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European Journal of Operational Research 191 (2008) 587–592

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Short Communication

Single-machine scheduling against due dates withpast-sequence-dependent setup times

Dirk Biskup *, Jan Herrmann

Bielefeld University, Department of Business Administration and Economics, P.O. Box 10 01 31, D-33501 Bielefeld, Germany

Received 21 February 2007; accepted 23 August 2007Available online 5 September 2007

Abstract

Recently Koulamas and Kyparisis [Koulamas, C., Kyparisis, G.J., in press. Single-machine scheduling with past-sequence-dependent setup times. European Journal of Operational Research] introduced past-sequence-dependent setuptimes to scheduling problems. This means that the setup time of a job is proportionate to the sum of processing timesof the jobs already scheduled. Koulamas and Kyparisis [Koulamas, C., Kyparisis, G.J., in press. Single-machine schedul-ing with past-sequence-dependent setup times. European Journal of Operational Research] were able to show for a numberof single-machine scheduling problems with completion time goals that they remain polynomially solvable. In this paperwe extend the analysis to problems with due dates. We demonstrated that some problems remain polynomially solvable.However, for some other problems well-known polynomially solution approaches do not guarantee optimality any longer.Consequently we concentrated on finding polynomially solvable special cases.� 2007 Elsevier B.V. All rights reserved.

Keywords: Scheduling; Setup times; Due dates

1. Introduction

Recently Koulamas and Kyparisis (in press) introduced the concept of past-sequence-dependent setuptimes. Past-sequence-dependent (psd) setup times occur, for example, in high tech manufacturing environ-ments, ‘‘in which a batch of jobs consists of a group of electronic components mounted together on an inte-grated circuit (IC) board’’ (Koulamas and Kyparisis, in press). In addition to this un-readiness of components,the wear-out of equipment (e.g. a drill) is another example where the sum of processing times of the prior jobsadds to the processing time of the actual job. Koulamas and Kyparisis (in press) concentrated on single-machine scheduling problems with completion time goals, namely minimizing maximum completion time,total completion time, total absolute differences in completion times, and a bi-criteria objective functionconsisting of the two last mentioned goals. They were able to show that all these problems remain polynomi-ally solvable when past-sequence-dependent setup times are included in the analysis. This note concentrates on

0377-2217/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2007.08.028

* Corresponding author.E-mail addresses: dbiskup@wiwi.uni-bielefeld.de (D. Biskup), jherrmann@wiwi.uni-bielefeld.de (J. Herrmann).

588 D. Biskup, J. Herrmann / European Journal of Operational Research 191 (2008) 587–592

single-machine scheduling problems with different due date based goals while past-sequence-dependent setuptimes are considered. After introducing the notation needed in the next section we will analyze problems withindividual due dates in Section 3. In Section 4 problems with common due dates are tackled. The paper con-cludes with some summarizing remarks in Section 5.

2. Notation and model formulation

There are n jobs available at time zero which have to be processed on a single machine. Preemption is notallowed and the machine is only able to process one job at a time. Let pi denote the processing time of the job i,i = 1, . . . , n. The setup time of a job, si, depends on the total length of the jobs already scheduled

s½i� ¼ cXi�1

r¼1

p½r�;

where [i] denotes the ith position in the sequence and c is a non-negative constant. For all jobs a due date di isgiven. Furthermore Ci, Ei = max{0, di � Ci}, Ti = max{0, Ci � di} and Li = Ci � di are the completion time,earliness, tardiness and lateness of job i, i = 1, . . . , n, respectively. A schedule, p, contains all the informationnecessary for the manufacturing of the jobs, that is, the sequence of the jobs and their starting times. Through-out the paper we will make use of the well-known three-field notation for scheduling problems introduced byGraham et al. (1979); past-sequence-dependent setup times will be abbreviated by psd.

