Modelling just in time purchasing in the ready mixed concrete industry 3

158

Chapter 6 JIT purchasing threshold value models for the RMC industry

6.1 Introduction
The objective of this chapter is to develop JIT Purchasing Threshold Value (JPTV)
models. These models consider the inventory physical storage cost, together with the
additional costs and benefits resulting from JIT purchasing, which have not been
considered by the models of Fazel (1997), Fazel et al. (1998) and Schniederjans and Cao
(2000, 2001) or the models in the previous chapters. These JPTV models are developed
particularly for the Ready Mixed Concrete (RMC) industry and are applicable for
boundary condition 2 (see Section 1.6 for its definition).

Section 6.2 expands the annual cost of carrying one unit of inventory in the classical
EOQ model (
h
) to include all the components of inventory physical storage cost. The
features of the expanded EOQ models are also discussed. Section 6.3 and Section 6.4
develop the EOQ-JIT cost indifference points under the revised EOQ models. These
EOQ-JIT cost indifference point models are the JPTV models.

6.2 Revised EOQ model
Chapter 3 argued that there were reasons to include the physical storage cost into the
annual cost of carrying one unit of inventory in the classical EOQ model (
h
). However,
to empirically examine the capability of an inventory facility to carry the EOQ-JIT cost
indifference point’s amount of inventories on the platform created by Fazel (1997), Fazel
(1998) and Schniederjans and Cao (2000, 2001), the physical plant space was treated as a
penalty cost and thus was still excluded from
h

when deriving the ultimate EOQ-JIT cost
159

indifference point in the previous chapters. To accurately capture the impact of inventory
purchasing policies on the selection of the inventory purchasing method and to develop
the JPTV models, the physical plant space, which is a component of the physical storage
costs, needs be included into
h
. This is to expand “
h
” to “
H
” which is the “expanded
annual cost of carrying one unit of inventory”. The classical EOQ model is thus to be
revised. The features of the revised EOQ model are discussed below.

6.2.1 Total annual cost under the revised EOQ model
When “
h
” is expanded to become “
H
”, the total costs under the revised EOQ model are
given by:
DP
HQ
Q
kD
TC
EEr
++=

2
(6.1)
where:
Er
TC
is total costs under the revised EOQ model.
“
H
”, which includes all the components of the inventory carrying costs, is greater than
“ h ” in Eq. (3.1). Accordingly,
Er
TC
is also greater than
E
TC
in Eq. (3.1) in the classical
EOQ model.

6.2.2 Optimal economic order quantity under the revised EOQ model
The optimum order quantity of the revised EOQ model derived from Eq. (6.1) is:
H
kD
Q
r
2
=
∗
(6.2)
where:
∗

r
Q is the optimum order quantity of the revised EOQ model.
160

The optimum order quantity of the revised EOQ model is significantly less than that of
the classical EOQ model, as
H
is substantially greater than
h
, assuming the values of
the other parameters, namely,
D
and
k
remain unchanged.

6.2.3 Total annual optimal cost under the revised EOQ model
The total annual optimal cost under the revised EOQ model derived based on Eq. (6.1)
and Eq. (6.2) is:
DPkDHDPkDHkDHTC
EEEr
+=++= 22
2
1
2
2
1
(6.3)
Eq. (6.3) is valid only when the inventory is ordered at its economic order quantity. This
means that the annual inventory ordering cost item (

Q
kD
) equals the annual inventory
carrying cost item (
2
QH
) as shown in Eq. (6.3).

To sum up, the revised EOQ model is different from the classical EOQ model on three
counts. First, the so called “fixed costs”, such as rental, utilities, personnel salaries, etc,
are considered in the inventory carrying cost item in the revised EOQ model. Hence, the
annual cost of holding one unit of inventory in the revised EOQ model is greater than that
in the classical EOQ model. The total annual cost under the revised EOQ model is also
greater than that under the classical EOQ model. Second, the revised EOQ model prefers
small lot sizes and frequent deliveries. Last, but not the least, the revised EOQ model
aims to reduce the actual total inventory ordering and carrying cost, while the classical
EOQ model aims to reduce the sum of the inventory ordering cost and a part of the
inventory carrying cost. The last point makes it very clear that the revised EOQ model is
161

more suitable than the classical EOQ model in representing the total annual cost under
the EOQ system when comparing the EOQ system with the JIT purchasing system.

