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Contents lists available at ScienceDirect
Journal of Manufacturing Systems
journal homepage: www.elsevier.com/locate/jmansys
Multi-objective particle swarm optimization for multi-workshop facility
layout problem
Chao Guan
a,b
, Zeqiang Zhang
a,b,⁎
, Silu Liu
a,b
, Juhua Gong
a,b
a
School of Mechanical Engineering, Southwest Jiaotong University, Chengdu 610031, China
b
Technology and Equipment of Rail Transit Operation and Maintenance Key Laboratory of Sichuan Province, Chengdu 610031, China
ARTICLE INFO
Keywords:
Multi-workshop facility layout problem
Multi-objective optimization
Particle swarm optimization
Two-stage approach
Discrete framework
ABSTRACT
The novel multi-workshop facility layout problem presented in this paper involves the placement of a group of
departments into several workshops; it deals with the distribution of departments over workshops and their
optimal exact coordinates without overlapping. In considering practical situations, the internal material hand-
ling flows and external transport flows are taken into account in the problem. In this study, the problem is first
formulated as a mixed integer linear programming model with three objectives: minimization of overall material
handling costs, minimization of number of workshops, and maximization of utilization ratio of workshop floor.
Thereafter, the proposed multi-objective particle swarm optimization algorithm with an innovative discrete
framework and incorporated with a two-stage approach is employed to search for feasible solutions locally and
globally. Finally, several benchmark instances derived from literature that satisfy our case requirements are
employed to evaluate the performance of the proposed method; highly preferable results are typically achieved.
1. Introduction
The facility layout problem (FLP) refers to the most profitable
physical arrangement of a set of given facilities with known dimensions
within the manufacturing system. In production organizations, the
configuration of facility layouts is particularly relevant to the material-
handling system, which involves the resources demanded for manu-
facture or delivery of service [1]. Nowadays, industrial companies
ameliorate the facility layout in order to efficiently utilize resources,
decrease unnecessary movements, improve internal logistics transpor-
tation, and consequently reduce costs [2]. Previous research found that
approximately 20%–50% of operating costs are constituted by the
material handling activities; moreover, 10%–20% of that cost can be
abated by a reasonable optimization of equipment layout [3,4].
In terms of practical applications, the FLP can be divided into three
broad categories: row facility layout problem (RFLP), unequal-area fa-
cility layout problem (UA–FLP), and multi-floor facility layout problem
(MFFLP) [1,5,6]. The classic FLP continues to be studied extensively
because of the remarkable reduction in investment and operating costs
that results from the adoption of a novel and efficient layout config-
uration.
In this study, an innovative multi-workshop facility layout problem
(MWFLP) is investigated; it takes into consideration that a group of
facilities (called departments in this work) can be arranged into more
than one workshop. Moreover, the MWFLP involves two types of phy-
sical flows: internal material handling flows and external work-in-pro-
cess transport flows. Internal flows, which are generally executed via
the material feeding system in the workshop, occurs inside each
workshop. On the other hand, there are critical external flows among
workshops that are dependent on extra transport vehicles (e.g., trucks
and trains). Therefore, the unit flow cost of the two different logistics
mentioned above is different; it can be embodied in the mathematical
model presented in this paper.
Because the MWFLP is classified with non-deterministic polynomial
time (NP)-hard problem, most studies have resolved these problems by
employing iterative meta-heuristic approaches, such as hybrid genetic
algorithm (HGA) [7–10], simulated annealing (SA) algorithm [11,12],
tabu search (TS) method [13], ant colony optimization (ACO) [14,15],
bacterial foraging optimization (BFO) [2], particle swarm optimization
(PSO) [3,16,17], and clonal selection algorithm (CSA) [18]. In this
article, a multi-objective particle swarm optimization (MOPSO) is
presented on the basis of our multi-objective model, with the mini-
mization of overall material handling cost, minimization of number of
workshops, and maximization of utilization ratio of workshop floor.
Additionally, because of the dual characteristics of combinatorial op-
timization and continuous optimization of the specific layout problem,
https://doi.org/10.1016/j.jmsy.2019.09.004
Received 23 May 2019; Received in revised form 7 September 2019; Accepted 9 September 2019
⁎
Corresponding author at: School of Mechanical Engineering, Southwest Jiaotong University, Chengdu 610031, China.
E-mail addresses: 17175371524@163.com (C. Guan), zhangzq@home.swjtu.edu.cn (Z. Zhang), liusilu_66@163.com (S. Liu),
gongjuhua@my.swjtu.edu.cn (J. Gong).
Journal of Manufacturing Systems 53 (2019) 32–48
0278-6125/ © 2019 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
T
Author's Personal Copy
a two-stage optimization method is designed and combined with the
MOPSO. In the first stage, a novel encoding measure is devised to
produce feasible MWFLP layout solutions; thus, all binary variables in
the proposed model can be uniquely determined. For the second stage,
a simplified linear model is implemented and solved by CPLEX opti-
mization software. Furthermore, several benchmark instances derived
from literature that satisfy our case requirements are employed to
evaluate the performance of MOPSO; highly preferable results are ty-
pically achieved.
The main contributions of our work are listed as follows:
•A novel multi-workshop facility layout problem is proposed here,
which involves the arrangement of departments in several available
workshops, besides, the internal and external logistics costs are also
considered according to practical production.
•A mixed-integer linear programming model is formulated for this
problem, and some instances can be solved to optimality within rea-
sonable times.
•The individual particle encoding and placement strategy for our
MWFLP are designed, which are implemented to represent and search
for all feasible solutions.
•On the basis of the combinatorial nature and intrinsic difficulty of
the problem, a multi-objective particle swarm optimization algorithm
with an embedded two-stage approach is proposed. Additionally,
comparison experiments with other methods indicate that the proposed
algorithm can achieve highly preferable results. Finally, some MWFLP
cases are solved by our proposed algorithm.
The next section of this article presents a brief review of related
literature; subsequently, a more detailed description of the MWFLP is
introduced. Moreover, a proposed mathematical model is formulated,
and a case study is introduced. Finally, the two-stage optimization
method and structure of MOPSO are described in detail; the remaining
section presents numerical experiments and certain comparisons.
2. Literature review
The FLP is among the most crucial classical problems in production
management and industrial engineering that has considerably attracted
the interest of scholars since the 1960s. Numerous variations of the FLP
have been considered and extensively investigated in literature, and a
wide variety of formulations and objectives has been proposed for each
variation. Meanwhile, various approaches have been successively pre-
sented in order to achieve effective solutions.
2.1. Facility layout problem and evaluation criteria
Facility layouts are directly influenced by the specifications of
manufacturing systems, and several types of physical configurations are
designed based on practical characteristics, such as product range and
production volumes. To date, in existing literature, most research works
focus on layouts in the single region [2,3,5–7,9,13–16,19–23]. In the
product layout, a set of rectangular machines and equipment are as-
signed to several rows depending on the procedure of operations re-
quired for the product; this is called row facility layout problem
[6,19,21]. Additionally, different from the aforementioned problem,
the unequal area facility layout problem considers the obtainment of an
excellent arrangement of given indivisible departments with a two-di-
mensional rectangle and diverse areas; all departments are located in
the same fixed region with no overlapping. The unequal area FLP has
received considerable interest since it was first proposed in 1963 [24],
it had several variants from then on. For example, an unequal area FLP
with a fixed size and shape has been conceptualized and investigated
[7,13]; geometric constraints for confirming the relative location of
departments have also been proposed to prevent overlapping. More-
over, a space-filling curve was utilized by connecting each unequal area
department to be continuously arranged without partitions [22]; this
technique was applied to manage the unequal area FLP with irregular
departments. Furthermore, the design of a cellular manufacturing
system (CMS) with an equal area FLP, which was a special case of the
unequal area FLP [23], was presented. On the other hand, the multi-
floor facility layout problem, which is relatively similar to the MWFLP
in this article, extended the unequal area FLP to different floors except
for a single building [5,23]. Recently, a novel multi-floor FLP was de-
veloped for real-life scenarios where a number of departments have to
be assigned to a multi-floor building with a fixed room configuration;
rooms under the same department were required to be adjacent [25]. A
further simplification of the multi-floor FLP was presented: the de-
partments were restricted to the same shape and assigned to special
locations in the building [26]. Apart from the above, an FLP in which
the departments were placed in a group of facilities was studied; sub-
sequently, because of its complexity, it was simplified into a quadratic
assignment problem [27,28].
Classically, a layout design should be evaluated from different
perspectives; in this regard, the optimization criteria in the FLP litera-
ture vary in form. The material handling cost minimization, which is
the most critical indicator for evaluating the performance of a layout, is
the most popular criterion in studies [2]. Additionally, minimizing the
re-layout cost is another type of objective that is considered a necessity
in dynamic layout design [2,16,23,27]. Other objectives that are stu-
died in existing research also include minimizing the distance travelled
[29,30], minimizing work in the process [31], maximizing satisfaction
[32], maximizing adjacency/closeness [32,33], and maximizing profits
[34,35]. In this work, two other functions are considered to measure
the performance of the existing MFFLP layouts: the number of work-
shops required to build for production and the utilization ratio of
workshop floor.
