A survey of results for sequencing problems with controllable processing times

Discrete Applied Mathematics 26 (1990) 271-287

North-Holland

271

A SURVEY OF RESULTS FOR SEQUENCING PROBLEMS WITH CONTROLLABLE PROCESSING TIMES

Eugeniusz NOWICKI and Stanislaw ZDRZALKA

Technical Universit]~ of Wrockrw, Institute ofEngineering Cybernetics, ul. Janiszewskiego 11117, 50-372 Wroclaw, Poland

Received 4 June 1987

Revised 30 June 1988

The paper deals with the sequencing problems in which job processing times, along with a pro-

cessing order, are decision variables having their own associated linearly varying costs. The existing

results in this area are surveyed and some new results are provided. In the paper, an attention

is focussed on the computational complexity aspects, polynomial algorithms and the worst-case

analysis of approximation algorithms.

1. Introduction

The problems of sequencing and scheduling have been the subject of a con-

siderable number of papers in the recent literature. Virtually all work in this area

has considered fixed job processing times or fixed processing time distributions (in

stochastic scheduling problems) as the models of jobs (operations). In the real-life

applications, apart from the machines, processing a job requires additional resources,

such as facilities, manpower, funds, and so on, which implies that jobs can often

be accomplished in shorter or longer durations by increasing or decreasing the addi-

tional resources. Generally, there are situations where compressing a job is possible,

but it entails an extra cost, and such an action would be rational only if this addi-

tional cost is compensated by the gains from job completion at an earlier time. This

fact is commonly recognized in the area of project planning and control, where pro-

blems of time-cost trade-offs and least cost scheduling have received a considerable

attention (see, e.g. [2,9]). Despite their importance and similarity to project planning,

the problems of sequencing and scheduling with variable but controllable job pro-

cessing times have been studied only in few papers. A close examination, including

the analysis of computational complexity, of the standard sequencing models with

controllable job processing times has been initiated by Vickson [19,20] earlier, such

a type of sequencing models has been considered by Lawler and Moore [IO].

In this paper we survey the existing results for the sequencing problems in which

job processing times are additional decision variables, paying special attention to the

computational complexity aspects, and optimization and approximation algorithms.

A second aim of the paper is an analysis of other sequencing problems of this type,

not investigated in the literature.

0166-218X/90/$3.50 0 1990, Elsevier Science Publishers B.V. (North-Holland)

212 E. Nowicki, S. Zdrzaika

2. Notation and preliminaries

In this section, we introduce basic notation and formulate the problems con-

sidered in this paper. Since each of them is a generalization of the corresponding

classical sequencing problem, we fully adopt notation, definitions and assumptions

exploited in this field, see survey of Graham et al. [4].

The results presented in this paper concern mainly single machine and permuta-

tion flow shop sequencing models and therefore we assume the latter as a basic

model for our further considerations; each of single machine models may be treated

as its special case.

Let J=(l,..., n} denote the set of jobs to be scheduled and M= { 1, . . . , m}, the

set of machines. It is assumed that each machine processes a job without interrup-

tion and that a processing time of job j on machine i is ai, -xjj, Olxijl uij, where

xti is the time by which the “normal” processing time aij has been compressed

(shortened), and uij is the maximum compression; uijs aij for all i and j. The usual

assumption of the flow shop model is that each job is performed first on machine

1, next on machine 2, and so on, until it completes execution on machine m. In the

permutation flow shop, we additionally assume that the job processing orders on

all the machines are identical. Let x = (x, 1, . . . , xln, . . . ,x, ,, . . . ,x,,) be a vector of

compressions, and 7c, a permutation of J defining the job processing order; n(i) denotes the job which is in position i in 7~. We denote by X the set of all feasible

compressions, X= {x: 0 <xtis uii, i E M, j E .I}, and by 17 the set of all permuta-

tions of J. We assume that a pair (x, 71) uniquely determines a completion time Cj

of each job j E J. This assumption is justified, since throughout this paper we shall

confine our attention to the regular measures of performance. For such measures,

defined as nondecreasing functions of job completion times, it can be shown that

there is no advantage in considering idle times. In consequence, each schedule is

completely defined by job processing times and a permutation of J. Two general performance measures are considered in the sequencing problems

with controllable job processing times.

The first one is based on the completion times Cj and therefore it will be called

a completion cost. Given x and 71, the completion cost is denoted by F,(x, x), and

defined as one of the standard measures used in the sequencing and scheduling

theory. Let fj be a real nondecreasing function of Cj, and

fm,,(G 3 . . . > Cn)=maXfj(Cj>, c “fj<c,, .“9 cn>= C fitcj). jcJ je/

Then, F, E {f,,, C fj}. The important completion costs in this paper are given by

the following functions fj :f,(Cj) = Cj, the completion time; J;(C’j) = Cj - dj 4 Lj,

the lateness; J;(Cj) = max{O,Lj} b Tj, the tardiness; dj is the due date of job j.

