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Single machine scheduling problems with deteriorating jobs
Applied Mathematics and Computation 161 (2005) 865–874 www.elsevier.com/locate/amc
Single machine scheduling problems with deteriorating jobs Chuan-li Zhao *, Heng-yong Tang College of Mathematics and Systems Sciences, Shenyang Normal University, Shenyang, Liaoning 110034, PeopleÕs Republic of China
Abstract This paper considers the single machine scheduling problems with deteriorating jobs, i.e. jobs whose processing times are a decreasing linear function of their starting time. It is assumed that jobs have the different basic processing time and same decreasing rate. Based on the analysis of problems, the optimal algorithms are presented for the problems to minimize the sum of earliness penalties subject to no tardy jobs, to minimize the resource consumption with makespan constraints and to minimize the makespan with the total resource consumption constraints. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Scheduling; Single machine; Deteriorating jobs; Earliness penalties; Resource constraints; Optimal algorithms
1. Introduction Machine scheduling problems with job processing times given as starting time dependent functions have received increasing attention in recent years [1]. In this model, the processing time of a job can be described by a basic processing time and an increasing (or decreasing) rate. For the model where job processing time is an increasing start time dependent function, there are many results in literatures [1]. Browne and Yechiali [9] consider stochastic single machine scheduling. Mosheiov [10–12]
*
Corresponding author. E-mail address: [email protected] (C.-l. Zhao).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.12.073
866
C.-l. Zhao, H.-y. Tang / Appl. Math. Comput. 161 (2005) 865–874
studied some spacial cases of this model. Bachman and Janiak [13] proved that the problem to minimize the maximum lateness with the arbitrary linear processing time is NP -complete. Mosheiov [14] analysis the complexity of job-shop problems. Apart from the model where job processing time is an increasing start time dependent function, there are also a model where job processing time is a decreasing start time dependent function. This model is introduced by Ho et al. [2]. Cheng and Ding [3,4] consider some problems with release times and deadlines. Ng et al. [5] consider the problems to minimize the total completion time. In this paper we consider the special case of the model where job processing time is a decreasing start time dependent function. It is assumed that jobs have the different basic processing time and same decreasing rate. For three single machine scheduling problems, the optimal algorithms are presented. The model can be described as follows. Suppose n jobs J1 ; J2 ; . . . ; Jn to be processed on a single machine. Associated with job Jj is a basic processing time aj and a decreasing rate b (without loss of generality we assume a1 P a2 P P an ). Let t be starting time of job Jj then the actual processing time of Jj is pj ¼ aj bt, j ¼ 1; . . . ; n. It is assumed that the decreasing rates satisfy the following conditions: ! n X 0 < b < 1 and b ai aj < aj ; j ¼ 1; . . . ; n: i¼1
The first condition ensures that the decrease of each job processing time is less than one unit for every unit delay in its starting moment. The second one ensures that all job processing times are positive in a feasible schedule [2].
2. Minimize the sum of earliness penalties In this section we consider the problem to minimize the sum of earliness penalties subject to no tardy jobs. For the classical scheduling problem, there are some results in [6]. We assume that all jobs have a common due-date d and release time is 0. For the schedule p ¼ ½J½1 ; J½2 ; . . . ; J½n , let C½j and E½j ¼ d C½j be the completion P time and earliness of job J½j in p. The sum of earliness penalties is EðpÞ ¼ nj¼1 gðE½j Þ, where g is a strictly increasing function. We denote all schedules by P. The problem is to find an optimal schedule p 2 P that minimizes the sum of earliness penalties under condition C½j 6 d. The problem be denoted as X 1jpj ¼ aj btj gðEj Þ: ð1Þ