2.2. Staged approaches for facility layout problem
It has been proven that the FLP is an NP-hard problem in general
[1,3,36]; hence, obtaining a global optimal solution within a reasonable
time is difficult. Actually, numerous restricted FLPs, in which the fixed
dimensions of the facilities are all equal and only the assignment of
facilities to designated locations is considered, are known as combi-
natorial optimization problems, which can be abstracted into discrete
quadratic programming formulations.
Furthermore, for some FLPs that merge the combinatorial nature
and characteristics of continuous optimization, the traditional exact
methods, such as branch-and-bound [37] and dynamic programming
[38,39], are unavailable for finding optimal results or even feasible
solutions because of their high computational cost. Therefore, several
multi-stage frameworks are presented to handle the composite opti-
mization FLPs. For example, Wang et al. [40] employed a methodology
that combines the SA with mathematical programming to resolve the
dynamic double-row layout problem (DDRLP). In the first stage, the
sequence and locations of departments were designed; then, the use of
SA was suggested to obtain the solution to the DDRLP. Actually, multi-
stage approaches are frequently applied to manage the multi-floor FLP,
as follows: in the first stage, the departments are generally distributed
over the floors; in the second stage, reasonable relative positions are
identified for the departments in each floor; in the last stage, no overlap
is enforced, and the exact coordinates of departments are determined
[41–43].
2.3. PSO and multi-objective processing methods
In this work, the multi-objective particle swarm optimization
(MOPSO) is modified and employed to solve the multi-objective
MWFLP. This is because the MOPSO is not only easy to implement
where there are few parameters to adjust, but it has good convergence
speed and is also one of the successfully established solution approaches
to multi-objective optimization problems. Liu et al. [3] proposed an
MPSO algorithm that combines the objective space division technique
C. Guan, et al. Journal of Manufacturing Systems 53 (2019) 32–48
33
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to optimize the multi-objective UAFLP. Asl and Wong [16] formulated
multi-objective models for unequal area static and dynamic facility
layout problems; therefore, a modified PSO was suggested to solve
them. Onut et al. [17] conceived a PSO to handle the multiple-level
warehouse layout design problem; the PSO was able to obtain near
optimal results in a short time. Zhang et al. [44] developed a new bare-
bone MOPSO algorithm to solve the environment/economic dispatch
problems, this algorithm was able to expand the search capability
without tuning up control parameters and a technique was introduced
to handle unfeasible solutions. Zhang et al. [45] proposed a multi-
swarm cooperative multi-objective particle swarm optimizer (MC-
MOPSO), which consisted of multiple slave and master swarms to im-
prove the performance of MOPSO. Mousavi et al. [46] examined a two-
echelon distribution supply chain network by presenting a modified
PSO to find the optimal location of manufacturers and retailers. Except
for the mentioned above, many new variants of MOPSO are in-
vestigated in recent years and are widely applied in many fields, such as
cost-based feature selection [47,48], robot path planning [49], equity
portfolio management [50], etc. Accordingly, in view of the extensive
applications mentioned above, it is appropriate to adopt the PSO based
on the nature of the MWFLP.
Extensive research concerning the multiple optimization objectives
of the FLP as a multi-objective optimization problem was implemented
to handle this type of problem effectively. In the existing literature, the
lexicographic ordering approaches are adopted to optimize each ob-
jective gradually [36,51–53]; however, it only optimizes some of the
objectives and ignores others. On the other hand, the weighted sum
approach is another common method employed [27,54–56]; it
straightforwardly transforms the multi-objective problem into one that
has a single objective by assigning weight values to each objective.
However, the balance among objectives is difficult to achieve because
the weight values are based on the subjective judgment of experts.
Additionally, the Pareto solution set, which can acquire a series of
balanced solutions that focuses on different objectives, is widely im-
plemented and popular in decision-making [3,15,25,33,35,57]. Based
on the foregoing method, in order to preserve the population diversity
and convergence to the Pareto front, the crowding distance strategy
proposed by Deb et al. [58] is applied in our algorithm. Compared with
other representative methods in the existing literature such as clus-
tering approach [59], the adaptive grid approach [60], the decom-
position-based archiving approach [61], the ε-dominance-based ap-
proach [62], and so on, it has been widely employed and was easy to
implement for multidimensional optimization problems to maintain the
diversity of the external archives.
3. Problem description and exact formulation
3.1. Multi-workshop facility layout problem
The extension of the proposed FLP mentioned above involves the
collection of fixed-shaped departments to multiple workshops, in which
overlaps among departments are forbidden in the horizontal and ver-
tical directions. Moreover, all the workshops have the same size and
interactive position via which the resources flow among different
workshops. Apart from the foregoing, based on the engineering of the
manufacturing factory, the internal and external flows require different
transportation expenses, and the external flows are extremely costly
because trucks or forklifts are used. More detailed descriptions per-
taining to the MWFLP are illustrated in Fig. 1; the figure indicates that
10 departments are allocated, and the interactive flows inside and
outside the site are considered. Being similar with the previous re-
search, the resource interaction point for each department is assumed to
be situated in the centroid of the rectangle, and the lower left corner of
each workshop is associated with a single location for entry and exit.
Additionally, because of the horizontal and vertical walking paths of
the automatic guided vehicle (AGV), the distance between two
departments is the Manhattan distance. For convenience in describing
the flow of communications among departments, Fig. 1 depicts two
resource transportation modes. The internal flows between departments
D-1 and D-2 are generated by means of material handling conveyances
(such as the AGV), which are appropriate for conveying goods inside
the workshop. On the other hand, transport conveyances, such as trucks
and forklifts, are employed to carry products among workshops. The
distance travelled by conveyances between departments D-1 and D-8 is
measured from 0
g
to 0
t
; the remaining movements from D-1 to D-8
involve the internal flows generated by means of the AGV. It should be
noted that in our MWFLP, it is necessary to optimize the precise loca-
tion of each department in order to minimize the weighted sum of
distances among departments. Moreover, the number of workshops
built for departments is necessarily considered to be reduced, then the
costs of establishment of all foundations are saved; besides, an adequate
concurrent workshop floor utilization is suggested, which is used to
minimize and balance the area of envelop rectangle among workshops,
thus there will be more available space for storage, transportation and
rest.
Moreover, as is shown in Fig. 1, the workshops placed along a
straight line can be regarded as a special case of the single-row equi-
distant facility layout problem [19]; the major difference among them
is that the number of workshops is not definite, and the allocation of
given departments in each workshop must be considered. Moreover,
note that the proposed problem has certain similarities with the multi-
floor facility layout problem; each workshop can be regarded as the
floor layout in the MFFLP, and the lower left corner can be treated as
elevators. However, compared with relevant MFFLP research [6,25,41],
the various transportation expenses of internal and external flows are
taken into account in this paper, and the number of workshops or floors
are not affirmatory beforehand. Thus, the MWFLP is a more popular
and complex case of the FLP.
3.2. Mathematical formulation
Based on the characteristics of the proposed problem and con-
sidering the convenience of description, an independent coordinate
system is defined for each workshop: the lower left corner is set as the
origin and is used as reference to calculate the positions of centroids of
departments; the Xand Yaxes coincide with the workshop length and
width edges, respectively. Accordingly, the constraints (such as no de-
partment overlapping, each department can be arranged only in one
workshop, and the department size fits into the workshop dimensions)
can be easily constructed as presented in this section.
First, certain parameters and variable notations are defined; sub-
sequently, the mixed integer linear programming (MILP) mathematical
model of the MWFLP is formulated.
Parameters and indices
n
: number of departments
m
: maximum number of available workshops
N
:set of given
n
departments
=Nn
(
{1, 2, ..., } )
M
:set of all numbers of given
m
available workshops (
=
M
m{1, 2, ..., }
)
i
js,, : indices for departments
k
gt,,
:indices for available workshops
L
F:length of workshop floor
W
F
: width of workshop floor
l
i:length of ith department
w
i: width of ith department
p
ij
: material flow value between departments iand j
f
i
j
:frequency of material flow between departments iand j
cIn
t
:handling cost of unit material per unit distance inside workshops
cEn
t
: transport cost of unit resources per unit distance among workshops
Decision variables
C. Guan, et al. Journal of Manufacturing Systems 53 (2019) 32–48
34
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′
L
k
:length of envelope rectangle, which can obtain all departments that are
assigned to kth workshop
′
W
k
:width of envelope rectangle, which can obtain all departments that are
assigned to kth workshop
d
Extij:Manhattan distance travelled by transport conveyances outside workshops
between departments iand j
d
Int
ij
x
:distance between departments iand jprojected in Xdirection that is
covered by material handling conveyances
d
Int
ij
y
:distance between departments iand jprojected in Ydirection that is
covered by material handling conveyances
x
ik
:X-coordinate of centroids of departments i, which are assigned to kth
workshop
yi
k
:Y-coordinate of the centroid of departments i, which are assigned to kth
workshop
α
β,
ij
kij
k
:sequence-pair variables that are used to determine the relative positions of
departments; for any departments iand j, the following conditions hold:
if =
α1
ij
kand
=
β1
ij
k
, then iprecedes jhorizontally;
if
=
α
0
ij
kand
=
β
0
ij
k
, then jprecedes ihorizontally;
if =
α1
ij
kand
=
β
0
ij
k
, then iprecedes jvertically;
if
=
α
0
ij
kand
=
β1
ij
k
, then jprecedes ivertically.
