Discrete OptimizationAnalysis of flow shop scheduling anomalies
Introduction
Scheduling anomalies (paradoxes) is a well-discussed topic among computer scientists. The popularity of the subject can be gauged by over eighteen hundred citations for the paper by Graham (1966) on multiprocessor anomalies. In the traditional shop scheduling literature, Graham's paper has been occasionally cited; see for example, Rustogi and Strusevich (2013). In static deterministic flow shop scheduling, reporting and analysis of anomalies has been rare. Only two instances of an anomaly have been reported so far. The first instance was observed by Abadi, Hall and Sriskandarajah (2000) and Spieksma and Woeginger (2005) and the second instance was observed by Kalczynski and Kamburowski (2007).
In the present paper, we formally define three types of anomalies in flow shop scheduling. Our objective is to analyze the sensitivity of permutation flow shops to these anomalies by demonstrating 12 new anomalies. We also show that when no-delay, non-permutation schedules are allowed, some flow shop models become sensitive to anomalies even though their permutation counterparts are not. Beyond its theoretical interest, the identification of these anomalies has managerial significance as well; we will present some discussion on the managerial significance of our findings in the final section.
The paper is organized as follows. In Section 2, we define the classical flow shop model and its variants along with their schematic representation. We also define three types of anomalies. In Section 3, we present relevant results from the literature along with a more detailed analysis of the flow shop models introduced in the previous section. Section 4 presents anomalies for many variants of the classical permutation flow shop. Section 5 presents propositions and corollaries identifying permutation flow shop models not sensitive to anomalies. In Section 6, we present anomalies for a non-permutation flow shop model. Some discussion and the conclusions of this research are summarized in Section 7.
Section snippets
Definitions and assumptions
We use the standard flow shop model assumptions (job availability at time zero, continuously available machines, no preemption, etc.). We assume positive processing times for all operations. This implies that all jobs must visit all machines (no missing operations are allowed). For the m-machine, n-job flow shop, let pi, j, Ci, j denote the processing time of job Jj () on machine Mi () and the corresponding completion time for a given sequence respectively. By definition, the n
Relevant results
The origin of multiprocessing anomalies discussed by Graham (1966) may be found in Richards (1960); see also discussion in Manacher (1966). As stated in the introduction, only two flow shop anomalies have been discussed in the literature, namely a Type 1 anomaly for the Fm|nwt|Cmax model by Abadi et al. (2000) and by Spieksma and Woeginger (2005) and a Type 1 anomaly for the Fm|nit, perm|Cmax model by Kalczynski and Kamburowski (2007).
In this section, we present some relevant results and
Permutation flow shop models sensitive to anomalies
In all examples presented in this section, we begin with an optimal permutation schedule for the objective function under consideration (makespan or total flow time). After increasing the workload, we show that the objective function value decreases.
Permutation flow shop models not sensitive to anomalies
The sensitivity of permutation flow shops to anomalies depends on the following two properties stated next; these properties may or may not apply to a specific model.
Property 1 A critical path (i.e. the longest path for an mxn permutation flow shop is a continuous path of individual processing time elements pij connected by precedence relationships (i.e. as in Fig. 1) and the sum of these elements is the makespan of the schedule.
Property 2 Let S(i, [j]), C(i, [j]) denote the start time and the completion time
Anomalies in non-permutation flow shop models
Throughout the paper we have assumed no-delay schedule. This assumption can lead to both permutation and non-permutation schedules. If two jobs are waiting for processing when a machine becomes available, the use of the “first come first served (fcfs)” priority rule results in a permutation schedule. But if the fcfs rule is not used, a non-permutation schedule will be created.
Discussion and conclusions
We introduced new type 2 and type 3 anomalies according to which the addition of new jobs/machines to an optimal schedule reduces the optimal objective function value for certain flow shop models. For each anomaly for the Cmax objective in a premutation flow shop, we were able to show a similar anomaly for the ∑Cj objective; we also showed that an anomaly for the ∑Cj objective is precluded when it is also precluded for the Cmax objective.
The insights we gained include the use of dualities in
Acknowledgement
We would like the thank both reviewers for their thorough reviews. In particular, we want to thank reviewer 1, whose insightful comments helped us not only in improving the earlier version of this paper; it substantially increased our understanding of the subject matter.
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