Information Processing Letters 65 (1998) 75-79

The complexity of scheduling starting time dependent with release times T.C.E. Cheng a**, Q. Dingb

tasks

a Office of the Vice President (Research & Postgraduate Studies), The Hong Kong Polytechnic University, Hung Horn, Kowloon, Hong Kong

b Department of Management, The Hong Kong Polytechnic University9 Hung Horn, Kowloon, Hong Kong

Received 18 August 1997; revised 18 November 1997

Communicated by W.M. Turski

Abstract

We consider a family of problems of scheduling a set of starting time dependent tasks with release times and linearly increasing/decreasing processing rates on a single machine to minimize the makespan. We first present an equivalence relationship between several pairs of problems. Based on this relationship, we show that the makespan problem with arbitrary release times and identical increasing processing rates is strongly NP-complete and the corresponding case with only one non-zero release time is at least NP-complete in the ordinary sense. On the other hand, the makespan problem with arbitrary release times and identical decreasing processing rates is solvable in 0( n6 log n) time by a dynamic programming algorithm. Using a different approach, we also show that, when the normal processing times are identical, the makespan problem with arbitrary release times and increasing/decreasing processing rates is strongly NP-complete and the corresponding case with only one non-zero release time is at least NP-complete in the ordinary sense. @ 1998 Elsevier Science B.V.

Keywords: Scheduling; Sequencing; Time dependence; Release time; Computational complexity

1. Introduction

Machine scheduling problems with starting time de- pendent tasks have received increasing attention in re- cent years. In this paper, we focus on a family of prob- lems with linear time dependent functions. Formally, these problems can be stated as follows. A task sys- tern consists of n independent tasks and is denoted by

TS = ({z}, {ri}, {di}, {ai}, {wi}). Each task Ti is as- sociated with a release time ri, a deadline di, a normal processing time ai > 0 and an increasing/decreasing processing rate Wi 2 0. If the task z is scheduled to

start at time si, then its actual processing time is pi = ai i wjsi > 0.

Given a nonpreemptive schedule S, S is feasible if each c is completely processed in the interval [ ri, di] .

A task system TS is feasible if there is a feasible schedule for it. Let G denote a given threshold. The makespan problem is to decide whether there is a fea- sible schedule for TS on a single machine with C,,,, =

maxi++ < G, where Ci is the completion time of 7;: in S. Adopting the three-field notation proposed by Graham et al. [ 71 to describe classical scheduling problems, we denote the makespan problem as

* Corresponding author. Email: mscheng@poIyu.edu.hk. l/pi = aj i wisi, t-j, c&/C,,.

0020-0190/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved.

PII SOO20-0190(97)00195-6

16 T.C.E. Cheng. Q. Ding/Information Processing Letters 65 II 998) 75-79

Similarly, the flow time problem is to decide whether there exists a feasible schedule for TS on a single machine with CF,, Ci < G and is denoted by

l/pi = Ui f WiSi, Tj, di/ C Ci.

The maximum lateness problem is to decide whether there exists a feasible schedule for TS on a single machine with L,,, = maxt<i+{Ci - di} f G and is

denoted by

l/pi = Ui * WiSi, Ti, di/Lm.

Let Vi = 0 if Ci 6 di, Ui = 1 otherwise. The number of late tasks problem is to decide whether there exists a nonpreemptive schedule for TS on a single machine with xy=, Ui 6 G and is denoted by

For the complexity of l/pi = ai f WiSi, ri, di/Cmax, since the classical problem l/ri, di/Cmax is known to be strongly NP-complete (see Graham et al. [ 71) , we only need to consider the computational complexity of the cases with identical release times or identical

deadlines. Several papers have studied the cases with identical release times. A complete classification of the

computational complexities of this family of problems is given by Cheng and Ding [ 51.

In this paper, we focus on the makespan problems with release times and linearly increasing/decreasing processing rates on a single machine. In Section 2, we present a symmetry reduction for several pairs of the makespan problems between the models pi = ~i + WiSi > 0 and pi = Ui - wisi 3 0. In Section 3, we show that the makespan problem with arbitrary re- lease times and identical increasing processing rates, denoted as l/pi = ai + wsi, ri, /C,,,,, is strongly NP- complete, and the corresponding case with only one non-zero release time, denoted as 1 /pi = ui + WSi, ri E

