Scheduling start time dependent tasks with deadlines and identical initial processing times on a single machine

Computers & Operations Research 30 (2003) 51–62www.elsevier.com/locate/dsw

Scheduling start time dependent tasks with deadlines andidentical initial processing times on a single machine

T.C.E. Cheng ∗, Q. Ding

Department of Management, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received 1 February 2000; received in revised form 1 December 2000

Abstract

In this paper, we study the feasibility problem of scheduling a set of start time dependent tasks ona single machine with deadlines, processing rates and identical initial processing times. First, we showthat the cases with arbitrary deadlines are strongly NP-complete. Second, we show that the cases withtwo distinct deadlines are NP-complete in the ordinary sense. Finally, we give an optimal polynomialalgorithm for the makespan problem with two distinct processing rates. We solve a series of openproblems in the literature and give a sharp boundary delineating the complexity of the problems.

Scope and purpose

There has been increasing interest in scheduling models where the job processing times are functionsof their starting times. We consider such a model where the job processing times change linearly withtheir starting times. The initial processing times are the same but the processing rates are arbitrary. Eachjob has a deadline, which cannot be exceeded. The feasibility problem is to 5nd a schedule satisfyingthe deadline constraints. The makespan problem is to 5nd a feasible schedule minimizing the makespan.The complexity of the feasibility problem is actually equivalent to that of the makespan problem. Weshow that the feasibility problem with arbitrary deadlines is strongly NP-complete and the case with twodistinct deadlines is NP-complete in the ordinary sense. We give also an optimal algorithm for a specialcase of the makespan problem. ? 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Sequencing; Time dependence scheduling; Computational complexity

∗ Corresponding author. Tel.: +852-2766-5215; fax: +852-2364-5245.E-mail address: [email protected] (T.C.E. Cheng).

0305-0548/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.PII: S 0305-0548(01)00077-6

52 T.C.E. Cheng, Q. Ding / Computers & Operations Research 30 (2003) 51–62

1. Introduction

Machine scheduling problems with start time dependent processing times have received in-creasing attention in recent years. The linear model is one of the most popular ones. Formally,these problems can be stated as follows: A task system consists of n independent tasks andis denoted by TS=({Ti}; {di}; {ai}; {bi}). Each task Ti is associated with a deadline di andcharacterized by an initial processing time ai¿ 0 and a processing rate bi¿ 0. Depending onthe task starting time si, the actual processing time of task Ti is pi= ai ± bisi¿ 0 and itscompletion time is Ci= si + pi= ai + (1 ± bi)si. For all tasks, the release time is 0. Since theprocessing times are not integers in many practical cases, values ai; bi and di are allowed to berational numbers. Adopting the three-5eld notation proposed by Graham et al. [1] to describe ascheduling problem, we denote the makespan problem as 1=pi= ai ± bisi; di=Cmax.

A nonpreemptive schedule is feasible if each Ti is completely processed in the interval [0; di].The feasibility problem is to decide whether there exists a feasible schedule forTS. Let Cmax =max16i6n {Ci} denote the maximum completion time of a schedule andG=max16i6n{di} ≡ dmax denote a threshold. The feasibility problem is equivalent to thedecision version of the makespan problem, which is to decide whether there exists a feasibleschedule for TS with Cmax6G.

Several papers consider the makespan problems of single machine scheduling with start timedependent tasks. For the model pi= ai+ bisi¿ 0 without deadlines, Gupta and Gupta [2] showthat the schedule of tasks arranged in nondecreasing order of the ratios ai=bi is optimal for themakespan problem. Meanwhile, Browne and Yechiali [3] obtain a similar result for a stochasticversion of the problem. Cheng and Ding [4] show that the makespan problem with arbitrarydeadlines is strongly NP-complete, and the case with bi= b can be solved in O(n5) time.Kononov [5] shows that the maximum lateness problem is NP-complete in the ordinary senseeven with ai=0 for all tasks except one, and the due dates di=d for all tasks with ai=0.