The second measure is the total cost of compressions

Sequencing problems with controllable processing times 273

where cij is the unit cost of compression of job j on machine i. Fz(x) will be called

a processing cost.

For a pair (x, n), a total scheduling cost K(x, rr) is defined by

K(x,n)=F,(x,n)+F2(x).

The main problem considered in this paper is formulated as follows.

Pl (least cost scheduling): Find x* EX and TC* E ZZ such that

K(x*,n*)=min{K(x,n): xEX, Nan}.

Other formulations are obvious:

P2 (least processing cost scheduling under limited completion cost): For given r,

find x*EX and rr*EZ7 such that

F,(x*)=min{F,(x): F,(x,7r)<r, XEX, nE17).

P3 (leas; completion cost scheduling under limited processing cost): For given K,

find x* E X and n * E 17 such that

Fi(x*,-n*)=min{F,(x,rc): F2(x)5~, XEX, n~fl}.

In order to formulate a more general problem, we need the following definitions.

Let D be a set of all ordered pairs (ai, Q) such that if (a,, az) E 52, then there exist

XE X and rc in such that (Y~ =F,(x, rr) and a2 =F2(x). We say that point (ai, a2) E Q

is efficient if there exists no point ((w;, ai) E B such that all Ui, i = 1,2, and o; < Q;

for at least one i. Denote by E the set of all efficient points in Q; the set E is often

called an efficient frontier in (F,,F,) space.

P4 (bicriterion scheduling): Find the set of all efficient points E and the set of

pairs (x, rr), XE X, II E I7, such that if (ai, al) E E, then a, = F,(x, n) and a2 =F2(x).

In the paper we are mainly concerned with the problem PI. Since each Pl pro-

blem is a generalization of certain classical scheduling problem, it is at least as hard,

in terms of the NP-completeness theory, as the original one. Therefore, one of the

purposes is to look for the cases of PI in which such a generalization “pushes the

problem through” from the polynomial class to the class of NP-hard problems. We

also indicate cases of Pl-P4 for which polynomial algorithms exist and show some

results of the worst-case analysis of approximation algorithms.

In the sequel, we say that we consider problem Pl (P2, P3, P4) for a scheduling

model . ) . ) . , where . ) . ) . is the three-field problem classification introduced in

[4], if the machine environment and job characteristics are given by the first and

the second field, respectively, and the completion cost Fi is of the form given by

the third field. For some instance of Pl, we denote by K* and KH the minimum

value of K and the value K(xH, nH) obtained by an approximation algorithm H.

3. Single machine scheduling

We begin with the problems Pl-P4 for single machine scheduling with completion

274 E. Nowicki, S. Zdrzalka

cost functions of the form f,,,. Here we suppress index i, i.e., we use the notation

aj, Xj, Uj, and Cj.

A crucial property of those models is that there exists a permutation of * E l7, in-

dependent of x, such that for each XE X, F,(x, II *) <F,(x, n) for all 7t E 17. In the

case of 1 / 1 T,,, and 1 / 1 L,,,, TI* is obtained by sequencing the jobs in the order

of nondecreasing due dates dj, and in 1 / rj 1 Cm,,, z * is given by a sequence arranged

in the order of nondecreasing release dates rj (Jackson’s rule [6]). This property im-

plies that the corresponding optimization problems Pl, P2, and P3 become linear

programs; Z* satisfying the above condition is an optimal permutation for PI, P2,

and P3. Moreover, 7t* generates the whole set of efficient points E in P4.

The problem Pl for 1 / I 7”,, has been discussed by Vickson [ 191. He provides an

algorithm with asymptotic worst-case running time 0(n2) basing on the fact that

the linear program min(K(x, n*): x E X} reduces to a production-inventory pro-

blem. He also shows that the two-mode version of this problem in which each job

can either be processed in a normal time at zero cost or in maximally compressed

time at positive cost is NP-hard.

An interesting feature of the models 1 1 I T,,,, 1 ) rj 1 C,,, and 1 ) ( L,,, is that

for each of them there exists a greedy algorithm which solves P4 in O(n’) time. In

fact, in these cases, the efficient frontier is a piecewise-linear curve with at most

n + 1 breakpoints, and the greedy algorithms compute only the breakpoints; the effi-

cient frontier may be obtained by connecting those points. After slight modifica-

tions, not increasing the computational complexity, those algorithms may be applied

for solving Pl, P2 and P3. Such an algorithm for 1 1 1 T,,, has been given by van

Wassenhove and Baker [21]. It can also be applied to the problem P4 (Pl, P2, P3)

for 1 / prec 1 T,,, by using updated due dates [8]. The proof, that the algorithm

traces out the efficient frontier, given by van Wassenhove and Baker is indirect.

