Applied Mathematics and Computation 174 (2006) 1245–1254

www.elsevier.com/locate/amc

Single machine scheduling with commondue date and controllable processing times

Ji-Bo Wang

Department of Science, Shenyang Institute of Aeronautical Engineering,

Shenyang 110034, People’s Republic of China

Abstract

In this paper, we consider a single machine scheduling problem with a common duedate. Job processing times are controllable to the extent that they can be reduced, up toa certain limit, at a cost proportional to the reduction. The common due date, alongwith the associated job schedule that minimizes a certain cost function, are to be deter-mined. This function is made up of costs associated with the common due date, process-ing time reduction as well as job absolute value in lateness. We show that the problemcan be formulated as an assignment problem and thus can be solved with well-knownalgorithms. For the case that where all the jobs have a common difference between nor-mal and crash processing time and an equal unit compression penalty, we present anO(n log n) algorithm to obtain the optimal solution.� 2005 Elsevier Inc. All rights reserved.

Keywords: Scheduling; Single machine; Just-in-time; Common due date; Controllable processingtimes

0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.amc.2005.05.046

E-mail address: wangjibo75@yahoo.com.cn

1246 J.-B. Wang / Appl. Math. Comput. 174 (2006) 1245–1254

1. Introduction

Due to the tremendous increase in international competition in the last twodecades, JIT production has proved to be an essential requirement of worldclass manufacturing. JIT philosophy seeks to identify and eliminate waste com-ponents as over production, waiting time, transportation, processing, inven-tory, movement, and defective products. Consequently, it is important thatthe area of scheduling contribute towards the realization of a JIT environment.JIT scheduling models assume the existence of job due dates and discourageearly as well as tardy jobs. If a job finishes before its due date, it incurs an ear-liness penalty such as holding cost. On the other hand, completing the job afterits due date can lead to such tardiness costs as late charges, express deliverycharges, or lost sales. A schedule that minimizes the sum of these penalties issaid to be a JIT-schedule.

In this paper, we consider the case in which job processing times are to bereduced, up to a limit, and with costs proportional to the amount of reductionin processing times. These costs will be offset by savings incurred due to theearly completion. This type of problem occurs frequently in project planning,see [8]. Its motive in the field of scheduling is of the same nature, that is, theassumption of controllable processing times is justified in the situations wherejobs can be accomplished in shorter or longer durations caused by the increas-ing or decreasing additional resources.

Works in the scheduling problem with controllable processing times andlinear cost functions are surveyed by Nowicki and Zdrzalka [13]. Vickson[16], who probably wrote one of the first papers on controllable processingtime scheduling problems, considered the objective of minimizing the totalflow time and the total processing cost incurred due to the job processing timecompression. Vickson [17] considered the single machine scheduling of mini-mizing the total flow and resource costs under the assumption that the jobflow costs are identical. Van Wassenhove and Baker [15] considered singlemachine scheduling problems in which the objective function is to minimizethe maximum completion penalty. They gave a bicriterion approach tosequencing with time/cost trade-offs which produces an efficient frontier ofthe possible schedules. Nowicki and Zdrzalka [12] considered a two-machineflow shop scheduling problem with controllable job processing times. They as-sumed that the cost of performing a job is a linear function of its processingtime, and the schedule cost to be minimized is the total processing cost plusmaximum completion time cost. They showed that the problem is NP-com-plete, and proposed two heuristic methods for solving the problem. Danielsand Sarin [7] considered single machine scheduling problem of joint sequenc-ing and resource allocation when the criteria is the number of tardy jobs.Zdrzalka [19] considered single machine scheduling problem in which eachjob has a release date, a delivery time and a controllable processing time.