(0, R}/Gnzm is at least NP-complete in the ordinary sense. We also give a dynamic programming algo- rithm which solves the makespan problem with arbi- trary release times and identical decreasing process- ing rates, denoted as l/pi = ui - wsi,ri/Cmax, in 0(n6 logn) time. Moreover, for the case with unit processing times, i.e. ai = 1, we demonstrate that the makespan problem with arbitrary release times and ar- bitrary increasing processing rates, denoted as l/pi =

1 +WiSi7 ri/Gax, is strongly NP-complete and the cor- responding case with only one non-zero release time,

denoted as l/pi = 1 + WiSi,ri E {O,R}/Cmm, is at least NP-complete in the ordinary sense. Similarly, the corresponding cases with decreasing processing rates, denoted as l/pi = 1 - WiSi, ri/C,, and l/pi = 1 - WiSi, ri E (0, R}/Cmax, are strongly NP-complete and at least NP-complete in the ordinary sense, re- spectively.

This family of scheduling problems has many real- world applications, including national defence, fire fighting, steel production, scheduling of resources to control epidemics, and maintenance scheduling, where any delay in commencing a task may result in an ad- dition or reduction in effort (time, cost, etc.) to ac-

complish the task (see, for example, [ 1,8,9] ) . Since the processing rates are not integers in many

practical cases, all parameters are allowed to be ratio- nal numbers. Without affecting the NP-completeness results, we assume that the threshold G satisfies the conditions rnaxtGi+(ri> < G < maxIGiG,( For the model pi = ui + wisi > 0, we assume wi 2 0. For the model pi = ai - WiSi 2 0, we assume 0 < Wj < 1 and 0 f widi < Ui < di (or wi cTzI uj < Ui, for the case with no deadline restriction). These assump-

tions are reasonable and indeed help eliminate some uninteresting cases (see [ 91) . Discussions and jus- tifications for these assumptions are provided in the remainder of this paper when necessary.

2. Symmetry reduction

In the classical scheduling theory, the problems

1 lrj/Gax and I/ /&,, are equivalent in the sense that an algorithm for one problem can be used to solve the other problem. This equivalence relationship can be generalized to the following scheduling problems with starting time dependent tasks.

Given a schedule S for an instance of l/pi = ui - wisi, ri/Cmax. it can be taken as a schedule for a corre- sponding instance of l/pi = ui + wisi, di/C,,,, viewed from the reverse direction. Based on this observa- tion, we show that there exists a symmetry reduc- tion between l/pi = LZ~ - wisi, ri/Cmax and l/pi = Ui + WiSi, di/Cma as fOllOWS.

Given an instance Z of 1 /pi = ai - wisi, ri/Cma con- sistingofatasksystemTS’= ({T~},{r~},{u~},{w~))

T.C.E, Cheng, Q. Ding/Information Processing Letters 65 (1998) 75-Z 77

and a threshold G > maxt<i+( ri), we construct an instance II of l/pi = ai + WiSi, d;/C,, consisting Of a task system TS” = ({T,“}, {r[‘}, {a;}, {WY}) and a threshold G, where the task T/’ corresponds to T”, the normal processing time is a; = (ai - w:G) /( 1 - w:) , the increasing processing rate is wr = wi/( 1 -w[) and the deadline is dj’ = G - r-i, for 1 < i < n. It is easy to see that the above construction can be achieved in polynomial time.

Given a nonpreemptive schedule S, if there are no idle times between the tasks in S, then S is called a consecutive schedule. Let (. . _) denote a schedule. Now we show that I has a solution if and only if II has a solution.

Ifpart: Given a solution SA = (T,‘, Ti, . . . , T,‘) for I, there may exist some idle times between the tasks

in Sb. Move the last task TL such that its comple- tion time is exactly at the time G and shift the other tasks from TL _ 1 to Ti to produce a consecutive sched- ule S’ = (T:, Ti, . . . , T,‘). It is easy to see that S’ is

also a solution for I. Let si and C: denote the start- ing time and completion time of T/ in S’, respectively. We have _ri > ri and .si + unzips = G. Further-

more,

”

c p;=G-si<G-ri, for1 <i<n. (1)

,j=i

Starting at time 0, we construct a consecutive sched-

ule S” = (T,“,T,“_,,. . . , Ti”) for II, where q’ is the

corresponding task of T/. Let sy and CF denote the starting time and the completion time of T” in S”, re- spectively. By induction, we show that p:’ = p[ and Ci” < d;’ as follows.

First, we consider the basic case i = n. Since sz = 0,

from the definition of ui and Ci = G = sk +a; -WAS;, it is easy to see that

a; - w;G pj’,‘=ar= 1_w,

n

= a; - wL(sL +a: - WLsL)

1 - w:,

=a: - w:sA =pA. (2)

From ( 1)) (2) and the definition of di, we obtain

C:=py=pL <G-rk=dF. (3)

Suppose that we have pfl = pi and Ci” < dr, for k + 1 < i < n. NOW WA show that the results remain true for the case i = k.

