Single machine scheduling problems with controllable processing times and total absolute differences penalties

European Journal of Operational Research 177 (2007) 638–645

www.elsevier.com/locate/ejor

Short Communication

Single machine scheduling problems with controllableprocessing times and total absolute differences penalties

Ji-Bo Wang a,*, Zun-Quan Xia b

a Department of Science, Shenyang Institute of Aeronautical Engineering, Shenyang 110034, People’s Republic of Chinab Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’s Republic of China

Received 3 March 2005; accepted 5 October 2005Available online 3 February 2006

Abstract

In this paper, we consider single machine scheduling problem in which job processing times are controllable vari-ables with linear costs. We concentrate on two goals separately, namely, minimizing a cost function containing totalcompletion time, total absolute differences in completion times and total compression cost; minimizing a cost functioncontaining total waiting time, total absolute differences in waiting times and total compression cost. The problem ismodelled as an assignment problem, and thus can be solved with the well-known algorithms. For the case where allthe jobs have a common difference between normal and crash processing time and an equal unit compression penalty,we present an O(n log n) algorithm to obtain the optimal solution.� 2006 Published by Elsevier B.V.

Keywords: Scheduling; Single machine; Controllable processing times

1. Introduction

Most of the scheduling literature examines regular measures of the performance, which are nondecreas-ing functions of job completion times. One of the most commonly occurring regular measures is the min-imization of mean completion times. Its attractiveness is perhaps due to its equivalence to mean waitingtime, mean lateness, and average in-process inventory. Yet in certain situations one is more interested inreducing the variability in the completion times, resulting in performance measures that are nonregular.

0377-2217/$ - see front matter � 2006 Published by Elsevier B.V.doi:10.1016/j.ejor.2005.10.054

* Corresponding author.E-mail address: [email protected] (J.-B. Wang).

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J.-B. Wang, Z.-Q. Xia / European Journal of Operational Research 177 (2007) 638–645 639

For instance, in a service-oriented environment, one might be interested in providing as much uniformquality of service as possible based on the customers’ waiting times in system.

Another example where one might be interested in a variability measure was given by Merten and Muller(1972) in the context of organization of computer databases. They noted that, in an organization of com-puter files in the large databases, it is desirable to provide uniform response time to users. The objectivethen is to determine the arrangement that minimizes variation of access time to different records in the file.

As measures of variation, Merten and Muller (1972) considered completion time variance (CTV) andwaiting time variance (WTV). These measures have also been used by Schrage (1975), Eilon and Chowdh-ury (1977), and Vani and Raghavachari (1987). Although several properties of the optimal schedules forthese measures have been established, no efficient procedure exists for solving these problems. Kanet(1981) proposed using total absolute differences in completion times (TADC) as an alternative measureof completion time variation, and presented an efficient algorithm for minimizing this measure. While Bag-chi (1989) proposed using total absolute differences in waiting times (TADW) as an alternative measure ofwaiting time variation, and presented an efficient algorithm for minimizing this measure.

In this paper, we consider the case in which job processing times are to be reduced, up to a limit, andwith costs proportional to the amount of reduction in processing times. These costs will be offset by savingsincurred due to the early completion. This type of problem occurs frequently in project planning, seeElmaghraby (1977). Its motive in the field of scheduling is of the same nature, that is, the assumption ofcontrollable processing times is justified in the situations where jobs can be accomplished in shorter orlonger durations caused by the increasing or decreasing additional resources. For example, in service sys-tems, if there are too many customers, it may be important to provide customers with identical or similarservice quality as well as to decrease everyone’s waiting time by increasing additional resources (otherwisethe customers who wait too long time will leave). We concentrate on two goals: minimizing a cost functioncontaining total completion time, total absolute differences in completion times and total compression cost;minimizing total waiting time, total absolute differences in waiting times and total compression cost.

