Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol68.Issue2.Nov2000:
Int. J. Production Economics 68 (2000) 137}149
Identi"cation of aggregate resource and job set characteristics
for predicting job set makespan in batch process industries
Wenny H.M. Raaymakers, Jan C. Fransoo*
Department of Operations Planning and Control, Eindhoven University of Technology, P.O. Box 513, Pav. F12, 5600 MB Eindhoven,
Netherlands
Received 27 March 1998; accepted 6 September 1999
Abstract
We study multipurpose batch process industries with no-wait restrictions, overlapping processing steps, and parallel
resources. To achieve high utilization and reliable lead times, the master planner needs to be able to accurately and
quickly estimate the makespan of a job set. Because constructing a schedule is time consuming, and production plans
may change frequently, estimates must be based on aggregate characteristics of the job set. To estimate the makespan of
a complex set of jobs, we introduce the concept of job interaction. Using statistical analysis, we show that a limited
number of characteristics of the job set and the available resources can explain most of the variability in the job
interaction. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Batch process industries; Multipurpose; Regression analysis; Order release; Scheduling
1. Introduction
Batch (chemical) processes exist in many industries, such as the food, specialty chemicals and
pharmaceutical industry, where production volumes of individual products do not allow continuous or semi-continuous processes. Batch processing
becomes more important because of the increasing
product variety and decreasing demand volumes
for individual products. Two basic types are distinguished. If products all follow the same routing, this
is called multiproduct. If products follow di!erent
routings, like in a discrete manufacturing job shop,
* Corresponding author. Tel.: #31-40-2472681; fax: #3140-2464596.
E-mail address: [email protected] (J.C. Fransoo).
it is called multipurpose. In this paper, we concentrate on multipurpose batch process industries.
Multipurpose batch process industries produce
a large variety of di!erent products that follow
di!erent routings through the plant. Considerable
di!erences may exist between products in the number and duration of the processing steps that are
required. Intermediate products may be unstable,
which means that a product needs to be processed
further without delay. These no-wait restrictions
and the large variety of products with di!erent
routings cause complex scheduling problems.
Namely, for each product a di!erent combination
of resources is required in a speci"c sequence and
timing due to these no-wait restrictions. Consequently, the capacity utilization realized by multipurpose batch process industries is generally low.
Furthermore, many of these companies operate in
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 1 1 1 - 5
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W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
highly variable and dynamic markets in which periods of high demand may be followed by periods of
low demand. Therefore, the amount and mix of
production orders may di!er considerably from
period to period. Consequently, bottlenecks may
shift over time due to variations in the mix of
production orders.
One of the primary di$culties for this type of
industries is to estimate the workload that can be
completed during a speci"c period. Namely, the
capacity utilization that can be realized in a planning period strongly depends on the current mix of
production orders [1]. It is important for planners
to accurately estimate what workload and mix of
production orders can be completed in a period, in
order to set reliable due dates to customers.
We assume a situation with hierarchical production control in which the (master) planner is responsible for selecting achievable sets of
production orders (jobs) per planning period. Hierarchical structures like this are very common in
industry and have been described in many (text)
books (e.g., [2,3]). A production plan for a period is
considered achievable if a schedule can be constructed that completes all production orders before the
end of that period. For the "rst period only, the job
set is released to a scheduler that constructs a detailed schedule. This schedule is then executed at
a production department. The planning horizon of
the scheduler is much shorter than the planning
horizon of the (master) planner. The (master) planner may evaluate the achievability of the job sets for
subsequent planning periods by constructing
a schedule for each planning period. This would
result in constructing a detailed schedule over a medium term horizon. The main disadvantage of this
approach is that it is very time consuming, since
a new schedule has to be constructed every time
when a change in the plan occurs. Therefore, we
prefer methods that assess the achievability of job
sets based on aggregate information.
In this paper, we investigate characteristics of the
job mix and of the resources in a production department, to identify the characteristics that signi"cantly in#uence the completion time (makespan) of
a fully speci"ed job set. These characteristics may
then be used to predict the makespan of a job set. In
that way, a job set is expected to be achievable if the
predicted makespan is shorter than the length of
a planning period. This provides an aggregate
method to assess the achievability of job sets. The
main advantage of this approach is that it provides
the opportunity to make customer order acceptance decisions quickly because computational
e!ort is small.
This paper is organized as follows. The following
section discusses the relevant literature. Section
3 gives the basic assumptions underlying the model.
Section 4 describes the factors that we expect to be
of in#uence on the job interaction. Then, the experimental design is described that is used to investigate if the interaction factors have a relation with
the idle time required in the schedule or not. Section 6 includes the experimental results and statistical analysis. The last section contains the
conclusions and suggestions for further research.
2. Literature
Makespan prediction is related to #ow time prediction, which has received considerable attention
in the literature. For an overview, we refer to the
paper by Cheng and Gupta [4]. Flow time is the
total throughput time of a job in a production
system, which consists of processing time and waiting time. Flow time prediction is generally used in
the determination of achievable due dates for customer orders. Upon arrival of a customer order, the
#ow time is estimated for the job related to that
customer order. The due date of the job is set to the
arrival date plus a #ow allowance, which is the
estimated #ow time. Several rules for determining
#ow time allowances are proposed in the literature.
A distinction can be made between rules that use
job characteristics only, and rules that also use
shop information. A simple and popular rule that
uses job characteristics only, is the total work-rule.
According to this rule, the #ow time allowance is
proportional to the total processing time of the job
[5}8]. Bertrand [9] introduced a rule that uses
both the total processing time of a job and the
number of processing steps to determine the #ow
time allowance. Other rules also use information on
the current shop status, which is generally modeled
as a queuing network. Some rules use the expected
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
waiting time per job or per processing step as
a basis for #ow time prediction [5,6]. Also, the
number of jobs in the queues can be used for #ow
time prediction. Some of these rules use information on the total workload or total number of jobs
in the shop, while others use only information of
the resources on the job's routing. Bertrand [9]
introduced the use of time-phased workload to
estimate the #ow time of a job. Vig and Dooley
[10] introduced the use of the average #ow time per
processing step of three recently completed jobs to
estimate the #ow time of a new job. The di!erent
job characteristics and shop congestion characteristics may also be used in combination to determine
the #ow time allowance. Some authors [10}12] use
regression analysis to determine the weighing coef"cients for the di!erent job and shop characteristics
included in the #ow time predictions. The characteristics included, and the corresponding coe$cients, depend on the dispatching rule that is used
for the execution of the jobs on the resources in the
shop. Vig and Dooley [12] showed that using combined static and dynamic #ow time prediction
methods yield #ow time predictions that are more
accurate and more robust to variable shop condition, than realized by dynamic #ow time prediction
methods.
A comparison of di!erent rules shows that using
information on the shop status improves the #ow
time prediction [5,6]. Furthermore, using information on the number of jobs or the workload in the
queues along the job's routing performs better than
using general shop congestion information [10,11].
From the literature, we conclude that both job
information, such as the number of processing
steps, and shop congestion information in#uence
the #ow time. Therefore, we investigate whether
these characteristics also in#uence the makespan of
a job set in multipurpose batch process industries.
However, two important di!erences exist between
the situation considered in the literature and the
situation considered in this paper. First, in the
literature, traditional job shops are considered,
whereas in this paper, we consider multipurpose
batch process industries. These di!er from traditional job shops in the no-wait restrictions between
processing steps and the possible overlap of processing steps. Second, in the literature, #ow times of
139
individual jobs are predicted, whereas in this paper,
we want to identify the characteristics that in#uence the makespan of a set of jobs. This is done
because we expect that the mix of jobs has an
important in#uence on the makespan rather than
an individual job.
3. Problem setting
It is obvious that the makespan of a job set is
in#uenced by the workload of the job set. More
speci"c, the workload on the bottleneck resource
puts a lower bound on the makespan of a job set.
However, the makespan of a job set is generally
longer than this lower bound. We argue that this
is caused by interactions between jobs at the
scheduling level. Job interaction occurs because
each job requires several resources at the same time
or consecutively without waiting time. Furthermore, in multipurpose batch process industries
each job may require a di!erent combination of
resources. To meet all no-wait restrictions, idle time
generally needs to be included on the resources.
Consequently, job interactions result in a makespan that is longer than the lower bound on the
makespan, which is based on the workload of the
job set.
