The Apparent Tardiness Cost (ATC) heuristic is one of the best performing dispatch rules for the weighted tardiness scheduling problem. This heuristic uses a lookahead parameter whose value must be specified. In this paper we develop a function that maps some instance statistics into an appropriate value for the lookahead parameter. This function is compared with some fixed values that have been previously used. The computational results show that the ATC heuristic performs better when the lookahead parameter value is determined by the proposed function.scheduling, weighted tardiness, dispatch rule, instance statistics

Jorge M. S. Valente

English

Research Papers in Economics

1
FACULDADE DE ECONOMIA
UNIVERSIDADE DO PORTO
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Tel . +351 225 571 100 - Fax. +351 225 505 050 - http://www.fep.up.pt
WORKING PAPERS DA FEP
USING INSTANCE STATICTICS TO 
DETERMINE THE LOOKAHEAD 
PARAMETER VALUE IN THE
ATC DISPATCH RULE
Making a good heuristic better
Jorge M. S. Valente
Investigação - Trabalhos em curso - nº 127, Abril de 2003
www.fep.up.ptUsing instance statistics to determine the
lookahead parameter value in the ATC dispatch
rule: Making a good heuristic better
Jorge M. S. Valente
F a c u l d a d ed eE c o n o m i ad oP o r t o
Rua Dr. Roberto Frias, 4200-464 Porto, Portugal
e-mail: jvalente@fep.up.pt
April 25, 2003
Abstract
The Apparent Tardiness Cost (ATC) heuristic is one of the best performing dispatch
rules for the weighted tardiness scheduling problem. This heuristic uses a lookahead pa-
rameter whose value must be speciﬁed. In this paper we develop a function that maps some
instance statistics into an appropriate value for the lookahead parameter. This function
is compared with some ﬁxed values that have been previously used. The computational
results show that the ATC heuristic performs better when the lookahead parameter value
is determined by the proposed function.
Keywords: scheduling, weighted tardiness, dispatch rule, instance statistics
Resumo
A Apparent Tardiness Cost (ATC) é uma das melhores heurísticas de construção para
o problema de sequenciamento que visa minimizar a weighted tardiness.E s t a h e u r í s t i c a
utiliza um parâmetro de pesquisa cujo valor é necessário especiﬁcar. Neste artigo é de-
senvolvida uma função que determina um valor apropriado para este parâmetro com base
em determinadas estatísticas da instância em causa. Esta função é comparada com al-
guns valores ﬁxos habitualmente utilizados. Os resultados computacionais mostram que a
função proposta permite melhorar o desempenho da heurística ATC.
1Palavras-chave: sequenciamento, problema weighted tardiness,h e u r í s t i c ad ec o n -
strução, estatísticas da instância
1 Introduction
In this paper we consider the single machine total weighted tardiness scheduling
problem. This problem can be stated as follows. A set of n independent jobs
{J1,J 2,···,J n} has to be scheduled without preemptions on a single machine that
can handle only one job at a time. The machine and the jobs are assumed to be
continuously available from time zero onwards. Job Jj,j =1 ,2,···,n,r e q u i r e sa
processing time pj and has a due date dj and a positive weight or penalty wj.F o r
any given schedule, the tardiness of Jj can be deﬁned as Tj =m a x {0,C j − dj},
where Cj is the completion time of Jj.T h e o b j e c t i v e i s t h e n t o ﬁnd the schedule
that minimizes the total weighted tardiness
Pn
j=1 wjTj.
Lenstra, Rinnooy Kan and Brucker [5] show that the total weighted tardiness
problem is strongly NP-hard. Several exact approaches used in solving this problem
have been surveyed and tested by Abdul-Razaq, Potts and Van Wassenhove [1].
Hoogeveen and Van de Velde [3] show that better Lagrangean lower bounds can be
obtained by reformulating the problem using slack variables. Akturk and Yildirim
[2] present a new dominance rule that can be used to eliminate nodes in an exact
procedure, as well as to improve both lower and upper bounding schemes. Several
heuristics and dispatch rules have also been proposed. The Apparent Tardiness Cost
(ATC) dispatch rule was developed by Rachamadugu and Morton [7], and several
computational studies ([6], [8]) have shown that it’s one of the best construction
heuristics available for the weighted tardiness problem.
The eﬀectiveness of the ATC heuristic depends on the value of a lookahead pa-
rameter. In previous studies, this parameter is set at a ﬁx e dv a l u e . I nt h i sp a p e r
we propose using certain instance statistics, or factors, to determine an appropri-