C.-l. Zhao, H.-y. Tang / Appl. Math. Comput. 161 (2005) 865–874
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Since the problem is to minimize earliness penalties, the jobs should be processed as late as possible. It is obviously that in the optimal schedule the completion time of the last job is d and there is not idle times between jobs. First we give some results of the problem to minimize makespan, it can be denoted as 1jpj ¼ aj btjCmax . Lemma 1. For the problem 1jpj ¼ aj btjCmax , the makespan is minimized by sequencing the jobs in non-increasing order of aj . Lemma 2. For the problem 1jpj ¼ aj btjCmax , if p ¼ ½J1 ; J2 ; . . . ; Jn and the starting time of first job is t0 , then makespan is n
Cmax ðt0 jJ1 ; J2 ; . . . ; Jn Þ ¼ t0 ð1 bÞ þ
n X
aj ð1 bÞ
nj
:
j¼1
Lemma 3. For the problem 1jpj ¼ aj btjCmax , if p ¼ ½J1 ; J2 ; . . . ; Jn and the makespan is C, then starting time of first job is !, n X nj n t0 ¼ C aj ð1 bÞ ð1 bÞ : j¼1
Based on the above lemmas, we have following theorem. P Theorem 1. For the problem 1jpj ¼ aj btj gðEj Þ, an optimal schedule can be obtained by sequencing the jobs in non-increasing order of aj . Where the starting time of the first job is !, n X nj n t0 ¼ C aj ð1 bÞ ð1 bÞ : j¼1
Proof (by contradiction). Consider an optimal schedule p. Assume under p, there two adjacent jobs, job Ji followed by job Jj , such that ai < aj . Denote the completion time of Jj is Cj ¼ C0 , then Ci ¼ ðC0 aj Þ=ð1 bÞ. Ej ¼ d C0 , Ei ¼ d ðC0 aj Þ=ð1 bÞ. Sj ¼ Ci , Si ¼ ðC0 aj ð1 bÞai Þ=ð1 bÞ2 . Per~. Under p ~, form a pair-wise interchange on jobs Ji and Jj , call new schedule p e i ¼ C0 , C e j ¼ ðC0 ai Þ=ð1 bÞ. E e i ¼ d C0 , E e j ¼ d ðC0 ai Þ=ð1 bÞ. C 2 e e j, e Si ¼ C S j ¼ ðC0 ai ð1 bÞaj Þ=ð1 bÞ . e j > Ci , e j < Ei e i ¼ Ej . then C E and E Hence Since ai < aj , e e gð E i Þ þ gð E j Þ < gðEi Þ þ gðEj Þ. The completion times of the jobs processed after jobs Ji and Jj is not affected by interchange, the jobs processed before jobs Ji and Jj is become later (Since e ~). Hence the sum S j > Si , then the starting times of these jobs are larger under p
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~ is strictly less than that of p. This contradicts of earliness penalties under p the optimality of p. h
3. Resource constrained problems In this section, the release time of job Jj is rj ¼ f ðuj Þ, u 6 uj 6 u, where uj is the amount of resource allocated to Jj , u, u are known constraint and f is a P strictly decreasing function (f ðuÞ > f ðuÞ > 0, 0 6 u 6 u). nj¼1 6 U , U is the global amount of continuously divisible non-renewable resource. For the problems with release time of a job is function of resource allocated, Cheng and Janiak [7] and Janiak [8] have considered the case where jobs have constant processing times (b ¼ 0). In this section we consider two problems with jobs have the different basic processing time and the same decreasing rate, the first is to minimize the resource consumption with makespan constraints while the second is to minimize the makespan with the total resource consumption constraints: X 1jpj ¼ aj bt; rj ¼ f ðuj Þ; Cmax 6 Cj uj ; ð2Þ X 1jpj ¼ aj bt; rj ¼ f ðuj Þ; uj 6 U jCmax : ð3Þ 3.1. Minimization of resource consumption We denotePthe set of all resource allocations u ¼ ½u1 ; u2 ; . . . ; un by e ðu 6 uj 6 u; n uj 6 U Þ. Given a schedule p ¼ ½J½1 ; J½2 ; . . . ; J½n and a reU j¼1 source allocations up ¼ ½u½1 ; u½2 ; . . . ; u½n , we have r½j ¼ f ðu½j Þ; j ¼ 1; 2; . . . ; n; C½1 ðp; uÞ ¼ a½1 þ ð1 bÞr½1 ; C½j ðp; uÞ ¼ maxfr½j ; C½j1 ðp; uÞg þ ð1 bÞ maxfr½j ; C½j1 ðp; uÞg ( ) j X jiþ1 jk ¼ max r½i ð1 bÞ þ a½k ð1 bÞ ; j ¼ 2; . . . ; n: 16i6j
k¼i
Hence Cmax ðp; uÞ ¼ max1 6 j 6 n fC½j ðp; uÞg;
U ðp; uÞ ¼
n X
u½j :
j¼1
Let C (C satisfying bðC aj Þ < aj , j ¼ 1; . . . ; n) be a given makespan, the e which minimizes the total resource problem (2) is to find p 2 P and u 2 U consumption i.e., U ðp ; u Þ ¼ minp2P min e fU ðp; uÞg. u2 U
C.-l. Zhao, H.-y. Tang / Appl. Math. Comput. 161 (2005) 865–874