We argue that di!erent job sets may have di!erent levels of interaction. In this paper, we identify
aggregate characteristics of the job set and of the
resources that may in#uence the amount job interaction. These characteristics will be used to obtain
a prediction model that provides accurate predictions of the makespan of a job set, based on the
expected interaction (IK ). The quality of this prediction is de"ned as the di!erence between the predicted and actual interaction. With the expected
interaction and a lower bound on the makespan we
may predict the makespan of a job set. Fig. 1 shows
the role of the prediction model. In this paper we
focus on identifying the aggregate characteristics,
which we call interaction factors, that in#uence the
amount of interaction. We only consider extreme
values to identify whether these factors have a signi"cant contribution to the interaction margin or
not, following basic principles from statistical theory of design of experiments.
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W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
Fig. 1. The role of the prediction model.
Fig. 2. Job de"nition.
The job sets and resource sets considered in this
paper originate from multipurpose production systems. Jobs di!er in the number of processing steps,
the resource types required, the sequence of processing steps and the processing times. For each
planning period, a job set consisting of J jobs has to
be completed on a given set of resources. Jobs are
de"ned as follows. Each job ( j) consists of a speci"c
number of processing steps (s ). These processing
j
steps may have an overlap in time. The start time of
each processing step is given by the time delay (d )
ij
and thus is "xed relative to the start time of the job.
Processing times (p ) are given for each step. The
ij
job duration (c ) is the time required from the start
j
of the "rst step until completion of the last step. An
illustration of a job is given in Fig. 2. The jobs have
to be completed on N resources of M di!erent
resource types. Resources of the same type are
considered identical.
The following assumptions regarding jobs and
resources are made:
f All jobs are available at the start of the planning
period.
f Resources are available from the start of the
planning period and without interruptions.
f No precedence relations exist between jobs.
f The processing times and time delay of each
processing step are given and "xed. This means
that if the start time of a job is determined the
start times of all processing steps are determined.
f Each processing step has to be performed without preemption on exactly one resource of a speci"c resource type.
f More than one processing step of a job may
require a resource of the same resource type.
These processing steps have to be performed on
di!erent resources of that type if they overlap.
A schedule is constructed for the set of J jobs to
be completed. We cannot expect to "nd an optimal
solution in reasonable time because job shop
scheduling problems with no-wait restrictions are
NP-hard [1]. Therefore, we have chosen simulated
annealing to obtain solutions to the scheduling
problems. A simulated annealing procedure that
aims at minimization of the makespan has been
implemented and tested for industrial instances [1].
Minimum makespan corresponds to minimum idle
time on the resources and hence, maximum overall
capacity utilization. Simulated annealing is a reasonably e!ective method to approximate these optima. We further assume that the jobs are executed
exactly according to the schedule.
4. Interaction factors
The workload that can be completed in a speci"c
planning period depends on the current mix of jobs.
We stipulate that this is caused by job interactions
at the scheduling level. The amount of job interaction depends on the job mix and the resource set of
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
a production department. Therefore, we investigate
which aggregate characteristics of the job set and
the resources in#uence the amount of job interaction. In this section, we de"ne the aggregate characteristics of the job set that we expect to in#uence the
job interaction.
The following notation is used:
J
number of jobs
N
number of resources
M number of resource types
n
number of resources of resource type m
m
¸
workload on resource type m
m
lb
lower bound on the makespan
I
interaction margin
C
makespan
.!9
For a speci"c job set, the total workload (¸ ) on
m
each resource type m is obtained by summing the
processing times of all processing steps that require
resource type m. The bottleneck resource is the
resource type with the maximum of the workload
divided by the number of resources of that type.
A lower bound (lb) on the makespan is obtained by
dividing the workload on the bottleneck resource
by the number of resources of that resource. Because all processing times are integer values and no
pre-emption is allowed, we may round this up.
Eq. (1) shows how this lower bound is obtained.
lb"v
maxM¸ /n , ¸ /n ,2, ¸ /n w
N .
(1)
1 1 2 2
M M
The lower bound is the amount of time required
to complete all jobs if the total processing time can
be distributed evenly over the resources of the
bottleneck resource type, and if these resources can
all process without interruption. If a feasible schedule is found with no idle time on any resource of the
bottleneck resource type, then the makespan is
equal to the lower bound. We observe that in all
other situations, when the makespan is longer than
the lower bound, idle time exists on the bottleneck
resource.
The interaction margin is de"ned as: the relative
diwerence between the makespan realized in the
schedule in which all no-wait restrictions are met and
the lower bound on the makespan. Eq. (2) gives the
interaction margin.
I"(C
.!9
!lb)/lb.
(2)
141
The interaction margin concept is related to the
concept of &solution gap'. We prefer to use the term
interaction margin to denote the speci"c e!ect
caused by interactions between the jobs in the situation we are considering. Job interaction results
from relations between capacity requirements on
di!erent resource types and from scarcity of capacity. We expect that the number of parallel resources in#uences the interaction margin because it
provides #exibility. The capacity requirements for
di!erent resources have a "xed o!set in time
for each job, because of the "xed time delay for
each processing step. We expect that the number
of processing steps of a job, and therefore the
number of resources required for a job, in#uences
the interaction margin. We also expect that the
amount of overlap of processing steps in#uences
the interaction margin. We further expect that
the workload balance of the resources is a factor
that may be of in#uence. Namely, when the workload is highly balanced, all resource types become
bottlenecks. A "nal interaction factor, which is
examined in this paper, is the variation in processing times. We will now discuss each of these factors
separately.
4.1. Number of parallel resources
If parallel resources are available for a processing
step, it is expected that this results in extra #exibility at the scheduling level, and hence in a decrease
of the interaction margin. Namely, when there is
more than one resource to choose from when
scheduling a processing step, it is likely that this
results in a schedule with less idle time. Therefore,
we include the average number of parallel resources
(k ) as an interaction factor:
!
k "N/M.
(3)
!
In industrial practice, generally, the number of
resources for each resource type may di!er. The
number of resources available of a resource type
depends on how frequently the resource type is
required and the processing times on this resource
type. In addition, the capital investment necessary
for purchasing a resource may in#uence the number of resources of a certain type. However, this is
142
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
not considered here. In this paper, we assume that
the number of parallel resources for each resource
type depend on the workload for that resource
type. Therefore, all resource types are potential
bottlenecks. The fact that for some processing
steps more parallel resources are available than
for others may in#uence the interaction
margin. The standard deviation of the number of
parallel resources (p ) is included as an interaction
!
factor:
p "J1/M +M (n !k )2 .
m/1 m
!
!
(4)
4.3. Standard deviation of processing time
The last interaction factor we consider is the
standard deviation of the processing time (p ):
p
(7)
p "J1/S +J +sj (p !k )2
p
j/1 i/1 ij
p
where S is the total number of processing steps,
with S"+J s , k is the average processing time
j/1 j p
over all processing steps, with k "(1/S) +J
j/1
p
+sj p , and p is the processing time of step i of
i/1 ij
ij
job j.
4.4. Overlap of processing steps
4.2. Number of processing steps per job
Job interaction arises from the fact that several
resources are required simultaneously or subsequently for each job. The start time of the job
determines the start time of each processing step,
because of the "xed time delay. It is expected that
jobs with more processing steps cause an increase
in the interaction margin because they require
more resources. Therefore, the average number of
processing steps is considered as an interaction
factor:
1 J
k" + s
s J
j
j/1
(5)
where k is the average number of processing steps
s
and s is the number of processing steps of job j.
j
In multipurpose production systems there exists
a considerable variety of jobs. Jobs consisting of
many processing steps are processed together with
jobs requiring few steps. In practice, schedulers use
this variation in the number of processing steps for
constructing a schedule. Schedulers start with the
job with the largest number of processing steps and
proceed in order of non-increasing number of processing steps. The jobs that require the smallest
number of resources are included last in the schedule, because these jobs &"t in' more easily. To investigate the in#uence of the variation in the number
of processing steps, we include the standard deviation of the number of processing steps (k ) as an
s
interaction factor:
p "J1/J +J (s !k )2.
j/1 j
s
s
(6)
The jobs considered also di!er in the level of
overlap of processing steps. Some jobs consist of
processing steps that all start at the same time.
However, there are also jobs for which all processing steps are performed consecutively. The smaller
the overlap, the larger the job duration (c ). This
j
job duration may in#uence the interaction margin.
For each job, the processing steps are put in
order of non-decreasing time delay. For each job,
the overlap g is computed as follows:
j
sj
d !d
1
i~1,j
(8)
+ 1! ij
g"
j s !1
p
i~1,j
j
i/2
where d is the time delay of step i of job j. Fig.
ij
3 shows three jobs with the same number of processing steps and processing times, but di!erent
overlap. To investigate the in#uence of the amount
of overlap on the interaction margin, both the average and the standard deviation of the overlap are
included.
