Optim LettDOI 10.1007/s11590-012-0579-0

ORIGINAL PAPER

Single machine scheduling with truncatedjob-dependent learning effect

Xue-Ru Wang · Ji-Bo Wang · Jian Jin · Ping Ji

Received: 26 March 2012 / Accepted: 1 September 2012© Springer-Verlag Berlin Heidelberg 2012

Abstract In this paper we consider the single machine scheduling problem withtruncated job-dependent learning effect. By the truncated job-dependent learningeffect, we mean that the actual job processing time is a function which depends notonly on the job-dependent learning effect (i.e., the learning in the production processof some jobs to be faster than that of others) but also on a control parameter. Theobjectives are to minimize the makespan, the total completion time, the total absolutedeviation of completion time, the earliness, tardiness and common (slack) due-datepenalty, respectively. Several polynomial time algorithms are proposed to optimallysolve the problems with the above objective functions.

Keywords Scheduling · Single machine · Learning effect

1 Introduction

In classical scheduling problems the processing time of a job is assumed to bea constant. However, in many realistic problems of operations management, bothmachines and workers can improve their performance by repeating the productionoperations. Therefore, the actual processing time of a job is shorter if it is scheduledlater in a sequence. This phenomenon is known as the “learning effect” in the literature[2]. Biskup [3] and Cheng and Wang [5] were among the pioneers that brought theconcept of learning into the field of scheduling, although it has been widely employed

X.-R. Wang (B) · J.-B. WangSchool of Science, Shenyang Aerospace University, Shenyang 110136, Chinae-mail: wxrlittle@163.com

J. Jin · P. JiDepartment of Industrial and Systems Engineering, The Hong Kong Polytechnic University,Hung Hom, Kowloon, Hong Kong, China

123

X.-R. Wang et al.

in management science since its discovery by Wright [14]. Biskup [3] assumed thatthe actual processing time of a job J j is pA

jr = p jra if it is scheduled in position rin a sequence, where p j is the normal processing time of job J j and a ≤ 0 is thelearning index. He showed that two important types of the single-machine problemremain polynomially solvable. For more details on scheduling with learning effects,the reader may refer to the recent surveys by Biskup [4] and Janiak and Rudek [8].

Mosheiov and Sidney [10] first considered job-dependent learning effects by assum-ing that the actual processing time of job J j is pA

jr = p jra j if it is scheduled in ther th position of a sequence, where a j ≤ 0 is the job-dependent learning index of jobJ j . They proved that several well-known scheduling problems remain polynomiallysolvable under this more realistic assumption.

However, the actual processing time of a given job drops to zero precipitouslyas the number of the jobs already processed increases in the job-dependent learningeffects proposed by Mosheiov and Sidney [10]. Motivated by this observation, in thispaper we propose a new learning model where the actual job processing time is afunction which depends not only on the job-dependent learning effect but also on acontrol parameter. The use of the truncated function can be justified on the groundsthat learning, like other human activities, is limited [12,13]. The remaining part ofthis paper is organized as follows. In Sect. 2 we formulate the model. In Sect. 3 weconsider several single machine scheduling problems. The last section presents theconclusions.

2 Problems description

There are given a single machine and n independent and non-preemptive jobs thatare immediately available for processing. The machine can handle one job at a timeand preemption is not allowed. Associated with each job J j ( j = 1, 2, ..., n) there is anormal processing time p j , a due date d j . In addition, let p[k] be the normal processingtime of a job if it is scheduled in the kth position in a sequence. Let pA

jr be the actualprocessing time of job J j if it is scheduled in position r in a sequence. In this paper,we consider a new learning effect model, i.e.,

pAjr = p j max{ra j , b}, r, j = 1, 2, . . . , n, (1)

where a j ≤ 0 is the learning index of job J j and b is a truncation parameter with0 < b < 1. It is easy to calculate the actual processing times for the different jobsaccording to their positions in a sequence (Table 1).