For the model pi= ai − bisi¿ 0, Ho et al. [6] show that the schedule of tasks arrangedin nonincreasing order of the ratios ai=bi is optimal for the makespan problem with identicaldeadlines. Cheng and Ding show that the case with bi= b and arbitrary deadlines is stronglyNP-complete [7], while the case with bi= b and two distinct deadlines is NP-complete in theordinary sense [8]. Woeginger [9] and Chen [10] give two dynamic programming algorithmsto solve the number of late task problem with identical deadlines in O(n3) and O(n2) time,respectively. The more recent results for scheduling models with time dependent processingtimes are surveyed in Alidaee and Womer [11].

The scheduling problem with identical processing times is an important branch in classicalscheduling theory (see the results surveyed in Tanaev et al. [12]). However, the correspondingproblem with time dependent processing times is a virtually new area of study. In this paper,we focus on the complexity of the feasibility problems with arbitrary processing rates andidentical initial processing times, i.e., ai= a. We show that the cases with arbitrary processingrates and arbitrary deadlines, denoted as 1=pi= a± bisi; di=Cmax, are strongly NP-complete. Wealso show that the cases with arbitrary processing rates and two distinct deadlines, denoted as1=pi= a ± bisi; di ∈{D1; D2}=Cmax, are NP-complete in the ordinary sense. Finally, we give anoptimal polynomial algorithm for the makespan problem with only two distinct processing rates,denoted as 1=pi= a± bisi; di; bi ∈{B1; B2}=Cmax.

T.C.E. Cheng, Q. Ding / Computers & Operations Research 30 (2003) 51–62 53

The considered models are rich in practical applications. Ho et al. [6] introduce an interestingmilitary application with negative processing rates. The task is to destroy an aerial threat and itsexecution time decreases with time as the longer the action is delayed, the closer the threat gets.This application is actually a problem with ai= a, considering that the eKcient scope of theaction is usually a sphere. For the case with positive processing rates and ai= a, we considermedical treatment as an application example. At the outset, the treatment is common to thepatients that take an identical processing time. However, if the treatment is delayed, additionaleLorts are needed for each treated individual, resulting in a longer time for each subsequenttreatment.

For the model pi= ai−bisi¿ 0, not only pi, but also Ci is decreasing in si if bi ¿ 1. In sucha case, some performance measures become non-regular. We assume 06 bi6 1 in this paper.Ho et al. [6] make some additional assumptions, such as bidi ¡ai6di, which are reasonableand indeed help eliminate some uninteresting cases. However, they are not necessary for theresults in this paper. On the other hand, a similar model with arbitrary processing rates isconsidered in Cheng et al. [13].

Furthermore, only schedules without idle time need to be considered. Without aLecting theresults of NP-completeness, in Sections 2 and 3, we assume that the identical initial processingtime is 1, i.e., a=1, is used in the formula describing the completion time of a task.

2. NP-completeness of the problems with arbitrary deadlines

The strongly NP-complete 3-Partition problem (see Garey and Johnson [14]) can be reducedto 1=pi=1− bisi; di=Cmax.

3-Partition. Given a list H = {h1; h2; : : : ; h3m} of 3m integers such that∑3m

i=1 hi=mB and B=4¡hi¡B=2 for each 16 i6 3m, can H be partitioned disjointedly into H1; H2; : : : ; Hm such that∑

hi∈Hjhi=B for each 16 j6m?

Given an instance I of 3-Partition with a list H = {h1; h2; : : : ; h3m} and B, de5ne q=2mB;v=32m2qB. Construct an instance II of 1=pi=1− bisi; di=Cmax as follows:

The set of tasks is TS=V ∪ R ∪ Q1 ∪ · · · ∪ Qm−1, where V = {T0;1; T0;2; : : : ; T0; v}, R={T1; T2; : : : ; T3m}, Qi= {Ti;1; Ti;2; : : : ; Ti;q}, for 16 i6m−1. De5ne A1 =3n3 and A2 =A3 =2nmB,where n= v+ 3m+ (m− 1)q is the number of tasks in TS.Given the threshold

G= n−m−1∑

k=1

q∑

l=1

v+ qk + 3k − lA1A3

−m−1∑

k=0

(v+ qk + 3k)BA1A2A3

and the constants

Di= v+ qi + 3i −i−1∑

k=1

q∑

l=1

v+ qk + 3k − lA1A3

−i−1∑

k=0

(v+ qk + 3k)BA1A2A3

; for 16 i6m− 1:


Fig. 1. The basic and standard schedule of II for 1=pi =1± bisi; di=Cmax.