A
B
1 J
(9)
k " + g,
j
g J
j/1
p "J(1/J) +J (g !k )2
(10)
j/1 j
g
g
where k is the average overlap and p the standard
g
g
deviation of the overlap.
4.5. Workload balance of resource types
Scarcity of capacity is a cause of job interaction.
If excess capacity exists for some of the resource
types, we expect that there will be less interaction,
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
143
5. Experimental design
Fig. 3. Overlap.
because it will be relatively easy to construct a feasible schedule for these resources. However, if capacity requirements are the same for all resource
types much interaction is expected to occur. This is
particularly the case, if after makespan minimization all resources have become bottlenecks. Therefore, an interaction factor is included for the
balance of the distribution of the total workload
over the di!erent resource types. If the workload
balance is low, then only few resource types have
signi"cantly higher workload than other resources.
These become clear bottlenecks for a given job set,
while the other resources will not cause many
scheduling interactions. If the balance is high, all
resource types have an equal workload, meaning
that all resources have become a bottleneck for
a given job set. We use the maximum utilization if
the makespan is equal to the lower bound as
a measure for the balance of the workload. The
maximum utilization (o ) of a job set gives the
.!9
utilization if a feasible schedule is constructed with
a makespan equal to the lower bound on this
makespan.
1 M
(11)
¸M " + ¸
m
N
m/1
where ¸M is the average resource workload, and
¸ the workload on resource type m.
m
o "¸M /lb
(12)
.!9
where o
is the workload balance. In this section,
.!9
we have discussed eight interaction factors.
The "rst two factors can be de"ned based on the
available resources. For a given production
department, they have the same value regardless of
the job set. The other factors are determined by the
current job set. In the next section, we will conduct
a number of experiments to investigate the in#uence of these interaction factors on the interaction
margin.
Job sets are generated randomly with the goal to
resemble realistic job sets from industry [14]. The
values of the interaction factors are varied on two
or three levels. In each experiment, a job set containing 50 jobs is generated and have to be completed on 10 resources.
To investigate the in#uence of the average and
standard deviation of the number of parallel resources, we consider four di!erent resource con"gurations. These con"gurations di!er in the average
and variation in the number of parallel resources.
The resource con"gurations used in the experiments are given in Table 1. Resource con"gurations
A and D are the extreme values when the average
number of parallel resources is considered. For
these extremes, it is not possible to include variation, because the total number of resources is kept
constant. To investigate the in#uence of this variation, we include con"gurations B and C. This
allows us to estimate the e!ects of both k and p ,
!
!
but not to estimate the two-way interaction
between these two factors.
For each job, the number of processing steps is
given. The number of processing steps per job cannot exceed the number of resources because each
processing step of a job has to be performed on
a di!erent resource if the processing steps overlap.
Three levels for both average and variation are
included in the experiments. For the average number of processing steps the values 1, 5 and 10
processing steps per job are included. Di!erent
levels of variation are considered for the medium
level. The values are given in Table 2.
To each processing step, a resource type is allocated at random. The probability of a speci"c resource type being allocated is equal to the number
of resources of that type available related to the
total number of resources available. For the "rst
processing step of each job, all resources are available. When the "rst processing step is allocated to
a speci"c resource type, the number of resources
available of that type decreases by one. The number
of processing steps of a speci"c job that are
allocated to the same resource type cannot be larger than the number of resources of that speci"c
type. In this way, each resource type has equal
144
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
Table 1
Number of parallel resources
Resource
con"guration
A
B
C
D
Table 3
Processing times
Number of
resource
types
Number of
resources
per type
k
!
10
5
5
1
1 of each type
2 of each type
4, 2, 2, 1, 1
10
1
2
2
10
p
!
0
0
1.2
0
p
p
Processing time
min.
max.
25
1
25
49
0
14
Table 4
Overlap
Table 2
Number of processing steps
Overlap
k
s
Number of steps
min.
max.
1
5
3
1
10
1
5
7
9
10
1
5
5
5
10
p
s
0
0
1.4
2.6
0
probability to become a bottleneck resource and
the balance of the workload of the resources is kept
high.
Next, a processing time is allocated to each
processing step. In the experiments, the average
processing time is kept constant. The variation in
processing times is measured on two levels, namely
no variation and processing times uniformly distributed between 1 and 49. The processing time
distribution is kept the same for all resource types.
The processing time always takes integer values.
Table 3 gives the values for the processing time
distribution.
Finally, the time delay of each processing step is
determined. The time delay of the "rst processing
step is zero for each job. The second and following
processing steps start when a given overlap of the
previous processing step has been processed. In the
experiments, both the average overlap and the variation in overlap are included. The di!erent values
used in the experiments for overlap are given in
Table 4.
min.
max.
0
0.5
0
1
0
0.5
1
1
k
g
p
g
0
0.5
0.5
1
0
0
0.3
0
Table 5
Experimental design
Parameters
Number of levels
Resource con"guration
Processing steps
Processing times
Overlap
4
5
2
4
The workload balance cannot be varied on distinct levels because it is the result of the random
allocation of processing times and resource types to
processing steps. This is a so-called random factor
[13], which cannot be included in the experimental
design.
A full factorial experimental design is constructed, resulting in 160 combinations (see Table 5).
Not all combinations are however feasible. If there
is only one processing step per job, overlap between
the processing steps is not possible and varying the
overlap is therefore not an option. Hence these
combinations have been removed from the experimental design. For each of the remaining 136 combinations, "ve replications have been constructed.
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
An additional exception needs to be made at this
stage. Namely, in the situations with resource con"guration D, when all resources are of the same
type and no variation exists in the number of processing steps, the overlap, and the processing times,
no degrees of freedom are left to generate di!erent
jobs. All parameters used to generate the job set
have been "xed beforehand. Therefore, each job set
generated with these parameter settings is identical
and a replication cannot take place. This leads to
a total number of 628 problems.
A schedule is obtained using a simulated annealing algorithm. This algorithm aims at minimization
of the makespan. We have to take into consideration that some variation in the interaction margin
results from the fact that the resulting schedule is
not necessarily optimal.
145
Fig. 4. In#uence of the number of parallel resources on the
interaction margin.
6. Experimental results
For each observation, the interaction margin is
obtained. The average interaction margin for the
di!erent parameter values is shown by Figs. 4}8.
The in#uence of the number of parallel resources
on the average interaction margin is given in Fig. 4.
We observe that the average interaction margin
becomes smaller if the number of parallel resources
increase. This corresponds with the intuition that
parallel resources provide scheduling #exibility,
and hence result in less job interaction. We further
observe that a small di!erence exists between con"gurations B and C, which have the same average
number of parallel resources. The di!erence in distribution of the resources over the resource types
for these two resource con"gurations may explain
this di!erence.
The in#uence of the number of processing steps
on the interaction margin is shown by Fig. 5. We
observe that an increase in the average number of
processing steps results in a considerable increase
in the interaction margin. This corresponds to the
intuition that an increase in the number of processing steps results in an increase in the job interaction, because more resources are required for each
job. We further observe that the variation in the
number of processing steps hardly in#uences the
average interaction margin. Namely, the three bars
Fig. 5. In#uence of the number of processing steps on the
interaction margin.
Fig. 6. In#uence of the variation in processing times on the
interaction margin.
in the middle all represent an average number of
processing steps of 5, with di!erent variation. This
indicates that in situations with variation in the
number of processing steps, the jobs with few
146
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
Fig. 7. In#uence of the overlap on the interaction margin.
Fig. 8. In#uence of the workload balance on the interaction
margin.
processing steps, which &"t in' easily, compensate
for the jobs with many processing steps, which are
more di$cult to schedule. The net e!ect of the
variation in the number of processing steps is therefore small.
The in#uence of the variation in the processing
times is shown by Fig. 6. We observe that higher
variation in the processing times results in a higher
average interaction margin. The in#uence of the
overlap on the interaction margin is shown by
Fig. 7. We observe that the interaction margin is
higher for an overlap of 0 than for an overlap of 1.
This is in line with our expectation, because the
smaller the overlap the longer the job completion
time. However, an average overlap of 0.5 results in
the highest average interaction margin, suggesting
increased scheduling complexity for these job structures. We further observe that the variation in the
overlap hardly in#uences the interaction margin.
Both center bars in the "gure represent an average
overlap of 0.5, with di!erent variation.
The in#uence of the workload balance on the
interaction margin is given in Fig. 8. This is repre-
sented by a scatter plot, because the workload
balance is a random factor. We observe that on
average, the interaction margin increases with an
increase in the workload balance. This corresponds
with our intuition, because a high workload balance means that resource types all have an equal
workload. Hence, all resource types have become
bottlenecks.