J.-B. Wang / Appl. Math. Comput. 174 (2006) 1245–1254 1247

He gave an approximation algorithm for minimizing the overall schedule cost.Panwalkar and Rajagopalan [14] considered the common due date assignmentand single machine scheduling problem in which the objective is the sum ofpenalties based on earliness, tardiness and processing time compressions.They reduced the problem to an assignment problem. Alidace and Ahmadian[1] extended the results of [14] to the parallel machine scheduling case. Chengand Janiak [5] further generalized the result to the case where the cost of com-pression is a general convex function of the amount of compression. Chenget al. [6] considered a due date assignment and single machine scheduling inwhich a penalty for due dates is added to the objective function which in-cludes the penalties for earliness, tardiness and processing time compressions.Biskup and Cheng [2] considered a due date assignment and single machinescheduling in which a penalty for completion times is added to the objectivefunction which includes the penalties for earliness, tardiness and processingtime compressions. Biskup and Jahnke [3] considered the problem of assign-ing a common due date to a set of jobs and scheduling them on a single ma-chine with jointly reducible processing times. Besides considering due dateassignment costs the first goal is to minimize the sum of earliness and tardi-ness penalties while the second one is to minimize the number of late jobs.For both cases polynomially solvable algorithms have been given. Hoogeveenand Woeginger [9] combined the resource allocation and the weighted flowtime costs to a single objective and proved that this problem is NP-hard.Ng et al. [10] considered the single machine problem with a variable commondue date. They presented polynomial time algorithms for minimizing a linearcombination of scheduling, due date assignment and resource consumptioncosts. Ng et al. [11] considered the single machine batch scheduling withjointly compressible setup and processing times. They presented polynomialtime algorithms to find an optimal batch sequence and optimal amounts ofresource consumption such that either total job completion time is minimized,subject to an upper bound on total weighted resource consumption, or totalweighted resource consumption is minimized, subject to an upper bound ontotal job completion time. Wang [18] considered single machine SLK due dateassignment scheduling problem in which job processing times are controllablevariables with linear costs. The objective is to determine the optimal sequence,the optimal common flow allowance and the optimal processing time com-pressions to minimize a total penalty function based on earliness, tardiness,common flow allowance and compressions. He solved the problem by formu-lating it as an assignment problem.

The rest of this paper is organized as follows. Notations and assumptionsare given in Section 2. In Section 3, we obtain optimal compressions for anygiven sequence. In Section 4, we show that the problem can be formulatedas an assignment problem. A special case that lends itself to an easy solutionis presented in Section 5. In Section 6, conclusions are presented.

1248 J.-B. Wang / Appl. Math. Comput. 174 (2006) 1245–1254

2. Notations and assumptions

Consider a set of n jobs J ¼ fJ 1; J 2; . . . ; Jng to be processed in a single ma-chine with the following assumptions:

• All jobs are available at time zero.• No job pre-emption and job splitting are allowed.• The machine is available at time zero and for the whole duration of timehorizon.

• The machine cannot process two or more jobs simultaneously.• After the process in the machine has started, no idle time can be inserted inthe schedule.

The following notations will be used throughout the paper:

r the sequence of jobs to be processed by the machiner(i) job in the ith positionti the normal processing time of job Jit0i the crash processing time of job JiGi the per time unit cost connected with the compression below ti of the

processing time of job Jimi ¼ ti � t0i the maximum reduction in processing time of job Jixi the compression of the processing time of job Ji which can take any

value in [0,mi]~x ¼ ðx1; x2; . . . ; xnÞ the vector of job compressionspi the actual processing time of job Ji, that is pi = ti � xid the common due dateCi the completion time of t job Jiwi the weights do not correspond with the jobs but with the positions in

which the jobs are scheduledjLr(i)j the absolute value in lateness, that is jLr(i)j = jCr(i) � dj

The problem can be stated as follows. n jobs are simultaneously available forprocessing at the time the schedule is to be made. Without loss of generality, weassume that jobs are numbered in nondecreasing order of normal processingtimes (t1 6 t2 6 � � � 6 tn). If a job is processed at its normal processing time,ti, it will incur no additional processing cost. On the other hand, a per unit timecost Gi is incurred if the processing time is reduced by one time unit. For job Ji,t0i denotes the crash (minimum allowable) processing time, mi denotes the max-imum reduction allowed (mi ¼ ti � t0i), and xi denotes the actual reduction inprocessing time (0 6 xi 6 mi). All jobs have a common due date d which is deci-sion variable to be determined. The objective is to determine the optimal com-pressions of the processing times and the optimal sequence of jobs in the

J.-B. Wang / Appl. Math. Comput. 174 (2006) 1245–1254 1249

machine so that the corresponding value of the following cost functions beoptimal:

f ðd; r;~xÞ ¼Xn

i¼1

wijLrðiÞj þ w0d þXn

i¼1

GrðiÞxrðiÞ; ð1Þ

where weights are position weights rather than weights associated with thejobs.