They show this using an equivalent linear programming formulation. In what

fohows, we present an algorithm for finding an efficient frontier E in sequencing

models 1 1 rj / C,,, and 1 ) / L,,,, and provide a direct proof based on the notion of

a supporting line.

The function Fr in 1 ( rj ( C,,, has the following form,

n

F, (x, TT) = max rncr)+ C (a,(j)-xz(j)) . j&i

1 ( / L,,,, we have

FI(x~ ~1~ ,z;$~ jfi, (an(j) -x,(j)) - 4(i) .

Assuming that D = maxj, J dj, rj = D - dj and a(j) = n(n + 1 -j), j E J, we get


FL@, n) = max (

‘0(i)+ i (“O(j)-xO(j)) -D, Isicn j=i 1

which implies that the problems Pl-P4 for 1 / ) L,,, are equivalent to the ones for

L l’j I Gmx* The same refers to the models 1 / prec 1 L,, and 1 ) rj, prec / Cm,,. Now we give an algorithm for finding efficient frontier E for 1 1 rj ( Cm,,.

Algorithm 1

Step 0. Renumber the jobs such that rt I r2 I ... I r, .

Step 1. Set x:=0, d;=ri+CJ=iaj for i= 1, . . ..n. TO=maxr,i,.d;, K’=O, and

k=max(i: di= TO}.

Step 2. If X: = Ui for i = k, . . . , n, then stop.

Step 3. Find the greatest 1 such that cl=min(ci: ksiln,x,?#ui}.

Step 4. If u,< T”-max,,i5nd,, then set XP :=u/, To := TO-X:, di :=di-X,? for

i=k ,...,1,Ko:=Ko+~~~~,andgotoStep2;otherwise,setx~:=To-max~,i~nni,

To := TO-x?, K” :=K’+c~x:, find k=max{i: f<iln, ni= To}, and go to Step 2.

Let r’(i), K’(i), x’(i) and k(i) be the values of To, K”, x0 and k, respectively,

generated in iteration i, i = 1, . . . , s; Step 1 of the algorithm is treated as iteration 1.

Denote by [(cx;, a;), (a:, rxi)] a line segment between the points ((r;, cr;), (a;, cr;) E fR2.

Theorem 1. The set of all efficient points E in the problem P4 for 1 1 rj 1 Cm,, is given by

S-l

E= U [(T’(i), K’(i)), (T’(i+ l), K’(i+ l))]. i=l

Proof. Let rc* be the permutation obtained by sequencing the jobs in the order of

nondecreasing rj, and assume that the jobs are renumbered such that rc * = (1, . . . , n). The algorithm starts from the point T’(1) =F,(O, 7c *), K’(1) = 0, and gradually

decreases F,(x, rc*) and increases F,(x) in order to find the next points (T’(i), K’(i)). The idea of the proof consists in showing that each pair of such consecutive points

determines a supporting line to Q.

Denote C,,,(x) =F,(x, n*) for XEX. The following properties follow from the

construction of the algorithm:

(i) To(i) = C,,,(x’(i)), K’(i) =F2(xo(i)), i= 1, . . . ,s;

(ii) To(s) = C,,&U), where u = (u,, . . . , u,);

(iii) each line segment [(T’(i),K’(i)),(T’(i+ l),K”(i+ l))], i= 1, . . ..s- 1 is a sub-

set of 52.

The properties (i) and (iii) are obvious. In order to show that (ii) holds it suffices

to note that, in accordance with Step 2 of the algorithm,

0 To@) = G&X 6)) = rkcs) (aj- uj) 5 CIII~X(~>~ s )

216 E. Nowicki, S. Zdrzdka

This and the inequality To(s) 2 Cmax(u) imply (ii).

In view of (i)-(iii) and the fact that for each (al, az) E Q, czI L C,,&U) and a2 L 0, in

order to prove the theorem it suffices to show that each pair of points (T’(i), K’(i)),

(T’(i+ l),K”(i+ 1)) determines a supporting line to the set Q. The line determined

by this pair of points is defined by the equality

where

(r2-K*(i)=cr(T0(i)-cxl), (1)

C,=IlliIl{Cj: k(i)ljlTZ, X,!(i)#Uj}.

By (i) and the inequality C,,,(x) <F,(x, n) for each XE X and n E 17, the line given

by (I) is a supporting line to the set Q if and only if

i Cj<xj - x,0(i)) 1 CAGdX”(i)) - Cm,,(X)) j=l

for each XEX. In what follows we show, by induction, that (2) is satisfied for each

i=l , . . . ..A?- 1.

Suppose that i= 1 and XEX. By the algorithm, x’(1) = 0, and in consequence

where the second inequality follows from the definition of I and the third one, from

the fact that the time by which &+.(x0(l)) is shortened does not exceed the sum of

compressions over the jobs satisfying j L k(1); the compression of any job j< k(1)

does not affect the completion time C,,,,,(x”(l)).

