Applied Mathematical Modelling 34 (2010) 294–300
Contents lists available at ScienceDirect
Applied Mathematical Modelling
journal homepage: www.elsevier.com/locate/apm
A note on single-machine scheduling with decreasing time-dependent
job processing times
Ji-Bo Wang
Operations Research and Cybernetics Institute, School of Science, Shenyang Institute of Aeronautical Engineering, Shenyang 110136, China
a r t i c l e
i n f o
Article history:
Received 27 November 2007
Received in revised form 13 April 2009
Accepted 29 April 2009
Available online 9 May 2009
Keywords:
Scheduling
Single-machine
Time-dependent processing times
Resource allocation
a b s t r a c t
In this note we consider some single-machine scheduling problems with decreasing timedependent job processing times. Decreasing time-dependent job processing times means
that its processing time is a non-increasing function of its execution start time. We present
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.
We also study two resource constrained scheduling problems under the same decreasing
time-dependent job processing times model and present algorithms to find their optimal
solutions.
Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction
Machine scheduling problems with start time dependent processing times 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 start time dependent processing times can be found in Alidaee and Womer [1], and Cheng et al. [2]. More 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. Gawiejnowicz et al. [6] considered two single-machine bicriterion scheduling problems with time-dependent job processing times. Wang et al. [7] considered two-machine flow-shop scheduling with simple linear job deterioration. Janiak and Kovalyov [8] considered the
problem of scheduling n jobs executed by a human in a contaminated area. 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. Gawiejnowicz [10] 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 et al. [11] considered single-machine scheduling with deteriorating
jobs in which the jobs are constrained by a series-parallel graph constraint. They proved that the problem can be solved in
polynomial time. Leung et al. [12] 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. Lee and Wu [13] considered multi-machine scheduling with deteriorating jobs and scheduled
E-mail address: [email protected]
0307-904X/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.apm.2009.04.018
J.-B. Wang / Applied Mathematical Modelling 34 (2010) 294–300
295
maintenance. Tang and Liu [14] considered two-machine flowshop scheduling problems involving a batching machine with
transportation or deterioration consideration.
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 note 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. [15]. The remaining part of this paper is organized as follows. In Section 2 we describe and formulate the problem. In Sections 3 and 4 we consider the sum of squared completion times minimization problem and sum of earliness penalties minimization problem, respectively. In Section 5 we consider two resource constrained scheduling problems under the
same decreasing time-dependent job processing times model and present algorithms to solve them. The last section is the
conclusion.
2. Model description
There are given a single machine and a set N ¼ fJ 1 ; J 2 ; . . . ; J n g of n independent and non-preemptive jobs, which are available for processing at some time t0 P 0. Each job J j has a normal processing time aj . Following Ho et al. [15] and Wang and
Xia [3], we assume that the actual processing time pj of job J j is a non-increasing linear function of the job’s starting time, i.e.,
pj ¼ aj ð1 ktÞ;
ð1Þ
where aj P 0 and t P t 0 is the job’s starting time. It is assumed that the decreasing rates satisfy the following condition:
k t0 þ
n
X
!
aj amin
< 1;
ð2Þ
j¼1
where amin ¼ mini¼1;2;...;n fai g. The condition ensures that all job processing times are positive in a feasible schedule (see also
[15,3] for detailed explanations).
For a given schedule p ¼ ð½1; ½2; . . . ; ½nÞ, C j ¼ C j ðpÞ represents the completion time of job J j . In the remaining part of the
paper, all the problems considered will be denoted using the three-field notation scheme ajbjc introduced by Graham et al.
[16].
3. Minimizing the sum of squared completion times
Townsend [17] considered a single machine scheduling problem with a quadratic cost function of completion times, i.e.,
P
the sum of the quadratic job completion times. He showed that the problem 1jj C 2j can be solved optimally by the shortest
processing time (SPT) rule. By using the job interchanging technique, we can show that the solution of Townsend’s still holds
P
for the problem 1jpj ¼ aj ð1 ktÞj C 2j .
Lemma 1. (Wang and Xia [3])For a given schedule p ¼ ð½1; ½2; . . . ; ½nÞ with job processing times in the form of pj ¼ aj ð1 ktÞ, if
P
job J ½1 starts at time 0 6 t0 < 1k nj¼1 aj þ amin , then the makespan is equal to
C max ¼
t0 n
1 Y
1
ð1 ka½i Þ þ ;
k i¼1
k
ð3Þ
if the makespan is C, then the starting time of the first job is
Y
n
1
1
t0 ¼ C ð1 ka½i Þ þ :
k
k
i¼1
P
Theorem 1. For the problem 1jpj ¼ aj ð1 ktÞj C 2j , an optimal schedule can be obtained by sequencing the jobs in non-decreasing order of aj (i.e., the smallest normal processing time (SPT) rule).