Since si = Cy=,+, py, si + x7=, pJ = G and p; =

p,‘, for k + 1 < i 6 n, from the definitions of ut and wz, similarly to (2)) we have

pp = u!J + w;s;

w;G --- 1 - w;

=p;. (4)

From ( l), (4) and the definition of d;, similarly

to (3)) we have C{ = Cy=, p/ 6 di.

By induction, we see that C(’ 6 di’, for n >, i > 1,

and C[ < G - r-i < G. Thus, we obtain that S” is a solution for II.

Only ifpart: Given a solution S” = (T:, T,“_, , . . . , Ti’) for ZZ, construct a consecutive schedule S’ =

(T,‘,T;, , . . , T,‘) for I such that the makespan of S’ is exactly G, where T/ is the corresponding task of T”.

Following the same approach as in the if part, we can show that S’ is a solution for I. Thus, we obtain the following lemma immediately.

Lemma 1. The problem 1 /pi = ai - WiSi, ri/Cmax cun

be reduced to the problem l/pi = ai + wisi, di/Cmax.

By the symmetry of the models, using the same ap- proach as in Lemma 1, we obtain the following lemma.

Lemma 2. Theproblem l/pi = ai+wisi, di/Cmax can

be reduced to the problem l/pi = ai - WiSi, ri/C,,.

The pair of problems mutually reducible to each other as stated in Lemmas 1 and 2 indeed possess an equivalence relationship in the sense that an algorithm for one problem can be used to solve the other prob- lem. We call such a relationship a symmetry reduction.

Thus, we obtain the following theorem.

Theorem 3. There exists a symmetry reduction be-

tween the problem l/pi = Ui - WiSi, ri/C,, and the problem 1 /pi = Ui + WiSi, dt/Cm,.

By the symmetry nature of the models, using the same approach as in Theorem 3, we can prove that

78 ZCX. Cheng, Q. Ding/Information Processing Letters 65 (1998) 75-79

there also exists a symmetry reduction between 1 /p; = Cheng and Ding [3] have given a dynamic pro-

U; + W;s;, r-i/C,, and l/p; = U; - W;s;, d;/Cmax. Note gramming algorithm (Rule DP) which solves the

that each increasing processing rate in II is determined problem 1 /p; = u; + Ws;, d;/C,,,, in 0( n5) time in

only by the corresponding decreasing rate in I. If the the following manner: Given an initial consecutive

decreasing processing rates in I are identical, then schedule obtained by the Shortest Processing Time the increasing processing rates in II are also identical. (SPT) rule, determine a task order and then obtain a Meanwhile, it is easy to see that the reduction used new consecutive iterative schedule using an approach

in Lemmas 1 and 2 can be adapted to the pair of similar to Smith’s backward rule for 1 /d;/ C C; (see

problems l/p; = a;fws;, r; E (0, R}/Cmax and l/p; = [7]). Repeat the iterative procedure until a feasible

a; T Ws;, d; E (01, D*}/Cmax. Thus from Theorem 3, schedule is found or it is confirmed that no solution we obtain the following corollary. exists.

Corollary 4. There exists a symmetry reduction be-

tween l/p; = a; + W;S;,r;/Cmax und l/p; = a; -

w;s;, d;/Cm;lx, l/p; = a;fws;, r;/Cmw and l/p; = a;+

WSir ~i/Gnax9 und l/p; = a; f Ws;, r; E {O, R}/Cmm

und l/p; = a; F ws;,d; E {Dl,&}/Cmax, respec- tively.

We may change iteratively the value of G and apply Rule DP to identify whether there exists a solution for TS’. When we obtain the minimum value of G for

which there exists a solution for TS’, the corresponding schedule constructed in the proof of Lemma 1 is a solution for TS.

Using the above results, we will easily establish the NP-completeness results for both l/p; = a; +

WSi9 ri/Gx and l/p; = U; + ws;yr; E {O,R}/C,,

and give a polynomial algorithm for l/p; = a; - ws;, r;/Cmax in the next section.

Let u be the denominator of w. Let B, and B2 de- note the lower bound and upper bound of u*-‘G, re- spectively. We propose the following algorithm.