It is convenient to introduce a dummy job 0 with actual processing timep0 = 0 and weight w0 which is always scheduled at time 0, i.e. we definedr(0) = 0. Then the objective function is

f ðd; r;~xÞ ¼Xn

i¼0

wijLrðiÞj þXn

i¼1

GrðiÞxrðiÞ.

We conclude that an optimal schedule is given by a schedule r(0),r(1), . . . ,r(n).

3. Optimal compressions

The objective function (1) without the possibility to compress the processingtimes has been considered by Brucker [4]. The following properties are used toformulate the problem as an assignment problem.

Property 1. For any given ~x, there exists an optimal sequence r* without any

machine idle time between the starting time of the first job and the completion time

of the last job. Furthermore, the first job in the sequence starts at time zero.

Proof. The proof can be found in [4]. h

Property 2. Given ~x, an optimal schedule exists, in which the kth job is completedat d, i.e. d ¼

Pki¼0pi, where k meetXk�1

j¼0

wj 6

Xn

j¼k

wj andXk

j¼0

wj PXn

j¼kþ1

wj.

Proof. The proof can be found in [4]. h

Property 3. Given ~x, let r(j) indicate the job scheduled at the jth position of a

schedule. Assuming that the waiting time of one job coincides with the common

flow allowance the objective function (1) can be rewritten as

f ðd; r;~xÞ ¼Xn

j¼1

kjprðjÞ; ð2Þ

1250 J.-B. Wang / Appl. Math. Comput. 174 (2006) 1245–1254

where kj ¼Pj�1

v¼0wv for j ¼ 1; 2; . . . ; k;Pnv¼jwv for j ¼ k þ 1; k þ 2; . . . ; n

�is the positional weight which

arises if a job occupies the jth position in a schedule.

Proof. The proof can be found in [4]. h

Property 4. For the controllable processing times problem with linear compres-

sion costs. There exists an optimal schedule such that there no partially com-

pressed jobs.

Proof. The proof is given by Panwalkar and Rajagopalan [14]. The fact thatthe positional weights kj are different from the weights used by Panwalkarand Rajagopalan [14] does not affect the validity of the results. h

Using Properties 2 and 3, and substituting CrðjÞ ¼Pj

i¼1prðiÞ, xr(j) = tr(j) �pr(j) and d ¼

Pki¼0pi into Eq. (1) and simplify, we have

f ðd; s;~xÞ ¼Xn

j¼1

kjprðjÞ þXn

j¼1

Gjtj;

where

kj ¼

Pj�1

v¼0

wv � GrðjÞ for j ¼ 1; 2; . . . ; k;

Pnv¼j

wv � GrðjÞ for j ¼ k þ 1; k þ 2; . . . ; n;

8>>><>>>:

ð3Þ

then kj, 1 6 j 6 n, represents the position weight of position r in the sequence r.Since

Pnj¼1Gjtj is a constant, for any sequence, the optimal processing time of a

job in a position with a negative position weight should be its normal processingtime, and the processing time of a job in a position with a positive positionweight should be its crash processing time. If a position j has a zero positionweight, then the optimal processing time of the job in this position may be anyvalue between t0j and tj. They can be written in the notational form as follows:

p�rðjÞ ¼trðjÞ; if kj < 0;

p0rðjÞ; if kj ¼ 0;

t0rðjÞ; if kj > 0;

8><>: ð4Þ

where t0rðjÞ 6 p0rðjÞ 6 trðjÞ and p�rðjÞ; 1 6 j 6 n, represents the optimal processingtime of the job in position j. Therefore, the optimal compressions can be ob-tained by

x�rðjÞ ¼ trðjÞ � p�rðjÞ; j ¼ 1; 2; . . . ; n. ð5Þ

Theorem 1. Given a sequence, for the model (1), the optimal compressions can be

determined as follows: the compression of the job in a negative-weight position iszero; the compression of the job in a positive-weight position is its upper bound; if

the position weight of a position is zero, then the compression of the job in his

position can be any vale between its lower bound and its upper bound.