For a given schedule π = [J1, J2, . . . , Jn], let C j = C j (π), E j = E j (π), andTj = Tj (π) denote the completion time, the earliness, and the tardiness of job J j ,respectively, where E j = max{0, d j − C j } and Tj = max{0, C j − d j } . In thispaper we will consider the minimization of the following objective functions: themakespan Cmax = max{C j | j = 1, 2, . . . , n}, the total completion time

∑C j , the

total absolute deviation in completion time T ADC = ∑ni=1

∑nj=i |C j −Ci |, the sum

of earliness, tardiness and common due-date penalty∑n

j=1(αE j +βTj +γ d j ), whereα, β and γ are the unit earliness, tardiness and due-date penalty. In the remaining part

123

Single machine scheduling with truncated job-dependent learning effect

Table 1 Matrix of processing times

p j r = 1 r = 2 . . . r = n

p1 p1 max{1a1 , b} p1 max{2a1 , b} . . . p1 max{na1 , b}p2 p2 max{1a2 , b} p2 max{2a2 , b} . . . p2 max{na2 , b}...

pn pn max{1an , b} pn max{2an , b} . . . pn max{nan , b}

of the paper, all the problems considered will be denoted using the three-field notationscheme introduced by Graham et al. [6].

3 A unified analysis for several single machine scheduling

Panwalkar et al. [11] introduced a CON (common) due-date assignment problem inscheduling problem, under which all jobs are assigned the same due date, i.e., d j = dfor j = 1, 2, . . . , n, where d ≥ 0 is a decision variable. The objective was to findan optimal common due date and an optimal schedule which minimizes the totalearliness, tardiness and due-date cost. Panwalkar et al. [11] provided some usefulresults as follows.

Lemma 1 (Panwalker et al. [11]) For the problem 1||∑(αE j +βTj +γ d) with fixedprocessing times, there holds the following properties:

1. There exist an optimal schedule π∗ without any machine idle time between thestarting time of the first job and the completion time of the last job. Furthermore,the first job in the schedule starts at time zero.

2. There exists an optimal schedule with the property d that coincides with thecompletion times of the K th, where K = �n(β − γ )/(α + β)�.

3. The optimal total cost can be written as: f (d, π) = ∑nr=1 μr p[r ], where the

positional weight of position r in the schedule is given by

μr = min{nγ + (r − 1)α, (n + 1 − r)β}, r = 1, 2, . . . , n.

Slack (SLK) due-date assignment method is a well-known due-date assignmentmethod, under which the jobs are given an equal flow allowance according to thefollowing equation, d j = p j + q for j = 1, 2, . . . , n, where slack q ≥ 0 is a decisionvariable. Adamopoulos and Pappis [1] provided some useful results as follows.

Lemma 2 (Adamopoulos and Pappis [1]) For the problem 1||∑(αE j + βTj + γ q)

with fixed processing times, there holds the following properties:

1. There exist an optimal schedule π∗ without any machine idle time between thestarting time of the first job and the completion time of the last job. Furthermore,the first job in the schedule starts at time zero.

2. There exists an optimal schedule with the property q that coincides with thecompletion times of the (K − 1)th, where K = � n(β−γ )

α+β�.

123

X.-R. Wang et al.

3. The optimal total cost can be written as: f (d, π) = ∑nr=1 νr p[r ], where the

positional weight of position r in the schedule is given by

νr = min{nγ + rα, β(n − r)}, r = 1, 2, . . . , n.

Theorem 1 The problem 1|pAjr = p j max{ra j , b}|ρ, ρ ∈ {Cmax,

∑C j , T ADC,

∑(αE j + βTj + γ d),

∑(αE j + βTj + γ q)} can be solved optimally in O(n3)

steps by an assignment problem.

Proof For a job sequence π = [J[1], J[2], . . . , J[n]], we have

Cmax = C[n] = p[1] max{1a[1] , b} + p[2] max{2a[2], b} + · · · + p[n] max{na[n] , b},

∑C j =

n∑

r=1

(n − r + 1)p[r ] max{ra[r ], b},

T ADC =n∑

r=1

(r − 1)(n − r + 1)p[r ] max{ra[r ], b} (Kanet [9]),

∑(αE j + βTj + γ d) =

n∑

r=1

μr p[r ] max{ra[r ], b},

where μr = min{nγ + (r − 1)α, (n + 1 − r)β} (Lemma 1), and

∑(αE j + βTj + γ q) =

n∑

r=1

νr p[r ] max{ra[r ], b},

where νr = min{nγ + rα, β(n − r)} (Lemma 2).For 1 ≤ j, r ≤ n, let us define

c jr = p j max{ra j , b} (2)

for the makespan objective function;

c jr = p j (n − r + 1) max{ra j , b} (3)

for the total completion time objective function;

c jr = p j (r − 1)(n − r + 1) max{ra j , b} (4)