Works in the scheduling problem with controllable processing times and linear cost functions are sur-veyed by Nowicki and Zdrzalka (1990). Vickson (1980a), who probably wrote one of the first papers oncontrollable processing time scheduling problems, considered the objective of minimizing the total flowtime and the total processing cost incurred due to the job processing time compression. Vickson(1980b) considered the single machine scheduling of minimizing the total flow and resource costs underthe assumption that the job flow costs are identical. Van Wassenhove and Baker (1982) considered singlemachine scheduling problems in which the objective function is to minimize the maximum completionpenalty. They gave a bicriterion approach to sequencing with time/cost trade-offs which produces an effi-cient frontier of the possible schedules. Nowicki and Zdrzalka (1988) considered a two-machine flow shopscheduling problem with controllable job processing times. They assumed that the cost of performing ajob is a linear function of its processing time, and the schedule cost to be minimized is the total processingcost plus maximum completion time cost. They showed that the problem is NP-complete, and proposedtwo heuristic methods for solving the problem. Daniels and Sarin (1989) considered single machine sched-uling problem of joint sequencing and resource allocation when the criteria is the number of tardy jobs.Zdrzalka (1991) considered single machine scheduling problem in which each job has a release date, adelivery time and a controllable processing time. He gave an approximation algorithm for minimizingthe overall schedule cost. Panwalkar and Rajagopalan (1992) considered the common due date assign-ment and single machine scheduling problem in which the objective is the sum of penalties based on ear-liness, tardiness and processing time compressions. They reduced the problem to an assignment problem.Alidaee and Ahmadian (1993) extended the results of Panwalkar and Rajagopalan (1992) to the parallelmachine scheduling case. Cheng and Janiak (1994) further generalized the result to the case where the costof compression is a general convex function of the amount of compression. Cheng et al. (1996) considereda due date assignment and single machine scheduling in which a penalty for due dates is added to the

640 J.-B. Wang, Z.-Q. Xia / European Journal of Operational Research 177 (2007) 638–645

objective function which includes the penalties for earliness, tardiness and processing time compressions.Alidaee and Kochenberger (1996) considered single and parallel machine scheduling problems in whichjob processing time of a job was assumed to depend on the position of the job in the schedule and isa function of units of resource applied for its processing. The processing time and the processing costfunctions are allowed to be nonlinear. For the single machine problem, the objective was minimizationof total compression costs plus a scheduling measure. The scheduling measures included makespan, totalflow time, total differences in completion times (TADC), total differences in waiting times (TADW), andtotal earliness and tardiness with a common due date for all jobs. Except for the total earliness and tar-diness measure, they solved all variations efficiently. Under an assumption that is typically satisfied in aJIT environment, they proved that the problem with total earliness and tardiness measure was also solvedefficiently. For a large class of processing time functions, they proved that the parallel machine problemwith total flow time, and total earliness and tardiness measure was solved efficiently. They reduced theproblems of all cases to a transportation problem which is known to be polynomially solvable. Biskupand Cheng (1999) considered a due date assignment and single machine scheduling in which a penaltyfor completion times is added to the objective function which includes the penalties for earliness, tardinessand processing time compressions. Biskup and Jahnke (2001) considered the problem of assigning a com-mon due date to a set of jobs and scheduling them on a single machine with jointly reducible processingtimes. Besides considering due date assignment costs the first goal is to minimize the sum of earliness andtardiness penalties while the second one is to minimize the number of late jobs. For both cases polyno-mially solvable algorithms have been given. Hoogeveen and Woeginger (2002) combined the resource allo-cation and the weighted flow time costs to a single objective and proved that this problem is NP-hard. Nget al. (2003) considered the single machine problem with a variable common due date. They presentedpolynomial time algorithms for minimizing a linear combination of scheduling, due date assignmentand resource consumption costs. Shabtay and Kaspi (2004) considered a single machine scheduling prob-lem with the minimum total weighted completion time criterion where the model of operations is assumedto be a specific convex function of the amount of resource consumed. They presented and analyzed somespecial cases that are solvable by using polynomial time algorithms. They also gave some heuristic algo-rithms for the general case. Ng et al. (2004) considered the single machine batch scheduling with jointlycompressible setup and processing times. They presented polynomial time algorithms to find an optimalbatch sequence and optimal amounts of resource consumption such that either total job completion timeis minimized, subject to an upper bound on total weighted resource consumption, or total weightedresource consumption is minimized, subject to an upper bound on total job completion time.

The rest of this paper is organized as follows. Notations and assumptions are given in Section 2. In Sec-tion 3, we obtain optimal compressions for any given sequence. In Section 4, we show that the problem canbe formulated as an assignment problem. A special case for which there is an easy solution is presented inSection 5. In Section 6, conclusions are presented.

