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Electronic version of an article published as [Lecture Notes in Computer Science, 2007,
No. 4705, p. 791-803] [DOI: http://dx.doi.org/10.1007/978-3-540-74472-6_65]
© [copyright Springer Verlag]
Evaluating optimization models to solve SALBP
*
Rafael Pastor, Laia Ferrer, Alberto García
Technical University of Catalonia, IOC Research Institute, Av. Diagonal 647, Edif.
ETSEIB, p.11, 08028 Barcelona, Spain
{rafael.pastor, laia.ferrer, alberto.garcia}@upc.edu
Abstract. This work evaluates the performance of constraint programming (CP)
and integer programming (IP) formulations to solve the Simple Assembly Line
Balancing Problem (SALBP) exactly. Traditionally, its exact solution by CP or
IP and standard software has been considered to be inefficient to real-world
instances. However, nowadays this is becoming more realistic thanks to recent
improvements both in hardware and software power. In this context, analyzing
the best way to model and to solve SALBP is acquiring relevance. The aim of
this paper is to identify the best way to model SALBP-1 (minimizing the
number of stations, for a given cycle time) and SALBP-2 (minimizing the cycle
time, for a given number of stations). In order to do so, a wide computational
experiment is carried out to analyze the performance of one CP and three IP
formulations to solve each problem. The results reveal which of the alternative
models and solution techniques is the most efficient to solve SALBP-1 and
SALBP-2, respectively.
Keywords: assembly line balancing
1. Introduction
An assembly line consists in set of workstations, through which the product to be
processed flows. In each workstation, a number of tasks are done, which are
characterized by their processing times and by a set of technological precedence
relations between them. The Simple Assembly Line Balancing Problem (SALBP)
consists of assigning a set of tasks to workstations in such a way that precedence
constraints are fulfilled, the total processing time assigned to a station do not exceed a
cycle time tc and a given efficiency measure is optimized. When the objective is to
minimize the number of workstations
m for a given cycle time tc, the problem is
usually referred to as SALBP-1; if the objective is to minimize tc given m, the
problem is called SALBP-2; and SALBP-F consists of finding a feasible solution,
given tc and m (see e.g. [1]).
The design of assembly lines has been extensively examined in the literature,
especially the SALB Problem. Several reviews have been published –the last is [2]–,
and a huge amount of specific research exists, both for heuristic and exact procedures.
*
This work is supported by the Spanish MCyT project DPI2004-03472, co-financed by FEDER.
Some of the exact procedures are based on mathematical programming and
different integer linear programming and mixed-integer linear programming models
(IP models) have been developed. Scholl highlights three basic formulations to solve
SALBP-F [1] (finding a feasible solution, given a cycle time and a number of
stations) based on different sets of assignment variables. Other exact procedures have
also used constraint programming (CP) [3].
Recent improvements both in software and hardware power have reduced
remarkably the computing time needed to solve combinatorial problems by constraint
programming or mathematical programming. Nowadays, these techniques are gaining
acceptance as a powerful computational tools [4]. In this context, analyzing the best
way to model and to solve combinatorial problems is acquiring relevance. Constraint
programming and mathematical programming can solve similar combinatorial
problems, but their effectiveness depends on the class of problems studied [5]. A
number of papers have compared the performance of CP and IP approaches for
solving different problems – for example, [6].
To our knowledge, the efficiency of a CP model and the IP enhanced by Scholl [1]
models has not been compared. In this work, a wide computational experiment is
carried out to analyze the performance of one CP model and three IP models to solve
SALBP-1 and SALBP-2. The results reveal which of the alternative models is the
most efficient to solve these SALBP problems.
The remaining paper is organized as follows. In Section 2 the different
formulations for SALBP-1 and SALBP-2 are presented. In Section 3 the results of the
computational experiment are analyzed and the performances of the models are
compared. Finally, in Section 4 the main conclusions of the study are summarized.
2. Models for SALBP
In this section four alternative formulations for SALBP-1 and SALBP-2 are
developed.
First, we present a CP model – constraint programming model-.
