Abstract
Chapter 2 provides in Section 2.1 the general assumptions and notations employed and introduces in Sections 2.2 – 2.5 four different problem classes of project scheduling problems. Each problem class is outlined in terms of a verbal description and a formal 0–1 programming model, respectively. Additionally, variants and special cases as well as complexity results are given.
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Generally, the term activities is used for project scheduling problems (cf. Moder et al. (1983) and Elmaghraby (1977)), the terms jobs and operations, respectively, are employed for (static) job shop and flow shop problems (cf. Baker (1974) and French (1982)), while tasks are referred to within processor scheduling (cf. Blazewicz et al. (1986, p. 229)).
It is assumed that once an activity is started, it has to be processed continuously until it is finished. Generally, preemption can be considered in two different ways: (i) In the discrete case, preemption is allowed at a priori defined points by splitting an activity into several subactivities with an integer duration of at least one period (cf. Davis / Heidorn (1971) and Remark 3.7). (ii) In the continuous case, preemption is allowed at any time moment, i.e. a continuous point in time (cf. Blazewicz et al. (1986, p. 163)).
More elaborated temporal constraints to represent minimum and maximum time lags between starting times and / or completion times of any two activities (cf. Bartusch et al. (1988)) are not treated in here.
Beside this activity-on-node (AON) representation, precedence relations can be depicted by the activity-on-arc (AOA) representation, which is not considered in here. Neither are the pros and cons of the two representations as well as the translation of AON to AOA networks subject of this work. For details refer to Elmaghraby (1977, p. 3), Kamburowski et al. (1993), Krishnamoorthy / Deo (1979), and Syslo (1984).
Note that in the literature the terms earliest and latest as well as early and late finish (start) times are used (cf. Moder et al. (1983, p. 74) and Talbot (1982)). Under the assumptions made, due to precedence constraints, no earlier (later) finish or start times exist when the project is started in r=0 or has to be finished at a pre-specified period. Hence, it is referred to “earliest” and “latest”.
Since the activities are topologically ordered, the earliest finish time of each activity preceding activity, j, 1 <j ≤ J, is determined. Because of transitivity of the precedence relations, only the subset of immediate predecessors has to be considered.
Note that since the modes are ordered by non-decreasing duration, the earliest finish times of the MMPSP example match those of the SMPSP, while the latest finish times of the MMPSP equal the latest finish times of the SMPSP example plus the difference between the upper bound of MMPSP and SMPSP, i.e. here 19–13=6.
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© 1995 Springer-Verlag Berlin Heidelberg
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Kolisch, R. (1995). Description of the Problems. In: Project Scheduling under Resource Constraints. Production and Logistics. Physica, Heidelberg. https://doi.org/10.1007/978-3-642-50296-5_2
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DOI: https://doi.org/10.1007/978-3-642-50296-5_2
Publisher Name: Physica, Heidelberg
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