3. Problems with individual due dates

In this section we will analyze different single-machine problems with individual due dates and past-sequence-dependent setup times. An excellent introduction to single-machine problems with individual duedates is given by Baker (1997).

3.1. Total lateness

The total lateness L can be calculated by

L ¼Xn

r¼1

C½r� � d ½r�� �

¼Xn

r¼1

Xr

i¼1

ðcðr � iÞ þ 1Þp½i� � d ½r�

" #:

Property 1. The shortest processing time (SPT) sequence leads to an optimal solution for 1/psd/P

Li.

Proof. Total lateness L is minimized if F ¼Pn

r¼1

Pri¼1ðcðr � iÞ þ 1Þp½i�

h iis minimized as

Pnr¼1d ½r� is a constant.

Rearranging F to F ¼Pn

r¼1 ðn� r þ 1Þ þ c=2ððn� rÞðn� r þ 1ÞÞð Þp½r� makes it obvious that it is the scalarproduct of two vectors, the wr = (n � r + 1) + c/2((n � r)(n � r + 1)) and p[r] vectors, respectively,(r = 1, . . . , n), (see Koulamas and Kyparisis, in press). As wr P wr+1, the SPT sequence leads to an optimalsolution (Hardy et al., 1967, p. 261). h

3.2. Total tardiness

The single-machine total tardiness problem, 1//P

Ti, is NP-hard (Du and Leung, 1990). Hence 1/psd/P

Ti

must be NP-hard, too. The total tardiness T can be calculated by

T ¼Xn

r¼1

max ½C½r� � d ½r��; 0� �

¼Xn

r¼1

maxXr

i¼1

ðcðr � iÞ þ 1Þp½i� � d ½r�

" #; 0

( ):

Property 2. For any pair of jobs, suppose that processing times and due dates are agreeable, i.e. pi 6 pk implies

di 6 dk. Then 1/psd/P

Ti is minimized by the SPT or, equivalently, by the earliest due date (EDD) sequence.

D. Biskup, J. Herrmann / European Journal of Operational Research 191 (2008) 587–592 589

Proof. The proof follows directly from the pairwise interchange analysis. Consider a schedule p, in which jobsi and k are adjacent in sequence at the positions x and x + 1, and the schedule p 0 that is identical to p exceptthat jobs i and k are interchanged. Let us assume that pi P pk and thus di P dk. For the comparison of p andp 0 it will suffice to compare the total tardiness that stems from the jobs i and k. The tardiness of the first x � 1jobs is obviously not affected by the interchange of job i and k. Let t = C[x�1] be the completion time of the(x � 1)th jobs. And as makespan is minimized by the SPT sequence (see Koulamas and Kyparisis, in press),the maximum tardiness among the last (n � x + 1) jobs in schedule p 0 cannot be higher than that of schedulep. The tardiness of the jobs i and k in schedule p, Ti,k(p), is calculated as the tardiness of job i, Ti(p) plus thetardiness of job k, Tk(p)

T i;kðpÞ ¼ T iðpÞ þ T kðpÞ ¼max tþXx�1

r¼1

cp½r� þ pi� di;0

( )þ max tþ 2

Xx�1

r¼1

cp½r� þ ð1þ cÞpiþ pk � dk;0

( ):

The calculation of the tardiness of the jobs k and i in schedule p 0, Tk,i (p 0), is very similar

T k;iðp0Þ ¼ T kðp0Þ þ T iðp0Þ ¼max tþXx�1

r¼1

cp½r� þ pk � dk;0

( )þ max tþ 2

Xx�1

r¼1

cp½r� þ ð1þ cÞpk þ pi� di;0

( ):

To compare p and p 0 we consider two cases. The first case is t þPx�1

r¼1cp½r� þ pi 6 di. Thus,

T i;kðpÞ ¼ max t þ 2Xx�1

r¼1

cp½r� þ ð1þ cÞpi þ pk � dk; 0

( );

