Abstract
This chapter is dedicated to the development of algorithms for the single-machine case (α = 1). Algorithms differ with respect to batching types and sequence dependent or independent setups. In Section 3.1, we describe the basic enumeration scheme which is used by the branch and bound (B&B) algorithms for the batching types ia-pb, ia-npb and ba. Section 3.2 deals with ia-pb, for which we present two exact and one heuristic algorithm and study their performance at the end of the section. Section 3.3 is concerned with ia-npb; in the B&B algorithm we must derive a minimum cost schedule from a job sequence (cf. Section 3.3.1.1), and dominance rules become more complicated than for ia-pb (cf. Section 3.3.1.2). For sequence independent setups a genetic algorithm handles batching type ia-npb (cf. Section 3.3.2) as well as ba (cf. Section 3.4.2). On the other hand, for ba in Section 3.4 the B&B algorithm again requires modification. We derive a late schedule from a sequence in Section 3.4.1.1 and state the dominance rule for ba in Section 3.4.1.2. Computational experiments are presented at the end of Sections 3.3 and 3.4, respectively. Successively solving the single-machine case, the genetic algorithm can also be used for the flow-shop case, which we describe in Section 3.4.4.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
If only the minimization of setup costs is considered, a forward sequencing scheme is more appropriate because there is an optimal schedule which forms a block (Theorem 2.4). Hence, we will not consider this objective in the B&B algorithms.
With this notation we refer to the concatenation of two vectors, and not to the mathematical definition where (ωs, σS) means a pair of vectors.
Especially for ba this may be suboptimal if a non-late schedule has lower costs.
The implementation uses ideas of Baker and Schrage [9].
We also implemented other storing schemes: (i) information is stored only if ASs is a subset SM of all jobs, and SM is determined such that the information of all ASs ∈ SM can be stored within the given memory limit, (ii) we just continued enumeration though no new information could be stored. The computational experience was disappointing in both cases. But in general, algorithms tend to exceed a certain time limit first before exceeding the memory limit, i.e., if we encounter memory problems for instances with large N, we encounter running time problems, too. Hence, we do not examine any other improved storage scheme.
But the time-cost tradeoff also allows for the new schedule to be stored though t(σs) >t(σ snew ) if c(σ snew ) ≪ c(σs). Then, there may be a feasible extension for σs but none for σ snew . Nevertheless, this “storing scheme” turned out to be more efficient.
B&B[loia-pb] can also be used as a heuristic if enumeration is stopped after a certain time limit, which is called truncated branch and bound (TBB). But the solution quality of TBB is disappointing, which suggests that a lot of time in B&B[loia-pb] is also needed to find the optimal solution and not only to prove its optimality.
The instance generator in [125] does not guarantee ni > 0 ∀i = 1,…, N. Therefore, we generated a larger number of instances and took only those with rij > 0 ∀i = 1,…, N.
CPU times for TSWS are reported on a Dell510 PC, which we assume to be 10 times slower than ours, so Ravg for TSWS is multiplied by 0.1 to adjust for this comparison.
Nevertheless, running times in Table 3.4 give the time for the minimization of setup costs. Setup costs are set equal to setup times so that the [loia-pb] algorithms B&;B, DP and C&I minimize total setup time.
Deriving a schedule from a sequence is called timetabling in French [51] (p. 26).
The model of ia-pb in Table 2.4 is a relaxation of the ia-npb model in Table 2.5 where constraints (2.7) and Yk are omitted. Thus, ZBSP(σ) ≤ ZBSP(σ)
E.g., for sequence independent setups, the computations in Table 3.5 can be restricted to gsk. Then, the maximal block size to be considered at position 3 is 3: the setup between (3,3) and (2,2) would be independent of the idle time between both jobs.
For a general introduction to genetic algorithms cf. e.g. Goldberg [57]. For the design of genetic algorithms for scheduling problems cf. Liepins and Hilliard [85]. In Rubin and Ragatz [107] an application to a problem with sequence dependent setups is presented.
Different concepts for encodings are presented e.g. in Bean [11].
This is also the reason why Aavg and Amax for GA[loia-npb,sti]-short is smaller than for GA[loia-npb,sti] in (ρ, θ) = (L,l). By chance, GA[loza-npb,sti]-short generated a better solution though running time is shorter than for GA[loia-npb,sti].
More precisely, consider σ3 as the late schedule in Figure 3.11. In σ3 we have C(2,2) = 7 but considering only a 3-partial schedule we have C(2,2) = 10 in σ2.
In Theorem 3.6 we do not calculate a late schedule when the bounding rule is applied but assume that the batch is enlarged.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Jordan, C. (1996). Methods for the Single-Machine Case. 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_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-48403-2_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61114-1
Online ISBN: 978-3-642-48403-2
eBook Packages: Springer Book Archive