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Scheduling with general position-based learning curves
Information Sciences 181 (2011) 5515–5522
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Scheduling with general position-based learning curves Wen-Chiung Lee Department of Statistics, Feng Chia University, Taichung, Taiwan
a r t i c l e
i n f o
Article history: Received 22 February 2010 Received in revised form 9 March 2011 Accepted 23 July 2011 Available online 11 August 2011 Keywords: Scheduling Learning curve Single-machine Two-machine flowshop
a b s t r a c t Learning effect in scheduling problems has received growing attention since Biskup [3] introduced the position-based model, where the learning curve is expressed as a power function of a job position. Hurley [11] pointed out that the actual processing time of a given job drops to zero precipitously as the number of jobs increases in the standard power model. Moreover, the learning rates show considerable variation within industries or firms. The variation extends not only across firms at a given time, but also within firms over time. For instance, the learning curves usually have an initial downward concavity, and no further improvements are made after some amount of production. Beside the standard power model, learning curve is seldom discussed in scheduling. In this paper, we offer a surprising simple yet realistic learning effect model which has the flexibility to describe different learning curves easily. For instance, the standard power model, the well-known S-shaped and the plateau functions are special cases of the proposed model. We then present the optimal solution for some scheduling problems. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Job processing times are assumed to be fixed and known in traditional scheduling problems. However, recent empirical studies in several industries have verified that unit costs decline as firms produce more of a product and gain knowledge or experience. For instance, Biskup [3] pointed out that repeated processing of similar tasks improves worker skills; workers are able to perform setup, to deal with machine operations and software, or to handle raw materials and components at a greater pace. Gawiejnowicz [10], Donheti and Mohanty [8], Biskup [3] and Cheng and Wang [5] were among the pioneers to bring the concept of learning effect into scheduling problems. Biskup [3] introduced the job-position-based learning effect model in which the actual job processing time is a power function of its position in schedule. He showed that two single-machine scheduling problems remain polynomially solvable. Mosheiov [16] gave several examples to demonstrate that the optimal schedules of some problems with learning effect may be different from those of the classical ones without learning consideration. Lee et al. [15] studied the single machine bi-criterion scheduling problem to minimize the sum of the total completion time and the maximum tardiness. Zhao et al. [22] developed the optimal solutions of some single machine and flowshop problems in some special cases. Koulamas and Kyparisis [13] expressed the learning effect as a function of the normal processing times of jobs already processed. They showed that the shortest processing time (SPT) sequence is optimal for the single-machine makespan and the total completion time problems. In addition, they also proved that the two-machine flowshop makespan and the total completion time problems are optimally solved by the SPT rule when the job processing times are ordered. Wang [18] considered some single-machine problems with the effects of learning and deterioration. Moreover, Biskup [4] provided a comprehensive survey of scheduling problems with learning effects. Recently, Janiak and Rudek [12]
E-mail address: [email protected] 0020-0255/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.07.051
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W.-C. Lee / Information Sciences 181 (2011) 5515–5522
considered a scheduling problem in which each job provides a different experience to the processor. They formulated the shape of the learning curve as a non-increasing k-stepwise function. Cheng et al. [6,7] considered some single-machine scheduling problems with specific forms of learning effect. Wu and Lee [19] considered the total completion time problem in the multiple machine permutation flowshop. Yin et al. [20] provided a general model with position-dependent and timedependent learning effects. Zhang and Yan [21] provided the optimal schedules for some single machine problems. Among the articles in scheduling with learning effects, it is assumed in the majority of models that the learning curve is expressed as a power function of its position in schedule. Despite its frequent use, Hurley [11] pointed out that the actual processing time of a given job drops to zero precipitously as the number of jobs increases in the standard power model. Moreover, Lapre et al. [14] claimed that the traditional power form does not accommodate two often observed patterns: initial downward concavity and the plateau effect where no further improvements are made after some amount of production. In this paper, we present a new model of the learning effect which has the shaping flexibility. The flexibility of the proposed model allows us to describe different learning curves or fit more complex environments easily. As pointed out in the Biskup [4] survey paper, it is a challenge to model with the learning effect as realistic as possible. Thus, the first contribution of this paper is to provide a surprisingly simple yet realistic model. It is completely unexpected in the sense that a simple model can describe many different learning curves in the literature, such as the well-known plateau or the S-shaped phenomena, which are more realistic. Moreover, Dutton and Thomas [9] found that the learning rates show considerable variation within industries or firms after a study of more than 200 learning curves. The variation extends not only across firms at a given time, but also within firms over time. Thus, the second contribution of the paper is to provide a more flexible model. The remainder of this note is organized as follows. The solution procedures for some single-machine problems are presented in the next section. In Section 3, the optimal solution for a two-machine flowshop problem is derived. The conclusion is given in the last section.
2. Single-machine single-criterion problems Formulation of the proposed learning effect model in the single-machine case is as follows. There are n jobs to be processed on a single machine. Each job j has a normal processing time pj, a weight wj, and a due date dj. Due to the learning effect, the actual processing time of job j is
pj½r ¼ pj
r1 Y
al ;
ð1Þ
l¼0
if it is scheduled in the rth position in a sequence, where a0 = 1 and 0 < al 6 1 for l = 1, . . . , n. Note that al denotes the learning Q impact on processing the lth position job and rl¼0 al denotes the cumulative level of learning effect after processing r jobs. The proposed model provides the shaping flexibility. For instance, the value of al can be set to a moderate large value for the stages of strong learning effects, and al is set to 1 or a value very close to 1 for the stages where the learning effect is weak. If Q al = 0.9 + 0.01l for l = 1, . . . , 10 and al = 1 for l = 11, . . . , 20, the learning curve rl¼0 al is the plateau function as shown in Fig. 1, where the actual processing time of a given job will not drop to zero precipitously as the number of jobs increases. If Q al = 0.999 0.05(0.1l 1)2 for l = 1, . . . , 20, the learning curve rl¼0 al is S-shaped as shown in Fig. 2. In fact, it reduces to b the Biskup [3] model if we choose al = (1 + 1/l) where b < 0. This flexibility allows us to fit more learning curves. For convenience, we denote the learning model by LE. Thus, using conventional notation for describing scheduling problems, the single-machine makespan problem is denoted as 1jLEjCmax, for instance.
Fig. 1. The plateau learning curve.
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Fig. 2. The S-shaped learning curve.
We will prove the following properties using the well-known pairwise interchange technique. Suppose that S and S0 are two job schedules and the difference between S and S0 is a pairwise interchange of two adjacent jobs i and j, i.e., S = (p, i, j, p0 ) and S0 = (p, j, i, p0 ), where p and p0 each denotes a partial sequence. Furthermore, we assume that there are r 1 scheduled jobs in p and B is the completion time of the last job in p. Under S, the completion times of jobs i and j are respectively
C i ðSÞ ¼ B þ pi
r1 Y
al
ð2Þ
l¼0
and
C j ðSÞ ¼ B þ pi
r1 Y
al þ pj
l¼0
r Y
al :
ð3Þ
l¼0
Similarly, the completion times of jobs j and i in S0 are respectively
C j ðS0 Þ ¼ B þ pj
r1 Y
al
ð4Þ
l¼0
and
C i ðS0 Þ ¼ B þ pj
r1 Y
al þ pi
l¼0
r Y
al :
ð5Þ
l¼0
Property 1. .For the 1jLEjCmax problem, the optimal schedule is obtained by sequencing jobs in the shortest processing time (SPT) order. Proof. Suppose that pi 6 pj. To show that S dominates S0 , it suffices to show that Cj(S) 6 Ci(S0 ). Taking the difference between Eqs. (3) and (5), we have
C i ðS0 Þ C j ðSÞ ¼ ðpj pi Þð1 ar Þ
r1 Y
al P 0;
l¼0
since pi 6 pj and 0 < al 6 1 for l = 1, . . . , r 1. Thus, S dominates S0 . Therefore, repeating this interchange argument for jobs not sequenced in the SPT order completes the proof of the property. h Property 2. For the 1jLEj
P
C j problem, the optimal schedule is obtained by sequencing jobs in the SPT order.
