# zbMATH — the first resource for mathematics

Scheduling jobs with release dates and tails on identical machines to minimize the makespan. (English) Zbl 0622.90049
The problem of scheduling jobs on m identical parallel machines is considered. Each job i has a release date $$r_ i$$ when it becomes available for processing, has a processing time $$p_ i$$ and has a tail $$q_ i$$ which represents the additional time that must elapse after processing to complete the job. The objective is to minimize the maximum completion time or makespan. A heuristic method is described in which, at each stage, an available job, chosen to have a tail as large as possible, is scheduled on the machine that completes its processing earliest. It is shown that the maximum deviation of the makespan given by this heuristic from the optimal value is less than twice the largest processing time. A branch and bound algorithm is proposed which applies the heuristic method at each node of the search tree. A special feature of the algorithm is the novel interval branching rule which is used. For this branching rule a deadline $$d_ i$$ on the completion of processing is required for each job i: initially it is computed using $$d_ i=UB-q_ i$$, where UB is the upper bound found from the heuristic. Branching is based on a time interval $$[r_ i,d_ i]$$ within which processing must be performed. Schedules are partitioned according to whether the processing for job i is performed in the interval $$[r_ i,r_ i+t+p_ i-1]$$ or in $$[r_ i+t,d_ i]$$ where, in this algorithm, $$t=\lceil (d_ i-r_ i-p_ i)/2\rceil$$. A lower bounding scheme is based on the result that by considering only jobs of a chosen subset, the sum of the m smallest release dates, the m smallest tails and all processing times is a lower bound on m times the optimal makespan. Computational results with the branch and bound algorithm for problems with up to 100 jobs and 8 machines are given.
Reviewer: C.N.Potts

##### MSC:
 90B35 Deterministic scheduling theory in operations research 68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
Full Text:
##### References:
 [1] Baker, K.R.; Su, Z.S., Sequencing with due dates and early start times to minimize tardiness, Naval research logistics quarterly, 21, 171-176, (1974) · Zbl 0277.90044 [2] Blazewicz, J., Deadline scheduling of tasks with ready times and resource constraints, Information processing letters, 2, 60-63, (1979) · Zbl 0401.90048 [3] Bratley, P.; Florian, M.; Robillard, P., On sequencing with earliest starts and due dates with applications to computing bounds for the n/m/G/FMAX problem, Naval research logistics quarterly, 20, 57-67, (1971) [4] Carlier, J., Problèmes d’ordonnancements à contraintes de ressources: algorithmes et complexité, Thèse d’état, (1984) [5] Carlier, J., Ordonnancements à contraintes disjonctives, Rairo, 12, 333-351, (1978) · Zbl 0401.90052 [6] Carlier, J., The one-machine sequencing problem, European journal of operational research, 11, 1, 42-47, (1982) · Zbl 0482.90045 [7] () [8] Garey, M.R.; Johnson, D.S., Computers and intractability: a guide to the theory of NP-completeness, (1979), Freeman San-Francisco · Zbl 0411.68039 [9] Grabowski, J., On two-machines scheduling with release and due dates to minimize maximum lateness, Opsearch, 17, (1980) · Zbl 0453.90043 [10] Grabowski, J.; Skubalska, E.; Smutnicki, C., On flow-shop with release and due-dates to minimize maximum lateness, Journal of the operational research society, 34, 615-620, (1983) · Zbl 0513.90042 [11] Jackson, J.R., Scheduling a production line to minimize maximum tardiness, () [12] Lageweg, B.J.; Lenstra, J.K.; Rinnooy Kan, A.H́.G., Minimizing maximum lateness on one machine: computational experience and some applications, Statistica neerlandica, 30, 25-41, (1976) · Zbl 0336.90029 [13] Lenstra, J.K., Sequencing by enumerative methods, (1976), Mathematisch Centrum Amsterdam · Zbl 0407.90025 [14] Mahon, G.B.Mc.; Florian, M., On scheduling with ready times and due dates to minimize maximum lateness, Operations research, 23, 475-482, (1975) · Zbl 0301.90024 [15] Potts, C.N., Analysis of a heuristic for one machine with release dates and delivery times, Operations research, 28, 1436-1441, (1980) · Zbl 0447.90041 [16] Rinnooy Kan, A.H.G., Machine scheduling problems: classification, complexity and computation, (1976), Nijhoff The Hague · Zbl 0366.90092 [17] Schrage, L., Obtaining optimal solutions to resource constrained scheduling problem, (1971), unpublished manuscript [18] Slowinski, R., Multiobjective network scheduling with efficient use of renewable and non-renewable resources, European journal of operational research, 7, 265-273, (1981) · Zbl 0455.90049 [19] Van Wassenhowe, L.; Gelders, L., Four solutions techniques for a general one machine scheduling problem: A comparative study, European journal of operational research, 2, 281-290, (1978) · Zbl 0378.90044 [20] Weglarz, J., Project scheduling with continuously-divisible, doubly constrained resources, Management science, 27, 1040-1053, (1981) · Zbl 0467.90033 [21] Schonhage, A.; Paterson, M.; Pippenger, M., Finding the Median, Journal of computer systems science, 13, 189-199, (1976) · Zbl 0335.68033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.