INDIAN INSTITUTE OF MANAGEMENT
AHMEDABAD • INDIA
Research and Publications
The Single Row Facility Layout Problem: State of the
Art
Ravi Kothari
Diptesh Ghosh
W.P. No. 2011-12-02
December 2011
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INDIAN INSTITUTE OF MANAGEMENT
AHMEDABAD – 380015
INDIA
W.P. No. 2011-12-02 Page No. 1
IIMA • INDIA
Research and Publications
The Single Row Facility Layout Problem: State of the
Art
Ravi Kothari
Diptesh Ghosh
Abstract
The single row facility layout problem (SRFLP) is a NP-hard problem concerned with the
arrangement of facilities of given lengths on a line so as to minimize the weighted sum of the
distances between all the pairs of facilities. The SRFLP and its special cases often arise while
modeling a large variety of applications. It was actively researched until the mid-nineties. It has
again been actively studied since 2005. Interestingly, research on many aspects of this problem
is still in the initial stages, and hence the SRFLP is an interesting problem to work on. In
this paper, we review the literature on the SRFLP and comment on its relationship with other
location problems. We then provide an overview of different formulations of the problem that
appear in the literature. We provide exact and heuristic approaches that have been used to
solve SRFLPs, and finally point out research gaps and promising directions for future research
on this problem.
1 Introduction
In this paper, we study the literature on the Single Row Facility Layout Problem (SRFLP). The
problem is known to be NP-hard Beghin-Picavet and Hansen (1982) and has attracted significant
attention in recent years. The problem is the following. Consider a set of rectangular facilities
in which each of the facilities differ in their lengths. Each facility communicates with every other
facility. The cost of communication between a pair of facilities is the product of the transmission
intensity between the two facilities and the distance between them. The transmission intensity can
be visualized as the number of times the facilities in the pair need to communicate with each other,
and the distance between the pair of facilities is measured as the distance between their centroids. So
the cost of communication between a pair of facilities is high either if the pair has a high transmission
intensity, i.e., they communicate frequently, or if the facilities are located far apart. The total cost
of transmission is the sum of the costs of transmission between every pair of facilities. The objective
of the SRFLP is to arrange the facilities in a single row along their lengths so that the total cost of
transmission is as low as possible. The size of a SRFLP is the number of facilities in the problem.
This problem was first proposed by Simmons Simmons (1969). Formally stated the SRFLP is defined
as follows:
Given: A set F = {1, 2, . . . , n} of n > 2 facilities, where facility j has length l
j
, and transmission
intensities c
ij
for each pair (i, j) of facilities, i, j ∈ F , i 6= j.
Objective: To find a permutation Π = (π
1
, π
2
, . . . , π
n
) of F that minimizes the expression
X
1≤i<j≤n
c
π
i
π
j
d
Π
π
i
π
j
where d
Π
π
i
π
j
= l
π
i
/2 +
P
i<k<j
l
π
k
+ l
π
j
/2 is the distance between the centroids of facilities π
i
and π
j
when the facilities in F are ordered as per the permutation Π.
W.P. No. 2011-12-02 Page No. 2
IIMA • INDIA
Research and Publications
The SRFLP has been used to model numerous practical situations. It has been a model for ar-
rangement of rooms in hospitals, departments in office buildings or in supermarkets Simmons (1969),
arrangement of machines in flexible manufacturing systems Heragu and Kusiak (1988), assignment
of files to disk cylinders in computer storage, and design of warehouse layouts Picard and Queyranne
(1981). Apart from these direct applications, there are a large number of applications of the special
case of the SRFLP in which the facilities have equal lengths. These include assignments of airplanes
to gates at an airport terminal Anjos and Yen (2009), triangulation problem of input output tables
in economics Laguna et al. (1999), and ranking of teams in sports Mart´ı and Reinelt (2011).
