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HEURISTICS AND HIPERHEURISTICS FOR SEQUENCING MIXED
MODEL ASSEMBLY LINES MINIMIZING WORK OVERLOAD
Joaquín Bautista Valhondo
NISSAN Chair UPC, – Escuela Técnica Superior de Ingeniería Industrial de Barcelona – Universitat Politècnica de
Catalunya, Barcelona, Spain
joaquin.bautista@upc.edu
Jaime Cano Belmán
Departament d’Estadística i Investigació Operativa – Universitat Politècnica de Catalunya
Barcelona, Spain
jaime.cano-belman@upc.edu
Abstract – There are different approaches for the
mixed model sequencing problem on assembly lines. In
this paper the goal of minimizing work overload is
treated. Since solve this problem optimally is difficult, we
test constructive procedures, local search and a new
hiperheuristic procedure.
Keywords: Sequencing, assembly line, work over-
load, local search, priority rules, JIT.
I. INTRODUCTION
Assembly lines are commonly used in automotive in-
dustry. Two important decisions for managing mixed
model assembly lines are to spread out work in stations
(balancing) and determine the sequence to introduce
cars in to the assembly line (sequencing). When the
medium term decision of balancing has been taken, se-
quencing decision must be considered. Depending on
the manufacturing environment it may be desirable
minimize or maximize some parameters or characteristics
of the line [1]. One of the two main criteria [2] for se-
quencing mixed models on assembly lines considers the
labelling of load on stations. The problem addressed in
this paper considers the objective of minimizing the work
overload.
Since a processing time for a product in a station can
be grater than cycle time, there is a maximum quantity of
those products that can be consecutively introduced in
the line without causing a delay in the finishing of
works. All those jobs with high and low work content
must be under control for avoiding excessive work load
and idle time. Stations are confined by upstream limit
and down stream limit. That implies that products are
mounted on an assembly line that moves at constant
speed and workers can do their job on products only
when they are inside their station. Products get in the
station at constant time intervals. Work overload occurs
when work on a product can not be finished before it
leaves the station.
Initial works on this criterion were carried out by [3]
or [4]. In this paper an extension of a procedure in [3] is
proposed, which considers not only two different
jobs/products (basic product and special product, dif-
ferentiable by their poor and rich work content respec-
tively), but also multiple products. Other literature re-
lated with the problem is [5], [6], [7] and recently [8].
Inspired on procedures from [4] and [9], four proce-
dures are proposed, which also consider multiple prod-
ucts and multiple stations. We use local search with
different neighborhoods for improving the solutions
obtained with constructive procedures. A new hiperheu-
ristic is tested with the sequencing problem.
This paper is organized as follows: section 2 contain
our constructive proposals, in section 3 we apply local
search, section 4 contains a proposal of a new hiperheu-
ristic procedure, section 5 shows computational experi-
ence for constructive procedures, local search and the
hiperheuristic. In section 6 conclusions are mentioned.
II. WORK OVERLOAD MEASUREMENT
In [3] a general formulation for measuring work over-
load is proposed. Work overload is measured in time
units and the time unit is the cycle time c (time between
product arrivals into the station). Let L denote the sta-
tion length (or time window), pik the processing time for
the job on the product i (i=1,…,I) in the station k
(k=1,…,K), st the starting instant of the job in the posi-
tion t (s1=0; st=max(t-1,ft-1)), and ft the finishing instant of
the job in position t (ft=min(st+pi,t-+L)). Given a se-
quence of size T (t=1,…,T), and considering only one
station, wot is the work overload obtained in position t
of the sequence: wot = [pi+st–(t-1+L)]+ where [x]+ =
max(0,x). The total work overload is z = ∑twot. Mathe-
matical programming formulations of the problem can be
found in [3] or [6]. The problem is difficult to solve due
to the lack of structural properties. The problem has a lot
of possible solutions and a big effort to evaluating those
solutions is required. As explained in [3], [6] or [8] the
problem is considered to be NP-hard.
