flow shop scheduling with multiple operations and time lags J. Riezebos and G.J.C. Gaalman Faculty of Management and Organization, University of Groningen J.N.D. Gupta Department of Management, Ball State University, Muncie April 1995 Publiced in the Journal of Intelligent Manufacturing, special issue on Production Planning and Scheduling, april 1995.
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flow shop scheduling with multiple operations and time lags
J. Riezebos and G.J.C. Gaalman
Faculty of Management and Organization, University of Groningen
J.N.D. Gupta
Department of Management, Ball State University, Muncie
April 1995
Publiced in the Journal of Intelligent Manufacturing, special issue on Production Planning and
Scheduling, april 1995.
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Abstract
A scheduling system is proposed and developed for a special type of flow shop. In this flow shop
there is one machine at each stage. A job may require multiple operations at each stage. The first
operation of a job on stage j cannot start until the last operation of the job on stage j-1 has
finished. Preemption of the operations of a job is not allowed.
The flow shop that we consider has another feature, namely time lags between the multiple
operations of a job. To move from one operation of a job to another requires a finite amount of
time. This time lag is independent of the sequence and need not be the same for all operations or
jobs. During a time lag of a job, operations of other jobs may be processed.
This problem originates from a flexible manufacturing system scheduling problem where,
between operations of a job on the same workstation, refixturing of the parts has to take place in a
load/unload station, accompanied by (manual) transportation activities.
In this paper a scheduling system is proposed in which the inherent structure of this flow shop is
used in the formulation of lowerbounds on the makespan. A number of lowerbounds are
developed and discussed. The use of those bounds makes it possible to generate a schedule that
minimizes makespan or to construct approximate solutions. Finally, some heuristic procedures for
this type of flow shop are proposed and compared with some well known heuristic scheduling
rules for job shop/flow shop scheduling.
1. INTRODUCTION
Flow shop scheduling problems can be encountered in scheduling flexible manufacturing
systems. To take advantage of the benefits of flexible manufacturing systems it is necessary to
give full attention to the scheduling of such systems.
The scheduling problem in this paper originates from an FMS scheduling problem where between
operations of a job on the same workstation refixturing of the part has to take place in a lo-
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ad/unload station accompanied by (manual) transportation activities. When a job has finished
processing on this station it has to be processed in the second machine cell of this FMS. This
particular FMS is described in (Aanen, 1988).
Seen as a flow shop, this FMS has some interesting characteristics:
1. A job may require multiple operations at a stage (e.g. workstation). These operations need not
be done consecutively: operations of other jobs may be processed in between two succeeding
operations of a job. Preemption of the operations of a job is not allowed.
2. Each operation of a job at a stage has to undergo some other activities that consume no
capacity at this stage, but take a finite amount of time. This time is independent of the job
sequence and need not be the same for all operations or jobs.
It is possible to model the activities to be performed during this time explicitly by defining the
operator, the load/unload station and other facilities used as resources for these activities. Because
of the complexity of the resulting scheduling problem and the fact that these resources are
normally not critical, a formulation of the problem with time lags between two succeeding
operations is often more convenient. A hierarchical approach can be used in which first the
scheduling is focused on a (constrained) number of critical capacities. After a schedule for these
resources has been constructed, it can be checked whether or not the other activities can be
performed within the available time.
In this paper we focus on the first part of the hierarchical approach. The type of relationship
between two succeeding operations of a job on the same workstation can be described with a
Finish-to-Start time lag. We are looking for a schedule where the time at which the last stage
finishes processing all jobs is as early as possible.
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The problem we consider can now be stated as follows:
Consider a flow shop with m stages, 1 machine at each stage. A job may require multiple
operations at each stage. The number of operations at a stage may vary between different stages
and jobs. The first operation of a job on stage j>1 cannot start until the last operation of the job on
stage j-1 has finished. Operations of other jobs may be processed in between two succeeding
operations of a job. Preemption of operations is not allowed.
To move from one operation of a job to the next operation of this job on the same stage requires a
finite amount of time ($ 0). This time lag is independent of the sequence and need not be the
same for all operations or jobs. The next operation of a job cannot start until the time lag after the
finish of the former operation has elapsed. Release times of jobs are treated as a special kind of
time lag. Objective of the scheduling is to minimize makespan.
It is well known that the flow shop and job shop scheduling problem with release times and
makespan minimization as well as the flow shop problem with variable time lags are NP Hard.
