Int J Adv Manuf Technol (2011) 54:743–748DOI 10.1007/s00170-010-2973-z

ORIGINAL ARTICLE

Single-machine scheduling with controllableprocessing times and learning effect

Na Yin · Xiao-Yuan Wang

Received: 1 July 2009 / Accepted: 29 September 2010 / Published online: 16 October 2010© Springer-Verlag London Limited 2010

Abstract In this paper, we consider single-machinescheduling problem with controllable processing timesand learning effect, i.e., processing times of jobs arecontrollable variables with linear costs and also aredefined as functions of positions in a schedule. Weconcentrate on two goals separately, namely minimiz-ing a cost function containing makespan, total com-pletion time, total absolute differences in completiontimes, and total compression cost and minimizing a costfunction containing makespan, total waiting time, totalabsolute differences in waiting times, and total com-pression cost. The problem is modeled as an assignmentproblem and thus can be solved with the well-knownalgorithms.

Keywords Scheduling · Single machine · Controllableprocessing times · Learning effect

1 Introduction

It is no doubt that learning can play a role in manu-facturing environments and learning effects have beenproven to exist by many empirical studies [1]. An ex-tensive review of research on scheduling with learningeffect could be found in Biskup [2]. More recent paperswhich have considered scheduling jobs with learningeffects include Wang et al. [3], Mosheiov [4], Toksar andGuner [5], Wang [6], Wang et al. [7, 8], Wang [9–11],

N. Yin (B) · X.-Y. WangSchool of Science, Shenyang Aerospace University,Shenyang, 110136, Chinae-mail: shenyinyin@126.com

Wang and Liu [12], Wang et al. [13], Eren and Guner[14], Toksari and Guner [15], Wang et al. [16], Wangand Guo [17], Wang et al. [18], Yin et al. [19], Wanget al. [20], Wang and Wang [21], Wang et al. [22], Yinet al. [23], Wang et al. [24, 25], Wang and Wang [26],Wang et al. [27], and Wang and Li [28]. Wang et al. [3]considered single-machine scheduling problems withtime-dependent learning effect. They proved that theweighted shortest processing time rule, the earliest duedate rule, and the modified Moore–Hodgson algorithmcan, under certain conditions, construct the optimalschedule for the problem to minimize the followingthree objectives: the total weighted completion time,the maximum lateness, and the number of tardy jobs,respectively. They also gave an error estimation foreach of these rules for the general cases. Mosheiov[4] considered the problem of minimizing total absolutedeviation of job completion times (TADC). Heshowed that with both extensions (simultaneously), i.e.,(a) position-dependent processing times and (b) paral-lel identical machines, the problem of minimizing thesum of the TADC values on all the machines remainspolynomially solvable. Toksar and Guner [5] consid-ered the parallel machine earliness/tardiness schedul-ing with simultaneous effects of learning and lineardeterioration, sequence-dependent setups, and a com-mon due date for all jobs. They introduced a mixednonlinear integer programming formulation for theproblem. Wang [6] considered single-machine schedul-ing with a sum-of-actual-processing-time-based learningeffect. He showed that the makespan minimization prob-lem, the total completion time minimization problem,and the total completion time square minimization prob-lem can be solved by the smallest (normal) processingtime first (SPT) rule. Wang et al. [7] considered some

744 Int J Adv Manuf Technol (2011) 54:743–748

single-machine scheduling problems with past-sequence-dependent setup times and the effects ofdeterioration and learning. They proved that the make-span minimization problem, the total completion timeminimization problem, and the sum of the δth (δ ≥ 0)power of job completion times minimization problemcan be optimally solved, respectively. They also provedthat some special cases of the total weighted completiontime minimization problem, the maximum latenessminimization problem, and the number of tardy jobsminimization problem can be solved in polynomialtime. Wang et al. [8] considered the single-machinescheduling problem with exponential time-dependentlearning effect and past-sequence-dependent setuptimes. They showed that the makespan minimizationproblem, the total completion time minimization prob-lem, and the sum of the quadratic job completion timesminimization problem can be solved by the SPT rule,respectively. Other types of learning effect jobs havealso been discussed; the reader is referred to papersby Wang [9–11], Wang and Liu [12], Wang et al. [13],Eren and Guner [14], Toksari and Guner [15], Wanget al. [16], Wang and Guo [17], Wang et al. [18], Yin et al.[19], Wang et al. [20], Wang and Wang [21], Wang et al.[22], Yin et al. [23], Wang et al. [24, 25], Wang andWang [26], Wang et al. [27], and Wang and Li [28].