Figs. 4}8 indicated the in#uence of the di!erent
interaction factors. An analysis of variance is performed to determine the contribution of the di!erent factors to the explanation of the variability in
the interaction margin. Main e!ects and two-way
interactions are included. The results are given in
Table 6. The results show that all main e!ects,
except for the standard deviation of the number of
parallel resources and the overlap, are signi"cant.
In addition, several two-way interactions are significant. Together, the signi"cant factors explain
a large part (91%) of the variability in the interaction margin. The proportion of the variability in
the interaction margin that is accounted for by
variation in the interaction factors is given in the
last column. The residual plots do not provide
reasons for concern. Considering these results, we
conclude that we have identi"ed the most important aggregate job set characteristics that are
required to predict the interaction.
The experimental results show that some factors
have a stronger in#uence on the interaction margin
than others. The most important variables are:
k : average number of parallel resources,
!
k : average number of processing steps,
s
p : standard deviation in processing time, and
p
k : average overlap.
g
In addition, we also include the workload balance (o ) in our further analysis, because in the
.!9
experiments we have generated job sets that all had
a high workload balance. Therefore, we have considered mostly values for o
that are close to 1.
.!9
For all other interaction factors, we have considered a wide range of values. We expect that the
in#uence of o
becomes considerably stronger if
.!9
a wider range of values is considered. It seems
reasonable to include the workload balance in
further analysis, because we have encountered
a wider range of values for o
in practice.
.!9
1.
2.
3.
4.
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
147
Table 6
ANOVA results for complete model
k
!
p
!
k
s
p
s
k
g
p
g
p
p
k )k
! s
k )p
! s
k )k
! g
k )p
! g
k )p
! p
p )k
! s
p )p
! s
p )k
! g
p )p
! g
p )p
! p
k )k
s g
k )p
s g
k )p
s p
p )k
s g
p )p
s g
p )p
s p
k )p
g p
p )p
g p
o
.!9
Error
Corrected total
Sum of squares
Degrees of freedom
Mean square
F-statistic
P-value
1.520
0.007
0.502
0.132
4.966
0.029
0.916
0.042
0.045
2.346
0.033
0.013
0.007
0.010
0.030
0.000
0.003
4.338
0.000
0.816
0.102
0.010
0.146
2.636
0.021
2.252
5.805
84.429
2
1
2
2
2
1
1
4
4
4
2
2
2
2
2
1
1
2
1
2
4
2
2
2
1
34
542
627
0.760
0.007
0.251
0.066
2.483
0.029
0.916
0.010
0.011
0.586
0.016
0.006
0.004
0.005
0.015
0.000
0.003
2.169
0.000
0.408
0.025
0.005
0.073
1.318
0.021
0.066
0.011
71.0
0.7
23.4
6.2
231.8
2.7
85.5
1.0
1.1
54.8
1.5
0.6
0.3
0.4
1.4
0.0
0.3
202.5
0.0
38.1
2.4
0.5
6.8
123.1
2.0
6.2
0.00
0.42
0.00
0.00
0.00
0.10
0.00
0.42
0.38
0.00
0.22
0.55
0.71
0.64
0.25
0.88
0.60
0.00
0.89
0.00
0.05
0.62
0.00
0.00
0.16
0.00
Sum of squares
Degrees of freedom
Mean square
F-statistic
P-value
1.811
1.045
12.678
1.021
0.149
2.551
0.022
6.719
0.754
4.487
2.312
6.666
84.429
2
2
2
1
4
4
2
2
2
2
34
570
627
0.906
0.523
6.339
1.021
0.037
0.638
0.011
3.360
0.377
2.243
0.068
0.012
77.5
44.7
542.1
87.3
3.2
54.5
1.0
287.3
32.2
191.8
5.8
0.00
0.00
0.00
0.00
0.01
0.00
0.39
0.00
0.00
0.00
0.00
Table 7
ANOVA results for reduced model
k
!
k
s
k
g
p
p
k )k
! s
k )k
! g
k )p
! p
ks ) k
g
k )p
s p
k )p
g p
o
.!9
Error
Corrected total
148
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
When the analysis of variance is repeated with
only these "ve variables and second-order interactions between these variables included, an R2 of
0.89 is obtained. Table 7 gives the results for the
reduced model. Considering these results, we conclude that a large part of the variability of the
interaction margin is explained by only a limited
number of interaction factors. All other signi"cant
factors and two-way interactions make small contributions.
7. Conclusions and directions for further research
In this paper, we studied multipurpose batch
process industries with overlapping processing
steps, parallel resources and no-wait restrictions
between processing steps. We have proposed
a method to estimate the feasibility of completing
a job set within a speci"c length of time. The
method is based on predicting job set makespan
using aggregate characteristics of the job set.
A lower bound on the makespan can be computed
easily by considering the workload on the resources. Because of job interaction, the minimal
makespan for which a feasible schedule is realized
will often be larger than the lower bound. Interaction at the scheduling level results from relations
between capacity requirements on di!erent resources and from scarcity of capacity.
In this paper, eight factors have been selected to
investigate how this interaction margin is determined. These interaction factors are:
1. average number of parallel resources,
2. standard deviation of the number of parallel
resources,
3. average number of processing steps per job,
4. standard deviation of the number of processing
steps per job,
5. average overlap of processing steps,
6. standard deviation of the overlap of processing
steps,
7. workload balance of resource types, and
8. standard deviation of the processing time.
Experiments in which each of these factors, except for the workload balance, were varied on two
or three levels showed that most factors have a
signi"cant in#uence on the interaction factor. The
standard deviation of the number of parallel resources was the only main e!ect that was not significant. Furthermore, several two-way interactions
proved to have a signi"cant in#uence on the interaction margin. Together these factors explain 91%
of the variation in the interaction margin. Some of
these factors have a much higher contribution to
the explanation of the variability in the interaction
margin than others. The average number of parallel
resources, the average number of processing steps,
the standard deviation in the processing times, the
average overlap and the workload balance are the
most important factors.
The experimental results provide a basis for further development of the prediction model. The prediction model may be used to support customer
order acceptance decisions. Namely, for a given job
set the makespan can be predicted by using the
lower bound and the interaction margin for the job
set. Orders can be accepted for a period until the
expected makespan exceeds the length of the period. The main advantage of this approach is that
realistic order sets are obtained for each time period with small computational e!orts.
Acknowledgements
The detailed comments on earlier versions of this
paper by one of the anonymous referees considerably improved its clarity and exposition.
References
[1] W.H.M. Raaymakers, J.A. Hoogeveen, Scheduling multipurpose batch process industries with no-wait restrictions
by simulated annealing, European Journal of Operational
Research 126 (1) (2000) 131}151.
[2] T.E. Vollmann, W.L. Berry, D.C. Whybark, Manufacturing Planning and Control Systems, 4th Edition, McGrawHill, New York, 1997.
[3] J.W.M. Bertrand, J.C. Wortmann, J. Wijngaard, Production Control: A Structural and Design Oriented Approach,
Elsevier, Amsterdam, 1990.
[4] T.C.E. Cheng, M.C. Gupta, Survey of scheduling research involving due date determination decisions,
European Journal of Operational Research 38 (1989)
156}166.
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
[5] S. Eilon, I.G. Chowdhury, Due dates in job shop scheduling, International Journal of Production Research 14
(1976) 223}237.
[6] J.K. Weeks, A simulation study of predictable due-dates,
Management Science 25 (1979) 363}373.
[7] K.R. Baker, J.W.M. Bertrand, An investigation of due-date
assignment rules with constrained tightness, Journal of
Operations Management 1 (1981) 109}120.
[8] K.R. Baker, J.W.M. Bertrand, A comparison of due-date
selection rules, AIIE Transactions 13 (1981) 123}131.
[9] J.W.M. Bertrand, The e!ects of workload dependent duedates on job shop performance, Management Science 29
(1983) 799}816.
[10] M.M. Vig, K.J. Dooley, Dynamic rules for due-date assignment, International Journal of Production Research
29 (1991) 1361}1377.
149
[11] M.M. Vig, K.J. Dooley, Mixing static and dynamic #owtime estimates for due-date assignment, Journal of Operations Management 11 (1993) 67}79.
[12] G.L. Ragatz, V.A. Mabert, A simulation analysis of due
date assignment rules, Journal of Operations Management
5 (1984) 27}39.
[13] D.C. Montgomery, E.A. Peck, Introduction to Linear Regression Analysis, Wiley, New York, 1992.
[14] W.H.M. Raaymakers, J.W.M. Bertrand, and J.C. Fransoo,
The performance of workload rules for order acceptance in batch chemical manufacturing, Journal of
Intelligent
Manufacturing
(1998)
accepted
for
publication.