123

Single machine scheduling with truncated job-dependent learning effect

for the TADC objective function;

c jr = μr p j max{ra j , b} (5)

for the∑

(αE j + βTj + γ d) objective function;

c jr = νr p j max{ra j , b} (6)

for the∑

(αE j + βTj + γ q) objective function. It is clear that c jr represents theminimum possible cost resulting from assigning job J j to position r in the sequence.

We define the 0–1 variables z jr ( j = 1, 2, . . . , n; r = 1, 2, . . . , n) suchthat z jr = 1 if job J j is processed in the r th position and z jr = 0 other-wise. As in Biskup [3], the scheduling problem 1|pA

jr = p j max{ra j , b}|ρ, ρ ∈{Cmax,

∑C j , T ADC,

∑(αE j + βTj + γ d),

∑(αE j + βTj + γ q)} can be formu-

lated as the following assignment problem:

minn∑

j=1

n∑

r=1

c jr z jr (7)

subject ton∑

r=1

z jr = 1, j = 1, 2, . . . , n,

n∑

j=1

z jr = 1, r = 1, 2, . . . , n,

z jr = 0 or 1, j, r = 1, 2, . . . , n.

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 foundin O(n3) time. �

From Lemma 1, Lemma 2 and Theorem 1, for the problem 1|pAjr = p j max{ra j ,

b}|ρ, ρ ∈ {∑(αE j +βTj +γ d),∑

(αE j +βTj +γ q)}, we can propose the followingoptimization algorithm:Algorithm 1.Step 1. Calculate K = �n(β − γ )/(α + β)�.Step 2. For the CON method, calculate the c jr by using Eq. (5). For the SLK method,calculate the c jr by using Eq. (6).Step 3. Solve the assignment problem (7) to determine the optimal job sequence.Step 4. For the CON method, set the optimal due date d equal to the completion time ofjob in position K . For the SLK method, set the optimal slack q equal to the completiontime of job in position K − 1.

123

X.-R. Wang et al.

Theorem 2 For the problem 1|pAjr = p j max{ra j , b}, a j = a|Cmax, an optimal

schedule can be obtained by sequencing the jobs in non-decreasing order of p j [thesmallest processing time (SPT) first rule].

Proof The proof follows directly from the pairwise interchange analysis. Let π and π ′be two job schedules where the difference between π and π ′ is a pairwise interchangeof two adjacent jobs J j and Jk , that is, π = [S1, J j , Jk, S2], π ′ = [S1, Jk, J j , S2],where S1 and S2 are partial sequences. Furthermore, we assume that there are r − 1jobs in S1. Thus, J j and Jk are the r th and the (r +1)th jobs with p j ≤ pk , respectively,in π . Likewise, Jk and J j are scheduled in the r th and the (r +1)th positions in π ′. Tofurther simplify the notation, let A denote the completion time of the last job in S1 andJh be the first job in S2. To show π dominates π ′, it suffices to show that the (r + 1)thjobs in π and π ′ satisfy the condition that Ck(π) ≤ C j (π

′) and Cu(π) ≤ Cu(π ′) forany Ju in S2. Under π , the completion times of jobs J j and Jk are

C j (π) = A + p j max{ra, b} (8)

and

Ck(π) = A + p j max{ra, b} + pk max{(r + 1)a, b}. (9)

Under π ′, the completion times of jobs Jk and J j are

Ck(π′) = A + pk max{ra, b} (10)

and

C j (π′) = A + pk max{ra, b} + p j max{(r + 1)a, b}. (11)

Based on Eqs. (9) and (11), we have

C j (π′) − Ck(π) = (pk − p j )[max{ra, b} − max{(r + 1)a, b}] ≥ 0. (12)

Note that

Ch(π ′) = C j (π′) + ph max{(r + 2)a, b} (13)

and

Ch(π) = Ck(π) + ph max{(r + 2)a, b}. (14)

From Eqs. (12)–(14), we have Ch(π ′) ≥ Ch(π) since C j (π′) ≥ Ck(π). In other

words, we have showed that the first job Jh in S2, which starts earlier in π than π ′,completes earlier in π . Similarly, we have Cu(π ′) ≥ Cu(π) for any Ju in S2.