Proof. The proof is omitted since it is similar to that of Property 1. h
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The well-known weighted smallest processing time (WSPT) rule provides the optimal schedule for the classical total weighted completion time problem. In the following example, we show that the WSPT rule does not provide the optimal P solution for 1jLEj wj C j . Example 1. n = 2, p1 = 1, p2 = 2, w1 = 5, w2 = 12, a1 = 0.8. The total weighted completion time of the WSPT sequence (2, 1) is 38, while the total weighted completion time of the alternative sequence (1, 2) is 36.2. P In the following, we will show that the WSPT rule still provides the optimal solution for 1jLEj wj C j if the processing times and the weights are agreeable, i.e., pj/pi P wj/wi P 1 for all jobs i and j. P Property 3. For the 1jLEj wj C j problem, the optimal schedule is obtained by sequencing jobs in non-decreasing order of pi/wi if the processing times and the weights are agreeable. Proof. Suppose that pj/pi P wj/wi P 1. Since pi 6 pj, it implies from Property 1 that Cj(S) 6 Ci(S0 ). Thus, to show that S dominates S0 , it suffices to show that wiCi(S) + wjCj(S) 6 wjCj(S0 ) + wiCi(S0 ). From Eqs. (2)–(5), we have
½wj C j ðS0 Þ þ wi C i ðS0 Þ ½wi C i ðSÞ þ wj C j ðSÞ ¼ ðwi pj wj pi Þ
r1 Y
al þ ðwj pj wi pi Þð1 ar Þ
l¼0
r1 Y
al :
ð6Þ
l¼0
Substituting k = pjwi/piwj and k = wj/wi into Eq. (6), we have
r1 Y 1 ð1 ar Þwj pi ½wj C j ðS0 Þ þ wi C i ðS0 Þ ½wi C i ðSÞ þ wj C j ðSÞ ¼ ½ðk 1Þ þ kk al P 0; k l¼0 since k = pjwi/piwj P 1,k P 1, and 0 < al 6 1 for l = 1, . . . , r. Thus, S dominates S0 . Repeating this interchange argument for jobs not sequenced in the WSPT order completes the proof of Property 3. It is known that the EDD rule provides the optimal schedule for the 1kLmax problem. The example below shows that this policy is not optimal for the 1jLEjLmax problem. h Example 2. n = 2, p1 = 1, p2 = 20, d1 = 30, d2 = 20, a1 = 0.8. The maximum lateness of the EDD sequence (2, 1) is 0, while the maximum lateness of the alternative sequence (1, 2) is 3. However, the EDD rule still provides the optimal solution for 1jLEjLmax if the processing times and the due dates are agreeable, i.e., di 6 dj implies pi 6 pj for all jobs i and j. Property 4. For the 1jLEjLmax and 1jLEjTmax problems, the optimal schedule is obtained by sequencing jobs in non-decreasing order of di if the job processing times and the due dates are agreeable. Proof. The proof is a result of the pairwise interchange technique. h In the following, we will show that the EDD rule provides the optimal solution for the total tardiness problem if the job processing times and the due dates are agreeable, i.e., di 6 dj implies pi 6 pj for all jobs i and j. P Property 5. For the 1jLEj T i problem, the optimal schedule is obtained by sequencing jobs in non-decreasing order of di if di 6 dj implies pi 6 pj for all jobs i and j. Proof. Suppose that di 6 dj. It also implies pi 6 pj. The total tardiness of the first r 1 jobs are the same since they are processed in the same order. It is easy to check that Cj(S) 6 Ci(S0 ). Thus, the total tardiness of partial sequence p0 in S will not be greater than that of partial sequence p0 in S0 . Therefore, to prove that the total tardiness of S is less than or equal to that of S0 , it suffices to show that Ti(S) + Tj(S) 6 Tj(S0 ) + Ti(S0 ). Q To compare the total tardiness of jobs i and j in S and S0 , we divide it into two cases. In the first case that B þ pj r1 l¼0 al 6 dj , 0 the total tardiness of jobs i and j in S and S are
( T i ðSÞ þ T j ðSÞ ¼ max B þ pi
r1 Y
) al di ; 0
( þ max B þ pi
l¼0
l¼0
and
( 0
0
T j ðS Þ þ T i ðS Þ ¼ max B þ pj
r1 Y l¼0
r1 Y
al þ pi
r Y l¼0
) al di ; 0 :
al þ pj
r Y l¼0
) al dj ; 0
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Suppose that neither Ti(S) nor Tj(S) is zero. Note that this is the most restrictive case since it comprises the case that either one or both Ti(S) and Tj(S) are zero. From pi 6 pj, we have
fT j ðS0 Þ þ T i ðS0 Þg fT i ðSÞ þ T j ðSÞg ¼ ðpj pi Þð1 ar Þ
r 1 Y
al þ dj pi
l¼0
r1 Y
al B P 0:
l¼0
Thus, {Tj(S0 ) + Ti(S0 )} {Ti(S) + Tj(S)} P 0 in the first case. Q 0 In the second case that B þ pj r1 l¼0 al > dj , the total tardiness of jobs i and j in S and S are
(
r1 Y
T i ðSÞ þ T j ðSÞ ¼ max B þ pi
) al di ; 0
( þ max B þ pi
l¼0
r1 Y
al þ pj
l¼0
r Y
) al dj ; 0
l¼0
and
T j ðS0 Þ þ T i ðS0 Þ ¼ 2A þ 2pj
r1 Y
al þ pi
l¼0
r Y
al di dj :
l¼0
Suppose that neither Ti(S) nor Tj(S) is zero. From Property 1 and pi 6 pj, we have
fT j ðS0 Þ þ T i ðS0 Þg fT i ðSÞ þ T j ðSÞg ¼ ðpj pi Þð2 ar Þ
r1 Y
al P 0:
l¼0
Thus, {Tj(S0 ) + Ti(S0 )} {Ti(S) + Tj(S)} P 0 in the second case. This completes the proof of the property. In the following, we will consider two single-machine multi-criteria problems. The first one is the due-date assignment problem proposed by Panwalkar and Smith [17], and the second one is the simultaneously minimization of the total completion time and the variation of the completion times proposed by Bagchi [1]. The time complexity of both problems is O(nlogn) when the learning effect is not present. In this section, we will show that the problems remain polynomially solvable, although they need more computational effort. The due-date assignment problem is described as follows. There are n jobs ready to be processed on a single machine. Each job j has a normal processing time pj. It is assumed that all the jobs have a common due date d, which needs to be determined. For a given schedule S, let Ci(S), Ei(S) = max{0, d Ci(S)}, Ti(S) = max{0, Ci(S) d} denote the completion time, the earliness and the tardiness of job j in sequence S. Furthermore, let w1, w2, and w3 denote the unit penalty for the due-date, the earliness and the tardiness, respectively. The objective function is to find both the due date d and the schedule S such that P the total penalty f ðd; SÞ ¼ nj¼1 ðw1 d þ w2 Ej þ w3 T j Þ is minimized. Panwalkar and Smith [17] derived two results for the classical problem. First, for any specified sequence, it is optimal to assign the due date to the completion time of the kth job, where k is the smallest integral value greater than or equal to n(w3 w1)/(w2 + w3). Second, the optimal schedule is obtained by matching the smallest positional penalty c with the largest processing time, the next smaller value of c with the next larger processing time, and so on, where the value of c is
