Complexity of mixed shop scheduling problems: A survey

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Abstract

We survey recent results on the computational complexity of mixed shop scheduling problems. In a mixed shop, some jobs have fixed machine orders (as in the job shop), while the operations of the other jobs may be processed in arbitrary order (as in the open shop). The main attention is devoted to establishing the boundary between polynomially solvable and NP-hard problems. When the number of operations per job is unlimited, we focus on problems with a fixed number of jobs.

Introduction

Optimization of a schedule plays a vital role in modern manufacturing, planning and control systems. Optimal multi-stage processing can be described in terms of a shop scheduling problem in which a set of jobs has to be sequentially allocated to a set of machines. In dependence on the processing discipline, multi-stage systems are classified as open shop (O), flow shop (F), and job shop (J) [23]. In an open shop, each job is processed on each machine exactly once and the order of operations is immaterial. In a flow shop, the operations of each job have to be done in the same given machine order, i.e. the jobs have to follow the same route through the machines. In a job shop, which is a generalization of the flow shop, the machine order is given for each job, but different jobs may have different routes.

Often, a real life planning and control system can be a mixture of the above `pure' shops. Therefore, a mixed shop (X) has been introduced recently, in which the routes are given for some jobs as in the flow shop or job shop, and the routes of the others are immaterial as in the open shop. Using the term `mixed shop' for this more general type of a multi-stage system, we follow [7] (meanwhile in Ref. [25], a mixture of a flow shop and an open shop was called a mixed shop). We introduce the following notations. Let N={J1,…,Jn} denote a set of jobs, M={M1,…,Mm} denote a set of machines, and ri, i=1,…,n, denote the number of operations of the job JiN. In the case of the open shop or flow shop, we have ri=m for i=1,…,n, while in the case of the job shop any job after processing on a certain machine may require the same machine later and so ri may be larger than m. Consequently, in a job shop and in a mixed shop, the parameter r=max{ri|i=1,…,n} may be arbitrarily large even if both n and m are fixed (i.e. if both n and m are positive constants).

In the mixed shop, the set of jobs N is split into two subsets N=NO∪NJ, where the jobs of the set NO are processed as in an open shop and the jobs of the set NJ are processed as in a job shop. We denote by nO and nJ the cardinality of the set NO and NJ, respectively. If the set of jobs NO is empty (nO=0), we have a job shop, if the set of jobs NJ is empty (nJ=0), we have an open shop, if NO≠∅ and NJ≠∅, we have a mixed shop. In this survey, we consider only deterministic scheduling problems, i.e. the processing times pij of all operations Oij are supposed to be known in advance (before scheduling). Hereafter, Oij denotes the operation of job Ji on machine MjM, if Ji∈NO, or the operation of job Ji at stage j=1,…,ri, if Ji∈NJ. For each job Ji∈NJ the technological route is given which defines a linear ordering of all its operations Oi1Oi2→⋯→Oi,ri. It implies that operation Oij cannot start before operation Oi,j−1 has been completed.

If preemption of operation is forbidden, a schedule is uniquely defined by the completion times Cij of operations Oij. If preemption is allowed, the processing of operation Oij may be interrupted at any time and resumed later on the same machine, the total duration of all parts of that operation being equal to pij. In both cases, the completion time Ci of job Ji is defined as the time when the last operation of the job is completed Ci=max{Cij|MjM} for Ji∈NO, and Ci=Ci,ri for Ji∈NJ. The makespan of the schedule is defined as the time when the last job is completed Cmax=max{Cij|iN}.

A shop scheduling problem is to find such a schedule which minimizes the value of a given objective function F=F(C1,…,Cn) which is assumed to be nondecreasing in the job completion times (such a criterion is called regular). To denote a scheduling problem, we follow the classification scheme α|β|γ [15], where α,β, and γ describe the machine environment, the job characteristics, and the objective function, respectively. The first field α denotes the type of the system O,F,J, or X, and for the problems with a fixed number of m machines, this type is followed by parameter m. If the number of jobs is fixed, then the corresponding equality for n is included in the second field β. If the job characteristic β includes the parameter pmtn, then operation preemption is allowed, otherwise preemption is forbidden. If characteristic β includes some expression with parameter ri, then there are additional restrictions on the number of operations per job. If parameter ri is omitted, the number of operations for the jobs from NJ is unlimited. The third field γ specifies the objective function. For instance, the problem of constructing a schedule with minimum makespan for nonpreemptive processing two jobs in a mixed shop is denoted by X|nJ=1,nO=1|Cmax and the same problem with preemption is denoted by X|nJ=1,nO=1,pmtn|Cmax.

The theoretical investigation of mixed shop problems has been initiated by Masuda et al. in 1985 [25] while constructing an O(nlogn) algorithm for the two-machine case. This algorithm may be considered as a generalization of the two well-known algorithms developed for the two-machine `pure' shops

  • an O(nlogn) algorithm by Johnson [18] for the flow shop problem, and

  • an O(n) algorithm by Gonzalez and Sahni [13] for the open shop problem.