Now suppose that (2) holds for iteration i- 1, where i I 2. This implies that

i Cj(Xj -xjow> 2 M&?,(xO(0) - Gl,,(x)> (3) j=l

for each XEX, where

cr=min{cj: k(i- l)~j<n, xg(i- l)+Uj}.

It follows from the construction of the algorithm that:

(iv) c,, I cl;

(v) if j 2 k(i), then either x:(i) = 0 or x;(i) = Uj;

(vi) k(i)rk(i-1) for i=2,...,s.

Suppose that XEX, and denote p= C,,,,,(x”(i))- C,,,(x). Let 0 be the difference

between the starting time of job k(i) and the release date rk(i), for the processing

times aj - Xj, j E J. It is clear that only the compression of job j with j 2 k(i) car

shorten the completion time Cm,,(xo(i)). Therefore,

i (Xj-Xjo(i))Zfi+O. j=k(i)

(4:


Let us define the sets:

B’={j: k(i)ljln, xj<Xp(i), Uj#O},

B”={j: k(i)rjsn, Xj>Xj”(i), Uj#O}.

In view of (v), forj E B’, x:(i) = Uj, and therefore, by (vi), cI > Cj forj E B’. Similarly,

for j E B”, x;(i) = 0, and by the definition of I, C,I cj for j E B”. This and (4) imply

the inequality:

i Cj(Xj-X~(i))'C,J~,(x/-X~(i))+ctj~,,(xI_-xY(')) j=k(i)

zc,p+ c,cT. (5)

Let ~j=Xj forj=l,..., k(i) - 1, and ij =x:(i) for j = k(i), . . . , n. By (3) and the fact

that Cm,,(xo(i>) - Cm,,(~) = -CT, we obtain

which together with (5) an (iv) shows that (2) holds for iteration i. This completes

the proof. 0

Modifying Step 2 in Algorithm 1 by introducing “If ci 2 1 or x1! = ui for kr is n, then stop” yields the algorithm for PI; in this case K* = To + K”. This is implied

by Theorem I and the form of the objective function in the problem PI.

Algorithm 1 can also be applied to the problem P4 (Pl, P2, P3) for 1 1 rj, prec / C,,,

after appropriate modification of the release dates; see [8].

In this scheduling problem,

Fl(X9 n)= max fn(i) f: (a,(j)-X,(j)) . lsi5n ( j=l >

The case when the functions fj satisfy the condition that there exists a permutation

n such that f,cl,(t)sfnm(t)s ..e sfnc,,(t) for all t has been studied by van

Wassenhove and Baker [21], see also [14]. They give an efficient algorithm for fin-

ding a solution to the problem P2 and next use it to generate the discrete points of

the set E in the problem P4. If the functions fj are linear or piecewise linear with

two different slopes, their procedure finds the whole set E in 0(n3) time.

There are no polynomial algorithms, in the literature, for Pl and P3 with general

(nondecreasing) functions fj. Tuzikov [18] presents an efficient algorithm for the

problem P2 and applies it for generating an s-approximation (c-kernel) of the set E. In what follows, we outline this approach. For arbitrary c>O, a set E, is called


an &-approximation of E if E, C 52 and for each (a,, a*) E 0, there exists (a;, LYE) E E,

such that cri( ai + E, i = 1,2.

For given r, let Dj = max(t: fj(t) I T), j E J (we assume here that fj are left con-

tinuous functions). Now problem P2 for 1 / If,,, can be stated as follows:

min l

i CjXj: XEX, there exists TC EJ~ such that CnciJ = j=I

j$, (an(j)-xn(,))~Dz(i)~ ieJ 1

.

For each XE X, if n E Z7 and Cn(j) I D,,j,, j E J, then Cn’(j) (D,,,j,, j E J, where

71’ is obtained by sequencing the jobs in accordance with nondecreasing Dj.

Therefore TI * = Z’ in the problem P2 for 1 ) ) f,,, . An optimum compression x* can

be obtained by applying an algorithm for the problem P2 for 1 ) 1 L,,, in which

dj=Dj, jEJ, and T=O.

Having an algorithm for P2 (for 1 / / f,,,), one can find the set E, in the following

way. Let Tr=min,6nF,(u,rr) and T2=minr,G,Fi(0,~). For k=O,l,...,s, find a

solution (xk, n k, to the problem P2 with r = Tk, where Tk+’ = Tk - E, To = T2, and

s is such that TS 5 T, + E. The set E, is given by E, = ((TO, F~(x’>), . . . , (T’, Fl(x’))} . The algorithm requires O(n 2(T2 - T,)/E) time.