To elucidate the reader on the work overload problem,
a single station illustrative example is shown. Four prod-
ucts are considered: A, B, C, and D, with the following
Knowledge and Decision Technologies
235
processing times (0.82, 0.94, 1.19, 1.15), and demands
(3,5,7,1). Station length L=1.2. Processing times and
station length are expressed in cycle time units (c=1).
Let us assume products are going to be introduced into
the assembly line in the following order:
CCCADCCBCABABCBB.
Figure 1- Worker movement diagram
Without loss of generality, the initial worker position
is assumed to be on the upstream limit of the station.
Figure 1 represents the movement of the worker during
his job on the products according to the sequence es-
tablished above. Arrows represent the processing times,
and dotted lines represent the worker displacement from
one finished product to the next product on the line.
Station length is limited by the down stream limit (dsl).
The first job is done on a product kind C, which requires
1.19 time units. When this job is finished, the worker
walks upstream for reaching the product C in position 2
of the sequence. We assume this time is negligible be-
cause the velocity of the worker is greater than the con-
veyor speed. Then, the worker starts the second job 0.19
units away from the upstream station limit and would
finish it 0.19+1.19 units away from the upstream limit.
Nevertheless, the worker can not go beyond the dsl, and
0.18 units of work must be left unfinished (w1). When
the worker reaches the dsl, he leaves his job and walks
upstream, and starts working on the product in position
3 of the sequence 0.20 units away from the upstream
limit. Again, the time allocated is not enough to finish
the job, and 0.19 units of work overload are produced
(w2). The product in position 4 requires 0.84 time units,
and the work on it is finished when the worker is
0.20+0.84 units away from the upstream limit. This time,
the job is completed. Products in positions 6, 7 and 9,
also produce work overload (w3=0.16, w4=0.19,
w5=0.13). The total work overload produced by the se-
quence is 0.85 cycle time units.
III. CONSTRUCTIVE PROCEDURES
In this section, four procedures are proposed inspired
on the works from [3], [4] y [9]. Our proposals assume
multiple products and only one station. Then those one
station procedures are used for determining work over-
load prediction values in a multi station procedure taken
from [3]. An original procedure (Y&R) from literature is
described to make more understandable the proposed
extension.
A. Y&R Procedure
The original procedure from [3] considers single sta-
tion and two kinds of products: products with optional
components and basic products. Let A denote special
products, which have processing time greater that c, and
B denote basic products with processing time smaller
than c. The procedure is based on the repetition of a
stable subsequence composed by ma units A, and mb
units B. Since processing times for special products pa is
greater than c, there is a limit on the number of units A
that can be consecutively sequenced, without causing
work overload. This limit (X) is the maximum integer
satisfying X = (L-c)/(pa-c), and is also the maximum value
ma can take. The procedure tries to regenerate, that is, to
bring the worker back to the beginning of the window
(its assumed original position), after a cycle composed
by ma units A and mb units B.
The subsequence of ma and mb units, can be deter-
mined with the equation ma⋅pa+mb⋅pb=ma+mb, where
ma≤X. Then, assuming there are integer values for ma
and mb, maximum utilization is achieved by solving the
next nonlinear MP model (1).
maximize )/()( babbaa mmmpmp+⋅+⋅ (1)
s.t. ma ≤ X
pa ⋅ ma + pb ⋅ mb = ma + mb
ma , mb ≥ 0, integers
Let ni the quantity to be produced of product i (T=∑ni),
and nc the maximum number of cycles composed by ma
units A, and mb units B. The sequence is built according
to the following steps: 1) assign the nc cycles, 2) assign
xa=min(na-nc⋅ma, ma) products A, 3) assign xb=nb-nc⋅mb
products B, and 4) if necessary assign na-nc⋅ma-xa prod-
ucts A.