Proof. Let p and p0 be two job schedules where the difference between p and p0 is a pair-wise interchange of two adjacent
jobs J i and J j , i.e., p ¼ ðS1 ; J i ; J j ; S2 Þ and p0 ¼ ðS1 ; J j ; J i ; S2 Þ, where S1 and S2 are partial sequences, such that ai 6 aj . Let t denote
the completion time of the last job in S1 . To show p dominates p0 , it suffices to show that C j ðpÞ 6 C i ðp0 Þ and
C 2i ðpÞ þ C 2j ðpÞ 6 C 2j ðp0 Þ þ C 2i ðp0 Þ. It is easy to derive the completion times of jobs J i and J j in p as
1
1
C i ðpÞ ¼ t þ ai ð1 ktÞ ¼ t ð1 kai Þ þ ;
k
k
ð4Þ
1
1
C j ðpÞ ¼ t ð1 kai Þð1 kaj Þ þ :
k
k
ð5Þ
and
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J.-B. Wang / Applied Mathematical Modelling 34 (2010) 294–300
Similarly, the completion times of jobs J j and J i in
p0 are
1
1
C j ðp0 Þ ¼ t ð1 kaj Þ þ ;
k
k
ð6Þ
and
C i ðp0 Þ ¼
1
1
t
ð1 kaj Þð1 kai Þ þ :
k
k
ð7Þ
Based on Eqs. (5) and (7), we have C j ðpÞ ¼ C i ðp0 Þ. In addition, from (4) and (6), and ai 6 aj , we have C i ðpÞ 6 C j ðp0 Þ. Hence
C 2i ðpÞ þ C 2j ðpÞ 6 C 2j ðp0 Þ þ C 2i ðp0 Þ; thus p dominates p0 . This completes the proof. h
Corollary 1. For the problem 1jpj ¼ aj ð1 ktÞj
non-decreasing order of aj (i.e., the SPT rule).
P
C hj , where h > 0; an optimal schedule can be obtained by sequencing the jobs in
4. Minimizing the sum of earliness penalties
In this section we consider the problem of minimization of the sum of earliness penalties subject to no tardy jobs under
the assumption that the actual processing times of the jobs follow the model given in (1). For the classical problem, there are
some results in Chang and Schneeberger [18], and Qi and Tu [19]. We assume that all the jobs have a common due date d and
P
gðEj Þ, where gðxÞ is a strictly
the release time is 0. Let Ej ¼ d C j be the earliness of job J j . The objective is to minimize
P
increasing function. The problem is denoted as 1jpj ¼ aj ð1 ktÞj gðEj Þ. A schedule is feasible if and only if there is no tardy
job in the schedule. For any optimal schedule, it is obvious that (i) the completion time of the last job is d, and (ii) there is no
idle time between the jobs, and idle time can only exist before the first job, i.e., the t0 is a decision variable.
P
Theorem 2. For the problem 1jpj ¼ aj ð1 ktÞj gðEj Þ, an optimal schedule can be obtained by sequencing the jobs in nonincreasing order of aj (i.e., the largest normal processing time (LPT) rule), where the starting time of the first job is
Qn
1
t0 ¼ d 1k
i¼1 ð1 kai Þ þ k :
Proof. Consider an optimal schedule p. Suppose there are two adjacent jobs J i and J j in p with job J i being followed by job J j
and ai < aj . Let the completion time of job J j be C j . Performing an adjacent pair-wise interchange of jobs J i and J j to get a new
schedule p0 , we have,
1
1
1
1
Cj ð1 kaj Þ þ ; Ej ¼ d C j ; Ei ¼ d C j ð1 kaj Þ ;
k
k
k
k
1
1
1
1
ð1 kai Þ þ ; E0i ¼ d C j ; E0j ¼ d C j ð1 kai Þ :
C 0i ¼ C j ; C 0j ¼ C j k
k
k
k
Ci ¼
From (2) and ai < aj , we have C j 1k < 0, E0j < Ei ; and E0i ¼ Ej : Hence gðE0i Þ þ gðE0j Þ < gðEj Þ þ gðEi Þ.
