Single machine scheduling with exponential time-dependent learning effect and past-sequence-dependent setup times

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Computers and Mathematics with Applications

journal homepage:www.elsevier.com/locate/camwa

Single machine scheduling with exponential time-dependent learning

effect and past-sequence-dependent setup times

Ji-Bo Wang

a,∗

, Dan Wang

a

, Li-Yan Wang

a,b

, Lin Lin

a

, Na Yin

a

, Wei-Wei Wang

a

a_{Department of Science, Shenyang Institute of Aeronautical Engineering, Shenyang 110136, People’s Republic of China} b_{Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’s Republic of China}

a r t i c l e i n f o Article history:

Received 7 May 2007

Received in revised form 17 August 2008 Accepted 10 September 2008 Keywords: Scheduling Single machine Learning effect Setup times a b s t r a c t

In this paper we consider the single machine scheduling problem with exponential time-dependent learning effect and past-sequence-dependent (p-s-d) setup times. By the exponential time-dependent learning effect, we mean that the processing time of a job is defined by an exponent function of the total normal processing time of the already processed jobs. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the quadratic job completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the quadratic job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.

1. Introduction

In classical scheduling problems the processing time of a job is assumed to be a constant. However, in many realistic problems of operations management, both machines and workers can improve their performance by repeating the production operations. Therefore, the actual processing time of a job is shorter if it is scheduled later in a sequence. This phenomenon is known as the ‘‘learning effect’’ in the literature [1].

Biskup [2] and Cheng and Wang [3] were among the pioneers that brought the concept of learning into the field of scheduling, although it has been widely employed in management science since its discovery by Wright [4]. Biskup [2] assumed that the processing time of a job is a log-linear learning curve, i.e., if jobJjis scheduled in positionrin a sequence, its actual processing time ispjr

=

pjra, wherepjis the normal processing time of jobJj,a

≤

0 is a constant learning effect. He proved that single machine scheduling problems to minimize the sum of job flow times and the total deviations of job completion times from a common due date are polynomial solvable. Cheng and Wang [3] considered a single machine scheduling problem in which the job processing times decrease as a result of learning. A volume-dependent piecewise linear processing time function was used to model the learning effect. The objective is to minimize the maximum lateness. They showed that the problem is NP-hard in the strong sense and then identified two special cases that are polynomially solvable. They also proposed two heuristics and analysed their worst-case performance. Later, Mosheiov [5,6] investigated several other single machine problems and the minimum total flow time problem on identical parallel machines. Lee and Wu [7] proposed a heuristic algorithm to solve the total completion time minimization problem in a two-machine flow shop with a

∗_{Corresponding author.}

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

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learning effect. Lee et al. [8] studied the learning effect in a bi-criterion single machine scheduling problem, with the objective of minimizing a linear combination of the total completion time and the maximum tardiness. They presented a branch-and-bound and a heuristic algorithm to search for optimal and near optimal solutions. Wang and Xia [9] considered flow shop scheduling problems with a learning effect. The objective was to minimize one of two regular performance measures, namely the makespan and the total flow time. They gave a heuristic algorithm with a worst-case error bound ofmfor each criterion, where mis the number of machines. They also found polynomial time solutions for two special cases of the problem, i.e., identical processing times on each machine and an increasing series of dominating machines. Wang [10] considered the same problem of Wang and Xia [9]. He suggested the use of Johnson’s rule as a heuristic algorithm for two-machine flow shop scheduling to minimize the makespan. He also developed polynomial time solution algorithms for some special cases of the following objective functions: the weighted sum of completion times and the maximum lateness. Wang and Xia [9] and Wang [10] also extended their results to the model: Pegels’ learning curve [11], i.e., if jobJjis scheduled in positionrin a sequence, its actual processing time ispjr

=

pj

[

α

ar−1

+

β

]

, where

α,

aand

β

are parameters obtained empirically. Mosheiov and Sidney [12] considered a job-dependent learning curve, where the learning rate of some jobs is faster than that of the others, i.e., if jobJjis scheduled in positionrin a sequence, its actual processing time ispjr

