Abstract
This chapter is devoted to a formal description and analysis of the Batch Sequencing Problem (BSP). We review the literature of batching and scheduling and define batching types in Section 2.1. The notation and a three-field descriptor for the BSP follow in Section 2.2; the descriptor will allow us to refer to the variants of the BSP in a short and concise manner. Formal models as well as an example for the single-machine case are presented in Section 2.3, multi-level and parallel machine models and examples in Section 2.4. We examine the complexity of the problems in Section 2.5, where it turns out that the BSP with nonzero setup times is NP-hard. Therefore, an analysis of structural properties in Section 2.6 is warranted. The results of Section 2.6 and the instance generator described in Section 2.7 are used when we present algorithms in Chapters 3 and 5.
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If setups are tool switches of a tool magazine, setups may be more complicated, cf. Locket and Muhlmann [88].
In the sequence independent case, setups can be associated with each job and batching is eliminated.
E.g., for the sum of completion times criterion, it is optimal to apply the shortest processing time (SPT) rule on both levels. First, jobs are ordered by SPT within the families, and then, with a sequence independent setup associated with each family, families are composite jobs. In the second step the families are ordered by SPT, giving the optimal schedule (cf. Webster and Baker [124]).
In the case of unit time jobs, i.e. P(i,j) = 1, this assumption is relaxed to d(i,j) +P(i,j+1) ≤d(i,j+i).
Processing times may now no longer be integer valued. However, integrality of the processing times is not needed to perform the next step of the algorithm.
For an introduction to the classical three-field descriptors for scheduling problems cf. e.g. Pinedo [99] or Domschke et al. [39].
Note that processing times must be divided by M = 2.
Echelon stock costs Wech in Figure 2.7 are We have d(2,1) = d(1,1) — P(1,1) and ―d(2,1) = C(1,1) — P(1,1) However, we can express Wech in terms of (physical) inventories as well, i.e. Now the earliness of (1,1) is weighted with (w(1,1) + w(2,1)) Equivalently, in geometric terms: the earliness minus inventory of (2,1) is the earliness of (1,1), see the dotted lines in Figure 2.7.
For the more general cases with uniform and heterogeneous machines, cf. e.g. Domschke et al. [39].
MAk is an (uppercase) decision variable in contrast to the (lowercase) parameter mai.
Cf. Figure 2.3 for an illustration. σ11 is feasible for ia-npb but not for ba.
We have no entry fam in β; consequently, the descriptor of Table 2.2 does not cover this case.
Cf. e.g. Domschke et al. [39].
Cf. also French [51] (Section 2.2) for the distinction between sequence and schedule.
Cf. also Bruno and Downey [22].
Consider e.g. the ia-pb schedule σ1 in Figure 2.3: leftshifting (3,2) and (3,3) to (3,1) as the first job in the group we obtain the ia-npb schedule σ11.
Each job can be represented as p(i j) unit time jobs with corresponding deadlines, cf. also p(i j)d(i,j) for BSPUT(..) in Figure 4.2, p. 102.
Order all jobs in the order of shortest weighted processing time (SWPT), cf. e.g. Domschke et al. [39].
Consequently, not every #b, vector is feasible (which leads to a problem only if N is small, e.g. N = 4). In the generators proposed in [121] and [125] the total number of batches is specified. But it is not guaranteed that the instances have a positive number of batches for each family. Small problems with a large number of families N and a small number of jobs J cannot be generated as proposed in [121] and [125].
Cf. Figure 2.5 for an illustration: families 4 and 5 go into family 3, thus, at least one batch of 4 and 5 must precede a batch of family 3.
And we cannot guarantee that the batch size is out of DU (#minjpb, #maxjpb).
Consequently, now deadlines may “interfere” and there is no longer d(i,j-1) < d(i j)-p(i j)
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© 1996 Springer-Verlag Berlin Heidelberg
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Jordan, C. (1996). Description of the Problems. In: Batching and Scheduling. Lecture Notes in Economics and Mathematical Systems, vol 437. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-48403-2_2
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DOI: https://doi.org/10.1007/978-3-642-48403-2_2
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