Applied Mathematical Modelling 35 (2011) 3509–3515
Contents lists available at ScienceDirect
Applied Mathematical Modelling
journal homepage: www.elsevier.com/locate/apm
Single-machine scheduling with linear decreasing deterioration
to minimize earliness penalties
Xue-Ru Wang ⇑, Xue Huang, Ji-Bo Wang
School of Science, Shenyang Aerospace University, Shenyang 110136, China
a r t i c l e
i n f o
Article history:
Received 13 October 2009
Received in revised form 2 January 2011
Accepted 11 January 2011
Available online 27 January 2011
Keywords:
Scheduling
Single-machine
Deteriorating jobs
SLK due date
a b s t r a c t
We consider a single-machine scheduling problem with linear decreasing deterioration in
which the due dates are determined by the equal slack (SLK) method. By the linear decreasing deterioration, we mean that the job’s processing time is a decreasing function of its
starting time. The objective is to minimize the total weighted earliness penalty subject
to no tardy jobs. We prove that two special cases of the problem remain polynomially solvable. The first case is the problem with equally weighted monotonous penalty objective
function and the other case is the problem with weighted linear penalty objective function.
Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction
Machine scheduling problems with deteriorating jobs have received increasing attention in recent years. Researchers
have formulated this phenomenon into different models and solved different versions of the problem for various criteria.
Extensive surveys of different scheduling models and problems involving deteriorating jobs can be found in Alidaee and Womer [1], and Cheng et al. [2].
Recently, Wang and Xia [3] considered various single-machine and flow-shop scheduling problems with decreasing linear
deterioration of job processing times. Wang and Xia [4,5] considered flow-shop problems involving job deterioration and
dominating machines. Wang et al. [6] considered two-machine flow shop scheduling to minimize total completion time with
simple linear deterioration. Gawiejnowicz et al. [7] considered two single-machine bicriterion scheduling problems with
time-dependent job processing times. Gawiejnowicz [8] considered two single-machine makespan minimization scheduling
problems with proportionally deteriorating jobs. In the first problem, the machine is not continuously available for processing but the number of non-availability periods, and the start time and the end time of each period are known in advance. In
the second problem, the machine is available all the time but for each job a ready time and a deadline are defined. He showed
that both problems are NP-hard. Wang [9] considered the general, no-wait and no-idle flow-shop scheduling problems with
deteriorating jobs. These studies assumed that the processing time of a job is a decreasing function of its starting time. Wu
et al. [10] considered single-machine total weighted completion time scheduling problem under linear deterioration. They
proposed a branch-and-bound method and several heuristic algorithms to solve the problem. Shiau et al. [11] considered
two-machine flowshop scheduling to minimize mean flow time with simple linear deterioration. Wang et al. [12] considered
the single-machine scheduling problems with deterioration jobs and group technology assumption. They showed that the
makespan minimization problem and the total weighted completion time minimization problem remain polynomially
⇑ Corresponding author.
E-mail address: [email protected] (X.-R. Wang).
0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.apm.2011.01.005
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X.-R. Wang et al. / Applied Mathematical Modelling 35 (2011) 3509–3515
solvable. Cheng et al. [13] considered some scheduling problems with the actual job processing time is a function of jobs
already processed. Leung et al. [14] considered the scheduling problem on parallel and identical machines where the jobs
are processed in batches and the processing time of each job is a step function of its waiting time. They showed that the
problem is NP-hard in the strong sense. Wu and Lee [15] considered the single-machine group scheduling problems with
deteriorating setup times and job processing times. Wang [16] considered the problems of scheduling jobs with start-time
decreasing processing times. The objective of the scheduling problems is to minimize the makespan. Under the parallel
chains and series–parallel graph precedence constraints assumption, they proved that the problems are polynomially solvable. Wang and Liu [17] investigated a two-machine flow shop scheduling problem with a proportional linear decreasing
deterioration. Wang et al. [18] considered some single-machine scheduling problems with the effects of deterioration and
learning. They proved that the makespan minimization problem, the total completion time minimization problem can be
optimally solved, respectively. They 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. Wang [19] considered some
single-machine scheduling problems with decreasing time-dependent job processing times. They presented polynomial
solutions for the sum of squared completion times minimization problem, and the sum of earliness penalties minimization
problem subject to no tardy jobs, respectively.
