Internutional
Journal
of Production
Economics,
30-31
(1993)
531
531-542
Elsevier
Determining economic
just-in-time production
A. Gunasekaran’,
S.K. Goyalb,
inventory
system
T. Martikainen”
policies in a multi-stage
and P. Yli-Olli”
aSchool qf Business Studies, Unirersit_v oj’ Vaasa, 65101
b Department
@‘Decision
Scrences
& MIS.
Concordia
Vaasa. Finland
Uhersity,
Montreul,
Que. H3G
Ih48,
Canada
Abstract
New manufacturmg concepts, such asJust-m-time
(JIT) productlon. and quality at source tremendously impact on productwty
and quality in many manufacturmg
systems. In order to implement the JIT manufacturing
concept. one has to analyze Its
consequence
on lot-slzmg and work-in-process
Inventory
Realising the significance of modelling such a sltuatlon, a mathematlcal model IS proposed to establish the relationshlp
between the quality at source. work-in-process
Inventory and lot-sizes m
a multi-stage JIT production
system The model along with a search method is used to determine the economx
production
quantltles (EPQs) by mmlmizmg the total system cost.
1. Introduction
Just-in-time
is a system that produces the
required item at the time and in the quantities
needed. Since the excellent work by Schonberger [l] on Japanese manufacturing
techniques,
a considerable progress has been made on the
methods and techniques related to the application of JIT concepts. Sugimori et al. [27] offered many useful insights into the Toyota JIT
manufacturing
system and its operational
issues. Lot-sizing problems are getting recognized all the time as they are practical and
exist almost in all types of production systems,
including JIT, flexible manufacturing
systems,
etc. Although, the batch size of one is preferable in JIT, but still there are set-up costs
associated with processing the products as well
as other technological
and operational
constraints. Nevertheless, they lead to a problem
of lot-sizing
in JIT production
systems.
Correspondence
to: A. Gunasekaran,
Studies, University
0925-5273/93/$06.00
School of Business
of Vaasa, 65101 Vaasa, Finland.
0
1993 Elsevier
Science Publishers
However, there have been relatively few models
and approaches
reported
for the lot-sizing
problems in JIT manufacturing
systems. Recently, Gunasekaran
et al. [3] reviewed the
available literature on JIT systems with an
objective of identifying the gap between theory
and practice based on suitable classification
criterion. They point out that there is a need
for mathematical
and simulation
models to
solve the problems of design, operational, and
justification in JIT manufacturing
systems. In
addition, they presented future research directions for modelling and analysis of JIT production systems.
Philipoom et al. [4] studied the factors that
influence the number of kanbans required in
the implementation
of JIT techniques and suggested in Ref. [S] a mathematical
programming approach for determining the economic
lot-sizes
in a JIT manufacturing
system.
Spence and Porteus [6] presented a model for
the set-up cost reduction.
The change in
inventory control concept has been suitably
motivated
and focussed
by Zangwill
[7].
Porteus [8-lo] presented a number of useful
B.V. All rights reserved.
ideas for developing various support systems
in JIT production
and quality control with
appropriate
mathematical
models. Funk [ 1 l]
presented attributes
of a JIT manufacturing
system, and compared various inventory cost
reduction
strategies in a JIT manufacturing
system. Bard and Golany
[12] formulated
a mixed integer programming
model to determine the number of kanbans required for each
product by minimizing the set-up cost, inventory holding cost, and shortage cost. However,
their model is applicable only for an assembly
system. Sipper and Shapira [ 131 developed
a decision rule to facilitate a priori classification of a production
system that would utilize
a JIT or WIP type inventory control policy.
Karmarkar
[ 141 investigated
and differentiated the push, pull, and hybrid control systems. Karmarkar
and Kekre [ 151 presented
a model to determine
the optimal batching
policies in a kanban system. Bitran and Chang
1161 offered a number of mixed integer programming formulations
to address the problems of product structure. Axsater and Rosling
[ 171 investigated
under what circumstances
(i.e., system configuration
and control rules)
policies based on echelon stocks are superior
to policies based on installation
stocks. Recently, Gunasekaran
et al. [IS] offered a model
for a multi-stage production
system to determine the optimal number of machines required
for achieving the JIT production.
