Operations Research Letters 10 (1991) 519-523 North-Holland

Scheduling jobs on a single machine with release dates, delivery times and controllable processing times: worst-case analysis

Stanistaw Zdrzatka Technical University of Wroctaw, Institute of Engineering Cybernetics, ul. Janiszewskiego 11/17, 50-372 Wroctaw, Poland

Received July 1990 Revised March 1991

The paper deals with the problem of scheduling jobs on a single machine, in which each job has a release date, a delivery time and a controllable processing time, having its own associated linearly varying cost. An approximation algorithm for minimizing the overall schedule cost is provided which has the performance guarantee of p + ½, where p is the worst-case performance bound of a procedure used in the proposed algorithm for solving the pure sequencing problem. The best approximation procedure known has p=~.

scheduling; programming; algorithms; heuristic

A set, J = {1 . . . . . n}, of jobs is to be processed without interruption on a single machine. Associated with each job j is a release date rj, the earliest time by which the processing can begin, a processing time defined as a difference aj - x j, 0 < xj < u j, where xj is the time by which the normal processing time aj is shortened (compressed) and uj is the maximum compression, and a delivery time qj; rj, qj >_ O, aj > uj >_ 0. It is assumed that a zero processing time corresponds to a job spending an arbitrary small time on a machine. Let 7r be a permutation of the set J, and 7r(i) the job which is in position i in permutation ~-; ~- denotes a processing order of jobs. Denote by C(x , 7r) the (earliest) completion time of all the jobs for compression times x = (Xl , . . . , x n) and processing order rr, then

C ( x , rr) = max r~(i,) + ~ (a~( j ) -x ,~( j ) ) +q,~(s2) • (1) l <-il <-i2<-n j=il

The total cost of compressions is equal to F.j ~ j c ix j , where cj is the cost per unit of compression, and the total scheduling cost for compressions x and processing order ~" is defined as

K(x , = C(x, + E (2) j ~J

Here, we assume, without loss of generality, that the cost units cj are chosen such that the cost per unit of the delivery time is equal to one. The problem is to find x* and ~-* minimizing K ( x , rr) under the constraints x ~ X = {x ~ ~: 0 _< xj < u j, j ~ J} and 7r ~ H, where H is the set of all permutations of J.

When all the processing times are fixed, the problem stated is equivalent to the well-known sequencing problem denoted as 1 ] rj, qj[Cmax, Graham et al. [5]; the equivalent variant of this problem with due dates, instead of delivery times, is denoted by 1 ] rj I Lmax. Since 11 5, qjlCma × is NP-hard [9], the problem stated remains also NP-hard.

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The models of jobs (operations, activities) in which processing times are decision variables influencing the overall project cost are commonly used in the area of project planning and control, e.g. [2]. Their motive in the field of sequencing and scheduling is of the same nature, that is, they are justified in situations where jobs can be accomplished in shorter or longer durations by increasing or decreasing additional resources. Studies on the standard sequencing models with controllable job processing times have been initiated by Vickson [14,15]. They have been continued by Van Wassenhove and Baker [16], Tuzikov [13], Ruiz Diaz and French [12], Ishii et al. [7], Grabowski and Janiak [3], Nowicki and Zdrzalka [10,11]. The last of the cited papers provides a survey of results in this area.

The sequencing problem 1 1 r i, qil Cm~x, of which an extension is discussed in this paper, has received considerable attention in the literature. Efficient branch-and-bound algorithms which are capable of solving problem instances with hundreds of jobs in very short time have been proposed by Carlier [1] and Grabowski et al. [4]. The best approximation algorithm known was given by Hall and Shmoys [6]. It guarantees the worst-case performance bound of 4 3. This relatively easy NP-hard sequencing problem has important applications: for scheduling jobs on a critical machine, as a simplified model of more complex scheduling situations in a sophisticated machine environment, and in the theoretical context, for computing lower bounds in flow shop and job shop problems.