3.3. 11 rj IL,,,

The problem 1 / rj (L,,, has an equivalent statement which is more convenient

for analysis, see, e.g. 181. Let M, and M3 be nonbottleneck machines of infinite

capacity and M2 be a bottleneck machine processing only one job at a time. Each

job j has to be processed on Ml, M2 and M3, in that order, and has to spend a time

from 0 to rj on Ml, a timePj on M2, and a time qj from Cj to C’+ qj on M3, where

qj = maxi _ci5n d, - dj and Cj is the completion time on M2. The objective is to find a

processing order on M2 that minimizes maxi ~j~n (Cj+ qj). This problem is denoted

by 1 ( rjT qJ / Cm,,. It has been shown in [ 111, that 1 / rj, qj [ C,,, is NP-hard in the

strong sense.

Here, we present an approximation algorithm for Pl and give its worst-case per-

formance ratio. Problem Pl for 1 1 rj, qj ) Cm,, can be stated as follows. Find X* and n* minimizing

K(x, n) = max l~i~~i~5rI (

‘n(il) + 5 (an(j)-xr(j)) + 4n(,z) j=il >

+ i CjXj j=l

subject to x E X and rr E 17.

Consider the following approximation algorithm H:

Step 1. Let 7rH be a permutation obtained by sequencing the jobs in the order of

nondecreasing rj.

Step 2. Find xH minimizing K(x, nH) subject to x E X.


The minimization problem in Step 2 is a linear program and can be solved using

standard procedures. This problem can also be solved by applying any procedure

for finding a time-cost trade-off curve in an activity network with linear cost-

duration functions, see e.g. [2].

Suppose that nH=(l , . . . , TZ) and denote qmax = maxjEJ qj and qmin = minj,J qj.

We have

KHlmin XEX

(6)

On the other hand,

minrj+ c jeI jet

and in consequence, taking ZE {I,, . . . . In,), where Z[= {li+ 1, . . . . n>, i= 1, . . . ,n, in-

stead of IcJ, we obtain

K* L min XEX

ri+~;aj-Xj)+~,Cjxj)+4,i,. (7)

BY (6) and (7), KH-K*sqmax - qmin, which in view of the fact that K*rqmax,

yields

KH/K*~2-qmin/qmax. (8)

Similar arguments show that ordering the jobs in accordance with nonincreasing qj

in Step 1 (algorithm H’) yields

KH’/Kh I 2 - r,,,in/rmax. (9)

The examples given in Tables 1 and 2 show that the bounds (8) and (9) are the

best possible. In Tables 1 and 2, 1 - l/A 5 c5 1, and A > 1. For the example of

Table 1, we have n H = (1,2)andKH=A+2+c(A-1). Sincex*=(2,1),x*=Oand

K* =A + 2, we get that KH/K* can be arbitrarily close to 2 as A tends to infinity.

Similarly, applying heuristic H’ to the example of Table 2, we get that KH’/K* can

be arbitrarily close to 2.

The best known approximation algorithm, in terms of the worst-case behaviour,

for Pl with fixed compression x (that is for the classical 1 1 rj, qj 1 C,,,,, problem)

has the worst-case performance ratio 1 [13]. In view of this, an interesting ques-

tion is whether there exists a polynomial approximation algorithm for the problem

considered with ratio less than 2.

Table 1 Table 2

i ‘J aj ‘j qj ‘J .i 5 aJ ‘j 4j cj

1 0 A A 0 C 1 A 1 0 1 c

2 1 1 0 A c 2 0 A A 0 c

280 E. Nowicki, S. Zdrzalka

3.4. 1 1 ) C Cj and 1 1 1 C WjCj

Problem Pl for 1 1 ) C C’j and 1 ( I C has studied

case of 1 ) ) 1 C,, it has shown that

ment problem time. The reduction based on the

following observation. For given

n> = Ii (tn -j + lb,(j) + tcx(j) - Cn -j + l))xx(j)). j=l

This function achieves minimum with respect to x for Xn(j) = urrtJJ, if en(j) < IZ -j + 1,

and Xn(j) = 0, otherwise. Thus the cost incurred by placing job i in position j, and

then selecting its processing time optimally, does not depend on the jobs which

precede or follow it in the sequence. Therefore, the problem can be formulated as

an assignment problem with cost kij for assigning job i to location j given by k, =

a,(n+j+ l)+min{O,uj(cj-(n-j+ 1))).

The computational complexity of Pl for 1 / I C WjCj remains an open problem.

In [20], an enumerative algorithm is presented which easily solves problems with up

to 100 jobs; an approximation algorithm is also given.

Ruiz Diaz and French [ 151 develop an exponential algorithm which finds the effi-

cient frontier E for the model 1 I I C Cj and Cj= 1, Jo J. The algorithm has been

tested for the instances with 30 jobs. The Vickson’s approach, described previously,

enables us to find particular points of E in this problem, however, a weighting

approach, applying Vickson’s algorithm, cannot be used here since, as it is shown

in [15], the efficient frontier is not, in general, convex. Ruiz Diaz and French show

also a special case in which the efficient frontier can be found in polynomial time.