We assume that the processing rates and the deadlines are

b0; i=0; d0; i= v for 16 i6 v;

bi; j=1

A1A3; di; j=Di for 16 i6m− 1 and 16 j6 q;

bi=hi

A1A2A3; di=G for 16 i6 3m:

Now, we analyze the structure of the feasible schedule for II (see Fig. 1). Let S be a givenfeasible schedule for II. Since d0; i= v, for 16 i6 v, the tasks in V should be scheduled in the5rst v positions in S. Since the tasks in Q1 ∪ · · · ∪ Qm−1 have identical processing rates, wecan arrange them in nondecreasing order of their deadlines and indices. The set of tasks afterT0; v and before T1; q consists of the tasks in Q1 and some tasks in R. All of their deadlinesare larger than D1. We re-arrange the task set in nonincreasing order of the processing rates.Since the processing rates of tasks in R are smaller than the processing rates of tasks in Q1, theresulting schedule is in the form of (R1; Q1). Since the schedule of tasks in nonincreasing orderof the processing rates minimizes the total processing time (see Ho et al. [6]), the resultingschedule is also feasible. Similarly, we can swap the tasks after Ti;q and before Ti+1; q in theform of (Ri+1; Qi+1) for 16 i6m− 2. Finally, we obtain a new feasible schedule in the formof (V; R1; Q1; R2; : : : ; Qm−1; Rm), which is called a basic schedule. Thus, we obtain the followinglemma:

Lemma 1. If there exists a feasible schedule for II; then there exists a feasible basic schedule.

In a basic schedule for II, if there are exactly three tasks in each Rj and∑

Ti∈Rj hi=B, for16 j6m, then we obtain an ideal schedule, which is called a standard schedule. Now, weillustrate that every standard schedule is feasible as follows:

Given a standard schedule S, since the processing rates are very small, all actual processingtimes are almost equal to 1. For any task Ti ∈Rj in S, the actual processing time is pi=1 − bisi=1 − bi[v + (j − 1)q] − biOsi, where Osi= si − v − (j − 1)q is much smaller than q.Since there are exactly three tasks in Rj and

∑Ti∈Rj hi=B, we have

∑

Ti∈Rjpi=3−

∑

Ti∈Rjbisi=3− [v+ (j − 1)q]B

A1A2A3−

∑

Ti∈RjbiOsi: (1)


De5ne PRj =3 − [v + (j − 1)q]B=A1A2A3, for 16 j6m. Since the error term∑

Ti∈Rj biOsi ismuch smaller than q=A1A2A3, from (1), we have

PRj −q

4mA1A2A36

∑

Ti∈Rjpi6PRj : (2)

If Qj is scheduled consecutively and completed exactly at Dj, then the total actual processingtime of Qj is its minimum, denoted as PQj , for 16 j6m− 1.

Note that D1 = v+PR1 +PQ1 ; that is, if Q1 is scheduled consecutively and completed exactlyat D1, then the length of the interval left for R1 is exactly PR1 . Such a sub-schedule (R1; Q1)is feasible. By induction, we can generalize the above results to the reminder of the scheduleand obtain the following lemma:

Lemma 2. If a schedule for II is standard; then it is feasible.

On the other hand, if a basic schedule is not standard, then we can show that it is notfeasible either. Given a basic schedule S, if there are more than 3j tasks in R1 ∪ · · · ∪ Rj, then,from (1), it is easy to see that

∑Ti∈R1∪···∪Rj pi ¿ 3j+ 1

2 ¿∑j

i=1 PRi . From the de5nition of Dj,the task set Qj cannot meet Dj. Similarly, if there are exactly 3j tasks in R1 ∪ · · · ∪ Rj and∑

Ti∈R1∪···∪Rj hi ¡ jB, then we also have that∑

Ti∈R1∪···∪Rj pi ¿∑j

i=1 PRi and Qj cannot meet Dj.Thus we obtain the following remark:

Remark 1. For a feasible basic schedule, there are no more than 3j tasks in R1 ∪ · · · ∪ Rj and∑Ti∈R1∪···∪Rj hi¿ jB, for 16 j6m.