Most of the scheduling of the frontal papers with deteriorating jobs literature examines regular measures of the performance, which are non-decreasing functions of job completion times. Yet in certain situations one is more interested in performance measures that are non-regular. To the best of our knowledge, there exist only a few research results on scheduling
models considering non-regular performance measures. Cheng et al. [20] considered single machine scheduling problem
with linear job-independent increasing deterioration. The problem was to determine an optimal combination of the due date
and schedule so as to minimize the sum of due date, earliness and tardiness penalties. They gave a polynomial time algorithm to solve this problem. Cheng et al. [21] considered the same model of Cheng et al. [20], but with linear job-independent
decreasing deterioration. Oron [22] considered a single machine scheduling problem with simple linear deterioration. The
objective function was to minimize the total absolute deviation of completion times (TADC). They proved some properties
of an optimal schedule, and introduced two heuristic algorithms to solve this problem. Wang and Guo [23] considered a
single-machine scheduling problem with the effects of learning and deterioration. The problem is to determine an optimal
combination of the due-date and schedule so as to minimize the sum of earliness, tardiness and due-date. They showed that
the problem remains polynomially solvable under the proposed model. Wang and Wang [24] studied the single-machine
scheduling problem with simple linear deterioration where the objective is to minimize total weighted earliness subject
to no tardy jobs. They considered the problem in which the due dates are determined by the equal slack (SLK) method.
For both the case with equally weighted monotonous penalty objective function and the case with weighted linear penalty
objective function, they proved that the problem can be solved in polynomial time. Huang et al. [25] extended the results of
[24], by considering a more general deterioration model that includes the one given in [24] as a special case.
Generally, there are two types of models describing this kind of processes. The first type is devoted to the problems in
which the job processing time is characterized by a non-decreasing function, and the second type concerns problems in
which the job processing time is given by a non-increasing function. In this paper we study the latter group of problems,
i.e., single-machine scheduling problems with decreasing time-dependent job processing times. This model was proposed
by Ho et al. [26]. In this paper we consider single machine scheduling with decreasing time-dependent job processing times.
The objective assumed is to minimize a non-regular performance measure, i.e., the total weighted earliness penalty subject
to no tardy jobs in which the due dates are determined by the equal slack (SLK) method. For the classical work on the SLK
method and the total weighted earliness penalty subject to no tardy jobs, the reader is referred to Chang and Schneeberger
[27], Qi and Tu [28], Gordon and Strusevich [29], Pathumnakul and Egbelu [30], and Wang and Wang [31].
The remaining part of this paper is organized as follows. In Section 2 we formulate the problem and give some basic results. In Sections 3 and 4 we consider two special cases of the scheduling problem, i.e., with equally weighted monotonous
penalty and weighted linear penalty, and also present algorithms to solve them. The last section is the conclusion.
2. Problem formulation and analysis
The single-machine scheduling problem is to schedule n jobs N = {J1, J2, . . . , Jn} on one machine. All the jobs are available
for processing at some time t0 P 0. Each job Jj has a normal processing time aj, a weight wj and a due date dj. Following Ho
et al. [25] and Wang and Xia [3], we assume that the actual processing time pj of job Jj is a non-increasing linear function of
the job’s starting time, i.e.,
pj ¼ aj ð1 btÞ;
ð1Þ
where aj P 0, b P 0 and t P t0 is the job’s starting time. It is assumed that the normal processing times satisfy the following
condition:
b t0 þ
n
X
j¼1
!
aj amin < 1;
ð2Þ
X.-R. Wang et al. / Applied Mathematical Modelling 35 (2011) 3509–3515
3511
where amin = mini=1,2,. . .,n{ai}. The condition ensures that all job processing times are positive in a feasible schedule (see also
[25,3] for detailed explanations).
