Priority Rules for the Single Machine Total Weighted ...onlinepresent.org/proceedings/vol141_2016/7.pdf · Tardiness Scheduling with Maximum Allowable Tardiness ... Single machine

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


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:



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.



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



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.



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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Priority Rules for the Single Machine Total Weighted ...onlinepresent.org/proceedings/vol141_2016/7.pdf · Tardiness Scheduling with Maximum Allowable Tardiness ... Single machine