The JPTV models are developed for two scenarios. The first scenario does not consider a
price discount. The second scenario considers a price discount.

6.3 EOQ-JIT cost indifference point under the revised EOQ model
6.3.1 EOQ-JIT cost difference function under the revised EOQ model
Based on assumption No. 8 in Table 1.1, Fazel (1997) and Fazel et al. (1998) suggested
that the total annual cost under the JIT purchasing system was the product of the unit

price under the JIT purchasing system and the annual demand. As suggested earlier, this
proposition did not consider the additional costs and benefits resulting from JIT
purchasing. Hence, the total annual cost under the JIT purchasing system for the JPTV
models is proposed to be the product of the unit price under the JIT purchasing system
(
J
P
) and the annual demand (
D
) plus the additional costs and benefits resulting from JIT
purchasing, given by:
ξ
+= DPTC
JJr
(6.4)
where:
Jr
TC is the total annual cost under the JIT purchasing system for the JPTV
models, and

ξ
is the sum of the additional costs and benefits resulting from JIT purchasing.
The additional costs are mainly the increased out-of-stock costs when inventory
is purchased in a JIT fashion. The additional costs of JIT purchasing contribute
positively to
ξ
. The additional benefits are mainly the improved quality, the
162

flexibility of production, reduced waste and increased organizational

competitiveness under the JIT purchasing system. The additional benefits of JIT
purchasing contribute negatively to
ξ
.
It is essential to highlight that the cost of the inventory physical plant space reduction in
the JIT purchasing system has been assumed to take its maximum value and is considered
in the total annual optimal cost under the revised EOQ system. This is because (a) the
maximum value of the inventory physical plant space reduction under the JIT purchasing
system is the inventory physical plant space under the EOQ system; and (b) the inventory
physical plant space under the EOQ system has been considered as a component of
inventory carrying costs and included in “
H
” in Eq. (6.3). The inventory physical plant
space reduction was proposed by Schniederjans and Olsen (1999) and Schniederjans and
Cao (2000, 2001).

The total annual optimal cost under the revised EOQ model where the price discount is
not considered (
Er
TC ) has been presented in Eq. (6.3). The cost difference between the
revised EOQ system and the JIT purchasing system is thus given by:
ξ
−−+= DPDPkDHZ
JEr
2
(6.5)
where:
r
Z
is the cost difference between the revised EOQ system and the JIT purchasing

system.

6.3.2 EOQ-JIT cost indifference points under the revised EOQ model
r
Z is continuous and differentiable as
D
is above zero. Taking the first order derivative
of
r
Z
with respect to
D
in Eq. (6.5), would result in:
163

JE
r
PPD
kH
dD
dZ
−+=
−
2
1
2
2
(6.6)
Taking the second order derivative of
r

Z with respect to
D
in Eq. (6.5), would result in:
2/3
2
2
4
2
−
−
= D
kH
dD
Zd
r
(6.7)
Note that
4
2
kH
is always positive.
2/3−
D
is also always positive, as
D
is above zero.
Hence
2
2
dD

Z
d
r
, the second order derivative of
r
Z
with respect to
D
, is always negative.

According to the theorem of the second derivative test for maxima and minima of
functions, two counts for the cost difference function can be concluded. First, the curve
of the cost difference function
r
Z
is concave downwards. Second, the cost difference
between the EOQ and the JIT system is maximized at the demand level, at which
0=
dD
dZ
r
. This demand level is the maximum cost advantage point and its value is given
by:
( )
2
max
2
EJ
PP
kH