Now, we analyze the converse cases. Given a basic schedule, suppose that there are onlytwo tasks in R1. Comparing with a standard schedule, the starting times of tasks in Q1 are allearlier by almost 1. The total actual processing time of tasks in Q1 is near PQ1 + (q=A1A2).Meanwhile, the total actual processing time of tasks in Qj is always larger than its minimumPQj . As to the tasks in R, similar to (2), we have

∑

Ti∈Rpi −

m∑

i=1

PRi 6∑

Ti∈Rbi(si − v)6

3mB2mqA1A2A3

: (3)

Since q=A1A2 is much larger than q(3mB2m+1)=A1A2A3, from (3) and the de5nition of G, weget Cmax¿G; that is, the schedule is not feasible. Thus, we have that there are exactly threetasks in R1. By induction and Remark 1, we obtain the following remark:

Remark 2. For a feasible basic schedule, there are exactly 3j tasks in R1 ∪ · · · ∪ Rj and∑Ti∈R1∪···∪Rj hi¿ jB, for 16 j6m.


Given a feasible basic schedule, from Remark 2, there are exactly 3j tasks in R1 ∪ · · · ∪ Rjand

∑Ti∈R1∪···∪Rj hi¿ jB, for 16 j6m. Suppose that∑

Ti∈R1

hi=B+ 1;∑

Ti∈R2∪···∪Rj−1

hi=B and∑

Ti∈Rjhi=B− 1: (4)

Similar to (1), we have∑

Ti∈R1

pi ≈ 3− v(B+ 1)A1A2A3

(5)

and∑

Ti∈Rjpi ≈ 3− [v+ (j − 1)q](B− 1)

A1A2A3: (6)

From (5) and (6), we have∑

Ti∈R1∪Rjpi − (PR1 + PRj) ≈ (j − 1)q

A1A2A3¿

q2A1A2A3

: (7)

If the other parts of the given schedule are in the standard form, then, from (2), we have∑T �∈R1∪Rj;Ti∈R pi −

∑i �=j;26j6m PRi ¿ − q=4A1A2A3. Further, from (7) and the de5nition of G,

we have that∑

Ti∈R pi −∑

16j6m PRi ¿ q=4A1A2A3 and Cmax¿G, contradicting the feasibilityassumption. Thus the given schedule is standard. If there exists another sub-schedule which isnot in the standard form, then, from Remark 2, the structure of this sub-schedule should ful5lthe assumption for the originally given schedule. Repeating the above analysis and results, weobtain the following lemma.

Lemma 3. If a basic schedule for II is feasible; then it is standard.

Given a solution for I, we can construct a corresponding standard schedule. From Lemma 2,it is feasible. Given a feasible schedule for II, from Lemmas 1 and 3, there exists a feasiblebasic schedule, which is standard and corresponds to a solution for I. A similar reduction witha detailed proof is presented in Cheng and Ding [7]. Thus, we obtain the following theorem:

Theorem 1. The problem of whether there exists a feasible schedule for the problem 1=pi=1− bisi; di=Cmax is strongly NP-complete.

A similar technique can be used to tackle 1=pi=1 + bisi; di=Cmax. Given an instance I of3-Partition with a list H = {h1; h2; : : : ; h3m} and B, de5ne q=32m2B, v=16m2qB and constructan instance II of 1=pi=1+ bisi; di=Cmax as follows:The set of tasks is TS=V ∪ R ∪ Q1 ∪ · · · ∪ Qm−1, where V = {T0;1; T0;2; : : : ; T0; v},

R= {T1; T2; : : : ; T3m} and Qi= {Ti;1; Ti;2; : : : ; Ti;q}, for 16 i6m − 1. De5ne E=4mnB andA=32n3E2, where n= v+ 3m+ (m− 1)q is the number of tasks in TS.