B. Y&R Procedure Extension
This extension considers not only two but also multi-
ple products. As in the original procedure, our extension
(YRx) builds sub-sequences composed by mi units of the
product i. While there are enough unscheduled units,
the sub-sequence is repeated. Otherwise, a new subse-
quence is calculated with the remaining products. Let A
the set of products with processing time greater than the
cycle time (pi>1), B the set of products with processing
time smaller than the cycle time (pi ≤ 1), and xi the maxi-
mum consecutive quantity of product i that can be
scheduled without causing overload or idle time.
maximize ∑∑ == ⋅I
ii
I
iii mmp11 (2)
s.t.
C C C A D C C B C A B A B C B B
w1 w2 w3 w4 w5
time
dsl
{
}
integers,0
,min
11
≥≤⋅
≤∑∑ ==
i
I
ii
I
iii
iii
m
mmp
dxm
L
1
0
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236
Model (2) is the extension proposal of the original
model (1), which finds the quantity of each product
contained in the sub-sequence.
In model (2), xi ≤ L-c/pi-c, ∀i∈A, and xi ≤ L-c/c-pi,
∀i∈B. di represents the remaining production of product
i. When t=0, di = ni. To arrange products in the subse-
quence, we consider alternatively the set A, and then the
set B. Let c denote the cycle time. The order in which
units will be incorporated into the sequence is done
according the diminishing value of index ri = mi⋅|c-pi|.
Thus, from the set A, the item with the bigger index ri (let
j) is selected and the mj products j are consecutively
assigned. Then, from the set B, the item with the bigger
index ri is selected (let l), and the ml corresponding
products l are assigned; and so on. Since it is easier to
solve a linear model than a non-linear model, we have
transformed the non-linear model (1) into a linear one.
Then we used the transformed model for finding subse-
quences.
C. Greedy Procedures
The proposed procedures try to favour the movement
of the worker in the station from the lower to the upper
limit of its station (Up-Down) as depicted in figure 2.
Arrows represent the processing times, and dotted lines
represent the worker displacement from one finished
product to the next product on the line. If a long arrow is
scheduled after the unit has been completed (in the
circle of figure 2), work overload is produced, then,
down step is required to avoid it.
With those two up-down cyclic steps, we attempt to
regenerate. Perfect regeneration is reached only for cer-
tain parameters values [3].
Figure 2- Up-down worker movement diagram
During the up step, only products with processing
time greater than c are considered. Products with proc-
essing time smaller than c are considered during the
down step.
For deciding the kind of product to sequence in the
period t, we make use of a dynamic index ri = di ⋅ |c-pi|,
where i belongs to the set A or B considered in the cur-
rent stage t. If t=0, di=ni. Since ri depends on the pend-
ing production of product i, it must be updated in each
phase. The product of the set under analysis with the
bigger index ri is selected to be assigned in period t. s1=0
is assumed.
Following those ideas, three procedures are pro-
posed: Ud, UdC and UdR. Ud allows commit in idle time
only ones in each cycle reaching the complete regenera-
tion; while it is possible; UdC forbids incurring in either
idle time or work overload, putting them off; UdR allows
a maximum quantity of work overload until the position t
of the sequence. This quantity is limited by lbw/T*t. lbw
is a lower bound of work overload considering multiple
stations.
( )
[
]
∑ ∑
=
+
=+−⋅−⋅=K
k
I
ikikiLTcpnlbw 1 1 )1( (3)
Where [x]+ is max(0,x). UdR tries to distribute or regu-
larize work load along the sequence.
D. Multi-station procedure
A multi-station procedure is taken from [3]. This pro-
cedure is based on single station procedures. Single
station procedures are used for determining work over-
load predictions. The sequence is built progressively in
such a way that for each sequencing instant t, the prod-
uct with remaining production and the best overload
predictor is selected and assigned in the tth position of
the sequence.
Let: wpik be the work overload predictor in station k
due to the assignment of product i, wi(k,t) be the work
overload obtained in station k in period t of the se-
quence due to the assignment of product i, skt be the
starting instant of job in station k in period t (or initial
worker position in station k in period t), and sci(t) be the
total work overload prediction produced by product i in
period t.
