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Common due date assignment for scheduling on a single machine with jointly reducible processing times

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Page 1: Common due date assignment for scheduling on a single machine with jointly reducible processing times

*Corresponding author. Tel.: #49-0521-106-3927; fax:#49-0521-106-6036.

E-mail address: [email protected] (H. Jahnke).

Int. J. Production Economics 69 (2001) 317}322

Common due date assignment for scheduling on a singlemachine with jointly reducible processing times

Dirk Biskup, Hermann Jahnke*

Faculty of Economics and Business Administration, University of Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany

Received 1 October 1999; accepted 15 March 2000

Abstract

This paper focuses on analyzing the problem of assigning a common due date to a set of jobs and scheduling them ona single machine. The processing times of the jobs are assumed to be controllable but contrary to former approaches weconsider a situation in which it is only possible to reduce all processing times by the same proportional amount. Thissituation, which is quite interesting from a practical point of view, has to the best of our knowledge never been understudy before. Besides the assignment of the common due date we concentrate on two goals, namely minimizing the sum ofearliness and tardiness penalties and minimizing the number of late jobs. ( 2001 Elsevier Science B.V. All rightsreserved.

Keywords: Scheduling; Earliness; Tardiness; Controllable processing times

1. Introduction

Meeting due dates is among the most importantgoals in scheduling practice, see, for example,[10,13]. Therefore, we concentrate on this objectivein the present paper. More speci"cally we considerthe scheduling of jobs against a common due date;this means the same due date has been assigned toall jobs. There are many practical situations inwhich a common due date exists, e.g. in a just-in-time-production environment, in the assemblyschedule or for a batch delivery. Furthermore, itmight be reasonable to assign a common due date

to a set of jobs to treat di!erent customers equally.Generally, two situations should be distinguished:On the one hand, a common due date may be(externally) given or agreed upon, on the other,a common due date can be a decision variable withits value speci"ed by the company. The latter situ-ation occurs, for example, at di!erent stages of theproduction process, where the timing of an assem-bly can be determined by the company. Anotherexample due to Cheng [5] is that of a shoemaker,who issues the same due date to all customerswaiting at a time. In both situations it is importantto quote realistic due dates. A long due date delaysthe production process but an unrealistic short duedate cannot speed it up (as some of the parts willnot be "nished) and causes high holding costs forthe parts waiting for assembly. A somewhat similarsituation exists for the shoemaker. Long due dates

0925-5273/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 5 - 5 2 7 3 ( 0 0 ) 0 0 0 4 0 - 2

Page 2: Common due date assignment for scheduling on a single machine with jointly reducible processing times

cannot serve as a marketing instrument but thepromise of a short due date which cannot be keptcauses dissatisfaction and leads } in the long run} to the loss of goodwill. Thus, the assignment ofrealistic common due dates is an important taskwhich will be considered in this paper. For anexcellent review on common due date research seeBaker and Scudder [2].

Scheduling problems with continuously com-pressible processing times have received consider-able attention during the last years, see[1,3,6,7,9,12,14,15]. The assumption in these papersis that the normal processing time of a job can bereduced continuously to a minimum value by in-curring higher processing costs. Chen et al. [4]appear to be the only researchers studying the caseof discretely controllable processing times.

In practice, there are situations in which process-ing times are not individually reducible. The onlyoption might be to reduce the processing times ofall jobs jointly, i.e. simultaneously by the samepercentage. A furnace, for example, might be heatedto a speci"ed temperature every day before theprocessing of the jobs starts. It is impossible (ora least not advantageous) to change the temper-ature for every single job. Note that, on the onehand, the temperature has an in#uence on the pro-cessing times of the jobs, but on the other, eachtemperature induces di!erent costs for using thefurnace. A similar situation might occur for a ma-chine at which speci"c tools have to be changedafter a "xed period of time, say a week. Each timea new tool is mounted, a decision about the toolcharacteristics, that is the productive power, needsto be made. For example a drilling machine mightrun with a diamond drill, a high- or a low-qualitysteel drill. If the diamond drill is set up, the process-ing of the jobs can be carried out faster than witha steel drill by incurring the highest costs. On theother hand, setting up the low-quality steel drillinduces the lowest costs and the highest processingtimes. Another example is the speed of an assemblyline which usually can be chosen out of a speci"edinterval (depending, for example, on the number ofworkers and tools available). With respect to thenew orders it may be necessary to decide on thespeed of the assembly line when production startswhile it is generally not possible or advantageous to

change the speed between di!erent jobs duringa day.