ate value for the lookahead parameter. The computational results show that this
approach improves the performance of the ATC heuristic.
The remainder of the paper is organized as follows. In section 2 we describe the
instance factors and the experiments performed to determine the function that maps
those instance statistics into a value for the lookahead parameter. Computational
results are reported in section 3. Finally, some concluding remarks are given in
section 4.
22 A function for determining the lookahead pa-
rameter value
T h eA T Ch e u r i s t i cu s e st h ef o l l o w i n gp r i o r i t yi n d e xIj (t) to determine the job Jj
to be scheduled at any instant t when the machine becomes available:
Ij (t)=
wj
pj
exp
µ
−
[dj − t − pj]
+
kp
¶
,
where p is the average processing time and k is the lookahead parameter. The value
of k should be releated to the number of competing critical and near-critical jobs.
We now describe the factors or statistics that characterize an instance and may
aﬀect the choice of k. The instance size n and variability of the processing times pj
and the penalties wj may inﬂuence the most eﬀective value of k. The remaining two
factors, which are associated with the due dates, are the tardiness factor TF and
the range of due dates RDD.T h ef a c t o rTF can be deﬁned as TF =1−
¡
d/Cmax
¢
,
where d is the average of the due dates and Cmax is the makespan. If the tardiness
factor is high, the average due date will be low, and most jobs will likely be tardy.
Conversely, when TF is low, most jobs should be completed on time. The factor
RDD, which is a measure of the due dates dispersion around their average, is deﬁned
as (dmax − dmin)/Cmax,w h e r edmax and dmin are, respectively, the maximum and the
minimum value of the due dates.
The experiments performed to determine the mapping function were similar to
those used by Lee, Bhaskaran and Pinedo [4] for the weighted tardiness problem
with sequence-dependent setups. A set of problems with 25, 50, 100, 250, 500 and
1000 jobs was randomly generated as follows. For each job Jj an integer processing
time pj and an integer penalty wj were generated from one of the two uniform
distributions [1, 10] and [1, 100], to create low and high variability, respectively.
For each job Jj, an integer due date dj is generated from the uniform distribution
[Cmax (1 − TF − RDD/2),C max (1 − TF + RDD/2)],w h e r eTF was set at 0.0, 0.2,
0.4, 0.6, 0.8 and 1.0, and RDD was set at 0.2, 0.4, 0.6 and 0.8. The values of the
factors involved in the instance generation process are summarized in Table 1. For
each combination of instance size, processing time and penalty variability, TF and
RDD, 20 instances were randomly generated.
An initial test was ﬁrst performed to determine, for each factor combination,
the range where the best values of k were concentrated. A more detailed test was
3Factors Settings
Number of jobs 25, 50, 100, 250, 500, 1000
Processing time and penalty variability [1, 10], [1, 100]
Tardiness factor 0.0, 0.2, 0.4, 0.6, 0.8, 1.0
Range of due dates 0.2, 0.4, 0.6, 0.8
Table 1: Experimental design
then performed on these ranges. In this test, we considered values of the lookahead
parameter ranging from the lower to the upper limit of the range, with 0.2 incre-
ments, and computed the total weighted tardiness for each instance and each value
of the lookahead parameter. For each instance we then identiﬁed all values of the
parameter k which led to objective function values which were less than or equal to
(1 + α)Z,w h e r eZ is the minimum of the computed objective function values and
α>0.T h eα is introduced so that a certain proportion of the lookahead parameter
values that lead to an objective function value near the minimum are considered,
instead of only the value (or values) that lead to the best schedule. The value of
α was selected so that, for each factor combination, an average of three values of
k were selected per instance. The method chosen for determining the value of α
automatically compensates for the diﬀerent variability the objective function value
may exhibit for diﬀerent factor combinations. The average of the lookahead pa-
rameter values selected for each instance is taken as the best estimate of k for that
instance. The best estimate of the parameter k for a given factor combination is
then given by the average of the best estimates for all twenty instances sharing that
factor combination.
From the results of these experiments we could conclude the following. The
processing time and penalty variability does not have a signiﬁcant impact on k.T h e