In this paper we propose an approximation algorithm for the problem 11 rj, qj [Cma x with controllable job processing times and show that it has the performance guarantee of p + ½, where p is the worst-case performance bound of a procedure used in our algorithm for solving the pure seqeuencing problem 1 [ rj, qj I Cmax (for some fixed x). Applying the heuristic of Hall and Shmoys [6], we get an algorithm with

1 4 1 performance guarantee p + g = 3 + g = 1.833.. . ; the best approximation algorithm before our result had 1 a worst-case bound equal to 2, [11]. It is shown that the bound p + 5 is tight for p = 1.

Approximation algorithm

First, we observe that if c i >_~ 1 for some i ~ J , then there exists an optimal schedule (x* , rr*) such that x* = 0. Indeed, let (x °, ~.0) be an optimal schedule with x ° > 0 and c i >_ 1 for some i. Define a schedule ( x * , r r * ) as follows: x * = x ° for j ~ J , j - ~ i , x * = 0 , and ~-*=Tr °. Since C ( x ° , r r ° ) - C ( x * , rr*)>_ - x ° and E j ~ j x ° c j - Ej~sX*Cj=X°Ci , we get the inequality K ( x °, r c ° ) - K ( x *, ~r*)>_ x°ici - x°i >_ o.

This observation enables us to assume in the sequel, without loss of generality, that cj < 1 for all j; if, in a problem instance, c i > 1 for some i, then we may set a new value for ci, less than one, setting simultaneously u i = O.

Since the problem considered is NP-hard, it is unlikely that an efficient algorithm will be found for obtaining the optimal schedule. Hence we focus our efforts on efficient approximation algorithms with acceptable guaranteed bounds on their performance. For fixed x, the problem becomes relatively easy (NP-hard) sequencing problem l lrj, qjlCma x, and for fixed 7r, it reduces to a linear programming problem. This suggests a straightforward approximation approach which can be sketched as follows: determine sequences of jobs for some fixed compressions, applying some heuristic method or exact algorithm, solve the optimization problem with continuous variables for each of those sequences, and then choose the best solution. Such an approach was applied in the two-machine flow shop problem with controllable processing times, [10], where only one linear programming problem had to be solved to obtain the worst-case performance bound of 3 2.

l = ( l _ c j ) u j , j ~ J , and x } = u j , j ~ J . Let xj

Algorithm A. begin

for i : = l t o 2 d o begin

find a processing order 7r i to the pure seqeuencing problem 1 L rj, qj I Cmax with processing times

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i a~ - xi, j ~ J, using an algorithm with worst-case bound p; determine Xop t ' minimizing K(x , zr ~) subject to x ~ X;

end choose the schedule (Xop t, zri), i = 1, 2, with the smaller value of the cost function;

end

The first stage of each iteration (i = 1, 2) requires finding a sequence for the problem 11 rj, qj[Cma x. To this end we can apply one of the existing exact algorithms, e.g. [1,4], or the polynomial approximation

4 The optimization problem algorithm of Hall and Shmoys [6] with worst-case performance bound p = 7- in the second stage can be reformulated as the following linear program (for zr i = (1,. , n)).

z ,x j = 1

i 1

s.t. rk+ ~ a j + q l < ~ x j + z , 1 < k _ < 1 , l < l < n , O<_xi<uj, l < j < _ n , O < z , j = k j = k

and can be solved using standard procedures. The optimization 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]. The latter approach is computationally more efficient since it exploits the natural structure of the problem; see [1] for the activity network corresponding to l lrj, q j l C m a x w i t h a fixed permutation of jobs.

Worst-case analysis

For some problem instance, let K A be the value of the objective function K obtained when Algorithm A is applied, and K* , the optimal value of K.