869
Subject to Cmax ðp ; u Þ 6 C. From Lemma 1, the minimum makespan of schedule is C ¼ Cmax ðf ðuÞjJ1 ; J2 ; . . . ; Jn Þ ¼ f ðuÞð1 bÞn þ
n X
aj ð1 bÞnj :
ð4Þ
j¼1
So a schedule p ¼ ½J½1 ; J½2 ; . . . ; J½n is feasible only if n
Cmax ðf ðuÞjJ½1 ; J½2 ; . . . ; J½n Þ ¼ f ðuÞð1 bÞ þ
n X
a½j ð1 bÞ
nj
6 C:
ð5Þ
j¼1
We denote a resource allocation by up with which the resource consumption is minimized, subject to a given C, i.e., U ðp; up Þ ¼ minfU ðp; uÞg: u2 e U Since releasing jobs sooner consumes more resource, jobs should be released as late as possible and the completion time of the last P job is C. If Cmax ðp; Þ ¼ C, then the starting time of the first job is t0 ¼ ðC nj¼1 a½j ð1 bÞnj Þ=ð1 bÞn . Hence the resource allocation up is determined as follows: Algorithm 1 (1) Let t0 ¼
C
n X
!, nj
a½j ð1 bÞ
ð1 bÞn :
j¼1
(2) If a½1 þ ð1 bÞt0 P f ðuÞ, then u½1 ¼ f 1 ðt0 Þ; u½j ¼ u; j ¼ 2; . . . ; n. Stop. Otherwise go to (3). (3) Let k be the maximum natural number satisfying t0 ð1 bÞk1 þ
k1 X
a½j ð1 bÞk1j 6 f ðuÞ
j¼1
(it is C½k1 6 f ðuÞ); then u½1 ¼ f 1 ðt0 Þ; u½j
¼f
1
t0 ð1 bÞ
j1
þ
j1 X i¼1
u½j ¼ u; j ¼ k þ 1; . . . ; n:
! j1i
a½i ð1 bÞ
;
j ¼ 2; . . . ; k;
870
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P Theorem 2. For the problem 1jpj ¼ aj bt; rj ¼ f ðuj Þ; Cmax 6 Cj uj , p 2 P e is obtained by sequencing the jobs in non-increasing order of aj and and u 2 U allocating the resource according to Algorithm 1. Proof. It is obviously that for given p, allocating the resource according to Algorithm 1 minimize the resource consumption. Suppose under p, there are two adjacent jobs, job Ji followed by job Jj , such that ai < aj . Denote the completion time of Jj is Cj ¼ C0 , then 2 rj ¼ ðC0 aj Þ=ð1 bÞ. ri ¼ ðC0 aj ð1 bÞai Þ=ð1 bÞ . Perform a pair-wise ~. Under p ~, interchange on jobs Ji and Jj , call new schedule p 2 rei ¼ ðC0 ai Þ=ð1 bÞ. rej ¼ ðC0 ai ð1 bÞaj Þ=ð1 bÞ . From ai < aj , we have rej > ri . Since f 1 is a strictly decreasing function, then ri Þ þ f 1 ð rej Þ < f 1 ðri Þ þ f 1 ðrj Þ: f 1 ðe The release times of the jobs processed after jobs Ji and Jj is not affected by interchange, the jobs processed before jobs Ji and Jj is become later ð rej > ri Þ. ~ is strictly less than that of p. This Hence the resource consumption under p contradicts the optimality of p. h If we are able to calculate f and f 1 in OðgðnÞÞ time, then find p 2 P and e needs time OðmaxfgðnÞ; n log ngÞ. u 2U
3.2. Minimization of makespan From Lemma 1 and results of Section 3.1, for the problem (3), we need only consider the schedule that the jobs sequenced in non-increasing order of aj , i.e., p ¼ ½J1 ; J2 ; . . . ; Jn . If uj ¼ u; j ¼ 1; 2; . . . ; n, then n
Cmax ðp; uÞ ¼ f ðuÞð1 bÞ þ
n X
nj
aj ð1 bÞ
:
j¼1
At this condition, the release times of all jobs equal to f ðuÞ, the makespan be constrained by r1 ¼ f ðuÞ. If we increase the resource allocated to J1 , then r1 will be smaller and makespan be smaller too. Let the maximal amount of resource max allocated to job J1 is umax ¼ minfU ðn 1Þu; ug. 1 , then u1 Let r1 ¼ f ðumax Þ, the completion time of J1 is a1 þ ð1 bÞr1 . 1 If a1 þ ð1 bÞr1 P f ðuÞ, the optimal resource allocation is u1 ¼ umax 1 ;
C.-l. Zhao, H.-y. Tang / Appl. Math. Comput. 161 (2005) 865–874
uj ¼ u;
871
j ¼ 2; . . . ; n; n
Cmax ¼ f ðumax 1 Þð1 bÞ þ
n X
aj ð1 bÞ
nj
:
j¼1
If a1 þ ð1 bÞr1 6 f ðuÞ, when the resource allocated to J1 reached the amount that the completion of J1 is f ðuÞ, the makespan be constrained by r2 ¼ f ðuÞ. At this time we should increase the resource allocated to J1 and J2 , and so on. This implies that there must be a natural number k, such that Cj ðp; up Þ 6 f ðuÞ;
j ¼ 1; . . . ; k 1;
Cj ðp; up Þ P f ðuÞ;
j ¼ k; . . . ; n;
uj ¼ u;
j ¼ k þ 1; . . . ; n:
Let d ¼ f ðuÞ Ck1 ðp; up Þ, from Lemma 3 r1
¼
f ðuÞ d
k1 X
!, aj ð1 bÞ
k1j
ð1 bÞ
k1
;
j¼1
r2 ¼ a1 þ ð1 bÞr1 ; ... rk1 ¼ r1 ð1 bÞ
k2
þ
k2 X
aj ð1 bÞ
k2j
;
j¼1
rk ¼ r1 ð1 bÞk1 þ
k1 X
aj ð1 bÞk1j ;
j¼1
¼ f ðuÞ d; uj ¼ f 1 ðrj Þ; uj ¼ u; k X
ð6Þ
j ¼ 1; . . . ; k;
j ¼ k þ 1; . . . ; n;
uj þ ðn kÞu ¼ U :
j¼1
k and d can be determined by Eq. (8). First we consider k.