The completion times of the jobs processed after jobs J i and J j are not affected by the interchange, and the completion
times of the jobs processed before jobs J i and J j are also not affected by the interchange either (from Lemma 1). 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
P
When gðxÞ ¼ x, and the objective function is
wj Ej , we have the following result:
P
Corollary 2. For the problem 1jpj ¼ aj ð1 ktÞj wj Ej , an optimal schedule can be obtained
the jobs in nonQn by sequencing
1
increasing order of aj =½wj ð1 kaj Þ, where the starting time of the first job is t 0 ¼ ðd 1kÞ
i¼1 ð1 kai Þ þ k .
P
P
P
P
Remark: When gðxÞ ¼ x, nj¼1 wj gðEj Þ ¼ d nj¼1 wj nj¼1 wj C j . Since d nj¼1 wj is a constant, the weighted sum of completion
times maximization problem can be solved by sequencing the jobs in non-increasing order of aj =½wj ð1 kaj Þ, while the
weighted sum of completion times minimization problem can be solved by sequencing the jobs in non-decreasing order
of aj =½wj ð1 kaj Þ (Bachman et al. [20]).
5. Resource constrained problems
Bachman and Janiak [21] first considered single-machine scheduling with job processing times dependent on the starting
moments of job execution and on the amounts of resource allocation to the jobs. They proved that the makespan minimization problem is NP-hard. They also gave some properties of the optimal resource allocation. Zhao et al. [22] considered
single-machine scheduling with deteriorating jobs where the release times of the jobs depend on the amounts of resource
allocation. For two resource constrained scheduling problems, they gave optimal algorithms to find the optimal resource
allocations. In this section we consider the following model:
J.-B. Wang / Applied Mathematical Modelling 34 (2010) 294–300
297
We assume that the release time of job J j is: r j ¼ f ðuj Þ; a 6 uj 6 b, j ¼ 1; 2; . . . ; n, where uj is the amount of resources allocated to job J j , with 0 6 a 6 uj 6 b; a and b are known technological constraints and f : Rþ ! Rþ (Rþ is the set of non-negative
real numbers) is a strictly decreasing continuous function with f ðaÞ P 0 and f ðbÞ P 0.
Let p denote any schedule of the jobs J 1 ; J 2 ; . . . ; J n and the job J ½j is in position j of p. We denote the set of all the feasible
e we denote the set of all possible resource allocations u ¼ ½u1 ; u2 ; . . . ; un satisfying the resource conschedules by P. By U
P
straint (i.e., a 6 uj 6 b; j ¼ 1; 2; . . . ; n; nj¼1 uj 6 U; where U is the total amount of a continuously divisible non-renewable
resource available, and U P naÞ.
We consider the problem of minimizing the total resource consumption with a makespan constraint and the problem of
minimizing the makespan with a resource consumption constraint. The two problems are denoted as 1jpj ¼ aj ð1 ktÞ;
P
P
r j ¼ f ðuj Þ; C max 6 Cj uj and 1jpj ¼ aj ð1 ktÞ; rj ¼ f ðuj Þ; uj 6 UjC max , respectively. For these two problems, the analysis is
similar to that of Janiak [23] and Cheng and Janiak [24] for the classical versions of the problems.
5.1. Minimizing the resource consumption
e for each job J , j ¼ 1; 2; . . . ; n, the completion time C ½j ðp; uÞ may be
Given a schedule p and a resource allocation u 2 U,
½j
calculated recursively as
r ½1 ¼ f ðu½1 Þ;
r ½j ¼ f ðu½j Þ;
1
1
C ½1 ðp; uÞ ¼ r ½1 ð1 ka½1 Þ þ ;
k
k
(
)
j
1 Y
1
r ½i ð1 ka½l Þ þ ;
C ½j ðp; uÞ ¼ max
16i6j
k l¼i
k
j ¼ 2; . . . ; n:
Thus the makespan is
C max ðp; uÞ ¼ maxfC ½j ðp; uÞg:
16i6n
Also, for
e denote the total resource consumption by
p 2 P and u 2 U,
Uðp; uÞ ¼
n
X
u½j :
j¼1
e that minimizes the total resource consumption, subLet C is a given makespan. The problem is to find p 2 P and u 2 U
ject to C max ðu ; u Þ 6 C.
From Lemma 1, the minimum makespan of the schedule is
C max ¼
f ðbÞ So, a schedule
p is feasible only if
C max ¼
n
1 Y
1
ð1 kai Þ þ :
k i¼1
k
f ðbÞ n
1 Y
1
ð1 kai Þ þ 6 C:
k i¼1
k
For p ¼ ð½1; ½2; . . . ; ½nÞ, we denote by u the resource allocation with which the resource consumption is minimized, subject to a given makespan C, i.e.,
Uðp; u Þ ¼ minfUðp; uÞg;
u2e
U
subject to C max ðp; u Þ 6 C.