=

pjraj, whereajis a negative job-dependent parameter. They showed that the makespan minimization problem, the total flow time minimization problem, a due date assignment problem, and total flow time minimization on unrelated parallel machines remain polynomailly solvable. Biskup and Simons [13] considered a scheduling problem where the processing times decrease according to a learning rate, which can be influenced by an initial cost-incurring investment. They presented a formulation of the common due date scheduling problem with autonomous and induced learning effects. They further proved some structural properties, which enable the development of a polynomial bound solution procedure. Kuo and Yang [14] considered a single machine scheduling problem with a time-dependent learning effect. The time-dependent learning effect of a job is assumed to be a function of the total normal processing time of the jobs scheduled in front of it, i.e., if jobJjis scheduled in positionrin a sequence, its actual processing time isp_jr

=

p_j

(

1

+

p[1]

+

p[2]

+ · · · +

p[r−1]

)

a

,

wherea

≤

0 is a constant learning index.

They showed that the SPT-sequence is the optimal sequence for the objective of minimizing the total completion time. Kuo and Yang [15] considered the single-machine group scheduling problem with a time-dependent learning effect. They showed that the single-machine group scheduling problem with a time-dependent learning effect remains polynomially solvable for the objectives of minimizing the makespan and minimizing the total completion time. Eren and Guner [16] considered a bicriteria single machine scheduling problem with a learning effect to minimize a weighted sum of total completion time and total tardiness. To solve this scheduling problem, they developed a mathematical programming model. Eren and Guner [17] considered the single machine total tardiness problem with a learning effect. They developed an integer programming model for the problem. Wang et al. [18] considered the same model of Kuo and Yang [14]. They proved that the weighted shortest processing time (WSPT) rule, the earliest due date (EDD) rule and the modified Moore–Hodgson algorithm can, under certain conditions, construct the optimal schedule for the problem to minimize the following three objectives: the total weighted completion time, the maximum lateness and the number of tardy jobs, respectively. They also gave an error estimation for each of these rules for the general cases. Koulamas and Kyparisis [19] considered a single machine scheduling problem with general learning functions. The learning is expressed as a function of the sum of the processing times of the already processed jobs, i.e., if jobJjis scheduled in positionrin a sequence, its actual processing time ispjr

=

pj

1

−

Pr−1 i=1p[i] Pn i=1pi

a

=

pj

Pn i=rp[i] Pn i=1pi

a

,

wherea

≥

1 is the learning index. They showed that the SPT-sequence is the optimal sequence for the objectives of minimizing the makespan and total completion time. They also considered two-machine flowshop scheduling with the general learning functions. Eren [20] considered a bicriteria parallel machine scheduling problem with a learning effect of setup times and removal times. The objective function of the problem is minimization of the weighted sum of total completion time and total tardiness. He developed a mathematical programming model for the problem which belongs to an NP-hard class. Extensive surveys of different scheduling models and problems involving jobs with learning effects can be found in [21,22].

On the other hand, it is reasonable and necessary to consider scheduling problems with setup times. There are two types of setup time or setup cost: sequence-independent and sequence-dependent. In the first case, the setup time/cost depends solely on the task to be processed, regardless of its preceding task. While in the sequence-dependent type, setup time/cost depends on both the task and its preceding task. In some environments, there may be different families or batches of tasks which involve (minor) setup times/costs among the tasks within a family and (major) setup times/costs among the task families. The family setup time/cost can be also sequence-independent or sequence-dependent [23,24]. Koulamas and Kyparisis [25] first introduced a scheduling problem with past-sequence-dependent (p-s-d) setup times, i.e., the setup time is dependent on all already scheduled jobs. They proved that the standard single machine scheduling with p-s-d setup times can be solvable in polynomial time when the objectives are the makespan, the total completion time and the total absolute differences in completion times, respectively. They also extended their results to nonlinear p-s-d setup times. For recent results and trends in scheduling problems with setup times or costs, the reader may refer to the review paper of Allahverdi et al. [24].