=⎧
⎨
⎩
Sk1,ifworkshop is employed
0,otherwise
k
=⎧
⎨
⎩
Yik1,ifdepartment is ass igned in workshop
0,otherwise
ik
=⎧
⎨
⎩
q
ij k1,ifdepartments and are placed in workshop together
0,otherwise
ij
k
In terms of the above notations, the MILP formulation of the MWFLP
can be expressed as follows.
Objective functions
∑∑ ∑∑=⋅⋅⋅++⋅⋅
⋅=
−
=+ =
−
=+
F p f cInt dInt dInt p f cExt
dExt
min ( )
MHC
i
n
ji
n
ij ij ij
xij
y
i
n
ji
n
ij ij
ij
1
1
11
1
1
(1)
∑
==
FSmin
NWS
k
m
k
1(2)
∑
⎜⎟
=⎛
⎝′⋅′
⋅⎞
⎠
=
FLW
LW nmin /
UR
k
m
kk
FF
1
2
(3)
The three primary objectives we focus on are as formulated above;
these duly reflect the multi-objective nature of the MWFLP. Objective
function (1) is developed to measure the overall material handling
costs, which is composed of internal and external transportation costs.
Objective function (2) minimizes the number of available workshops
required for departments. Objective function (3) is presented to opti-
mize the the utilization ratio of the shop floor; it involves the use of the
workshop floor occupied by departments. Given the foregoing, the in-
terrelated constraints of the proposed MILP model can be formulated as
follows:
Subject to
−⋅⋅− − + ≥
−⋅ ≤<≤ ∀ ∈ ≠M
mL YYdExtk
gL i j n kg k g
(1)(2 ) (
),1 ; , ;
Fikj
gij
F
(4)
−⋅⋅− − + ≥
−⋅ ≤<≤ ∀ ∈ ≠M
mL YYdExtg
kL i j n kg k g
(1)(2 ) (
),1 ; , ;
Fikj
gij
F
(5)
∑
⋅− + ≥ − ≤<≤ ∈
=
M
LqdIntxxijnk(1 ) , 1 ;
F
t
m
ij
tij
xikj
k
1
(6)
∑
⋅− + ≥ − ≤<≤ ∈
=
M
LqdIntxxijnk(1 ) , 1 ;
F
t
m
ij
tij
xj
kik
1
(7)
∑
⋅− + ≥ − ≤<≤ ∈
=M
W
qdIntyy ijnk(1 ) , 1 ;
F
t
m
ij
tij
y
i
k
j
k
1(8)
∑
⋅− + ≥ − ≤<≤ ∈
=M
W
qdIntyy ijnk(1 ) , 1 ;
F
t
m
ij
tij
y
j
k
i
k
1(9)
∑
⋅⋅ + ≥ + ≤<≤ ∀ ∈ ≠
=MLqdIntxx ijnkg kg2,1;,;
F
t
m
ij
tij
xikj
g
1(10)
∑
⋅⋅ + ≥ + ≤<≤∀ ∈ ≠
=MWqdIntyy ijnkg kg2,1;,;
F
t
m
ij
tij
y
i
k
j
g
1(11)
≥≤<≤
d
Int dInt i j
n
,0,1
ij
xij
y(12)
≥≤<≤
d
Ext i j n0, 1
ij (13)
As mentioned above, constraints (4)–(13) determine the horizontal
and vertical distances among various departments. As for constraints
(4) and (5), the distance between two adjacent workshops is assumed to
be
L
F
. Thus, the translation path from 0
k
to 0
g
is −⋅kgL
|
|F; it can be
linearized as constraints (4) and (5). Constraints (6)–(9) define the
distances in the horizontal and vertical directions among departments
that are assigned to the same workshop. Finally, for the departments
that are assigned to separate workshops, the distances among them can
be measured by constraints (10) and (11).
∑
=∀∈
=
N
Yi1,
k
m
ik
1(14)
≥∀∈
−
SSk m, {2, 3, ..., }
kk1(15)
Fig. 1. Schematic of multi-workshop facility layout problem.
C. Guan, et al. Journal of Manufacturing Systems 53 (2019) 32–48
35
Author's Personal Copy
∑≥∀∈
=
M
YSk,
i
n
ikk
1(16)
∑
⋅≥ ∀∈
=
M
Sn Y k,
k
i
n
ik
1(17)
≤⋅ + ≤<≤ ∀∈
Mq
YY ijnk
1
2(),1 ;
ij
kikj
k
(18)
+≥ + ≤<≤ ∀∈
Mq
YY ijnk1,1 ;
ij
kikj
k(19)
∈∀∈
M
Sk{0, 1},
k(20)
∈∀∈∀∈N
M
Yik{0, 1}, ;
ik
(21)
∈≤≤≠∀∈
Mq
ij ni j k{0, 1}, 1 , ; ;
ij
k(22)
In constraints (14)–(22), the binary variables (
YS,
ik
k
and
q
ij
k
) are
confirmed; these involve the distribution of a given number of de-
partments. Constraints (14) guarantee that each department is only
arranged to one workshop. Constraints (15)–(17) ensure that the binary
values of S
k
are attached to
Y
i
k
. Constraints (15) guarantee that the
workshops are sequentially employed; this prevents having an empty
workshop in the layout. Furthermore, the relationship between S
k
and
Y
i
k
are expressed by constraints (16) and (17). Note that
q
ij
k
, which is
defined to simplify the expressions of various distances, should satisfy
constraints (18) and (19).
⋅⋅ ≤ ∀ ∈ ∀ ∈N
M
lY x i k
1
2,;
iikik
(23)
≤−⋅⋅∀∈∀∈N
M
xL lYi k(1
2), ;
ikFi
ik
(24)
⋅⋅ ≤ ∀∈ ∀ ∈N
M
wY y i k
1
2,;
iik
i
k
(25)
≤−⋅⋅∀∈∀∈N
M
yW wYi k(1
2), ;
i
kFi
ik
(26)
As indicated by constraints (23)–(26), it is necessary for the co-
ordinates of the centroids of departments to satisfy these constraints.
Based on the coordinate system we developed, the ranges of the ab-
scissa and ordinate of department i, which is assigned to workshop k,
are limited in order to guarantee that this department is located inside
the workshop with no overlapping.
+= ≤<≤∀∈M
α
αijnk1, 1 ;
ij
kji
k
(27)
+= ≤<≤∀∈
Mβ
βijnk1, 1 ;
ij
k
ji
k(28)
+−≤∀ ∈∀∈N
Mα
αα ijs k1, , , ;
is
ksj
kij
k(29)
+−≤∀ ∈∀∈N
Mβ
ββ ijs k1, , , ;
is
k
sj
k
ij
k
(30)
∈≤≤≠∀∈
Mα
βijnijk,{0,1},1,;;
ij
k
ij
k(31)
Constraints (27)–(31) in this model are frequently applied to the
UA–FLP for confirming the relative positions of departments and re-
presenting valid sequences. A pair of variables,
α
ij
k
and
β
ij
k
, encode the
positional relationship. As mentioned above, constraints (27) and (28)
ensure that each department appears exactly once when used, and
constraints (29)–(30) are transitivity constraints for the two pairs of
variables [1].
+⋅⋅ ≤ −⋅⋅ + ⋅−
−∀∈≠∀∈N
M
xlYxlYLα
βij ijk
1
2
1
2(2
), , ; ;
ikiikj
kjj
kFij
k
ij
k(32)
+⋅⋅ ≤ −⋅⋅ + ⋅+
−∀∈≠∀∈N
M
ywYywYWα
βij ijk
1
2
1
2(1
), , ; ;
i
kiik
j
kjj
kFij
k
ij
k
(33)
−⋅+
′≥−+⋅+∀∈∀∈N
M
qL L x x l l ij k
(
1) ( )
1
2(),, ;
ij
kFkikj
kij (34)
−⋅+
′≥−+⋅+∀∈∀∈N
M
qW W y y w w ij k
(
1) ( )
1
2(),,;
ij
kFk
i
k
j
kij
(35)
′′
≥∀∈
M
LW k,0,
kk
(36)
In view of variables
α
ij
k
and
β
ij
k
, constraints (32) and (33) are em-
ployed to prevent the overlapping of departments in the horizontal and
vertical directions when these are allocated in the same workshop. As
for constraints (32), if department iis on the left of department j, and
the value of
α
ij
k
and
β
ij
k
together is 1, then the minimum interval be-
tween iand jshould be
+ll
(
)/
2
ij
. Similarly, if department jprecedes i
vertically, then =
α
0
ij
kand =
β1
ij
k; the distance between two centroids
should not be less than
+ww
(
)/
2
ij
, as presented in constraints (33). The
envelope rectangle of kth workshop can be determined by constraints
(34) and (35).
In general, it is evident that the MILP model built for the MWFLP is
also able to represent the UA-FLP with a fixed shape; it is only required
to set the value of parameter mto 1. Furthermore, the model in this
study can also be regarded as a special MFFLP, which assign the shape-
fixed departments to several floors; the heights of adjacent floors are
considered to be the same. Moreover, the handling costs consisting of
internal and external flows are separately taken into account; this is
more realistic and practical.
