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An optimal batch size for a
JIT manufacturing System
指導老師：楊金山
博士
學生：黃惠卿
中華民國九十三年五月五日
Outline
• Abstract
• Introduction
• Model formulation
– Assumptions
– Case I : JIT delivery system
– Case II: JIT Supply and delivery system
• Solution methodology
– Problem I
– Problem II
• Results and discussions
• Conclusions
Abstract
• Production batch sizes for a JIT
delivery system.
• Incorporate a JIT raw material supply
system.
• Compute the batch sizes for both
manufacturing and raw material
purchasing policies.
Introduction
• The manufacturing lot size is dependent
on the retailer’s sales volume, unit product
cost, set-up cost, inventory holding cost
and transportation cost.
• The raw material purchasing lot size is
dependent on raw material requirement in
the manufacturing system, unit raw
material cost, ordering cost and inventory
holding cost.
Introduction (cont.)
Introduction (cont.)
• Case I：
– The ordering quantity of raw material is
assumed to be equal to the raw material
required for one batch of the production
system.
– Can be fitted in the JIT supply system.
• Case II：
– Ordering quantity of a raw material to be n
times the quantity required of one lot of a
product, where n is an integer.
– Is not favourable for the JIT environment.
Model formulation
• 2.1 Assumptions
– To simplify the analysis, we make the
following assumptions:
• There is only one manufacturer and only
one raw material supplier for each item.
• The production rate is uniform and finite.
• There are no shortages.
• The delivery of the product is in a fixed
quantity at a regular interval.
• The raw material supply is available in fixed
quantity whenever required.
• The producer is responsible to transport the
product to the retailers’ location.
Model formulation (cont.)
Model formulation (cont.)
• 2.2 Case I：JIT delivery system
– The total cost function, for case I, can be
expressed as follows：
D p  Qr
Dr
TC1 
Ar 

Qr
P  2
where
Dp

Ap  IPavg H p
 Hr 
Qp

1
Dr
Ar =ordering cost of raw materials
Qr
Dp  Qr

P  2
Dp
Qp

 Hr

=inventory holding cost of raw materials
Ap =set-up cost of finished products
IPavg H p =inventory holding cost of finished products
Model formulation (cont.)
f r  Dr / D p  Qr / Q p
Dp  f r Qr 
Dp
TC1  Ar  
 H r  Ap  IPavg H p
Qp
P 2 
Qp
Dp
IPavg
 Dp   m  1 
 Qp 1 

x
 2P   2 
 3
 2
Model formulation (cont.)
• Using the relationship in Eq.(3),the
total cost in Eq.(1) may be written
as
Dp
Dp  f r Qp 
 Dp 
 m 1 
TC1   Ar  Ap   
 H r  QP 1   H p  
 xH P  4
Qp
P 2 
 2 
 2P 
Dp
xH p
 Dp
 xH p
TC1   Ar  Ap   Qp  f r H r  H p  H p  
m
Qp
2P
2
 2P
 2
Dp
 5
Model formulation (cont.)
• 2.3 Case II：JIT Supply and delivery system
– The total cost function, for case II, can
be expressed as follows：
Dp  fr Qr 
 Dp 
Dr
 m 1 
TC2  Ap  Qp 1   H p  Ar  
 xH p
 Hr  
Qp
Qr
P 2 
 2 
 2P 
Dp
 6
where
Dp
Qp
Ap =set-up cost of finished products
 Dp 
 m 1 
Qp  1 
Hp 
 xH p
2
P
2




=inventory holding cost of finished products
Dr
Ar =ordering cost of raw materials
Qr
Dp  f r Qr 

 H r =inventory
P  2 
holding cost of raw materials
Model formulation (cont.)
Qp  mx
Qp  nQr
Qr  PL
TC2 
xH p
f r Dp
Dp  f r PL 
 Dp 
xH P
Ap  Qp 1 
H

m

A

H

 p
r

 r
Qp
2
P
2
PL
P
2
2




Dp
7
Solution methodology
• 3.1 Problem I
Dp
Hp
 Dp
TC1 
Ar  Ap   Qp 
fr H r 
Hp 

Qp
2P
2
 2P
Dp
Q 
*
p
D p  Ap  Ar 
9
KK
where
KK 
Dp
2P
fr H r 
Dp
2P
Hp 
Hp
2
 xH p

2

8
Solution methodology (cont.)
• Algorithm I: finding batch size
– Step0 Initialize and store DP, P, AP, Ar, Hp and Hr .
– Step1 Compute the number of batch size Q*p
using Eq.(9)
 Q

.
m is an integer, then
– Step2 Compute m   x If
stop.
– Step3 Compute TC1 using Eq.(5)
for m  m and m .
*
– Choose the m  m that gives minimum TC1
– Stop
*
p
Solution methodology
• 3.2 Problem II
– Substituting the integer variable, we can
rewrite Eq.(7) as follow:
TC2 
Hp
Dp  f r PL 
 Dp 
Dr
xH P
Ap  Qp 1 
H

Q

A

H

 p
p
r

 r
Qp
2
PL
P  2 
2
 2P 
Dp
Q*p 
D p Ap
H p  Dp 
1 

2 
P 
11
10
Solution methodology
• Algorithm II: finding batch size
– Step0 Initialize and store DP, P, AP, Ar, Hp and Hr
*
– Step1 Compute the number of batch size Qp
using Eq.(11)
 Q

– Step2 Compute m   x  . If m is an integer, then
stop.
– Step3 Compute TC2 using Eq.(7) for m  m and m .
and Choose the m*  m that gives minimum TC1
– Stop
*
p
Results and discussions
Results and discussions
Dp  2400 units / year
P  3600 units / year
Ap  $300 / set  up
Ar  $200 / order
H p  $2 / unit / year
H r  $1/ unit / year
fr  1
x  100
Conclusions
• A supplier to the JIT buyer is expected to
synchronize his production capacity with the
buyer’s demand so that the inventory in the
supply pipeline is reduced and eventually
eliminated.
• Total cost function is convex for a given m .
• The quality of solution is discussed and sensitivity
analysis is provided.
• In problem II, it is assumed that
Qp  nQr where Qr  PLTo generalize the problem ,
the relation Qr  PL needs to be relaxed.