T k;iðp0Þ ¼ max t þXx�1

r¼1

cp½r� þ pk � dk; 0

( )þmax t þ 2

Xx�1

r¼1

cp½r� þ ð1þ cÞpk þ pi � di; 0

( ):

Let us assume that neither term in Tk,i (p 0) is zero. Then

T i;kðpÞ � T k;iðp0Þ ¼ cðpi � pkÞ þ di � t þXx�1

r¼1

cp½r� þ pk

!P cðpi � pkÞ þ di � t þ

Xx�1

r¼1

cp½r� þ pi

!P 0:

Note that for the sake of comparing Ti,k (p) and Tk,i (p 0) the assumption that neither term in Tk,i (p 0) is zero isthe most restrictive one. It comprises the case that either one or both terms of Tk,i (p 0) are zero. Therefore, thefirst case yields Ti,k (p) P Tk,i (p 0). The second case consequently is di < t þ

Px�1r¼1cp½r� þ pi

T i;kðpÞ ¼ 2t þ 3Xx�1

r¼1

cp½r� þ ð2þ cÞpi � di þ pk � dk;

T k;iðp0Þ ¼ max t þXx�1

r¼1

cp½r� þ pk � dk; 0

( )þ t þ 2

Xx�1

r¼1

cp½r� þ ð1þ cÞpk þ pi � di;

T i;kðpÞ � T k;iðp0Þ ¼ t þXx�1

r¼1

cp½r� þ ð1þ cÞpi � cpk � dk �max t þXx�1

r¼1

cp½r� þ pk � dk; 0

( ):

Let us assume the maximum in the last term is positive, then

T i;kðpÞ � T k;iðp0Þ ¼ t þXx�1

r¼1

cp½r� þ ð1þ cÞpi � cpk � dk � t �Xx�1

r¼1

cp½r� � pk þ dk ¼ ð1þ cÞðpi � pkÞP 0:

Therefore, the second case yields Ti,k (p) P Tk,i (p 0), too. This completes the proof. h

3.3. Maximum lateness and maximum tardiness

Both, the single-machine maximum lateness problem, 1//Lmax and the single-machine maximum tardinessproblem, 1//Tmax are minimized by the EDD sequence (see Smith, 1956 or Baker, 1997). The following

590 D. Biskup, J. Herrmann / European Journal of Operational Research 191 (2008) 587–592

example shows that the EDD rule does not necessarily lead to an optimal solution for the single-machine max-imum lateness problem with past-sequence-dependent setup times.

Example 1. Let p1 = 100, p2 = 1, d1 = 101, d2 = 102, and c = 1. Then the EDD sequence (S = (1, 2)) leads toLmax = 99 and the sequence S = (2, 1) leads to Lmax = 1.

Note that the example holds for 1/psd/Lmax as well as 1/psd/Tmax. However, the following property can beestablished:

Property 3. The EDD sequence leads to an optimal schedule for 1/psd/Lmax and 1/psd/Tmax if the due dates and

the processing times are agreeable, i.e. if pi 6 pk implies di 6 dk.

Proof. Let p be a schedule in which the jobs i and k are sequenced on the positions x and x + 1, respectively,and let di > dk. We will create a new schedule p 0 by interchanging the jobs i and k. Hence the lateness of jobs i

and k in schedule p can be calculated as

LiðpÞ ¼ t þXx�1

r¼1

cp½r� þ pi � di;

LkðpÞ ¼ t þ 2Xx�1

r¼1

cp½r� þ ð1þ cÞpi þ pk � dk:

Similarly, the lateness of jobs k and i in schedule p 0 can be calculated as

Lkðp0Þ ¼ t þXx�1

r¼1

cp½r� þ pk � dk;

Liðp0Þ ¼ t þ 2Xx�1

r¼1

cp½r� þ ð1þ cÞpk þ pi � di:

Interchanging the jobs i and k has no impact on the maximum lateness among the first (x � 1) jobs. As make-span is minimized by the SPT sequence (see Koulamas and Kyparisis, in press), the maximum lateness amongthe last (n � x + 1) jobs in schedule p 0 cannot be higher than that of schedule p. To prove the property it isthus sufficient to show that Lk(p) P max{Lk(p 0), Li(p 0)}.