Moreover, lot-sizing problem in JIT manufacturing systems considering
the quality at
the source has not been given due consideration. The quality at the source involves controlling the process whenever it drifts from
normal process condition.
It requires shutdown and start-up of the machines in order to
bring the process to normal operating conditions. The related inventory costs and service
costs that are arising from these activities must
be considered while determining the economic
lot-sizes. Furthermore,
the purpose of balancing the production
rates between stages is predominant in any JIT production
systems, and
this is an important
problem at the planning
level in order to identify the bottle-neck operations, balancing of production,
and number of
kanbans required to achieve the JIT production. A model is developed
in this paper
to determine
the optimal batch sizes considering the impact of the process control
which would lead to a minimum total system
cost.
The rest of this paper is as follows: The
mathematical
model is presented in Section 2.
An example problem is presented in Section 3.
Section 4 provides the details on the results
obtained and the corresponding
analysis. Section 5 offers discussions
on the model
developed
and its limitations.
Finally, the
conclusions
of this research work are presented in Section 6.
2. Mathematical
model
The proposed mathematical
model here estimates the total system cost in a JIT manufacturing system as a function of the batch sizes at
each stage and for all products.
2.1.
i
j
‘j
Dl
A,j
Qii
t
LJ
Cl,
ciO
product index (i = 1, 2,. . . , M),
stage index (j = 1. 2,. . . , N),
number of machines at stage j,
demand for product i per unit time or per
year,
set up cost per set-up for product i at
stage j,
batch size for product i at stagej (decision
variable),
priority assigned in processing (capacity
allocation) product i at stage j,
penalty cost due to imbalance in production rates between stages j and j + 1,
mean process drift rate while processing
product i at stage j,
mean service rate for bringing the process
to normal operating condition for product i at stage j,
processing time per unit of product i at
stage j.
cost per unit product i after processing at
stage j,
raw material cost per unit product i,
A. Gunasekaran
H
Rij
Tij
Li,
Gij
/I
d:
Z
et al.lDetermming
inventory
cost per unit investment
per
unit time period,
number of production cycles for the given
demand of product i at stage j,
processing time for a batch of product i at
stage j,
average completion
time for a batch of
product i at stage j,
average cost per unit of product i between
stages j and j + 1,
production
rate for product i at stage j,
lower bound on the value of batch size for
product i,
total system cost.
2.2. Assumptions
The following assumptions are made in developing the model:
(i) Demand
for each product
is uniform,
deterministic,
and known.
(ii) Set-up c OSt per set-up is constant, independent of set-up sequence, and batch
sizes.
(iii) For any process, the process drifts follow
Poisson distribution,
and the service time
for each drift follows exponential distribution with average drift and service rates,
respectively.
(iv) Once the process goes out of control, the
machine automatically
stops producing
defective products.
(v) There is no finished product inventory
cost as the products will be dispatched
once the processing is completed at the
final stage.
2.3. The basic model
The total system cost consists of the following costs: (1) set-up cost, (2) cost due to process
control, and (3) cost due to imbalance in production rates. These costs are derived hereunder.
2.3.1.
Set-up cost
The total set-up cost considering
ucts and stages is given by
all prod-
economx
inventory policies
533
(1)
2.3.2. Cost due to process control for quality
at the source
This cost arises from waiting of batches due
to stopping and restarting the machines for
controlling
the process. This process control
supports the “quality at the source” which
undoubtedly improves the quality and reduces
the level of scrappage or defective products. In
most of the occasions, the start-up and shut
down of the machines for controlling the process lead to an in-process inventory carrying
cost due to waiting for the process being
brought to the normal operating condition.
But at the same time, the scrappage level can
be reduced.
The processing time for a batch is given by
TLj= Qlj X
t,j.
(2)
It has been assumed here that the time between two successive drifts of the process at
a particular machine follows exponential distribution. Hence, the number of drifts per unit
time follows Poisson process with mean rate of
drifts. The drift rate is a function of the machine age, motivation
of the workers in performing the process, and the nature of the
process. All these decide the drift rate for a particular product at a stage. Depending upon the
nature of process control required and the skill
of the worker, the service time required to
bring the process to a normal working condition varies. Therefore, it is assumed that the
service time required for each process control
task follows exponential
distribution
with
mean service rate.