In this paper we survey the literature on the SRFLP. Section 2 places the problem in a hierarchy
of related problems. Section 3 presents several ways in which the problem has been formulated in
the literature. Sections 4 and 5 present different solution methods that have been used to solve the
problem, with the former section dealing with exact methods which guarantee optimal solutions and
the latter section dealing with heuristic solution procedures. Finally, Section 6 points out gaps in
the literature and suggests promising directions for future research on this problem.
2 Relations to other problems
The SRFLP can be placed in a hierarchy of problems dealt with in the literature, some of whose
members are special cases of other members. We present this hierarchy below, starting from the
most general problem in the hierarchy and ending with the most specialized problem.
Space allocation problem: The objective in a space allocation problem (SAP) is to shape and
orient facilities on a floor plan so as to achieve a given efficiency measure. Such a problem arises
when an architect is trying to arrange facilities of equal area but unspecified shape so as to minimize
a given linear combination of distances between all pairs of facilities (see Simmons (1969)). This is
not strictly a facility location problem, because in a facility location problem there are pre-defined
set of locations which are separated by a fixed distance and it is only required to determine which
facilities are to be located at which locations. In the SAP the locations are not defined and it is up
to the architect to design the entire floor plan.
Single row facility layout problem: The single row facility layout problem (SRFLP) is a special
case of the SAP where facilities have fixed lengths and are to be arranged in a linear fashion. It is an
ordering problem with facilities of non-uniform length being arranged along a single row (see, e.g.,
Anjos et al. (2005)). The SRFLP is also known as the one dimensional space allocation problem
(ODSAP).
Generalized linear ordering problem: An interesting location problem that is closely related to
SRFLP is the generalized linear ordering problem (GLOP). Here one is given n facilities, n locations
along a line, and the transmission intensity between each pair of facilities, and one needs to find the
one-one assignment of n facilities to the n locations so as to minimize the total transmission cost
(see, e.g., Picard and Queyranne (1981)).
Linear ordering problem: The linear ordering problem (LOP) is a special case of both the
SRFLP and the GLOP. It can be described as a SRFLP in which all facilities have the same length.
It can also be seen as a GLOP with regularly placed locations separated by the common length of
the facilities. The LOP is also referred to as the linear arrangement problem in the literature (see,
e.g., Picard and Queyranne (1981)).
W.P. No. 2011-12-02 Page No. 3
IIMA • INDIA
Research and Publications
3 Formulations of SRFLP
We now describe the different integer and mixed integer programming formulations of the SRFLP
that are available in the literature. In this section, we describe these formulations in chronological
order. For consistency, we use the same notation as used in the formal description of the SRFLP in
Section 1.
Note that the term c
ij
has been used in different connotations in the literature. It has been
defined as cost by Simmons Simmons (1969), frequency of travel by Heragu and Kusiak Heragu
and Kusiak (1988), affinity by Romero and S´anchez-Flores Romero and S´anchez-Flores (1990), and
transition probabilities by Picard and Queyranne Picard and Queyranne (1981).
The first formulation of the problem which was proposed by Simmons Simmons (1969) is the fol-
lowing.
Minimize
n−1
X
i=1
n
X
j=i+1
c
ij
d
ij
.
The paper showed that this formulation is equivalent to
Minimize
n−1
X
i=1
n
X
j=i+1
c
ij
b
ij
,
where b
ij
is sum of the lengths of all facilities between facility i and facility j in a permutation. The
formulation used n(n − 1)/4 variables. The constraints were not explicitly formulated in the paper.
Love and Wong Love and Wong (1976) provided the first linear mixed integer programming
formulation for the SRFLP. The formulation was as follows.