J.-B. Wang / Appl. Math. Comput. 174 (2006) 1245–1254 1251

Proof. The proof follows from the analysis above. h

4. Optimal sequences

Now we discuss the determination of optimal sequences for the model (1). Inview of the analysis in the previous sections, where the optimal processingtimes and compressions can be computed for any given sequence, the problemreduces to a pure sequencing problem. In order to obtain the optimal sequence,we formulate the model (1) as an assignment problem.

For the model (1), let

kij ¼

Pi�1

v¼0

wv � Gj for i ¼ 1; 2; . . . ; k;

Pnv¼i

wv � Gj for i ¼ k þ 1; k þ 2; . . . ; n

8>>><>>>:

and

pij ¼tj; if kij < 0;

p0j; if kij ¼ 0;

t0j; if kij > 0;

8><>: ð6Þ

where t0j 6 p0j 6 tj. Furthermore, let xij be a 0/1 variable such that xij = 1 if jobJi is scheduled in position j, and xij = 0, otherwise. As in [14], the optimalmatching of jobs to positions requires a solution for the following assignmentproblem:

minXn

i¼1

Xn

j¼1

kijpijxij

subject toXn

i¼1

xij ¼ 1; j ¼ 1; 2; . . . ; n;

Xn

j¼1

xij ¼ 1; i ¼ 1; 2; . . . ; n;

xij ¼ 0 or 1; i; j ¼ 1; 2; . . . ; n.

ð7Þ

1252 J.-B. Wang / Appl. Math. Comput. 174 (2006) 1245–1254

Recall that solving an assignment problem of size n requires an effort of O(n3)(using the well-known Hungarian method), hence the optimal sequence can befound in polynomial time.

5. A special case

We now consider a special case in each model in which Gi = G andti � t0i ¼ m; i ¼ 1; 2; . . . ; n. This represents the case in practice where thesame means is employed to compress the processing time of each job. Itcan be shown that the optimal solution can be found in O(n log n) time in thiscase.

Theorem 1. For the model (1) in which Gi = G and ti � t0i ¼ m, i = 1,2, . . . , n, theoptimal sequence r* is the sequence obtained from matching the position weights

in descending order with the normal processing times in ascending order.

Proof. The idea of the proof is as follows. In an optimal sequence r*, if thereare two adjacent positions j and j + 1 such that kr(j) < kr(j+1) but tr(j) < tr(j+1),then interchanging the two jobs in positions j and j + 1, we can obtain anothersequence r 0 which is better than r*, which contradicts the fact that r* is an opti-mal sequence. Here, for both problems, kr(j) and kr(j+1) may be positive, nega-tive or zero. For brevity, we only show a case for the model (1) in whichkr(j) < 0 and kr(j+1) > 0. Interchanging the two jobs in positions j and j + 1,we obtain another sequence r 0. The change in the value of the objective func-tion is given by

f ðr0; xiÞ � f ðr�; xiÞ ¼ krðjÞtrðjþ1Þ þ krðjþÞt0rðjÞ � ðkrðjÞtrðjÞ þ krðjþ1Þt0rðjþ1ÞÞ¼ krðjÞtrðjþ1Þ þ krðjþ1ÞðtrðjÞ � mÞ � krðjÞtrðjÞ

� krðjþ1Þðtrðjþ1Þ � mÞ¼ ðtrðjþ1Þ � trðjÞÞðkrðjÞ � krðjþ1ÞÞ < 0.

The other cases can be proved in a similar manner. h

In the following, we present an O(n log n) algorithm for this special case.

Algorithm 1

Step 1. Weight n positions by using Eq. (3) for the model (1).Step 2. Rank the position weights ki for the model (1) in descending order of

magnitude such that the largest ki is ranked 1 and smallest ki is rankedn. Break ties arbitrarily.

J.-B. Wang / Appl. Math. Comput. 174 (2006) 1245–1254 1253

Step 3. Find the optimal sequence by matching the position weights indescending order with the jobs in ascending order of their normal pro-cessing times.

Step 4. Calculate the optimal processing times by using Eq. (4) for the model(1).