cj ¼
nw1 þ ðj 1Þw2
1 6 j 6 k;
ðn þ 1 jÞw3
k þ 1 6 j 6 n:
Now consider the same objective function while the processing time might vary due to the learning effect as given in Eq. (1). Using the notation as in Biskup [3], let xjr be the decision variable which assumes the value of 1 if job j is scheduled in the rth position and 0 otherwise. The objective is to find a schedule that
minimizes
n X n X j¼1
subject to
n X
cr pjr xjr ;
r¼1
xjr ¼ 1;
r ¼ 1; . . . ; n; ð7Þ
j¼1 n X
xjr ¼ 1;
j ¼ 1; . . . ;n ;
r¼1
xjr 2 f0; 1g;
j; r ¼ 1; . . . ; n:
Since the time complexity to solve an assignment problem is O(n3), the due-date assignment problem remains polynomially solvable. In order to demonstrate the procedure, we solve the example introduced by Panwalkar and Smith [17] under the proposed model. h
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W.-C. Lee / Information Sciences 181 (2011) 5515–5522 Table 1 The optimal solutions of the due-date assignment problem under various patterns. Pattern
(a1, a2, a3, a4, a5, a6)
Optimal due-date
Optimal sequence
Total cost
No LE Constant K-shaped V-shaped Increasing Decreasing Plateau
( 1, 1, 1, 1, 1, 1) (0.95, 0.95, 0.95, 0.95, 0.95, 0.95) (0.85, 0.90, 0.95, 0.95, 0.90, 0.85) (0.95, 0.90, 0.85, 0.85, 0.90, 0.95) (0.70, 0.75, 0.80, 0.85, 0.90, 0.95) (0.95, 0.90, 0.85, 0.80, 0.75, 0.70) (0.85, 0.90, 0.95, 1, 1, 1)
34.00 28.73 24.34 20.30 12.73 20.30 26.89
(6, 4, 2, 1, 3, 5, 7) (5, 4, 2, 1, 3, 6, 7) (5, 3, 2, 1, 4, 6, 7) (4, 3, 2, 1, 5, 6, 7) (1, 2, 3, 4, 5, 6, 7) (4, 3, 2, 1, 5, 6, 7) (5, 4, 2, 1, 3, 6, 7)
2664.00 2293.92 1975.61 1937.81 1258.37 1752.38 2132.61
Table 2 The optimal solutions for the bi-criteria problem of Bagchi [1] under various patterns. Pattern
(a1, a2, a3, a4, a5, a6)
Optimal sequence
Total cost
No LE Constant K-shaped V-shaped Increasing Decreasing Plateau
( 1, 1, 1, 1, 1, 1) (0.95, 0.95, 0.95, 0.95, 0.95, 0.95) (0.85, 0.90, 0.95, 0.95, 0.90, 0.85) (0.95, 0.90, 0.85, 0.85, 0.90, 0.95) (0.70, 0.75, 0.80, 0.85, 0.90, 0.95) (0.95, 0.90, 0.85, 0.80, 0.75, 0.70) (0.85, 0.90, 0.95, 1, 1, 1)
(7, 5, 3, 1, 2, 4, 6) (7, 4, 2, 1, 3, 5, 6) (6, 4, 2, 1, 3, 5, 7) (6, 3, 1, 2, 4, 5, 7) (6, 2, 1, 3, 4, 5, 7) (6, 3, 1, 2, 4, 5, 7) (7, 5, 3, 1, 2, 4, 6)
1141 977.10 826.36 815.69 575.09 664.76 933.81
Example 3. n = 7, p1 = 3, p2 = 4, p3 = 6, p4 = 9, p5 = 14, p6 = 18, p7 = 20, w1 = 5, w2 = 11, w3 = 18. The optimal value of k is 4. The positional penalties are c1 = 35, c2 = 46, c3 = 57, c4 = 68, c5 = 54, c6 = 36, c7 = 18. Now, assume that a1 = 0.95, a2 = 0.95, a3 = 0.95, a4 = 0.95, a5 = 0.95, a6 = 0.95. The matrix of cjpjr values can then be derived. The optimal value of k is 4 and the optimal sequence of this assignment problem is (5, 4, 2, 1, 3, 6, 7). The optimal due date is at time 28.73, and the total cost is 2293.92. The results for several different patterns of learning effects, including the K-shaped, V-shaped, increasing, decreasing, and the plateau function are given in Table 1. The second multi-criteria problem is the simultaneously minimization of the total completion time and the variation of the completion times proposed by Bagchi [1]. It is to find a schedule that minimizes the mean and the variation of the P completion times. For a given schedule S, let Cj(S) denote the completion time of job j in S, TCðSÞ ¼ nj¼1 C j ðSÞ denote the total Pn Pn completion time in S, and TADCðSÞ ¼ j¼1 k¼1 C j ðSÞ C k ðSÞ denote the total absolute differences in completion times in S. Furthermore, let 0 6 a 6 1. The objective function is to find a schedule S such that f(S) = a(TC) + (1 a)(TADC) is minimized. Bagchi [1] showed that the optimal schedule is obtained in O(n log n) time by matching the smallest positional penalty c with the largest processing time, the next smaller value of c with the next larger processing time, and so on, where
cr ¼ ð2a 1Þðn þ 1Þ þ r½2 3a þ nð1 aÞ r2 ð1 aÞ; r ¼ 1; . . . ; n:
ð8Þ
Now consider the same problem under the proposed model given in Eq. (1), it also becomes an assignment problem as in Eq. (7) with positional penalty given in Eq. (8). In order to demonstrate the procedure, we solve the example introduced by Bagchi [1] under the proposed learning effect model. Example 4. n = 7, p1 = 2, p2 = 3, p3 = 6, p4 = 9, p5 = 21, p6 = 65, p7 = 82, a = 0.5. The positional penalties are c1 = 3.5, c2 = 6, c3 = 7.5, c4 = 8, c5 = 7.5, c6 = 6, c7 = 3.5. Now, assume that a1 = 0.95, a2 = 0.95, a3 = 0.95, a4 = 0.95, a5 = 0.95, a6 = 0.95. The optimal sequence of this assignment problem is (7, 4, 2, 1, 3, 5, 6) with a total cost is 977.1. The results for several different patterns of learning effects, including the K-shaped, V-shaped, increasing, decreasing, and the plateau function are summarized in Table 2.
3. A two-machine flowshop problem The formulation of the proposed model in the off-line two-machine permutation flowshop case is similar to that of the single-machine case and stated as follows. There are n jobs to be processed on machines M1 and M2. Each job j consists of two operations to be processed on one machine each. Operation of job j on machine M2 can start only operation on machine M1 is completed. A machine can handle one job at a time and preemption is not allowed. The actual processing time pkj[r] of job j on machine Mk is
pkj½r ¼ pkj
r1 Y
al ;
l¼0
if it is scheduled in the rth position in a sequence where 0 < al 6 1 for l = 1, . . . , n.