For a given schedule p = [J1, J2, . . . , Jn], Cj = Cj(p) represents the completion time of job Jj. Let Ej = dj Cj be the earliness of
P
job Jj. The objective is to minimize nj¼1 wj gðEj Þ, where g(x) is a strictly increasing function. A schedule is feasible if and only if
there is no tardy job in the schedule. In this paper we assume that the due dates are determined according to the so-called
SLK rule, i.e., for each job the due date is obtained by adding a positive slack q to the processing times: dj = pj + q, j = 1, 2, . . . , n.
Extending the stand scheme for scheduling notation (Graham et al. [32] and Gawiejnowicz [33]), we refer to this due date
P
assignment problem as 1jpj ¼ aj ð1 btÞ; C j 6 dj ¼ pj þ qj nj¼1 wj gðEj Þ.
We first give some lemmas, which are useful for the following theorems.
Lemma 1. For a given schedule p = [J1, J2, . . . ,Jn], if job J1 starts at time t0 P 0, then job Jj’s processing time pj is equal to
Y
j1
1
pj ¼ b t0 aj ð1 bai Þ:
b
i¼1
ð2Þ
Q
1
Proof. For a given scheduling p = [J1, J2, . . . , Jn], the completion time Cj1 of job Jj1 is C j1 ðpÞ ¼ t 0 1b j1
i¼1 ð1 bai Þ þ b
(Wang and Xia [3]), hence,
Y
j1
1
pj ¼ aj ð1 bC j1 Þ ¼ b t0 aj ð1 bai Þ:
b
i¼1
This completes the lemma. h
Lemma 2. For a given schedule p = [J1, J2, . . . , Jn] for the problem 1jpj = aj(1 bt)jCmax, if the makespan is C, then the starting time
Sj of job Jj is
Sj ¼
C
n
1 Y
1
= ð1 bai Þ þ :
b
b
i¼j
Proof. Similar to the proof of Lemma 1. h
Theorem 1. In a feasible schedule, there exists at most one on-time job.
Proof. Suppose p is a feasible schedule and Jp(j) is an on-time job in p, where p(j) denote jth job in the schedule p,
Q
C pðjÞ ¼ dpðjÞ ¼ b t 0 1b apðjÞ j1
i¼1 ð1 bapðiÞ Þ þ q. Consider the job Jp(k).
Q
If k > j, then SpðkÞ P C pðjÞ ¼ b t 0 1b apðjÞ j1
i¼1 ð1 bapðiÞ Þ þ q; and we have
j1
Y
1
C pðkÞ ¼ SpðkÞ þ ppðkÞ P b t 0 apðjÞ ð1 bapðiÞ Þ þ q þ ppðkÞ > dpðkÞ :
b
i¼1
So job Jp(k) is tardy and p is not a feasible schedule.
For all k < j,
C pðkÞ 6 SpðjÞ ¼ C pðjÞ ppðjÞ ¼ q < dpðkÞ ;
hence job Jp(k) is an early job. So job Jp(j) is the only on-time job in p.
h
Corollary 1. There exists an optimal schedule where the last job is the on-time job.
Proof. In fact, if the last job is not the on-time job, it is an early job. Since it is the last job, we can move the job to be the ontime job so that the objective value will not increase. h
Theorem 2. There exists an optimal schedule with no idle time between the jobs. The idle times can only exist before the first job of
the schedule.
P
Proof. Since the objective function nj¼1 wj gðdj C j Þ are non-increasing function of the completion time. If there exists idle
time, the jobs can be moved later without increasing the objective value. h
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X.-R. Wang et al. / Applied Mathematical Modelling 35 (2011) 3509–3515
3. Case with equally weighted function
In this section we consider the case of minimizing the total earliness penalties (i.e., wj = 1) subject to no tardy jobs under
the assumption that the actual processing times of the jobs follow the model given in (1).
