determining economic inventory policies in a multi-stage just-in-time
TRANSCRIPT
Internutional Journal of Production Economics, 30-31 (1993) 531-542 Elsevier
531
Determining economic inventory policies in a multi-stage just-in-time production system
A. Gunasekaran’, S.K. Goyalb, T. Martikainen” and P. Yli-Olli”
a School qf Business Studies, Unirersit_v oj’ Vaasa, 65101 Vaasa. Finland b Department @‘Decision Scrences & MIS. Concordia 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 mathemat-
lcal 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 Schonber- ger [l] on Japanese manufacturing techniques, a considerable progress has been made on the methods and techniques related to the applica- tion of JIT concepts. Sugimori et al. [27] of- fered many useful insights into the Toyota JIT manufacturing system and its operational issues. Lot-sizing problems are getting recog- nized 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 prefer- able in JIT, but still there are set-up costs associated with processing the products as well as other technological and operational con- straints. Nevertheless, they lead to a problem of lot-sizing in JIT production systems.
Correspondence to: A. Gunasekaran, School of Business Studies, University of Vaasa, 65101 Vaasa, Finland.
However, there have been relatively few models and approaches reported for the lot-sizing problems in JIT manufacturing systems. Re- cently, 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 direc- tions for modelling and analysis of JIT produc- tion systems.
Philipoom et al. [4] studied the factors that influence the number of kanbans required in the implementation of JIT techniques and sug- gested in Ref. [S] a mathematical program- ming 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
0925-5273/93/$06.00 0 1993 Elsevier Science Publishers 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 deter- mine the number of kanbans required for each product by minimizing the set-up cost, inven- tory 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 classifica- tion of a production system that would utilize a JIT or WIP type inventory control policy. Karmarkar [ 141 investigated and differenti- ated the push, pull, and hybrid control sys- tems. 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 pro- gramming formulations to address the prob- lems 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. Re- cently, Gunasekaran et al. [IS] offered a model for a multi-stage production system to deter- mine the optimal number of machines required for achieving the JIT production.
Moreover, lot-sizing problem in JIT manu- facturing systems considering the quality at the source has not been given due considera- tion. The quality at the source involves con- trolling the process whenever it drifts from normal process condition. It requires shut- down and start-up of the machines in order to bring the process to normal operating condi- tions. 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 balanc- ing the production rates between stages is pre- dominant in any JIT production systems, and this is an important problem at the planning level in order to identify the bottle-neck opera- tions, balancing of production, and number of
kanbans required to achieve the JIT produc- tion. A model is developed in this paper to determine the optimal batch sizes con- sidering 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. Sec- tion 5 offers discussions on the model developed and its limitations. Finally, the conclusions of this research work are pres- ented in Section 6.
2. Mathematical model
The proposed mathematical model here es- timates the total system cost in a JIT manufac- turing 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 produc- tion 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 prod- uct 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 et al.lDetermming economx inventory policies 533
H
Rij
Tij
Li,
Gij
/I d:
Z
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 de- veloping the model:
(i) Demand for each product is uniform, deterministic, and known.
(ii) Set-up c OS per set-up is constant, inde- t pendent 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 distribu- tion 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 follow- ing costs: (1) set-up cost, (2) cost due to process control, and (3) cost due to imbalance in pro- duction rates. These costs are derived here- under.
2.3.1. Set-up cost
The total set-up cost considering all prod- ucts and stages is given by
(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 pro- cess 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 be- tween two successive drifts of the process at a particular machine follows exponential dis- tribution. Hence, the number of drifts per unit time follows Poisson process with mean rate of drifts. The drift rate is a function of the ma- chine age, motivation of the workers in per- forming the process, and the nature of the process. All these decide the drift rate for a par- ticular 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 condi- tion 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 queu- ing 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 repre- sented by
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 cost due to process drift- ing is estimated as
(Rt, Qij Tij Gij) (Q
2.3.3. Cost due to irnbalmce irz productiort rates het,\?een successive stages
The main aim of any production system which intends to implement JIT is to balance the production rates between successive pro- duction 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 actu- ally used for a product. If more than one machine is used for a product, then the pro- duction 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 prod- uct 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
C &,j = 1, forj = 1, 2,. . ., N. i=l
The parameter ~,j indicates the priority as- signed 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 perfor- mances. It has been assumed here that the machines at each stage have identical capac- ities.