3.3. Case study
In this section, several trials are performed to evaluate the effec-
tiveness of our MILP model; moreover, certain benchmark instances are
applied and analyzed for the newly defined problem. However, the lack
of test problems that satisfy our case requirements leads us to find
benchmark instances that can be employed for the UA–FLP with some
parameter modifications. Accordingly, seven benchmark instances are
used for optimizing objectives (1)–(3), as listed in Table 1. In the first
column in Table 1, the digits indicate the number of departments in a
particular instance; the literature cited are listed in the second column.
Detailed data including the number of available workshops, dimensions
of each workshop floor, and cost of internal and external flows are also
listed in Table 1. Moreover, the additional matrix of flow frequency is
created for these instances and attached to the appendix.
As mentioned above, all trials are performed with IBM ILOG CPLEX
V12.8 optimization software using the pre-processing values; and exe-
cuted in Windows 10 environment on a desktop PC configured with
Intel(R) Core(TM) i5-7400 CPU E6700 3.2-GHz and 4-GB RAM. Finally,
accurate results are derived and summarized in Table 2. It is note-
worthy that objective (3) in Section 3.2 involves a quadratic term in the
formula; it is also nonconvex and cannot be managed via CPLEX. Ac-
cordingly, Table 2 only lists the optimal solutions to the two other
objectives; the optimal layouts and coordinates of each department are
Table 1
Case study data.
Instance name Reference nmL
F
W
F
cInt cEnt
Das04 Das [63]44203025
Das06 Liu et al. [3]662525410
Das08 Liu et al. [3]88403079
SFLP08 Asl and Wong [16]8 8 7 6 1 5
FLP09 Zhang et al. [64]995449
Das10 Liu et al. [3] 10 5 40 30 6 13
SFLP11 Asl and Wong [16] 11 5 8 8 10 21
C. Guan, et al. Journal of Manufacturing Systems 53 (2019) 32–48
36
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listed in columns 6 and 7, whereas column 8 lists the runtimes of in-
stances. In Table 2, symbols (a) and (b) indicate the MWFLP criterion
with the best material handling cost and the MWFLP criterion with the
minimum number of workshops, respectively.
It can be observed in Table 2 that the departments are divided into
several workshops, and the parameters that are set make it impossible
to allocate all departments to only one workshop. On the other hand,
the runtime rapidly increases with the scale of benchmark instances;
this is one of the characteristics of combinatorial optimization pro-
blems. In the real-life situations, compromising plans are also required
such that decision-makers can choose their preferred layout. Thus, in
order to solve the MWFLP more effectively, the MOPSO is proposed and
described with an embedded two-stage optimization frame.
4. Methodology for multi-workshop facility layout problem
Notice that certain sub-problems are required to handle our pro-
posed problem. One key point is that it is necessary to determine the
number of workshops available for layout before the departments are
assigned to them; this renders the problem difficult to handle through
exact methods because of the combinatorial explosion of allocations.
Hence, the meta-heuristic MOPSO is proposed.
In this paper, a novel discrete framework for the evolutionary
computation of each particle is presented; the emulating mechanism of
each particle to the global optimal swarm (called Gbest) and to the best
position in the past of every particle (called Pbest) is redefined for the
multi-workshop facility layout problem. Additionally, a two-stage ap-
proach is incorporated to search for feasible solutions; the Pareto
principle is applied to handle the multi-objective nature of the problem.
4.1. Individual particle encoding
According to the characteristics of the MWFLP, the simple se-
quences of department indices cannot be employed to represent intact
and feasible layouts of the problem that, do not specify the information
concerning department arrangement and exact locations of depart-
ments. On the other hand, different from the established bay structure
developed in [65] and the flexible bay structure [66] that are frequently
deployed to define the solutions to UA–FLP/MFFLP only when the aisles
or floors are predefined and the shape unfixed, the constructive ap-
proach for our MWFLP should indicate all possible combinations re-
gardless of the number of workshops employed. Meanwhile, reasonable
positions for the departments that satisfy the formulations mentioned in
Section 3.2 are supposed to be presented in encoding.
Therefore, as depicted in Fig. 2, the solutions to the problem can be
encoded by vectors with four segments. The first nelements represent
the identifier of departments to be laid out; these are used to obtain the
placing sequence in Segment 1. Moreover, Segment 2 is used to obtain
the number of departments assigned to each workshop. For example, if
=Γ
4
k, then the departments from
∑
=
−Γ
t
kt
1
1+1 to
∑
=Γ
t
kt
1in Segment 1
are to be allocated to the kth workshop, and the number of operational
workshops in this layout will be
∑
=S
k
m
k
1. As for Segment 3, based on the
allocation plans of Segment 2, the relative positions for every pair of
departments are determined and shown in this segment. Finally, Seg-
ment 4 is used to obtain the two-dimensional coordinates of each de-
partment that are subject to the constraints in this study.
4.2. Placement strategy
For each individual particle, decoding the placing sequence can be
accomplished by using the placement strategy described in this section.
While attempting to place the departments into the workshops, the
number of workshops used and the relative positions of pairs of de-
partments assigned to the same workshop are noted; thus, Segments 2
and 3 can be acquired according to this strategy.
The placement strategy follows a sequential process, which arranges
Table 2
Computational results of seven benchmark instances obtained by CPLEX.
Instance name MWFLP Objective function values Solutions from proposed MILP model
FMH
C
FNW
S
F
UR
Optimal assignment for workshop layout Location coordinates {(∑=x
k
mik
1,
∑
=
y
k
m
i
k
1
)},
=−
i
nn1, 2, ..., 1,
Running time (s)
Das04 (a) 9130.00 2 0.5069 {1, 2, 4}; {3} {(10, 5), (10, 12.5), (10, 22.5)}; {(5, 5)} 0.11
(b) 16 164.00 2 0.5973 {2, 4}; {1, 3} {(15.5, 27.5), (10, 7.5)}; {(9, 5), (5, 25)} 0.05
Das06 (a) 42 944.00 2 0.3720 {1, 3, 4, 5, 6}; {2} {(5, 4), (16, 13), (16, 4), (5, 18), (5, 11)}; {(10, 7.5)} 1.97
(b) 80 782.00 2 0.5031 {4}; {1, 2, 3, 5, 6} {(19, 21)}; {(5, 4), (15, 9.5), (16, 19), (14, 4), (20.5, 11)} 0.17
Das08 (a) 110 893.50 2 0.3322 {1, 2}; {3, 4, 5, 6, 7, 8} {(7, 5), (7, 13.5)}; {(5, 6.5), (28, 7.5), (27, 21.5), (14, 7.5), (13.5, 14.5), (10.5, 23)} 109.66
(b) 241 989.00 2 0.3830 {2, 3, 5, 6, 7, 8}; {1, 4} {(5, 11.5), (35, 16), (19, 23.5), (12, 4), (35.5, 3), (23.5, 5.5)}; {(7, 20), (10, 7.5)} 0.89
SFLP08 (a) 6018.00 4 0.4842 {7}; {1, 3, 4, 6}; {2, 8}; {5} {(2, 2)}; {(1, 4.5), (3, 5), (1.5, 1.5), (5, 2),}; {(5, 2.5), (1.5, 2)}; {(1, 2)} 87.11
(b) 7415.00 3 0.5602 {1, 3, 4, 6}; {5, 7}; {2, 8} {(6, 4.5), (1, 1), (5.5, 1.5), (2, 4)}; {(1, 4), (4, 4)}; {(5, 2.5), (1.5, 2)} 0.67
FLP09 (a) 11 863.80 5 0.5742 {8}; {4, 9}; {1, 2, 3, 5}; {6}; {7} {(2.5, 1.5)}; {(1, 0.9), (3.5, 2)}; {(3.95, 2.4), (1.5, 2.5), (1, 0.5), (3.25, 0.75)}; {(1.5, 1.5)}; {(1.5, 1.4)} 2760.42
(b) 20 908.00 5 0.6809 {1, 2, 3, 5}; {4, 6}; {9}; {8}; {7} {(0.95, 2.1), (3.5, 3), (1, 3.5), (3.75, 1.25)}; {(1, 3.1), (3.5, 2.5)}; {(3.5, 2)}; {(2.5, 2.5)}; {(1.5, 1.4)} 12.47
Das10 (a) 289 462.00 3 0.3526 {2}; {1, 3, 4, 5, 6, 8, 9}; {7, 10} {(6, 4)}; {(7.5, 25), (27.5, 6), (6.5, 15.5), (15.5, 9), (27.5, 21), (16, 3.5), (5.5, 5.5)}; {(10, 21), (9.5, 6.5)} 706.72
(b) 518 600.00 2 0.4586 {2, 3, 5, 7, 10}; {1, 4, 6, 8, 9} {(25, 4), (32.5, 24), (23.5, 11), (15, 22), (9.5, 7.5)}; {(7.5, 19), (4.5, 9.5), (27.5, 20), (4, 2.5), (14.5, 5.5)} 0.53
SFLP11 (a) 174 132.00 3 0.3792 {7, 11}; {1, 2, 3, 4, 5, 6, 8, 9}; {10} {(2, 6.5), (2.5, 2.5)}; {(2, 6), (3.5, 1), (3.5, 3), (6.4, 2.5), (1.5, 1), (4.7, 2.5), (1.7, 3), (6, 6.4)}; {(2, 3.5)} 2996.55
(b) 288 906.00 3 0.5039 {1, 2, 3, 4, 5, 6, 8}; {7, 9, 11}; {10} {(4.6, 4), (0.5, 1), (7.5, 7), (1, 4.5), (5.1, 1), (7.3, 3.5), (5.7, 7)}; {(6, 6.5), (2, 6.4), (5.5, 2.5)}; {(2, 3.5)} 49.00
C. Guan, et al. Journal of Manufacturing Systems 53 (2019) 32–48
37
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Fig. 2. Individual particle encoding.