Identi"cation of aggregate resource and job set characteristics
for predicting job set makespan in batch process industries
Wenny H.M. Raaymakers, Jan C. Fransoo*
Department of Operations Planning and Control, Eindhoven University of Technology, P.O. Box 513, Pav. F12, 5600 MB Eindhoven,
Netherlands
Received 27 March 1998; accepted 6 September 1999
Abstract
We study multipurpose batch process industries with no-wait restrictions, overlapping processing steps, and parallel
resources. To achieve high utilization and reliable lead times, the master planner needs to be able to accurately and
quickly estimate the makespan of a job set. Because constructing a schedule is time consuming, and production plans
may change frequently, estimates must be based on aggregate characteristics of the job set. To estimate the makespan of
a complex set of jobs, we introduce the concept of job interaction. Using statistical analysis, we show that a limited
number of characteristics of the job set and the available resources can explain most of the variability in the job
interaction. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Batch process industries; Multipurpose; Regression analysis; Order release; Scheduling
1. Introduction
Batch (chemical) processes exist in many industries, such as the food, specialty chemicals and
pharmaceutical industry, where production volumes of individual products do not allow continuous or semi-continuous processes. Batch processing
becomes more important because of the increasing
product variety and decreasing demand volumes
for individual products. Two basic types are distinguished. If products all follow the same routing, this
is called multiproduct. If products follow di!erent
routings, like in a discrete manufacturing job shop,
* Corresponding author. Tel.: #31-40-2472681; fax: #3140-2464596.
E-mail address: [email protected] (J.C. Fransoo).
it is called multipurpose. In this paper, we concentrate on multipurpose batch process industries.
Multipurpose batch process industries produce
a large variety of di!erent products that follow
di!erent routings through the plant. Considerable
di!erences may exist between products in the number and duration of the processing steps that are
required. Intermediate products may be unstable,
which means that a product needs to be processed
further without delay. These no-wait restrictions
and the large variety of products with di!erent
routings cause complex scheduling problems.
Namely, for each product a di!erent combination
of resources is required in a speci"c sequence and
timing due to these no-wait restrictions. Consequently, the capacity utilization realized by multipurpose batch process industries is generally low.
Furthermore, many of these companies operate in
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 1 1 1 - 5
138
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
highly variable and dynamic markets in which periods of high demand may be followed by periods of
low demand. Therefore, the amount and mix of
production orders may di!er considerably from
period to period. Consequently, bottlenecks may
shift over time due to variations in the mix of
production orders.
One of the primary di$culties for this type of
industries is to estimate the workload that can be
completed during a speci"c period. Namely, the
capacity utilization that can be realized in a planning period strongly depends on the current mix of
production orders [1]. It is important for planners
to accurately estimate what workload and mix of
production orders can be completed in a period, in
order to set reliable due dates to customers.
We assume a situation with hierarchical production control in which the (master) planner is responsible for selecting achievable sets of
production orders (jobs) per planning period. Hierarchical structures like this are very common in
industry and have been described in many (text)
books (e.g., [2,3]). A production plan for a period is
considered achievable if a schedule can be constructed that completes all production orders before the
end of that period. For the "rst period only, the job
set is released to a scheduler that constructs a detailed schedule. This schedule is then executed at
a production department. The planning horizon of
the scheduler is much shorter than the planning
horizon of the (master) planner. The (master) planner may evaluate the achievability of the job sets for
subsequent planning periods by constructing
a schedule for each planning period. This would
result in constructing a detailed schedule over a medium term horizon. The main disadvantage of this
approach is that it is very time consuming, since
a new schedule has to be constructed every time
when a change in the plan occurs. Therefore, we
prefer methods that assess the achievability of job
sets based on aggregate information.
In this paper, we investigate characteristics of the
job mix and of the resources in a production department, to identify the characteristics that signi"cantly in#uence the completion time (makespan) of
a fully speci"ed job set. These characteristics may
then be used to predict the makespan of a job set. In
that way, a job set is expected to be achievable if the
predicted makespan is shorter than the length of
a planning period. This provides an aggregate
method to assess the achievability of job sets. The
main advantage of this approach is that it provides
the opportunity to make customer order acceptance decisions quickly because computational
e!ort is small.
This paper is organized as follows. The following
section discusses the relevant literature. Section
3 gives the basic assumptions underlying the model.
Section 4 describes the factors that we expect to be
of in#uence on the job interaction. Then, the experimental design is described that is used to investigate if the interaction factors have a relation with
the idle time required in the schedule or not. Section 6 includes the experimental results and statistical analysis. The last section contains the
conclusions and suggestions for further research.
2. Literature
Makespan prediction is related to #ow time prediction, which has received considerable attention
in the literature. For an overview, we refer to the
paper by Cheng and Gupta [4]. Flow time is the
total throughput time of a job in a production
system, which consists of processing time and waiting time. Flow time prediction is generally used in
the determination of achievable due dates for customer orders. Upon arrival of a customer order, the
#ow time is estimated for the job related to that
customer order. The due date of the job is set to the
arrival date plus a #ow allowance, which is the
estimated #ow time. Several rules for determining
#ow time allowances are proposed in the literature.
A distinction can be made between rules that use
job characteristics only, and rules that also use
shop information. A simple and popular rule that
uses job characteristics only, is the total work-rule.
According to this rule, the #ow time allowance is
proportional to the total processing time of the job
[5}8]. Bertrand [9] introduced a rule that uses
both the total processing time of a job and the
number of processing steps to determine the #ow
time allowance. Other rules also use information on
the current shop status, which is generally modeled
as a queuing network. Some rules use the expected
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
waiting time per job or per processing step as
a basis for #ow time prediction [5,6]. Also, the
number of jobs in the queues can be used for #ow
time prediction. Some of these rules use information on the total workload or total number of jobs
in the shop, while others use only information of
the resources on the job's routing. Bertrand [9]
introduced the use of time-phased workload to
estimate the #ow time of a job. Vig and Dooley
[10] introduced the use of the average #ow time per
processing step of three recently completed jobs to
estimate the #ow time of a new job. The di!erent
job characteristics and shop congestion characteristics may also be used in combination to determine
the #ow time allowance. Some authors [10}12] use
regression analysis to determine the weighing coef"cients for the di!erent job and shop characteristics
included in the #ow time predictions. The characteristics included, and the corresponding coe$cients, depend on the dispatching rule that is used
for the execution of the jobs on the resources in the
shop. Vig and Dooley [12] showed that using combined static and dynamic #ow time prediction
methods yield #ow time predictions that are more
accurate and more robust to variable shop condition, than realized by dynamic #ow time prediction
methods.
A comparison of di!erent rules shows that using
information on the shop status improves the #ow
time prediction [5,6]. Furthermore, using information on the number of jobs or the workload in the
queues along the job's routing performs better than
using general shop congestion information [10,11].
From the literature, we conclude that both job
information, such as the number of processing
steps, and shop congestion information in#uence
the #ow time. Therefore, we investigate whether
these characteristics also in#uence the makespan of
a job set in multipurpose batch process industries.
However, two important di!erences exist between
the situation considered in the literature and the
situation considered in this paper. First, in the
literature, traditional job shops are considered,
whereas in this paper, we consider multipurpose
batch process industries. These di!er from traditional job shops in the no-wait restrictions between
processing steps and the possible overlap of processing steps. Second, in the literature, #ow times of
139
individual jobs are predicted, whereas in this paper,
we want to identify the characteristics that in#uence the makespan of a set of jobs. This is done
because we expect that the mix of jobs has an
important in#uence on the makespan rather than
an individual job.
3. Problem setting
It is obvious that the makespan of a job set is
in#uenced by the workload of the job set. More
speci"c, the workload on the bottleneck resource
puts a lower bound on the makespan of a job set.
However, the makespan of a job set is generally
longer than this lower bound. We argue that this
is caused by interactions between jobs at the
scheduling level. Job interaction occurs because
each job requires several resources at the same time
or consecutively without waiting time. Furthermore, in multipurpose batch process industries
each job may require a di!erent combination of
resources. To meet all no-wait restrictions, idle time
generally needs to be included on the resources.
Consequently, job interactions result in a makespan that is longer than the lower bound on the
makespan, which is based on the workload of the
job set.
We argue that di!erent job sets may have di!erent levels of interaction. In this paper, we identify
aggregate characteristics of the job set and of the
resources that may in#uence the amount job interaction. These characteristics will be used to obtain
a prediction model that provides accurate predictions of the makespan of a job set, based on the
expected interaction (IK ). The quality of this prediction is de"ned as the di!erence between the predicted and actual interaction. With the expected
interaction and a lower bound on the makespan we
may predict the makespan of a job set. Fig. 1 shows
the role of the prediction model. In this paper we
focus on identifying the aggregate characteristics,
which we call interaction factors, that in#uence the
amount of interaction. We only consider extreme
values to identify whether these factors have a signi"cant contribution to the interaction margin or
not, following basic principles from statistical theory of design of experiments.