P
Theorem 3. For the problem 1jpj ¼ aj ð1 btÞ; C j 6 dj ¼ pj þ qj nj¼1 gðEj Þ, there exists an optimal schedule in which the sequence
of early jobs can be obtained by sequencing the jobs in non-increasing order of aj (i.e., the largest normal processing time (LPT)
rule).
Proof. Consider an optimal schedule p. Suppose there are two adjacent jobs Ji and Jj in p with job Ji being followed by job Jj
and ai < aj. Let the start time of job Ji be t, we have
f ðpÞ ¼ gðdi C i Þ þ gðdj C j Þ ¼ gðpi þ q t pi Þ þ gðpj þ q t pi pj Þ ¼ gðq tÞ þ gðq t ai ð1 btÞÞ:
Performing an adjacent pair-wise interchange of jobs Jj and Ji to get a new schedule p0 , we have
f ðp0 Þ ¼ gðq tÞ þ gðq t aj ð1 btÞÞ:
Since ai < aj, we have g(q t aj(1 bt)) < g(q t ai(1 bt)) and f(p0 ) < f(p). The completion times of the jobs processed
after jobs Ji and Jj are not affected by the interchange, and the completion times of the jobs processed before jobs Ji and Jj are
also not affected by the interchange either. Hence the value of the objective function under p0 is strictly less than that under
p. This contradicts the optimality of p and proves the theorem. h
From Theorem 3, the optimal schedule can be found through two steps. First determine the on-time job and second, sequence the other jobs as LPT order in front of the on-time job. For the on-time job, we have the following result.
Theorem 4. In an optimal schedule p of the problem 1jpj ¼ aj ð1 btÞ; C j 6 dj ¼ pj þ qj
the largest normal processing time.
Pn
j¼1 gðEj Þ,
the on-time job is the job with
Proof. Suppose p is an optimal schedule in which the on-time job is Jp(n), i.e., Cp(n) = dp(n) = pp(n) + q. Then for job Jp(j), j = 1,
2, . . . ,n 1, we have
C pðjÞ ¼ C pðnÞ n
X
ppðkÞ ¼ q ðppðjþ1Þ þ ppðjþ2Þ þ þ ppðn1Þ Þ;
k¼jþ1
and it follows that
dpðjÞ C pðjÞ ¼ ppðjÞ þ q ðq ðppðjþ1Þ þ ppðjþ2Þ þ þ ppðn1Þ ÞÞ ¼ ppðjÞ þ ppðjþ1Þ þ þ ppðn1Þ :
Therefore the objective function is
f ð pÞ ¼
n1
X
gðdpðjÞ C pðjÞ Þ ¼
j¼1
¼
n1
X
j¼1
n1
X
j¼1
g
gðppðjÞ þ ppðjþ1Þ þ þ ppðn1Þ Þ ¼
n1
X
gðC pðn1Þ C pðj1Þ Þ
j¼1
!
n1
j1
1 Y
1 Y
t0 ð1 bapðkÞ Þ t 0 ð1 bapðkÞ Þ :
b k¼1
b k¼1
ð3Þ
From (3), g(x) is a strictly increasing function, ðt0 1bÞ < 0, and 0 < (1 bap(k)) 6 1 (k 2 {1, 2, . . . ,n}), we know that ap(n) does
not affect the objective function f(p), it is always better to schedule the on-time job with the largest normal processing
time. h
Using Theorems 3 and 4, a simple algorithm to determine an optimal schedule of the problem 1jpj ¼ aj ð1 btÞ;
P
C j 6 dj ¼ pj þ qj nj¼1 gðEj Þ is developed as follows:
Algorithm 1
Step 1. Sequence the jobs in non-decreasing order of aj (i.e., the SPT rule) to get a sequence p = [Jp(1), Jp(2), . . . ,Jp(n)].
Q
Step 2. The optimal schedule is p⁄ = [Jp(n1), Jp(n2), . . . ,Jp(1), Jp(n)] and Sp(n) = q, SpðjÞ ¼ ðq 1bÞ= ji¼1 ð1 bapðiÞ Þ þ 1b ; j ¼
1; 2; . . . ; n 1.