Since releasing jobs sooner consumes more resources, jobs should be released as
as possible and the completion time
late
Qn
1
of the last job is C. If C max ðp; Þ ¼ C, then the starting time of the first job is t0 ¼ C 1k
i¼1 ð1 ka½i Þ þ k :
Hence, the resource allocation u that minimizes the total resource consumption, subject to C max ðp; u Þ 6 C, is determined
as follows:
Algorithm 1
Qn
1
(1) Set t0 ¼ C 1k
i¼1 ð1 ka½i Þ þ k :
(2) If t0 P f ðaÞ, then u½j ¼ a; j ¼ 1; . . . ; n: Stop. Otherwise, go to (3).
(3) t 0 1k ð1 ka½1 Þ þ 1k P f ðaÞ, then u½1 ¼ f 1 ðt0 Þ; u½j ¼ a; j ¼2; . . . ; n: Stop. Otherwise, go to (4).
Q
1
(4) Let m be the maximum natural number satisfying t0 1k m1
i¼1 ð1 ka½i Þ þ k 6 f ðaÞ; i.e., C ½m1 6 f ðaÞ: Then
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J.-B. Wang / Applied Mathematical Modelling 34 (2010) 294–300
u½1 ¼ f 1 ðt 0 Þ;
!
j1
1 Y
1
; j ¼ 2; . . . ; m;
u½j ¼ f 1
t0 ð1 ka½i Þ þ
k i¼1
k
u½j ¼ a; j ¼ m þ 1; . . . ; n:
P
Theorem 3. For the problem 1jpj ¼ aj ð1 ktÞ; rj ¼ f ðuj Þ; C max 6 Cj uj , an optimal schedule p can be obtained by sequencing
jobs in non-increasing order of their normal processing times (the LPT rule), and an optimal resource allocation u can be calculated
by using Algorithm 1.
Proof. For a given schedule, by contradiction, an optimal resource allocation u can be easily obtained by using Algorithm 1.
Consider an optimal schedule p. Suppose there are two adjacent jobs J i and J j in p with job J i being followed by job J j and
ai < aj . Let the completion time of job J j be C j . Performing a adjacent pair-wise interchange on jobs J i and J j to get a new schedule p0 , we have
1
1
1
1
ð1 kaj Þ þ ; r i ¼ C j ½ð1 kai Þð1 kaj Þ þ ;
Cj k
k
k
k
1
1
1
1
C 0i ¼ C j ; r 0i ¼ C 0 ð1 kai Þ þ ; r 0j ¼ C j ½ð1 kaj Þð1 kai Þ þ :
k
k
k
k
rj ¼
Since ai < aj , C j 1k < 0; we have r 0i > rj ; rj 0 ¼ ri ; hence f 1 ðr0i Þ þ f 1 ðr0j Þ < f 1 ðrj Þ þ f 1 ðr 0i Þ.
The release times of the jobs processed after jobs J i and J j are not affected by the interchange, and the release times of the
jobs processed before jobs J i and J j are unchanged. Hence the value of the objective function under p0 is strictly less than
under p. This contradicts the optimality of p and proves the theorem. h
5.2. Minimizing the makespan
P
From Lemma 1 and the results of Section 5.1, for the problem 1jpj ¼ aj ð1 ktÞ; r j ¼ f ðuj Þ; uj 6 UjC max , we also only need
to consider the schedule in which the jobs are processed in non-increasing order of aj (i.e., the LPT schedule).
Without loss of generality, let p ¼ ð½1; ½2; . . . ; ½nÞ be the schedule in which the jobs are processed in non-increasing order
of aj , if uj ¼ a; j ¼ 1; 2; . . . ; n. Then
n
1 Y
1
C max ðp; aÞ ¼ f ðaÞ ð1 kai Þ þ :
k i¼1
k
Under this condition, the release times of all the jobs are equal to f ðaÞ, and the makespan is constrained by r1 ¼ f ðaÞ. If we
increase the amount of resources allocated to J 1 , then the release time r1 will be smaller and the makespan will be smaller,
, i.e., umax
¼ minfU ðn 1Þa; bg, we have r1 ¼ f ðumax
Þ,
too. Let the maximum amount of resources allocated to job J 1 be umax
1
1
1
1
1
is
r
Þ
þ
.