0. Initialize s=0.
for ( t =1 to T )
for ( i=1 to I )
1. Assume sequence(t) ← i, (di-1).
2. Compute predictor sci(t) for each station.
3. di+1.
end for
4. sequence(t) ←
{
}
)(min)(:**tsctscii
Ii
i∈
=.
5. Update data.
end for
wpik is obtained with single station procedures. The
work overload produced in period t due to the assign-
ment of product i is obtained by (5) and (6).
(
)
∑=⋅+= K
kkiikisctkwwptsc 1),()( (4)
wi(k,t) = [skt+pik - ((t-1)⋅c)-Lk]+ ∀ i ∈ A (5)
wi(k,t) = [(t⋅c)-skt-pik] + ∀ i ∈ B (6)
IV. LOCAL SEARCH
For improving the solutions obtained with the four
constructive procedures described in the previous sec-
tion, local search is applied. Two well known kinds of
neighborhoods had been used in the search: swap and
insertion.
up down
up-down regenated cycle
L
0
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237
A. Swaps
Swaps of two and three elements of the sequence had
been considered. Swap of two elements (2S) is simple.
One solution can just produce a new one. Nevertheless,
swap of three elements have five possible neighbors
solutions. From those five possibilities, only in two of
them the three elements considered take a new position
in the sequence: (b,c,a) and (c,a,b) in figure 3.
Figure 3- 3 swap neighbors
Then, we had used two different 3-swap neighbor-
hoods: 3S(a) and 3S(b). 3S(a) considers the tow changes
where all of the elements considered take a new position
in the sequence. In the other hand, 3S(b) take into ac-
count the five possibilities. We had also considered the
idea of applying 3 swap after no improvement can be
found with 2 swap, we call that 2-3S(b).
B. Insertion
By the insertion, a new neighbor from the current fea-
sible sequence is obtained getting a segment of certain
size from the solution, and then it is inserted in a differ-
ent position of the sequence. Even tough size of seg-
ments (let or) can take the value 1 ≤ or ≤ T, in the com-
putational experience only had been tested 2 ≤ or ≤ 10.
The size of the Insertion neighborhood is smaller than
the Swap neighborhood; therefore, the computational
effort is smaller too.
Figure 4- Insertion neighbors
In all the local search experience, the maximum num-
ber if iterations with out improvement has been estab-
lished to T.
V. PRIORITY RULES
In this section a hiperheuristic is described. The pro-
cedure is inspired on the Scatter Search (SS) Meta heu-
ristic. Instead of using feasible solutions for producing
new ones, our proposal use priority rules chains. The
objective value of a chain is obtained getting the corre-
sponding solution sequence. That is done using a con-
structive procedure of rules combination (PCCR), and
measuring its work overload value.
A. PCCR
PCCR is a greedy constructive procedure based in the
combination of priority rules. The assignation of a prod-
uct in certain position of a sequence depends on the
priority rules. A set of rules R={r1,r2,r3,...,rR} establishes
the order of the products in a sequence. The chain (se-
quence of priority rules) will have the same number of
rules as positions have a product solution sequence (T).
The rule in the position t of the chain, determines from a
set of products, the product i that best satisfies the rule
r. Given a rules chain size T, the determination of the
product that best satisfies the rule r of the position t of
the chain is obtained as follows:
- For each candidate product, compute the value of
the application of the rule r.
- Select the product i with the best value for the rule
r.
- Assign the product i in the position t of the prod-
ucts sequence.
- Update pending demand for product i.
B. Hiperheuristic
Similarly to SS [10], our proposal is an evolutionary
algorithm that creates new elements combining the exis t-
ing ones, improving this way the criterion used to evalu-
ate the elements. Our proposal operates on a Reference
Set (RefSet). But, instead of a reference set of solutions,
we use a reference set of rules chains. Combining those
rules chains, new rules chains are created. A typical
RefSet size in SS is 20 or less, while the size of our RefSet
is in function of the number of rules R considered.