In order to achieve a balance among pro- duction 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 535
production rates, production rates given by
The cost due to imbalance in between stages j and j + 1 is
(10)
The penalty cost due to imbalance in produc- tion 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 sys- tems, 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
i-M N-l 1
i c
1=1
C IAij - )Lij+ 1 I (11) j=l
2.3.4. 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 formula- tion of the lot-sizing problem can be given as
Minimize
+ 5 i {Rij Qij Tij G,j} i=l ix1
M N-’
+ C C I3bij - Aij+ 11 <Dij (12) i=l i=l
Subject to:
Bij > &j, for all i and j, (13)
Qij d Dij, for all i and j, (14)
Qij 3 di, for all i and j. (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 required can be estimated [2] by
Yij =
(Dl(l + Pii)>, ~~ij X qij)
(16)
where ‘I’iJ is the total number of kanbans re- quired, pij the safety coefficient, and U],j = the container capacity. Here, the values of ~ij and V]ij are assumed to be deterministic and known.
3. An example
A four stage JIT production system manu- facturing three products is considered to explain the application of the model. The ob- jective 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 non- linear nature and, hence, the conventional op- timization technique may not be suitable for the objective function (Z). Therefore, a direct pattern search method (DPSM) is used for this purpose [21].
4. Analysis of the results
The results obtained by the DPSM are pres- ented 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 pro- vided 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 re- quired (Qij) for each product at each stage. Also, the model considers the batch splitting and forming in order to achieve both balanc- ing 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 im- balance 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 $16481).
4.2. Comparison of the results without and with process control
The optimal batch sizes and the correspond- ing costs obtained for the cases with and with- out process control are presented in Table 3. The quality at the source concept which embo- dies the process control results in a significant savings in the scrappage of items and in turn its associated costs such as investment in ma- terials, labour, and products. Also, the reduc- tion in scrappage leads to an increase in the utilization level of the facilities.
The optimal batch sizes obtained with pro- cess control facilitate the use of smaller batch sizes as compared to those of without process control. Also, they lead to an effective balanc- ing of production rates as compared to the larger batch sizes in the case of without process control.
4.3. Sensitivity unulysis
To study the behaviour of the model, a sensitivity analysis has been conducted. The
Tab
le
2 T
he
resu
lts
obta
ined
by
th
e se
arch
m
etho
d
Iter
atio
n B
atch
si
zes
Set-
up
cost
C
ost
due
to
proc
ess
Cos
t du
e im
bala
nce
Tot
al
cost
(Z
) A
ctua
l im
bala
nce
in
+
(Q,,
j =
1,
N;
i =
1. M
) ($
xl
O1)
co
ntro
l ($
x10
”)
in p
rodu
ctio
n ($
xlO
S)
prod
uctio
n (b
atch
esi’
Q
ra
tes
($ x
104
) un
it tim
e)
T
E,
1 40
0,
400.
40
0,
400,
k
300.
30
0,
300,
30
0,
20.6
667
41.1
375
64.4
191
126.
2233
10
.736
5 e
500,
50
0,
500,
50
0 f R
400,
60
0,
200,
40
0,
e 21
50
0,
500.
10
0,
300,
23
.169
0 30
.262
5 41
.777
8 95
.209
3 6.
9630
b
700,
30
0,
300.
50
0 2
500,
50
0,
200,
30
0,
2.
68
600,
40
0,
100,
20
0,
23. I
667
30.2
348
16.5
968
69.9
983
2.76
61
3.
800.
30
0.