ð7Þ ð8Þ
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C.-l. Zhao, H.-y. Tang / Appl. Math. Comput. 161 (2005) 865–874
When rk ¼ f ðuÞ, suppose the release time of Jj is rj , j ¼ 1; . . . ; k, then !, k1 X k1j r1 ¼ f ðuÞ aj ð1 bÞ ð1 bÞk1 ; j¼1
r2 ¼ a1 þ ð1 bÞr1 ; ... rk1 ¼ r1 ð1 bÞk2 þ
k2 X
aj ð1 bÞk2j ;
j¼1 k1
rk ¼ r1 ð1 bÞ
þ
k1 X
k1j
aj ð1 bÞ
;
j¼1
ð9Þ
¼ f ðuÞ; uj ¼ f 1 ðrj Þ;
j ¼ 1; . . . ; k 1;
uj ¼ u; j ¼ k; . . . ; n; k1 X
uj þ ðn ðk 1ÞÞu 6 U :
ð10Þ
j¼1
From Eq. (10), we can get k and then d can be calculated by Eq. (8). Based on above analysis, we have Algorithm 2 (1) Sequencing the jobs in non-increasing order of aj , let umax ¼ 1 minfU ðn 1Þu; u, r1 ¼ f ðumax Þ. 1 (2) If a1 þ ð1 bÞr1 P f ðuÞ, then u1 ¼ umax 1 , uj ¼ u; j ¼ 2; . . . ; n. Stop. Otherwise go to (3). (3) Let k be the maximum natural number such that k1 X
uj þ ðn ðk 1ÞÞu 6 U :
j¼1
uj ¼ f 1 ðrj Þ, rj satisfying form (9). (4) The resource allocation is uj ¼ f 1 ðrj Þ; uj ¼ u;
j ¼ 1; . . . ; k;
j ¼ k þ 1; . . . ; n;
where rj can be determined by form (6) and d satisfying the equation
C.-l. Zhao, H.-y. Tang / Appl. Math. Comput. 161 (2005) 865–874 k X
873
uj þ ðn kÞu ¼ U :
j¼1
Theorem 3. For the problem 1jpj ¼ aj bt; rj ¼ f ðuj Þ; mal schedule can be obtained by Algorithm 2.
P
uj 6 U jCmax , an opti-
If we are able to calculate f , f 1 and d in OðgðnÞÞ time, then the complexity of Algorithm 2 is OðmaxfgðnÞ; n log ngÞ.
4. Conclusion We studied single machine scheduling problems under assumption that job processing times are a function of their starting time. There are many practical scheduling problems that can be modelled in this way. We consider a special case of such model where the processing time of each job is given by a decreasing linear function of its starting time. For the problems to minimize the sum of earliness penalties subject to no tardy jobs, to minimize the resource consumption with makespan constraints and to minimize the makespan with the total resource consumption constraints, the optimal algorithms are presented, respectively.
References [1] B. Alidaee, N.K. Womer, Scheduling with time dependent processing times: review and extensions, Journal of Operational Research Society 50 (1999) 711–720. [2] K.I.-J. Ho, J.Y.-T. Leung, W.-D. Wei, Complexity of scheduling tasks with time-dependent execution times, Information Processing Letters 48 (1993) 315–320. [3] T.C.E. Cheng, Q. Ding, Single machine scheduling with deadlines and increasing rates of processing times, Acta Informatica 36 (2000) 673–692. [4] T.C.E. Cheng, Q. Ding, The complexity of scheduling starting time dependent tasks with release dates, Information Processing Letters 65 (1998) 75–79. [5] C.T. Ng, T.C.E. Cheng, A. Bachman, A. Janiak, Three scheduling problems with deteriorating jobs to minimize the total completion time, Information Processing Letters 81 (2002) 327– 333. [6] S. Chang, H. Schneeberger, Single machine scheduling to minimize weighted earliness subject to no tardy jobs, European Journal of Operational Research 34 (1988) 221–230. [7] T.C.E. Cheng, A. Janiak, Resource optimal control in some single-machine scheduling problems, IEEE Transactions on Automatic Control 39 (1994) 1243–1246. [8] A. Janiak, Time-optimal control in a single machine problems with resource constraints, Automatica 22 (1986) 745–747. [9] S. Browne, U. Yechiali, Scheduling deteriorating jobs on a single processor, Operations Research 38 (1990) 495–498.
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[10] G. Mosheiov, V-shaped policies for scheduling deteriorating jobs, Operations Research 39 (1991) 979–991. [11] G. Mosheiov, K-shaped policies for scheduling deteriorating jobs, Journal of Operational Research Society 47 (1996) 1184–1191. [12] G. Mosheiov, Scheduling jobs under simple linear deterioration, Computer and Operations Research 21 (1994) 653–659. [13] A. Bachman, A. Janiak, Minimizing maximum lateness under linear deterioration, European Journal of Operational Research 126 (2000) 557–566. [14] G. Mosheiov, Complexity analysis of job-shop scheduling with deteriorating jobs, Discrete Applied Mathematics 117 (2002) 195–209.
