Computing 60, 73-79 (1998)
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© Springer-Verlag 1998 Printed in Austria
Short Communication
Makespan Minimization in Preemptive Two Machine Job Shops S. v. Sevastianov* and G. J. Woeginger**, Graz Received August 30, 1996; revised April 18, 1997
Abstract
In this note we investigate the NP-complete problem of minimizing the makespan in a preemptive two machine job shop. We present a polynomial time approximation algorithm with worst case ratio 3/2 for this problem, and we also argue that this is the best possible result that can be derived via our line of approach.
AMS Subject Classifications: 68Q25, 68C15, 90B35. Key words: Scheduling, approximation algorithm, worst-case analysis, job shop.
1. Introduction In the preemptive two machine job shop problem, there are two machines, M A and MB, and n jobs J = {Jl,--., Jn}- For 1 < i < n, the job Ji consists of a chain of r i > 1 operations Oij where 1 < j < r i. Let r = ET=lrs denote the total number of operations. The processing of operation Oij requires pij time units; every operation O~j is either assigned to machine MA or to machine M B where it has to be processed. For 1 < i < n and 1 < j < rg - 1, the processing of operation Ogj has to be completed before the processing of operation O~,j+1 may start. In processing the operations, preemption is allowed, i.e. one may interrupt an operation and resume its processing lateron. At any time, every job can be processed by at most one machine and every machine can process at most one job. The goal is to compute a schedule that minimizes the makespan, i.e. the maximum completion time of all jobs. The optimum makespan is denoted by Cmax. In the standard notation of Lawler, Lenstra, Rinnooykan and Shmoys [8], this scheduling problem is denoted by J2[pmtnlCma x.
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S.V. Sevastianov and G. J. Woeginger
The problem J21pmtnlCma x is NP-hard in the strong sense (Gonzalez and Sahni [3]). Only very recently, Brucker, Kravchenko and Sotskov [2] proved that even the subproblem J21n = 3, pmtnlCma x is NP-hard in the ordinary sense, i.e. the subproblem where there are only three jobs! We only know of three polynomially solvable special cases of problem J21pmtnlCmax: the special case where there are only two jobs (cf. Akers [1]); the special case where every job has at most two operations (Jackson [7]); the special case where all operations Oij have unit processing time and where operations Oq and O~,j+1 are always assigned to different machines (Hefetz and Adiri [6]).
Hence, the problem arises of obtaining polynomial time approximation algorithms for this problem, i.e. fast algorithms which construct schedules whose makespan is not too far away from the optimum makespan. If for some real p > 1, an approximation algorithm always delivers a solution with makespan at m o s t pCma x then we say that its worst case ratio (or performance ratio) equals p. For the non-preemptive job shop problem Jml" ICmax with a constant number m of machines, Shmoys, Stein and Wein [10] provide approximation algorithms with worst case ratios (2 + e), for arbitrary e > 0. It is easily verified that the analysis in [10] also applies to the preemptive version JmlpmtntCma x, for which it yields the same worst case guarantee (2 + e). For the case m = 2, this result is not impressive: Any reasonable schedule (that does not introduce simultaneous idle time on both machines) has a makespan of at most 2Cmax. * In this note, we derive a polynomial time approximation algorithm with worst case ratio 3 / 2 for the problem J21pmtnlCm~ x. This is the first non-trivial approximation result for this problem. We will also argue that this is the best possible result that can be derived via our line of approach.
Related Results Closely related to the job shop problem are the flow shop and the open shop problem. In both of these problems, every job consists of m operations where the j-th operation is assigned to the j-th machine. In the flow shop, every job must go through the machines in the same ordering; in the open shop, the processing order of the operations is immaterial. Although these three scheduling problems seem to be pretty similar to each other at first sight, the job shop turned out to be more difficult to attack than the open shop and the flow shop: If the number of machines is bounded, the flow shop (Hall [5]) and the open shop (Sevastianov and Woeginger [9]) possess polynomial time approximation schemes, i.e. families of polynomial time (1 + e)-approximation algorithms over
Makespan Minimization in Preemptive Two Machine Job Shops
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all e > 0. For the job shop with a bounded number of machines, deciding the existence of such an approximation scheme is an outstanding open problem.