lookahead parameter is sensitive to the remaining instance factors (instance size, TF
and RDD). The value of k is non-decreasing in the instance size, non-increasing in
RDD and symmetric around TF =0 .5. These factors, however, are interrelated,
since their eﬀect on k depends on the values of the other factors. A simple formula
would then fail to capture the variety of eﬀects and interactions among the factors,
and we therefore decided to develop separate formulas for diﬀerent TF values. This
approach yields a mapping function that is more accurate, even though it’s also
somewhat more cumbersome. In table 2 we present formulas that determine the
value of k as a function of n and RDD for TF values of 0.0, 0.2 and 0.4. Given
the symmetry in the lookahead parameter values, any TF >0.5 is converted to an
4equivalent value of 1 − TF. Linear interpolation is then used to determine k for
other TF values. When the tardiness factor is equal to 0.2 or 0.4, we present the
formulas for selected RDD values. The lookahead parameter is linear (or piecewise
linear) in the RDD factor for these two TF values, but the straight line expression
is unwieldy, so we chose to present these simpler expressions. The experiments also
showed that the best values of k are rarely below 0.6. Therefore, if the function ever
returns a value lower than 0.6, k is set to 0.6 instead.
TF RDD k
0.0 all 0.6
0.2
0.2
≥ 0.4
0.115n0.8
(0.22 − 0.175(RDD − 0.4))ln(n)
0.4
0.2
0.8
0.07n
0.31n0.35
Table 2: Mapping function
3 Computational results
In this section we present the results from the computational tests. The set of test
problems was generated as described in the previous section. The algorithms were
coded in Visual C++ 6.0 and executed on a Pentium IV-1700 personal computer.
Previous studies have indicated that k should be set at a value between 0.5 and
2.0. Therefore, we compared the function-based version of the ATC heuristic with
the ﬁxed values 0.5, 1.0, 1.5 and 2.0. In the following, and for each combination of
instance size and processing time and penalty variability, we only present the results
for the best ﬁxed value (best k). Computational times will not be presented, since
the increased computational eﬀort required by the mapping function could hardly
be noticed, even for the largest instances.
In table 3 we present the average objective function values (mean ofv) and the
relative diﬀerence in average weighted tardiness, calculated as F−K
K ∗ 100,w h e r eF
and K are the average weighted tardiness values of the function and ﬁxed value
versions, respectively. Table 4 gives the number of instances for which the function-
based version performs better (<), equal (=) or worse (>)t h a nt h eﬁxed value
version. We also performed a test to determine if the diﬀerence between the two
versions is statistically signiﬁcant. Given that the heuristics were used on exactly
5t h es a m ep r o b l e m s ,ap a i r e d - s a m p l e st e s ti sa p p r o p r i a t e .S i n c es o m eo ft h eh y p o t h -
esis of the paired-samples t-test were not met, the non-parametric Wilcoxon test
was selected. Table 4 also includes the signiﬁcance (sig.) values of this test, i.e.,
the conﬁdence level values above which the equal distribution hypothesis is to be
rejected. From the results presented in these two tables, we can conclude that the
function-based version outperforms its ﬁxed value counterpart, since the average
objective function value is lower and it provides better results for most of the test
instances. The Wilcoxon test values also indicate that the diﬀerences in distribution
between the two versions is statistically signiﬁcant. The relative improvement over
the ﬁxed value version increases with the instance size and the processing time and
penalty variability.
var. 1-10 var. 1-100
mean ofv mean ofv
n function best k % ofv ch. function best k % ofv ch.
25 1759 1764 -0,27 138001 138119 -0,09
50 6532 6582 -0,75 501234 505127 -0,77
100 25114 25520 -1,59 1895957 1930072 -1,77
250 152441 156304 -2,47 11630510 11986324 -2,97
500 600742 618276 -2,84 46154087 47787819 -3,42
1000 2382689 2454444 -2,92 182334086 189148723 -3,60
Table 3: Average weighted tardiness and relative diﬀerence
var. 1-10 var. 1-100
function vs best k function vs best k
n < = > sig. < = > sig.
25 163 263 54 0,000 187 215 78 0,000
50 250 152 78 0,000 241 149 90 0,000
100 300 130 50 0,000 310 131 39 0,000
250 304 137 39 0,000 302 142 36 0,000
500 302 133 45 0,000 312 131 37 0,000
1000 305 135 40 0,000 307 130 43 0,000