Given the threshold

G= n+m−1∑

k=0

3E(v+ qk + 3k + 1)A

+m−1∑

k=0

B(v+ qk + 3k + 1)A

+2mBA

and the constants

Di= v+ qi + 3i +i−1∑

k=0

3E(v+ qk + 3k + 1)A

+i−1∑

k=0

B(v+ qk + 3k + 1)A

+2mBA

;

for 16 i6m− 1. We assume that the processing rates are

b0; i=0; d0; i= v for 16 i6 v;

bi; j=0; di; j=Di for 16 i6m− 1 and 16 j6 q;

bi=E + hiA

; di=G for 16 i6 3m:

Let S be a given feasible schedule for II . Since d0; i= v, for 16 i6 v, the tasks in V shouldbe scheduled in the 5rst v positions. Since the schedule of tasks arranged in increasing pro-cessing rates minimizes the total processing time (see Gupta and Gupta [2]) and the increasingprocessing rates of tasks in R are larger than those of tasks in Q1 ∪ · · · ∪ Qm−1 , the tasks inR should be scheduled as early as possible. By swapping tasks in Q1 ∪ · · · ∪ Qm−1 and R, weobtain a feasible schedule in the form of (V; R1; Q1; R2; : : : ; Qm−1; Rm). Moreover, the number oftasks in each Rj is exactly three and the sum of the processing rates is exactly (3E + B)=A.Similar to Theorem 1, we obtain the following theorem:

Theorem 2. The problem of whether there exists a feasible schedule for the problem 1=pi=1+bisi; di=Cmax is strongly NP-complete.

3. NP-completeness of the problems with two distinct deadlines

Now we discuss the complexity issue for cases with two distinct deadlines. The NP-completepartition problem (Garey and Johnson [14]) can be reduced to 1=pi=1−bisi; di ∈{D1; D2}=Cmax.

Partition. Given a list H = {h1; h2; : : : ; hm} of m integers such that∑m

i=1 hi=2B, can H bepartitioned disjointedly into H1 and H2 such that

∑hi∈H1

hi=∑

hi∈H2hi=B?

Given an instance I of Partition with a list H = {h1; h2; : : : ; hm} and B, construct an instanceII of 1=pi=1− bisi; di ∈{D1; D2}=Cmax as follows:

The set of tasks is TS=R0∪R1∪· · ·∪Rm, where Ri= {Ti;0; Ti;1; Ti;2; : : : ; Ti;m+1}, for 06 i6m.De5ne A1 =4n3 and A2 =A3 =2m+1mmn2B, where n=(m+1)(m+2) is the number of tasks in


Fig. 2. The basic and standard schedule of II for 1=pi =1± bisi; di ∈{D1; D2}=Cmax.

TS. We assume that the processing rates are

b0;0 = b0;1 =0; b0; j=1

A1A3for 26 j6m+ 1;

bi;0 =2imiB− hi

(i + 1)A1A2A3; bi; j=

2imiB(i + 1)A1A2A3

for 16 i6m and 16 j6m+ 1:

De5ne

D1 =2m+ 2−m∑

i=1

(i + 1)bi;1 −m+1∑

j=2

(m+ j)b0; j +B

A1A2A3+

12A1A2A3

and

D2 = n−m∑

i=1

(i + 1)bi;1 −m+1∑

j=2

(m+ j)b0; j −m∑

i=1

(i + 1)(m+ 1)bi;0

−m∑

i=1

m∑

j=1

[(i + 1)(m+ 1) + j]bi; j+1 − mBA1A2A3

+1

2A1A2A3:

The deadlines are d0; j=D1 and di; j=D2, for 16 i6m and 06 j6m + 1. The threshold isG=D2.If there exists a feasible schedule, then, using a strategy similar to the above two reductions,

we get a basic feasible schedule in the form of R0;1(R1;1; R2;1; : : : ; Rm;1)R0;2(R1;2; R2;2; : : : ; Rm;2),where Ri;1 ∪ Ri;2 =Ri, for 06 i6m. Moreover, if a basic schedule is feasible, then we haveR0;1 = {T0;0; T0;1} and there is exactly one task, either Ti;0 or Ti;1, in Ri;1. That is, the scheduleis in a standard form such as (T0;0; T0;1)(T1; k1 ; T2; k2 ; : : : ; Tm;km)(R0;2)(R1;2; R2;2; : : : ; Rm;2), ki=0 or1 (see Fig. 2). The total actual processing time of a schedule with ki=0 is diLerent fromthat of the counterpart case with ki=1 by a multiple of hi. The multiple for each index i isequal, ensured by the structure of II. For the case