D
−
= (6.8)
where:
max
D is the maximum cost advantage point.
The maximum cost advantage of using a JIT purchasing system over an EOQ system can
be derived by substituting
max
D for
D
in Eq. (6.5). and its value is given by:
( )
ξ
−
−
=
EJ
r
PP
kH
Z
2
max
(6.9)
where:
r
Z
max
is maximum cost advantage of using a JIT purchasing system over an EOQ

system.
164


It should be noted that Eq. (6.9) is applicable for computing the maximum cost advantage
of using the JIT purchasing system over the EOQ system only if the order size in the
EOQ system equals the optimal economic order quantity. Should the order quantity in the
EOQ system not follow the optimal economic order quantity, the cost advantage of using
the JIT purchasing system over the EOQ system would be given by:

ξ
−−++= DPDP
HQ
Q
kD
Z
JEr
2
(6.10)

Setting
r
Z in Eq. (6.5) to zero, the roots of
(
)
0=DZ
r
are the revised EOQ-JIT cost
indifference points. Their values are given by:
(

)
( )
EJ
EJ
indr
PP
PPkHkH
D
−
−−−
=
2
422
1
ξ
(6.11)
where:
1indr
D is the lower EOQ-JIT cost indifference point under the revised EOQ
system.
The value of
1indr
D is given by:
(
)
(
)
( )
2
22

1
2
EJ
EJEJ
indr
PP
PPkHHkPPkH
D
−
−−−−−
=
ξξ
(6.12)
The value of
2indr
D
is given by:
(
)
( )
EJ
EJ
indr
PP
PPkHkH
D
−
−−+
=
2

422
2
ξ
(6.13)
where:
2indr
D
is the upper EOQ-JIT cost indifference point under the revised EOQ
system.
165

The value of
2indr
D is given by:
(
)
(
)
( )
2
22
2
2
EJ
EJEJ
indr
PP
PPkHHkPPkH
D
−

−−+−−
=
ξξ
(6.14)

6.3.3 Discussion
Eq. (6.12) and Eq. (6.14) indicate that the sum of the additional costs and benefits
resulting from JIT purchasing (
ξ
) is an important factor that affects the lower and upper
EOQ-JIT cost indifference points. It should be noted that
ξ
varies from industry to
industry and it may not be a constant even when the annual demand remains unchanged
for the same manufacturers. However, there can be three scenarios for
ξ
. 1)
ξ
is equal to
zero; 2)
ξ
is greater than zero; and 3)
ξ
is less than zero. Hence, the discussion on the
lower and upper EOQ-JIT cost indifference points is presented below for the three
scenarios.

6.3.3.1
ξ
is equal to zero

When the sum of the additional costs and benefits of JIT purchasing equals zero, Eq.
(6.12) suggests that the lower EOQ-JIT cost indifference point (
1indr
D ) equals zero, and
Eq. (6.13) suggests the upper EOQ-JIT cost indifference point (
2indr
D ) is greater than
zero. Meanwhile, the curve of the EOQ-JIT cost difference function is concave
downwards. Hence, it can be concluded that an EOQ system is preferred to the JIT
purchasing system, provided that the annual demand is greater than the upper EOQ-JIT
cost indifference point.

166

When the sum of the additional costs and benefits of JIT purchasing equals zero, the
upper EOQ-JIT cost indifference point (
2indr
D
) equals
( )
2
2
EJ
PP
kH
−
. Let
indr
D
represents

( )
2
2
EJ
PP
kH
−
. Fazel’s (1997) and Schniederjans and Cao’s (2001) studies were also focused
on the scenarios where
ξ
was equal to zero. It is thus essential to compare the present
study with that of Fazel’s (1997) and Schniederjans and Cao’s (2001) models.

A comparison of the present study with that of Fazel’s (1997) model
( )
2
2
EJ
PP
kH
−
is similar to the EOQ-JIT cost indifference point
indF
D
(see Eq. (3.7)), which
is given by
( )
2
2
EJ

PP
kh
−
. Since
H
is significantly greater than
h
, the revised EOQ-JIT
cost indifference point (
2
indr
D ) is greater than the
indF
D , provided that the values of the
other parameters, namely,
k
,
J
P and
E
P remain unchanged. This is because some of the
inventory physical storage costs were not accounted for in
indF
D . It should be noted that
the EOQ-JIT cost indifference point proposed by Fazel (1997) is even less than
indF
D .
This has been explained in the explanation notes for Eq. (3.7) in Section 3.3.1. Hence, the
revised EOQ-JIT cost indifference point (
2

indr
D ) is substantially greater than the EOQ-
JIT cost indifference point proposed by Fazel (1997). Again, this finding suggests that the
JIT system can still remain cost effective even at a high level of annual demand, thus
invalidating Fazel’s (1997) conclusion that JIT was cost effective only at low level of
annual demand (Schniederjans and Cao, 2001). This conclusion is in line with what was
reached in Chapter 3.
167