However, to the best of our knowledge, apart from the recent paper of Kuo and Yang [26], it has not been investigated the scheduling models considering the setup times and learning effect at the same time. Kuo and Yang [26] considered single machine scheduling with past-sequence-dependent setup times and job-independent (job-dependent) learning effect. The

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phenomena of past-sequence-dependent setup times occurring can be found in many real-life situations. For example, in high tech manufacturing environments, in which a batch of jobs consists of a group of electronic components mounted together on an integrated circuit (IC) board. These jobs must be processed one-by-one by a machine while they are mounted together on the board. The machine’s operation on any of these components has an adverse effect on the ‘‘readiness’’ of all the other components which have not yet been processed due to the flow of electrical current through the IC board while the machine is operating. Once a component is fully processed, its condition is not affected by the subsequent operation of the machine even if it remains mounted on the IC board. The degree of ‘‘un-readiness’’ of any component is proportional to the amount of time it has been exposed to the machine’s operation on other components. Consequently, prior to a component’s processing, a setup operation, proportional to the degree of ‘‘un-readiness’’ of the respective component, is needed to restore it to ‘‘full-readiness’’ status; this setup operation has no effect on the ‘‘readiness’’ of the remaining unprocessed components. The overall manufacturing process is completed when all components on the IC board have been processed by the machine [25]. The exponential time-dependent learning effect can be described by the following example. If human interactions have a significant impact during the processing of the job, the processing time will add to the workers’ experience and cause learning effects. In this paper we consider the single machine scheduling problem with exponential time-dependent learning effect and past-sequence-dependent setup times.

The remaining part of this paper is organized as follows. In Section2we formulate the model. In Section3we consider several single-machine scheduling problems. The last section presents the conclusions.

2. Problems description

There are given a single machine andn independent and non-preemptive jobs that are immediately available for processing. The machine can handle one job at a time and preemption is not allowed. Associated with each jobJj

(

j

=

1

,

2

, . . . ,

n

)

there is a normal processing timepj, a due datedjand a weight

w

j. In addition, letp[k]andpA[k]be the normal

processing time and actual processing time of a job if it is scheduled in thekth position in a sequence, respectively. LetpA_jr

be the actual processing time of jobJjif it is scheduled in positionrin a sequence. In this paper, we consider a new learning effect model, i.e.,

pA_jr

=

pj





α

a r−1 P i=1 p[i]

+

β





,

r

,

j

=

1

,

2

, . . . ,

n

,

(1) where

α

≥

0

, β

≥

0 and 0

<

a

≤

1 are parameters obtained empirically, and

α

+

β

=

1. As in [25], we assume that the p-s-d setup time of jobJ[r]if it is scheduled in positionris given as follows:

s[1]

=

0 and s[r]

=

b

r−1

X

i=1

pA_[_i_]

,

(2)

whereb

≥

0 is a normalizing constant.

For a given schedule

π

= [

J1,J2, . . . ,Jn

]

. LetCj

=

Cj

(π)

represent the completion time of jobJj. Also letCmax

=

max

{

Cj

|

j

=

1

,

2

, . . . ,

n

}

,

P

Cj,

P

Cj2,

P

w

jCjandLmax

=

max

{

Cj

−

dj

|

j

=

1

,

2

, . . . ,

n

}

represent the makespan, the total completion time, the sum of the quadratic job completion times, the total weighted completion time and the maximum lateness of a given permutation, respectively. Obviously, the completion ofJ[j]isC[j]

(π)

=

P

j

i=1(s[i]

+

pA[i]

)

=

P

j

i=1(b

(

j

−

i

)

+

1

)

pA[i]. For

convenience, we denote the learning effect given in(1)byLE. In the remaining part of the paper, all the problems considered will be denoted using the three-field notation scheme

α

|

β

|

γ

introduced by Graham et al. [27].

3. Single machine scheduling problems

First, we give some lemmas; they are useful for the following theorems.

Lemma 1. x2_ax_ln_a

−

_xax

+

_x

≥

₀_if ₀

<

_a

≤

₁_{, and x}

≥

₀_.

Proof. Leth

(

a

)

=

x2_ax_ln_a

−

_xax

+

_x_{. Then we have}_h0

(

_a

)

=

_x3_ax−1_ln_a

≤

_{0 for 0}

<

_a

≤

_{1, and}_x

≥

_{0. Hence}_h

(

_a

)

_is decreasing on 0

<

a

≤

1, andx

≥

0 forh

(

a

)

≥

h

(

1

)

=

0. This completes the proof.

Lemma 2. x

(

aλx

−

1

)

−

λ

x

(

ax

−

1

)

≥

0if

λ

≥

1,0

<

a

≤

1, and x

≥

0.