The following is a pseudo code of the proposed pro-
cedure:
start
0.1 Create RefSet static and dynamic.
0.2 Initialize frequency matrix, Fr.
while ( Diversifications < Max Diversifications )
1. Combine rules of the RefSet.
2. Regenerate RefSe.t
if ( RefSet state is not improved )
3. Diversify RefSet.
end if
end while
The RefSet is conformed by two tiers: the static sub-
set (RSs) and the dynamic (operative) subset (RSd). The
size of both RSs and RSd is R.
RSs is called “static”, because it is not modified dur-
ing the search process. In RSs the rules chain 1 contains
only the rule 1, the rules chain 2 contains only the rule 2,
and so on. In this way the procedure ensures the con-
sideration of all the priority rules in the combination
phase.
}:{),...,,,(RrcrRSsrrrrcr rr ∈=→= (7)
a b c
or
=2
T
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238
The dynamic RefSet changes in each iteration of the
search process. The rules chains of the initial RSd are
generated randomly. RSd contains the elite group of
rules chains. RSd is updated in each iteration of the
process taking into account the new chains with the
best values of work overload obtained by the last com-
bination.
1 ≤ RSdit ≤ R (8)
The work overload value for a chain is obtained ap-
plying the PCCR. In this work, 20 priority rules had been
used. If during the PCCR procedure, one or more prod-
ucts have the best value for the rule in period t of the
sequence, the tie is eliminated taking only the products
in the tie and applying the rules in order (starting with
rule 1), until the tie disappears.
Different criteria are considered in the priority rules
used in the procedure. Rules 1-4 decide which product is
going to be assigned using the processing times data.
Rules 5 and 6 select the product with the bigger and
smaller pending demand respectively. Rules 7,8,14 and
15 differentiate the products making a relation of pend-
ing production and the difference between the process-
ing time and cycle time. The displacement of the workers
in the stations is used for selecting the product in rules 9
and 10. Bottleneck station processing times are consid-
ered in rules 11 and 12. Rules 16 and 18 use the work
overload caused by the assignment of a product. Rules
17 and 19 use idle time. Rule 13 select a product using
the measurement of the regularization of the load along
the sequence. Similarly, rule 20 select according the
regularization of idle time.
In each iteration of the process, a frequency matrix is
obtained Fr(r,t) , which contains the number of times
that a rule r is in the position t of the chains in the RSd.
Since RSs do not change, it is not necessary consider it
for computing Fr. The Fr matrix is used in the combina-
tion of the rules chains in the RefSet. Fr is also used in
the diversification phase.
When two rules chains (parents) are combined, a new
one (son) is obtained.
The element in the position t of the son chain is de-
termined according to the frequency that the rule r has
in the position t in the Fr matrix.
cp(t) if cp(t) = cq(t)
cr(t) = cq(t) if Fr(cp(t),t) ≥ Fr(cq(t),t)
cp(t) otherwise
Once all the son chains had been obtained, the PCCR
is used for getting the work overload value for the new
chains. Those with the best values are considered for
updating the RSd. That is, the RSd is regenerated.
Three regeneration alternatives are analyzed in this
paper:
- The RefSet is regenerated with the best chains,
considering both, the parents set and the sons set.
- The RefSet is regenerated with the R best chains in
the sons set.
- The worst αR chains in the RefSet are regenerated
by the αR best chains in the parents set.
In the regeneration process, duplication must be
avoided. Then, all the chains in the RefSet have different
work overload values.
When the regeneration process does not produce im-
provements, the RefSet must be diversified. Diversifica-
tion is done in two steps: 1) creation of diversified
chains, and 2) selection of those diversified chains
which are the least similar to each other.
Step 1 of diversification is done using the informa-
tion contained in the frequency matrix Fr. When diversi-
fication is necessary, the combination of rules is done in
a different way. Given two parent chains p and q, one
diversified son chain is obtained. The difference in the
way chains are combined in the diversification phase is
the following: in position t of the new diversified son
chain, the rule of the parent chain that has the smaller
value in the frequency matrix Fr is assigned. The idea is
to create chains containing rules with inferior frequency
in the Fr, so the space of solutions explored is changed.