In the next section we introduce some assump-tions and the notation used. Section 3 deals withthe problem of jointly minimizing due date assign-ment costs and the sum of earliness and tardinesspenalties. In Section 4 we analyze an objectivefunction consisting of costs for assigning a due dateand "nishing jobs late.

2. Assumptions and notation

There are n jobs available at time zero. Each jobhas a normal processing time and the jobs areindexed according to the shortest (normal) process-ing time (SPT) sequence, i.e. p

1)p

2)2)p

n. If

the jobs are processed with their normal processingtimes, the costs K for operating the machine areinduced. Note, that these operating costs are usu-ally neglected in scheduling theory, as they areassumed to be "xed and thus have no in#uence onan optimal schedule.

If the speed of the machine is continuously con-trollable, x will be the proportion of the processingtime reduction of all jobs, 0)x)x

.!9(1. That

is, for a given x the normal processing times arereduced to p

1(1!x), p

2(1!x),2, p

n(1!x). For

the sake of simplicity, the operating costs k(x) withk(0)"K are given as a monotonous increasingfunction in x. For example, k(x)"K#ax is a lin-ear function of operating costs, with the intercept ofthe ordinate K and the slope a, and k(x)"K#ax2

describes a quadratic operating cost function. Con-vex cost functions, however, seem to "t most situ-ations in practice. For the sake of simplicity weassume K"0.

3. Minimizing earliness and tardiness penalties

It has been widely known in industrial practicethat "nishing jobs too early will result in holdingcosts. These costs could be handling-costs depend-ing on the quantity stored and holding-costs likeinsurance premiums, imputed interests and taxes,which depend on the value of the inventory. Depre-ciations may be caused by technical progress

318 D. Biskup, H. Jahnke / Int. J. Production Economics 69 (2001) 317}322

Page 3: Common due date assignment for scheduling on a single machine with jointly reducible processing times

and/or changing markets. Furthermore, early jobstie up capital, so that opportunity costs have to beconsidered. On the other hand, the e!ects of tardyjobs are dissatis"ed customers and thus, in the longrun, the loss of goodwill and reputation. Conse-quently, orders may be canceled and/or no moreorders will be placed in the future so that tardinesscan reduce the sales volume in the long run. Inaddition, contractual penalties for late deliveriesmight have been agreed upon with some of thecustomers. Comparing the consequences of earli-ness and tardiness, tardy jobs are usually con-sidered more undesirable than early jobs.

The remainder of this section mainly consists oftwo parts: Firstly, the problem to jointly minimizeearliness, tardiness and due date assignment costswith "xed (normal) processing times is introducedand some of its theoretical results together with anoptimizing algorithm are cited from the literature.Secondly, we extend the problem by the possibilityof continuously reducing processing times andpresent a new solution procedure for this situation.

Let Ci, E

i"maxM0, d!C

iN and ¹

i"maxM0,

Ci!dN be the completion time, earliness and tardi-

ness of job i, i"1,2, n, respectively. Further, leta and b be the penalties for earliness and tardinessand c be the due date assignment cost per time unitrepresenting the notion that an early due date isdesirable to the customers. The general objective isto "nd a feasible schedule S and a common duedate d which jointly minimize

f (S, d)"n+i/1

(aEi#b¹

i#cd). (1)

For example, with a"b"1 and c"0, the objec-tive function (1) becomes that for the well-knownproblem of Kanet [8]. Clearly, an optimal scheduledoes not contain any idle time between any pair ofconsecutive jobs. Note that a reduction of the pro-cessing times results in lower due date assignmentcosts as well as lower earliness and tardiness costs,since all jobs are completed closer to the commondue date.

Lemma 1. An optimal schedule exists, in which thebth job is completed at d, where b is the smallestinteger greater than or equal to n(b!c)/(a#b).

