Priority Rules for the Single Machine Total Weighted
Tardiness Scheduling with Maximum Allowable
Tardiness
Jae-Gon Kim1, June-Young Bang2†, Seung-Kil Lim2, Joung-Yun Lee1
1 Dept. of Industrial & Management Engineering, Incheon National University
119 Aacdemy-ro Yeonsu-gu, Incheon, 406-772, Korea 2 Dept. of Industrial and Management Engineering, Sungkyul University
53 Sungkyuldaehak-ro Manan-gu, Anyang-city, Gyeonggi-do 430-742, Korea † Corresponding author ([email protected])
Abstract. In many manufacturing or service industries, there exists maximum
allowable tardiness for orders. Customers cancel their orders when delivery
time exceeds the maximum allowable tardiness whereas they allow delayed
delivery of orders within the maximum allowable tardiness. In this study, we
consider the single machine total weighted tardiness scheduling problem with
maximum allowable tardiness. Two kinds of penalty costs are considered, i.e.
one for tardy jobs and another for canceled jobs. We extend well-known
priority rules for the single machine total tardiness scheduling to solve the
considered problem. Computational experiments on 270 test instances show
that the suggested priority rules work much better than existing ones.
Keywords: Single machine total weighted tardiness scheduling, Maximum
allowable tardiness, Priority rules, Algorithm
1 Introduction
In many manufacturing or service industries, there exists maximum allowable
tardiness for orders. Here, the maximum allowable tardiness is the delayed time over
the due date within which customer can allow delayed delivery. That is, customers
cancel their orders only when the delayed time over the due date will exceed the
maximum allowable tardiness. In this study, we consider the single machine total
weighted tardiness scheduling problem with maximum allowable tardiness
(SMTWTSP-MAT). We study the SMTWTSP-MAT with the objective of
minimizing total penalty cost where two kinds of penalty costs are considered, i.e.
tardiness penalty costs for allowable tardy jobs and those for cancelled jobs, which
are called lost-sale penalty costs.
There is a limited literature which considers maximum allowable tardiness in
machine scheduling problems. Smith (1956) and Chand and Schneeberger (1986)
considered minimization of weighted completion time subject to the constraint that
the tardiness for any job does not exceed a pre-specified maximum allowable
Advanced Science and Technology Letters Vol.141 (GST 2016), pp.31-38
http://dx.doi.org/10.14257/astl.2016.141.07
ISSN: 2287-1233 ASTL Copyright © 2016 SERSC
tardiness. Seo et al. (2001) focused on minimizing mean squared deviation of
completion times with maximum tardiness constraint. Mönch et al. (2006) studied the
single burn-in oven scheduling problem with the objective of minimizing the sum of
the absolute deviations of completion times from the due date of all jobs under the
constraint that the maximum tardiness should be less than or equal to the maximum
allowable time value. Mzadeh et al. (2010) proposed tabu search algorithms for the
single machine total weighted tardiness scheduling problem, where each job has two
different due-dates, i.e. ordinary due-date and drop dead date. Recently, Koulamas
and Panwalkar (2015) suggested the optimal algorithms for single-machine
scheduling problems with earliness criteria and job rejection. The SMTWTSP-MAT
considered in this study is more realistic than that of Mzadeh et al. (2010) in that lost-
sale penalty costs of jobs are not dependent on the job’s completion time but fixed to
given values once they occur.
2 Problem Description
We use the following notation throughout the paper:
i index for jobs (1, …, n)
id due date of job i
id cancellation deadline for job i
i tardiness of job i
i maximum allowable tardiness of job i , i.e., iii dd .
ip processing time of job i
i tardiness penalty cost per unit time for job i
i lost-sale penalty cost for job i
The objective function of the problem is to minimize the total weighted penalty
cost of all orders. The cost can be calculated with tardiness penalty costs for allowable
tardy jobs and lost-sale penalty costs for cancelled jobs, as depicted in Figure 1.
