Complexity results for flow-shop problems with a single server

New complexity results are derived. It is shown that the two machine flow shop problem with a single server is NP-hard if either the setup times or the processing times are constant. Furthermore, polynomial algorithms are given for special cases in which both setup times and processing times are constant. Flow-shop problems with a single server are generalizations of classical flow-shop problems and can be formulated as follows. We are given m machines M1,...,Mm and n jobs j=1,...,n. Each job j consists of m operations Oij (i=1,...m) which have to be processed in the order O1j → O2j → ... → Om.j. Operation Oij has to be processed on machine Mi without preemption for pij ≥ 0 time units. Each machine can only process one operation at a time. Immediately before processing an operation Oij it has to be prepared for processing on the corresponding machine, which takes a setup time of sij time units. During such a setup the machine is also occupied for sij time units, i.e. no other job can be processed on it. The setup times are assumed to be separable from the processing times, i.e. a setup on a subsequent machine may be performed while the job is still processed on the preceding machine. All setups have to be done by a single server which can perform at most one setup at a time. The goal is to determine a feasible schedule which minimizes a given objective function. To specify classes of scheduling problems we use the well-known α|β|γ-notation, which has been extended to server problems by [6]. In the α-field, indicating the machine environment, we write F since we are given a flow-shop environment. If the number m of machines is fixed (i.e. it is not a part of the input), we write Fm. Additionally, the entry S1 indicates the presence of a single server. In the β-field, indicating the job characteristics, we write pij=p if constant processing times are given. The entry rj denotes that release dates rj are given for the first setups of the jobs (i.e. the setup for the first operation O1j of job j cannot be started before time rj). Additionally, we add some information on the setup times and distinguish arbitrary setup times sij, constant setup times sij=s and unit setup times sij=1. We assume that sij=0 does not mean that the setup for operation Oij is missing, but that the server is required for an arbitrary small time to perform this setup. As common in scheduling theory all numerical data are supposed to be non-negative integers. If weights wj are given, we additionally assume wj ≥ 0. The γ-field contains one of the well-known objective functions Cmax, Lmax, Σ Cj, Σ wjCj, Σ Uj, Σ wjUj, Σ Tj, and Σ wjTj. Complexity results for flow-shop problems with separable setup times (but with sufficient servers, i.e. all setups can be performed simultaneously) were obtained by [9]. For a survey on problems with setups cf. [1] and [3]. Complexity results for shop problems with a single server were obtained by [4] who studied two-machine flow shops where the server moves in some predetermined patterns. [5] studied a more general environment with dedicated machines. Parallel machine problems with a single server were considered by [6], [7], [2], [8]. In this paper we derive some structural properties and new complexity results for the flowshop problems with a single server. The following two theorems provide some general reductions. Theorem 1 shows that flow-shop problems with a single server, arbitrary processing times and unit setup times are at least as difficult as the corresponding classical flow-shop problems without setup times. Theorem 2 shows that flow-shop problems with a single server, unit processing times, but arbitrary setup times are at least as difficult as the corresponding single-machine problems with arbitrary processing times. Theorem 1: For α∈{m,ο}, arbitrary processing times pij, β∈{rj,ο} and all objective functions γ∈{Cmax, Lmax, Σ Cj, Σ wjCj, Σ Uj, Σ wjUj, Σ Tj, Σ wjTj} problem Fα|β|γ reduces polynomially to Fα, S1|β, sij=1|γ. Thus, if the classical flow-shop problem Fα|β|γ is NP-hard, also problem Fα, S1|β, sij=1|γ is NP-hard. Theorem 2: For α∈{m,ο}, β∈{rj, ο} and all objective functions γ∈{Cmax, Lmax, ΣCj, Σ wjCj, Σ Uj, Σ wjUj, Σ Tj, Σ wjTj} problem 1|β|γ reduces polynomially to Fα, S1|pij=1, β|γ. By this general reduction several NP-hardness results for problems with a single server can be derived. Problems Fm,S1|pij=1 Σ wjUj and Fm, 1|pij=1|Σ Tj are NP-hard for each fixed number m≥ 2 of machines, since the corresponding single-machine problems are NP-hard. Furthermore, problems Fm,S1|pij=1, rj|Σ Cj, Fm,S1|pij=1, rj|Lmax, and Fm, S1|pij=1 |Σ wjTj are strongly NP-hard for each fixed m≥ 2, since the corresponding single-machine problems are strongly NP-hard. Further NP-hardness results are provided by the next two theorems. Theorem 3: Problem F2, S1|sij=s|Cmax is NP-hard in the strong sense. This result is derived by a reduction from numerical matching with target sums. Theorem 4: Problem F2, S1|pij=p|Cmax is NP-hard. The corresponding reduction is from the partitioning problem. Finally, we derived polynomial algorithms for the following problems. F2,S1 | pij=1,rj |Cmax F2,S1 | pij=p, sij=s,rj |Cmax F,S1 | pij=1, sij=s, rj |Cmax F,S1 | pij=p, sij=s | Cmax F2,S1 | pij =p, sij=s, rj | Σ Cj F,S1 | pij=1, sij=s, rj | Σ Cj F2,S1 | pij=1 |Σ wjCj F2,S1 | pij=p, sij=1,rj | Σ wjCj F2,S1 | pij=1 | Σ Uj F2,S1 | pij=p, sij=1,rj | Σ wjUj F2,S1 | pij =p, sij=s |Σ wjUj F,S1 | pij =1, sij=s |Σ wjUj F2,S1 | pij =p, sij=1, rj |Σ Tj F2,S1 | pij =p, sij=s | Σ wjTj F,S1 | pij =1, sij=s |Σ wjTj A complete overview of complexity results for flow-shop problems with a single server as well as a list of still open problems can be found at the web-site http://www.mathematik.uni-osnabrueck.de/research/OR/class/.