300,
50
0 2 2
593,
55
6,
188,
24
8.
s
328
546,
40
0,
100,
18
4,
22.9
882
32.1
262
4.82
88
59.9
432
0.80
48
:. ‘)
842,
26
6,
300,
56
9 5’
59
3,
556,
18
3,
260.
:
378
533,
40
0,
100,
17
1,
23.2
702
32.1
347
2.35
18
57.7
567
0.39
20
5 85
4.
291.
31
2,
593
G
%
593,
55
6.
187,
26
0,
=:
2.
403
533,
40
0,
100,
17
1.
23.0
670
32.3
096
1.64
81
57.0
247
0.27
47
2 85
3,
278.
29
9.
569
Savi
ngs
in t
otal
sy
stem
co
st
=
[(12
6.22
33-5
7.02
47),
12
6.22
33)]
xl
00
=
54.8
2%.
Tab
le
3 C
ompa
riso
n of
the
res
ults
with
out
and
with
pro
cess
con
trol
Situ
atio
n O
ptim
al
batc
h si
zes
(Q,]
, j =
1, N
: i
= 1
, M)
Set-
up
cost
C
ost
due
to
Cos
t du
e to
im
bala
nce
(S x
lOA
’)
Tot
al
cost
(2)
A
ctua
l im
bala
nce
in
proc
ess
cont
rol
m p
rodu
ctio
n ra
tes
(S x
lOS)
(S
x10
”)
($ x
104)
pr
oduc
tion
(bat
ches
, un
it tim
e)
With
out
1094
, 69
4,
302.
34
5,
proc
ess
cont
rol
800,
50
0,
100,
30
0,
18.0
797
~.O
O~
41.5
190
59.5
987
6.91
98
997,
62
5,
300,
50
0
with
pro
cess
59
3,
556,
18
7,
260,
co
ntro
l 53
3.
400,
10
0,
171,
23
.067
0 32
.309
6 85
3.
278,
29
9,
569
1.64
806
57.0
241
0.27
47
Tab
le
4 R
esul
ts
obta
ined
by
the
sen
sitiv
ity
anal
ysis
Para
met
ers!
O
ptim
al
batc
h si
zes
Set-
up
cost
C
ost
due
to
Cos
t du
e im
bala
nce
Tot
al
cost
(2)
A
ctua
l im
bala
nce
m
5’
Var
iabl
es
(Q,,,
j =
1, N
; i
= 1
, M)
($ X
lOj)
2 pr
oces
s co
ntro
l in
pro
duct
ion
($ X
lOj)
(S
xlO
1)
prod
uctto
n (b
atch
es,
rate
s ($
x 1
0”)
unit
time)
5 .;
H =
0.3
0 59
3,
556,
f8
7,
260,
B
Z
533,
40
0,
100,
17
1,
23.0
670
32.3
096
c;
1.64
81
51.0
247
0.27
41
E
853,
27
8.
299,
56
9.
0.
H = 0
.10
597,
62
5,
200,
27
8,
533,
40
0,
100.
17
1,
22.1
736
11.2
977
2.08
80
35.5
593
0.34
80
862.
29
6,
319,
59
7.
N =
0.2
0 59
7,
625,
20
0,
278,
53
3,
400,
10
0,
171,
22
300
0 22
.4 1
69
8.53
. 29
0,
313,
59
4.
46.7
454
0.33
81
A =
2.0.4
597,
62
5,
200,
27
8,
533
400,
10
0,
171.
36
.643
1,
38.2
924
4.88
85
79.8
240
0.81
48
853,
29
0,
313.
59
4.
A =
1.OA
A =
OSA
cl
= l.O
a
5( =
0.5
ct
a =
0.2
5~
D =
l.O
D
D =
0.5
D
D =
2.O
D
/? =
1.
og
b =
2.
08
/I =
4.o
p
593,
55
6,
187,
26
0.
