Abstract
This chapter deals with artificial intelligence approaches for solving the BSP. We compare two solvers based on the artificial intelligence paradigm and a mixed-integer programming (MIP) solver with each other.
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The model in Table 2.4 also allows schedules where jobs are not in EDDWF; it is the result of the analysis in Section 2.6 that only EDDWF schedules need to be considered.
E.g. a feasibility bound as in Theorem 3.1 would represent algorithmic knowledge.
There is also a broad range of research activities in this field, for recent developments cf. e.g. Mitra and Maros [91].
A MIP solver works effectively if the MIP model generates “tight” LP-bounds. A lot of research (e.g. parts of polyhedral optimization) deals with the development of “tight” MIP models, cf. e.g. Eppen and Martin [47]. General statements on how to develop a “tight” MIP model systematically seem to be difficult to derive. As for specialized algorithms, the development of a “tight” MIP model is based on the analysis of the structural properties of the problem at hand.
Constraint propagation may be illustrated with the following (very simple) example: consider the domain variables Y, X, Y in 2..5, X in 1..4, and the constraint Y < X. Constraint propagation then reduces the domains to Y in 2..3 and X in 3..4. For each X (Y) there is always an Y (X) so that the constraint is satisfied.
If jobs are ordered in EDDWF, we can access or identify a job by an s-partial sequence of families. Consequently, if jobs are indexed with (i, j) we must employ either a backward or forward scheduling scheme according to an EDDWF precedence graph (cf. Section 3.1). But since we do not want to specify “how to solve the problem”, we must be able to access a job by a single index.
For a similar idea cf. Woodruff and Spearman [125].
E.g. with the query “d(l, 2, X).” the variable X is unified with d(1,2).
In Prolog, variables must start with an uppercase letter, also lists may be variables. In L = [] the variable L is the empty list, in L = [Head ∣ Tail] the variable Head is unified with the first element of the list, and Tail with the rest of the list. The comma operator ‘,’ represents the logical AND, and a period ‘,’ terminates a predicate.
More precisely, member( [F2, N2], ELJobs) chooses job [F2, N2] as the next job to schedule, and delete(ELJobs, [F2, N2], ELJobsDel) determines the jobs list ELJobsDel without [F2, N2]. If [F2, N2] is the last job of family N2, the deletion is correct, i.e. ELJobsNew=ELJobsDel, otherwise the next job of family N2 must be added so that ELJobsNew=[[F2,Nl] ∣ ELJobsDel]. Then, the attributes of [F2,N2] are determined and we can state the demand constraint Completion2 in O..Deadline and the sequencing constraint lessequal (Completion1,Completion2,Processing+Setup).
And then, for ELJobs=[], the first solve predicate succeeds.
E.g., the definition of constraint lessequal () uses the keyword “in”.
LINGO generates so called MPS files. MPS is a standard format for the representation of the coefficients of a MIP model. MPS files can be used as input for any MIP solver.
CPU times on the workstation and on the PC are in the same order of magnitude so that we do not use a correction factor for the slower PC.
We generated the instances similar to the generator in Section 2.7. The 3 instances in each problem class have different setup structures: sequence independent si, sequence sq with setup costs proportional to setup times and sequence sq but arbitrary setup costs.
At least not with the current implementation. For the instances in Table 3.2 (p. 66), clp(FD) solves the feasibility problem in a maximum time of 175 sec for (N, J) = (3, 25), but none of the (N, J) = (5, 18) instances is solved — even in a CPU time of several hours.
In Table 6.4, the construction of a schedule is formulated in terms of a rule, but in the same way we could include algorithmic knowledge, e.g. a feasibility bound.
But also for CHARME this depends on the way a problem is modeled. Due to modeling freedom, each MIP model can also be implemented in CHARME (but not vice versa, see above). However, implementing a MIP model in CHARME, the LP-based B&B algorithm solves the problem more efficient. One has to find a concise model to employ CHARME effectively.
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© 1996 Springer-Verlag Berlin Heidelberg
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Jordan, C. (1996). Artificial Intelligence Approaches. 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_6
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DOI: https://doi.org/10.1007/978-3-642-48403-2_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61114-1
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