Suppose, the operator is an M/M/l server
(or a server when there are Sj machines at stage
j). Every time a machine drifts it has to be
serviced by the operator. From M/M/l queuing theory, the total time spent (waiting time
plus service time per drift) by a batch per drift
can be estimated using the M/M/l
(infinite
source) queuing formula (see Ref. [ 191).
(3)
The average time spent by the batch due to
process drifts while processing that batch can
be obtained as
(4)
The average total time required (processing
time for batch + average time spent by the
batch due to process drifts) for product i at
stagej to complete the processing of a batch is
given by
Lij = T,,
(3
The number of production
cycles per unit
time (per year) for product i at stage j is represented by
2.3.3. Cost due to irnbalmce
rates het,\?een successive stages
irz productiort
The main aim of any production
system
which intends to implement JIT is to balance
the production
rates between successive production stages. The aspects of quality at the
source have been modelled
by Goyal and
Gunasekaran
[20]. However, they considered
solely a traditional production
system in their
article.
The production rate for a particular product
depends upon the number of machines actually used for a product.
If more than one
machine is used for a product, then the production rate will be much higher than that of
with only one machine. However, the required
production
rate to balance the production
between consecutive
stages is an important
subject matter. The production rate for a product at a stage is a function of the total process
completion time of a batch, priority assigned
to the product for processing, and the number
of machines used for processing the product at
the stage, etc.
The production rate for product i from stage
j can be calculated using Eq. (8) as
(9)
where
Since a process drift may occur at any time
during the processing of a batch of a product,
the value of the per unit product at which the
drift occurs may be difficult to obtain. Hence,
the average cost per unit product has been
accounted to compute the cost due to process
control. This cost can be calculated as
(7)
The total inventory
ing is estimated as
cost due to process drift-
(Rt, Qij Tij Gij)
(Q
C &,j = 1,
forj
= 1, 2,. . ., N.
i=l
The parameter
~,j indicates the priority assigned for processing product i at stage j. This
parameter selects the number of machines to
be assigned for a particular product at a stage.
In practice the value of &ij is being estimated
based on the marketing and financial performances. It has been assumed here that the
machines at each stage have identical capacities.
In order to achieve a balance among production rates, we have considered
a penalty
cost (amplified one) which encompasses all the
relevant costs associated with imbalance in the
A. Gunasekaran
et al..lDetermining economic inventory policies
rates, The cost due to imbalance in
rates between stages j and j + 1 is
production
production
given by
(10)
The penalty cost due to imbalance in production rates (Qij) may include the following costs:
(i) inventory cost due to waiting of batches, (ii)
shortage costs, (iii) cost due to idleness of the
facilities, etc. Depending upon the production
configuration,
for example in assembly systems, there may be integer multiple processing
rates at successive stages.
Total cost due to imbalance in production
rates considering
all products and stages is
given by
N-l
i-M
i
1
c j=lC
IAij
-
)Lij+
I
1
(11)
1=1
2.3.4.
Minimize
+
5i
i=l
{Rij Qij Tij G,j}
ix1
M
N-’
C C
+
i=l
(15)
Constraint (13) indicates that the service rate
for the process control must be greater than
the drift rates for a product at any stages.
Constraint
(14) gives upper bound on the
batch sizes. The lower bound on the batch
sizes is represented by constraint (15).
2.3.5. Number of kanbam
required
The number of kanbans
estimated [2] by
required
(Dl(l + Pii)>,
Yij
=
~~ij
X
can be
(16)
qij)
where ‘I’iJ is the total number of kanbans required, pij the safety coefficient, and U],j= the
container capacity. Here, the values of ~ij and
V]ijare assumed to be deterministic and known.
I3bij
-
Aij+
11
<Dij
(12)
&j,
Qijd Dij,
3. An example
A four stage JIT production
system manufacturing
three products
is considered
to
explain the application of the model. The objective here is to determine the optimal batch
sizes and the kanbans required by minimizing
the total system cost. The input to the example
problem is presented in Table 1. The value of
sij has been determined
here arbitrarily.
But
usually, this value is determined
based on
economical and technical considerations.