Hence, repeating this argument will lead to the optimality of the SPT sequence forthe problem 1|pA

jr = p j max{ra j , b}, a j = a|Cmax. �

123

Single machine scheduling with truncated job-dependent learning effect

Theorem 3 For the problem 1|pAjr = p j max{ra j , b}, a j = a| ∑ C j , an optimal

schedule can be obtained by sequencing the jobs in non-decreasing order of p j (theSPT rule).

Proof Here, we still use the same notations as in the Proof of Theorem 2. In order toshow π dominates π ′, it suffices to show that (i) Ck(π) ≤ C j (π

′) and (ii) C j (π) +Ck(π) ≤ Ck(π

′) + C j (π′).

The proof of part (i) is given in Theorem 2. In addition, from p j ≤ pk , we haveC j (π) ≤ Ck(π

′), hence

C j (π) + Ck(π) ≤ Ck(π′) + C j (π

′).

This completes the Proof of part (ii) and thus of the theorem. �Theorem 4 The problem 1|pA

jr = p j max{ra j , b}, a j = a|T ADC can be solved inO(n log n) time.

Proof From Kanet [9] we have

T ADC =n∑

r=1

(r − 1)(n − r + 1)p[r ] max{ra, b}. (15)

Equation (15) can be viewed as the scalar product of two vectors, the wr =(r − 1)(n − r + 1) max{ra, b} and p[r ] vectors respectively (r = 1, 2, . . . , n). Itis well known ([7], P261) that Eq. (15) is minimized by sorting the elements of thewr and p[r ] vectors in opposite orders. Consequently, an optimal sequence for the1|pA

jr = p j max{ra j , b}, a j = a|T ADC problem can be derived in O(n log n) timeby implementing a simple sorting procedure. �Theorem 5 The problem 1|pA

jr = p j max{ra j , b}, a j = a| ∑nj=1(αE j +βTj +γ d)

can be solved in O(n log n) time.

Proof From Lemma 1 and Eq. (1) we have

n∑

j=1

(αE j + βTj + γ d) =n∑

r=1

μr p[r ] max{ra, b}. (16)

Equation (16) can be viewed as the scalar product of two vectors, the θr =μr max{ra, b} and p[r ] vectors, respectively (r = 1, 2, . . . , n). It is well known([7], P261) that Eq. (16) is minimized by sorting the elements of the θr and p[r ]vectors in opposite orders. Consequently, an optimal sequence for the 1|pA

jr =p j max{ra j , b}, a j = a| ∑n

j=1(αE j +βTj +γ d)problem can be derived in O(n log n)

time by implementing a simple sorting procedure, i.e., the optimal sequence can beobtained in the following way: assign the job with the largest p j value to the positionwith the smallest value of θr , the job with the second largest p j value to the positionwith the second smallest value of θr , and so on. �

123

X.-R. Wang et al.

Table 2 Summary of complexity results obtained in this paper

Problem Complexity Reference

1|pAjr = p j max{ra j , b}|Cmax O(n3) Theorem 1

1|pAjr = p j max{ra j , b}|∑ C j O(n3) Theorem 1

1|pAjr = p j max{ra j , b}|T ADC O(n3) Theorem 1

1|pAjr = p j max{ra j , b}|∑(αE j + βTj + γ d) O(n3) Theorem 1

1|pAjr = p j max{ra j , b}|∑(αE j + βTj + γ q) O(n3) Theorem 1

1|pAjr = p j max{ra j , b}, a j = a|Cmax O(n log n) Theorem 2

1|pAjr = p j max{ra j , b}, a j = a| ∑ C j O(n log n) Theorem 3

1|pAjr = p j max{ra j , b}, a j = a|T ADC O(n log n) Theorem 4

1|pAjr = p j max{ra j , b}, a j = a| ∑(αE j + βTj + γ d) O(n log n) Theorem 5

1|pAjr = p j max{ra j , b}, a j = a| ∑(αE j + βTj + γ q) O(n log n) Theorem 6

Theorem 6 The problem 1|pAjr = p j max{ra j , b}, a j = a| ∑n

j=1(αE j + βTj + γ q)

can be solved in O(n log n) time.