Then, the three SALBP-F models presented in Scholl (1999) are adapted to
SALBP-1 and SALBP-2. In sum, the main difference between these three linear
models is the definition of the assignment variables used:
- impulse variables based model: binary variables x
ij
take value 1 if and only if
task
i is assigned to workstation j (see also [7] and [8])
- step variables based model: binary variables x
ij
take value 1 if and only if task
i
is assigned to workstation
j or earlier (see also [7] and [8]).
- mixed-integer variables model: integer variables z
i
denotes the number of the
station to which task i is assigned.
In the following sections the formulations for the four models are detailed. In each
section, first the model for SALBP-1 is presented. Then the model for SALBP-2 is
explained highlighting the new data, the new variables and the changes to be done in
the formulation with respect to the model for SALBP-1.
2.1. The constraint programming models
Next, the constraint programming model for SALBP-1 (SALBP-1-c) is presented and
the changes for SALBP-2 (SALBP-2-c) are explained.
SALBP-1-c
Data:
Note subindexes i and k are related with tasks and subindex j with workstations.
n
Number of tasks
()
1,...,in= .
m
max
Upper bound on the number of workstations
max
( 1,..., )jm= .
m
min
Lower bound on the number of workstations.
t
i
Processing time of task i .
TC Cycle time.
P Set of pairs of tasks
()
,ik such that there is immediate precedence between
them.
S Set of tasks without any successive task.
E
i
Earliest possible workstation for task i.
L
i
Latest possible workstation for task i, given a value of
max
m .
Before a task is assigned the total processing time of the tasks that precede it
must be assigned, and afterwards the total time of the tasks that follow it; as
a result, the range of workstations [E
i
, L
i
] to which each task can be assigned
is obtained and the number of binary variables is reduced (see, for example,
[1]).
Variables:
ws Number of workstations used.
z
i
Number of the workstation to which task i is assigned
[
]
(
)
;,
iii
iz E L∀∈
Model SALBP-1-c:
[
]
=
M
IN Z ws
(
1)
max ( )
i
iSws z ∈=
(
2)
[]
,()
ii i
i
ij E L z j
tTC j
∀∈ =∧
≤
∀
∑
(
3)
(
)
,
ik
ik Pzz≤∀∈
(
4)
The objective function (1) consists in minimizing the number of workstations,
which is calculated in (2); constraints (3) ensures that the total task processing time
assigned to workstation
j does not exceed the cycle time; constraint set (4) imposes
the technological precedence conditions.
SALBP-2-c
Data:
The model uses the same data of SALBP-1-c; furthermore we redefine:
m Number of workstations
(
)
1,...,jm= .
C Upper bound on the cycle time.
E
i
Earliest possible workstation for task i, given a value of C.
L
i
Latest possible workstation for task
i
, given a value of C.
Variables:
tc
Cycle time.
Model SALBP-2-c:
[
]
M
IN Z tc
=
(
5)
[]
,()
ii i
i
ij E L z j
ttc j
∀∈ =∧
≤
∀
∑
(
6)
Constraint (4) has to be added.
The objective function (5) minimizes the cycle time and constraint set (6) ensures
that the total task processing time assigned to workstation
j
does not exceed the
cycle time.
2.2. The impulse variables based models
Next, the impulse variables based model for SALBP-1 (SALBP-1-i) is presented and
the changes for SALBP-2 (SALBP-2-i) are explained.
SALBP-1-i
Data:
The data used in this model is the same as the previous one.
Variables:
{
}
0,1
ij
x ∈ 1, if and only if task i is assigned to workstation
j
, value 0 otherwise
(
)
; ,...,
ii
ij E L∀= .
{
}
0,1
j
y ∈
1, if and only if any task is assigned to workstation j
min max
( = m ,..., )1 mj + .
Model SALBP-1-i:
[]
max
min
1
m
j
jm
M
IN Z j y
=+
=
⋅
∑
(
7)
1
i
i
ij
L
jE
x i
=
=
∀
∑
(
8)
[]
min
,
1,...,
ii
iij
ij E L
tx TC j m
∀∈
⋅≤ =
∑
(
9)
[]
min max
,
1,...,
ii
iij j
ij E L
tx TCy j m m
∀∈
⋅≤ ⋅ = +
∑
(
10)
()
,
ik
ik
LL
ij kj
jE jE
jx jx ik P
==
⋅≤ ⋅ ∀ ∈
∑∑
(
11)
The objective function (7) consists in minimizing the number of workstations;
constraint set (8) implies that each task i is assigned to one and only one workstation;
constraints (9) and (10) are equivalent to (3) and they ensure the cycle time is not
exceeded; constraint set (11) replaces (4) and imposes the precedence conditions.