A comparison of the present study with that of Schniederjans and Cao’s (2001) model
The concept of the carrying capacity of an inventory facility, which has been developed
in Chapter 3, can assist to compare the present study with that of Schniederjans and Cao’s
(2001) model. Eq. (3.12) suggests that the carrying capacity of an inventory facility is
governed by
α
E
h
N
Q = . When selecting an inventory purchasing approach, it is possible
to design the size of the inventory facility proportionate to the optimal economic order
quantity amount of inventory, or
∗
=
rh
bQQ , where
b
is the stock flexibility parameter
and has been explained in Chapter 3 and Chapter 4. Substituting
∗

=
rh
bQQ
into
h
E
Q
N
=
α
,
would result in
∗
=
rE
bQN
α
. This is the formula of the floor area of an inventory facility
determined by the inventory optimal economic order quantity. Substituting
( )
2
2
EJ
indr
PP
kH
D
−
=
into Eq. (6.2), the optimal economic order quantity at the revised

EOQ-JIT cost indifference point (
∗
rind
Q ) can be derived as
EJ
rind
PP
k
Q
−
=
∗
2
. Substituting
∗
rind
Q for
∗
r
Q in
∗
=
rE
bQN
α
, would result in
Eind
N the minimum area of the inventory
facility that can accommodate the EOQ-JIT cost indifference point’s amount of inventory
as

EJ
Eind
PP
bk
N
−
=
α
2
. Hence, when a) the inventory space reaches
EJ
PP
bk
−
α
2
, and b) the
annual demand reaches
( )
2
2
EJ
PP
kH
−
, the total cost under the revised EOQ system will be
equal to the total annual cost under the JIT purchasing system. It should be noted that the
cost of the physical inventory plant space has been considered in the total cost under the
revised EOQ system. Meanwhile, the physical inventory plant space under the revised
168

EOQ system can accommodate the EOQ-JIT cost indifference point’s amount of
inventory. The cost of the inventory physical plant space under the revised EOQ system
can be balanced by the JIT purchasing system. For example, if
F
represents the annual
cost to own and maintain a square meter of physical inventory plant space,
bF
α
2
is then
a component of
H
. This again suggests that Schniederjans and Cao (2001) overlooked
that it was possible for an inventory facility to hold the EOQ-JIT cost indifference point’s
amount of inventory when the floor area of an inventory facility reached
Eind
N
: the
minimum area of the inventory facility to house the EOQ-JIT cost indifference point’s
amount of inventory. This conclusion is in line with what was reached in Chapter 3.
Another expression of this finding is that an EOQ based system can be more cost
effective than a JIT purchasing system when the floor area of the inventory facility is
above
Eind
N and the magnitude of the annual demand is above the EOQ-JIT cost
indifference point, which equals
( )
2
2

EJ
PP
kH
−
.

6.3.3.2
ξ
is greater than zero
Eqs. (6.11) to (6.14) can have three implications when the additional costs resulting from
JIT purchasing are greater than the additional benefits resulting from JIT purchasing.
First, Eqs. (6.11) and (6.12) suggest that the JIT purchasing approach may not be a cost
effective alternative for inventory purchasing, provided the annual demand of the
inventory is less than the lower EOQ-JIT cost indifference point. Second, Eqs (6.13) and
(6.14) suggest that the additional costs resulting from JIT purchasing shift the upper
EOQ-JIT cost indifference point (
2
indr
D ) to be less than
( )
2
2
EJ
PP
kH
−
. Third, the EOQ
169

approach can always be preferred to the JIT purchasing approach, provided that the

additional costs resulting from JIT purchasing are substantially high. The analysis of the
third implication is presented below.