W.-C. Lee / Information Sciences 181 (2011) 5515–5522
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It is known that Johnson’s rule provides the optimal sequence for the F2kCmax problem, however, it fails to provide the optimal solution for the F2jLEjCmax problem, as shown in the following example. Example 5. n = 2, p11 = 3, p12 = 4, p21 = 10, p22 = 4, a1 = 0.8. The makespan of Johnson’s sequence (1, 2) is 16.2, while the makespan of the alternative sequence (2, 1) is 16. In the following, we will show that the SPT rule provides the optimal solution for F2jLEjCmax under the assumption of ordered processing times, i.e., p1j 6 p2j for all jobs j and with p2i 6 p2j whenever p1i 6 p1j for any two jobs i and j. Property 6. For the F2jLEjCmax problem, the optimal schedule is obtained by sequencing jobs in non-decreasing order of p1j under the assumption of ordered processing times. Proof. Suppose that p1i 6 p1j. It implies that p2i 6 p2j. Let S and S0 be two job schedules and the difference between S and S0 is a pairwise interchange of two adjacent jobs i and j, i.e., S = (p, i, j, p0 ) and S0 = (p, j, i, p0 ), where p and p0 each denotes a partial sequence. Furthermore, we assume that there are r 1 scheduled jobs in p. It is easy to check that C1j(S) 6 C1i(S0 ). Thus, to show that S dominates S0 , it suffices to show that
C 2j ðSÞ 6 C 2i ðS0 Þ:
ð8Þ
As in Baker [2], the completion times of the (r+1) th job in S and S0 are
C 2½rþ1 ðSÞ ¼
rþ1 X
p2½l ðSÞ
l¼1
l1 Y
ak þ
rþ1 X
k¼0
I2½l ðSÞ
l¼1
and
C 2½rþ1 ðS0 Þ ¼
rþ1 X
p2½l ðS0 Þ
l1 Y
l¼1
ak þ
rþ1 X
k¼0
I2½l ðS0 Þ;
l¼1
where p2[l](S) denote the normal processing time of the lth job in S and I2[l](S) denote the idle time occurring on M2 immediately prior to the lth job in S. P Ql1 Prþ1 0 Ql1 Since p2i 6 p2j, we have that rþ1 l¼1 p2½l ðSÞ k¼0 ak 6 l¼1 p2½l ðS Þ k¼0 ak . Thus, to show Eq. (8) hold, it suffices to show that rþ1 X
I2½l ðSÞ 6
rþ1 X
l¼1
I2½l ðS0 Þ:
ð9Þ
l¼1
As in Baker [2], the sum of the first q idle times in S and S0 are q X
I2½l ðSÞ ¼ maxðY ½t ðSÞÞ 16t6q
l¼1
and q X
I2½l ðS0 Þ ¼ maxðY ½t ðS0 ÞÞ 16t6q
l¼1
for q = 1, . . . , r + 1 where Y ½t ðSÞ ¼
Y ½t ðS0 Þ ¼
t X
p1½l ðS0 Þ
l¼1
l1 Y
Pt
ak
k¼0
Ql1
l¼1 p1½l ðSÞ
t1 X
p2½l ðS0 Þ
l¼1
k¼0 ak
l1 Y
Pt1
l¼1 p2½l ðSÞ
Ql1
k¼0 ak
and
ak :
k¼0
It is noted that Y[t](S) = Y[t](S0 ) for t = 1, . . . , r 1. Thus, to show Eq. (9) hold, it suffices to show that
maxfY ½r ðSÞ; Y ½rþ1 ðSÞg 6 Y ½r ðS0 Þ: Since p1i 6 p1j, we have
Y ½r ðS0 Þ Y ½r ðSÞ ¼ ðp1j p1i Þ
l1 Y k¼0
ak P 0:
ð10Þ
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Since p1i 6 p2i and ar 6 1, we have
Y ½r ðS0 Þ Y ½rþ1 ðSÞ ¼ ðp2i p1i Þ
r1 Y k¼0
ak þ p1j ð1 ar Þ
r1 Y
ak P 0:
k¼0
Thus, S dominates S0 . Therefore, repeating this interchange argument for jobs not sequenced in the SPT order completes the proof of the property. h
4. Conclusions In this paper, we proposed a new model of the learning effect. For the single-machine case, we showed that the problem to minimize the total tardiness is polynomially solvable under certain agreeable conditions. Moreover, we provided the optimal solutions for two single-machine multiple-criteria problems. For the case of two-machine permutation flowshop, we presented polynomial-time optimal solutions for the makespan problem under the assumption of ordered processing times. For future research, it is interesting to study the problems without the agreeable condition, to consider the multi-machine model with multi-criteria problems of scheduling, taking into account the effect of learning for the case of the machines in on-line mode, or to integrate the proposed approaches based on the effect of learning, and approaches based on Soft Computing technologies for solving multi-criteria problems of scheduling works on the multiple machines. Acknowledgements The author is grateful to the editor and the referees, whose constructive comments have led to a substantial improvement in the presentation of the paper. This work was supported by the NSC of Taiwan, ROC, under NSC 98-2221-E-035-033-MY2. References [1] U. Bagchi, Simultaneous minimization of‘mean and variation of flow time and waiting time in single machine systems, Operations Research 37 (1989) 118–125. [2] K. Baker, Introduction to Sequencing and Scheduling, John Wiley, New York, 1974. [3] D. Biskup, Single-machine scheduling with learning considerations, European Journal of Operational Research 115 (1999) 173–178. [4] D. Biskup, A state-of-the-art review on scheduling with learning effect, European Journal of Operational Research 188 (2008) 315–329. [5] T.C.E. Cheng, G. Wang, Single machine scheduling with learning effect considerations, Annals of Operations Research 98 (2000) 273–290. [6] T.C.E. Cheng, C.C. Wu, W.C. Lee, Some scheduling problems with sum-of-processing-times-based and job-position-based learning effects, Information Sciences 178 (2008) 2476–2487. [7] T.C.E. Cheng, P.J. Lai, C.C. Wu, W.C. Lee, Single-machine scheduling with sum-of-logarithm-processing-times-based learning considerations, Information Sciences 179 (2009) 3127–3135. [8] V.R. Donheti, B.B. Mohanty, Impact of learning and fatigue factors on single machine scheduling with penalties for tardy jobs, European Journal of Operational Research 105 (1998) 509–524. [9] J.M. Dutton, A. Thomas, Treating progress function as a managerial opportunity, Academy of Management 9 (1984) 235–247. [10] S. Gawiejnowicz, A note on scheduling on a single processor with speed dependent on a number of executed jobs, Information Processing Letters 56 (1996) 297–300. [11] W.J. Hurley, When are we going to change the learning curve lecture?, Computers & Operations Research 23 (1996) 509–511 [12] A. Janiak, R. Rudek, A new approach to the learning effect: beyond the learning curve restrictions, Computers & Operations Research 35 (2008) 3727– 3736. [13] C. Koulamas, G.J. Kyparisis, Single-machine and two-machine flowshop scheduling with general learning function, European Journal of Operational Research 178 (2007) 402–407. [14] M.A. Lapre, A.S. Mukherjee, L.N.V. Wassenhove, Behind the learning curve: linking learning activities to waste reduction, Management Science 46 (2000) 597–611. [15] W.C. Lee, C.C. Wu, H.J. Sung, A bi-criterion single-machine scheduling problem with learning considerations, Acta Informatica 40 (2004) 303–315. [16] G. Mosheiov, Scheduling problems with a learning effect, European Journal of Operational Research 132 (2001) 687–693. [17] S.S. Panwalkar, M.L. Smith, Common due date assignment to minimize total penalty for the one machine scheduling problem, Operations Research 30 (1982) 391–399. [18] J.B. Wang, Single-machine scheduling problems with the effects of learning and deterioration, OMEGA-The International Journal of Management Science 35 (2007) 397–402. [19] C.C. Wu, W.C. Lee, A note on the total completion time problem in a permutation flowshop with a learning effect, European Journal of Operational Research 192 (2009) 343–347. [20] Y. Yin, D. Xu, K. Sun, H. Li, Some scheduling problems with general position-dependent and time-dependent learning effects, Information Sciences 179 (2009) 2416–2425. [21] X.G. Zhang, G.L. Yan, Machine scheduling problems with a general learning effect, Mathematical and Computer Modelling 51 (2010) 84–90. [22] C.L. Zhao, Q.L. Zhang, H.Y. Tang, Machine scheduling problems with learning effects, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis 11 (2004) 741–747.