Single machine scheduling problems with deteriorating jobs Chuan-li Zhao *, Heng-yong Tang College of Mathematics and Systems Sciences, Shenyang Normal University, Shenyang, Liaoning 110034, PeopleÕs Republic of China
Abstract This paper considers the single machine scheduling problems with deteriorating jobs, i.e. jobs whose processing times are a decreasing linear function of their starting time. It is assumed that jobs have the different basic processing time and same decreasing rate. Based on the analysis of problems, the optimal algorithms are presented for the problems to minimize the sum of earliness penalties subject to no tardy jobs, to minimize the resource consumption with makespan constraints and to minimize the makespan with the total resource consumption constraints. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Scheduling; Single machine; Deteriorating jobs; Earliness penalties; Resource constraints; Optimal algorithms
1. Introduction Machine scheduling problems with job processing times given as starting time dependent functions have received increasing attention in recent years [1]. In this model, the processing time of a job can be described by a basic processing time and an increasing (or decreasing) rate. For the model where job processing time is an increasing start time dependent function, there are many results in literatures [1]. Browne and Yechiali [9] consider stochastic single machine scheduling. Mosheiov [10–12]
*
Corresponding author. E-mail address: [email protected] (C.-l. Zhao).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.12.073
866
C.-l. Zhao, H.-y. Tang / Appl. Math. Comput. 161 (2005) 865–874
studied some spacial cases of this model. Bachman and Janiak [13] proved that the problem to minimize the maximum lateness with the arbitrary linear processing time is NP -complete. Mosheiov [14] analysis the complexity of job-shop problems. Apart from the model where job processing time is an increasing start time dependent function, there are also a model where job processing time is a decreasing start time dependent function. This model is introduced by Ho et al. [2]. Cheng and Ding [3,4] consider some problems with release times and deadlines. Ng et al. [5] consider the problems to minimize the total completion time. In this paper we consider the special case of the model where job processing time is a decreasing start time dependent function. It is assumed that jobs have the different basic processing time and same decreasing rate. For three single machine scheduling problems, the optimal algorithms are presented. The model can be described as follows. Suppose n jobs J1 ; J2 ; . . . ; Jn to be processed on a single machine. Associated with job Jj is a basic processing time aj and a decreasing rate b (without loss of generality we assume a1 P a2 P P an ). Let t be starting time of job Jj then the actual processing time of Jj is pj ¼ aj bt, j ¼ 1; . . . ; n. It is assumed that the decreasing rates satisfy the following conditions: ! n X 0 < b < 1 and b ai aj < aj ; j ¼ 1; . . . ; n: i¼1
The first condition ensures that the decrease of each job processing time is less than one unit for every unit delay in its starting moment. The second one ensures that all job processing times are positive in a feasible schedule [2].
2. Minimize the sum of earliness penalties In this section we consider the problem to minimize the sum of earliness penalties subject to no tardy jobs. For the classical scheduling problem, there are some results in [6]. We assume that all jobs have a common due-date d and release time is 0. For the schedule p ¼ ½J½1 ; J½2 ; . . . ; J½n , let C½j and E½j ¼ d C½j be the completion P time and earliness of job J½j in p. The sum of earliness penalties is EðpÞ ¼ nj¼1 gðE½j Þ, where g is a strictly increasing function. We denote all schedules by P. The problem is to find an optimal schedule p 2 P that minimizes the sum of earliness penalties under condition C½j 6 d. The problem be denoted as X 1jpj ¼ aj btj gðEj Þ: ð1Þ