Fig. 3. Placement strategy example on decoding individual particles.
C. Guan, et al. Journal of Manufacturing Systems 53 (2019) 32–48
38
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a single department at every step. The selected department is obtained
sequentially according to the placing sequence in Segment 1. In the
initial stage, the first workshop is employed, and the entire floor space
is available; thereafter, the first single department is placed in the lower
left corner of the workshop. To generate feasible solutions to the pro-
blem, the remaining empty spaces are updated after every placement of
a single department. Accordingly, it is only necessary to find the
available empty space where the department being placed fits; the next
department is placed immediately along the X-axis. Moreover, once
there is no adequate empty space available for the following depart-
ment, a new workshop is to be used.
An example of the special placing process is shown in Fig. 3, where
benchmark instance SFLP11 mentioned in Section 3.3 is used; the
placing sequence is Sol={6, 3, 7, 5, 1, 10, 9, 8, 2, 11, 4}. As depicted in
Fig. 3a, initially, the first department, 6, is placed on the lower left
corner of the workshop. Therefore, the next departments, 3 and 7, are
placed sequentially close to the previous one, and their relative posi-
tions to each other are determined concurrently. Department 5 can be
placed above departments 3 and 7 or on top of this aforementioned
location (locations 1 and 2 in Fig. 3d) because there is no available
empty space for it if it is placed behind department 7. As for the pla-
cement of department 1, another workshop is required for it if de-
partment 5 is placed in location 2; this may make three objectives in-
ferior because an extra workshop is used, and additional distances
among the departments have to be travelled. Thus, department 5 is
placed in location 1, and the first five departments are arranged in one
workshop, as shown in Fig. 4e. Department 10 is placed in the corner of
another workshop, and the rest are placed following the arrangement
mentioned above.
The final result of the aforementioned example is illustrated in
Fig. 3f. In this example, 11 departments are distributed to three
workshops, and the individual particles can be represented as {6, 3, 7,
5, 1, 10, 9, 8, 2, 11, 4, [5,4,2],
=
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
α
Dep 63751
6 01101
3 00100
7 00000
5 10001
1 01100
,
ij
i
1=
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
β
Dep 63751
6 01111
3 00101
7 00001
5 00001
1 00000
;
ij
i
1
=
α
ij
2
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
Dep 10 9 8 2
10 0 111
9 0 000
8 0 101
2 0 100
,
i=⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
β
Dep 10 9 8 2
10 0 111
9 0 011
8 0 001
2 0 000
;
ij
i
2=
α
ij
3⎡
⎣
⎢
⎢⎤
⎦
⎥
⎥
Dep 11 4
11 0 1
400
,
i
=⎡
⎣
⎢
⎢⎤
⎦
⎥
⎥
β
Dep 11 4
11 0 1
400
,
ij
i
3
∑
∑∈Nxyi,;
gig
gi
g
33
} according to the proposed
placement strategy. The coordinate values in the last segment of en-
coding are determined using the approach described in the next section.
4.3. Two-stage approach
The stage approach concept is based on the attractor–repeller
technique developed in the study of Anjos and Vannelli [67] for very-
large-scale integration floorplanning, which was used to convert the
nonconvex optimization problem into a convex version step by step. In
this paper, an optimization-based approach is presented to efficiently
find the unique optimal solution for each individual particle by it
combining two stages. The idea of the two-stage approach is that, for
every individual particle, the search for the relative locations of de-
partments is implemented in the first stage; thereafter, their exact co-
ordinates are determined in the second stage. In particular, decision
variables
S
k
,
Y
i
k
,
α
ij
k
, and
β
ij
k
mentioned in the mathematical model can
be obtained using the placement strategy introduced in Section 4.2.
Consequently, the approximate positions of departments can be en-
forced, and the formulation pertaining to the optimal coordinates can
be simplified to a linear programming (LP) model.
The resulting second-stage LP model is given by
objective functions
(1)–(2)
subject to:
(4)–(13), (23)–(26), and (32)–(36).
In general, the placing sequences decoded in the first stage are
generated by the operations of MOPSO; thereafter, the above LP model
is solved in CPLEX software (except for objective 3 because it is non-
convex). Hence, the optimal objective values of individual particles are
acquired and subsequently compared.
4.4. Local and global searches
Note that the original particle swarm optimization is initially used
to solve continuous optimization problems, which is supposed to be
discretized for the MWFLP. In this section, the velocity and position of
particles are redefined; their cognitive activities and social commu-
nications are also discretized according to the formulas in Liu et al. [3].
In our work, the process with which particles broadcast their
neighborhood space and learn about the Gbest individual is abstracted
as the local search of MOPSO; it is employed to search the solution
Fig. 4. Local and global search architecture.
C. Guan, et al. Journal of Manufacturing Systems 53 (2019) 32–48
39
Author's Personal Copy
space adequately and extensively. Moreover, the global search com-
bines the parents of the current individual particle and elite individuals
in the non-dominated solution set obtained from the local research. The
architecture of the local and global procedures of MOPSO is illustrated
in Fig. 4. The remainder of this section describes the details of the
aforementioned procedures.
4.4.1. Neighborhood search
In our MOPSO, suppose that the position of the ith individual par-
ticle,
X
it=(
Dep Dep Dep, , ...,
i
t
i
t
n
i
t
12
) represents a configuration of the
MWFLP in the tth interaction. The neighborhood search space around
the current individual particle is regarded here as a series of countable
solutions with considerably similar sequences. Thus, the 2-opt proce-
dure [6] is suggested to generate some offspring particles from the
current individual. A valid 2-opt move consists of swapping the posi-
tions of two elements in the current sequence; this propels the search
away from the current individual particle; however, it should not be
extremely far. Moreover, it can be useful to execute several samplings in
the neighborhood search space that strive to traverse all feasible solu-
tions and find one that is preferable.
The specific executive process of the neighborhood search in the
MOPSO is as follows.
2-opt_ NeighborSearch (Current individual, Xit; number of departments n)
Set
P
neighbo
r
is employed to record offspring particles;
Set
P
′
neighbo
r
represents the non-dominated solutions;
SetFneighborindicates the objective function values of
P
neighbo
r
;
Set
F′
neighbo
r
expresses the objective function values of
P
′
neighbo
r
;
repeat
for i=1 to
≤i
⌊n/2⌋
a,b:=random index between 1 and n;
while a=bdo
a,b:=random index between 1 and n;
end while
swap positions of elements aand bin Xit, and an offspring is generated;
calculate the objective function values of the offspring by applying the placement
strategy and two-stage approach;
record the novel solution in set
P
neighbo
r
, and add the objective function values to set
Fneighbor;
end for
P
neighbo
r
is filtered to obtain the non-dominated solution set
P
′
neighbo
r
;
F′
neighbo
r
:=f(
P
′
neighbo
r
, placement strategy, two-stage approach);
return (
P
′
neighbo
r
and
F′
neighbo
r
)
end
4.4.2. Social communications with Gbest
Apart from exploring the neighborhood solution space, it is neces-
sary for all particles to learn from the global optimal swarm to accel-
erate the convergence process in the local search. Accordingly, the re-
structuring of the formula that describes the social communications
with Gbest is presented in this section; it is expressed as Eq. (37). In this
equation,
X
Gbest
t
denotes the best position that global particles in the
population have obtained thus far; it is replaced by the elite individual
selected at random from the non-dominated solution set. The specific
discrete processes of this equation are introduced in the following
section.
=⊕ ⊗X′XXXΘ(rand(0, 1) ( ))
itititGbest
t(37)
The special definition and discrete arithmetic of the procedure are
described in detail as follows.