Contents lists available at SciVerse ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
Scheduling with general position-based learning curves Wen-Chiung Lee Department of Statistics, Feng Chia University, Taichung, Taiwan
a r t i c l e
i n f o
Article history: Received 22 February 2010 Received in revised form 9 March 2011 Accepted 23 July 2011 Available online 11 August 2011 Keywords: Scheduling Learning curve Single-machine Two-machine flowshop
a b s t r a c t Learning effect in scheduling problems has received growing attention since Biskup [3] introduced the position-based model, where the learning curve is expressed as a power function of a job position. Hurley [11] pointed out that the actual processing time of a given job drops to zero precipitously as the number of jobs increases in the standard power model. Moreover, the learning rates show considerable variation within industries or firms. The variation extends not only across firms at a given time, but also within firms over time. For instance, the learning curves usually have an initial downward concavity, and no further improvements are made after some amount of production. Beside the standard power model, learning curve is seldom discussed in scheduling. In this paper, we offer a surprising simple yet realistic learning effect model which has the flexibility to describe different learning curves easily. For instance, the standard power model, the well-known S-shaped and the plateau functions are special cases of the proposed model. We then present the optimal solution for some scheduling problems. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Job processing times are assumed to be fixed and known in traditional scheduling problems. However, recent empirical studies in several industries have verified that unit costs decline as firms produce more of a product and gain knowledge or experience. For instance, Biskup [3] pointed out that repeated processing of similar tasks improves worker skills; workers are able to perform setup, to deal with machine operations and software, or to handle raw materials and components at a greater pace. Gawiejnowicz [10], Donheti and Mohanty [8], Biskup [3] and Cheng and Wang [5] were among the pioneers to bring the concept of learning effect into scheduling problems. Biskup [3] introduced the job-position-based learning effect model in which the actual job processing time is a power function of its position in schedule. He showed that two single-machine scheduling problems remain polynomially solvable. Mosheiov [16] gave several examples to demonstrate that the optimal schedules of some problems with learning effect may be different from those of the classical ones without learning consideration. Lee et al. [15] studied the single machine bi-criterion scheduling problem to minimize the sum of the total completion time and the maximum tardiness. Zhao et al. [22] developed the optimal solutions of some single machine and flowshop problems in some special cases. Koulamas and Kyparisis [13] expressed the learning effect as a function of the normal processing times of jobs already processed. They showed that the shortest processing time (SPT) sequence is optimal for the single-machine makespan and the total completion time problems. In addition, they also proved that the two-machine flowshop makespan and the total completion time problems are optimally solved by the SPT rule when the job processing times are ordered. Wang [18] considered some single-machine problems with the effects of learning and deterioration. Moreover, Biskup [4] provided a comprehensive survey of scheduling problems with learning effects. Recently, Janiak and Rudek [12]
E-mail address: [email protected] 0020-0255/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2011.07.051
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considered a scheduling problem in which each job provides a different experience to the processor. They formulated the shape of the learning curve as a non-increasing k-stepwise function. Cheng et al. [6,7] considered some single-machine scheduling problems with specific forms of learning effect. Wu and Lee [19] considered the total completion time problem in the multiple machine permutation flowshop. Yin et al. [20] provided a general model with position-dependent and timedependent learning effects. Zhang and Yan [21] provided the optimal schedules for some single machine problems. Among the articles in scheduling with learning effects, it is assumed in the majority of models that the learning curve is expressed as a power function of its position in schedule. Despite its frequent use, Hurley [11] pointed out that the actual processing time of a given job drops to zero precipitously as the number of jobs increases in the standard power model. Moreover, Lapre et al. [14] claimed that the traditional power form does not accommodate two often observed patterns: initial downward concavity and the plateau effect where no further improvements are made after some amount of production. In this paper, we present a new model of the learning effect which has the shaping flexibility. The flexibility of the proposed model allows us to describe different learning curves or fit more complex environments easily. As pointed out in the Biskup [4] survey paper, it is a challenge to model with the learning effect as realistic as possible. Thus, the first contribution of this paper is to provide a surprisingly simple yet realistic model. It is completely unexpected in the sense that a simple model can describe many different learning curves in the literature, such as the well-known plateau or the S-shaped phenomena, which are more realistic. Moreover, Dutton and Thomas [9] found that the learning rates show considerable variation within industries or firms after a study of more than 200 learning curves. The variation extends not only across firms at a given time, but also within firms over time. Thus, the second contribution of the paper is to provide a more flexible model. The remainder of this note is organized as follows. The solution procedures for some single-machine problems are presented in the next section. In Section 3, the optimal solution for a two-machine flowshop problem is derived. The conclusion is given in the last section.
2. Single-machine single-criterion problems Formulation of the proposed learning effect model in the single-machine case is as follows. There are n jobs to be processed on a single machine. Each job j has a normal processing time pj, a weight wj, and a due date dj. Due to the learning effect, the actual processing time of job j is
pj½r ¼ pj
r1 Y
al ;
ð1Þ
l¼0
if it is scheduled in the rth position in a sequence, where a0 = 1 and 0 < al 6 1 for l = 1, . . . , n. Note that al denotes the learning Q impact on processing the lth position job and rl¼0 al denotes the cumulative level of learning effect after processing r jobs. The proposed model provides the shaping flexibility. For instance, the value of al can be set to a moderate large value for the stages of strong learning effects, and al is set to 1 or a value very close to 1 for the stages where the learning effect is weak. If Q al = 0.9 + 0.01l for l = 1, . . . , 10 and al = 1 for l = 11, . . . , 20, the learning curve rl¼0 al is the plateau function as shown in Fig. 1, where the actual processing time of a given job will not drop to zero precipitously as the number of jobs increases. If Q al = 0.999 0.05(0.1l 1)2 for l = 1, . . . , 20, the learning curve rl¼0 al is S-shaped as shown in Fig. 2. In fact, it reduces to b the Biskup [3] model if we choose al = (1 + 1/l) where b < 0. This flexibility allows us to fit more learning curves. For convenience, we denote the learning model by LE. Thus, using conventional notation for describing scheduling problems, the single-machine makespan problem is denoted as 1jLEjCmax, for instance.
Fig. 1. The plateau learning curve.
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Fig. 2. The S-shaped learning curve.
We will prove the following properties using the well-known pairwise interchange technique. Suppose that S and S0 are two job schedules and the difference between S and S0 is a pairwise interchange of two adjacent jobs i and j, i.e., S = (p, i, j, p0 ) and S0 = (p, j, i, p0 ), where p and p0 each denotes a partial sequence. Furthermore, we assume that there are r 1 scheduled jobs in p and B is the completion time of the last job in p. Under S, the completion times of jobs i and j are respectively
C i ðSÞ ¼ B þ pi
r1 Y
al
ð2Þ
l¼0
and
C j ðSÞ ¼ B þ pi
r1 Y
al þ pj
l¼0
r Y
al :
ð3Þ
l¼0
Similarly, the completion times of jobs j and i in S0 are respectively
C j ðS0 Þ ¼ B þ pj
r1 Y
al
ð4Þ
l¼0
and
C i ðS0 Þ ¼ B þ pj
r1 Y
al þ pi
l¼0
r Y
al :
ð5Þ
l¼0
Property 1. .For the 1jLEjCmax problem, the optimal schedule is obtained by sequencing jobs in the shortest processing time (SPT) order. Proof. Suppose that pi 6 pj. To show that S dominates S0 , it suffices to show that Cj(S) 6 Ci(S0 ). Taking the difference between Eqs. (3) and (5), we have
C i ðS0 Þ C j ðSÞ ¼ ðpj pi Þð1 ar Þ
r1 Y
al P 0;
l¼0
since pi 6 pj and 0 < al 6 1 for l = 1, . . . , r 1. Thus, S dominates S0 . Therefore, repeating this interchange argument for jobs not sequenced in the SPT order completes the proof of the property. h Property 2. For the 1jLEj
P
C j problem, the optimal schedule is obtained by sequencing jobs in the SPT order.