LkðpÞP Lkðp0Þ ()Xx�1

r¼1

cp½r� þ ð1þ cÞpi P 0;

LkðpÞP Liðp0Þ () cðpi � pkÞP dk � di:

As di > dk implies pi P pk, schedule p 0 is not inferior to schedule p.Maximum tardiness Tmax is defined as Tmax = max{0, Lmax}. The results of Property 3 can be transferred

directly to the case of 1/psd/Tmax. This completes the proof. h

3.4. Number of tardy jobs

The number of tardy jobs on a single machine, 1//P

Ui, can be minimized by the polynomially boundedalgorithm presented by Moore (1968). If past-sequence-dependent setup times are included into the analysis,Moore’s algorithm does not necessarily deliver optimal results anymore, see the following example:

Example 2. Let p1 = 1, p2 = 2, p3 = 3, with d1 = 12, d2 = 11, d2 = 10 and c = 1. Moore’s algorithm starts withthe EDD sequence (3,2,1) and, because job 1 is late, moves job 3 to the set of tardy jobs. As within (2, 1,3) thejobs 2 and 1 are not late, this is the final sequence with job 3 being tardy. However, the SPT sequence (1,2,3)leads to an optimal schedule without tardy jobs.

4. Common due date problems

In this section we tackle common due date problems with past-sequence-dependent setup times. An excel-lent introduction to common due date problems is given by Baker and Scudder (1990), for a literature review

D. Biskup, J. Herrmann / European Journal of Operational Research 191 (2008) 587–592 591

(see Gordon et al., 2002). For all problems discussed in this section a non-restrictive common due date d isgiven. A common due date is non-restrictive if it is either a decision variable or if it is sufficiently large; withsufficiently large we mean that d P C[n] holds.

We first consider the famous problem of Kanet (1981): 1=psd; di ¼ d=PðEi þ T iÞ. Let b denote the number

of jobs that are completed before or in the due date and let a denote the number of jobs that are finished afterthe due date, a + b = n. For the objective function the following holds:

f ðpÞ ¼Xn

i¼1

ðEi þ T iÞ ¼Xb

r¼1

werp½r� þ

Xn

r¼bþ1

wtrp½r�;

with wer ¼ ðr � 1þ ðn� rÞcÞ and wt

r ¼ ðn� r þ 1þ ðn� rÞcÞ.

Property 4. If c 6 1, for 1=psd; di ¼ d=PðEi þ T iÞ an optimal schedule exists that is V-shaped. V-shaped means

that the first b jobs are scheduled in non-increasing order of their processing times and that the last a jobs are

scheduled in non-decreasing order of their processing times.

Proof. For any b 2 {1, . . . , n} the following holds: wer 6 we

rþ1; r ¼ 1; . . . ; b� 1 and wtr P wt

rþ1; r ¼ bþ 1; . . . ;n� 1 if c 6 1. An optimal schedule can be constructed by assigning the longest job to the position with thesmallest weight, the second longest job to the position with the second smallest weight (see Hardy et al.,1967, p. 261). This leads to a V-shaped schedule as defined above. h

Property 5. If c 6 1, for 1=psd; di ¼ d=PðEi þ T iÞ an optimal schedule exists where the bth job is completed in

the due date with b = max{1 6 r 6 njr � 1 + (n � r)c 6 n � r + 1 + (n � r)c}.

Proof. Obvious from the objective function and Property 4.