Lemma 2. Let [r] indicate the job scheduled at therth position of a schedule. Assuming that the comple-tion of one job coincides with the due date theobjective function (1) can be rewritten as

f (S, d)"n+r/1

urp*r+

,

where

ur"minM(r!1)a#nc, (n!r#1)bN

is the positional weight which arises if a job occupiesthe rth position in a schedule.

The proofs of Lemmas 1 and 2 can be foundin Panwalkar et al. [11], see also Baker andScudder [2]. Problem (1) can now be solved in twosteps. Firstly, a simple matching algorithm is ap-plied to obtain an optimal sequence of the jobs: Thelongest job is placed at the position with thesmallest u

r, the second longest job at the position

with the second smallest ur, etc. Secondly, the due

date is determined by d"+bi/1

p*i+

and the process-ing of the "rst job starts at time zero. This proced-ure is called Algorithm K in the following, as it isoriginally due to Kanet [8]. Furthermore, letf H denote the objective function value minimizing(1).

Including the possibility to decrease the process-ing times continuously by incurring higher operat-ing costs according to k(x), the following lemmaholds.

Lemma 3. If the processing times are reduced by x,the objective function value of an optimal sequencedecreases to f H

x"(1!x) f H.

Proof. By reducing the processing times of all jobsby the same proportion, the initial SPT orderingstill holds. Consequently, the "rst step of the Algo-rithm K leads to the same job sequence. Accordingto Lemma 2 the new objective function value is

f Hx"

n+r/1

ur(1!x)p

*r+

"(1!x)n+r/1

urp*r+"(1!x) f H. h

D. Biskup, H. Jahnke / Int. J. Production Economics 69 (2001) 317}322 319

Page 4: Common due date assignment for scheduling on a single machine with jointly reducible processing times

As a consequence of Lemma 3, the optimalsequence S and the optimal reduction of the pro-cessing times can be obtained by the followingprocedure:

Step 1: For the case of normal processing timesdetermine f H and an optimal sequence of the jobsSH by the Algorithm K.

Step 2: Find the optimal reduction of the process-ing times xH by solving the following program:

Minimize (1!x) f H#k(x)

subject to 0)x)x.!9

.

Step 3: If xH'0, reduce the processing times topHi"(1!xH)p

ifor i"1,2, n and calculate the

optimal due date dH by the second step of theAlgorithm K.

For the special case of a linear cost function k(x),the following lemma holds.

Lemma 4. If k(x) is linear in x, say k(x)"ax, anoptimal sequence exists, in which the compression ofthe job processing times is x"0 if a*f H orx"x

.!9if a)f H.

Proof. The objective function (1!x) f H#ax"f H#(a!f H)x is linear in x, hence a global min-imum exists at the border of the solution space, thatis at x"0 if a*f H or at x"x

.!9if a)f H. K

Example 1. Let n"5 jobs be given with the nor-mal processing times p

1"1, p

2"2, p

3"3,

p4"4 and p

5"5. The earliness, tardiness and due

date assignment penalties are a"3, b"4 andc"2, respectively. The processing times are con-tinuously reducible to 60% of their normal values,thus 0)x)0.6, by inducing the operating costsk(x)"13x#106x2.

Step 1: With b"2 the positional penalties are:u

1"10, u

2"13, u

3"12, u

4"8, u

5"4. An

optimal sequence is SH"(3, 1, 2, 4, 5) withf H"119.

Step 2: Minimizing 119(1!x)#13x#106x2

subject to 0)x)0.6 gives xH"0.5.Step 3: With xH"0.5 the optimal processing

times are: pH1"0.5, pH

2"1, pH

3"1.5, pH

4"2 and

pH5"2.5. The optimal due date is dH"pH

3#pH

1"2.

The minimal total costs induced are 92.5.

4. Minimizing the number of tardy jobs

Another objective in connection with the assign-ment of common due dates is to minimize thenumber of tardy jobs, see Cheng [5]. This goalshould be considered if the costs induced by "nish-ing a job late do not depend on the absolute valueof its tardiness; if, for example, a promised deliverydate is not kept, a stipulated penalty might occur orthe order might be canceled. Similar to Section3 the original problem (with normal processingtimes) is at "rst presented and some of its theorywill be repeated. Afterwards, the possibility ofreducing the processing times is included in theanalysis.