Let i+ if aisaj and ai-u;~aj-uj. If there exists a permutation 71 of J such that

7c(l)~71(2)4 ... en(n), then the efficient frontier may be generated by sequencing

the jobs in the order n and then crashing the jobs in the following way: 7c( 1) is crashed

first until it is fully crashed, then 77(2), and finally r(n). In this special case the effi-

cient frontier is a piecewise-linear, convex curve with n + 1 breakpoints. They also

present two approximation algorithms and give results of the experimental analysis.

4. Single machine scheduling with controllable release dates

In this section, we consider single machine scheduling problem 1 1 rj ( C,,, in

which release dates are given in the form of rj - Xj , 0 ‘Xj 5 Uj, j E J, where Xj is the

time by which the “normal” release date rj has been compressed and Uj is the

maximum compression; Uj’rj for all j. We assume here that job processing times

are fixed and given by Pj, j E 1. A compression cost for job j is equal to CjXj , where

cjZ0; and the total compression cost is given by Fz(x) = CjeJCjXj.

In this case, problem Pl can be stated as follows. Find z *EZ~ and x* EX mini-

mizing

Sequencing problems with controllable processing times

K(X~ n)= max I<i<n

‘n(i)-X,(i) + i Pn(j) + f CjXj j=i > j=l

281

(10)

subject to n E 17 and XEX. The decision form of this problem is: DPl: Given a set of jobs J= { l,..., n>, nonnegative integers Pj, rj, Uj, where 1 Uj each j J, and rationals , j E J, y, determine whether there

exists a permutation 7~ on J and rationals

Theorem 2. The decision form of Pl for the problem 1 j rj ) C,, with controllable release dates is NP-complete in the strong sense.

Proof. By (lo), it is clear that DPl ENP. We shall show that the decision form of the problem 1 ) 1 C WjTj, which is strongly NP-complete [ll], is reducible to DPl. The decision form of 1 1 ) C WjTj is as follows:

DP2: Given a set of jobs J’= { 1, . . . , t}, integers qj, dj, Wj, Jo J’, Z, determine whether there exists a permutation 71 on J’ such that

L(n) 2 f: wn(i) max I 0, f: qn(j)-dn(;)

3 SZ.

i=l j=l

Note that without loss of generality we may assume that djS CL= 1 qk for all j. Denote q=Cf=tqi and w=CfzI Wje

An instance of DP 1 is defined as follows: n = t, pj = qj , rj = q - dj, Uj = rj = q - dj ,

Cj= Wj/W, jc J, y=q+Z/w.

For this instance of DP 1, for each rz and each x satisfying the condition 0 I Xj 5 Uj ,

je J, the following inequalities hold:

= max max Ilist ( L

0, f: 4x(j) -d*(i) -Xx(i) j=i 11

+ $

t

,F;, wz(i)xi7(i) + 4 I

5 I

1

z- W i=l

wn(i) max Xx(i), C 4x(j) -d,(i) + 4 j=i I

5 >-- wrr(i) max 0, I? qn(j,- dn(i) + 4 W i=i j=i 1

‘i =-- W i=l

b(i)max 4 f: qo(j)-da(i) j=l I +q=iQo)+q, (11)

where o is defined by o(i) = Ir(t + 1 -i), i= 1, . . . , t. Moreover, if X~(i) = max{O,Cf=iq,(j)-d,(,,}, i=l,..., t, then

K(x, 71) = -_:L(o) + q. (12)

282 E. Nowicki, S. Zdrzuika

(a) If for each rc, L(n) > z, then by (1 l), for each TI and each x satisfying the condi-

tion OlXj’Uj, jEJ’, K(x,z)>z/w+~=~.

(b) Suppose that DP2 has a solution xc, i.e., Liz. Then, in view of (12), setting

o(i)=n(t+l-i), i=l,..., t, and x0(;) = max{O, CiZi qocj) -do,,,}, i = 1, . . . , t, we ob-

tain that K(x, o) 5 z/w + q =y.

This completes the proof. q

It follows from the proof that the decision versions of P, and P, (in which

F,(x, n) sq and F2(x)~zz/w) are also NP-complete in the strong sense.

In what follows, we discuss approximation algorithms for P,.

First note that we may assume without loss of generality that Cj< 1 for jE J. It

can be shown that if Cj L 1 for some j, than there exists an optimal solution (x*, n *)

in which x1? = 0. This implies that we can confine our considerations to a modified

problem in which for each j with Cj’ 1, new data Uj=O and Cj< 1 are set.

Consider the following general approximation algorithm H:

Step 1. Choose a permutation 71~.

Step 2. Determine xH minimizing K(x, nH) subject to XEX.