On the other hand, works in the scheduling prob-lem with controllable processing times and linear costfunctions are surveyed by Nowicki and Zdrzalka [29].Zdrzalka [30] considered single-machine schedulingproblem in which each job has a release date, a deliverytime, and a controllable processing time. He gave anapproximation algorithm for minimizing the overallschedule cost. Panwalkar and Rajagopalan [31] con-sidered the common due date assignment and single-machine scheduling problem in which the objective isthe sum of penalties based on earliness, tardiness, andprocessing time compressions. They reduced the problemto an assignment problem. Alidace and Ahmadian [32]extended the results of Panwalkar and Rajagopalan[31] to the parallel machine scheduling case. Cheng andJaniak [33] further generalized the result to the casewhere the cost of compression is a general convex func-tion of the amount of compression. Cheng et al. [34]considered a due date assignment and single-machinescheduling in which a penalty for due dates is addedto the objective function which includes the penaltiesfor earliness, tardiness, and processing time compres-sions. Alidaee and Kochenberger [35] considered singleand parallel machine scheduling problems in which jobprocessing time of a job was assumed to depend on theposition of the job in the schedule and is a functionof units of resource applied for its processing. The

processing time and the processing cost functions are al-lowed to be nonlinear. For the single-machine problem,the objective was minimization of total compressioncosts plus a scheduling measure. Biskup and Cheng [36]considered a due date assignment and single-machinescheduling in which a penalty for completion timesis added to the objective function which includes thepenalties for earliness, tardiness, and processing timecompressions. Biskup and Jahnke [37] considered theproblem of assigning a common due date to a set ofjobs and scheduling them on a single machine withjointly reducible processing times. Besides consideringdue date assignment costs, the first goal is to minimizethe sum of earliness and tardiness penalties while thesecond one is to minimize the number of late jobs. Forboth cases, polynomially solvable algorithms have beengiven. Hoogeveen and Woeginger [38] combined theresource allocation and the weighted flow time coststo a single objective and proved that this problem isNP-hard. Ng et al. [39] considered the single-machineproblem with a variable common due date. They pre-sented polynomial time algorithms for minimizing alinear combination of scheduling, due date assignment,and resource consumption costs. Shabtay and Kaspi[40] considered a single-machine scheduling problemwith the minimum total weighted completion time cri-terion where the model of operations is assumed to be aspecific convex function of the amount of resource con-sumed. They presented and analyzed some special casesthat are solvable by using polynomial time algorithms.They also gave some heuristic algorithms for the gen-eral case. Ng et al. [41] considered the single-machinebatch scheduling with jointly compressible setup andprocessing times. They presented polynomial time al-gorithms to find an optimal batch sequence and optimalamounts of resource consumption such that either totaljob completion time is minimized, subject to an upperbound on total weighted resource consumption, or totalweighted resource consumption is minimized, subjectto an upper bound on total job completion time. Wang[42] considered single-machine common due date as-signment scheduling problem in which job processingtimes are controllable variables with linear costs. Theobjective is to determine the optimal sequence, theoptimal common due date, and the optimal processingtime compressions to minimize a total penalty functionbased on the common due date, job absolute valuein lateness, and compressions. He proved that theproblem can be solved in polynomial time. Wang [43]considered single-machine slack due date assignmentscheduling problem in which job processing times arecontrollable variables with linear costs. The objectiveis to determine the optimal sequence, the optimal

Int J Adv Manuf Technol (2011) 54:743–748 745

common flow allowance, and the optimal processingtime compressions to minimize a total penalty functionbased on earliness, tardiness, common flow allowance,and compressions. He solved the problem by formu-lating it as an assignment problem. Wang and Xia [44]considered single-machine scheduling with controllableprocessing times. The objective function is to minimizea cost function containing total completion (waiting)time, total absolute differences in completion (waiting)times, and total compression cost. They solved theproblem by formulating it as an assignment problem.Tseng et al. [45] considered the single-machine totaltardiness problem with controllable processing times.They proposed a mixed-integer programming model tofind the optimal solution. They also proposed both alinear programming model and a net benefit of com-pression algorithm to obtain a set of optimal amountsof compression for a given sequence.

In this paper, we consider single-machine schedulingproblem with controllable processing times and learn-ing effect at the same time. The phenomena of jobswith controllable processing times and learning effectsoccurring simultaneously can be found in many real-life systems. For example, processing times of jobs arecontrollable by using limited disposal resources such asfinancial budget, energy, fuel, catalyzer, subcontract-ing, or manpower. On the other hand, the learningeffects reflect that the workers become more skilledto operate the machines through experience accumu-lation. Thus, the scheduler has to considering the joblearning effects and resource allocation in job schedul-ing.

The rest of this paper is organized as follows: Nota-tions and assumptions are given in Section 2. In Sec-tion 3, we obtain optimal compressions for any givensequence. In Section 4, we show that the problem canbe formulated as an assignment problem. In Section 5,conclusions are presented.