2. Notations and assumptions

Consider a set of n jobs J ¼ fJ 1; J 2; . . . ; J ng to be processed in a single machine with the followingassumptions:

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


The following notations will be used throughout the paper:

s the sequence of jobs to be processed by the machine[i] job in the ith positionti the normal processing time of job Ji

t0i the crash processing time of job Ji

Gi the per time unit cost associated with the compression below ti of the processing time of job Ji

mi ¼ ti � t0i the maximum reduction in processing time of job Ji

xi the compression of the processing time of job Ji which can take any value in [0, mi]pi the actual processing time of job Ji, that is pi = ti � xi

Ci the completion time of t job Ji

Wi the waiting time of job Ji, that is Wi = Ci � pi

TC the total completion times, that is TC ¼Pn

i¼1Ci

TW the total waiting times, that is TW ¼Pn

i¼1W iTADC the total absolute differences in completion times, that is

Xn Xn

TADC ¼i¼1 j¼i

jCi � Cjj.

TADW the total absolute differences in waiting times, that is

TADW ¼Xn

i¼1

Xn

j¼i

jW i � W jj.

In the PERT/CPM terminology, the maximum processing time ti is called the normal processing time,and the minimum time t0i as the crash time. The objective is to determine the optimal compressions of theprocessing times and the optimal sequence of jobs in the machine so that the corresponding value of thefollowing cost functions be optimal:

f ðs; xiÞ ¼ d1TCþ d2TADCþ d3

Xn

i¼1

Gixi; ð1Þ

f ðs; xiÞ ¼ d1TWþ d2TADWþ d3

Xn

i¼1

Gixi; ð2Þ

where weights d1 P 0, d2 P 0 and d3 P 0 are given constants (the decision-maker selects the weights d1, d2, d3).

3. Optimal compressions

For the model (1), if we substitute, C½j� ¼Pj

i¼1p½i�, TC ¼Pn

j¼1C½j�, TADC ¼Pn

j¼1ðj� 1Þðn� jþ 1Þp½j�(Kanet, 1981) and x[j] = t[j] � p[j] into (1) and simplify, we have

f ðs; xiÞ ¼ d1

Xn

j¼1

ðn� jþ 1Þp½j� þ d2

Xn

j¼1

ðj� 1Þðn� jþ 1Þp½j� þ d3

Xn

j¼1

G½j��

t½j� � p½j��

¼Xn

j¼1

½ðd1 � d2Þðnþ 1Þ þ jððnþ 2Þd2 � d1Þ � j2d2 � d3G½j��p½j� þ d3

Xn

j¼1

Gjtj.


Let

kj ¼ ðd1 � d2Þðnþ 1Þ þ jððnþ 2Þd2 � d1Þ � j2d2 � d3G½j�; 1 6 j 6 n ð3Þ

then kj, 1 6 j 6 n, represents the position weight of position j in the sequence s. Since d3

Pnj¼1Gjtj is a con-

stant, for any sequence, the optimal processing time of a job in a position with a negative position weightshould be its normal processing time, 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 position weight, then the optimal pro-cessing time of the job in this position may be any value between t0i and ti. These can be written in the nota-tional form as follows:

p�½j� ¼t½j� if kj < 0;

p0½j� if kj ¼ 0;

t0½j� if kj > 0;

8><>:

ð4Þ

where t0½j� 6 p0½j� 6 t½j� and p�½j�, 1 6 j 6 n, represents the optimal processing time of the job in position j.Therefore, the optimal compressions can be obtained by

x�½j� ¼ t½j� � p�½j�; j ¼ 1; 2; . . . ; n. ð5Þ

For the model (2), if we substitute, W ½j� ¼Pj�1

i¼1 p½i�, TW ¼Pn

j¼1W ½j�, TADW ¼Pn

j¼1jðn� jÞp½j� (Bagchi,1989) and x[j] = t[j] � p[j] into (2) and simplify, we have

f ðs; xiÞ ¼Xn

j¼1

½d1nþ jðnd2 � d1Þ � j2d2 � d3G½j��p½j� þ d3

Xn

j¼1

Gjtj.

Let kj and p�½j�, 1 6 j 6 n, denote the position weight of position j and the optimal processing time of the jobin position j, respectively, then

kj ¼ d1nþ jðnd2 � d1Þ � j2d2 � d3G½j�; 1 6 j 6 n; ð6Þ

p�½j� ¼t½j�; if kj < 0;

p0½j�; if kj ¼ 0;

t0½j�; if kj > 0;

8>><>>:

ð7Þ

where t0½j� 6 p0½j� 6 t½j�. Using the same argument as for model (1), we see that the optimal compressions canbe obtained by

x�½j� ¼ t½j� � p�½j�; j ¼ 1; 2; . . . ; n. ð8Þ

Theorem 1. Given a sequence, for the two models, the optimal compressions can be determined as follows: The

compression of the job in a negative-weight position is zero; the compression of the job in a positive-weight

position is its maximum reduction in the processing time; if the position weight of a position is zero, then thecompression of the job in this position can be any value between zero and its maximum reduction in processing

time.