Note that the minimization problem in Step 2 is the following linear program

(assume that the jobs are renumbered such that nH = (1,2, . . . , n)):

subject to T+Xj’rjf i pi’Aj, j=l,...,n, i=J

O~Xj'Uj, j= l,..., n, TzO.

Denote by o the permutation of J such that A,(l)gA~(2)~...2A.(,), and by I,

the maximum integer less or equal to n such that Cl=, C~cj,’ 1; I= 0 if co(,)> 1.

Let Tmin = max{Aj - uj > and Aocn + 1) - -0; note that each feasible solution to the

above linear program satisfies TZ Tmin. It is not difficult to verify that x,?=

maxj.J{O,Aj- TO}, jgJ, T”=max{A,(,+r,, Tmin} is an optimal solution to this

problem.

We now show results of the worst-case analysis for various variants of the algo-

rithm H, obtained from the general scheme by applying various rules for choosing

nH in Step 1. Suppose that rc H is chosen according to one of the following rules:

arbitrary permutation (HO), nondecreasing rj (Hl), nondecreasing rj - uj (H2), non-

decreasing rj - (1 - Cj)Uj (H3), nondecreasing rj +pj (H4), nondecreasing rj +Pj - Uj

(H5), nondecreasing rj +pj - (1 - Cj)Uj (H6), nonincreasing Pj (H7), nonincreasing

rj+Pj (H8), nondecreasing (rj-(I -cj)Uj)/~~j (H9). If there is a choice in Hl-H3,

then take the job with the longest processing time. The rules H3, H6, H9 are justified

Sequencing problems with controllable processing times 28:

Table 3

j

1

n-l

n

‘J

R-@-l)&

R-e R

MJ

R-(n-l)&

R-E R

Pj

&

& R

cJ

l/n

l/n

l/n

by the following lower bound. Using the well-known inequality “min maxr

maxmin”, we obtain from (lo),

K(X*,71*)Zmin max ncIl lsisn

T,(i)-X,(i)+ i Pn(j)+ i C,(j)X,(j) jzi j=l

=min max (

‘n(i) - (l - c7r(i))u77(i) + i Pn(j) . nEnI5isn j=i )

Thus the lower bound on K* can be obtained by sequencing the jobs in the order

of nondecreasing r-j’ = rj - (1 - Cj)Uj, in the classical problem 1 ) rj’ / Cm,, .

Theorem 3. KH/K*r2 and this bound is the best possible for each HE {HO,Hl,..., H9).

Proof. Denote C,,,(X, 72) = maxi ci<n (r,(i) -Xx(i) + CS=iP,(j,). Let IT(X) be a per-

mutation minimizing C,,,(x, 7~) over II en. Observe that for any n E I7,

C,,,(X> 71) 5 2 C,,,(& 7-c (4). (13)

Indeed, for any TC E Z7, C&,(X, X) I maxi <j< n (rj - Xj) + CJ= I pj, and C&,(X, 71 (x)) L

max,,j5n(rj-xj), C,,,(X, 7r(X)) 1 Es= 1 Pj * Since K(X*, z*) = K(X*, n(X*)), we get

from (13) that

5 2 Gnax (X*9 X(X*))+ f CjXj' j=l

~2K(x*,n*)=2K*.

Table 4

‘J

(n - 1)R 0

0

pJ

R R--E

R - (n - 1)~

eJ

I/n

I/n

l/n


Table 5

‘i

m

R

R

Now consider the example given in Table 3. It is assumed that O< E < l/n*. For HE {HO,Hl, . . . , H6}, the algorithm generates rcH= (1, . . . , n), and in consequence, KH=2R. For rc”=(n,1,2 ,..., n-l) and xi=R, x,?=O for j=l,...,n-I, we get K*sK(x”,no)=R+(n-l)&+(l/n)R. Since ccl/n*; then for fixed R, KH/K*+2

as n+w. In the example given in Table 4, O< E< l/n*. For HE (H7, HS}, the algorithm

generates rrH=(l, . . . . n), which yields KH > (n - l)R -E + n(R - (n - 1)~) + s/n. Let x0=(2 ,..., n,l) and x7=0 for j=l,..., n. Then K*sK(x’, no)= nR. Again, by E< l/n*, for fixed R, KH/K*+2 as n-t w.

In the example of Table 5, n > R > 1. Here, the algorithm H= H9 generates 7?=(1,..., n), which yields KH=2m+ 1 -p. Let 7c”=(2, . . ..n. 1) and xj”=O for j= 1, . . . . n. Then K*r K(x”, no) = R + 1 + /%-J/R%, and in consequence KH/K* + 2 as n --f 03. This completes the proof. 0

It remains an open question whether there exists a polynomial approximation algorithm with worst-case performance ratio less than 2. The fact that for fixed x, the problem is easily solvable by arranging the jobs in the order of nondecreasing rj- Xj, and when 7r is fixed, then the optimal solution has even an analytic form, leads us to conjecture that such an algorithm can be found. However, the results of Theorem 3 suggest that we should take into consideration more involved method for choosing rrH than simple priority rules.