Step 5. Calculate the optimal compressions by using Eq. (5) for the model (1).

To determine the computational complexity of Algorithm 1, we note thatStep 2 can be completed in O(n log n) time and Steps 1, 4 and 5 can be com-pleted in O(n) time. Hence, the overall time complexity of this algorithm isO(n log n).

Theorem 2. Algorithm 1 delivers an optimal solution to the problems described in

Theorem 1 and the complexity of the algorithm is O(n log n).

Proof. The proof of Theorem 2 follows from the analysis above. h

6. Conclusions

In this paper, we consider the problem of single machine common due datescheduling with controllable processing times. The objective is to determine theoptimal sequence, the optimal common due date and the optimal processingtime compressions to minimize a total penalty function based on the commondue date, job absolute value in lateness and compressions. The problem is mod-elled as an assignment problem. An O(n log n) algorithm is proved to obtainedthe optimal solution for a special case.

References

[1] B. Alidaee, A. Ahmadian, Two parallel machine sequencing problems involving controllablejob processing times, European Journal of Operational Research 70 (3) (1993) 335–341.

[2] D. Biskup, T.C.E. Cheng, Single machine scheduling with controllable processing times andearliness, tardiness and completion time penalties, Engineering Optimization 31 (1999) 329–336.

[3] D. Biskup, H. Jahnke, Common due date assignment for scheduling on a single machine withjointly reducible processing times, International Journal of Production Economics 69 (2001)317–322.

[4] P. Brucker, Scheduling Algorithms, Third ed., Springer, 2001.[5] T.C.E. Cheng, A. Janiak, Resource optimal control in some single machine scheduling

problems, IEEE Transactions on Automatic Control 39 (1994) 1243–1246.[6] T.C.E. Cheng, C. Oguz, X.D. Qi, Due-date assignment and single machine scheduling with

compressible processing times, International Journal of Production Economics 43 (1996) 29–35.

1254 J.-B. Wang / Appl. Math. Comput. 174 (2006) 1245–1254

[7] R.L. Daniels, R.K. Sarin, Single machine scheduling with controllable processing times andnumber of jobs tardy, Operations Research 37 (1989) 981–984.

[8] S.E. Elmaghraby, Activity Network, Wiley, New York, 1977.[9] H. Hoogeveen, G.J. Woeginger, Some comments on sequencing with controllable processing

times, Computing 68 (2002) 181–192.[10] C.T.D. Ng, T.C.E. Cheng, M.Y. Kovalyov, S.S. Lam, Single machine scheduling with a

variable common due date and resource-dependent processing times, Computers andOperations Research 30 (2003) 1173–1185.

[11] C.T.D. Ng, T.C.E. Cheng, M.Y. Kovalyov, Single machine batch scheduling with jointlycompressible setup and processing times, European Journal of Operational Research 153(2004) 211–219.

[12] E. Nowicki, S. Zdrzalka, A two machine flow shop scheduling problem with controllable jobprocessing times, European Journal of Operational Research 34 (1988) 208–220.

[13] E. Nowicki, S. Zdrzalka, A survey of results for sequencing problems with controllableprocessing times, Discrete Applied Mathematics 26 (1990) 271–287.

[14] S.S. Panwalkar, R. Rajagopalan, Single machine sequencing with controllable processingtimes, European Journal of Operational Research 59 (1992) 298–302.

[15] L.N. Van Wassenhove, K.R. Baker, A bicriterion approach to time/cost trade-offs insequencing, European Journal of Operational Research 11 (1982) 48–52.

[16] R.G. Vickson, Choosing the job sequence and processing times to minimize total processingplus flow cost on a single machine, Operations Research 29 (1980) 1155–1167.

[17] R.G. Vickson, Two single-machine sequencing problems involving controllable job processingtimes, AIIE Transactions 12 (1980) 258–262.

[18] J.-B. Wang, Single machine common flow allowance scheduling with controllable processingtimes, Journal of Applied Mathematics & Computing, in press.

[19] S. Zdrzalka, Scheduling jobs on a single machine with relese dates, delivery times andcontrollable processing times: worst-case analysis, Operations Research Letters 10 (1991)519–524.