The velocity, Ii, in the MOPSO is defined as the pairs of distinct
departments generated from solutions
X
itand
X
Gbest
t
according to the
subtraction term XXΘ
(
)
itGbest
tin Eq. (37). Suppose that
X
Gbest
t
=
(Dep Gbest
t
1,Dep Gbest
t
2,…,
Dep
nGbest
t
), which is a valid swapping pair, is
composed of different elements in the same dimension. If the ath ele-
ment,
Dep
ai
t, in sequence
X
itis distinct from element Dep
aGbest
tin
sequence
X
Gbest
t
, then the ath swapping pair, IX X(, )
a
iitGbest
t, will be
Dep Dep
(
,)
ai
t
aGbest
t; otherwise, =
∅
IX X(, )
a
iitGbest
t. After executing the ar-
ithmetic subtraction of particle i, the multiplication term (rand(0,1)
⊗
Ii) is used to choose the swapping pairs randomly ⌊rand(0,1)
×
|I|
i
⌋;
rand(0,1) is a uniform random number between 0 and 1. Moreover, as
for the arithmetic addition (
⊕
) operation, the novel placing sequences
can be acquired in each addition operation on particle iwith every
swapping pairs selected by arithmetic multiplication. Here, the swap-
ping pair Dep Dep
(
,)
ai
t
aGbest
tis considered as an example. The specific
multiplication process exchanges the positions of
Dep
ai
tand Dep
aGbest
t
departments in the current placing sequence. It is worthwhile to explain
that swapping pairs Dep Dep
(
,)
ai
t
aGbest
tand Dep Dep
(
,)
aGbest
t
ai
tlead to the
same new particle; hence, duplicate checking work is essential to avoid
replicating calculations after the subtraction operation. Finally, the
non-dominated solution set,
P
′
Gbest
, is obtained by filtering the new
solutions using the Pareto principle.
In general, the current particle,
X
it, is guided to converge with non-
dominated solutions according to Eq. (37); this is used to improve the
convergence speed and ameliorate the individual’s quality.
4.4.3. Global search and update of current particle
After each particle searches its neighborhood space and learns about
the elite individual, preferable solutions that are used to update the
current position in the global search probably exist. Accordingly, off-
spring particles that dominate other particles from solution set
∪
P
′P′
Gbest neighbo
r
are first found. Thereafter, the partial mapped cross-
over (PMX) [68] operation is applied to combine particle iand each
non-dominated individual particle,
X
local
t, from
P
′
local
; the concrete
process of the PMX procedure is shown in Fig. 5. A simple case is also
illustrated in this figure: the continuous random substring is determined
firstly in the sequence, which is called mapping section. Here, the
substring composed of elements 3–5 in the two-parent solutions is se-
lected; thereafter, the mapping section is exchanged to generate two
new sequences. However, duplicated elements in the new sequences
may probably exist; thus, the conflict detection operation has to be
executed to guarantee the feasibility of solutions. For example, for the
elements outside the two mapping sections, the second reiterated ele-
ment, “5”, of sequence in the third row can be replaced by element “1”
according to the mapping list; this is similarly the case with elements
“4”and element “6”. Consequently, a feasible offspring solution, X′it
,2,is
produced. As mentioned above, the profitable mapping sections from
elite individuals are available for the PMX to use; it can evolve the
current particle to approach the Pareto front.
Note that the similarity between offspring solutions and elite in-
dividuals is dependent upon the mapping section length. If the offspring
solutions consist of larger fragments of elite individuals, it is probable
that better features will be inherited. Thus, an adaptable method that
can adjust the length of the mapping section is proposed for the PMX
operation. The size of the mapping section is calculated by ⌊×
pn
pmx ⌋,
where
p
pm
x
indicates the probability obtained by the equation below. In
Eq. (38),
p
pmx
max and
p
pmx
min express the upper and lower bounds of
p
pm
x
,
respectively, and
P
Paret
o
represents the global non-dominated solution
set; Zis the number of objectives considered in Section 3.2. Moreover,
F
b
i
indicates the bth objective function value of the ith particle in
P
′
local
;
Fb
max and Fb
max represent the maximum and minimum of the bth ob-
jective function value in
P
Paret
o
, respectively. The term
⋅∑=
P′d
1
/| | P′
local ii
1
||
2
local in Eq. (38) describes the affinity between
P
′
local
and
P
Paret
o
; the smaller its value, the closer
P
′
local
is to the global non-
inferior solution set.
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∑
∑
=+ −⋅
⎛
⎝
⎜−⋅ ⎞
⎠
⎟
=⎧
⎨
⎩
−
−⎫
⎬
⎭
=
∈=
P′
pp p p d
dFF
FF
()1
1
||
where
min ( )
P′
P
pmx pmx pmx pmx local i
i
ijb
Z
b
ib
j
bb
min max min
1
||
2
{1,2, ..., | |} 1
max min
2
local
Pareto (38)
Accordingly, a series of offspring solutions are generated by
adopting the PMX operation; some solutions that dominate particle iare
saved and used to update themselves.
4.5. Generation of new population and update of external archive
After each particle complete its local and global searches, there
exists probably some preferable solutions that dominate itself.
Thereafter, these preferable solutions are assembled with the primary
particle population to update
P
Paret
o
and produce the next generation
population. In our MOPSO, noP particles are supposed to compose a
population. To guarantee the population quality with each iteration, an
individual in the Pareto front is first selected at random. If the popu-
lation is not complete after all the Pareto optimal solutions have been
selected, then the mutation procedure on randomly selected optimal
particles is exploited to increase population diversity. The aforemen-
tioned procedure is applied with the selection of two elements from one
solution and then inserting them into any other positions in the re-
maining sequence.
After each iteration, an external archive is employed to store the
global optimal Pareto solutions, i.e., solutions that can achieve ad-
vantageous objectives are added to the external archive. Because the
size of the external archive is limited, the crowding distance measure
[58] is applied to control the number of solutions in the archive to be
less than Na.
4.6. Description of MOPSO
The entire process of the MOPSO algorithm is described below. The
parameter
maxIter
mentioned as below represents the maximum
number of algorithm iterations.
Algorithm: multi-objective particle swarm optimization
Input problem information.
Set algorithm parameters.
Randomly generate noP feasible particles, and calculate the objective function
values of all particles using the placement strategy and two-stage approach. Find
the global optimal Pareto solutions,
P
Paret
o
, and let =
t1
.
Let
=
i1
.
Execute neighborhood sampling ⌊n/2⌋times for the
i
t
h
particle, Xit, and gain new
configurations,
P
′
neighbo
r
.
Choose an optimal individual,
X
Gbes
t
t
, from the current
P
Paret
o
; thereafter compute
the velocity IX X(,
)
iitGbest
tand novel positions,
P
′Gbes
t
, according to Eq. (37).
Filter ∪
P
′P′
neighbor Gbest based on the Pareto principle to acquire non-dominated
individual particles,
P
′loca
l
, obtained by the local search; the PMX operation is
applied to Xitand each sequence in
P
′loca
l
.
From the offspring solutions obtained in Step 7, select and record solutions that
dominate particleXit.
Let =+
i
i
1
.If
≤
i
no
P
, then return to Step 5; otherwise, proceed to the next step.
Update the external archive,
P
Paret
o
, of all particles.
Generate the new population for the next iteration.
Execute
=+
t
t1
, and return to Step 4 until >
t
maxIte
r
.
Terminate the MOPSO, and export the final external archive
P
Paret
o
and
P
f
(
Pareto
placement strategy, two-stage approach).
5. Numerical experiments and analysis
To assess the performance of our proposed MOPSO, a series of
computational experiments are executed on a computer with the same
configuration as that mentioned in Section 3.3. The MOPSO algorithm
is coded in MATLAB 2016b, and the linear programs in Section 4.3 are
achieved with IBM ILOG CPLEX V12.8 optimization software. In Sec-
tion 5.1, two different groups of instances are investigated, and com-
parisons with other algorithms are implemented. For each instance, the
MOPSO is run 30 times independently; which is similar with other
studies reported in literature. However, detailed sample data of re-
peated calculations obtained by other existing methods are unable to be
found in their papers, which makes it impossible to use the paired-
sample ttest or other statistical methods. On the other hand, there are
some processed statistical results reported in their works such as the
best, worst and average objective function values/running time, which
were all compared in our work.
Subsequently, several benchmark instances are applied to solve the
MWFLP using the MOPSO; the applicability of the proposed method
and the Pareto optimal solutions for each instance are demonstrated in
Section 5.2.
Fig. 5. Partial Mapped Crossover (PMX) operation.
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5.1. Comparison experiments for UA–FLP
In order to evaluate the behavior of our method objectively, the
shape-fixed UA–FLP instances are first solved using MOPSO and
compared with the results obtained by other algorithms. These UA–FLP
problem instances are selected because there is no other problem in-
stance in literature that is available for our MWFLP.
Table 3
Comparison of results between proposed method and modified PSO for UA–FLP.
Problem instance Method F
MHC
(min) Running time(s)
Best Worst Average Time for the best result Time for the worst result
IM08 Modified PSO 193.7488 220.3437 208.74165 220.69 205.67
Proposed MOPSO 191.5000 191.5000 191.5000 71.62 78.22
IM11 Modified PSO 1286.1069 1371.3264 1335.6385 888.32 919.74
Proposed MOPSO 1259.2000 1293.3000 1274.0600 134.29 125.91
MHI20 Modified PSO 1206.6489 1315.2316 1264.2131 2352.12 2250.87
Proposed MOPSO 1205.5000 1250.5000 1234.0000 747.11 800.95
Fig. 6. Best solutions obtained by the proposed method for datasets IM08, IM11, and MHI20.