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867
Since the problem is to minimize earliness penalties, the jobs should be processed as late as possible. It is obviously that in the optimal schedule the completion time of the last job is d and there is not idle times between jobs. First we give some results of the problem to minimize makespan, it can be denoted as 1jpj ¼ aj btjCmax . Lemma 1. For the problem 1jpj ¼ aj btjCmax , the makespan is minimized by sequencing the jobs in non-increasing order of aj . Lemma 2. For the problem 1jpj ¼ aj btjCmax , if p ¼ ½J1 ; J2 ; . . . ; Jn and the starting time of first job is t0 , then makespan is n
Cmax ðt0 jJ1 ; J2 ; . . . ; Jn Þ ¼ t0 ð1 bÞ þ
n X
aj ð1 bÞ
nj
:
j¼1
Lemma 3. For the problem 1jpj ¼ aj btjCmax , if p ¼ ½J1 ; J2 ; . . . ; Jn and the makespan is C, then starting time of first job is !, n X nj n t0 ¼ C aj ð1 bÞ ð1 bÞ : j¼1
Based on the above lemmas, we have following theorem. P Theorem 1. For the problem 1jpj ¼ aj btj gðEj Þ, an optimal schedule can be obtained by sequencing the jobs in non-increasing order of aj . Where the starting time of the first job is !, n X nj n t0 ¼ C aj ð1 bÞ ð1 bÞ : j¼1
Proof (by contradiction). Consider an optimal schedule p. Assume under p, there two adjacent jobs, job Ji followed by job Jj , such that ai < aj . Denote the completion time of Jj is Cj ¼ C0 , then Ci ¼ ðC0 aj Þ=ð1 bÞ. Ej ¼ d C0 , Ei ¼ d ðC0 aj Þ=ð1 bÞ. Sj ¼ Ci , Si ¼ ðC0 aj ð1 bÞai Þ=ð1 bÞ2 . Per~. Under p ~, form a pair-wise interchange on jobs Ji and Jj , call new schedule p e i ¼ C0 , C e j ¼ ðC0 ai Þ=ð1 bÞ. E e i ¼ d C0 , E e j ¼ d ðC0 ai Þ=ð1 bÞ. C 2 e e j, e Si ¼ C S j ¼ ðC0 ai ð1 bÞaj Þ=ð1 bÞ . e j > Ci , e j < Ei e i ¼ Ej . then C E and E Hence Since ai < aj , e e gð E i Þ þ gð E j Þ < gðEi Þ þ gðEj Þ. The completion times of the jobs processed after jobs Ji and Jj is not affected by interchange, the jobs processed before jobs Ji and Jj is become later (Since e ~). Hence the sum S j > Si , then the starting times of these jobs are larger under p
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~ is strictly less than that of p. This contradicts of earliness penalties under p the optimality of p. h
3. Resource constrained problems In this section, the release time of job Jj is rj ¼ f ðuj Þ, u 6 uj 6 u, where uj is the amount of resource allocated to Jj , u, u are known constraint and f is a P strictly decreasing function (f ðuÞ > f ðuÞ > 0, 0 6 u 6 u). nj¼1 6 U , U is the global amount of continuously divisible non-renewable resource. For the problems with release time of a job is function of resource allocated, Cheng and Janiak [7] and Janiak [8] have considered the case where jobs have constant processing times (b ¼ 0). In this section we consider two problems with jobs have the different basic processing time and the same decreasing rate, the first is to minimize the resource consumption with makespan constraints while the second is to minimize the makespan with the total resource consumption constraints: X 1jpj ¼ aj bt; rj ¼ f ðuj Þ; Cmax 6 Cj uj ; ð2Þ X 1jpj ¼ aj bt; rj ¼ f ðuj Þ; uj 6 U jCmax : ð3Þ 3.1. Minimization of resource consumption We denotePthe set of all resource allocations u ¼ ½u1 ; u2 ; . . . ; un by e ðu 6 uj 6 u; n uj 6 U Þ. Given a schedule p ¼ ½J½1 ; J½2 ; . . . ; J½n and a reU j¼1 source allocations up ¼ ½u½1 ; u½2 ; . . . ; u½n , we have r½j ¼ f ðu½j Þ; j ¼ 1; 2; . . . ; n; C½1 ðp; uÞ ¼ a½1 þ ð1 bÞr½1 ; C½j ðp; uÞ ¼ maxfr½j ; C½j1 ðp; uÞg þ ð1 bÞ maxfr½j ; C½j1 ðp; uÞg ( ) j X jiþ1 jk ¼ max r½i ð1 bÞ þ a½k ð1 bÞ ; j ¼ 2; . . . ; n: 16i6j
k¼i
Hence Cmax ðp; uÞ ¼ max1 6 j 6 n fC½j ðp; uÞg;
U ðp; uÞ ¼
n X
u½j :
j¼1
Let C (C satisfying bðC aj Þ < aj , j ¼ 1; . . . ; n) be a given makespan, the e which minimizes the total resource problem (2) is to find p 2 P and u 2 U consumption i.e., U ðp ; u Þ ¼ minp2P min e fU ðp; uÞg. u2 U
C.-l. Zhao, H.-y. Tang / Appl. Math. Comput. 161 (2005) 865–874
869
Subject to Cmax ðp ; u Þ 6 C. From Lemma 1, the minimum makespan of schedule is C ¼ Cmax ðf ðuÞjJ1 ; J2 ; . . . ; Jn Þ ¼ f ðuÞð1 bÞn þ