Eq. (6.5) can be rewritten as:
( )
( )
( )






−
−
+



















−−
−
−=
ξ
EJ
EJ
EJ
r
PP
kH
DPP
PP
kH
Z
2
2
2
(6.15)
The first term on the right hand side of Eq. (6.15) is always negative. The second term on
the right hand side of Eq. (6.15) is the maximum cost advantage of using a JIT
purchasing system over an EOQ system, which is given in Eq. (6.9). When the additional
costs and benefits resulting from JIT purchasing are greater than
( )
EJ
PP
kH

−2
,
r
Z is
always negative; the EOQ approach is thus always preferable to the JIT purchasing
approach. Comparing Eqs. (6.9), (6.12) and (6.14), it can also be found that the condition
for the lower cost indifference point and the upper cost indifference point to be real is that
the maximum cost difference between the EOQ and the JIT system is above zero.

6.3.3.3
ξ
is less than zero
Eq. (6.11) suggests that
1
indr
D is not a feasible cost indifference point, when the sum of
the additional costs and benefits resulting from JIT purchasing (
ξ
) is less than zero. This
is because
1
indr
D is negative when
ξ
is less than zero. However,
2
indr
D , which is given
by Eq. (6.14), is still a feasible EOQ-JIT cost indifference point. Eqs. (6.13) and (6.14)
suggest that the additional benefits resulting from JIT purchasing shift

2
indr
D to be a
170

value that is greater than
( )
2
2
EJ
PP
kH
−
. This suggests that JIT purchasing probably can be
adopted in a much wider range than that is stipulated by
( )
2
2
EJ
PP
kH
−
, provided that the
additional benefits resulting from JIT purchasing, such as the improvement in quality and
the flexibility in production, exceed the costs of not having stocks.

6.4 EOQ-JIT cost indifference point under the revised EOQ with a price discount
model for the RMC industry
6.4.1 Existing EOQ with a price discount models
The cost incurred by suppliers is usually a decreasing function of the size of the delivery

lot; the delivery price of an inventory is thus usually a decreasing function of the order
quantity (Fazel et al., 1998). Goyal and Gupta (1989) concluded that there were three
basic types of price discount models, namely, Two-Part Tariff, Two-Block Tariff and All
Unit Quantity Discount. In the Two-Part Tariff price discount model, “the buyer is
required to pay a fixed charge and a uniform price
p
for all units purchased. Although
the buyer pays the same marginal unit price for all quantities, the average price paid is a
monotonically decreasing function of quantity purchased” (Goyal and Gupta, 1989,
p.263). In the Two-Block Tariff price discount model, “the price of a unit,
1
p , is
maintained up to a quantity
x
, the per unit price
2
p is charged for all units in excess of
quantity (
1
p
>
2
p
)” (Goyal and Gupta, 1989, p.263). In the
All Unit Quantity Discount
model, when “a buyer buys less than a quantity
x
, the price of all units is decreased”
(Goyal and Gupta, 1989, p.263). Britney et al. (1983a, b), Dolan (1987) and Wilcox et al.
(1987) suggested several variations within each of the three basic types of price quantity

171

discount models. The price discount scheme proposed by Fazel et al. (1998) was one
variation of the All Unit Quantity Discount model, as the buyer paid the same unit price
for every unit purchased (Fazel et al., 1998).

6.4.2 Critics on the price discount scheme of Fazel et al. (1998)
As stated in Chapter 4, the EOQ-JIT cost difference functions of Schniederjans and Cao
(2000) and the EOQ-JIT cost difference functions Fazel et al. (1998) were both based on
a price discount scheme proposed by Fazel et al. (1998). The price discount scheme had
two assumptions. a) For quantities below a certain level (
max
Q ) the delivery price was a
decreasing, continuous and linear function of the order quantity. b) Beyond
max
Q ,
however, the price stayed at its minimum (
min
E
P ), which was the lowest price the supplier
would charge, no matter how large the order quantity was. The discount functions are
mathematically defined as:







≤−=−

=
=
max
min
max
0
0


0
QQP
QQ
dQ
dP
orQP
QP
P
E
E
E
EE
E
E

ππ
(6.16)
where
0
E
P

, Q and
E
π
were explained in Chapter 4

It seems that the price discount scheme proposed by Fazel et al. (1998) may not fit well
into the reality in the RMC industry on at least two counts. Firstly, the initial condition
for the first-order differential equation
dQ
dP
E
=
E
π
− in the price discount model (
0
E
P ), may
not even be a feasible price, as there may be a minimum order size so that an
172

infinitesimally small order size is not possible in RMC industry. The lowest order
quantity (
min
Q ) which one can order must be one with an unit price of
1
E
P , if the
inventory is not divisible, for example, one truck of bulk Portland cement. Secondly, the

intensive site studies by the author and the in-depth discussion with the general manager
of RMC supplier L in Chongqing, China and the production manger of
S
I in Singapore
suggested that a greater price discount can usually be offered by the raw materials
suppliers, if the order quantity was to be increased. There is no such things as
min
E
P
.
Nevertheless, there was a maximum order quantity limit for an individual RMC supplier.
The maximum order quantity limit for each individual RMC suppliers was usually
governed by either the inventory carrying capacity of the inventory facility in the RMC
batching plant or the production capacity of the raw material supplier, whichever was less.
The general manager of RMC supplier L and the production manager of
S
I made their
suggestions based on their work experience in the RMC industry.

6.4.3 A price discount scheme for the RMC industry
Based on the above analysis, to fit the RMC industry, a new price discount scheme that
incorporate the reality in the RMC industry is suggested based on the price discount
scheme proposed by Fazel et al. (1998) and is given below. The delivery price per unit
starts from
min
Q
E
P , where
min
Q is the lowest order quantity that can be placed. The delivery

price is then a decreasing, continuous, and linear function of the order quantity for the
rest of the order quantity that is below a certain level (
max
Q ).
max
Q is determined by the
inventory carrying capacity of the RMC supplier and the production capacity of the raw
173

material supplier, whichever is less. This discount scheme is shown graphically in Figure
6.1.

Figure 6.1 The EOQ price discount scheme proposed for the RMC industry

The discount function can be mathematically presented as:
(
)
min
min
QQPP
E
Q
EE
−−=
π
for
maxmin
QQQ ≤≤ (6.17)
where:
min

Q is the lowest order quantity that can be placed,
max
Q is the maximum order quantity that can be placed,
min
Q
E
P is the purchase price when order quantity is
min
Q , and

E
π
is a constant.

6.4.4 Revised EOQ with a price discount model for the RMC industry
The revised EOQ with a price discount model for the RMC industry is developed by
incorporating the price discount scheme of the RMC industry in Eq. (6.17) into the
max
Q
Q
E
P
min
Q
min
Q
E
P
174

revised EOQ model in Eq. (6.1). The total annual cost under the revised EOQ with a
price discount model is the sum of the inventory ordering cost, the expanded inventory
carrying cost and the cost of the actual purchased units, and is thus given by:
(
)
DQQP
QH
Q
kD
TC
EE
Q
EErd
ππ
−+++=
min
min
2
for
maxmin
QQQ ≤≤ (6.18)
where:
Erd
TC
is the total annual cost under the revised EOQ with a price discount model.

The optimum order quantity which minimizes the total cost under the revised EOQ with a
price discount system is thus:
DH
kD

Q
E
rd
π
2
2
−
=
∗
for
maxmin
QQQ ≤≤ (6.19)
where:
∗
rd
Q is the optimum order quantity which minimizes the total cost under the
revised EOQ with a price discount system.
It should be highlighted that the optimum order quantity of the revised EOQ with a price
discount model (
∗
rd
Q
) in Eq. (6.19) is significantly less than the optimum order quantity
of the EOQ with a price discount model (
∗
r
Q ) in Eq. (4.3), which was proposed by Fazel
et al. (1998). This is because the annual cost of carrying one unit of inventory in the
revised EOQ with a price discount model is substantially greater than that in the models
of Fazel et al. (1998), provided the values of the other parameters, namely,

D
,
k
and
E
π

remain unchanged.