Obviously, the time complexity of Algorithm 1 is O(nlogn). Now we demonstrate the result of Algorithm 1 in the following
example.
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X.-R. Wang et al. / Applied Mathematical Modelling 35 (2011) 3509–3515
Example 1. n = 4, a1 = 1, a2 = 2, a3 = 3, a4 = 4, b = 0.1, q = 5.464. The objective function is f ðpÞ ¼
Pn
j¼1 ðdj
C j Þ2 .
Solution. According to Algorithm 1, the optimal job sequence is p⁄ = [J3, J2, J1, J4]. The starting time of each job S1 = 4.96,
S2 = 3.7, S3 = 1, S4 = 5.464. The due dates are d1 = 5.968, d2 = 6.724, d3 = 8.164, d4 = 7.2784. The optimal value of the objective
function is f(p⁄) = 23.293.
4. Case with linear penalty function
P
In this section we consider the problem 1jpj ¼ aj ð1 btÞ; C j 6 dj ¼ pj þ qj nj¼1 wj Ej under the assumption that the actual
processing times of the jobs follow the model given in (1). Similar to the Section 3, we have the following results.
P
Theorem 5. For the problem 1jpj ¼ aj ð1 btÞ; C j 6 dj ¼ pj þ qj nj¼1 wj Ej , there exists an optimal schedule in which the sequence
of early jobs can be obtained by sequencing the jobs in non-increasing order of aj/[wj(1 baj)].
Thus we can find the optimal schedule by determining the on-time job. Once the on-time job has been decided, other jobs
can be sequenced in non-increasing order of aj/[wj(1 baj)] in front of the on-time job.
Since there are at most n possibilities of the on-time job, therefore, all jobs can be considered at the last sequence position
to generate n schedules. The one with the minimum total weighted earliness among these n schedules is an optimal schedule. But we have a simpler procedure. In order to find the on-time job, we first sequence the jobs in non-decreasing order of
aj/[wj(1 baj)] and get a sequence p = [Jp(1), Jp(2), . . . ,Jp(n)]. For each job Jp(j), j = 1, 2, . . . , n, let
ApðjÞ
0
B
1
B
ð1 bapðjÞ ÞB1 j
¼ wpðjÞ q @
b
Q
1
1
ð1 bapðkÞ Þ
n
C
X
C
wpðkÞ :
C þ apðjÞ ð1 bqÞ
A
k¼jþ1
k¼1
Then the on-time job can be obtained by calculating Ap(j). We have the following result.
Theorem 6. For the problem 1jpj ¼ aj ð1 btÞ; C j 6 dj ¼ pj þ qj
job is the job with the maximum Ap(j).
Pn
j¼1 wj Ej ,
there exists an optimal schedule in which the on-time
Proof. Similar to the proof of Qi and Tu [27] (page 506, Theorem 6). We assume that Jp(j) is the on-time job of an optimal
schedule, j = 1, 2, . . . ,n, and it is in the candidate set. Then according to Theorem 5, the schedule p⁄ = [Jp(n), Jp(n1),
. . . ,Jp(j+1), Jp(j1), . . . ,Jp(1), Jp(j)] will be optimal. The completion times of the jobs in p⁄ will be the following.
For i, 1 6 i 6 j 1,
C pðiÞ ¼ dpðjÞ ðppðjÞ þ ppð1Þ þ þ ppði1Þ Þ ¼ q ðppð1Þ þ þ ppði1Þ Þ;
dpðiÞ C pðiÞ ¼ ppð1Þ þ þ ppðiÞ :
For i, j + 1 6 i 6 n,
C pðiÞ ¼ dpðjÞ ðppðjÞ þ ppð1Þ þ þ ppðj1Þ þ ppðjþ1Þ þ þ ppði1Þ Þ ¼ q ðppð1Þ þ þ ppðj1Þ þ ppðjþ1Þ þ þ ppði1Þ Þ;
dpðiÞ C pðiÞ ¼ ppð1Þ þ þ ppðj1Þ þ ppðjþ1Þ þ . . . þ ppðiÞ :
Then, from Qi and Tu [27], we have
f ðp Þ ¼
n
X
wpðiÞ
i¼1
i
X
!