SALBP-2-i
Data:
The data used in this model is the same as the previous ones.
Variables:
The variables have been defined in the previous models.
Model SALBP-2-i:
[
]
M
IN Z tc
=
(
5)
[]
,
ii
iij
ij E L
tx tc j
∀∈
⋅
≤∀
∑
(
12)
Constraints (8) and (11) have to be added.
The objective function (5) minimizes the cycle time; constraint set (12) is
equivalent to (6) and ensures that the total task processing time assigned to
workstation
j
does not exceed the cycle time.
2.3. The step variables based models
Next, the step variables based model for SALBP-1 (SALBP-1-s) is presented and the
changes for SALBP-2 (SALBP-2-s) are explained.
SALBP-1-s
Data:
The data used in this model is the same as the previous ones.
Variables:
The variables used in the step variables based models are the same of the impulse
variables based models but
ij
x
are redefined:
{
}
0,1
ij
x ∈ 1, if and only if task i is assigned to workstation j or earlier, 0 otherwise
(
)
; ,..., 1
ii
ij E L∀= −. Note that
,
1
i
iL
x
=
and it is not defined.
{
}
0,1
j
y ∈ 1, if and only if any task is assigned to workstation j, 0 otherwise
min max
( = m ,..., )1 mj + .
Model SALBP-1-s:
[]
max
min
1
m
j
jm
M
IN Z j y
=+
=
⋅
∑
(
7)
,1
;,...,2
ij i j i i
xx ijE L
+
≤∀= −
(
13)
()
[]
()
,1 ,1
1, 1
min
1;
1,...,
iii i
iij i ij ij i ij
ij E ij E L ij L
tx t x x t x TC
jm
−−
∀= ∀∈ + − ∀=
⋅+ ⋅ − + ⋅− ≤
=
∑∑ ∑
(
14)
()
[]
()
,1 ,1
1, 1
min max
1
1,...,
iii i
iij i ij ij i ij j
ij E ij E L ij L
tx t x x t x TCy
jm m
−−
∀= ∀∈ + − ∀=
⋅+ ⋅ − + ⋅− ≤ ⋅
=+
∑∑ ∑
(
15)
()
[
]
[
]
,; ,1,1
kj ij i i k k
xx ikPjEL EL
≤
∀∈∀∈ −∩ −
(
16)
Constraint sets (13), (14) and (15) are equivalent to constraint sets (8), (9) and (10),
respectively. Now, the technological precedence conditions –constraint set (4) or
(11)– is modeled by (16).
SALBP-2-s
Data:
The data used in this model is the same as the previous ones.
Variables:
The variables have been defined in the previous models.
Model SALBP-2-s:
[
]
M
IN Z tc
=
(
5)
()
[]
()
,1 ,1
1, 1
1
iii i
iij i ij ij i ij
ij E ij E L ij L
tx t x x t x tc j
−−
∀= ∀∈ + − ∀=
⋅+ ⋅ − + ⋅− ≤ ∀
∑∑ ∑
(
17)
Constraints (13) and (16) have to be added. Constraint set (17) is equivalent to (6)
and (12).
2.4. The mixed-integer variables based models
Next, the mixed-integer variables based model for SALBP-1 (SALBP-1-m) is
presented and the changes for SALBP-2 (SALBP-2-m) are explained.
SALBP-1-m
Data:
The model uses the same data of the previous ones; furthermore we define:
P* Set of pairs of tasks
(, )ik
such that there is an immediate or transitive
precedence between them.
T Upper-bound of the total time of the workstations.
Variables:
This formulation introduces continuous non-negative variables b
i
for the clock time at
which task i is started and binary variables w
ik
:
{
}
0,1
i
b ∈
Clock time at which task i is started (measured in the time elapsed since
entering the first workstation).
{
}
0,1
ik
w ∈
1, if and only if task i is performed before task k, value 0 otherwise
[
]
[
]
(
)
;(, ) *; , ,
ii kk
ikik P EL EL<∉ ∩ ≠∅.
ws Number of workstations used.
z
i
Number of the workstation to which task i is assigned
[
]
(
)
;,
iii
iz E L∀∈
.