175

6.4.5 EOQ-JIT cost indifference point under the revised EOQ with a price discount
model for the RMC industry
Eq. (6.19) results in the total annual optimal cost under the revised EOQ with a price
discount system as:
D
DH
kD
QP
DH
kD
H
kD
DH
kDTC
E
EE

Q
E
E
E
Erd






−
−++
−
+
−
=
π
ππ
π
π
2
2
2
2
2
2
2
min
min

for
maxmin
QQQ ≤≤ (6.20)
Eq. (6.20) is valid only when the inventory is ordered at its optimal economic order
quantity. The total annual cost under the JIT system (
Jr
TC ) is still the same as given in
Eq. (6.4). The difference between the total annual optimal cost under the revised EOQ
with a price discount system in Eq. (6.20) and the total annual cost under the JIT
purchasing system in Eq. (6.4) is thus:
ξ
π
ππ
π
π
−−






−
−++
−
+
−
= DPD
DH

kD
QP
DH
kD
H
kD
DH
kDZ
J
E
EE
Q
E
E
E
rd
2
2
2
2
2
2
2
min
min

for
maxmin
QQQ ≤≤
(6.21)

where:
rd
Z is the difference between the total annual optimal cost under the revised EOQ
with a price discount system and the total annual cost under the JIT purchasing
system.

Setting
rd
Z to zero, the roots of
(
)
0=DZ
rd
are the revised EOQ-JIT cost indifference
points. Their values are given by:
(
)
(
)
( )
kQPP
kQPPkHHkQPPkH
D
E
E
Q
EJ
EE
Q
EJE

Q
EJ
indrd
ππ
ξππξπξ
4
42
2
min
2
min
22
min
1
min
minmin
+−−
−−−−−−−−
=
176

for
maxmin
QQQ ≤≤ (6.22)
where:
1
indrd
D is the lower EOQ-JIT cost indifference point under the revised EOQ with a
price discount system for the RMC industry.
(

)
(
)
( )
kQPP
kQPPkHHkQPPkH
D
E
E
Q
EJ
EE
Q
EJE
Q
EJ
indrd
ππ
ξππξπξ
4
42
2
min
2
min
22
min
2
min
minmin

+−−
−−−−+−−−
=

for
maxmin
QQQ ≤≤
(6.23)
where:
2indrd
D
is the upper EOQ-JIT cost indifference point under the revised EOQ with
a price discount system for the RMC industry.
The models in Eq. (6.12), Eq. (6.14), Eq. (6.22) and Eq. (6.23) are the JIT purchasing
threshold value (JPTV) models suggested in this study.

6.4.6 Discussion
Eq. (6.22) and Eq. (6.23) also indicate that the sum of the additional costs and benefits
resulting from JIT purchasing (
ξ
) is an important factor that affects the lower and upper
EOQ-JIT cost indifference points under the revised EOQ with a price discount system.
As suggested earlier,
ξ
is not a constant and may have three scenarios: 1)
ξ
is equal to
zero; 2)
ξ
is greater than zero; and 3)

ξ
is less than zero. The lower and upper EOQ-JIT
cost indifference points under the revised EOQ system are thus also discussed below for
the three scenarios.



177

6.4.6.1
ξ
is equal to zero
Eq. (6.22) suggests that
1indrd
D equals zero and Eq. (6.23) suggests
2indrd
D is positive,
provided that
ξ
equals zero. When
ξ
equals zero and when the price discount rate (
E
π
)
is small, it can be proved that the EOQ system is more cost effective than the JIT
purchasing system, provided that the annual demand is greater the upper EOQ-JIT cost
indifference point (
2indrd
D ).

It can be proved that the EOQ system is more cost effective than the JIT purchasing
system when the annual demand is above the upper EOQ-JIT cost indifference point
(
2indrd
D
) and if the price discount rate (
E
π
) is small. It can also be proved that the JIT
purchasing system is more cost effective than the EOQ system when the annual demand
is below
2indrd
D and if the price discount rate (
E
π
) is small.