∑i∈{j:kj=0} hi ¿B, the sum of processing

rates before R0;2 is so small and the total actual processing time before R0;2 is so large thatR0;2 cannot meet D1. On the other hand, for the case

∑i∈{j:kj=0} hi ¡B, the sum of processing

rates after R0;2 is so small and the total actual processing time after R0;2 is so large that wehave Cmax¿G=D2. Each standard feasible schedule for II corresponds to a solution for I. Asimilar reduction with a detailed proof is presented in Cheng and Ding [8]. Thus, similar toTheorems 1 and 2, we obtain the following theorem:


Theorem 3. The problem of whether there exists a feasible schedule for the problem 1=pi=1− bisi; di ∈{D1; D2}=Cmax is NP-complete in the ordinary sense.

An instance II of 1=pi=1 + bisi; di ∈{D1; D2}=Cmax is constructed as below: TS consists ofn=(m+2)(m+1) tasks: TS= {T0;0; T0;1; : : : ; T0;m+1}∪{T1;0; T1;1; : : : ; T1;m+1}∪···∪{Tm;0; Tm;1; : : : ;Tm;m+1}. De5ne E= n222mm2mB and A=16n3E2. The processing rates are

b0;0 = b0;1 =2EA; b0; j=0 for 26 j6m+ 1;

bi;0 =E + 22m−2i+2m2m−2i+2B+ hi

(i + 1)A; bi; j= bi;0 − hi

(i + 1)Afor 16 i6m; 16 j6m+ 1:

De5ne

D1 =2m+ 2+4E − 2B+ 1

2A+

m∑

i=1

(i + 1)bi;0

and

D2 = n+4E + 2mB+ 1

2A+

m∑

i=1

(i + 1)bi;0 +m∑

i=1

m∑

j=0

[(i + 1)(m+ 1) + j]bi; j+1:

The deadlines are d0; j=D1 and di; j=D2, for 16 i6m and 06 j6m + 1. The threshold isG=D2. Similar to Theorem 3, we obtain the following theorem:

Theorem 4. The problem of whether there exists a feasible schedule for the problem 1=pi=1+ bisi; di ∈{D1; D2}=Cmax is NP-complete in the ordinary sense.

4. Solvable cases

Now we present a polynomial optimal algorithm for the makespan problem 1=pi= a ±bisi; di; bi ∈{B1; B2}=Cmax. In this algorithm, we try to generate a feasible schedule, in which thetasks with the smaller processing rate B1 are scheduled as early as possible and the tasks withthe larger processing rate B2 are scheduled as late as possible (see Fig. 3).Given an instance I of 1=pi= a − bisi; di; bi ∈{B1; B2}=Cmax with m distinct deadlines

D1¡D2¡ · · ·¡Dm. A schedule is called canonical, if it satis5es that the tasks with the samebi are in the earliest due date (EDD) order and the tasks in Ri= {Ti: Ci ∈ (Di−1; Di]}, where16 i6m and D0 =0, are in nondecreasing bi order. If there exists a feasible schedule for agiven instance, then, similar to Lemma 1, we can get the following lemma:

Lemma 4. If there exists a feasible (optimal) schedule for an instance of 1=pi= a−bisi; di; bi ∈{B1; B2}=Cmax; then there exists a canonical feasible (optimal) schedule.