Proof. The proof follows from the analysis above. h


4. Optimal sequences

Now we discuss the determination of optimal sequences for the two models. In view of the analysis in theprevious sections, where we provided the expressions for computing the optimal processing times and com-pressions for any given optimal sequence, the problem reduces to a pure sequencing problem. In order toobtain the optimal sequence, we formulate the models (1) and (2) as an assignment problem, respectively.

For the model (1), let

kij ¼ ðd1 � d2Þðnþ 1Þ þ iððnþ 2Þd2 � d1Þ � i2d2 � d3Gj; i; j ¼ 1; 2; . . . ; n

and

pij ¼tj; if kij < 0;

p0j; if kij ¼ 0;

t0j; if kij > 0;

8><>:

ð9Þ

where t0j 6 p0j 6 tj. Furthermore, let xij be a 0/1 variable such that xij = 1 if job Jj is scheduled in position i,and xij = 0, otherwise. As in Panwalkar and Rajagopalan (1992), the optimal matching of jobs to positionsrequires a solution for the following assignment problem:

minXn

i¼1

Xn

j¼1

kijpijxij ð10Þ

s.t.

Xn

i¼1

xij ¼ 1; i ¼ 1; 2; . . . ; n;

Xn

j¼1

xij ¼ 1; j ¼ 1; 2; . . . ; n;

xij ¼ 0 or 1; i; j ¼ 1; 2; . . . ; n.

For the model (2), let

kij ¼ d1nþ iðnd2 � d1Þ � i2d2 � d3Gj; i; j ¼ 1; 2; . . . ; n;

p�ij ¼tj; if kij < 0;

p0j; if kij ¼ 0;

t0j; if kij > 0;

8>><>>:

ð11Þ

where t0j 6 p0j 6 tj. The optimal sequence is obtained, as the same assignment problem (10).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 be found in polynomial time.

5. A special case

We now consider a special case in each model in which Gi = G and ti � t0i ¼ m, i = 1, 2, . . . , n. This rep-resents the case in practice where the same means is employed to compress the processing time of each job.It can be shown that the optimal solution can be found in O(n log n) time in this case.


Theorem 2. For the models (1) and (2) in which Gi = G and ti � t0i ¼ m, i = 1, 2, . . . , n, the optimal sequence s*

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 s*, if there are two adjacent Ji and Jj

which are scheduled in the rth and (r + 1)th positions such that kr < kr+1 but ti < tj, then interchangingthe two jobs in positions r and r + 1, we can obtain another sequence s 0 which is better than s*, which con-tradicts the fact that s* is an optimal sequence. Here, for both problems, kr and kr+1 may be positive, neg-ative or zero. For brevity, we only show a case for the model (1) in which kr < 0 and kr+1 > 0. Interchangingthe two jobs in positions r and r + 1, we obtain another sequence s 0. The change in the value of the objec-tive function is given by

f ðs0; xiÞ � f ðs�; xiÞ ¼ krtj þ krþ1t0i � ðkrti þ krþ1t0jÞ ¼ krtj þ krþ1ðti � mÞ � krti � krþ1ðtj � mÞ¼ ðtj � tiÞðkr � krþ1Þ < 0. �

The other cases can be proved in a similar manner.

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) and (6) for the model (2).Step 2. Rank the position weights ki for the model (1) and ki for the model (2) in descending order of

magnitude such that the largest kiðkiÞ is ranked 1 and smallest kiðkiÞ is ranked n. Break tiesarbitrarily.

Step 3. Find the optimal sequence by matching the position weights in descending order with the jobs inascending order of their normal processing times.

Step 4. Calculate the optimal processing times by using Eq. (4) for the model (1) and (7) for the model (2).Step 5. Calculate the optimal compressions by using Eq. (5) for the model (1) and (8) for the model (2).

To determine the computational complexity of Algorithm 1, we note that Step 2 can be completed inO(n log n) time and Steps 1, 4 and 5 can be completed in O(n) time. Hence, the overall time complexityof this algorithm is O(n log n).

Theorem 3. Algorithm 1 delivers an optimal solution to the problems described in Theorem 3 and the

complexity of the algorithm is O(n log n).