Proof From Lemma 2 and Eq. (1) we have

n∑

j=1

(αE j + βTj + γ q) =n∑

r=1

νr p[r ] max{ra, b}. (17)

Equation (17) can be viewed as the scalar product of two vectors, the ηr =νr max{ra, b} and p[r ] vectors, respectively (r = 1, 2, . . . , n). It is well known([7], P261) that Eq. (17) is minimized by sorting the elements of the ηr and p[r ]vectors in opposite orders. Consequently, an optimal sequence for the 1|pA

jr =p j max{ra j , b}, a j = a| ∑n

j=1(αE j +βTj +γ q)problem can be derived in O(n log n)

time by implementing a simple sorting procedure, i.e., the optimal sequence can beobtained in the following way: assign the job with the largest p j value to the positionwith the smallest value of θr , the job with the second largest p j value to the positionwith the second smallest value of θr , and so on. �

4 Conclusions

In this paper we considered single machine scheduling problems with truncated job-dependent learning effect. We proved that the makespan minimization problem, thetotal completion time minimization problem, the total absolute deviation of completiontime minimization problem, and the earliness, tardiness and CON (SLK) due-datepenalty minimization problem can be solved in polynomial time (see Table 2). Severaldifferent directions for the future research may focus on studying the other objectivefunctions, or designing the efficient heuristics and approximation algorithms for thehard cases.

123

Single machine scheduling with truncated job-dependent learning effect

Acknowledgements The authors are grateful for the editor and two anonymous referees for their helpfulcomments on earlier version of the article. This research was supported by the National Natural ScienceFoundation of China (Grant No. 11001181), The Hong Kong Polytechnic University (PolyU’s projectG-YL27) and the Program for Liaoning Excellent Talents in University (Grant No. LJQ2011014).

References

1. Adamopoulos, G.I., Pappis, C.P.: Single machine scheduling with flow allowances. J. Oper. Res. Soc.47, 1280–1285 (1996)

2. Badiru, A.B.: Computational survey of univariate and multivariate learning curve models. IEEE Trans.Eng. Manag. 39, 176–188 (1992)

3. Biskup, D.: Single-machine scheduling with learning considerations. Eur. J. Oper. Res. 115, 173–178(1999)

4. Biskup, D.: A state-of-the-art review on scheduling with learning effects. Eur. J. Oper. Res. 188,315–329 (2008)

5. Cheng, T.C.E., Wang, G.: Single machine scheduling with learning effect considerations. Ann. Oper.Res. 98, 273–290 (2000)

6. Graham, R.L., Lawler, E.L., Lenstra, J.K.: Optimization and approximation in deterministic sequencingand scheduling: a survey. Ann. Discret. Math. 5, 287–326 (1979)

7. Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1976)8. Janiak, A., Rudek, R.: Experience-based approach to scheduling problems with the learning effect.

IEEE Trans. Syst. Man Cybern. A 39, 344–357 (2009)9. Kanet, J.J.: Minimizing variation of flow time in single machine systems. Manag. Sci. 27, 1453–1459

(1981)10. Mosheiov, G., Sidney, J.B.: Scheduling with general job-dependent learning curves. Eur. J. Oper. Res.

147, 665–670 (2003)11. Panwalkar, S.S., Smith, M.L., Seidmann, A.: Common due date assignment to minimize total penalty

for the one machine scheduling problem. Oper. Res. 30, 391–399 (1982)12. Wang, J.-B., Wang, X.-Y., Sun, L.-H., Sun, L.-Y.: Scheduling jobs with truncated exponential learning

functions. Optim. Lett. doi:10.1007/s11590-011-0433-913. Wu, C.-C., Yin, Y., Cheng, S.-R.: Some single-machine scheduling problems with a truncation learning

effect. Comput. Ind. Eng 60, 790–795 (2011)14. Wright, T.P.: Factors affecting the cost of airplanes. J. Aeronaut. Sci. 3, 122–128 (1936)

123