Proof. The proof is omitted since it is similar to that of Property 1. h
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The well-known weighted smallest processing time (WSPT) rule provides the optimal schedule for the classical total weighted completion time problem. In the following example, we show that the WSPT rule does not provide the optimal P solution for 1jLEj wj C j . Example 1. n = 2, p1 = 1, p2 = 2, w1 = 5, w2 = 12, a1 = 0.8. The total weighted completion time of the WSPT sequence (2, 1) is 38, while the total weighted completion time of the alternative sequence (1, 2) is 36.2. P In the following, we will show that the WSPT rule still provides the optimal solution for 1jLEj wj C j if the processing times and the weights are agreeable, i.e., pj/pi P wj/wi P 1 for all jobs i and j. P Property 3. For the 1jLEj wj C j problem, the optimal schedule is obtained by sequencing jobs in non-decreasing order of pi/wi if the processing times and the weights are agreeable. Proof. Suppose that pj/pi P wj/wi P 1. Since pi 6 pj, it implies from Property 1 that Cj(S) 6 Ci(S0 ). Thus, to show that S dominates S0 , it suffices to show that wiCi(S) + wjCj(S) 6 wjCj(S0 ) + wiCi(S0 ). From Eqs. (2)–(5), we have
½wj C j ðS0 Þ þ wi C i ðS0 Þ ½wi C i ðSÞ þ wj C j ðSÞ ¼ ðwi pj wj pi Þ
r1 Y
al þ ðwj pj wi pi Þð1 ar Þ
l¼0
r1 Y
al :
ð6Þ
l¼0
Substituting k = pjwi/piwj and k = wj/wi into Eq. (6), we have
r1 Y 1 ð1 ar Þwj pi ½wj C j ðS0 Þ þ wi C i ðS0 Þ ½wi C i ðSÞ þ wj C j ðSÞ ¼ ½ðk 1Þ þ kk al P 0; k l¼0 since k = pjwi/piwj P 1,k P 1, and 0 < al 6 1 for l = 1, . . . , r. Thus, S dominates S0 . Repeating this interchange argument for jobs not sequenced in the WSPT order completes the proof of Property 3. It is known that the EDD rule provides the optimal schedule for the 1kLmax problem. The example below shows that this policy is not optimal for the 1jLEjLmax problem. h Example 2. n = 2, p1 = 1, p2 = 20, d1 = 30, d2 = 20, a1 = 0.8. The maximum lateness of the EDD sequence (2, 1) is 0, while the maximum lateness of the alternative sequence (1, 2) is 3. However, the EDD rule still provides the optimal solution for 1jLEjLmax if the processing times and the due dates are agreeable, i.e., di 6 dj implies pi 6 pj for all jobs i and j. Property 4. For the 1jLEjLmax and 1jLEjTmax problems, the optimal schedule is obtained by sequencing jobs in non-decreasing order of di if the job processing times and the due dates are agreeable. Proof. The proof is a result of the pairwise interchange technique. h In the following, we will show that the EDD rule provides the optimal solution for the total tardiness problem if the job processing times and the due dates are agreeable, i.e., di 6 dj implies pi 6 pj for all jobs i and j. P Property 5. For the 1jLEj T i problem, the optimal schedule is obtained by sequencing jobs in non-decreasing order of di if di 6 dj implies pi 6 pj for all jobs i and j. Proof. Suppose that di 6 dj. It also implies pi 6 pj. The total tardiness of the first r 1 jobs are the same since they are processed in the same order. It is easy to check that Cj(S) 6 Ci(S0 ). Thus, the total tardiness of partial sequence p0 in S will not be greater than that of partial sequence p0 in S0 . Therefore, to prove that the total tardiness of S is less than or equal to that of S0 , it suffices to show that Ti(S) + Tj(S) 6 Tj(S0 ) + Ti(S0 ). Q To compare the total tardiness of jobs i and j in S and S0 , we divide it into two cases. In the first case that B þ pj r1 l¼0 al 6 dj , 0 the total tardiness of jobs i and j in S and S are
( T i ðSÞ þ T j ðSÞ ¼ max B þ pi
r1 Y
) al di ; 0
( þ max B þ pi
l¼0
l¼0
and
( 0
0
T j ðS Þ þ T i ðS Þ ¼ max B þ pj
r1 Y l¼0
r1 Y
al þ pi
r Y l¼0
) al di ; 0 :
al þ pj
r Y l¼0
) al dj ; 0
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Suppose that neither Ti(S) nor Tj(S) is zero. Note that this is the most restrictive case since it comprises the case that either one or both Ti(S) and Tj(S) are zero. From pi 6 pj, we have
fT j ðS0 Þ þ T i ðS0 Þg fT i ðSÞ þ T j ðSÞg ¼ ðpj pi Þð1 ar Þ
r 1 Y
al þ dj pi
l¼0
r1 Y
al B P 0:
l¼0
Thus, {Tj(S0 ) + Ti(S0 )} {Ti(S) + Tj(S)} P 0 in the first case. Q 0 In the second case that B þ pj r1 l¼0 al > dj , the total tardiness of jobs i and j in S and S are
(
r1 Y
T i ðSÞ þ T j ðSÞ ¼ max B þ pi
) al di ; 0
( þ max B þ pi
l¼0
r1 Y
al þ pj
l¼0
r Y
) al dj ; 0
l¼0
and
T j ðS0 Þ þ T i ðS0 Þ ¼ 2A þ 2pj
r1 Y
al þ pi
l¼0
r Y
al di dj :
l¼0
Suppose that neither Ti(S) nor Tj(S) is zero. From Property 1 and pi 6 pj, we have
fT j ðS0 Þ þ T i ðS0 Þg fT i ðSÞ þ T j ðSÞg ¼ ðpj pi Þð2 ar Þ
r1 Y
al P 0:
l¼0
Thus, {Tj(S0 ) + Ti(S0 )} {Ti(S) + Tj(S)} P 0 in the second case. This completes the proof of the property. In the following, we will consider two single-machine multi-criteria problems. The first one is the due-date assignment problem proposed by Panwalkar and Smith [17], and the second one is the simultaneously minimization of the total completion time and the variation of the completion times proposed by Bagchi [1]. The time complexity of both problems is O(nlogn) when the learning effect is not present. In this section, we will show that the problems remain polynomially solvable, although they need more computational effort. The due-date assignment problem is described as follows. There are n jobs ready to be processed on a single machine. Each job j has a normal processing time pj. It is assumed that all the jobs have a common due date d, which needs to be determined. For a given schedule S, let Ci(S), Ei(S) = max{0, d Ci(S)}, Ti(S) = max{0, Ci(S) d} denote the completion time, the earliness and the tardiness of job j in sequence S. Furthermore, let w1, w2, and w3 denote the unit penalty for the due-date, the earliness and the tardiness, respectively. The objective function is to find both the due date d and the schedule S such that P the total penalty f ðd; SÞ ¼ nj¼1 ðw1 d þ w2 Ej þ w3 T j Þ is minimized. Panwalkar and Smith [17] derived two results for the classical problem. First, for any specified sequence, it is optimal to assign the due date to the completion time of the kth job, where k is the smallest integral value greater than or equal to n(w3 w1)/(w2 + w3). Second, the optimal schedule is obtained by matching the smallest positional penalty c with the largest processing time, the next smaller value of c with the next larger processing time, and so on, where the value of c is