1 Theorem. For every problem instance, K A / K * <_ p + -~.

Proof. We shall make use of the following bounds on K*. By the inequality "min max > max min", we have

K * = min min m a x rrr(i p + ~ a¢u ) - x¢(j) + qT.r(i2 ) + CiX i 7r x l <il <_i2 <_n j = i l i=1

>_ min max m i n rrr(i 0 + E a~r(j) - xrr( j ) + qrr(iz) + CiXi rr l <_il <_i2 <~n x j = i l i=1

= min max r~(it ) + _~ a=(j) - ,(1 - c~u))u~o + q~-(i2) ~c l<il<_i2<_n J=im

1 = m i n C ( x 1, 7r) >_ - C ( x 1, 3T1), (3)

p

It is clear that for each x and 7r,

C(x, ~r) >_ E a j - x j . (4) jeJ

Since C is a nonincreasing function with respect to each x j, for fixed remaining variables, we get that for each x and 7r,

1 C(x, >_ C(x 2, >_ - C ( x L

P

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Renumber the jobs such that rr z = (1 . . . . , n), and denote by k and l the jobs for which 1

C(x 2, rr 2) = r k + )-". aj- -u j+ql . (5) j = k

This, together with the previous inequality, yields 1

C( x, rr) > - ( r k + q,) (6) P

for each x and 7r. By (4) and (6),

/ ( 1 } ) m a x a i x j , _ r k _ __ c j x j K * > rain - - ( + q t ) + x ~ I, j E J t ° j ~ J

1 ) > rain a ~ - x j + ~-~o(rk+qt) + ~ c j x i

X j E J

1 = - - 2(a~ - u i ) + c i u i } . (7) 2p ( rk + q t ) + E min{laj , '

j ~ J

We shall now derive an upper bound on K A. It follows from (3) that

K A < mxin ( C ( x , 7r') + ~ c, xj} < C(x 1, 7r l) + ~ c i ( 1 - c , ) u , j e J " j e J

< o K * + E c j ( 1 - c j ) u j , (s) j ~ J

and similarly, by (5),

KA < C( x2, 7/'2) -~- E CjUj <__ r e + qt + ~-, ai -- (1 -- cj)u i. (9) j E J j E J

1 1 Adding (8) and (9) with weights ~ and ~, respectively, we obtain

KA<_TpK + ~., a j - u i - ( c i ) 2 u j +ciu i +7 ( rk+q t ) , j E J

which together with (7) and the obvious inequality

min{½aj, 1 1 1 2 - u j ) + cjui} > - " i ) + c ju j - cj) uj

yields

1 * ½ P - 1 K A < ~ p K + K * + ( r k + q , ) < _ % ½ p g * + g * + ½ ( p - 1 ) K * = ( p + ½ ) K * ,

P where the last inequality follows from (6). This completes the proof.

1 We now show that the bound P + 7 is the best possible one when p -- 1. To this end consider the instance specified by the data in Table 1. It is assumed that 0 < c < d < 1 and 0 < e < B. We have (al - x l , a2 -X12, a3 - x ~ ) = ( B + e ( 1 - d ) , 1, B + e ( 1 - c ) ) , and (a~ - x 2, a 2 - - x 2, a 3 - x 2 ) = ( e , 1, s).

Table 1

j 1 2 3 rj 0 e 0 a i B / d 1 B / c uj B / d - e 0 B / c - e % 0 B 0 cj d 0 c

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For both compressions x t and x z, the permutation (1, 2, 3) is an optimal sequence for the sequencing problem 11 Q, qi] Cmax; in both cases, the length of seqeuence (1, 2, 3) is equal to the lower bound on the minimum length. According to Algorithm A, we now solve the optimization problem m i n x K ( x , ~r i) for i = 1, 2, where er a = er2 = (1, 2, 3). It is clear that K A = e + 1 + B + ( B / d - e ) d + ( B / c - B ) c = e(1 - d) + 1 + 3 B - Bc . A n optimal permutation is (3, 2, 1), for which we have K * -- (e(1 - c) + 1 + 3B - Bd;

this can be verified by examining all the possible permutations. Assuming d = 1 - l / B , c = l / B , 3 e = l / B , we get that K A / K * ~ ~ as B ---> oo.

It follows from the proof of the Theorem that the optimization step of the Algorithm A is not 1 necessary to guarantee the bound p + 3. In order to maintain such an accuracy it is enough to solve the

problem 11 ri, q i l Cma x twice, for compressions x 1 and x 2, and then choose the better (lower value of K) of (x l, erz) and (X 2, '/7"2).

Acknowledgment

I thank the referee for his careful reading of the text and for his remarks.

References

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