C. Guan, et al. Journal of Manufacturing Systems 53 (2019) 32–48
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5.1.1. Computational results for problem instances IM08, IM11, and
MHI20
First, three problem instances with different scales are solved using
the MOPSO. These three instances, namely IM08, IM11, and MHI20, are
from Asl and Wong [16]. The computational results of the MOPSO and
modified PSO in literature are summarized in Table 3. In this table, the
parameters for our method are set as follows: noP = 30, maxIter = 100,
Na = 15,
p
pmx
max =0.6, and
p
pmx
min =0.5. Because only a single objective,
FMH
C
, is considered in Asl and Wong [16], the best results of FMH
C
obtained by MOPSO will be selected from the external archive and used
for comparison, as listed in Table 3; the numbers written in bold denote
the best values of these results.
As summarized in Table 3, the overall best objective values obtained
by the MOPSO are uniformly better than those yielded by the modified
PSO; thus, the same applied to the worst and average objective values.
This is particularly the case with instance IM08; the optimal objective
of 191.50 is achieved at every run. Furthermore, in terms of running
time, the times spent for obtaining the best or worst results using our
proposed method are considerably less than those of the modified PSO.
This is because, for Modified PSO, the optimization for the exact co-
ordinates of departments is integrated in the intelligent algorithms in-
stead of solving the exact model via optimizers. For verification con-
venience, the optimal results of each instance are illustrated in Fig. 6.
Additionally, the comparison of MOPSO with other methods is listed
in Table 4. It is evident that the MOPSO has improved the best objective
function, FMH
C
, compared with other methods; the improvement over
other methods is measured by the relative deviation (Dev) criteria,
which are listed in the last row of Table 4. The symbol “+”indicates
that the average results obtained by the MOPSO are superior to those of
the other four methods. Accordingly, the advantages of the proposed
MOPSO for solving the instances above are verified.
5.1.2. Comparison of methods with five instances (from n = 15 to n = 50)
The other five available benchmark instances for the UA–FLP are
also employed to check the effectiveness of our proposed MOPSO fur-
ther. The detailed data of these instances are found in Scholz et al. [13]
Table 4
Comparison of results between our proposed method and other methods for MHI20 benchmark.
Problem Evaluation criteria Our method-
MOPSO
Modified PSO [16] TOPOPT method [69] FLOAT method [70] HOT method [71]
MHI20 Best 1205.5000 1206.6489 1320.72 1264.94 1225.40
Average 1234.0000 1264.2131 1395.64 1333.81 1287.29
Dev (%) —+2.3899 +11.5818 +7.4831 +4.1397
Table 5
Comparison between previous approaches and our MOPSO for instances
=
n15, 20, and 30.
Problem Objective function value F
MHC
/Running time (s)
MOPSO HGA [8]GA[7] HGA [7]TS[13]FA[20]
n=15 2546.61/6.11 6813/81.21 9120/28.2 6941.4/34.1 6615.81/17 5956.1/—
n=20 4946.70/16.98 13 190/167.59 21 885/40.6 14 696/50.7 13 198.40/50 11 505/—
n=30 11 538.88/86.66 35 358/284.01 50 492/67.7 32 386/83.9 33 721.20/95.4 29 115/—
Table 6
Best cost solutions for instances
=
n15, 20, and 30.
Problem Optimal layout Center-coordinates of departments {(location coordinates along X-axis, location coordinates along Y-
axis)}
n= 15 {3, 4, 12, 9, 11, 13, 7, 8, 5, 14, 15, 2, 1, 10, 6} {(17.15, 11.00), (12.85, 9.85), (3.38, 4.23), (6.70, 2.60), (17.50, 6.10), (7.45, 14.10), (6.80, 5.60),
(12.85, 6.00), (13.20, 2.00), (3.26, 12.10), (17.70, 2.00), (9.20, 2.00), (9.20, 5.10), (8.20, 9.10), (4.70,
9.10)}
n= 20 {11, 10, 3, 9, 8, 16, 4, 5,
, 17, 20, 2, 1, 15, 7, 6, 18, 12, 14, 13}
{(2.15, 12.15), (12.25, 9.60), (12.63, 3.88), (7.50, 6.85), (17.45, 8.35), (16.15, 12.95), (7.60, 9.85),
(2.58, 7.35), (17.45, 4.25), (7.86, 2.85), (3.43, 3.35), (8.70, 12.45), (12.90, 12.95), (8.70, 16.55),
(5.50, 10.05), (5.50, 6.20), (5.50, 8.05), (5.80, 12.55), (9.40, 6.50), (9.30, 9.30)}
n= 30 {3, 26, 24, 18, 14, 10, 16, 12, 6, 17, 27, 23, 20, 7, 30, 25, 1,
4, 11, 22, 19, 13, 15, 28, 29, 8, 21, 2, 9, 5}
{(3.00, 16.00), (7.85, 19.95), (2.33, 9.33), (13.00, 15.00), (8.65, 23.90), (20.64, 13.00), (9.10, 12.40),
(11.93, 19.00), (20.07, 20.50), (7.09, 9.40), (16.00, 16.50), (17.49, 11.10), (19.50, 17.00), (21.39,
8.00), (11.00, 16.00), (15.21, 10.55), (11.00, 12.00), (15.83, 8.00), (9.10, 14.65), (15.13, 12.95),
(15.71, 20.65), (6.55, 16.00), (6.70, 12.90), (11.93, 8.40), (11.00, 14.00), (8.65, 6.55), (13.00, 12.00),
(22.50, 17.00), (8.98, 17.00), (17.32, 13.60)}
Table 7
Results of n= 40 and n= 50 facility layout problems using the firefly algorithm.
Method Trial noP maxIter n =40 n=50
F
MHC
(min) Running time F
MHC
(min) Running time
Best Average Average Best Average Average
FA[20] 1 100 500 488 611 499 546.3 198.0 2 056 984.0 2 084 621.9 300.0
2 100 1000 486 505 498 544.2 396.0 2 036 950.0 2 076 700.2 648.0
3 200 500 485 736 497 569.8 498.0 2 033 821.0 2 074 298.3 738.0
4 200 1000 485 150 495 940.0 798.0 1 995 866.0 2 046 281.0 1494.0
MOPSO 1 5 30 376 274 383 609.3 551.2 1 598 395.5 1 661 332.7 1406.7
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Fig. 7. Best layout obtained by our method for instances n= 40 and 50.
C. Guan, et al. Journal of Manufacturing Systems 53 (2019) 32–48
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and Ingole and Singh [20]; all instances have been tested for optimizing
objective FMH
C
.
Table 5 summarizes the best cost results obtained by the MOPSO
and five other approaches from literature for typical instances n= 15,
20, and 30. The parameter combination for our MOPSO is set as
noP =5,maxIter = 10, Na = 15,
p
pmx
max =0.6, and
p
pmx
min =0.2. As can be
observed in Table 5, the objective function values of MOPSO are sig-
nificantly better than those of the HGA (hybrid genetic algorithms), GA
(genetic algorithms), TS (tabu search), and FA (firefly algorithm); this
could be attributed to the different encoding and decoding methods
used for the shape-fixed UA–FLP. Given that the departments/facilities
are arranged in the considered bay from the top left to the horizontal
direction to avoid vertical overlapping among them in any other ap-
proaches, the remaining empty spaces among the bays are ignored for
placement; this leads to redundant distances among the departments.
On the other hand, the running times of the MOPSO (except for the
instance n= 30) are less than those of the other methods in this study.
As for instance n= 30, the GA method developed in Lee and Lee [7]
only spent 67.7 s to solve the problem; however, the result is con-
siderably inferior compared to that of MOPSO as well as those of the
other approaches. The best cost solutions obtained by the MOPSO for
these three instances are listed in Table 6.
The two larger extra benchmark instances, n= 40 and 50 (provided
by Ingole and Singh [20]), are used here to test the performance of the
MOPSO. The statistics of the computational results are summarized in
Table 7. The table lists the best and average costs found by the MOPSO;
the average value of running times and four other parameter combi-
nation trials of the FA are also listed. In this table, it is evident that the
results obtained by the MOPSO are better than those obtained by the
FA; this suggests that the performance of the proposed algorithm is
preferable to other related algorithms. It should also be noted that the
values of parameters noP and maxIter decided for the MOPSO are
considerably less than those for the FA in spite of the longer running
times of the MOPSO. This may be explained by the multiple complex
operations embedded in the MOPSO that would consume a consider-
able amount of time with each iteration for every particle. Finally, the
best solutions generated by the MOPSO for instances n= 40 and 50 are
shown in Fig. 7.
5.2. Solutions to multi-workplace facility layout problem
As mentioned above, the performance of the MOPSO is demon-
strated by several different instances that are available in literature. The
proposed algorithm is significantly superior to the previous methods
even though it is forced to be compared with single objective optimi-
zation algorithms. Thus, the MOPSO is implemented to solve the
MWFLP here; for a few of the problem instances, it is attempted to
achieve some excellent Pareto solutions. As presented in this section,
the distances outside the workshops among departments which are
assigned to different workshops are presumed to be the same, i.e.,
=
d
Ext L
ij
F
when
∑
=
q
k
m
ij
k
1
is not equal to one; here, a trivial simplifica-
tion of the multi-workplace facility layout problem is made for the
MOPSO.
The three problem instances that are adopted for the MWFLP are
Das6, SFLP11, and SFLP20; the detailed data of instances Das6 and
SFLP11 are described in Section 3.3, whereas instance SFLP20 with 20
departments is obtained from Mir and Imam [71]. For problem instance
SFLP20, the default dimensions of each workshop floor are
×
1
010
, and
the unit moving costs inside and outside the workshops are set to 8 and
14, respectively; the additional dataset of frequency of material flows is
provided in the appendix.