Proof. Letf

(λ)

=

x

(

aλx

−

₁

)

−

λ

_x

(

_ax

−

₁

)

_{. Then we have}

f0

(λ)

=

x2aλxlna

−

x

(

ax

−

1

)

and

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Hence,f0

(λ)

is increasing on

λ

≥

1, 0

<

a

≤

1, andx

≥

0 forf00

(λ)

≥

0. In addition, fromLemma 1, we have

f0

(

1

)

=

x2axlna

−

xax

+

x

≥

0

.

Therefore,f0

(λ)

≥

_f0

(

₁

)

≥

_{0 for}

λ

≥

_{1, 0}

<

_a

≤

_{1, and}_x

≥

_0.

Hence,f

(λ)

is increasing on

λ

≥

1, 0

<

a

≤

1, andx

≥

0. Also,f

(λ)

≥

f

(

1

)

=

0 for

λ

≥

1, 0

<

a

≤

1, andx

≥

Theorem 1. For the problem1

|

LE

,

spsd

|

Cmaxwhen pAjris given by(1), an optimal schedule can be obtained by sequencing the jobs

in non-decreasing order of pj(the SPT rule).

Proof. Let

π

and

π

0_{be two job schedules where the difference between}

π

_and

π

0_{is a pairwise interchange of two adjacent}

jobsJjandJk, that is,

π

= [

S1,Jj

,

Jk

,

S2

]

, π

0

= [

S1,Jk

,

Jj

,

S2

]

, whereS1andS2are partial sequences. Furthermore, we assume that there arer

−

1 jobs inS1. Thus,JjandJkare therth and the

(

r

+

1

)

th jobs, respectively, in

π

and withpj

≤

pk. Likewise,Jk andJjare scheduled in therth and the

(

r

+

1

)

th positions in

π

0. To further simplify the notation, lettdenote the completion time of the last job inS1. To show

π

dominates

π

0, it suffices to show that the

(

r

+

1

)

th jobs in

π

and

π

0satisfy the condition thatCk

(π)

≤

Cj

(π

0

)

. Under

π

, the completion times of jobsJjandJkare

Cj

(π)

=

r−1

X

i=1

(

b

(

r

−

i

)

+

1

)

pA[i]

+

pj





α

a r−1 P i=1 p[i]

+

β





.

Ck

(π)

=

r−1

X

i=1

(

b

(

r

+

1

−

i

)

+

1

)

pA_[_i_]

+

(

b

+

1

)

pj





α

a r−1 P i=1 p[i]

+

β





+

pk





α

a r−1 P i=1 p[i]+pj

+

β





.

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Whereas under

π

0, they are

Ck

(π

0

)

=

r−1

X

i=1

(

b

(

r

−

i

)

+

1

)

pA_[_i_]

+

pk





α

a r−1 P i=1 p[i]

+

β





and Cj

(π

0

)

=

r−1

X

i=1

(

b

(

r

+

1

−

i

)

+

1

)

pA_[_i_]

+

(

b

+

1

)

pk





α

a r−1 P i=1 p[i]

+

β





+

pj





α

a r−1 P i=1 p[i]+pk

+

β





.

(4)

Based on Eqs.(3)and(4), we have

Cj

(π

0

)

−

Ck

(π)

=

b





α

a r−1 P i=1 p[i]

+

β





(

pk

−

pj

)

+

α

a r−1 P i=1 p[i]

pj apk

−

1

−

pk apj

−

1

.

Let

λ

=

pk

/

pj, thenpj

(

apk

−

1

)

−

pk

(

apj

−

1

)

can be rewritten as

pj

(

aλpj

−

1

)

−

λ

pj

(

apj

−

1

).

Letx

=

pj

≥

0. Then fromLemma 2, if

λ

=

pk

/

pj

≥

1 and 0

<

a

≤

1, we havepj

(

apk

−

1

)

−

pk

(

apj

−

1

)

≥

0. In addition

pj

≤

pk, we haveb

(α

a Pr−1

i=1p[i]

+

β)(

_p

k

−

pj

)

≥

0, henceCj

(π

0

)

−

Ck

(π)

≥

|

LE

,

spsd

|

P

Cjwhen pAjris given by(1), an optimal schedule can be obtained by sequencing the

jobs in non-decreasing order of pj(the SPT rule).