3. Complexities

Using an approach similar to that of Ho et al. [ 91, Cheng and Ding [ 21 have shown that l/p; = a; -

WS;, d;/Cm, is strongly NP-complete. Using an ap-

proach similar to that of Du and Leung [ 61, Cheng and Ding [4] have shown that l/p; = a; - Ws;, d; E

(01, D2}/Cmax is at least NP-complete in the ordinary sense. From these results and Corollary 4, we obtain the following theorem immediately.

Algorithm A. For I/p; = a; - WS;, r;/Cmax 1. BI := max(r;) and

B2 :=u “-‘(Cyzl ai +ma(Q));

2. WHILE B2 - Bl > 1 DO BEGIN

3. G’ := [(Bt + B2)/2]; G := G’/u”-t; 4. Use Rule DP to find an optimal schedule

for TS’ and G,

5. IF there exists an optimal schedule THEN B, := G’

ELSE B2 := G’ END;

6. G := B2/u n-1. ,

7. STOP

Theorem 5. The problem 1 /p; = u; + ws;, r;/Cmax is

strongly NP-complete, and the problem l/p; = u; +

ws;, r; E (0, R}/Cmax is at least NP-complete in the

ordinary sense.

Now we introduce a dynamic programming algo- rithm for 1 /p; = u; - WS;, r;/Cmax. Given a task sys- tem TS = ({2;:}, {r;}, {a;}, {w;}) of l/p; = a; - ws;,

ri/Gax7 from Corollary 4, there exists a correspond-

ing task system TS’ of l/p; = a; + ws;, d;/C,, such that TS has a feasible schedule with C,, < G if and only if TS’ has a feasible schedule with C,, < G.

The value of G can be determined in log,( B2 -

B1 ) time in Step 2, which needs at most 0( n log n) time. In each iteration, at most 0( n’) time is needed to determine whether there exists a feasible schedule for TS’ and G (see [3]). Therefore, the total time complexity of Algorithm A is 0( n6 log n) .

Theorem 6. The problem l/p; = a; - WS;, r;/Cmax can be solved in O( n6 log n) time.

In the reduction presented in Section 2, the normal processing time of a task in II depends on both the

T.C.E. Cheng, Q. Ding/information Processing Letters 65 (1998) 75-79 19

normal processing time and the decreasing processing rate of its corresponding task in I. Thus, the normal processing times of the tasks in II are not identical, even if the normal processing times of the tasks in I are identical. Thus, the symmetry reduction cannot be adopted for the problems with identical normal pro- cessing times. Therefore, we have to use a different approach to show their NP-completeness results.

Cheng and Ding [ 51 have given an intricate reduc-

tion to show that both 1 /pi = 1 - WSi, di/Cm, and

l/pi = 1 + WiSiv di/Cmax are strongly NP-complete.

They also gave another very complicated reduction to show that both l/pi = 1 + Wisi, di E {Dt , &)/Cmx and l/pi = 1 - Wisi,di E (01, &}/Cmax are at least

NP-complete in the ordinary sense. By some adapta- tion of those proofs, we can obtain the following the- orem.

Theorem 7. Both l/pi = 1 + WiSi, ri/Cmax and

l/pi = 1 - WiSirri/Cmax are strongly NP-complete.

Both l/pi = 1 + Wisi,ri E (0, R}/Cm,x and l/pi = 1 - Wisi, ri E (0, R}/C,,, are at least W-complete in the ordinary sense.

4. Conclusions

In this paper, we have examined the single machine makespan scheduling problems involving starting time dependent tasks with release times and linearly in- creasing/decreasing processing rates. First, we have presented a symmetry reduction for several pairs of the problems between the models pi = ai f WiSi 2 0 and pi = ai - wisi 3 0. Based on this property, we have

shown that l/pi = ai + WS~, ri/C,, is strongly NP-

complete and l/pi = ai + WSi, ri E (0, R}/Cmax is at least NP-complete in the ordinary sense. We have also

given a dynamic programming algorithm for solving

l/pi = Ui - WSi, ri/C,, in 0( n6 log n) time. Using

a different approach, we have shown that both l/pi =

1 + WiSi, ri/Gax and l/pi = 1 - wsi, ri/Cmax are strongly NP-complete and both l/pi = 1 + WiSi, ri E (0, R}/C’,,, and l/pi = 1 -WiSi, ri E (0, R}/Cmm are

at least NP-complete in the ordinary sense. Thus, we have solved a series of open problems in the literature and given a sharp boundary delineating the complex- ity of the problems.

Acknowledgements

The research was supported in part by The Hong Kong Polytechnic University under grant number 3501239. We are grateful to an anonymous referee for his helpful comments on an earlier version of the paper.

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