To achieve balance between solving quality and efficiency, nu-
merous experimental tests are conducted; the preferable algorithm
parameter combinations are listed in Table 8. For the instances con-
sidered, the initial attempt is to solve their objective, F
MHC
, via CPLEX
optimization software; however, this attempt failed to find optimal
results except for instance Das06. Thereafter, the final suboptimal so-
lutions, running time, and MIP gap tolerances are obtained, as listed in
columns 2–4ofTable 9. Moreover, the best Pareto optimal solution set
obtained by the MOPSO is extracted from the results of ten independent
operations; these are also listed in Table 9.
As listed in Table 9, it would require an enormous computational
cost to acquire the exact optimal solutions through accurate methods.
In the table, instance Das06 is easily solved in 1.78 s, whereas SFLP11
and SFLP20 are barely calculated even after a long time. Thus, as the
size of problem instances increases, it also becomes increasingly diffi-
cult to achieve effective solutions using accurate methods. For com-
parison convenience, the obtained cost results and relative tolerances of
instances SFLP11 and SFLP20 are listed in the table after a long running
time. On the other hand, as can be observed from the computational
statistics of the MOPSO, a cluster of better solutions is obtained within
reasonable times. As for instance Das06, the two best Pareto solutions
could be gained at every turn in 10 repeated runs; the best cost ob-
jective value of 42944 obtained by the MOPSO is equal to the exact
result but achieved within a shorter time. For instances SFLP11 and
SFLP20, 15 and 6 Pareto optimal solutions are achieved, respectively.
The best objective function values of F
MHC
are all better than those
Table 8
Parameter values for MOPSO method.
Instance name noP maxIter
p
pmx
max
p
pmx
min
Na
Das06 10 10 0.8 0.3 15
SFLP11 30 100 0.8 0.3 15
SFLP20 30 200 0.8 0.3 15
Table 9
Computational results of instances Das 6, SFLP11, and SFLP20 obtained by MOPSO.
Instance name Solved by CPLEX software Pareto optimal solution set obtained by MOPSO
F
MHC
Running time (s) MIP Gap tolerances (%) Objective function values: [F
MHC
,F
NWS
,F
UR
] Running time (s)
Das06 42 944 1.78 0.00 [42 944, 2, 0.372] [71 956, 2, 0.362] 0.62
SFLP11 185 234 854.45 99.98 [171 276, 3, 0.379] [243 064, 4, 0.298]
[224 320, 5, 0.270] [222 584, 5, 0.286]
[219 430, 3, 0.356] [188 060, 4, 0.357]
[227 820, 4, 0.307] [239 978, 4, 0.299]
[207 544, 4, 0.319] [201 074, 4, 0.331]
[212 246, 3, 0.358] [207 024, 3, 0.371]
[194 252, 4, 0.339] [204 942, 4, 0.329]
[189 896, 4, 0.348]
306.02
SFLP20 495 140 8661.61 100 [123 052, 2, 0.224] [125 660, 2, 0.215]
[129 848, 2, 0.204] [166 316, 2, 0.185]
[299 404, 2, 0.184] [302 448, 2, 0.180]
1177.25
C. Guan, et al. Journal of Manufacturing Systems 53 (2019) 32–48
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obtained by CPLEX; moreover, the running times of the MOPSO are
considerably shorter. In summary, our proposed MOPSO is convin-
cingly a more effective approach to solve a complex problem. Mean-
while, the various solutions with a different emphasis can be employed
by organization leaders to aid in decision-making.
In order to demonstrate the performance of our MOPSO more
clearly, Fig. 8 illustrates the specifics of the particle convergence pro-
cess in the MOPSO method for instance SFLP20. In the figure, genera-
tions 1, 20, 50, 100, and 200 in the convergence process are selected
and described; the scatter points in the figure represent individual
particles, and the Pareto front of each generation is indicated by the
curves created by Pareto optimal solutions (expressed by solid points).
Only two objectives, F
MHC
and F
UR
, are shown in Fig. 8 because the
objective values of F
NWS
for most Pareto optimal solutions is 2. As can
be observed in the figure, the particle population is concentrated near
the contemporary Pareto front except for the initial generation; this
illustrates that the MOPSO method has an excellent convergence. Fur-
thermore, there are several Pareto optimal solutions distributed uni-
formly on the front; this demonstrates the good dispersion capability of
the MOPSO method.
6. Conclusions and future research directions
This study presents a multi-workshop facility layout problem, which
involves the physical organization of a group of departments inside
several workshops. This problem has two aspects: arrangement of de-
partments within the workshops and exact locations of departments
without overlapping. In this study, a mixed integer linear programming
model is developed for the problem relative to three objectives: the
minimization of overall material handling costs including internal and
external flow costs, minimization of the number of workshops, and
maximization of the utilization ratio of the workshop floor. In order to
test our proposed model, several benchmark instances from literature
are employed and exact solutions are obtained using CPLEX.
Moreover, a discrete framework for particle swarm optimization is
developed to discretize this method and handle the NP-hard problem. In
the proposed MOPSO, an individual particle encoding approach and a
placement strategy are proposed according to the characteristics of the
MWFLP. Thereafter, the local search that is executed by individual
particles through several samplings in the neighborhood search space is
implemented by a 2-opt operation. As for the social communications
among particles, the redefinition of velocity and discrete arithmetic
according to the equation are developed to accelerate convergence
speed. Furthermore, the two-stage approach for the MOPSO is pre-
sented to efficiently find the unique optimal solution for each individual
particle. Finally, several available problem instances are applied; the
contrastive results suggest the preferable performance of the proposed
algorithm.
In future studies, the third objective F
UR
should be relaxed or re-
formulated to make it tractable for exact methods. Afterwards, complex
obstacle avoidance routes of the AGV/trucks that may influence flow
costs will be considered in the manufacturing system. Additionally,
more valuable objectives should be considered in line with the actual
production situation. On the other hand, more efficient methods have
to be developed to enhance the solution quality for the problem.
Acknowledgments
This research was partially supported by the National Natural
Science Foundation of China [No.51205328, 51675450]; the Youth
Foundation for Humanities, Social Sciences of Ministry of Education of
China [No. 18YJC630255]; Sichuan Science and Technology Program
[No. 2019YFG0285], and Doctoral Innovation Fund Program of
Southwest Jiaotong University [No. G-CX201910].
Fig. 8. Population convergence process of MOPSO method for instance SFLP20.
C. Guan, et al. Journal of Manufacturing Systems 53 (2019) 32–48
46
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Appendix A. Frequency of flow matrix for benchmark instances
Instance
name
Frequency of material flows
Das04 =∀ ∈ ≤ ≤ ≠
f
ij N ij i j1, , { *|1 , 4 ; }
ij
Das06 =∀ ∈ ≤ ≤ ≠
f
ij N ij i j1, , { *|1 , 6 ; }
ij
Das08 =∀ ∈ ≤ ≤ ≠
f
ij N ij i j1, , { *|1 , 8 ; }
ij
SFLP08 {0 6 8 7 2154;60342568;83032851;74303867;22230384;15883016;56568106;48174660}
Das10 =∀ ∈ ≤ ≤ ≠
f
ij N ij i j1, , { *|1 , 10 ; }
ij
SFLP11 {0 10 8 10 5 9 0 7 4 5 1; 10 0 7 7 5 11 5 0 1 10 3; 8 7 0 2 1 8 3 1 7 6 2; 10 7 2 0 6 4 2 4 9 10 7; 5 5 1 6 0 6 2 6 5 7 9; 9 11 8 4 6 0 5 7 111 4; 0 5 3 2 2 5 0 4 3 3 9; 7 0 1 4 6
740277;41795132030;51061071137307;13279497070}
SFLP20 {0 5 9 11 14 15 16 11 12 15 3 19 3 20 18 9 15 11 15 10; 5 0 13 13 2 8 19 2 20 12 4 2 1 11 2 2 0 10 5 3; 9 13 0 11 11 14 7 2 11 19 1 2 19 14 15 20 1 18 15 16; 11 13 11
0111413310121336201571316192;14211110991416063661161215176;1581414901710521112784111125;16197139170410
17 14 11 9 9 12 6 15 1 7 11; 11 2 2 3 14 10 4 0 18 10 10 1 13 15 6 1 14 2 7 2; 12 20 11 10 16 5 10 18 0 17 11 19 1 16 3 10 16 16 14 8; 15 12 0 12 0 2 17 10 17 0 9 15
172415619122;34113611141011901511418131414162;19223312111191515017712113716;31196679131171117015281446;
20 11 14 20 6 8 9 15 16 2 4 7 1 0 1 16 1 2 2 12; 18 2 15 15 11 4 12 6 3 4 18 1 5 1 0 18 16 2 15 19; 9 2 20 7 6 1 6 1 10 15 13 2 2 16 18 0 7 13 4 9; 15 0 1 13 12 11 15 14
16614118116707814;11101816151121619143142213701115;15515191727714121674215481109;103162651128221661219
9141590}
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