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74 European Journal of Operational Research 72 (1994) 74-81 North-Holland

Theory and Methodology

An approximation algorithm for a single-machine scheduling problem w th release times, delivery times and controllable processing times

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

Received April 1991; revised January 1992

Abstract: The paper deals with a single-machine scheduling problem with release times, delivery times and controllable job processing times. It is assumed that the cost of performing a job is a linear function of its processing time, and the total schedule cost to be minimized is the total processing cost plus maximum completion time cost. In consequence, the processing order of jobs and their processing times are decision variables. A (p + 1/3)-approximation algorithm for the problem is provided, where p is the worst-case performance bound of a procedure for solving the pure sequencing problem.

Keywords: Single-machine scheduling; Approximation algorithms; Worst-case analysis

1. Introduction and problem formulation

In the real-life applications of scheduling research, apart from the machines, processing a job requires additional resources, such as facilities, manpower, funds, and so, which implies that jobs can often be accomplished in shorter or longer durations by increasing or decreasing the additional resources. Generally, this situation is usually very difficult to analyze. One of the simplest and most basic forms of resource allocation is represented by the t ime/cost trade-off, [3]. There are situations where compressing a job is possible, but it entails extra costs, and such an action would be rational only if these additional costs are compensated by the gains from job completion at an earlier time. In the t ime/cost model, the time required to perform a job can be reduced by the application of additional nonrenewable resources (measured by their cost). In consequence, job processing time can be considered as a decision variable.

Studies on the standard sequencing problems with controllable job processing times have been initiated by Vickson [12]. Vickson has analyzed the single-machine sequencing problem denoted as 1//)Zw~Tj (see [6] for notation). He has assumed that the cost of performing each job is a linear function of its processing time and a schedule cost is equal to the total processing cost plus a cost associated with

Correspondence to: Eugeniusz Nowicki, Technical University of Wrockaw, Institute of Engineering Cybernetics, ul. Janiszewskiego 11/17, 50-372 Wroctaw, Poland.

0377-2217/94/$07.00 1994 - Elsevier Science B.V. All rights reserved SSDI 0377-2217(92)00163-V

E. Novicki / A single-machine scheduling problem 75

the job completion times (the total delay cost). The problem is to find a job sequence and processing times minimizing the schedule cost. Other sequencing problems with controllable job processing times have been considered e.g. in [2,4,8,9,13]. Recently, in [10] a review of current results has been presented.

In this paper we consider the well-known sequencing problem denoted as 1/rj ,qj/Cma with control- lable job processing times. The problem is formulated as follows:

Each of n jobs from the job set J = {1, 2 . . . . . n} has to be processed on one machine. Jobs preemption is not allowed. For each job j, we define:

(i) ready time rj, rj > 0, (ii) processing time aj - x j, 0 _< xj < u j, where xj is the time by which the normal processing time a i is

shortened (compressed) and uj is the maximum compression, aj > u j, (iii) cost per unit of compression c j, cj > 0, (iv) delivery time q j, qj >_ 0.

At most one job can be processed at a time, but all jobs may be simultaneously delivered. Let ~r be a permutation of the job set J, and /7, the set of all permutations; ~- denotes a processing order of jobs. Denote by Cmax(X, ~-) the maximum completion time for compressions x=(x~, x 2 . . . . . x , , )~X, and processing order ~- e H, where X = {x ~ ~" : 0 _< xj < u j, j ~ J}. It is easy to verify that

C .... (x, 7r) = max r~(il)+ ~ (%(i) -x,~(i)) +q=(i:) . l

76 E. Novicki / A single-machine scheduling problem

worst-case performance bound of a procedure used for solving the pure sequencing problem l/r~, q//Cma x. In this paper we present an approximation algorithm with the worst-case performance bound

1 equal to p + 7.

2. Approximation algorithm

First, we observe (as in [10,11]) that if c /> 1 for some j E J, then there exists an optimal schedule (x*, 7r*) such that x* = 0 (the function K(x, rr) is nondecreasing in the variable x/). Hence, if c/> 1 for some j, then we may set c /= 0, simultaneously setting u/= 0. This observation enables us to assume in what follows, without loss of generailty, that c/< 1, j ~J.