Fig. 1. The penalty cost function
Advanced Science and Technology Letters Vol.141 (GST 2016)
32 Copyright © 2016 SERSC
3 Priority Rules
In many industries, operation managers require practical tools that can solve large-
sized SMWTTSP-MAT quickly, rather than time-consuming optimization tools. In
this paper, we suggest several priority rules for the SMWTTSP-MAT, which can be
easily applied to practice. Various priority rules exist for the SMTTSP (Koulamas
2010, Vepsalainen and Morton 1987) and five priority rules (EDD, MDD, SLACK,
COVERT, and ATC) among them are most widely used in practice and are known to
perform better than others in terms of the total tardiness (Moon and Christy 1998,
Chiang and Fu 2007). In this section, we extend these five priority rules to those for
the SMWTTSP-MAT by considering the maximum allowable tardiness.
DOP rule: In the EDD (Earliest Due Date) rule, jobs with earlier due dates have
higher priorities and are processed before those with later due dates. The EDD rule is
represented by EDD: )(min ii
d . For the SMWTTSP-MAT, we develop a new rule by
considering penalty costs for tardy and cancelled jobs as well as the due date and
maximum allowable tardiness. The new rule is called DOP (Deadline Over Penalty)
rule and represented by DOP: )/ ,/min(min iiiii
dd , where iii dd . Note that
id and id can be considered as the original deadline and the cancellation deadline
for job i, respectively. In the DOP rule, iid / and iid / represent deadline over
penalty for job i in terms of the original and cancellation deadline, respectively. Jobs
with smaller deadline over penalties have higher priorities and are processed before
those with larger deadline over penalties in the DOP rule.
MDOP rule: The MDD rule was originally suggested by Baker and Bertrand (1982)
for the SMTTSP. In the rule, the modified due date of a job is defined as the larger
value of the due date of the job and the earliest possible completion time of the job.
The MDD rule selects the job with the least modified due date among the unselected
jobs to be processed next. The modified due dates of jobs are updated each time a new
job is selected because the earliest possible completion times of the unselected jobs
depend on those of selected ones. In this regard, the MDD rule is called the dynamic
priority rule. The MDD rule is represented by MDD: ) ,max(min iii
ptd .
We develop a new rule, called MDOP (Modified Deadline Over Penalty) rule, for
the SMTTSP-MAT by extending the MDD rule. In the MDOP rule, modified deadline
over penalty of job i, denoted by iMDOP , is defined as below. Here, M is a big
constant.
if
if /
if /
i
i
ii
iiii
iii
i
dptM
dptdd
dptd
MDOP
(1)
Note that ipt is the earliest possible completion time of job i at time t in equation
Advanced Science and Technology Letters Vol.141 (GST 2016)
Copyright © 2016 SERSC 33
(1). The MDOP can be represented by MDOP: )(min ii
MDOP
, where is a set of
unselected jobs. Jobs with smaller modified deadline over penalties have higher
priorities and are processed before those with larger modified deadline over penalties
in the MDOP rule.
MSOP rule: In the SLACK rule, the job with the least slack time has the highest
priority. The slack time of a job is defined as the maximum available time to delay the
completion of the job without violating its due date. The job slack time is computed
as the difference between the job due-date and the earliest possible completion time
for the job. As with the MDD rule, the SLACK rule is also a dynamic priority rule
because the job slack time is not static but should be updated each time a new job is
selected for processing. The SLACK rule is represented by SLACK: )(min iii
ptd
where is a set of unselected jobs.
We develop a new rule, called MSOP (Modified Slack Over Penalty) rule, for the
SMTTSP-MAT by extending the SLACK rule. In the MSOP rule, modified slack
over penalty of job i, denoted by iMSOP , is defined as below.
if
if /)(
if /)(
ii
iiiiii
iiiii
i
dptM
dptdptd
dptptd
MSOP
(2)
The MSOP rule can be represented by MSOP: )(min ii
MSOP
. Jobs with smaller
modified slack over penalties have higher priorities and are processed before those
with larger modified slack over penalties in the MSOP rule.
