceptible to be lost by the application of genetic operators but potentially useful in later
generations. By incorporating the unexpressed genes, the algorithm enjoys the advantage
of being able to maintain diversity of the population preventing premature convergence.
The authors have shown that their algorithm outperforms neighbourhood search
algorithms.
Gungor and Gunes (2001) discuss an algorithm related to the problem of how many
various main populations the samples come from; in other words in how many different
clusters these samples can band together. This algorithm has pursued a solution to the
problem of clustering of cities in Turkey based on sunbathing data. At the same time it
appears that applied algorithm, which performs clustering processes, is compatible with
significance tests.
8 Conclusion
Facility location problem is a challenging one in industrial as well as service organisations.
Among different types of facility location problem, covering problem aims to minimise
the total number of sites to which facilities are to be assigned to cover all the customers.
Covering problem in operations management 281
In this paper, a comprehensive review of literature of the covering problem and its related
techniques is presented. The review covers different types of covering problem, viz. total
covering, partial covering, simple set covering, maximal covering, minimal covering and
other types of covering, namely special covering. The techniques to solve the covering
problem have been classified into exact procedures, heuristics, meta-heuristics, hybrid
heuristics and special techniques. In each of these categories, an extensive review of
literature of the techniques have been studied and presented.
Due to globalisation, the level of competition is very high. In this context, every
organisation tries to cut cost and explore the possibilities to maintain the quality
standards and offer products/services with less cost to stay in the market. This paper
would help a company to select a suitable technique to obtain the best solution for
locating its facilities to cover all its customers. Further, it will help the researchers to
proceed from the latest benchmarked approaches to obtain solution for the covering
problem.
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