2 Notations and assumptions

Consider a set of n jobs J = {J1, J2, . . . , Jn} to beprocessed in a single machine with the following as-sumptions:

• 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 si-

multaneously.

• After the process in the machine has started, no idletime can be inserted in the schedule.

• Processing costs are decreasing linear functions ofprocessing times.

The following notations will be used throughout thepaper:

s The sequence of jobs to be processed bythe machine

[ j] The job in the jth positionti The normal processing time of Ji

t′i The crash processing time of Ji

Gi The per time unit cost associated with thecompression below ti of the processingtime of job Ji

mi = ti − t′i The maximum reduction in processingtime of job Ji

xi The compression of the processing time ofjob Ji which can take any value in [0, mi]

pir the actual processing time of job Ji inposition r in a sequence, that is pir = (ti −xi)ra, where a ≤ 0 is a learning effect [46]

p[i] the actual processing time of the job, sayjob Ji, in position r in a sequence, that isp[i] = pir

Ci The completion time of job Ji

Wi The waiting time of job Ji

Cmax The makespan of all jobs, that is Cmax =max{C j| j = 1, 2, . . . , n}

TC The total completion times, that isTC=

∑ni=1 Ci

TW The total waiting times, that isTW=

∑ni=1 Wi

TADC The total absolute differences in comple-tion times that is

TADC =n∑

i=1

n∑

j=i

|Ci − C j|

TADW The total absolute differences in waitingtimes, that is

TADW =n∑

i=1

n∑

j=i

|Wi − W j|

The objective is to determine the optimal compres-sions of the processing times and the optimal sequenceof jobs in the machine so that the corresponding valueof the following cost functions be optimal:

f (s, xi) = δ1Cmax + δ2TC + δ3TADC + δ4

n∑

i=1

Gixi, (1)

746 Int J Adv Manuf Technol (2011) 54:743–748

f (s, xi) = δ1Cmax + δ2TW + δ3TADW + δ4

n∑

i=1

Gixi,

(2)

where weights δ1 ≥ 0, δ2 ≥ 0, δ3 ≥ 0, and δ4 ≥ 0 aregiven constants (the decision maker selects the weightsδ1, δ2, δ3, δ4).

3 Optimal compressions with a learning effect

For the model 1, if we substitute, C[ j] = ∑ ji=1 p[i],

Cmax = ∑ni=1 p[i], TC=

∑nj=1 C[ j], TADC=

∑nj=1( j −

1)(n − j + 1)p[ j] [47], and x[ j] = t[ j] − p[ j] j−a into Eq. 1and simplify, we have

f (s, xi) = δ1

n∑

j=1

p[ j] + δ2

n∑

j=1

(n − j + 1)p[ j]

+ δ3

n∑

j=1

( j − 1)(n − j + 1)p[ j]

+ δ4

n∑

j=1

G[ j](t[ j] − p[ j] j−a)

=n∑

j=1

[δ1 + δ2(n + 1 − j) + δ3( j − 1)(n − j + 1)

− δ4G[ j] j−a]p[ j] + δ4

n∑

j=1

G jt j.

Let

λ j = δ1 + δ2(n + 1 − j) + δ3( j − 1)(n − j + 1)

− δ4G[ j] j−a, 1 ≤ j ≤ n, (3)

then λ j, 1 ≤ j ≤ n, represents the position weight ofposition j in the sequence s. Since δ4

∑nj=1 G jt j is a con-

stant, for any sequence, the optimal processing time of ajob in a position with a negative position weight shouldbe its normal processing time, and the processing timeof a job in a position with a positive position weightshould be its crash processing time. If a position j has azero position weight, then the optimal processing timeof the job in this position may be any value between t′jand t j. These can be written in the notational form asfollows:

p∗[ j] =

⎧⎨

⎩

t[ j] ja, if λ j < 0,

p′[ j] ja, if λ j = 0,

t′[ j] ja, if λ j > 0,

(4)

where t′[ j] ≤ p′[ j] ≤ t[ j] and p∗

[ j], 1 ≤ j ≤ n, represents theoptimal processing time of the job in position j. There-fore, the optimal compressions can be obtained by

x∗[ j] = t[ j] − p∗

[ j] j−a, j = 1, 2, . . . , n. (5)

For the model 2, if we substitute, Cmax = ∑ni=1 p[i],

W[ j] = ∑ j−1i=1 p[i], TW=

∑nj=1 W[ j], TADW=

∑nj=1 j(n −

j)p[ j] [48], and x[ j] = t[ j] − p[ j] j−a into Eq. 2 and sim-plify, we have

f (s, xi) =n∑

j=1

[δ1 + δ2(n − j) + δ3 j(n − j)

− δ4G[ j] j−a]p[ j] + δ4

n∑

j=1

G jt j.