Model SALBP-1-m:
[
]
=
M
IN Z ws
(
1)
≥∀
i
ws z i
(
18)
(1)
ii
bTCz i≥−∀
(
19)
ii i
btTCz i
+
≤∀
(
20)
[
]
[
]
(1 ) , ( , ) *, , ,
ik k i i i i k k
wTbbtikikPEL EL−⋅+≥+ < ∉ ∩ ≠∅
(
21)
[
]
[
]
,( , ) *, , ,
ik i k k i i k k
wTb b t ikik P EL EL⋅+ ≥ + < ∉ ∩ ≠∅
(
22)
(, ) ,
ii k i k
bt b ik PL E+≤ ∈ ≥
(
23)
ii
E
zi
≤
∀
(
24)
ii
zL i
≤
∀
(
25)
The objective function (1) consists in minimizing the number of workstations
calculated by constraint set (18); constraint sets (19) and (20) ensure that each task i is
fully performed within one workstation; the disjuntive constraints (21) and (22)
guarantee that for each pair of tasks, which are not related by precedence and may
interfere which each other, either task i is completely processed before task k , or vice
versa; constraint set (23) ensure the fullfilment of the precedence constraints; the
assignment task is restricted to the possible workstation interval by (24) and (25).
SALBP-2-m
The adaptation of mixed-integer variables based model for SALBP-F to SALBP-2
produces a non-linear model since the variable cycle time tc replaces data TC in
constraints (19) and (20). This non-linear formulation of SALBP-2-m is linearised as
follows.
Variables:
p
i
not negative real variable that indicates the total time of the workstations
until the workstation in which task i is assigned (this one also included)
{
}
0,1
ij
r ∈ 1, if and only if task i is assigned to workstation
j
, value 0 otherwise
(
)
; ,...,
ii
ij E L∀= .
Model SALBP-2-m:
[
]
M
IN Z tc
=
(
5)
ii
btcp i
+
≥∀
(
26)
ii i
bt p i
+
≤∀
(
27)
i
i
L
iij
jE
zjr i
=
=
⋅∀
∑
(
28)
1
i
i
L
ij
jE
ri
=
=
∀
∑
(
29)
(1 ) , ,...,
iij ii
p
jtc r T i j E L−⋅ ≤ − ⋅ ∀ =
(
30)
(1 ) , , ...,
iij ii
jtc p r T i j E L⋅− ≤− ⋅ ∀ =
(
31)
The real variables p
i
replaces the product
i
tc z
⋅
in (19) and (20) obtaining (26) and
(27); the variables z
i
are expressed as shown in (28); constraint sets (29), (30) and (31)
are added.
Constraint sets (21)-(23) need to be added too.
3. Computational experiment
A computational experiment is carried out to compare the efficiency of the models.
The basic data used for the experiment are all the well-known instances available
in the assembly line balancing research homepage (www.assembly-line-
balancing.de). A total of 269 instances for SALBP-1 and 302 for SALBP-2 were
used.
The CP models were solved using ILOG Solver 6.0 and the MILP models were
solved by CPLEX 9.0, with a PC Pentium IV at 3.4 GHz and with 512 Mb of RAM.
A maximum computing time of 2,000 seconds was set.
The analysis of the results of the computational experiment starts with a initial
comparison of the performance of the models in terms of the type of the solutions
obtained: whether the model finds a solution or not and whether this solution is
optimal or feasible. This initial analysis identifies the best models to be analyzed in
detail. Next, the computing time used by these models is studied, focusing on the
instances in which the optimal solution is found. Next, the solutions obtained in the
instances in which the optimality is not guaranteed are presented. Finally, considering
all these aspects, a detailed analysis of the performance of the different models is
carried out.
3.1. Results of the type the solutions
Table 1 and table 2 show the results of the computational experiment for SALBP-1
and SALBP-2, respectively, focusing on the type of the solutions obtained. For each
model, the following information is summarized:
-
the number of instances with a proved optimal solution
(
)
Opt prov− : an
optimal solution is found and the solving software guarantees it.
-
the number of instances in which an unproved optimal solution
()
Opt prov− : an optimal solution is found but the solving software does not
guarantee its optimality. The optimal solution of the instances is available in
the assembly line balancing research homepage.