Now, we introduce the algorithm. Schedule the tasks with the same bi in EDD order andwe get two task chains S1 and S2. Insert the tasks in S2 into [0; Dm] from backward as fol-lows: Schedule the tasks in S2 with the largest deadline Dm as a consecutive subschedule S2;m


Fig. 3. The middle and 5nal schedule of Algorithm A.

completed at Dm. Schedule the tasks in S2 with Dm−1 as a consecutive subschedule S2;m−1completed at the minimum of Dm−1 and the stating time of S2;m. By induction, we can insertthe remainder tasks in S2 and generate a schedule in the form of (I1; M1)(I2; M2) · · · (Ik ;Mk),where Ii is the ith idle time and Mi is the ith consecutive subschedule, k6m. This sched-ule is called the middle schedule, denoted as M . If some tasks in S2 cannot be inserted inM , then put them in a late task set, denoted as L. Then, we insert the tasks in S1 intothe idle times in M from forward. Starting from the beginning of I1, insert the tasks ac-cording to the order in S1, until the spare space of I1 cannot hold the next task. If aninserted task cannot meet its deadline, then take it out of the schedule and put it in L.Then, insert the following task continuously. Let S1;1 denote the inserted consecutive sub-schedule. Shift M1 forward to connect with S1;1. Apply the same operations to the remain-ing schedule. We get a 5nal schedule F in the form of (S1;1; M1)(S1;2; M2) · · · (S1; k ; Mk) anda late task set L. The following two lemmas are used to show that the above algorithmis optimal:

Lemma 5. For an instance of 1=pi= a−bisi; di; bi ∈{B1; B2}=Cmax; the 5nal schedule is optimalif the late task set is empty.

Proof. Since L is empty, all tasks with bi=B2 are scheduled in M . From the generation of F; Fis a feasible schedule for II and the tasks in M and S1 are inserted in EDD order, respectively.From Lemma 4, there exists a canonical optimal schedule for II, denoted as S. In S, the taskswith the same bi are also in EDD order. In S before the last task in M1, the tasks with bi=B1are no more than the tasks in S1;1. Otherwise, from the generation of S1;1, the last task in M1cannot meet its deadline in S, a contradiction.In S, advancing the tasks in S1;1 to the beginning of schedule, the resulting schedule S∗ starts

with (S1;1; M1). Since the tasks with the smaller processing rate bi=B1 always shift forward,the makespan of S∗ is no larger than the makespan of S. A task with bi=B2 before a task inS1;1 may shift backward, but its completion time is still no larger than that in F . Hence, S∗ isalso an optimal schedule. Applying the above policy to the remaining part of S, we 5nally getan optimal schedule with the same task order as F . That is, the 5nal schedule F is optimal.

Lemma 6. If the late task set is not empty, then an instance of 1=pi= a−bisi; di; bi ∈{B1; B2}=Cmax does not have a feasible schedule.

Proof. Let T ∗ denote the 5rst late task generated by the algorithm. For the case with b∗=B2,there exists a continuous sub-schedule consisting of (M1; : : : ; Mj), which cannot be inserted in


[0; Dj], according to the generation of M and L. The makespan of this sub-instance is inde-pendent of the order of tasks, since they have identical initial processing times and processingrates. Thus the instance has no feasible schedule.

For the case with b∗=B1, according to the algorithm, the tasks with bi=B2 and the tasksbefore T ∗ in S1 have been inserted in a partial schedule F∗. However, T ∗ cannot be inserted inF∗. On the other hand, if the instance has a feasible schedule, then it has a canonical feasibleschedule F , in which the task order is the same as that of the tasks in F∗, according to theproof in Lemma 5. This means that T ∗ can be inserted in F∗, a contradiction.

From Lemmas 5 and 6, we obtain the following optimal algorithm for 1=pi= a− bisi; di; bi ∈{B1; B2}=Cmax:

Algorithm A. For 1=pi= a− bisi; di; bi ∈{B1; B2}=Cmax:

1. Establish task chains S1 and S2.2. Establish the middle schedule M , the 5nal schedule F and the late task set L.3. If L is empty, then F is optimal. Otherwise, there exists no feasible schedule.

Since each step in Algorithm A can be completed in O(n log n) time, the total running timeof Algorithm A is O(n log n). This method can be readily adapted to deal with 1=pi= a +bisi; di; bi ∈{B1; B2}=Cmax. Thus, we obtain the following theorem:

Theorem 5. The makespan problem 1=pi= a ± bisi; di; bi ∈{B1; B2}=Cmax can be solved inO(n log n) time.