The
resulting total system cost, eq. (12) is of nonlinear nature and, hence, the conventional
optimization technique may not be suitable for
the objective function (Z). Therefore, a direct
pattern search method (DPSM) is used for this
purpose [21].
i=l
Subject to:
>
for all i and j.
Problem formulation
Now the problem of lot-sizing in JIT is to
determine
the optimal batch sizes for each
product at each stage which would lead to
a minimum total system cost. This total system
cost can be obtained by the summation of the
cost equations (l), (8). and (11). The formulation of the lot-sizing problem can be given as
Bij
Qij 3 di,
535
4. Analysis of the results
for all i and j,
(13)
for all i and j,
(14)
The results obtained by the DPSM are presented in Table 2. The results corresponding
to
the cases with and without process control
A. Gunasekaran
et al.iDetermning
economic
incentory policies
(quality at the source) are compared in Table 3.
The results of the sensitivity analysis are provided in Table 4.
4.1. Optimul results obtuined b?) the model
The functioning of the DPSM is illustrated
in Table 2. The direct pattern search method
starts from some initial values (i.e., uniform
batch sizes for each product at all stages) for
the decision variables, i.e., the batch size required (Qij) for each product at each stage.
Also, the model considers the batch splitting
and forming in order to achieve both balancing the production
rates and minimum total
system cost. The optimal batch sizes obtained
from the model leads to a savings in total
system cost of about 54% as compared to the
uniform batch sizes. Also, the cumulative imbalance in production
rates has come down
from 10.7365 to 0.2747. Table 2 also presents
the savings in the cost due to imbalance in
production
rates (from $644191 to $16481).
4.2. Comparison
process control
of the results without and with
The optimal batch sizes and the corresponding costs obtained for the cases with and without process control are presented in Table 3.
The quality at the source concept which embodies the process control results in a significant
savings in the scrappage of items and in turn
its associated costs such as investment in materials, labour, and products. Also, the reduction in scrappage leads to an increase in the
utilization level of the facilities.
The optimal batch sizes obtained with process control facilitate the use of smaller batch
sizes as compared to those of without process
control. Also, they lead to an effective balancing of production
rates as compared
to the
larger batch sizes in the case of without process
control.
4.3. Sensitivity
unulysis
To study the behaviour
of the model, a
sensitivity analysis has been conducted.
The
556.
400,
278.
593,
533,
853,
556,
593,
266,
556,
400,
291.
500,
400,
300.
500,
600,
800.
842,
593,
533,
854.
300,
700,
400,
600,
500.
400,
500,
546,
400.
300,
500,
400,
300.
500,
187,
100,
299.
300,
183,
100,
312,
100,
188,
200,
100,
300,
300.
200,
100,
400,
300,
500,
Batch sizes
(Q,, j = 1, N; i = 1. M)
260,
171.
569
569
260.
171,
593
184,
248.
300,
200,
500
500
400,
300,
400,
300,
500
by the search method
32.3096
32.1347
32.1262
30.2348
30.2625
41.1375
Cost due to process
control ($ x10”)
126.2233)] xl00 = 54.82%.
23.0670
23.2702
22.9882
23. I667
23.1690
20.6667
Set-up cost
($ xlO1)
Savings in total system cost = [(126.2233-57.0247),
403
378
328
68
21
1
Iteration
Table 2
The results obtained
1.6481
2.3518
4.8288
16.5968
41.7778
64.4191
Cost due imbalance
in production
rates ($ x 104)
57.0247
57.7567
59.9432
69.9983
95.2093
126.2233
Total cost (Z)
($ xlOS)
0.2747
0.3920
0.8048
2.7661
6.9630
10.7365
Actual imbalance in
production (batchesi’
unit time)
:.
‘)
5’
:
5
G
%
2.
=:
2
2.
3.
2
2
s
e
b
2
+
Q
T
E,
k
e
f
R
187,
100,
299,
260,
171,
569
A = 2.0.4
N = 0.20
533
853,
597,
625,
400,
290,
400,
278.
625,
400,
296,
625,
400,
290,
533,
853,
597,
533,
862.
597,
533,
8.53.
H = 0.10
556,
593,
H = 0.30
200,
100,
313.
100,
299,
200,
100.