n X
aj ð1 bÞnj :
ð4Þ
j¼1
So a schedule p ¼ ½J½1 ; J½2 ; . . . ; J½n is feasible only if n
Cmax ðf ðuÞjJ½1 ; J½2 ; . . . ; J½n Þ ¼ f ðuÞð1 bÞ þ
n X
a½j ð1 bÞ
nj
6 C:
ð5Þ
j¼1
We denote a resource allocation by up with which the resource consumption is minimized, subject to a given C, i.e., U ðp; up Þ ¼ minfU ðp; uÞg: u2 e U Since releasing jobs sooner consumes more resource, jobs should be released as late as possible and the completion time of the last P job is C. If Cmax ðp; Þ ¼ C, then the starting time of the first job is t0 ¼ ðC nj¼1 a½j ð1 bÞnj Þ=ð1 bÞn . Hence the resource allocation up is determined as follows: Algorithm 1 (1) Let t0 ¼
C
n X
!, nj
a½j ð1 bÞ
ð1 bÞn :
j¼1
(2) If a½1 þ ð1 bÞt0 P f ðuÞ, then u½1 ¼ f 1 ðt0 Þ; u½j ¼ u; j ¼ 2; . . . ; n. Stop. Otherwise go to (3). (3) Let k be the maximum natural number satisfying t0 ð1 bÞk1 þ
k1 X
a½j ð1 bÞk1j 6 f ðuÞ
j¼1
(it is C½k1 6 f ðuÞ); then u½1 ¼ f 1 ðt0 Þ; u½j
¼f
1
t0 ð1 bÞ
j1
þ
j1 X i¼1
u½j ¼ u; j ¼ k þ 1; . . . ; n:
! j1i
a½i ð1 bÞ
;
j ¼ 2; . . . ; k;
870
C.-l. Zhao, H.-y. Tang / Appl. Math. Comput. 161 (2005) 865–874
P Theorem 2. For the problem 1jpj ¼ aj bt; rj ¼ f ðuj Þ; Cmax 6 Cj uj , p 2 P e is obtained by sequencing the jobs in non-increasing order of aj and and u 2 U allocating the resource according to Algorithm 1. Proof. It is obviously that for given p, allocating the resource according to Algorithm 1 minimize the resource consumption. Suppose under p, there are two adjacent jobs, job Ji followed by job Jj , such that ai < aj . Denote the completion time of Jj is Cj ¼ C0 , then 2 rj ¼ ðC0 aj Þ=ð1 bÞ. ri ¼ ðC0 aj ð1 bÞai Þ=ð1 bÞ . Perform a pair-wise ~. Under p ~, interchange on jobs Ji and Jj , call new schedule p 2 rei ¼ ðC0 ai Þ=ð1 bÞ. rej ¼ ðC0 ai ð1 bÞaj Þ=ð1 bÞ . From ai < aj , we have rej > ri . Since f 1 is a strictly decreasing function, then ri Þ þ f 1 ð rej Þ < f 1 ðri Þ þ f 1 ðrj Þ: f 1 ðe The release times of the jobs processed after jobs Ji and Jj is not affected by interchange, the jobs processed before jobs Ji and Jj is become later ð rej > ri Þ. ~ is strictly less than that of p. This Hence the resource consumption under p contradicts the optimality of p. h If we are able to calculate f and f 1 in OðgðnÞÞ time, then find p 2 P and e needs time OðmaxfgðnÞ; n log ngÞ. u 2U
3.2. Minimization of makespan From Lemma 1 and results of Section 3.1, for the problem (3), we need only consider the schedule that the jobs sequenced in non-increasing order of aj , i.e., p ¼ ½J1 ; J2 ; . . . ; Jn . If uj ¼ u; j ¼ 1; 2; . . . ; n, then n
Cmax ðp; uÞ ¼ f ðuÞð1 bÞ þ
n X
nj
aj ð1 bÞ
:
j¼1
At this condition, the release times of all jobs equal to f ðuÞ, the makespan be constrained by r1 ¼ f ðuÞ. If we increase the resource allocated to J1 , then r1 will be smaller and makespan be smaller too. Let the maximal amount of resource max allocated to job J1 is umax ¼ minfU ðn 1Þu; ug. 1 , then u1 Let r1 ¼ f ðumax Þ, the completion time of J1 is a1 þ ð1 bÞr1 . 1 If a1 þ ð1 bÞr1 P f ðuÞ, the optimal resource allocation is u1 ¼ umax 1 ;
C.-l. Zhao, H.-y. Tang / Appl. Math. Comput. 161 (2005) 865–874
uj ¼ u;
871
j ¼ 2; . . . ; n; n
Cmax ¼ f ðumax 1 Þð1 bÞ þ
n X
aj ð1 bÞ
nj
:
j¼1
If a1 þ ð1 bÞr1 6 f ðuÞ, when the resource allocated to J1 reached the amount that the completion of J1 is f ðuÞ, the makespan be constrained by r2 ¼ f ðuÞ. At this time we should increase the resource allocated to J1 and J2 , and so on. This implies that there must be a natural number k, such that Cj ðp; up Þ 6 f ðuÞ;
j ¼ 1; . . . ; k 1;
Cj ðp; up Þ P f ðuÞ;
j ¼ k; . . . ; n;
uj ¼ u;
j ¼ k þ 1; . . . ; n:
Let d ¼ f ðuÞ Ck1 ðp; up Þ, from Lemma 3 r1
¼
f ðuÞ d
k1 X
!, aj ð1 bÞ
k1j
ð1 bÞ
k1
;
j¼1
r2 ¼ a1 þ ð1 bÞr1 ; ... rk1 ¼ r1 ð1 bÞ
k2
þ
k2 X
aj ð1 bÞ
k2j
;
j¼1
rk ¼ r1 ð1 bÞk1 þ
k1 X
aj ð1 bÞk1j ;
j¼1
¼ f ðuÞ d; uj ¼ f 1 ðrj Þ; uj ¼ u; k X
ð6Þ
j ¼ 1; . . . ; k;
j ¼ k þ 1; . . . ; n;
uj þ ðn kÞu ¼ U :
j¼1
k and d can be determined by Eq. (8). First we consider k.