Proof. Here, we still use the same notations as in the proof ofTheorem 1. In order to show

π

dominates

π

0_{, it suffices to}

show that (i)Ck

(π)

≤

Cj

(π

0

)

and (ii)Cj

(π)

+

Ck

(π)

≤

Ck

(π

0

)

+

Cj

(π

0

)

.

The proof of part (i) is given inTheorem 1. In addition, frompj

≤

pk, we haveCj

(π)

≤

Ck

(π

0

)

, hence

Cj

(π)

+

Ck

(π)

≤

Ck

(π

0

)

+

Cj

(π

0

).

This completes the proof of part (ii) and thus of the theorem.

Townsend [28] studied the single machine scheduling with quadratic objective. He showed that the problem 1

||

P

C2 j can be solved optimally by the SPT rule. By the similar proof ofTheorem 2, we can show that the solution of Townsend’s still holds for the problem 1

|

LE

,

spsd

|

P

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Theorem 3.For the problem1

|

LE

,

spsd

|

P

C_j2 when pA_jr is given by(1), there exists an optimal schedule in which jobs are sequenced in non-decreasing order of their basic processing times (the SPT rule).

Proof. Here, we still use the same notations as in the proof ofTheorem 1. In order to show

π

dominates

π

0_{, it suffices to}

show that (i)Ck

(π)

≤

Cj

(π

0

)

and (ii)Cj2

(π)

+

C 2 k

(π)

≤

C 2 k

(π

0

)

+

C_j2

(π

0

)

.

The proof of part (i) is given inTheorem 1. In addition, frompj

≤

pk, we haveCj

(π)

≤

Ck

(π

0

)

, hence

C_j2

(π)

+

C_k2

(π)

≤

C_k2

(π

0

)

+

C_j2

(π

0

).

This completes the proof of part (ii) and thus of the theorem.

For the other two objective functions, i.e., minimizing the total weighted completion time and minimizing maximum lateness, we show that the problems can be solved in polynomial time under some special conditions.

Lemma 3. 1

−

λ1

at

+

_t

λ2

_at_ln_a

≥

₀_if ₀

<

_a

≤

₁_,₀

≤

λ1

≤

λ2

≤

₁_{and t}

≥

₀_.

Proof. Obviously, 1

−

λ1

at

+

_t

λ2

_at_ln_a

≥

₁

−

λ2

_at

+

_t

λ2

_at_ln_a_{for 0}

≤

λ1

≤

λ2

≤

_{1. Let}_h

(

_t

)

=

₁

−

λ2

_at

+

_t

λ2

_at_ln_a_{. Then} we haveh0

(

t

)

=

t

λ2

atln2a

≥

0 for 0

<

a

≤

1, 0

≤

λ1

≤

λ2

≤

1 andt

≥

0. Henceh

(

t

)

is increasing on the value oft. Since

h

(

t

)

≥

h

(

0

)

=

1

−

λ2

≥

Lemma 4.

λ(

1

−

λ1

at

)

−

(

₁

−

λ2

_aλt

)

≥

₀_if ₀

<

_a

≤

₁_,₀

≤

λ1

≤

λ2

≤

₁_,

λ

≥

₁_{and t}

≥

₀_.

Proof. Letf

(λ)

=

λ(

1

−

λ1

at

)

−

(

₁

−

λ2

_aλt

)

_{. Then we have}

f0

(λ)

=

1

−

λ1

at

+

t

λ2

aλtlna

and

f00

(λ)

=

t2

λ2

aλtln2a

≥

0

.

Hence,f0

(λ)

_{is increasing on 0}

<

_a

≤

_{1, 0}

≤

λ1

≤

λ2

≤

_1,

λ

≥

_{1 and}_t

≥

_{0 for}_f00

(λ)

≥

_{0. In addition, from}_{Lemma 3}_{, we}

have

f0

(

1

)

=

1

−

λ1

at

+

t

λ2

atlna

≥

0

.

Therefore,f0

(λ)

≥

_f0

(

₁

)

≥

_{0 for 0}

<

_a

≤

_{1, 0}

≤

λ1

≤

λ2

≤

_1,

λ

≥

_{1 and}_t

≥

_0.