140
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
Fig. 1. The role of the prediction model.
Fig. 2. Job de"nition.
The job sets and resource sets considered in this
paper originate from multipurpose production systems. Jobs di!er in the number of processing steps,
the resource types required, the sequence of processing steps and the processing times. For each
planning period, a job set consisting of J jobs has to
be completed on a given set of resources. Jobs are
de"ned as follows. Each job ( j) consists of a speci"c
number of processing steps (s ). These processing
j
steps may have an overlap in time. The start time of
each processing step is given by the time delay (d )
ij
and thus is "xed relative to the start time of the job.
Processing times (p ) are given for each step. The
ij
job duration (c ) is the time required from the start
j
of the "rst step until completion of the last step. An
illustration of a job is given in Fig. 2. The jobs have
to be completed on N resources of M di!erent
resource types. Resources of the same type are
considered identical.
The following assumptions regarding jobs and
resources are made:
f All jobs are available at the start of the planning
period.
f Resources are available from the start of the
planning period and without interruptions.
f No precedence relations exist between jobs.
f The processing times and time delay of each
processing step are given and "xed. This means
that if the start time of a job is determined the
start times of all processing steps are determined.
f Each processing step has to be performed without preemption on exactly one resource of a speci"c resource type.
f More than one processing step of a job may
require a resource of the same resource type.
These processing steps have to be performed on
di!erent resources of that type if they overlap.
A schedule is constructed for the set of J jobs to
be completed. We cannot expect to "nd an optimal
solution in reasonable time because job shop
scheduling problems with no-wait restrictions are
NP-hard [1]. Therefore, we have chosen simulated
annealing to obtain solutions to the scheduling
problems. A simulated annealing procedure that
aims at minimization of the makespan has been
implemented and tested for industrial instances [1].
Minimum makespan corresponds to minimum idle
time on the resources and hence, maximum overall
capacity utilization. Simulated annealing is a reasonably e!ective method to approximate these optima. We further assume that the jobs are executed
exactly according to the schedule.
4. Interaction factors
The workload that can be completed in a speci"c
planning period depends on the current mix of jobs.
We stipulate that this is caused by job interactions
at the scheduling level. The amount of job interaction depends on the job mix and the resource set of
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
a production department. Therefore, we investigate
which aggregate characteristics of the job set and
the resources in#uence the amount of job interaction. In this section, we de"ne the aggregate characteristics of the job set that we expect to in#uence the
job interaction.
The following notation is used:
J
number of jobs
N
number of resources
M number of resource types
n
number of resources of resource type m
m
¸
workload on resource type m
m
lb
lower bound on the makespan
I
interaction margin
C
makespan
.!9
For a speci"c job set, the total workload (¸ ) on
m
each resource type m is obtained by summing the
processing times of all processing steps that require
resource type m. The bottleneck resource is the
resource type with the maximum of the workload
divided by the number of resources of that type.
A lower bound (lb) on the makespan is obtained by
dividing the workload on the bottleneck resource
by the number of resources of that resource. Because all processing times are integer values and no
pre-emption is allowed, we may round this up.
Eq. (1) shows how this lower bound is obtained.
lb"v
maxM¸ /n , ¸ /n ,2, ¸ /n w
N .
(1)
1 1 2 2
M M
The lower bound is the amount of time required
to complete all jobs if the total processing time can
be distributed evenly over the resources of the
bottleneck resource type, and if these resources can
all process without interruption. If a feasible schedule is found with no idle time on any resource of the
bottleneck resource type, then the makespan is
equal to the lower bound. We observe that in all
other situations, when the makespan is longer than
the lower bound, idle time exists on the bottleneck
resource.
The interaction margin is de"ned as: the relative
diwerence between the makespan realized in the
schedule in which all no-wait restrictions are met and
the lower bound on the makespan. Eq. (2) gives the
interaction margin.
I"(C
.!9
!lb)/lb.
(2)
141
The interaction margin concept is related to the
concept of &solution gap'. We prefer to use the term
interaction margin to denote the speci"c e!ect
caused by interactions between the jobs in the situation we are considering. Job interaction results
from relations between capacity requirements on
di!erent resource types and from scarcity of capacity. We expect that the number of parallel resources in#uences the interaction margin because it
provides #exibility. The capacity requirements for
di!erent resources have a "xed o!set in time
for each job, because of the "xed time delay for
each processing step. We expect that the number
of processing steps of a job, and therefore the
number of resources required for a job, in#uences
the interaction margin. We also expect that the
amount of overlap of processing steps in#uences
the interaction margin. We further expect that
the workload balance of the resources is a factor
that may be of in#uence. Namely, when the workload is highly balanced, all resource types become
bottlenecks. A "nal interaction factor, which is
examined in this paper, is the variation in processing times. We will now discuss each of these factors
separately.
4.1. Number of parallel resources
If parallel resources are available for a processing
step, it is expected that this results in extra #exibility at the scheduling level, and hence in a decrease
of the interaction margin. Namely, when there is
more than one resource to choose from when
scheduling a processing step, it is likely that this
results in a schedule with less idle time. Therefore,
we include the average number of parallel resources
(k ) as an interaction factor:
!
k "N/M.
(3)
!
In industrial practice, generally, the number of
resources for each resource type may di!er. The
number of resources available of a resource type
depends on how frequently the resource type is
required and the processing times on this resource
type. In addition, the capital investment necessary
for purchasing a resource may in#uence the number of resources of a certain type. However, this is
142
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
not considered here. In this paper, we assume that
the number of parallel resources for each resource
type depend on the workload for that resource
type. Therefore, all resource types are potential
bottlenecks. The fact that for some processing
steps more parallel resources are available than
for others may in#uence the interaction
margin. The standard deviation of the number of
parallel resources (p ) is included as an interaction
!
factor:
p "J1/M +M (n !k )2 .
m/1 m
!
!
(4)
4.3. Standard deviation of processing time
The last interaction factor we consider is the
standard deviation of the processing time (p ):
p
(7)
p "J1/S +J +sj (p !k )2
p
j/1 i/1 ij
p
where S is the total number of processing steps,
with S"+J s , k is the average processing time
j/1 j p
over all processing steps, with k "(1/S) +J
j/1
p
+sj p , and p is the processing time of step i of
i/1 ij
ij
job j.
4.4. Overlap of processing steps
4.2. Number of processing steps per job
Job interaction arises from the fact that several
resources are required simultaneously or subsequently for each job. The start time of the job
determines the start time of each processing step,
because of the "xed time delay. It is expected that
jobs with more processing steps cause an increase
in the interaction margin because they require
more resources. Therefore, the average number of
processing steps is considered as an interaction
factor:
1 J
k" + s
s J
j
j/1
(5)
where k is the average number of processing steps
s
and s is the number of processing steps of job j.
j
In multipurpose production systems there exists
a considerable variety of jobs. Jobs consisting of
many processing steps are processed together with
jobs requiring few steps. In practice, schedulers use
this variation in the number of processing steps for
constructing a schedule. Schedulers start with the
job with the largest number of processing steps and
proceed in order of non-increasing number of processing steps. The jobs that require the smallest
number of resources are included last in the schedule, because these jobs &"t in' more easily. To investigate the in#uence of the variation in the number
of processing steps, we include the standard deviation of the number of processing steps (k ) as an
s
interaction factor:
p "J1/J +J (s !k )2.
j/1 j
s
s
(6)
The jobs considered also di!er in the level of
overlap of processing steps. Some jobs consist of
processing steps that all start at the same time.
However, there are also jobs for which all processing steps are performed consecutively. The smaller
the overlap, the larger the job duration (c ). This
j
job duration may in#uence the interaction margin.
For each job, the processing steps are put in
order of non-decreasing time delay. For each job,
the overlap g is computed as follows:
j
sj
d !d
1
i~1,j
(8)
+ 1! ij
g"
j s !1
p
i~1,j
j
i/2
where d is the time delay of step i of job j. Fig.
ij
3 shows three jobs with the same number of processing steps and processing times, but di!erent
overlap. To investigate the in#uence of the amount
of overlap on the interaction margin, both the average and the standard deviation of the overlap are
included.
A
B
1 J
(9)
k " + g,
j
g J
j/1
p "J(1/J) +J (g !k )2
(10)
j/1 j
g
g
where k is the average overlap and p the standard
g
g
deviation of the overlap.