ð7Þ ð8Þ
872
C.-l. Zhao, H.-y. Tang / Appl. Math. Comput. 161 (2005) 865–874
When rk ¼ f ðuÞ, suppose the release time of Jj is rj , j ¼ 1; . . . ; k, then !, k1 X k1j r1 ¼ f ðuÞ aj ð1 bÞ ð1 bÞk1 ; j¼1
r2 ¼ a1 þ ð1 bÞr1 ; ... rk1 ¼ r1 ð1 bÞk2 þ
k2 X
aj ð1 bÞk2j ;
j¼1 k1
rk ¼ r1 ð1 bÞ
þ
k1 X
k1j
aj ð1 bÞ
;
j¼1
ð9Þ
¼ f ðuÞ; uj ¼ f 1 ðrj Þ;
j ¼ 1; . . . ; k 1;
uj ¼ u; j ¼ k; . . . ; n; k1 X
uj þ ðn ðk 1ÞÞu 6 U :
ð10Þ
j¼1
From Eq. (10), we can get k and then d can be calculated by Eq. (8). Based on above analysis, we have Algorithm 2 (1) Sequencing the jobs in non-increasing order of aj , let umax ¼ 1 minfU ðn 1Þu; u, r1 ¼ f ðumax Þ. 1 (2) If a1 þ ð1 bÞr1 P f ðuÞ, then u1 ¼ umax 1 , uj ¼ u; j ¼ 2; . . . ; n. Stop. Otherwise go to (3). (3) Let k be the maximum natural number such that k1 X
uj þ ðn ðk 1ÞÞu 6 U :
j¼1
uj ¼ f 1 ðrj Þ, rj satisfying form (9). (4) The resource allocation is uj ¼ f 1 ðrj Þ; uj ¼ u;
j ¼ 1; . . . ; k;
j ¼ k þ 1; . . . ; n;
where rj can be determined by form (6) and d satisfying the equation
C.-l. Zhao, H.-y. Tang / Appl. Math. Comput. 161 (2005) 865–874 k X
873
uj þ ðn kÞu ¼ U :
j¼1
Theorem 3. For the problem 1jpj ¼ aj bt; rj ¼ f ðuj Þ; mal schedule can be obtained by Algorithm 2.
P
uj 6 U jCmax , an opti-
If we are able to calculate f , f 1 and d in OðgðnÞÞ time, then the complexity of Algorithm 2 is OðmaxfgðnÞ; n log ngÞ.
4. Conclusion We studied single machine scheduling problems under assumption that job processing times are a function of their starting time. There are many practical scheduling problems that can be modelled in this way. We consider a special case of such model where the processing time of each job is given by a decreasing linear function of its starting time. For the problems to minimize the sum of earliness penalties subject to no tardy jobs, to minimize the resource consumption with makespan constraints and to minimize the makespan with the total resource consumption constraints, the optimal algorithms are presented, respectively.
References [1] B. Alidaee, N.K. Womer, Scheduling with time dependent processing times: review and extensions, Journal of Operational Research Society 50 (1999) 711–720. [2] K.I.-J. Ho, J.Y.-T. Leung, W.-D. Wei, Complexity of scheduling tasks with time-dependent execution times, Information Processing Letters 48 (1993) 315–320. [3] T.C.E. Cheng, Q. Ding, Single machine scheduling with deadlines and increasing rates of processing times, Acta Informatica 36 (2000) 673–692. [4] T.C.E. Cheng, Q. Ding, The complexity of scheduling starting time dependent tasks with release dates, Information Processing Letters 65 (1998) 75–79. [5] C.T. Ng, T.C.E. Cheng, A. Bachman, A. Janiak, Three scheduling problems with deteriorating jobs to minimize the total completion time, Information Processing Letters 81 (2002) 327– 333. [6] S. Chang, H. Schneeberger, Single machine scheduling to minimize weighted earliness subject to no tardy jobs, European Journal of Operational Research 34 (1988) 221–230. [7] T.C.E. Cheng, A. Janiak, Resource optimal control in some single-machine scheduling problems, IEEE Transactions on Automatic Control 39 (1994) 1243–1246. [8] A. Janiak, Time-optimal control in a single machine problems with resource constraints, Automatica 22 (1986) 745–747. [9] S. Browne, U. Yechiali, Scheduling deteriorating jobs on a single processor, Operations Research 38 (1990) 495–498.
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[10] G. Mosheiov, V-shaped policies for scheduling deteriorating jobs, Operations Research 39 (1991) 979–991. [11] G. Mosheiov, K-shaped policies for scheduling deteriorating jobs, Journal of Operational Research Society 47 (1996) 1184–1191. [12] G. Mosheiov, Scheduling jobs under simple linear deterioration, Computer and Operations Research 21 (1994) 653–659. [13] A. Bachman, A. Janiak, Minimizing maximum lateness under linear deterioration, European Journal of Operational Research 126 (2000) 557–566. [14] G. Mosheiov, Complexity analysis of job-shop scheduling with deteriorating jobs, Discrete Applied Mathematics 117 (2002) 195–209.