The second step of the diversification process itera-
tively looks for diversified chains that are different with
respect to the chains inside the current RSd. The grade
of differentiation between two chains is measured with
the number of coincidences. A coincidence exists if in
the same position t, both of the chains have the same
rule r.
VI. COMPUTATIONAL EXPERIENCE
The proposed procedures are tested with the battery
of problems designed by [6]. In all the instances c=90
time units. Instances do not consider weight (cost) by
incurring in work overload or idle time in stations. For
measuring the quality of the solutions obtained by the
proposed procedures we use the same global index used
by the battery designers.
Originally, instead of weak lower bounds, the objec-
tive function values wo*h of the best known solution for
an instance h is used when comparing procedures. Since
this measure is not defined for wo*h = 0, the following
aggregated relative deviation is used:
(
)
∑∑ ∑ =∗
= = ∗⋅−100
1
100
1
100
1%100:.hh
h h hh wowowoworel (9)
We have added two more indexes for aggregated rela-
tive deviation. The original index (9) is represented by
rel.wo2. In rel.wo1 the value wo*h is the lower bound
obtained by (3). In rel.wo3 the value wo*h is the best
solution founded considering the results obtained with
CPLEX after 15 minutes of search. Computations are
performed in a Pentium 4 CPU 2.4 GHz, 512 MB RAM
under a system Microsoft windows XP professional
2002. CPLEX 7.5 was used in YR-x procedure, and for
searching the optimum during 15 minutes.
Knowledge and Decision Technologies
239
Table I shows the results for the three indexes evalu-
ated for the four constructive procedures described in
section III. The best indexes are reached by the proce-
dure that tries to spread out the work overload along the
sequence. In the other hand, our extension proposal for
Y&R procedure produces not the expected results.
Table I- Global results for initial procedures
Index Ud UdC UdR YRx
rel.wo1 45.64 41.82 40.97 69.25
rel.wo2 5.49 2.73 2.11 27.90
rel.wo3 6.00 3.22 2.59 28.51
#best 19 42 44 2
Cpu 3.99 4.96 4.56 4276
Local search was also applied on the results of con-
structive procedures. Table II shows the value for the
index rel.wo1 after LS. 2S, 3S(a), 3S(b) and 2-3S(b) corre-
spond to Swap neighborhoods. 2-10 Ins are segment
insertion neighborhoods with segment size from 2 until
10. In the search by three elements sweeping, the com-
putational effort is much bigger than the effort required
in 2 elements sweep due to the size of the neighborhood.
Applying 3 swap elements after 2 swap can improve the
solutions.
Table II- Global results after LS, rel.wo1
Rel.wo1 (%)
Initial Procedure
Ud UdC UdR YRx
Original 45.64 41.82 40.97 69.25
2S
3S(a)
3S(b)
2-3S(B)
24.32
35.16
32.99
23.96
24.30
33.03
31.86
23.88
24.53
34.84
30.55
24.09
24.54
55.39
44.15
24.06
2Ins
3Ins
4Ins
5Ins
6Ins
7Ins
8Ins
9Ins
10Ins
23.88
23.51
23.34
23.15
23.01
22.98
22.82
22.72
22.56
23.64
23.35
23.20
23.00
22.95
22.87
22.71
22.61
22.65
23.80
23.34
23.19
23.00
22.99
22.77
22.63
22.60
22.44
24.38
24.22
24.25
24.02
24.03
23.91
23.34
23.35
23.29
The neighborhood size when segment insertion is
used, is much smaller, and requires less computational
effort than swapping. Furthermore, according to global
indexes, work overload solutions are better. Cpu time
limit was imposed in the search of local optimums. The
limit for LS with Swaps was 3600 seconds, except for 2-
3S(b), which had 3600 for the 2Swap search, and 1800
seconds for the 3S(b) search. In segment insertion
search, time limit was 1800 seconds.