Hence,f

(λ)

is increasing on 0

<

a

≤

1, 0

≤

λ1

≤

λ2

≤

1,

λ

≥

1 andt

≥

0. Also,f

(λ)

≥

f

(

1

)

=

(λ2

−

λ1)

at

≥

_{0 for} 0

<

a

≤

1, 0

≤

λ1

≤

λ2

≤

1,

λ

≥

1 andt

≥

Theorem 4.For the problem1

|

LE

,

spsd

|

P

w

jCjwhen pAjris given by(1), if the jobs have agreeable weights, i.e., pj

≤

pkimplies

w

j

≥

w

kfor all the jobs Jjand Jk, an optimal schedule can be obtained by sequencing the jobs in non-decreasing order of pj

/w

j

(the WSPT rule).

Proof (By Contradiction). Consider an optimal schedule

π

that does not follow the WSPT rule. In this schedule there must be at least two adjacent jobs, say jobJjfollowed by jobJk, such thatpj

/w

j

>

pk

/w

k, which impliespj

≥

pk. Assume that jobJj is scheduled in positionr. Perform an adjacent pair-wise interchange of jobsJjandJk. Whereas under the original schedule

π

jobJjis scheduled in positionrand jobJkis scheduled in positionr

+

1, under the new schedule jobJkis scheduled in positionrand jobJjis scheduled in positionr

+

1. In addition, lett be the starting time for jobJjin

π

. All the other jobs remain in their original positions. Call the new schedule

π

0_{. The completion times of the jobs processed before jobs}_J

jand

Jkare not affected by the interchange. Furthermore, the completion times of the jobs processed after jobsJjandJkare not increased by the interchange sincepj

≥

pk. Under

π

, the completion times of jobsJjandJkare

Cj

(π)

=

r−1

X

i=1

(

b

(

r

−

i

)

+

1

)

pA[i]

+

pj





α

a r−1 P i=1 p[i]

+

β





.

Ck

(π)

=

r−1

X

i=1

(

b

(

r

+

1

−

i

)

+

1

)

pA_[_i_]

+

(

b

+

1

)

pj





α

a r−1 P i=1 p[i]

+

β





+

pk





α

+

β





whereas under

π

0, they are

Ck

(π

0

)

=

r−1

X

i=1

(

b

(

r

−

i

)

+

1

)

pA_[_i_]

+

pk





α

a r−1 P i=1 p[i]

+

β





(6)

and Cj

(π

0

)

=

r−1

X

i=1

(

b

(

r

+

1

−

i

)

+

1

)

pA_[_i_]

+

(

b

+

1

)

pk





α

a r−1 P i=1 p[i]

+

β





+

pj





α

+

β





.

So we have

w

jCj

(π)

+

w

kCk

(π)

−

w

kCk

(π

0

)

−

w

jCj

(π

0

)

=

b

(w

k

−

w

j

)

r−1

X

i=1 pA_[_i_]

+





b





α

a r−1 P i=1 p[i]

+

β





+

β





(w

kpj

−

w

jpk

)

+

(w

_j

+

w

_k

)(

pj

−

pk

)





α

a r−1 P i=1 p[i]





+

w

_kpk





α





−

w

jpj





α





Let

λ1

=

_wjwj +wk

, λ2

=

wk wj+wk

,

x

=

P

r−1 i=1p[i]

,

t

=

pkand

λ

=

pj pk. Then

(w

j

+

w

k

)(

pj

−

pk

)

α

aPri=−11p[i]

+

w

kpk

α

aPri=−11p[i]+pj

−

w

_jpj

α

aPri=−11p[i]+pk

can be rewritten as

(w

j

+

w

k

)(

pj

−

pk

)





α

a r−1 P i=1 p[i]





+

w

kpk





α





−

w

jpj





α





=

α(w

_j

+

w

_k

)

a r−1 P i=1 p[i] pk

[

λ(

1

−

λ1

at

)

−

(

1

−

λ2

aλt

))

]

≥

0 (fromLemma 4)

.

Frompj

/w

j

>

pk

/w

k, we have

b

α

aPri=−11p[i]

+

β

+

β

(w

kpj

−

w

jpk

)

≥

0. In addition, frompj

≥

pkwhich implies

w

j

≤

w

k, we haveb

(w

k

−

w

j

)

P

r−1

i=1pA[i]

≥

0. Hence,

w

jCj

(π)

+

w

kCk

(π)

≥

w

kCk

(π

0

)

+

w

jCj

(π

0

)

. It follows that the weighted sum of completion times under

π

0is less than that under

π

. This contradicts the optimality of

π

and proves the theorem.