Minimize
n−1
X
i=1
n
X
j=i+1
c
ij
(R
ij
+ L
ij
)
Subject to R
ij
− L
ij
= x
i
− x
j
+
1
2
(l
j
− l
i
)
x
i
− x
j
+ M(α
ij
) ≥ l
i
x
j
− x
i
+ M(1 − α
ij
) ≥ l
j
l
i
≤ x
i
≤
n
X
i=1
l
i
α
ij
= 0, 1
R
ij
, L
ij
, x
1
, ..., x
n
≥ 0,
where x
i
is the endpoint of facility i farthest from the origin; α
ij
is 1 if facility i is to the left of
facility j, and 0 otherwise; R
ij
is the distance between the centroid of facility i and centroid of
facility j if α
ij
= 0, and 0 otherwise; L
ij
is the distance between centroid of facility i and centroid
of facility j if α
ij
= 1 and 0 otherwise; and M is an arbitrarily large number.
This formulation used n(n − 1)/2 binary variables and had 3n(n − 1)/2 constraints excluding the
non-negativity constraints and lower and upper bounds on x
i
.
Picard and Queyranne Picard and Queyranne (1981) proposed the the same formulation as by
Simmons Simmons (1969). However, their paper was the first to explicitly state the idea that every
solution to the SRFLP is in fact a permutation of the set of facilities.
W.P. No. 2011-12-02 Page No. 4
IIMA • INDIA
Research and Publications
Ravi Kumar et al. Ravi Kumar et al. (1995) presented a quadratic assignment problem formu-
lation for the GLOP, which is a special case of the SRFLP in which a number of distinct facilities
are to be assigned to an equal number of equidistant locations which lie on a straight line. The
mathematical formulation was the following.
Minimize
n
X
i=1
n
X
j=1
n
X
k=1
n
X
k
0
=1
x
ik
x
jk
0
c
ij
|k − k
0
|
Subject to
n
X
i=1
x
ik
= 1, k = 1, 2, ..., n
n
X
k=1
x
ik
= 1, i = 1, 2, ..., n
x
ik
∈ 0, 1 i, k = 1, 2, ..., n,
where x
ik
is 1 if facility i is assigned to location k, and 0 otherwise. This formulation used n
2
binary
variables and had 2n constraints.
Amaral Amaral (2006) introduced a new mixed integer linear programming model for the SRFLP.
This formulation used additional variables of the form α
ij
which assume a value of 1 if facility i is
to the left of facility j, and 0 otherwise. It also used two polytopes, H
n
= {α ∈ <
n(n−1)/2
+
: 0 ≤
α
ij
+α
jk
−α
ik
≤ 1, 1 ≤ i < j < k ≤ n} and D
n
= {d ∈ <
n(n−1)/2
: d
ij
≥ (l
i
+l
j
)/2, 1 ≤ i < j ≤ n}.
The formulation was as follows.
Minimize
n−1
X
i=1
n
X
j=i+1
c
ij
d
ij
Subject to d
ij
≥
X
k<i
l
k
α
ki
+
X
k>i
l
k
(1 − α
ik
) −
X
k<j
l
k
α
kj
−
X
k>j
l
k
(1 − α
kj
) + (l
i
− l
j
)/2 1 ≤ i < j ≤ n
d
ij
≥
X
k<i
l
k
α
ki
−
X
k>i
l
k
(1 − α
ik
) +
X
k<j
l
k
α
kj
+
X
k>j
l
k
(1 − α
kj
) + (l
j
− l
i
)/2 1 ≤ i < j ≤ n
α ∈ H
n
d ∈ D
n
α
ij
∈ 0, 1 1 ≤ i < j ≤ n.
The formulation used n(n − 1)/2 binary variables and n(n − 1)/2 continuous variables. There were
n(n − 1)(2n − 1)/3 constraints excluding the non-negativity constraints and the bounds on the
variables.
Amaral Amaral (2009) proposed a new “betweenness” based formulation for the SRFLP. The
formulation defined a set U
n
as a convex hull of all vectors ζ = (ζ
ijk
) where ζ
ijk
is 1 when facility k
lies between facilities i and j, and 0 otherwise. The formulation was the following.
W.P. No. 2011-12-02 Page No. 5