-
the number of instances with a feasible but not optimal solution
(
)
F
ea opt− .
-
the number of instances in which the solving software does not find any
solution
()
Sol .
Table 1. Results of the computational experiment for SALBP-1
SALBP-1 SALBP-2
c i s m c i s m
Opt prov
−
98 136 123 51 55 84 122 0
Opt prov−
12 17 5 24 0 16 12 4
F
ea opt−
2 19 14 14 199 174 168 64
Sol
157 97 127 180 48 28 0 234
The results show that the performance of the mixed-integer based model is worse
than the performance of the constraint programming model, the impulse variables and
the step variables based models. For SALBP-1, SALBP-1-m obtains 51 proved
optimal solutions; nearly half of the optimal solutions reached by SALBP-1-c, SALBP-
1-i or SALBP-1-s (98, 136 and 123, respectively). For SALBP-2, SALBP-2-m does not
obtain any proved optimal solution. Moreover, the mixed-integer variables based
models do not reach a feasible solution in more instances than the other models, both
for SALBP-1 and for SALBP-2.
Due to clear inferiority of the mixed-integer variables based model, we focus the
detailed comparison of the results only in the constraint programming, the impulse
variables and the step variables based models. We analyze the percentage of proved
optimal solutions depending on the the number of tasks (NT) and the order strength
(OS = number of all precedence relations / (NT * (NT - 1))) of the instances. We
classify: i) Low-OS
(
)
22.49 25.80OS≤≤
, Middle-OS
(
)
40.38 60.0OS≤≤
and High-
OS
(
)
70.95 83.82OS≤≤
; ii) Low-NT
(
)
745NT≤≤
, Middle-NT
(
)
53 111NT≤≤
and
High-NT
(
)
148 297NT≤≤
. Table 2 shows the percentage of proved optimal solutions
obtained with the constraint programming (c), impulse variables (i) and step variables
(s) based models.
Table 2. Percentage of proved optimal solutions depending on OS and NT
SALBP-1 SALBP-2
c i s c i s
Low-OS 10.77 29.23 12.31 6.061 21.21 25.76
Middle-OS 40.79 55.92 53.95 18.68 28.02 36.81
High-OS 55.77 61.54 63.46 31.48 35.19 70.37
Low-NT 100.00 98.72 98.72 77.50 97.50 100.00
Middle-NT 14.62 43.85 33.85 10.71 19.90 36.73
High-NT 1.64 3.28 3.28 4.55 9.09 15.15
3.2. Results of the computing time
We compare the computing time used by the constraint programming, the impulse
variables and the step variables based models when all of them obtain a proved
optimal solution (95 instances in SALBP-1 and 48 instances in SALBP-2). Table 3
shows, for each model: the number of instances with the minimum calculation time
(in seconds) to obtain a proved optimal solution (Best time); the total of time used by
these instances (Total time); and the number of instances in which the time used by
the model is less than 75% of the time used by each of the other two models
(time(a/b)<0.75).
Table 3.
Results when the 3 models find an optimal solution.
SALBP-1 SALBP-2
c i s c i s
Best-time 26 47 22 27 15 7
Total time 2084.4 3302.4 2824.7 2491.3 6608.2 508.9
time(a/b)<0.75 9 1 2 8 0 5
3.3. Results of the solutions with no optimality guaranteed
Next, we summarize the results obtained when the constraint programming, the
impulse variables and the step variables based models find a feasible solution but
none of them guarantees optimality. This situation occurs in 1 instance for SALBP-1,
and in 119 instances for SALBP-2: in 10 of them SALBP-2-c obtains the best solution,
SALBP-2-i in 9 instances and SALBP-2-s in 82. When SALBP-2-s obtains a better
solution, the average solution is 95.5% and 71.1% of the average obtained by SALBP-
2-i and SALBP-2-c, respectively.
3.4. Analysis of the performance of models
In this section, a detailed analysis of the performance of the models is carried out.
First, we study the results for SALBP-1 and then a similar study is presented for
SALBP-2. Each study starts with a brief final conclusion to facilitate the
comprehension of the analysis of the results. These conclusions are justified through a
detailed analysis that compares the type of solutions obtained, the computing time
used and the results of the solutions in which their optimality is not guaranteed. Due
to clear inferiority of the performance of the mixed-integer variables based model
(Section 3.1), these analyses focus on the constraint programming, the impulse
variables and the step variables based models.
