3.2 Set Covering Model
A set covering model is used in cases such as service facility locating problems,
airline crews on a flight, assigning customers to delivery routes, and military
logistics problems (Karasakal and Karasakal 2004; Lanza-Gutierrez et al. 2017). It
is a zero-one linear programming model. Covering problems hold a central place in
the location theory (Farahani and Asgari 2007). In these problems, we are given a set
of demand points and potential sites for locating facilities. A demand point is
claimed to be covered by a facility if it lies within a pre-specified distance of that
facility. Covering problems are divided into two main classes, namely, set (total)
covering problems, which cover all demand points with a minimum number of
facilities, and maximal (partial) covering problems, which cover a maximum number
of demand points with a fixed number of facilities (Church and Velle 1974). Based
on the information provided in this section, the set covering model for the problem is
as follows:
Max Z 1 ¼
X m
i¼1
V i y i
ð5:6Þ
Min Z 2 ¼
X m
i¼1
y i
ð5:7Þ
Subject to:
X m
i¼1
a di y i ! 1
for all d
ð5:8Þ
y i 2 0, 1
f g
ð5:9Þ
The first objective function (Eq. 5.6) represents the utility of the chosen places for
establishing bioethanol distribution centers. As the number of centers decreases, the
second objective function (Eq. 5.7) in the suggested model decreases the cost
allocated to establish distribution centers. In this model, m is the number of candidate
places for locating bioethanol distribution centers, and n is the number of places that
should be covered by distribution centers. y i is a binary variable; it is one if the
candidate i is suitable for establishing bioethanol distribution centers and zero
otherwise. V i is the output of MCDM for candidate alternatives. The coverage matrix
(a di ) is a binary parameter so that if the distance between the candidate place i and
service applicant d is less than or equal to coverage radius, a di is one and otherwise it
is equal to zero. The coverage matrix is formed based on the coverage radius, and the
coverage radius is the maximum distance to a center which can provide services for
service applicants. In other words, it acts as its supporter. The constraint 8 presented
in the model shows places satisfying coverage radius.
5 Location Selection of Bioethanol Distribution Centers
81
A set covering model is used in cases such as service facility locating problems,
airline crews on a flight, assigning customers to delivery routes, and military
logistics problems (Karasakal and Karasakal 2004; Lanza-Gutierrez et al. 2017). It
is a zero-one linear programming model. Covering problems hold a central place in
the location theory (Farahani and Asgari 2007). In these problems, we are given a set
of demand points and potential sites for locating facilities. A demand point is
claimed to be covered by a facility if it lies within a pre-specified distance of that
facility. Covering problems are divided into two main classes, namely, set (total)
covering problems, which cover all demand points with a minimum number of
facilities, and maximal (partial) covering problems, which cover a maximum number
of demand points with a fixed number of facilities (Church and Velle 1974). Based
on the information provided in this section, the set covering model for the problem is
as follows:
Max Z 1 ¼
X m
i¼1
V i y i
ð5:6Þ
Min Z 2 ¼
X m
i¼1
y i
ð5:7Þ
Subject to:
X m
i¼1
a di y i ! 1
for all d
ð5:8Þ
y i 2 0, 1
f g
ð5:9Þ
The first objective function (Eq. 5.6) represents the utility of the chosen places for
establishing bioethanol distribution centers. As the number of centers decreases, the
second objective function (Eq. 5.7) in the suggested model decreases the cost
allocated to establish distribution centers. In this model, m is the number of candidate
places for locating bioethanol distribution centers, and n is the number of places that
should be covered by distribution centers. y i is a binary variable; it is one if the
candidate i is suitable for establishing bioethanol distribution centers and zero
otherwise. V i is the output of MCDM for candidate alternatives. The coverage matrix
(a di ) is a binary parameter so that if the distance between the candidate place i and
service applicant d is less than or equal to coverage radius, a di is one and otherwise it
is equal to zero. The coverage matrix is formed based on the coverage radius, and the
coverage radius is the maximum distance to a center which can provide services for
service applicants. In other words, it acts as its supporter. The constraint 8 presented
in the model shows places satisfying coverage radius.
5 Location Selection of Bioethanol Distribution Centers
81