Using the similar method ofTheorem 4, the following corollaries can be easily obtained.

Corollary 1. For the problem1

|

LE

,

spsd,pj

=

p

|

P

w

jCjwhen pAjr is given by(1), an optimal schedule can be obtained by

sequencing the jobs in non-increasing order of

w

j.

|

LE

,

spsd, wj

=

kpj

|

P

w

jCjwhen pAjr is given by(1), an optimal schedule can be obtained by

sequencing the jobs in non-decreasing order of pj(the SPT rule).

|

LE

,

spsd

|

Lmaxwhen pAjris given by(1), if the jobs have agreeable conditions, i.e., pi

≤

pjimplies

di

≤

djfor all the jobs i and j, an optimal schedule can be obtained by sequencing the jobs in non-decreasing order of dj, (the EDD

rule).

Proof. We still use the same notations mentioned above. Now we use the job interchanging technique to prove the theorem. From the proof ofTheorem 1, under

π

, the lateness of the jobsJjandJkare

Lj

(π)

=

r−1

X

i=1

(

b

(

r

−

i

)

+

1

)

pA[i]

+

pj





α

a r−1 P i=1 p[i]

+

β





−

dj

,

Lk

(π)

=

r−1

X

i=1

(

b

(

r

+

1

−

i

)

+

1

)

pA_[_i_]

+

(

b

+

1

)

pj





α

a r−1 P i=1 p[i]

+

β





+

pk





α

+

β





−

dk

,

whereas under

π

0_{, they are}

Lk

(π

0

)

=

r−1

X

i=1

(

b

(

r

−

i

)

+

1

)

pA_[_i_]

+

pk





α

a r−1 P i=1 p[i]

+

β





−

dk

,

(7)

Lj

(π

0

)

=

r−1

X

i=1

(

b

(

r

+

1

−

i

)

+

1

)

pA_[_i_]

+

(

b

+

1

)

pk





α

a r−1 P i=1 p[i]

+

β





+

pj





α

+

β





−

dj

.

Ifdj

≥

dk, we haveLk

(π)

≥

Lj

(π)

. In addition, ifdj

≥

dkandpj

≥

pk, fromTheorem 1, we haveLk

(π)

≥

Lj

(π

0

)

and

Lk

(π)

≥

Lk

(π

0

)

. Therefore, we have

max

{

Lj

(π

0

),

Lk

(π

0

)

} ≤

max

{

Lj

(π),

Lk

(π)

}

.

This completes the proof of the theorem.

Using the similar method ofTheorem 5, the following corollaries can be easily obtained.

|

LE

,

spsd,pj

=

p

|

Lmaxwhen pAjris given by(1), an optimal schedule can be obtained by sequencing

the jobs in non-decreasing order of dj(the EDD rule).

|

LE

,

spsd,dj

=

d

|

the jobs in non-decreasing order of pj(the SPT rule).

|

LE

,

spsd,dj

=

kpj

|

the jobs in non-decreasing order of dj(the EDD rule).

4. Conclusions

In this paper we considered single machine scheduling problems with p-s-d setup times and exponential time-dependent learning effect. We proved that the makespan minimization problem, the total completion time minimization problem and the sum of the quadratic job completion times minimization problem can be solved by the SPT rule, respectively. We also proved that some special cases of the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time. However, the computational complexity of the total weighted completion time minimization problem and the maximum lateness minimization problem are still open. It is suggested for future research to investigate these open problems, consider the other objective functions, use the property for solving the single machine problems with p-s-d setup times and exponential time-dependent learning effect in the context of other more complicated scheduling settings, or propose more general and practical models.

Acknowledgements

We are grateful to two anonymous referees for their helpful comments on an earlier version of this paper. This research was supported by the Science Research Foundation of the Educational Department of Liaoning Province, China, under grant number 20060662, and the National Natural Science Foundation of China. LYW was supported by the Science Research Foundation of Shenyang Institute of Aeronautical Engineering, China, under grant number 200712Y. NY was supported by the Science Research Foundation of Shenyang Institute of Aeronautical Engineering, under grant number 200817Y.

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ScienceDirect

doi:10.1016/j.camwa.2008.09.025

doi:10.1016/j.apm.2008.01.010.