5. Conclusions

In this paper, we have examined the single machine scheduling problems with start timedependent tasks that have identical initial processing times. We have shown that the feasibil-ity problems, 1=pi=1 ± bisi; di=Cmax, are strongly NP-complete, and the feasibility problemswith two distinct deadlines, 1=pi=1± bisi; di ∈{D1; D2}=Cmax, are NP-complete in the ordinarysense. As usually the case, all these results can be adapted to their corresponding schedul-ing problems with the popular regular performance measures of total Pow time and maximumlateness.

On the other hand, we have shown that the makespan problem 1=pi= a±bisi; di; bi ∈{B1; B2}=Cmax can be solved in O(n log n) time. For the general makespan problem 1=pi= a±bisi; di=Cmax,it is not diKcult to develop a heuristic based on the intuition of Algorithm A or designan exact algorithm by enumerating all canonical schedules. The combination of these ap-proaches should be eLective on average, but it is hard to get an ideal upper bound for theerror or running time. For further research, it would be interesting to design approximationalgorithms with a constant worst-case bound for the problems or prove infeasibility of suchapproximations.


Acknowledgements

We are grateful to two anonymous referees for their constructive comments on an earlier ver-sion of this paper. This research is supported in part by The Hong Kong Polytechnic Universityunder grant number 350=239. The 5rst author is partially supported by the Croucher Foundationunder a Croucher Senior Research Fellowship.

References

[1] Graham RL, Lawler EL, Lenstra JK, Rinnooy Kan AHG. Optimization and approximation in deterministicsequencing and scheduling: a survey. Annals of Discrete Mathematics 1976;5:287–326.

[2] Gupta JND, Gupta SK. Single facility scheduling with nonlinear processing times. Computers and IndustrialEngineering 1988;14:387–93.

[3] Browne S, Yechiali U. Scheduling deteriorating jobs on a single processor. Operations Research 1990;38:495–8.

[4] Cheng TCE, Ding Q. Single machine scheduling with deadlines and increasing rates of processing times. ActaInformatica 2000;36:673–92.

[5] Kononov AV. Scheduling problems with linear increasing processing times. Operations Research Proceedings1996: Selected Papers of the Symposium on Operations Research (SOR ‘96), Braunschweig, September3–6, 1996. Berlin: Springer, 1997, pp. 208–12.

[6] Ho KI-J, Leung JY-T, Wei W-D. Complexity of scheduling tasks with time-dependent execution times.Information Processing Letters 1993;48:315–20.

[7] Cheng TCE, Ding Q. The time dependent machine makespan problem is strongly NP-complete. Computersand Operations Research 1999;26:749–54.

[8] Cheng TCE, Ding Q. The complexity of single machine scheduling with two distinct deadlines and identicaldecreasing rates of processing times. Computers and Mathematics with Applications 1998;35:95–100.

[9] Woeginger GJ. Scheduling with time dependent execution times. Information Processing Letters 1995;54:155–6.

[10] Chen Z-L. A note on single-processor scheduling with time-dependent execution times. Operations ResearchLetters 1995;17:127–9.

[11] Alidaee B, Womer NK. Scheduling with time dependent processing times: review and extensions. Journal ofthe Operational Research Society 1999;50:711–20.

[12] Tanaev VS, Gordon VS, Shafransky YM. Scheduling theory, single-stage systems. Dordrecht: KluwerAcademic Publishers, 1994.

[13] Cheng TCE, Ding Q, Kovalyov MY. Scheduling jobs with linearly decreasing processing times. WorkingPaper, Faculty of Business and Information Systems. Hong Kong: The Hong Kong Polytechnic University,1998.

[14] Garey MR, Johnson DS. Computers and intractability: a Guide to the Theory of NP-completeness. New York:W.H. Freeman and Co., 1979.

T.C. Edwin Cheng received his Ph.D. in Operational Research from the University of Cambridge, England. Heis Chair Professor of Management at The Hong Kong Polytechnic University, Hong Kong. His research interests arein operations management and operations research, and has regularly published papers in this and other academicjournals.

Q. Ding received his Ph.D. in Management from The Hong Kong Polytechnic University. His research interestsare in scheduling and optimization. He has published several papers in this and other academic journals.

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Scheduling start time dependent tasks with deadlines and identical initial processing times on a single machine