Organization of the Note
Section 2 defines the notation and gives some preliminaries. In Section 3, we introduce a simple pairing procedure that is called COMBINE for the preemptive two machine job shop problem. In Section 4, we design and analyze our approximation algorithm with worst case ratio 3 / 2 for J21pmtnlCm,x; this algorithm uses procedure CoMmr,m as a subprogram. Section 5 closes the note with a short discussion.
2. Preliminaries
Without loss of generality assume that the processing times Pij of all operations Oij are integer. For every job ~, 1 < i < n, denote by a~ the total time it has to be processed on machine MA, by b i the total time it has to be processed on machine Me, and by Pi = ai + bi its overall processing time on both machines. For J c f any subset of the jobs, define a(J) = Z l i e j a i , b(J) = E li~jbi and p ( J ) = Eji~jpi. Moreover, we will simply write A , B and P short for a ( f ) , b ( J ) , and p ( f ) , respectively. By Pmax= max~ p~ denote the length of the longest job. Finally, define (1)
A - max{ A , B, Pmax} ,
and observe that the inequality A< -- C* max
(2)
holds. In fact, we will show that also Cm, * x < 3/2A holds. For an instance I of J2]pmtnlCma x construct its corresponding normalized instance I* as follows. (1) If /)max> max{A, B} holds, then let Jk denote the uniquely defined job with p~ =Pmax- Introduce two dummy jobs Jn + 1 and Jn+2 with an+ a = ak + bk - A , bn+ 1 = 0, an+ 2 = 0 and bn+ 2 = a k + b~ - B. It is easy to see that the dummy jobs cannot increase the optimum makespan: In any feasible schedule for I, job Jk is processed for b k time units on machine M B. During these b k time units, machine M A is busy for at most A - a k time units. Hence, the dummy operation an+ a can be processed during the remaining time units. For dummy operation bn+ 2, one argues symmetrically. (2) If A > max{B, Pmax}, then introduce a dummy job Jn +1 with a n + 1 = 0 and bn + 1 A - B , and if B > max{A, Pmax}, then introduce a dummy job Jn+ a with an+ 1 = B - A and bn+ 1 = 0. Also these dummy jobs do not increase the optimum makespan. For the resulting normalized instance I* =
Pmax _
max4_ 0?
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References
[1] Akers, S. B.: A graphical approach to production scheduling problems. Oper. Res. 4, 244-245 (1956). [2] Brucker, P., Kravchenko, S. A., Sotskov, Y. N.: Preemptive job-shop scheduling problems with a fixed number of jobs. Osnabriicker Schriften zur Mathematik, 184, Universit~it Osnabriick, Germany (1997). [3] Gonzalez, T., Sahni, S.: Flowshop and jobshop schedules: complexity and approximation. Oper. Res. 26, 36-52 (1978). [4] Graham, R. L.: Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math. 17, 416-429 (1969). [5] Hall, L. A.: Approximability of flow shop scheduling. In: Proceedings of 36th IEEE Symposium on Foundations of Computer Science, pp. 82-91 (1995). [6] Hefetz, N., Adiri, I.: An efficient optimal algorithm for the two-machine, unit-time, jobshop, schedule-length problem. Math. Oper. Res. 7, 354-360 (1982). [7] Jackson, J. R.: An extension of Johnson's result on job lot scheduling. Naval. Res. Log. Q. 3, 201-203 (1956). [8] Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G., Shmoys, D. B.: Sequencing and scheduling: algorithms and complexity. In: Handbooks in Operations Research and Management Science, Vol. 4, pp. 445-522. Amsterdam: North Holland, 1993. [9] Sevastianov, S. V., Woeginger, G. J.: Makespan minimization in open shops: a polynomial time approximation scheme. Math. Program. to appear (1997). [10] Shmoys, D. B., Stein, C., Wein, J.: Improved approximation algorithm for shop scheduling problems. SIAM. J. Comput. 23, 617-632 (1996). S. V. Sevastianov Institute of Mathematics Siberian Branch of the Russian Academy of Sciences Universitetskii pr. 4 630090 Novosibirsk-90 Russia e-mail: [email protected]
G. J. Woeginger Institut fiir Mathematik Graz University of Technology Steyrergasse 30 A-8010 Graz Austria e-mail: [email protected]
