The theory of deterministic sequencing and scheduling has expanded rapidly during the past years. In this paper we survey the state of the art with respect to optimization and approximation algorithms and interpret these in terms of computational complexity theory. Special cases considered are single machine scheduling, identical, uniform and unrelated parallel machine scheduling, and open shop, flow shop and job shop scheduling. We indicate some problems for future research and include a selective bibliography.

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Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey

NP-complete problems form an extensive equivalence class of combinatorial problems for which no nonenumerative algorithms are known. Our first result shows that determining a shortest-length schedule in an m-machine flowshop is NP-complete for m ≥ 3. For m = 2, there is an efficient algorithm for finding such schedules. The second result shows that determining a minimum mean-flow-time schedule in an m-machine flowshop is NP-complete for every m ≥ 2. Finally we show that the shortest-length schedule problem for an m-machine jobshop is NP-complete for every m ≥ 2. Our results are strong in that they hold whether the problem size is measured by number of tasks, number of bits required to express the task lengths, or by the sum of the task lengths.

https://www.jstor.org/stable/pdfplus/3689278.pdf

The Complexity of Flowshop and Jobshop Scheduling

In a general flow-shop situation, where all the jobs must pass through all the machines in the same order, certain heuristic algorithms propose that the jobs with higher total process time should be given higher priority than the jobs with less total process time. Based on this premise, a simple algorithm is presented in this paper, which produces very good sequences in comparison with existing heuristics. The results of the proposed algorithm have been compared with the results from 15 other algorithms in an independent study by Park [13], who shows that the proposed algorithm performs especially well on large flow-shop problems in both the static and dynamic sequencing environments.

A heuristic algorithm for the m-machine, n-job flow-shop sequencing problem

We survey and extend the results on the complexity of machine scheduling problems. After a brief review of the central concept of NP-completeness we give a classification of scheduling problems on single, different and identical machines and study the influence of various parameters on their complexity. The problems for which a polynomial-bounded algorithm is available are listed and NP-completeness is established for a large number of other machine scheduling problems. We finally discuss some questions that remain unanswered.

/pdf/complexity-of-machine-scheduling-problems-3kmjz4hlnd.pdf

Complexity of machine scheduling problems

Various optimizers for single‐stage production

Each of a collection of items are to be produced on two machines (or stages). Each machine can handle only one item at a time and each item must be processed through machine one and then through machine two. The setup time plus work time for each item for each machine is known. A simple decision rule is obtained in this paper for the optimal scheduling of the production so that the total elapsed time is a minimum. A three-machine problem is also discussed and solved for a restricted case.

/pdf/optimal-two-and-three-stage-production-schedules-with-setup-1o4p1q37d5.pdf

Optimal two- and three-stage production schedules with setup times included

: An improved solution is presented for the problem of finding the smallest number of direct pairwise comparisons which will always suffice to rank n objects according to some transitive characteristic. In his book, Mathematical Snapshots, Steinhaus discusses the problem of ranking n objects according to some transitive characteristic, by means of successive pairwise comparisons. In this paper, the terminology was adopted of a tennis tournament by n players. The problem may be briefly stated: 'What is the smallest number of matches which will always suffice to rank all n players.'

Selmer Martin Johnson

Papers

Optimal two- and three-stage production schedules with setup times included

A Tournament Problem