(Kern and Nawijn, 1991) proved that also the single machine scheduling problem with 2
operations per job and a time lag in between those operations is NP Hard when makespan is the
optimization criterion. Therefore, we can conclude that for our problem no polynomially bounded
algorithm can be found, so we are interested in lowerbounds on the makespan and approximate
solutions for this problem. In the formulation of these lowerbounds and heuristics we aim to
make use of the inherent structure of this flow shop.
The organization of this paper is as follows: an overview of literature on this subject is given in
the next section. Section 3 gives an example problem and the mathematical formulation of the
multiple operations flow shop problem with time lags. In section 4 a number of lowerbounds on
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the makespan are developed and illustrated with examples. In section 5 approximate solutions for
this problem are proposed. The performance of these heuristics and some well known heuristics
for flow shop/job shop scheduling is compared. Finally some directions for future research are
given and conclusions are presented (section 6).
2. LITERATURE REVIEW
The first research on time lags in flow shop problems was by (Mitten, 1958). The time lags he
considered were of the Start-to-Start type combined with Finish-to-Finish lags. Processing of a
job on the next stage can start ai periods after processing of the job on the current stage started.
The time lag ai need not be larger than the processing time on the current stage. Multiple
operations of a job on the same stage were not considered by Mitten. His results are an extension
of the research of (Johnson, 1954) on two-machine flow shop problems. The constructive
algorithm of Mitten generates an optimal permutation schedule for the two machine flow shop
problem with variable time lags. The class of permutation schedules, however, need not contain
the optimal schedule for his problem.
(Szwarc, 1983) generalized this model to cover m-machine flow shop problems with variable
time lags as well as problems where setup, processing and release times are separated. The
general m-machine flow shop problem with variable time lags is NP hard, so approximate
solutions and lowerbounds for the completion times were developed.
In (Cao and Bedworth, 1992) a flow shop problem with transfer times and setup times is
considered. In their description of the model it is assumed that, after processing an operation on
the current stage, transportation of the part to the next stage takes a finite amount of time (e.g. the
transfer time). During this time no other job can be processed on the current stage. The reason
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why processing times and transfer times are separated is not made clear. Multiple operations of a
job on the same stage were not considered.
(Aanen et al., 1993) gave a description of a flexible manufacturing scheduling problem with two
workstations. Some jobs require multiple operations on a workstation, accompanied with time
lags, and there are sequence dependent setup times on the first machine.
(Monma, 1979) constructed an algorithm for flow shop problems with parallel-chain and series-
parallel precedence constraints between the jobs. In his model multiple occurrences of the same
job were considered, but it is assumed that these jobs are identical. The algorithm he presented
cannot be used for the multiple operations flow shop problem with time lags.
From this literature review we can conclude that some research has be done on the subject of time
lags, but the combination of multiple operations and time lags in a flow shop is not found in the
literature on flow shop scheduling.
3. MATHEMATICAL FORMULATION AND EXAMPLE PROBLEM
The multiple operations flow shop scheduling problem with time lags and makespan
minimization can mathematically be formulated as follows:
Indices
i job i=1..n
j stage j=1...m
k number of operation of a job k=1...Kij
with Kij : total number of operations of job i on stage j
Constants
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Oijk kth operation of job i on stage j
Preijk Predecessor of operation Oijk : Oi j k-1 if k>1
Oi j-1 Ki j-1 if k=1, j>1
0 if k=1, j=1
Tijk Processing time of Oijk
Dijk Time lag between operation Oijk and its predecessor Preijk
IF Preijk = 0 THEN Dijk = Di11 = Release time of job i
Variables
C(Oijk) Completion time of operation Oijk in the schedule for stage j
Bj Sequence of all operations {Oijk }, i=1...n, k=1...Kij on stage j:
Bj = < Bj [1] , Bj [2] ,..., Bj [3i=1...n Kij ] >
Rijk Position of operation Oijk in the sequence Bj
Bj [ Rijk ] = Oijk Operation Oijk placed at the Rijkth position in the sequence Bj .
The problem is to find sequences Bj for all stages j=1...m such that Rijk > RPreijk if Preijk needs
processing on the same stage as Rijk (Technological restrictions) and the completion time of the
last operation on the last stage C(Bm [3i=1...n Kim ]) is minimized, where