Rel.wo2 index is shown in table III. Since better solu-
tions had been founded during LS, the original rel.wo2
value for some initial procedures is different to those
values in table I. In general, better results are obtained
when LS is applied on the UdR initial procedure, and for
our problem, segment insertion neighborhood over-
whelms swapping neighborhoods.
Table III- Global results after LS, rel.wo2
rel.wo2 (%)
Initial Procedure
Ud UdC UdR YRx
Original
20.12 16.97 16.37 40.16
2S
3S(a)
3S(b)
2-3S(B)
2.53
11.47
9.68
2.23
2.51
9.72
8.75
2.17
2.70
11.1
7.67
2.34
2.71
28.16
18.89
2.32
2Ins
3Ins
4Ins
5Ins
6Ins
7Ins
8Ins
9Ins
10Ins
2.17
1.86
1.73
1.57
1.45
1.43
1.29
1.21
1.08
1.97
1.74
1.61
1.45
1.40
1.34
1.21
1.12
1.16
2.11
1.72
1.60
1.44
1.44
1.26
1.14
1.12
0.98
2.58
2.45
2.48
2.28
2.29
2.20
1.73
1.73
1.69
After LS, all results are better than those obtained 15
minutes of search using the optimization software
CPLEX 7.5, then, the index rel.wo3 loss worth.
Table IV- Results for HH1, rel.wo1
rel.wo1 (%)
Regeneration type
Max
Diversif. Reg1 Reg2 Reg3
D3
D4
D5
D6
42.81
42.72
42.64
42.53
42.79
42.74
42.64
42.55
42.54
42.45
42.47
42.33
A computational experience has also been done for
the hiperheuristic proposed procedure. This experience
is not related with the solutions values from previous
experiences. Our analysis considers the combination of
three kind of regeneration used in the procedure and the
maximum number of diversifications (first column of
tables). Table V- Results for HH1, rel.wo3
rel.wo3 (%)
Regeneration type
Max
Diversif.
Reg1 Reg2 Reg3
D3
D4
D5
D6
1.85
1.78
1.73
1.65
1.84
1.80
1.73
1.66
1.66
1.60
1.54
1.51
Table IV shows the global results for the index
rel.wo1. The performance of the three regeneration
methods is similar. As can be expected, when more di-
versifications are applied, results are improved; never-
Knowledge and Decision Technologies
240
theless, this improvement is small. Index rel.wo3, consid-
ers also the CPLEX result after 15 minutes of search.
Work overload solutions are almost as good as the solu-
tions obtained considering CPLEX. In average, 156 sec-
onds are needed by the procedure to finishing the
search.
VII. CONCLUSIONS
This work treats with a variant of the problem of se-
quencing products (mixed models) on a paced assembly
line. We consider the approach in which a product de-
mands a component (attribute), which has different ver-
sions, and requires different processing times in the
application of each. The aim of these procedures is to
minimize work overload (lost work) in all the stations of
the assembly line due to the limited time spared in the
stations and to the work loads along a given sequence.
Both boundaries of stations are closed, and we assume
as in [3], and [6], the displacement time the worker need
to go from one product in to the next, is negligible. We
compiled some procedures founded in literature: [3], [4]
and [9], and we propose greedy procedures inspired on
the previous procedures. Procedures consider more than
two different products and multi stations.
Results obtained for constructive procedures dealing
with the deviation in relation to a bound, are highly
satisfactory, results also show CPU time required for the
greedy proposed procedures is acceptable for big in-
stances, and when work overload is spread our along
the sequence, better results are obtained.
When LS is applied on the constructive procedures
results, index rel.wo1 is improved to almost the half.
Local search with segment insertion outperforms the
elements swap neighborhood, requiring less computa-
tional effort.
The hiperheuristic proposed procedure get good re-
sults which are comparable with those obtained with the
constructive procedures.
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