Let ¸(i) be the indicator function assuming thevalue 1 if the job i is late, i.e.

¸(i)"G1 if C

i'd,

0 if Ci)d,

and g be the penalty cost for "nishing a job late.The goal is to jointly minimize the costs of assign-ing a long due date and completing jobs late, i.e.

f (S, d)"n+i/1

(g¸(i)#cd). (2)

Lemma 5. The optimal solution is obtained, if thejobs are sequenced according to the SPT rule andthe assigned due date coincides with the completionof the bth job, d"C

b, so that the inequality

pb)g/nc)p

b`1holds.

The proof of Lemma 5 can be found in Cheng[5]. Now the situation with jointly controllableprocessing times will be considered. If all process-ing times are reduced by the proportion x, b andhence d depend on x since g/nc is a constant. There-fore, a trade-o! between continuously increasingoperating costs, stepwise decreasing lateness costsand due date assignment costs has to be taken intoaccount. Exactly one job less will be "nished late ifthe processing times are reduced by the proportion

320 D. Biskup, H. Jahnke / Int. J. Production Economics 69 (2001) 317}322

Page 5: Common due date assignment for scheduling on a single machine with jointly reducible processing times

x so that

pb`1

(1!x)"gnc

or x"1!g

ncpb`1

holds. Note that this reduction of processing timesleads to a new due date of C

b`1(1!x) altering

the due date assignment costs by nc[Cb`1

(1!x)!Cb], with C

band C

b`1representing the

completion time of the jobs b and b#1 for the caseof normal processing times, respectively. Hence thereduction of processing times should be realized, ifthe incurred costs are compensated for by the re-duced lateness, i.e. if k(x)#nc[C

b`1(1!x)!

Cb](g holds. Of course, the second term on the

left-hand side of this inequality might be negative.The comparison between increased operating anddecreased lateness costs has to be carried out withall jobs b#1, b#2,2 until a further reduction ofprocessing times is not possible. Note that onlya subset of all possible reductions of the processingtimes have to be considered, namely those forwhich p

b`i(1!x

i)"g/nc for i"1, 2,2 with

xi)x

.!9holds. Thus, the following algorithm can

be applied to solve problem (2) for the case that theprocessing times are jointly reducible:

Step 1: Sequence the jobs according to the SPTrule (SH).

Step 2: Find the optimal due date d"Cb

withpb)g/nc)p

b`1.

Step 3: Find iH3M1, 2,2, n!bN so that * f Hi"

k(xHi)#nc[C

b`iH (1!x

iH)!C

b]!iHg for x

iH"

1!g/ncpb`i

H is minimal with respect to 0)xiH

)x.!9

.Step 4: If * f

iH(0 reduce the processing times by

xiH and assign the optimal due date d"C

b`iH .

Example 1 (continued). Let c"2 and g"28.Step 1: SH"(1, 2, 3, 4, 5).Step 2: As p

2)g/nc"2.8)p

3it follows that

d"C2"3 (b"2) with f (S, d)"28 ) 3#5 ) 3 ) 2"

114.Step 3: x

1"1!g/ncp

3"1!28/(05 ) 2 ) 3)" 1

15and * f

1"k( 1

15)#5 ) 2[C

3(1! 1

15)!C

2]!1 ) 28"

!0.6622.x2"0.3 and * f

2"!2.56.

x3"11

25and * f

3"6.2416.

iH"2.

Step 4: Since * f2(0 it follows that pH

1"

0.7, pH2"1.4, pH

3"2.1, pH

4"2.8 and pH

5"3.5,

dH"C4"7 with the minimal costs of 111.44.

5. Conclusions

In this paper we studied the problem of assigninga common due date to a set of simultaneouslyavailable jobs for which the processing times arejointly reducible. Within this setting we concen-trated on two kinds of objective functions. Besidesconsidering due date assignment costs the "rst goalwas to minimize the sum of earliness and tardinesspenalties while the second one was to minimize thenumber of late jobs. For both cases polynomiallysolvable algorithms have been found and demon-strated by an example.

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