319,
200,
100,
313,
f87,
Optimal batch sizes
(Q,,,j = 1, N; i = 1, M)
Parameters!
Variables
278,
171.
594.
171,
569.
278,
171,
597.
278,
171,
594.
260,
Table 4
Results obtained by the sensitivity analysis
556,
400,
278,
36.643 1,
22 3000
22.1736
23.0670
Set-up cost
($ XlOj)
23.0670
593,
533.
853.
with process
control
345,
300,
500
18.0797
302.
100,
300,
1094,
800,
997,
Without
process control
694,
500,
625,
Set-up cost
(S xlOA’)
Optimal batch sizes
(Q,], j = 1, N: i = 1, M)
Situation
Table 3
Comparison of the results without and with process control
38.2924
22.4 169
11.2977
32.3096
Cost due to
process control
(S xlO1)
32.3096
~.OO~
Cost due to
process control
(S xlOS)
4.8885
2.0880
1.6481
Cost due imbalance
in production
rates ($ x 10”)
1.64806
41.5190
Cost due to imbalance
m production rates
($ x104)
79.8240
46.7454
35.5593
51.0247
Total cost (2)
($ XlOj)
57.0241
59.5987
Total cost (2)
(S x10”)
0.8148
0.3381
0.3480
0.2741
Actual imbalance m
productton (batches,
unit time)
0.2747
6.9198
Actual imbalance in
production (batches,
unit time)
B
Z
c;
E
0.
5’
2
5
.;
l.Oa
/I = 4.op
b = 2.08
/? = 1.og
D = 2.OD
D = 0.5D
D = l.OD
a = 0.25~
5( = 0.5ct
cl =
A = OSA
A = 1.OA
556,
400,
278,
556,
400,
290,
556,
400,
253.
556,
400,
278,
625,
479,
520,
654,
547,
927,
593,
533,
853,
797,
647,
897,
1424,
1263,
1525,
556,
400,
278,
625,
479,
520,
654,
547,
927,
593,
533,
853,
791,
647,
897,
1424,
1263,
1525,
593,
533
853,
593,
533
853,
593,
533
853,
556,
400,
278,
455,
400,
278,
593,
533
853,
485,
533
853,
187,
100,
299,
200,
100,
294,
208,
113,
486,
187,
100,
299,
187,
100,
312,
187,
100,
273,
187,
100,
299,
200,
100,
294,
208,
113,
486,
187,
100,
299,
165,
100,
299,
260,
171,
569.
231.
214,
504.
234,
318,
824.
260,
171,
569.
260,
171,
598
260,
171,
512
260,
171,
569.
231,
274,
504.
234,
318,
824.
260.
171,
569
229,
171,
569
15.6429
20.6624
23.0670
47.4958
11.3905
23.0670,
15.6429
20.6624
23.0670
12.1178
23.0670
3.4748
6.0495
32.3096
61.0853
16.5638
32.3096
3.4748
6.0495
32.3096
31.2539
32.3096
20.4191
30.4490
1.6481
2.9897
1.1409
1.6481
20.4191
30.4490
1.6481
3.1958
1.6481
39.5368
57.1609
57.0241
111.5708
29.0952
51.0241
39.5368
57.1609
57.0247
46.5675
51.0241
3.4032
5.0748
0.2747
0.4983
0.1901
0.2741
3.4032
5.0748
0.2141
0.5326
0.2747
details of the results obtained
for different
levels of inventory holding rate (H), mean drift
rate (‘XX).
demand rate (D), and process control
rate (p) are reported in Table 4.
The variation
in inventory
holding rate
leads to significant changes in the cost due to
process control and set-up cost. For instance,
lowering the inventory holding rate (H) from
0.30 to 0.10 results in a reduction in set-up cost
($230670 to $221736). This indicates that the
model selects larger batch sizes when H = 0.10
compared to those of when H = 0.30 in order
to save some set-up cost. Because of the larger
batch sizes, the cost due to cumulative imbalance in production
rates has risen sharply.
This has been revealed
by the increased
level of cumulative imbalance from 0.2747 to
0.3480.