4.5. Workload balance of resource types
Scarcity of capacity is a cause of job interaction.
If excess capacity exists for some of the resource
types, we expect that there will be less interaction,
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
143
5. Experimental design
Fig. 3. Overlap.
because it will be relatively easy to construct a feasible schedule for these resources. However, if capacity requirements are the same for all resource
types much interaction is expected to occur. This is
particularly the case, if after makespan minimization all resources have become bottlenecks. Therefore, an interaction factor is included for the
balance of the distribution of the total workload
over the di!erent resource types. If the workload
balance is low, then only few resource types have
signi"cantly higher workload than other resources.
These become clear bottlenecks for a given job set,
while the other resources will not cause many
scheduling interactions. If the balance is high, all
resource types have an equal workload, meaning
that all resources have become a bottleneck for
a given job set. We use the maximum utilization if
the makespan is equal to the lower bound as
a measure for the balance of the workload. The
maximum utilization (o ) of a job set gives the
.!9
utilization if a feasible schedule is constructed with
a makespan equal to the lower bound on this
makespan.
1 M
(11)
¸M " + ¸
m
N
m/1
where ¸M is the average resource workload, and
¸ the workload on resource type m.
m
o "¸M /lb
(12)
.!9
where o
is the workload balance. In this section,
.!9
we have discussed eight interaction factors.
The "rst two factors can be de"ned based on the
available resources. For a given production
department, they have the same value regardless of
the job set. The other factors are determined by the
current job set. In the next section, we will conduct
a number of experiments to investigate the in#uence of these interaction factors on the interaction
margin.
Job sets are generated randomly with the goal to
resemble realistic job sets from industry [14]. The
values of the interaction factors are varied on two
or three levels. In each experiment, a job set containing 50 jobs is generated and have to be completed on 10 resources.
To investigate the in#uence of the average and
standard deviation of the number of parallel resources, we consider four di!erent resource con"gurations. These con"gurations di!er in the average
and variation in the number of parallel resources.
The resource con"gurations used in the experiments are given in Table 1. Resource con"gurations
A and D are the extreme values when the average
number of parallel resources is considered. For
these extremes, it is not possible to include variation, because the total number of resources is kept
constant. To investigate the in#uence of this variation, we include con"gurations B and C. This
allows us to estimate the e!ects of both k and p ,
!
!
but not to estimate the two-way interaction
between these two factors.
For each job, the number of processing steps is
given. The number of processing steps per job cannot exceed the number of resources because each
processing step of a job has to be performed on
a di!erent resource if the processing steps overlap.
Three levels for both average and variation are
included in the experiments. For the average number of processing steps the values 1, 5 and 10
processing steps per job are included. Di!erent
levels of variation are considered for the medium
level. The values are given in Table 2.
To each processing step, a resource type is allocated at random. The probability of a speci"c resource type being allocated is equal to the number
of resources of that type available related to the
total number of resources available. For the "rst
processing step of each job, all resources are available. When the "rst processing step is allocated to
a speci"c resource type, the number of resources
available of that type decreases by one. The number
of processing steps of a speci"c job that are
allocated to the same resource type cannot be larger than the number of resources of that speci"c
type. In this way, each resource type has equal
144
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
Table 1
Number of parallel resources
Resource
con"guration
A
B
C
D
Table 3
Processing times
Number of
resource
types
Number of
resources
per type
k
!
10
5
5
1
1 of each type
2 of each type
4, 2, 2, 1, 1
10
1
2
2
10
p
!
0
0
1.2
0
p
p
Processing time
min.
max.
25
1
25
49
0
14
Table 4
Overlap
Table 2
Number of processing steps
Overlap
k
s
Number of steps
min.
max.
1
5
3
1
10
1
5
7
9
10
1
5
5
5
10
p
s
0
0
1.4
2.6
0
probability to become a bottleneck resource and
the balance of the workload of the resources is kept
high.
Next, a processing time is allocated to each
processing step. In the experiments, the average
processing time is kept constant. The variation in
processing times is measured on two levels, namely
no variation and processing times uniformly distributed between 1 and 49. The processing time
distribution is kept the same for all resource types.
The processing time always takes integer values.
Table 3 gives the values for the processing time
distribution.
Finally, the time delay of each processing step is
determined. The time delay of the "rst processing
step is zero for each job. The second and following
processing steps start when a given overlap of the
previous processing step has been processed. In the
experiments, both the average overlap and the variation in overlap are included. The di!erent values
used in the experiments for overlap are given in
Table 4.
min.
max.
0
0.5
0
1
0
0.5
1
1
k
g
p
g
0
0.5
0.5
1
0
0
0.3
0
Table 5
Experimental design
Parameters
Number of levels
Resource con"guration
Processing steps
Processing times
Overlap
4
5
2
4
The workload balance cannot be varied on distinct levels because it is the result of the random
allocation of processing times and resource types to
processing steps. This is a so-called random factor
[13], which cannot be included in the experimental
design.
A full factorial experimental design is constructed, resulting in 160 combinations (see Table 5).
Not all combinations are however feasible. If there
is only one processing step per job, overlap between
the processing steps is not possible and varying the
overlap is therefore not an option. Hence these
combinations have been removed from the experimental design. For each of the remaining 136 combinations, "ve replications have been constructed.
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
An additional exception needs to be made at this
stage. Namely, in the situations with resource con"guration D, when all resources are of the same
type and no variation exists in the number of processing steps, the overlap, and the processing times,
no degrees of freedom are left to generate di!erent
jobs. All parameters used to generate the job set
have been "xed beforehand. Therefore, each job set
generated with these parameter settings is identical
and a replication cannot take place. This leads to
a total number of 628 problems.
A schedule is obtained using a simulated annealing algorithm. This algorithm aims at minimization
of the makespan. We have to take into consideration that some variation in the interaction margin
results from the fact that the resulting schedule is
not necessarily optimal.
145
Fig. 4. In#uence of the number of parallel resources on the
interaction margin.
6. Experimental results
For each observation, the interaction margin is
obtained. The average interaction margin for the
di!erent parameter values is shown by Figs. 4}8.
The in#uence of the number of parallel resources
on the average interaction margin is given in Fig. 4.
We observe that the average interaction margin
becomes smaller if the number of parallel resources
increase. This corresponds with the intuition that
parallel resources provide scheduling #exibility,
and hence result in less job interaction. We further
observe that a small di!erence exists between con"gurations B and C, which have the same average
number of parallel resources. The di!erence in distribution of the resources over the resource types
for these two resource con"gurations may explain
this di!erence.
The in#uence of the number of processing steps
on the interaction margin is shown by Fig. 5. We
observe that an increase in the average number of
processing steps results in a considerable increase
in the interaction margin. This corresponds to the
intuition that an increase in the number of processing steps results in an increase in the job interaction, because more resources are required for each
job. We further observe that the variation in the
number of processing steps hardly in#uences the
average interaction margin. Namely, the three bars
Fig. 5. In#uence of the number of processing steps on the
interaction margin.
Fig. 6. In#uence of the variation in processing times on the
interaction margin.
in the middle all represent an average number of
processing steps of 5, with di!erent variation. This
indicates that in situations with variation in the
number of processing steps, the jobs with few
146
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
Fig. 7. In#uence of the overlap on the interaction margin.
Fig. 8. In#uence of the workload balance on the interaction
margin.
processing steps, which &"t in' easily, compensate
for the jobs with many processing steps, which are
more di$cult to schedule. The net e!ect of the
variation in the number of processing steps is therefore small.
The in#uence of the variation in the processing
times is shown by Fig. 6. We observe that higher
variation in the processing times results in a higher
average interaction margin. The in#uence of the
overlap on the interaction margin is shown by
Fig. 7. We observe that the interaction margin is
higher for an overlap of 0 than for an overlap of 1.
This is in line with our expectation, because the
smaller the overlap the longer the job completion
time. However, an average overlap of 0.5 results in
the highest average interaction margin, suggesting
increased scheduling complexity for these job structures. We further observe that the variation in the
overlap hardly in#uences the interaction margin.
Both center bars in the "gure represent an average
overlap of 0.5, with di!erent variation.
The in#uence of the workload balance on the
interaction margin is given in Fig. 8. This is repre-
sented by a scatter plot, because the workload
balance is a random factor. We observe that on
average, the interaction margin increases with an
increase in the workload balance. This corresponds
with our intuition, because a high workload balance means that resource types all have an equal
workload. Hence, all resource types have become
bottlenecks.