ppðkÞ
ApðjÞ ;
k¼1
where
ApðjÞ ¼ wpðjÞ
j
X
i¼1
ppðiÞ þ ppðjÞ
n
X
i¼jþ1
wpðiÞ
0
B
1
B
ð1 bapðjÞ ÞB1 j
¼ wpðjÞ q @
b
Q
1
1
ð1 bapðkÞ Þ
n
C
X
C
wpðkÞ ;
C þ apðjÞ ð1 bqÞ
A
k¼jþ1
ð4Þ
k¼1
The first term of f(p⁄) is a constant for all j. For each possibility of on-time job Jp(j), f(p⁄) depends on Ap(j) and the job with
maximum Ap(j) has the minimum f(p⁄). So the on-time job should be the job with the maximum Ap(j) in the candidate set. h
Using Theorems 5 and 6, a simple algorithm to determine an optimal schedule of the problem 1jpj ¼ aj ð1 btÞ; C j 6
P
dj ¼ pj þ qj nj¼1 wj Ej is developed as follows:
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X.-R. Wang et al. / Applied Mathematical Modelling 35 (2011) 3509–3515
Algorithm 2
Step
Step
Step
Step
1.
2.
3.
4.
Sequence the jobs in non-decreasing order of aj/[wj(1 baj)] to get a sequence p:p = [Jp(1), Jp(2), . . . , Jp(n)].
Calculate Ap(j) for each job Jp(j), j = 1, 2, . . . ,n.
Find the job Jp(j) with the maximum Ap(j). Suppose it is Jp(k).
Q
The optimal schedule is p⁄ = [Jp(n), Jp(n1), . . . ,Jp(k+1), Jp(k1), . . . ,Jp(1), Jp(k)] and Sp(k) = q, SpðjÞ ¼ ðq 1bÞ= 16i6j;i–k
ð1 bapðiÞ Þ þ 1b for j – k.
As in Qi and Tu [28], the time complexity of Algorithm 2 is O(n2). Now we demonstrate the result of Algorithm 2 in the
following example.
Example 2. n = 4, a1 = 1, a2 = 2, a3 = 3, a4 = 4, w1 = 3, w2 = 2, w3 = 2, w4 = 1, b = 0.1, q = 8. The objective function is
P
f ðpÞ ¼ nj¼1 wj ðdj C j Þ.
Solution. According to Algorithm 2, we solve Example 2 as follows:
Step 1: We sequence the jobs in non-decreasing order of aj/[wj(1 baj)] to get a sequence p = [J1, J2, J3, J4].
Steps 2 and 3. Calculate Ap(j) for each job Jp(j). We have Ap(1) = A1 = 1.6000, Ap(2) = A2 = 2.4444, Ap(3) = A3 = 3.3556,
Ap(4) = A4 = 2.7683. So the job J3 is the on-time job.
Step 4. The optimal schedule is p⁄ = [J4, J2, J1, J3] and S1 = 7.7778, S2 = 7.2222, S3 = 8, S4 = 5.3704. The due dates are
d1 = 8.2222, d2 = 8.5556, d3 = 8.6, d4 = 9.8518. The optimal value of the objective function is f(p⁄) = 4.8518.
5. Conclusions
In this paper we considered a single-machine scheduling problem under the assumption that the processing times are not
constant over starting time. We assumed that job processing times are decreasing functions of their starting time. The objective is to minimize total weighted earliness subject to no tardy jobs. We considered the problem in which the due dates are
determined by the SLK method. For the case with equally weighted monotonous penalty objective function and the case with
weighted linear penalty objective function, we proved that the problem can be solved in polynomial time, respectively.
Acknowledgements
We are grateful to an anonymous referee for his/her helpful comments on an earlier version of this paper. This research
was supported by the National Natural Science Foundation of China under Grant No. 11001181, and the Science Research
Foundation of Shenyang Aerospace University, under Grant No. 201011Y.
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