For SALBP-1:
In sum, in terms of number of optimal and feasible solutions the results of SALBP-1-i
are better than results of SALBP-1-c and SALBP-1-s. However, concerning the time
used, SALBP-1-c is the quickest model.
In terms of the type of solutions obtained (Table 1), the number of proved and
unproved optimal solutions obtained by SALBP-1-i is higher than those obtained by
SALBP-1-c and those obtained by SALBP-1-s (136 and 17, 98 and 12, 123 and 5,
respectively). Moreover, SALBP-1-i does not obtain a feasible solution in less
instances than SALBP-1-c and SALBP-1-s (97, 157 and 127, respectively) The results
of SALBP-1-c are worse than the results of the other models, in particular, for
instances with middle and high levels of NT; the performance of SALBP-1-i is
especially the best for instances with low OS (Table 2).
In terms of the computing time (Table 3), when the three models guarantee the
optimal solution, SALBP-1-i need less time in more instances than SALBP-1-c and
SALBP-1-s (47, 26 and 22, respectively). Nevertheless, for solving all 95 instances the
time needed by SALBP-1-c is considerably less than the time required by SALBP-1-i
and by SALBP-1-s (2084.4 s, 3302.64 s and 2824.7 s, respectively). Moreover, among
the 26 instances where SALBP-2-c is quicker, there the 9 instances in which the time
used is less than 75% of the time needed by each of the other two models, whereas
this difference only occurs in 1 instance for SALBP-1-i and 2 for SALBP-1-c.
For SALBP-2:
In sum, the results of SALBP-2-s are much better than the results of SALBP-2-c and
SALBP-2-i, in terms of optimal and feasible solutions obtained and total computing
time. SALBP-2-c is only superior to SALBP-2-s in the number of instances that use the
minimum time.
In terms of the type of solutions obtained (Table 1), the SALBP-2-s model obtains
more proved solutions than SALBP-2-c and SALBP-2-i: (122, 55 and 84,
respectively). The total number of optimal solutions (proved and unproved) is also
superior for SALBP-2-s than for SALBP-2-c and SALBP-2-i (134, 55 and 100,
respectively). In addition, SALBP-2-s always obtains a feasible solution whereas
SALBP-2-c does not obtain a feasible solution in 48 instances and SALBP-2-i in 28.
The influence of NT is similar in the 3 models, SALBP-1-c is remarkably the best
model for instances of low OS (Table 2).
In terms of the computing time (Table 3), when the three models guarantee an
optimal solution, SALBP-2-c uses less time in more instances than SALBP-2-i and
SALBP-2-s (27, 15 and 7, respectively). Moreover, the time used by SALBP-2-c is in
8 instances less than 75% of the time used by the other models (SALBP-2-i does not
have this difference in any instance and SALBP-2-s only in 5). However, for solving
all the instances in which the three models guarantee an optimal solution, the total
time used for SALBP-2-s is much less than the time used by SALBP-2-c and SALBP-
2-i (508.9 s, 2492.4 s and 6608.2 s, respectively).
Finally, when none of the models guarantees the optimal solution, SALBP-2-s
obtains a better solution in considerably more instances than the others.
4. Conclusions
The SALB Problem has been extensively examined in the literature and different and
equivalent CP models and IP models have been developed in order to solve it.
However, their efficiency has not been compared and the best one is not known. The
best way to model and to solve the hard combinatorial problems has a high relevance.
The use of constraint programming or mathematical programming techniques to solve
these problems is becoming more realistic thanks to recent improvements both in
software and hardware power.
This paper focus on comparing one CP formulation –constraint programming
model- and three IP formulations that were highlighted by Scholl [1] -the impulse
variables, the step variables and the mixed-integer variables based model-. A wide
computational experiment is carried out to compare the efficiency of these models,
both for SALBP-1 and SALBP-2.
The analysis of the results shows the bad performance of the mixed-integer
variables models. For SALBP-1, the impulse variables based model obtains the best
solutions although constraint programming model is the quickest. The step variables
based model obtains the best results for SALBP-2.
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