cj ¼
nw1 þ ðj 1Þw2
1 6 j 6 k;
ðn þ 1 jÞw3
k þ 1 6 j 6 n:
Now consider the same objective function while the processing time might vary due to the learning effect as given in Eq. (1). Using the notation as in Biskup [3], let xjr be the decision variable which assumes the value of 1 if job j is scheduled in the rth position and 0 otherwise. The objective is to find a schedule that
minimizes
n X n X j¼1
subject to
n X
cr pjr xjr ;
r¼1
xjr ¼ 1;
r ¼ 1; . . . ; n; ð7Þ
j¼1 n X
xjr ¼ 1;
j ¼ 1; . . . ;n ;
r¼1
xjr 2 f0; 1g;
j; r ¼ 1; . . . ; n:
Since the time complexity to solve an assignment problem is O(n3), the due-date assignment problem remains polynomially solvable. In order to demonstrate the procedure, we solve the example introduced by Panwalkar and Smith [17] under the proposed model. h
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W.-C. Lee / Information Sciences 181 (2011) 5515–5522 Table 1 The optimal solutions of the due-date assignment problem under various patterns. Pattern
(a1, a2, a3, a4, a5, a6)
Optimal due-date
Optimal sequence
Total cost
No LE Constant K-shaped V-shaped Increasing Decreasing Plateau
( 1, 1, 1, 1, 1, 1) (0.95, 0.95, 0.95, 0.95, 0.95, 0.95) (0.85, 0.90, 0.95, 0.95, 0.90, 0.85) (0.95, 0.90, 0.85, 0.85, 0.90, 0.95) (0.70, 0.75, 0.80, 0.85, 0.90, 0.95) (0.95, 0.90, 0.85, 0.80, 0.75, 0.70) (0.85, 0.90, 0.95, 1, 1, 1)
34.00 28.73 24.34 20.30 12.73 20.30 26.89
(6, 4, 2, 1, 3, 5, 7) (5, 4, 2, 1, 3, 6, 7) (5, 3, 2, 1, 4, 6, 7) (4, 3, 2, 1, 5, 6, 7) (1, 2, 3, 4, 5, 6, 7) (4, 3, 2, 1, 5, 6, 7) (5, 4, 2, 1, 3, 6, 7)
2664.00 2293.92 1975.61 1937.81 1258.37 1752.38 2132.61
Table 2 The optimal solutions for the bi-criteria problem of Bagchi [1] under various patterns. Pattern
(a1, a2, a3, a4, a5, a6)
Optimal sequence
Total cost
No LE Constant K-shaped V-shaped Increasing Decreasing Plateau
( 1, 1, 1, 1, 1, 1) (0.95, 0.95, 0.95, 0.95, 0.95, 0.95) (0.85, 0.90, 0.95, 0.95, 0.90, 0.85) (0.95, 0.90, 0.85, 0.85, 0.90, 0.95) (0.70, 0.75, 0.80, 0.85, 0.90, 0.95) (0.95, 0.90, 0.85, 0.80, 0.75, 0.70) (0.85, 0.90, 0.95, 1, 1, 1)
(7, 5, 3, 1, 2, 4, 6) (7, 4, 2, 1, 3, 5, 6) (6, 4, 2, 1, 3, 5, 7) (6, 3, 1, 2, 4, 5, 7) (6, 2, 1, 3, 4, 5, 7) (6, 3, 1, 2, 4, 5, 7) (7, 5, 3, 1, 2, 4, 6)
1141 977.10 826.36 815.69 575.09 664.76 933.81
Example 3. n = 7, p1 = 3, p2 = 4, p3 = 6, p4 = 9, p5 = 14, p6 = 18, p7 = 20, w1 = 5, w2 = 11, w3 = 18. The optimal value of k is 4. The positional penalties are c1 = 35, c2 = 46, c3 = 57, c4 = 68, c5 = 54, c6 = 36, c7 = 18. Now, assume that a1 = 0.95, a2 = 0.95, a3 = 0.95, a4 = 0.95, a5 = 0.95, a6 = 0.95. The matrix of cjpjr values can then be derived. The optimal value of k is 4 and the optimal sequence of this assignment problem is (5, 4, 2, 1, 3, 6, 7). The optimal due date is at time 28.73, and the total cost is 2293.92. The results for several different patterns of learning effects, including the K-shaped, V-shaped, increasing, decreasing, and the plateau function are given in Table 1. The second multi-criteria problem is the simultaneously minimization of the total completion time and the variation of the completion times proposed by Bagchi [1]. It is to find a schedule that minimizes the mean and the variation of the P completion times. For a given schedule S, let Cj(S) denote the completion time of job j in S, TCðSÞ ¼ nj¼1 C j ðSÞ denote the total Pn Pn completion time in S, and TADCðSÞ ¼ j¼1 k¼1 C j ðSÞ C k ðSÞ denote the total absolute differences in completion times in S. Furthermore, let 0 6 a 6 1. The objective function is to find a schedule S such that f(S) = a(TC) + (1 a)(TADC) is minimized. Bagchi [1] showed that the optimal schedule is obtained in O(n log n) time by matching the smallest positional penalty c with the largest processing time, the next smaller value of c with the next larger processing time, and so on, where
cr ¼ ð2a 1Þðn þ 1Þ þ r½2 3a þ nð1 aÞ r2 ð1 aÞ; r ¼ 1; . . . ; n:
ð8Þ
Now consider the same problem under the proposed model given in Eq. (1), it also becomes an assignment problem as in Eq. (7) with positional penalty given in Eq. (8). In order to demonstrate the procedure, we solve the example introduced by Bagchi [1] under the proposed learning effect model. Example 4. n = 7, p1 = 2, p2 = 3, p3 = 6, p4 = 9, p5 = 21, p6 = 65, p7 = 82, a = 0.5. The positional penalties are c1 = 3.5, c2 = 6, c3 = 7.5, c4 = 8, c5 = 7.5, c6 = 6, c7 = 3.5. Now, assume that a1 = 0.95, a2 = 0.95, a3 = 0.95, a4 = 0.95, a5 = 0.95, a6 = 0.95. The optimal sequence of this assignment problem is (7, 4, 2, 1, 3, 5, 6) with a total cost is 977.1. The results for several different patterns of learning effects, including the K-shaped, V-shaped, increasing, decreasing, and the plateau function are summarized in Table 2.
3. A two-machine flowshop problem The formulation of the proposed model in the off-line two-machine permutation flowshop case is similar to that of the single-machine case and stated as follows. There are n jobs to be processed on machines M1 and M2. Each job j consists of two operations to be processed on one machine each. Operation of job j on machine M2 can start only operation on machine M1 is completed. A machine can handle one job at a time and preemption is not allowed. The actual processing time pkj[r] of job j on machine Mk is
pkj½r ¼ pkj
r1 Y
al ;
l¼0
if it is scheduled in the rth position in a sequence where 0 < al 6 1 for l = 1, . . . , n.