A decrease in set-up cost per set-up ( l.OA to
OSA) may lead to a substantial
reduction in
the total
set-up
cost (from $230670
to
$121178). Owing to the reduction
in set-up
cost, the model selects smaller batch sizes. This
obviously results in a lower cost due to process
control when A is at 0.5A ($323096 to $3 12539)
as compared to that of l.OA. Corresponding
to
this, the cost due to imbalance in production
rates has increased from $1648 1 to $3 1958.
This implies that the model looks for higher
savings in set-up cost when A is equal to 0.5.4.
Under these circumstances, the increase in cost
due to the imbalance in production
rates may
not be significant as compared to the savings
in total set-up cost. Hence, the model prefers
smaller batches to achieve a reasonable reduction in cost due to process control.
A reduction in the average drift rate (1.0~ to
0.5~~)will allow the model to select larger batch
sizes. This perhaps results in a lower total
set-up cost ($230670 to $206624). Obviously,
the reduction in drift rate leads to a corresponding reduction in cost due to process control, from $323096 to $60495. However, the
increase in batch sizes causes problems for
balancing
the production
rates among the
stages. This can be noticed from the increase in
cost due to imbalance
in production
rates
($1648 1 to $304490). This indicates the influence of the process control.
Table 5
Number of kanbans
Stage
Number
of kanbans
product
1
I
7
7
7
8
X
__-
;
4
required for the data given m Table 1
reqmred
product
5
5
5
5
2
(TNK)
product
3
13
13
I2
12
The optimal batch sizes obtained for different levels of demand (1 .OD, 0.5D, and 2.00) are
also presented in Table 4. The analysis of the
results reveals the significance of the optimal
batch sizes and the capacity
available
in
achieving the balancing of production
rates.
Also. the results corresponding
to different
levels of fls are presented in order to gain more
insights into the characteristics
and application of the model. The number of kanbans
required corresponds
to the safety coefficient
value of 0.20 and constant container capacity
of 200 for all the products and at all stages are
presented in Table 5.
5. Discussion on the model and results
The main purpose of our modelling effort is
to explain the relationship
between lot-sizing
policies, quality at the source (process control),
and balancing of production
in a JIT production system. Moreover,
it motivates the lotsizing policies which are based on batch
splitting and forming with a view to attain the
balancing among the production
rates. The
proposed model is a planning model and it
helps to derive overall ideas about the batching policies wherein the set-up cost is quite
accountable as compared to the inventory cost
due to process control and imbalance in production rates. Nevertheless,
there are many
other operational,
and process constraints
such as material handling capcity, capacity of
the machines, various priority
rules in sequencing and scheduling, etc. are to be considered in the batch size optimization.
A. Gunasekaran
et al.,‘Drterrnining
The results obtained
indicate the significance of process control which rather supports
the quality at the source as noted earlier. The
assumptions
that the drifts rates follow
Poisson and the service time follows exponential require further investigation
in order to
confirm the application
with real life JIT
situations. Besides, a reduction in set-up cost
demonstrates
the potential in achieving the
balancing among the production stages. In addition to this, investing in quality control leads
to a significant improvement
in achieving the
JIT material flow. Furthermore,
various insights could be derived from the model for
different production situations on different related issues like investing in quality control,
set-up reduction programme, process control,
etc., and their implications on the performance
of the JIT system. The trade-off between the
value of the scrap and the cost related to process control would be an interesting problem
to pursue in future. The study reported here
may be useful for a time-phased implementation of JIT in manufacturing
organizations.
6. Conclusions
A mathematical
model has been developed
for determining economic production quantities in JIT manufacturing
systems by minimizing the total system cost. The main objective of
the model is to determine the optimal batch
sizes incorporating
the process control so that
there is a smooth material flow among the
stages that support the JIT production. A direct pattern search method has been employed
for determining
the economic
production
quantities. However, the model developed is
based on a number of assumptions
and approximations.
This implies that there are avenues for further investigation
to enhance the
accuracy of modelling the JIT systems.
Acknowledgements
The authors
Meester
and
are grateful to Professors G.J.
John Miltenberg,
and three
economic
rnrrrrtor~ policies
541
anonymous referees for their extremely useful
and helpful comments on the earlier version of
this manuscript. Also, the authors thank Professor Ilkka Virtanen, Rector, University
of
Vaasa, Finland, for his extended co-operation
in their research projects.
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