Proof. The proof of Theorem 3 follows from the analysis above. h

6. Conclusions

The problem of scheduling n jobs with controllable processing times has been studied. The objectivefunction is to minimize a cost function containing total completion (waiting) time, total absolute differ-ences in completion (waiting) times and total compression cost. We have solved the problem by formulatingit as an assignment problem. An O(n log n) algorithm is proved to obtain the optimal solution for a specialcase.


Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments on an earlierversion of this paper. The research was partially supposed by the foundation of Shenyang Institute of Aero-nautical Engineering under Grant Number: 05YB08.

References

Alidaee, B., Ahmadian, A., 1993. Two parallel machine sequencing problems involving controllable job processing times. EuropeanJournal of Operational Research 70 (3), 335–341.

Alidaee, B., Kochenberger, G., 1996. A framework for machine scheduling problems with controllable processing times. Productionand Operations Management 5 (4), 391–405.

Bagchi, U.B., 1989. Simultaneous minimization of mean and variation of flow-time and waiting time in single machine systems.Operations Research 37, 118–125.

Biskup, D., Cheng, T.C.E., 1999. Single machine scheduling with controllable processing times and earliness, tardiness and completiontime penalties. Engineering Optimization 31, 329–336.

Biskup, D., Jahnke, H., 2001. Common due date assignment for scheduling on a single machine with jointly reducible processing times.International Journal of Production Economics 69, 317–322.

Cheng, T.C.E., Janiak, A., 1994. Resource optimal control in some single machine scheduling problems. IEEE Transactions onAutomatic Control 39, 1243–1246.

Cheng, T.C.E., Oguz, C., Qi, X.D., 1996. Due-date assignment and single machine scheduling with compressible processing times.International Journal of Production Economics 43, 29–35.

Daniels, R.L., Sarin, R.K., 1989. Single machine scheduling with controllable processing times and number of jobs tardy. OperationsResearch 37, 981–984.

Eilon, S., Chowdhury, I.E., 1977. Minimizing waiting time variance in the single machine problem. Management Science 23, 567–575.Elmaghraby, S.E., 1977. Activity Network. Wiley, New York.Hoogeveen, H., Woeginger, G.J., 2002. Some comments on sequencing with controllable processing times. Computing 68, 181–192.Kanet, J.J., 1981. Minimizing variation of flow time in single machine systems. Management Science 27 (12), 1453–1459.Merten, A.G., Muller, M.E., 1972. Variance minimization in single machine sequencing problems. Management Science 18, 518–528.Ng, C.T.D., Cheng, T.C.E., Kovalyov, M.Y., Lam, S.S., 2003. Single machine scheduling with a variable common due date and

resource-dependent processing times. Computers and Operations Research 30, 1173–1185.Ng, C.T.D., Cheng, T.C.E., Kovalyov, M.Y., 2004. Single machine batch scheduling with jointly compressible setup and processing

times. European Journal of Operational Research 153, 211–219.Nowicki, E., Zdrzalka, S., 1988. A two machine flow shop scheduling problem with controllable job processing times. European

Journal of Operational Research 34, 208–220.Nowicki, E., Zdrzalka, S., 1990. A survey of results for sequencing problems with controllable processing times. Discrete Applied

Mathematics 26, 271–287.Panwalkar, S.S., Rajagopalan, R., 1992. Single machine sequencing with controllable processing times. European Journal of

Operational Research 59, 298–302.Schrage, L., 1975. Minimizing the time-in-system variance for a finite jobset. Management Science 21, 540–543.Shabtay, D., Kaspi, M., 2004. Minimizing the total weighted flow time in a single machine with controllable processing times.

Computers and Operations Research 31, 2279–2289.Van Wassenhove, L.N., Baker, K.R., 1982. A bicriterion approach to time/cost trade-offs in sequencing. European Journal of

Operational Research 11, 48–52.Vani, V., Raghavachari, M., 1987. Deterministic and random single machine sequencing with variance minimization. Operations

Research 35, 111–120.Vickson, R.G., 1980a. Choosing the job sequence and processing times to minimize total processing plus flow cost on a single machine.

Operations Research 29, 1155–1167.Vickson, R.G., 1980b. Two single-machine sequencing problems involving controllable job processing times. AIIE Transactions 12,

258–262.Zdrzalka, S., 1991. Scheduling jobs on a single machine with release dates, delivery times and controllable processing times: Worst-case

analysis. Operations Research Letters 10, 519–524.

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Single machine scheduling problems with controllable processing times and total absolute differences penalties