Figs. 4}8 indicated the in#uence of the di!erent
interaction factors. An analysis of variance is performed to determine the contribution of the di!erent factors to the explanation of the variability in
the interaction margin. Main e!ects and two-way
interactions are included. The results are given in
Table 6. The results show that all main e!ects,
except for the standard deviation of the number of
parallel resources and the overlap, are signi"cant.
In addition, several two-way interactions are significant. Together, the signi"cant factors explain
a large part (91%) of the variability in the interaction margin. The proportion of the variability in
the interaction margin that is accounted for by
variation in the interaction factors is given in the
last column. The residual plots do not provide
reasons for concern. Considering these results, we
conclude that we have identi"ed the most important aggregate job set characteristics that are
required to predict the interaction.
The experimental results show that some factors
have a stronger in#uence on the interaction margin
than others. The most important variables are:
k : average number of parallel resources,
!
k : average number of processing steps,
s
p : standard deviation in processing time, and
p
k : average overlap.
g
In addition, we also include the workload balance (o ) in our further analysis, because in the
.!9
experiments we have generated job sets that all had
a high workload balance. Therefore, we have considered mostly values for o
that are close to 1.
.!9
For all other interaction factors, we have considered a wide range of values. We expect that the
in#uence of o
becomes considerably stronger if
.!9
a wider range of values is considered. It seems
reasonable to include the workload balance in
further analysis, because we have encountered
a wider range of values for o
in practice.
.!9
1.
2.
3.
4.
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
147
Table 6
ANOVA results for complete model
k
!
p
!
k
s
p
s
k
g
p
g
p
p
k )k
! s
k )p
! s
k )k
! g
k )p
! g
k )p
! p
p )k
! s
p )p
! s
p )k
! g
p )p
! g
p )p
! p
k )k
s g
k )p
s g
k )p
s p
p )k
s g
p )p
s g
p )p
s p
k )p
g p
p )p
g p
o
.!9
Error
Corrected total
Sum of squares
Degrees of freedom
Mean square
F-statistic
P-value
1.520
0.007
0.502
0.132
4.966
0.029
0.916
0.042
0.045
2.346
0.033
0.013
0.007
0.010
0.030
0.000
0.003
4.338
0.000
0.816
0.102
0.010
0.146
2.636
0.021
2.252
5.805
84.429
2
1
2
2
2
1
1
4
4
4
2
2
2
2
2
1
1
2
1
2
4
2
2
2
1
34
542
627
0.760
0.007
0.251
0.066
2.483
0.029
0.916
0.010
0.011
0.586
0.016
0.006
0.004
0.005
0.015
0.000
0.003
2.169
0.000
0.408
0.025
0.005
0.073
1.318
0.021
0.066
0.011
71.0
0.7
23.4
6.2
231.8
2.7
85.5
1.0
1.1
54.8
1.5
0.6
0.3
0.4
1.4
0.0
0.3
202.5
0.0
38.1
2.4
0.5
6.8
123.1
2.0
6.2
0.00
0.42
0.00
0.00
0.00
0.10
0.00
0.42
0.38
0.00
0.22
0.55
0.71
0.64
0.25
0.88
0.60
0.00
0.89
0.00
0.05
0.62
0.00
0.00
0.16
0.00
Sum of squares
Degrees of freedom
Mean square
F-statistic
P-value
1.811
1.045
12.678
1.021
0.149
2.551
0.022
6.719
0.754
4.487
2.312
6.666
84.429
2
2
2
1
4
4
2
2
2
2
34
570
627
0.906
0.523
6.339
1.021
0.037
0.638
0.011
3.360
0.377
2.243
0.068
0.012
77.5
44.7
542.1
87.3
3.2
54.5
1.0
287.3
32.2
191.8
5.8
0.00
0.00
0.00
0.00
0.01
0.00
0.39
0.00
0.00
0.00
0.00
Table 7
ANOVA results for reduced model
k
!
k
s
k
g
p
p
k )k
! s
k )k
! g
k )p
! p
ks ) k
g
k )p
s p
k )p
g p
o
.!9
Error
Corrected total
148
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
When the analysis of variance is repeated with
only these "ve variables and second-order interactions between these variables included, an R2 of
0.89 is obtained. Table 7 gives the results for the
reduced model. Considering these results, we conclude that a large part of the variability of the
interaction margin is explained by only a limited
number of interaction factors. All other signi"cant
factors and two-way interactions make small contributions.
7. Conclusions and directions for further research
In this paper, we studied multipurpose batch
process industries with overlapping processing
steps, parallel resources and no-wait restrictions
between processing steps. We have proposed
a method to estimate the feasibility of completing
a job set within a speci"c length of time. The
method is based on predicting job set makespan
using aggregate characteristics of the job set.
A lower bound on the makespan can be computed
easily by considering the workload on the resources. Because of job interaction, the minimal
makespan for which a feasible schedule is realized
will often be larger than the lower bound. Interaction at the scheduling level results from relations
between capacity requirements on di!erent resources and from scarcity of capacity.
In this paper, eight factors have been selected to
investigate how this interaction margin is determined. These interaction factors are:
1. average number of parallel resources,
2. standard deviation of the number of parallel
resources,
3. average number of processing steps per job,
4. standard deviation of the number of processing
steps per job,
5. average overlap of processing steps,
6. standard deviation of the overlap of processing
steps,
7. workload balance of resource types, and
8. standard deviation of the processing time.
Experiments in which each of these factors, except for the workload balance, were varied on two
or three levels showed that most factors have a
signi"cant in#uence on the interaction factor. The
standard deviation of the number of parallel resources was the only main e!ect that was not significant. Furthermore, several two-way interactions
proved to have a signi"cant in#uence on the interaction margin. Together these factors explain 91%
of the variation in the interaction margin. Some of
these factors have a much higher contribution to
the explanation of the variability in the interaction
margin than others. The average number of parallel
resources, the average number of processing steps,
the standard deviation in the processing times, the
average overlap and the workload balance are the
most important factors.
The experimental results provide a basis for further development of the prediction model. The prediction model may be used to support customer
order acceptance decisions. Namely, for a given job
set the makespan can be predicted by using the
lower bound and the interaction margin for the job
set. Orders can be accepted for a period until the
expected makespan exceeds the length of the period. The main advantage of this approach is that
realistic order sets are obtained for each time period with small computational e!orts.
Acknowledgements
The detailed comments on earlier versions of this
paper by one of the anonymous referees considerably improved its clarity and exposition.
References
[1] W.H.M. Raaymakers, J.A. Hoogeveen, Scheduling multipurpose batch process industries with no-wait restrictions
by simulated annealing, European Journal of Operational
Research 126 (1) (2000) 131}151.
[2] T.E. Vollmann, W.L. Berry, D.C. Whybark, Manufacturing Planning and Control Systems, 4th Edition, McGrawHill, New York, 1997.
[3] J.W.M. Bertrand, J.C. Wortmann, J. Wijngaard, Production Control: A Structural and Design Oriented Approach,
Elsevier, Amsterdam, 1990.
[4] T.C.E. Cheng, M.C. Gupta, Survey of scheduling research involving due date determination decisions,
European Journal of Operational Research 38 (1989)
156}166.
W.H.M. Raaymakers, J.C. Fransoo / Int. J. Production Economics 68 (2000) 137}149
[5] S. Eilon, I.G. Chowdhury, Due dates in job shop scheduling, International Journal of Production Research 14
(1976) 223}237.
[6] J.K. Weeks, A simulation study of predictable due-dates,
Management Science 25 (1979) 363}373.
[7] K.R. Baker, J.W.M. Bertrand, An investigation of due-date
assignment rules with constrained tightness, Journal of
Operations Management 1 (1981) 109}120.
[8] K.R. Baker, J.W.M. Bertrand, A comparison of due-date
selection rules, AIIE Transactions 13 (1981) 123}131.
[9] J.W.M. Bertrand, The e!ects of workload dependent duedates on job shop performance, Management Science 29
(1983) 799}816.
[10] M.M. Vig, K.J. Dooley, Dynamic rules for due-date assignment, International Journal of Production Research
29 (1991) 1361}1377.
149
[11] M.M. Vig, K.J. Dooley, Mixing static and dynamic #owtime estimates for due-date assignment, Journal of Operations Management 11 (1993) 67}79.
[12] G.L. Ragatz, V.A. Mabert, A simulation analysis of due
date assignment rules, Journal of Operations Management
5 (1984) 27}39.
[13] D.C. Montgomery, E.A. Peck, Introduction to Linear Regression Analysis, Wiley, New York, 1992.
[14] W.H.M. Raaymakers, J.W.M. Bertrand, and J.C. Fransoo,
The performance of workload rules for order acceptance in batch chemical manufacturing, Journal of
Intelligent
Manufacturing
(1998)
accepted
for
publication.