533
400,
10
0,
171,
85
3,
278,
29
9,
569
485,
45
5,
165,
22
9,
533
400,
10
0,
171,
85
3,
278,
29
9,
569
593,
55
6,
187,
26
0,
533,
40
0,
100,
17
1,
853,
27
8,
299,
56
9.
791,
62
5,
200,
23
1,
647,
47
9,
100,
27
4,
897,
52
0,
294,
50
4.
1424
, 65
4,
208,
23
4,
1263
, 54
7,
113,
31
8,
1525
, 92
7,
486,
82
4.
593,
55
6,
187,
26
0,
533
400,
10
0,
171,
85
3,
278,
29
9,
569.
59
3,
556,
18
7,
260,
53
3 40
0,
100,
17
1,
853,
29
0,
312,
59
8 59
3,
556,
18
7,
260,
53
3 40
0,
100,
17
1,
853,
25
3.
273,
51
2
593,
55
6,
187,
26
0,
533,
40
0,
100,
17
1,
853,
27
8,
299,
56
9.
797,
62
5,
200,
23
1.
647,
47
9,
100,
21
4,
897,
52
0,
294,
50
4.
1424
, 65
4,
208,
23
4,
1263
, 54
7,
113,
31
8,
1525
, 92
7,
486,
82
4.
23.0
670
32.3
096
1.64
81
51.0
241
0.27
47
12.1
178
31.2
539
3.19
58
46.5
675
0.53
26
23.0
670
32.3
096
1.64
81
57.0
247
0.21
41
20.6
624
6.04
95
30.4
490
57.1
609
5.07
48
15.6
429
3.47
48
20.4
191
39.5
368
3.40
32
23.0
670,
32
.309
6 1.
6481
51
.024
1 0.
2741
11.3
905
16.5
638
1.14
09
29.0
952
0.19
01
47.4
958
61.0
853
2.98
97
111.
5708
0.
4983
23.0
670
32.3
096
1.64
81
57.0
241
0.27
47
20.6
624
6.04
95
57.1
609
5.07
48
15.6
429
3.47
48
30.4
490
20.4
191
39.5
368
3.40
32
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 imbal- ance 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 reduc- tion 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 corres- ponding reduction in cost due to process con- trol, 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 influ- ence of the process control.
Table 5 Number of kanbans required for the data given m Table 1
Stage Number of kanbans reqmred (TNK)
product 1 product 2 product 3
I 7 5 13 7
; 7 5 13 8 5 I2
4 X 5 12 __-
The optimal batch sizes obtained for differ- ent 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 applica- tion 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 produc- tion system. Moreover, it motivates the lot- sizing 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 batch- ing policies wherein the set-up cost is quite accountable as compared to the inventory cost due to process control and imbalance in pro- duction rates. Nevertheless, there are many other operational, and process constraints such as material handling capcity, capacity of the machines, various priority rules in se- quencing and scheduling, etc. are to be con- sidered in the batch size optimization.
A. Gunasekaran et al.,‘Drterrnining economic rnrrrrtor~ policies 541
The results obtained indicate the signifi- cance 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 exponen- tial 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 ad- dition to this, investing in quality control leads to a significant improvement in achieving the JIT material flow. Furthermore, various in- sights could be derived from the model for different production situations on different re- lated 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 pro- cess control would be an interesting problem to pursue in future. The study reported here may be useful for a time-phased implementa- tion of JIT in manufacturing organizations.
6. Conclusions
A mathematical model has been developed for determining economic production quantit- ies in JIT manufacturing systems by minimiz- ing 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 dir- ect pattern search method has been employed for determining the economic production quantities. However, the model developed is based on a number of assumptions and ap- proximations. This implies that there are av- enues for further investigation to enhance the accuracy of modelling the JIT systems.
Acknowledgements
The authors are grateful to Professors G.J. Meester and John Miltenberg, and three
anonymous referees for their extremely useful and helpful comments on the earlier version of this manuscript. Also, the authors thank Pro- fessor Ilkka Virtanen, Rector, University of Vaasa, Finland, for his extended co-operation in their research projects.
References
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