C-MAT rule: The original COVERT priority index represents the expected tardiness
cost per unit of imminent processing time, or cost over time (Vepsalainen and
Morton, 1987). The COVERT rule is a ratio-based priority rule which combines the
SLACK rule and the SPT (Shortest Processing Time) rule. It puts the job with the
largest COVERT ratio in the first position of the job sequence.
The COVERT ratio of a job is computed by dividing a derived urgency ratio of the
job by the pseudo-processing time of the job. The COVERT rule is represented by
COVERT:
i
ii
i kp
ptd
p
) ,0max(1 ,0max
1max , where k is a parameter. The value
of k is usually determined through experimental analysis. The COVERT rule is also a
dynamic priority rule. It is known that it performs well on the due date-based
objectives, especially on the total tardiness measure for the SMTTSP (Russell et al.,
1987).
We develop a new rule, called C-MAT (COVERT-MAT), for the SMTTSP-MAT
by extending the COVERT rule. In the C-MAT rule, a modified covert ratio of job i,
denoted by iMCR , is defined as below:
Advanced Science and Technology Letters Vol.141 (GST 2016)
34 Copyright © 2016 SERSC
ii
iii
i
ii
i
i
ii
i
ii
i
i
i
dptM
dptdpk
ptd
p
dptpk
ptd
p
MCR
if
if 1 0,max
if 1 0,max
(3)
The C-MAT rule can be represented by C-MAT: )(max ii
MCR
where is a set
of unselected jobs. Jobs with larger modified covert ratios have higher priorities and
are processed before those with smaller modified covert ratios in the C-MAT rule.
A-MAT rule: The ATC rule was developed based on the COVERT rule for the
SMTTSP (Vepsalainen and Morton, 1987). The basic concept of the ATC rule is the
same with the COVERT rule with two main differences. First, the ATC rule uses an
exponential function rather than linear one to emphasize the part of slack. Second, the
ratio is calculated by dividing the job slack by the average processing time, instead of
the job processing time. The ATC rule is represented by ATC:
pk
ptd
p
ii
i
) ,0max(exp
1max , where k is a parameter and
i
ipp where
is a set of unselected jobs.
We develop a new rule, called A-MAT (ATC-MAT), for the SMTTSP-MAT by
extending the ATC rule. In the A-MAT rule, a modified ATC ratio of job i, denoted
by iMAR , is defined as below:
ii
iiiii
i
i
iiii
i
i
i
dptM
dptdpk
ptd
p
dptpk
ptd
p
MAR
if
if ) ,0max(
exp
if ) ,0max(
exp
(4)
The A-MAT rule can be represented by A-MAT: )(max ii
MAR
. Jobs with larger
modified ATC ratios have higher priorities and are processed before those with
smaller modified ATC ratios in the A-MAT rule.
Note that the DOP rule is a static rule, and the rest rules are dynamic ones. In the
static rules, the order priority index values do not change over time, whereas they
might change over time in the dynamic ones.
Advanced Science and Technology Letters Vol.141 (GST 2016)
Copyright © 2016 SERSC 35
4 Computational Experiments
To test performances of the suggested priority rules, we randomly generated 270 test
instances varying the numbers of jobs, due-date tightness and range of allowable
tardiness. We considered 27 combinations: three levels for the number of jobs, three
levels for the due-date tightness, and three levels for the range of allowable tardiness,
and generated 10 problem instances for each combination to obtain the 270 test
instances. In each problem instance, relevant data were generated as follows. Here,
DU(a, b) and U(a, b) denote random numbers generated from the discrete and
continuous uniform distributions with a range [a, b], respectively.
1) The number of jobs is 10, 20 or 30.
2) Job processing time was set to DU(1,10).