With Property 4 and 5 an optimal solution to 1=psd; di ¼ d=PðEi þ T iÞ with c 6 1 can be constructed as

follows: The number of non-tardy jobs b is determined and the weights for the positions are calculated. Thenthe longest job is assigned to the position with the smallest weight, the second longest job is assigned to theposition with the second smallest weight, etc. Ties can be broken arbitrarily. This matching procedure requiresO(n logn) time. After constructing the solution as described, either d = C[b] (if d is a decision variable) or thestart time of the first job is adjusted to s½1� ¼ d �

Pbi¼1ðs½i� þ p½i�Þ (if a sufficiently large d is given).

With c > 1 the analysis becomes slightly more complicated as the problem is no longer V-shaped. Nowwe

r > werþ1 and wt

r > wtrþ1 holds for non-tardy as well as for tardy jobs. This means that b cannot be determined

analytically by Property 5 but depends on the actual value of c. However, for a given b all weights can be easilycalculated and the jobs can be assigned to positions as described above. Therefore, a polynomially boundedsolution procedure for this case is given by the following Algorithm 1:

Algorithm 1

Step 1: Set b = 1.Step 2: Calculate the positional weights and assign the longest job to the position with the smallest weight, the

second longest job to the position with the second smallest weight, etc. Ties can be broken arbitrarily.Store the solution.

Step 3: If b 6 n calculate b:¼b + 1 and return to Step 2.Step 4: The best solution of the n stored solutions is an optimal solution.

Algorithm 1 requires O(n2 logn) time. The above analysis can easily be extended to all common due date prob-lems that consist of positional weights. These are• Different but not job-individual earliness and tardiness penalties a and b

f ðpIÞ ¼Xn

i¼1

ðaEi þ bT iÞ ¼Xb

r¼1

weI;rp½r� þ

Xn

r¼bþ1

wtI;rp½r�

with weI;r ¼ ðr � 1Þaþ ðn� rÞc and wt

I;r ¼ ðn� r þ 1Þbþ ðn� rÞc.

• A penalty for assigning a late due date v

592 D. Biskup, J. Herrmann / European Journal of Operational Research 191 (2008) 587–592

f ðpIIÞ ¼Xn

i¼1

ðaEi þ bT i þ vdÞ ¼Xb

r¼1

weII;rp½r� þ

Xn

r¼bþ1

wtII;rp½r�

with weII ;r ¼ ðr � 1Þaþ nvþ ðn� rÞc and wt

II ;r ¼ ðn� r þ 1Þbþ nvþ ðn� rÞc.• A penalty on the completion time of the jobs h

f ðpIIIÞ ¼Xn

i¼1

ðaEi þ bT i þ hCiÞ ¼Xb

r¼1

weIII;rp½r� þ

Xn

r¼bþ1

wtIII;rp½r�

with weIII;r ¼ ðr � 1Þaþ ðn� r � 1Þhþ ðn� rÞc and wt

III;r ¼ ðn� r þ 1Þbþ ðn� r � 1Þhþ ðn� rÞc.

These three extensions remain polynomially solvable as Algorithm 1 delivers an optimal solution for eachof them. h

5. Conclusion

In this paper we considered single-machine scheduling problems with past-sequence-dependent setup timesand due dates. We were able to show that some problems remain polynomially solvable even when the setuptimes are dependent on the sequence of the already scheduled jobs. Namely, these problems are minimizingtotal lateness, total tardiness (with agreeable due dates), maximum lateness (with agreeable due dates), max-imum tardiness (with agreeable due dates) and the problem of Kanet (1981). For other problems, namely min-imizing the number of tardy jobs, maximum lateness, and maximum tardiness well-known solution proceduresthat find the optimal solution if past-sequence-dependent setup times do not need to be considered are nolonger valid.

Acknowledgements

We wish to thank three anonymous referees for their remarks that helped to improve an earlier version ofthis note. We further wish to thank Prof. Hermann Jahnke, University of Bielefeld, for intensive discussions.

References

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