Table 4: Comparison of heuristic average weighted tardiness values and statistical
test
In table 5 we present the eﬀect of TF and RDD on the relative diﬀerence in
average weighted tardiness for instances with 100 jobs and low processing time and
penalty variability. The results for other instances are similar to those presented.
Both versions of the heuristic generate a schedule with no tardy jobs for all instances
6with TF =0 .0 or TF =0 .2 and RDD =0 .6;0.8. The weighted tardiness problem
is easier when TF is equal to 0.0 (1.0), since most jobs will be on time (tardy).
For these instances, there is usually little room for improvement, as can be seen
from the results in table 5. As the tardiness factor comes closer to 0.5, the problem
becomes harder, and the improvement provided by the function version increases
signiﬁcantly, particularly for a range of due dates of 0.2 or 0.4.
RDD
TF 0.2 0.4 0.6 0.8
0.0 – – – –
0.2 -11,04 -29,11 – –
0.4 -25,95 -11,30 -1,24 -5,91
0.6 -15,34 -10,12 -3,94 0,22
0.8 -2,13 0,05 -0,45 -0,29
1.0 -0,10 -0,11 -0,14 -0,12
T a b l e5 : R e l a t i v ed i ﬀerence in average weighted tardiness for instances with 100
jobs and low processing time and penalty variability
We also compared the function and ﬁxed value versions on instances with TF
and RDD values of 0.1, 0.3, 0.5, 0.7 and 0.9. The results for these instances are
presented in tables 6 and 7. The function-based heuristic once again outperforms
the ﬁxed value approach, and the relative improvement is even higher for this second
set of instances. These results conﬁrm the validity of the proposed mapping function
(and the linear behaviour it assumes between TF values), and illustrate its improved
performance over a wide range of test instances.
var. 1-10 var. 1-100
mean ofv mean ofv
n function best k % ofv ch. function best k % ofv ch.
25 1456 1460 -0,31 106896 107154 -0,24
50 5187 5230 -0,82 398953 401601 -0,66
100 19897 20422 -2,57 1426281 1464787 -2,63
250 119601 124866 -4,22 8994431 9470246 -5,02
500 470984 497043 -5,24 35676414 38045419 -6,23
1000 1880019 1992808 -5,66 140870384 151089892 -6,76
Table 6: Average weighted tardiness and relative diﬀerence (second set of instances)
7var. 1-10 var. 1-100
function vs best k function vs best k
n < = > sig. < = > sig.
25 209 209 82 0,000 196 221 83 0,000
50 240 165 95 0,000 233 161 106 0,000
100 299 129 72 0,000 294 123 83 0,000
250 303 120 77 0,000 311 119 70 0,000
500 309 120 71 0,000 312 120 68 0,000
1000 314 120 66 0,000 317 120 63 0,000
Table 7: Comparison of heuristic average weighted tardiness values and statistical
test (second set of instances)
4C o n c l u s i o n
The ATC heuristic is one of the best dispatch rules available for the weighted tar-
diness scheduling problem, as several studies have shown. This heuristic uses a
lookahead parameter whose value must be speciﬁed. Previously, this parameter
had been set at a ﬁxed value. We developed a function that maps some instance
statistics into an appropriate value for the lookahead parameter. This approach was
compared with some ﬁxed values that have been recommended. The computational
results showed that using the mapping function improves the performance of the
heuristic over a wide range of instances.
References
[1] Abdul-Razaq, T. S., Potts, C. N., and Van Wassenhove, L. N. A
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[3] Hoogeveen, J. A., and Van de Velde, S. L. Stronger lagrangian bounds
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8[4] Lee, Y. H., Bhaskaran, K., and Pinedo, M. A heuristic to minimize the
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[5] Lenstra, J. K., Rinnooy Kan, A. H. G., and Brucker, P. Complexity of
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9

A new lower bounding scheme for the total weighted tardiness problem.

A survey of algorithms for the single machine total weighted tardiness scheduling problem.

Complexity of machine scheduling problems.

Improved heuristics for the n-job singlemachine weighted tardiness problem.

Myopic heuristics for the single machine weighted tardiness problem.

Single machine tardiness sequencing heuristics.

Stronger lagrangian bounds by use of slack variables: Applications to machine scheduling problems.

Using Instance Statistics to Determine the Lookahead Parameter Value in the ATC Dispatch Rule: Making a good heuristic better

http://www.fep.up.pt/investigacao/workingpapers/wp127.pdf

Using Instance Statistics to Determine the Lookahead Parameter Value in the ATC Dispatch Rule: Making a good heuristic better

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