3) Due dates of jobs were set to [λ ∙ DU(1,∑𝑝𝑖)] for each job, where λ is a
due-date tightness factor which is set to 0.5, 1.0 and 1.5, and [●] is the
closest integer to ●. Lager λ generates tighter due-dates.
4) Maximum allowable tardiness of job i, 𝜏�̅�, was set to [𝜇 ∙ 𝑑𝑖], where 𝜇 is a
factor for controlling the range of allowable tardiness which was set to 0.5,
1.0 and 1.5. Lager 𝜇 generates bigger allowable tardiness.
5) Tardiness penalty cost of job i, 𝛼𝑖, was set to U(1.0, 3.0)
6) Lost-sale penalty cost of job i, 𝛽𝑖 = 3.0 ∙ 𝛼𝑖 ∙ 𝜏�̅�,.
The five new priority rules developed in this study were compared with
corresponding original priority rules for SMTTSP (i.e., EDD, MDD, SLACK,
COVERT, and ATC rule) and CPLEX 12.1, the commercial optimization solver. In
the COVERT and C-MAT rules, the parameter k was set to 4, 8 and 12 for 10-, 20-
and 30-job test instances, respectively, and it was set to 2, 4 and 6 in the ATC and A-
MAT, after experimental analysis through preliminary test.
Fig. 2. The total number of the best solutions found by each rule
Advanced Science and Technology Letters Vol.141 (GST 2016)
36 Copyright © 2016 SERSC
Figure 2 shows the average number of best solutions found by each rule for each
job size. Result of the test with 10 jobs shows that there are no outstanding rules for
finding best solutions with comparison to other rules except CPLEX. CPLEX found
optimal solutions for all the 10-job test instances within 2~28 sec, while all the
priority rules found the best solutions for only less than the half of 90 instances.
Except for CPLEX, slack-based rules (such as A-MAT, MSOP, and ATC) found
more number of best solutions than due date based rules (such as EDD and DOP).
(For the large-sized problems in which the order sizes are 20 and 30, C-MAT and
MDOP give the best performance for the number of the best solutions found. For
these problems, CPLEX found no optimal solutions within 1,000 sec, and the
performance rank is behind C-MAT and MDOP. The DOP rule gives worse
performance as the EDD rule does not give good performance for the SMTTSP. As
shown in Figure 2, the suggested rules outperformed the existing rules in terms of the
total number of best solutions found (NBF) regardless of the tightness of due date or
allowable tardiness.
Fig. 3. The average relative deviation index of each rule
The relative deviation index (RDI) of solutions obtained by each rules summarized
in Figure 3 shows the average relative deviation index (ARDI) of solutions obtained
by each rule. Note that the range of RDI is between 0 and 1, where 0 and 1 denote the
best and the worst solutions, respectively. The suggested rules in which the deadline
for the lost sales is considered consistently outperformed the existing rules. In terms
of the solution quality, the suggested rules were much better than their corresponding
existing rules. Among the suggested rules, C-MAT and MDOP were best and A-MAT
follows them. The outperformance of the suggested rules becomes more evident when
the order size becomes larger as shown in Figure 3 regardless of the tightness of due-
date or allowable tardiness.
Advanced Science and Technology Letters Vol.141 (GST 2016)
Copyright © 2016 SERSC 37
5 Conclusion
In this study, we suggested five priority rules for the SMWTTSP-MAT by extending
corresponding priority rules for the SMTTSP. The extended priority rules performed
much better than original ones and the commercial optimizing solver, CPLEX, in the
computation experiments. Among the suggested rules, C-MAT and MDOP rules were
best, and A-MAT also worked well, and slack-based rules gives better performance
for SMWTTSP-MAT than due date-based rules (such as MDD) which give generally
good performances for SMTTSP. Due to the simplicity and fastness of suggested
priority rules, they can be easily used in practice and in simulation models of
production scheduling.
Acknowledgements: This work was supported by the Incheon National University
(International Cooperative) Research Grant in 2016.
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38 Copyright © 2016 SERSC