When all the processing times are fixed (i.e. x is fixed), the optimization problem (1)-(2) reduces to the pure sequencing problem l/r/, qJCma x (see [1,5,6]); the problem is strongly NP-hard, but in practice it seems to be relatively easy to solve. On the other hand, when the permutation ~- is fixed, the optimization problem (1)-(2) reduces to a linear programming problem. This suggests the following heuristic approach to the problem (1)-(2). First, for the given x' ~X determine ~r r' ~H, minimizing Cmax(X', 7r), and then determine xHeX, minimizing K(x, ~.H) subject to x EX. Obviously, the heuristic quality strongly depends on the selection of x'. As was mentiond in [11], the above heuristic algorithm has been run twice: first with x] = (1 - c/)%, and then with xj = uj. We propose a single run of the algorithm with

x j=min{1,}(1 -c / )}u / , j~ J . (3)

Thus we have the following approximation algorithm.

Algorithm H Step 1. Find a processing order T/"H for the pure sequencing problem 1/r/, qj / /Cma x with processing

times a / -x ] , j ~ J, using an algorithm with worst-case bound p. H H Step 2. Determine x minimizing K(x, Tr ) subject to x ~ X.

In Step 1 we can apply one of existing exact (then p = 1) branch-and-bound algorithms, e.g. [1,5], polynomial approximation schemes [7], or the polynomial approximation algorithm with worst-case

4 performance bound p = ~ [7] (the algorithm of Hall and Shmoys). The minimum x H in Step 2 can be found using the simplex method. This 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. [3] (certain modifications of the stopping condition in the procedure is necessary).

Now we give certain intuition explaining the selection of x'. It is quite natural to select x' 'close' to (unknown) x *. Let us consider the selection xj = z(c)uj, where z is a function from [0,1] on [0,1] (then x' ~X) . There exists an optimal schedule (x*, 7r*) such that x* = u~ if c /= 0 and x* = 0 if c /= 1. Hence we obtain that z(0) = 1 and z(1) = 0. Moreover, it is reasonable to take z as a nonincreasing function. These facts and the form of lower bound introduced in the next section suggest jointly the choice of the function z in the form (3) (i.e. as z(%) = rain{l, 4(1 - %)}).

3. Lower bounds

In this section we derive various lower bounds for K(x*, 7r*) -~ K *. For a ~ [0,1] we define

LB( a) ~- a min Cmax( X', ~') ,rr E II

+ E ra in{(1 -a )a j , (1 -oQ(a j -u j ) + rain{c/, c , (1 -4a) + a}u/}. j~ J

E. Novicki / A single-machine scheduling problem 77

Lemma. LB(a) __af(x, 7r) +g(x , a), (4) j E J

where

f (x , ~)~ Coax(X, ~') + 2 max{0, 3c i -~}x j j~ J

and

g(x , ~) ~ E [(1 - o0(a j -xa) + min{cj, cj(1 - ~a) + a}xj]. j c J

Applying the machine-based bound Coax(X, 7r)> Ej e g(a s -x i ) , we obtain

Cmax(X , 77") + ECjX j~OlCmax(X , 77") +(1 --O~) E (a j -x j ) + ~_,cjxj. (5) j~ J j~ J j~ J

1 Using the equality max{0, y} + min{0, -y} = 0 for y = (3cj - ~)axj, we have

4 cjxj = cjxj + a max{0, ~cj4 _ .~}xj~ + rain{0, ~ - ~cj}ax j

=amax{0, 4c j -1}x ,+min{c j , c j (1 -~a)+ a}xj, j~ J . (6)