Since the `pure' shop problems become NP-hard for m=3, the two-machine mixed shop problem is the maximal polynomially solvable problem for rim (provided that P≠NP). A review of the results on mixed shop problems under the assumption rim is given in Section 2.

In 3 Nonpreemptive case with, 4 Preemptive case with, we summarize the results on mixed shop problems under the alternative assumption ri>m, when the number of operations per job is unlimited. Due to the NP-hardness of these problems for m=2, we focus on a mixed shop problem with a fixed number of jobs. It should be noted that the investigation of `pure' shops with a fixed number of jobs has been started in the paper by Akers and Friedman [1] with presenting a geometrical approach to solve the job shop problem with two jobs. Later a number of publications appeared with polynomial algorithms based on this approach (see e.g. 5, 16, 29, 30, 38). For different `pure' shops with fixed number of jobs, a boundary between polynomially solvable and NP-hard problems has been established in 6, 8, 9, 19, 20, 28, 30, 32 and surveyed in 31, 33.

In Section 5, we observe the complexity results on mixed shop problems under precedence constraints given on the set of jobs along with a linear ordering of the operations per job. In Section 6, we give an overview of the known results for other regular criteria, e.g. for minimizing the total flow time ∑Ci=∑i=1nCi and minimizing the maximum lateness Lmax=max{CiDi|i=1,…,n}, where Di is a given due date for the completion of job JiN. Some concluding remarks are given in Section 7.

Section snippets

Mixed shop problem with rim

The paper by Masuda et al. [25] deals with the two-machine problem with the set of jobs N=NO∪NJ, all jobs of the subset NJ having the same routes (M1,M2), i.e. the jobs from the set NJ are processed as in a flow shop. In that paper, an O(nlogn) algorithm was proposed to construct a schedule with minimum makespan. This result was improved by Strusevich in 34, 36, an O(nO+nJlognJ) algorithm was developed for the two-machine mixed shop problem with jobs from the set NJ having different routes (M1,M

Nonpreemptive case with ri>m

As it was observed in Section 2, the three-machine mixed shop problem is unary NP-hard. Moreover, the two-machine problem with ri>m is unary NP-hard if at least one parameter nO or nJ is unlimited:

  • the unary NP-hardness of the two-machine problem with nO=0 and unlimited nJ was proved by Lenstra and Rinnooy Kan [23], and

  • the unary NP-hardness of the two-machine problem with nJ=1 and unlimited nO was proved by Shakhlevich et al. [27].

Thus, to establish a boundary between NP-hard and polynomially

Preemptive case with ri>m

Now we consider the preemptive case of mixed shop problems with an unlimited number of operations per job. While the nonpreemptive case of the two-machine problem with ri>m is NP-hard for unlimited nO or nJ, the preemptive case of the same problem is polynomially solvable for nJ=2 and unlimited parameter nO. An O(r3+nO) algorithm for the latter problem was proposed by Shakhlevich et al. [27]. The main idea of the algorithm is to split the initial problem X2|nJ=2,pmtn|Cmax into two `pure'

Precedence constraints

A comprehensive study of the two-machine mixed shop problem under precedence constrains has been carried out by Strusevich. We resume the results presented in 35, 37, where two types of precedence constraints given on the set of jobs have been introduced:

  • for the first type, job Ji precedes job Jj if and only if the processing of job Jj on any machine cannot start before the processing of job Ji is completed on all the machines;

  • for the second type, job Ji precedes job Jj if and only if the

Other regular criteria

In 2 Mixed shop problem with, 3 Nonpreemptive case with, 4 Preemptive case with, 5 Precedence constraints, we have given a complexity classification of mixed shop problems with the objective function Cmax. For mixed shop problems with other criteria essentially less complexity results are known.

Mixed shop problems with the objective functions ∑Cj and Lmax are NP-hard if m=2 since the corresponding `pure' shop problems are NP-hard (see Table 7Table 8). Since the mixed shop problem with m=2 is NP

Conclusion

Most real life scheduling problems are essentially more complicated than those considered in scheduling theory. In this paper, we surveyed recent results on a scheduling environment when some jobs have fixed technological routes (like in a job or flow shop) while the others may follow an arbitrary machine order (like in an open shop). Since a mixture of problem settings is more general, usually it is more difficult than a `pure' shop problem. However, the main issue of this survey is that the

Acknowledgements

This survey was prepared under support of the International Association for the Promotion of Cooperation with Scientists from the Independent States of the Former Soviet Union (projects INTAS 93-257 and INTAS 96-0820) and by Deutsche Forschungsgemeinschaft (project ScheMA). The second author was supported also by the International Science and Technology Center (project ISTC B-104-98). The authors wish to thank the anonymous referees for their comments and suggestions.

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