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It is known that Johnson’s rule provides the optimal sequence for the F2kCmax problem, however, it fails to provide the optimal solution for the F2jLEjCmax problem, as shown in the following example. Example 5. n = 2, p11 = 3, p12 = 4, p21 = 10, p22 = 4, a1 = 0.8. The makespan of Johnson’s sequence (1, 2) is 16.2, while the makespan of the alternative sequence (2, 1) is 16. In the following, we will show that the SPT rule provides the optimal solution for F2jLEjCmax under the assumption of ordered processing times, i.e., p1j 6 p2j for all jobs j and with p2i 6 p2j whenever p1i 6 p1j for any two jobs i and j. Property 6. For the F2jLEjCmax problem, the optimal schedule is obtained by sequencing jobs in non-decreasing order of p1j under the assumption of ordered processing times. Proof. Suppose that p1i 6 p1j. It implies that p2i 6 p2j. Let S and S0 be two job schedules and the difference between S and S0 is a pairwise interchange of two adjacent jobs i and j, i.e., S = (p, i, j, p0 ) and S0 = (p, j, i, p0 ), where p and p0 each denotes a partial sequence. Furthermore, we assume that there are r 1 scheduled jobs in p. It is easy to check that C1j(S) 6 C1i(S0 ). Thus, to show that S dominates S0 , it suffices to show that
C 2j ðSÞ 6 C 2i ðS0 Þ:
ð8Þ
As in Baker [2], the completion times of the (r+1) th job in S and S0 are
C 2½rþ1 ðSÞ ¼
rþ1 X
p2½l ðSÞ
l¼1
l1 Y
ak þ
rþ1 X
k¼0
I2½l ðSÞ
l¼1
and
C 2½rþ1 ðS0 Þ ¼
rþ1 X
p2½l ðS0 Þ
l1 Y
l¼1
ak þ
rþ1 X
k¼0
I2½l ðS0 Þ;
l¼1
where p2[l](S) denote the normal processing time of the lth job in S and I2[l](S) denote the idle time occurring on M2 immediately prior to the lth job in S. P Ql1 Prþ1 0 Ql1 Since p2i 6 p2j, we have that rþ1 l¼1 p2½l ðSÞ k¼0 ak 6 l¼1 p2½l ðS Þ k¼0 ak . Thus, to show Eq. (8) hold, it suffices to show that rþ1 X
I2½l ðSÞ 6
rþ1 X
l¼1
I2½l ðS0 Þ:
ð9Þ
l¼1
As in Baker [2], the sum of the first q idle times in S and S0 are q X
I2½l ðSÞ ¼ maxðY ½t ðSÞÞ 16t6q
l¼1
and q X
I2½l ðS0 Þ ¼ maxðY ½t ðS0 ÞÞ 16t6q
l¼1
for q = 1, . . . , r + 1 where Y ½t ðSÞ ¼
Y ½t ðS0 Þ ¼
t X
p1½l ðS0 Þ
l¼1
l1 Y
Pt
ak
k¼0
Ql1
l¼1 p1½l ðSÞ
t1 X
p2½l ðS0 Þ
l¼1
k¼0 ak
l1 Y
Pt1
l¼1 p2½l ðSÞ
Ql1
k¼0 ak
and
ak :
k¼0
It is noted that Y[t](S) = Y[t](S0 ) for t = 1, . . . , r 1. Thus, to show Eq. (9) hold, it suffices to show that
maxfY ½r ðSÞ; Y ½rþ1 ðSÞg 6 Y ½r ðS0 Þ: Since p1i 6 p1j, we have
Y ½r ðS0 Þ Y ½r ðSÞ ¼ ðp1j p1i Þ
l1 Y k¼0
ak P 0:
ð10Þ
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Since p1i 6 p2i and ar 6 1, we have
Y ½r ðS0 Þ Y ½rþ1 ðSÞ ¼ ðp2i p1i Þ
r1 Y k¼0
ak þ p1j ð1 ar Þ
r1 Y
ak P 0:
k¼0
Thus, S dominates S0 . Therefore, repeating this interchange argument for jobs not sequenced in the SPT order completes the proof of the property. h
4. Conclusions In this paper, we proposed a new model of the learning effect. For the single-machine case, we showed that the problem to minimize the total tardiness is polynomially solvable under certain agreeable conditions. Moreover, we provided the optimal solutions for two single-machine multiple-criteria problems. For the case of two-machine permutation flowshop, we presented polynomial-time optimal solutions for the makespan problem under the assumption of ordered processing times. For future research, it is interesting to study the problems without the agreeable condition, to consider the multi-machine model with multi-criteria problems of scheduling, taking into account the effect of learning for the case of the machines in on-line mode, or to integrate the proposed approaches based on the effect of learning, and approaches based on Soft Computing technologies for solving multi-criteria problems of scheduling works on the multiple machines. Acknowledgements The author is grateful to the editor and the referees, whose constructive comments have led to a substantial improvement in the presentation of the paper. This work was supported by the NSC of Taiwan, ROC, under NSC 98-2221-E-035-033-MY2. References [1] U. Bagchi, Simultaneous minimization of‘mean and variation of flow time and waiting time in single machine systems, Operations Research 37 (1989) 118–125. [2] K. Baker, Introduction to Sequencing and Scheduling, John Wiley, New York, 1974. [3] D. Biskup, Single-machine scheduling with learning considerations, European Journal of Operational Research 115 (1999) 173–178. [4] D. Biskup, A state-of-the-art review on scheduling with learning effect, European Journal of Operational Research 188 (2008) 315–329. [5] T.C.E. Cheng, G. Wang, Single machine scheduling with learning effect considerations, Annals of Operations Research 98 (2000) 273–290. [6] T.C.E. Cheng, C.C. Wu, W.C. Lee, Some scheduling problems with sum-of-processing-times-based and job-position-based learning effects, Information Sciences 178 (2008) 2476–2487. [7] T.C.E. Cheng, P.J. Lai, C.C. Wu, W.C. Lee, Single-machine scheduling with sum-of-logarithm-processing-times-based learning considerations, Information Sciences 179 (2009) 3127–3135. [8] V.R. Donheti, B.B. Mohanty, Impact of learning and fatigue factors on single machine scheduling with penalties for tardy jobs, European Journal of Operational Research 105 (1998) 509–524. [9] J.M. Dutton, A. Thomas, Treating progress function as a managerial opportunity, Academy of Management 9 (1984) 235–247. [10] S. Gawiejnowicz, A note on scheduling on a single processor with speed dependent on a number of executed jobs, Information Processing Letters 56 (1996) 297–300. [11] W.J. Hurley, When are we going to change the learning curve lecture?, Computers & Operations Research 23 (1996) 509–511 [12] A. Janiak, R. Rudek, A new approach to the learning effect: beyond the learning curve restrictions, Computers & Operations Research 35 (2008) 3727– 3736. [13] C. Koulamas, G.J. Kyparisis, Single-machine and two-machine flowshop scheduling with general learning function, European Journal of Operational Research 178 (2007) 402–407. [14] M.A. Lapre, A.S. Mukherjee, L.N.V. Wassenhove, Behind the learning curve: linking learning activities to waste reduction, Management Science 46 (2000) 597–611. [15] W.C. Lee, C.C. Wu, H.J. Sung, A bi-criterion single-machine scheduling problem with learning considerations, Acta Informatica 40 (2004) 303–315. [16] G. Mosheiov, Scheduling problems with a learning effect, European Journal of Operational Research 132 (2001) 687–693. [17] S.S. Panwalkar, M.L. Smith, Common due date assignment to minimize total penalty for the one machine scheduling problem, Operations Research 30 (1982) 391–399. [18] J.B. Wang, Single-machine scheduling problems with the effects of learning and deterioration, OMEGA-The International Journal of Management Science 35 (2007) 397–402. [19] C.C. Wu, W.C. Lee, A note on the total completion time problem in a permutation flowshop with a learning effect, European Journal of Operational Research 192 (2009) 343–347. [20] Y. Yin, D. Xu, K. Sun, H. Li, Some scheduling problems with general position-dependent and time-dependent learning effects, Information Sciences 179 (2009) 2416–2425. [21] X.G. Zhang, G.L. Yan, Machine scheduling problems with a general learning effect, Mathematical and Computer Modelling 51 (2010) 84–90. [22] C.L. Zhao, Q.L. Zhang, H.Y. Tang, Machine scheduling problems with learning effects, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis 11 (2004) 741–747.