Finally, applying (6) the definition of the f (x , 7r) and g(x, a), we obtain (4) from (5). Next, we derive a lower bound on min,~xf (x , rr) and we calculate the value of min~xg(X , a). The

following inequality holds

min f (x , ~') >__ Coax(X', 7r). (7) x~)(

Indeed, employing the definition of Coax(X, rr) and inequalities

1 max{0, 4 -~}x=o)>0, j~ J -{ i I i 1+1, i2} ~C ~(j) _ , . . . . ,

we have

f (x , =

where

>_

i2 1 max r~(i,)+ E (a~(j)--X~r(j))+qr(i2) + E max{0, 43crr(j)-5)x~r(i)

1

78 E. Novicki / A single-machine scheduling problem

Applying the definition of the set X, inequalities

min{1,3(1-c=( j ) )}>0, j~{i l , i , +1 . . . . . i2}

and (3), we obtain

i2 rain h(x, 7, il, i2) =r=(il)+ ~ min xEX J =il x~r(J)~ [O'u~(D]

i2

= r~(i,)+ 2 [a~(j)- min{1, 4 (1 - c~(j))}u~(j)] + q~r(i2 ) j=i 1

i2

= rrr(ip + E (art(j) -- X~(j)) + qlr(iz ). j=i l

Finally, from (8), the well-known property

rain max h(x, 7, i1, i2) > max min h(x, 7, il, i2) x~X l rain min[af(x, 7) +g(x, a)] rr~H x~X ~rEH x~X

> ~nmin t[aminx~x f(x, 7) + x~xmin g(x, a)] > =~umin tl~Cm~x(X" 7) + x_ ~-Cmax(X', "ITH).

E. Novicki / A single-machine scheduling problem 79

From there, and by applying the lemma with a = 1 we have

K* >_ LB( I ) = Cmax(X', ~-') + E min{0, xj' - (1 -cy)uy} = Cm~x(X', ~"1 j~J

1 >__ - -Cmax(X t, 7FH) .

p

3 3 r Similarly, for a = ~ and from inequalities aj >_ u i, ~xj - (1 - cj)uj LB(~) 3 3 ' - (1 - %)uj] __ 7Cmax(X ' , gl") -'}'- E [ l aj -}- gXj je J

3 t >__ ~-Cmax(X , 'w ' ) ~'- E min{ {, c,}uj

je J

1 3- -Cmax(X ' , 3"1 ".1) -t- }7~min{, cj}uj.

P jc J

Lower bounds (11) and (12)will be employed in the worst-case analysis of Algorithm H. Let consider another lower bound for K* (see lemma)

(11)

(12)

LB ~ max LB(a) . e~ ~ [(),1]

Next we will show how to calculate LB. First note, that the function LB(a) is piecewise linear and concave. Let % ~ max{ 3, 1 - %}, j E J and a ,+ 1 ~ 1. It follows from the definition of the xj' and aj, that

cLx~!=max{,1-%}min{1,4(1 -%)}u j=(1-c j )uy , j~ J .

Hence and using the inequality

3 __ X! 1 3 , (1 - cy)u)] = E [a j - (1 - c~)uj] : LB(O) j~J j~J j~J

we obtain the simple formula for the lower bound: LB = max I ~j_

80

Table 1

E. Novicki / A single-machine scheduling problem

j 1 2 3 4

rj 1+3e 0 0 0 ay e l+e l+e l+e uj 0 1 l l

1

qj 2 + 2e 0 0 0

From (13), (14), (12) and (11) we obtain

1 4 1 v t g H _~< Crnax ( x ' , "/7- H) q- ~ E min{cj, ~}uj < Cmax( X , 77" H) -t- 4K * - -Cmax( x , 77- H)

j e J P

< 4K* + 1 - oK*= (O + -~)K ,

which completes the proof of the Theorem. []

1 We now show that the bound p + ~ is the best possible when p = 1. To this end consider the instance specified by the data in Table 1. It is assumed that e is an arbitrarily small positive number. From (3), we

! ! v t 2 have x t = 0, x e = x 3 = x 4 ~. Applying Algorithm H, we get rr n = (2, 3, 4, 1) in Step 1 and next, in Step 1 .2 . The optimal permutation is 2, we obtain x~=0, x~=l , x~=l , x4 n=0 and K H=3+6e+7

4 (2, 1, 3, 4), for which we have x =x~ =x~' =x~' = 0 and K* = 3 + 6e. Hence KH/K * ~ ~ as e ~ 0. It follows from the proof of the Theorem that the optimization Step 2 of Algorithm H is not necessary

1 t to guarantee the bound p + 7. It is enough to set x~: = xj, j ~ J (see (13)). However, the performance of Step 2 usually decreases the goal function value and does not need much computation time.