and the
completion
time
of
J
ð1
ka
1
1
1
k
k
If r 1 1k ð1 ka1 Þ þ 1k P f ðaÞ; then the optimal resource allocation is
u1 ¼ umax
1 ;
uj ¼ a; j ¼ 2; . . . ; n;
n
1 Y
1
C max ¼ r 1 ð1 kai Þ þ :
k i¼1
k
If ðr 1 1kÞð1 ka1 Þ þ 1k < f ðaÞ; then there must be a natural number m such that
C j ðp; up Þ 6 f ðaÞ; j ¼ 1; 2; . . . ; m 1
C j ðp; up Þ > f ðaÞ; j ¼ m; . . . ; n;
uj ¼ a; j ¼ m þ 1; . . . ; n;
where up is the optimal resource allocation for the schedule
Let d ¼ f ðaÞ C k1 ðp; up Þ. From Lemma 1
mY
1
1
1
r1 ¼ f ðaÞ d ð1 kai Þ þ ;
k
k
i¼1
1
1
ð1 ka1 Þ þ ;
r2 ¼ r 1 k
k
... ;
p.
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J.-B. Wang / Applied Mathematical Modelling 34 (2010) 294–300
Y
1 m2
1
ð1 kai Þ þ ;
k i¼1
k
m1
Y
1
1
ð1 kai Þ þ ¼ f ðaÞ d;
rk ¼ r 1 k i¼1
k
rk1 ¼
r1 uj ¼ f 1 ðr j Þ;
ð8Þ
j ¼ 1; 2; . . . ; k;
uj ¼ a; j ¼ k þ 1; . . . ; n:
k
X
ð9Þ
uj þ ðn kÞa ¼ U:
ð10Þ
j¼1
d can be determined by Eq. (10).
Now, we consider m. When r m ¼ f ðaÞ, suppose the release time of J j is r j , j ¼ 1; 2; . . . ; m. Then
m1
Y
1
1
r1 ¼ f ðaÞ ð1 kai Þ þ ;
k
k
i¼1
1
1
r2 ¼ r1 ð1 ka1 Þ þ ;
k
k
... ;
Y
1 m2
1
ð1 kai Þ þ ;
rm1 ¼ r 1 k i¼1
k
mY
1
1
1
ð1 kai Þ þ ¼ f ðaÞ;
rm ¼ r1 k i¼1
k
uj ¼ f 1 ðr j Þ;
uj ¼ a;
m1
X
ð11Þ
j ¼ 1; 2; . . . ; m;
j ¼ m þ 1; . . . ; n;
uj þ ðn ðm 1ÞÞa 6 U:
ð12Þ
j¼1
From Eq. (12), we can get m and d can be calculated by Eq. (10).
P uj 6 U, is
Based on the above analysis, the optimal resource allocation uj that minimizes the makespan, subject to
determined as follows:
Algorithm 2
¼ minfU ðn 1Þa; bg; r 1 ¼ f ðumax
Þ:
(1) Sequencing the jobs in non-increasing order of aj , and let umax
1
1
1
1
max
(2) If r 1 k ð1 ka1 Þ þ k P f ðaÞ, then u1 ¼ u1 ; uj ¼ a; j ¼ 2; . . . ; n; and stop. Otherwise, go to (3).
(3) Let m be the maximum natural number satisfying the following inequality:
m1
X
uj þ ðn ðm 1ÞÞa 6 U;
j¼1
where uj ¼ f 1 ðr j Þ, and r j satisfies Eq. (11).
(4) The resource allocation is
uj ¼ f 1 ðrj Þ;
uj ¼ a;
j ¼ 1; 2; . . . ; m;
j ¼ m þ 1; . . . ; n;
where rj can be determined by form (8) and d satisfies Eq. (10).
Theorem 4. For the problem 1jpj ¼ aj ð1 ktÞ; rj ¼ f ðuj Þ;
tion u can be obtained by Algorithm 2.
P
uj 6 UjC max , an optimal schedule
p and an optimal resource alloca-
6. Conclusions
In this note we considered some single-machine scheduling problems with decreasing time-dependent job processing
times. For the sum of squared completion times minimization problem, and the sum of earliness penalties minimization
problem subject to no tardy jobs, we proved that the problems can be solved in polynomial time. We also studied two re-
300
J.-B. Wang / Applied Mathematical Modelling 34 (2010) 294–300
source constrained problems under the same job deterioration model. We presented an algorithm which can obtain the optimal schedule and the optimal resource allocation, respectively. Future research may consider more general time-dependent
job processing times types, or study the other objective functions.
Acknowledgement
We are grateful to the editor and an anonymous referee, whose constructive comments have led to a substantial improvement in the presentation of the paper. This research was supported by the Science Research Foundation of the Educational
Department of Liaoning Province, China, under grant number 20060662.
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