5. Two-machine flow shop

Problem Pl for F2 ) ( C,, (in permutation version) can be stated as follows. Find x*, II* minimizing

K(x, n) =F,(x, 7r) +F2(x)

subject to x E X and rc E 17. It can be shown, applying similar arguments as in the classical case, that for the problem allowing different processing orders on machines


1 and 2 there exists an optimal solution with identical processing orders on both machines. Here, similarly as in the problem considered in Section 4, we may assume that cij< 1 for all i and j.

The problem PI has been studied by Nowicki and Zdrzalka [12]. In [12], it is shown that the decision form of the above problem is NP-complete, even when the processing times on one machine are fixed and all the processing cost units are identical. This paper provides also the following approximation algorithm H:

Step 1. Find rcH minimizing maximum completion time for job processing times aii-(l-cij)uij, i=l,2;j=l,..., n.

Step 2. Determine xH minimizing K(x, rcH) subject to XEX.

It has been shown that KH/K*r+ and that this bound is the best possible. The bounds depending on cti, given in [12], indicate that the algorithm behaves better in particular classes of instances. For example, if U2j = 0 and Clj = c for all j, then

KH/K% 1

93 -4, C&

l+c(l-c), +5c<1

and this bound is the best possible. Note that the problem in Step 1 of this procedure can be solved in O(n log n) time

by Johnson’s algorithm [7], and the problem to be solved in Step 2 is a linear program. The processing times used in Step 1 is justified by the lower bound K*> min x E nF, (x’, or), where xi;. = (1 - cti)uii for all i and j, which can be derived in a similar way as the lower bound given in Section 4.

Results obtained for the two-machine case indicate that applying this approach to the m-machine permutation problem, with a suitable heuristic method for obtain- ing a “good” processing order in Step 1, can give good results.

6. Other multimachine scheduling problems

Problem P3 for the job shop scheduling problem with completion cost function C max has been treated by Grabowski and Janiak [3]. They propose a branch-and- bound algorithm based on certain elimination properties of a critical path, and report results of numerical experiments for the test problems with up to 36 operations.

Ishi et al. [5] consider the problem with parallel uniform machines in which the speed of machine i is a continuous nonnegative variable Xi and the processing cost of machine i is equal to hit iEM. They assume that preemptions are allowed and consider the deadline problem with minimum processing cost Cy=,fi(xi), and the problem of minimizing&(T) + Cy= 1 fi(Xi), where Tis the minimum completion time for the given speed vector (x1,x2, . . . , x,,,). Based on very strong assumptions con- cerning the functions h, i = 41, . . . , m, they derive polynomial algorithms for both problems. The cases in which these problems become NP-hard are also identified,

E. Nowicki, S. Zdrzalka

7. Concluding remarks

In this paper, we have focused our attention on the class of sequencing problems

in which job processing times are continuous variables and the processing cost func-

tions are linear. Other models of scheduling with controllable processing times have

been studied mainly in the project scheduling context, see, e.g. [l, 2,16,17].

The problems of least cost scheduling (Pl) considered in the paper are simple

generalizations of the corresponding classical scheduling problems. Therefore, it has

been interesting to note the cases in which such a generalization moves the problem

from the class of tractable to intractable ones. These are a two-machine flow shop

problem and a single machine problem with controllable release dates; both with a

completion cost given by the makespan. It is obvious that the cases in which an

optimal sequence does not depend on the processing times remain, after generaliza-

tion, in the polynomial class. The only nontrivial case from those considered up to

now, which exhibits the same property, is the single machine sequencing problem

with completion cost function given by the flow time (with all weights equal to one).

The actual results indicate that the tractable cases will be the rare exceptions and

therefore, it seems that the main emphasis should be put on the construction of

approximation algorithms. One straightforward approach for the class of problems

Pl has been investigated in this paper. The method can be sketched as follows:

Determine the best sequence of jobs for some fixed compression, applying some

heuristic method or exact algorithm, and then, basing on that sequence, solve the

optimization problem for continuous variables. This approach yields the worst-case

performance ratio 3 in the two-machine flow shop, and 2, in the single machine

scheduling problem with release dates and due dates, which are better than the ratios

of worst to best schedules in the corresponding purely sequencing problems; the

latter ones are the same as in the generalized problems. An exception is the single

machine sequencing problem with controllable release dates where a number of dif-

ferent sequencing rules have been applied and all of them have shown the same

worst-case behaviour as in the purely sequencing problem (Theorem 3).

Acknowledgment

We wish to thank the referees for their helpful suggestions and comments regard-

ing an earlier version of this paper.

This research was supported by R.P.I.02 “Theory of control and optimization of

continuous dynamic systems and discrete processes”.

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Documents

A survey of results for sequencing problems with controllable processing times