Let λ j and p∗[ j], 1 ≤ j ≤ n, denote the position weight of

position j and the optimal processing time of the job inposition j, respectively, then

λ j = δ1 + δ2(n − j) + δ3 j(n − j) − δ4G[ j] j−a, 1 ≤ j ≤ n,

(6)

p∗[ j] =

⎧⎪⎨

⎪⎩

t[ j] ja, if λ j < 0,

p′[ j] ja, if λ j = 0,

t′[ j] ja, if λ j > 0,

(7)

where t′[ j] ≤ p′[ j] ≤ t[ j]. Using the same argument as for

model 1, we see that the optimal compressions can beobtained by

x∗[ j] = t[ j] − p∗

[ j] j−a, j = 1, 2, . . . , n. (8)

Theorem 1 Given a sequence, for the model, the optimalcompressions with a learning ef fect can be determinedas follows: The compression of the job in a negative-weight position is zero; the compression of the job ina positive-weight position is its maximum reduction inthe processing time with a learning ef fect; if the positionweight of a position is zero, then the compression of thejob in this position can be any value between zero and itsmaximum reduction in processing time.

Proof The proof follows from the analysis above. ��

We demonstrate the result of Theorem 1 in thefollowing example:

Example 1 Consider the model f (s, xi) = δ1Cmax +δ2TC + δ3TADC + δ4

∑ni=1 Gixi, where n = 4, t1 = 2,

t2 = 3, t3 = 4, t4 = 5, t′1 = 1, t′2 = 2, t′3 = 3, t′4 = 4, G1 = 3,G2 = 5, G3 = 4, G4 = 2, δ1 = δ2 = δ3 = δ4 = 1, a = −0.5.

Int J Adv Manuf Technol (2011) 54:743–748 747

For a given sequence s = [J1, J2, J3, J4], from Eqs. 3and 4, we have λ1 = 2, λ2 = −0.0711, λ3 = 0.0718,λ4 = 1, p∗

1 = 1 ∗ 1−0.5 = 1, p∗2 = 3 ∗ 2−0.5 = 2.1213, p∗

3 =3 ∗ 3−0.5 = 1.7321, p∗

4 = 4 ∗ 4−0.5 = 2.

4 Optimal sequences with a learning effect

Now we discuss the determination of optimal sequencesfor the model. In view of the analysis in the previoussections, where we provided the expressions for com-puting the optimal processing times and compressionsfor any given optimal sequence with a learning effect,the problem reduces to a pure sequencing problem. Inorder to obtain the optimal sequence, we formulate themodels (Eq. 1) as an assignment problem, respectively.

For the model 1, let

λij = δ1 + δ2(n + 1 − j) + δ3( j − 1)(n − j + 1)

− δ4Gi j−a, i, j = 1, 2, . . . , n,

and

pij =⎧⎨

⎩

ti ja, if λij < 0,

p′i ja, if λij = 0,

t′i ja, if λij > 0,

(9)

where t′i ≤ p′i ≤ ti. Furthermore, let zij be a 0/1 variable

such that zij = 1 if job Ji is scheduled in position j andzij = 0, otherwise. As in Panwalkar and Rajagopalan[31], the optimal matching of jobs to positions requiresa solution for the following assignment problem:

minn∑

i=1

n∑

j=1

λij pijzij (10)

subject to

n∑

i=1

zij = 1, i = 1, 2, . . . , n,

n∑

j=1

zij = 1, j = 1, 2, . . . , n,

zij = 0 or 1, i, j = 1, 2, . . . , n.

For the model 2, let

λij = δ1 + δ2(n − j) + δ3 j(n − j) − δ4Gi j−a, i,

j = 1, 2, . . . , n.

p∗ij =

⎧⎨

⎩

ti, if λij < 0,

p′i, if λij = 0,

t′i, if λij > 0,

(11)

where t′i ≤ p′i ≤ ti. The optimal sequence is obtained, as

the same assignment problem (Eq. 10).Recall that solving an assignment problem of size

n requires an effort of O(n3) (using the well-knownHungarian method); hence, the optimal sequence canbe found in polynomial time.

5 Conclusions

The problem of scheduling n jobs with controllableprocessing times and learning effect has been studied.The objective function is to minimize a cost func-tion containing makespan, total completion (waiting)time, total absolute differences in completion (waiting)times, and total compression cost. We have solved theproblem by formulating it as an assignment problem.In future research, we plan to explore more generallearning effects, consider different resource consump-tion functions, and extend the problems to multiplemachine settings.

Acknowledgements We are grateful to the editor and twoanonymous referees for their helpful comments on an earlierversions of this paper. This research was supported by the Na-tional Natural Science Foundation of China under grant number11001181.

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