Algorithm H and Algorithm Z (taken from [11]) were coded in Pascal and run on an IBM PC AT. In Step 1 of both algorithms the procedure of Hall and Shmoys [7] was implemented. Test instances were generated in the way described by Carlier [1] (for the pure sequencing problem 1/rs, q JC~x) . For each n e {50, 100, 200, 300, 400, 500} and k e {17, 18, 19, 20, 21} samples of 20 instances were obtained (values rj, at, ui, qj, cj were chosen with uniform distribution between 1 and rmax, amax, Umax, qmax, Cmax, respectively). We set ama x = 50, Uma x = 25 , rma x = qmax = nk, Cma x = 100 and c = 100 (see (1) for the definition of c). Thus, 600 instances were tested. It should be stressed that instances with k ~ {18, 19, 20} were reported by Carlier as the hardest one. For each instance we calculated following values: LB, K n and r/A = ((K A _ LB) /LB)* 100%, A ~ {Z, H} (it is quite obvious that ~TA >_ ( (K A _ K * ) /K *)* 100%, A e {Z, H}). The computational results are presented in Table 2.

The results form Table 2 show the dominance of Algorithm H. Namely, on 600 tested instances we obtained average value of r/H and ~7 z equal respectively to 1.25 and 6.68%. We found K H < K z in 547

Table 2 Computational results

Number Number Average Average Number of instances for which Average comput. of of of of K u < K z K H = K z K n > K z time of Alg. H jobs instances r/H ~7 z (in sec)

50 100 1.21 6.53 90 5 5 2 100 100 1.12 7.61 89 6 5 7 200 100 1.26 5.98 92 6 2 14 300 100 1.34 6.73 89 4 7 24 400 100 1.27 6.42 93 2 5 42 500 100 1.31 6.81 94 3 3 63

E. Novicki / A single-machine scheduling problem 81

cases, K H =K z in 26 cases and KH>K z in 27 cases only. Moreover, the computat ion t ime of A lgor i thm Z is by its structure twice longer than that of A lgor i thm H (Algor i thm Z is run twice for two various x).

Up to now there is not known in the l i terature any exact algor ithm for the prob lem (1)-(2). Though in theory such an algor i thm could be constructed based on branch-and-bound techniques, nevertheless the computat ion t ime of this algor ithm can be evaluated with respect to the computat ional t ime of an exact algor ithm for the pure sequencing prob lem 1/r j , qj /Cma x and therefore it could appear ineffective. For example, for est imation purposes we have employed the best known exact a lgor i thm (Carl ier [1]) for the prob lem 1/r j , qj /Cma x in calculat ion of LB. For n = 500 the average computat ional t ime of Car l ier 's algor ithm was 668 sec (12 instances from 100 tested have not been solved in t ime limit 1800 sec). It seems that an exact algor ithm for the prob lem (1)-(2) would work significantly longer.

Taking all these arguments into account, it seems that approximat ion A lgor i thm H can be recom- mended for pract ical appl ications.

Acknowledgements

We wish to thank the referees for their helpful suggestions and comments regarding an ear l ier version of this paper.

References

[1] Carlier, J. (1982), "The one-machine sequencing problem", European Journal of Operational Research 11, 42-47. [2] Daniels, R.L. (1990), "A multi-objective approach to resource allocation in single machine scheduling", European Journal of

Operational Research 48, 226-241. [3] Elmaghraby, S.E., (1977), Activity Networks, Wiley, New York. [4] Grabowski, J., and Janiak, A. (1986), "Job-shop scheduling with resource-time models of operations", European Journal of

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