When
ξ
equals zero,
2indrd
D can be simplified as
( )
kQPP
kH
E
E
Q
EJ

ππ
4
2
2
min
min
+−−
. If
min
Q is
also equal to zero,
2indrd
D can be further simplified as
(
)
kPP
kH
E
O
EJ
π
4
2
2
+−
, where
min
0
min
QPP

E
Q
EE
π
+= . The cases where both
ξ
and
min
Q are equal to zero was the scenario
that had been studied by Fazel et al. (1998) and Schniederjans and Cao (2000). Hence, it
is essential to compare this present study with the studies of Fazel et al. (1998) and
Schniederjans and Cao (2000).


178

A comparison of this present study with the study of Fazel et al. (1998)
(
)
kPP
kH
E
O
EJ
π
4
2
2
+−
is similar to EOQ-JIT cost indifference point

∗
inddF
D
(see Eq. (4.18)),
which is given by
∗
inddF
D =
(
)
kPP
kh
E
O
EJ
π
4
2
2
+−
. Since
H
is significantly greater than
h
, the
revised EOQ-JIT cost indifference point (
2indrd
D
) is thus greater than
∗

inddF
D , assuming
the values of the other parameters, namely,
k
,
J
P ,
0
E
P and
E
π
are the same. This is
because plant space reduction was not accounted for in
indF
D . It should be noted that the
EOQ-JIT cost indifference point proposed by Fazel et al. (1998) is even lower than
∗
inddF
D . This has been explained in the explanation notes for Eq. (4.18). Hence, the
revised EOQ-JIT cost indifference point (
2indrd
D ) is significantly greater than the EOQ-
JIT cost indifference point proposed by Fazel et al. (1998). Again, this finding suggests
that the JIT purchasing system can still be cost effective even at a high level of annual
demand, thus modifying the conclusion of Fazel et al. (1998) that JIT purchasing was
cost effective only at a low level of annual demand, when a price discount was available.
This conclusion is in line with what was reached in Chapter 4.

A comparison of this present study with the study of Schniederjans and Cao (2000)

Again, the concept of the carrying capacity of an inventory facility can assist to compare
the present study with the study of Schniederjans and Cao (2000). As stated earlier, when
selecting an inventory purchasing approach, it is possible to design the size of the
inventory facility proportionate to the optimal economic order quantity of the inventory,
or
∗
=
rdh
bQQ .
b
is the stock flexibility parameter.
∗
rd
Q is the optimal economic order
179

quantity under the revised EOQ with a price discount model which is given in Eq. (6.19),
h
Q is the carrying capacity of an inventory facility.
h
Q is given in Eq. (3.12) or
h
Q =
α
E
N
, where
E
N is the floor area of an inventory facility under an EOQ system.

Substituting
∗
=
rdh
bQQ into
h
E
Q
N
=
α
, would result in
∗
=
rdE
bQN
α
, namely, the formula
of the area of an inventory facility governed by its optimal economic order quantity under
the revised EOQ with a price discount system. Substituting the simplified formula of
2indrd
D , which is
(
)
kPP
kH
E
O
EJ
π

4
2
2
+−
, into Eq. (6.2),
∗
rdind
Q the optimal economic order
quantity at the revised EOQ-JIT cost indifference point (
2indrd
D ) can be derived as
0
2
EJ
rdind
PP
k
Q
−
=
∗
. Substituting
∗
rdind
Q for
∗
rd
Q in
∗
=

rdE
bQN
α
, would result in the
minimum area of the inventory facility under the revised EOQ with a price discount
system that can accommodate the EOQ-JIT cost indifference point’s amount of inventory
(
Edind
N ) as
0
2
EJ
Edind
PP
bk
N
−
=
α
. Hence, the total cost under the revised EOQ with a price
discount system will be equal to the total cost under the JIT purchasing system, provided
that (1) the inventory space reaches
0
2
EJ
PP
bk
−
α
, and (2) the annual demand reaches

(
)
kPP
kH
E
EJ
π
4
2
2
0
+−
, and (3) the order quantity is below
max
Q . Meanwhile, the physical
inventory plant space under the revised EOQ with a price discount system can
accommodate the EOQ-JIT cost indifference point’s amount of inventory. The cost of the
physical inventory plant space under the revised EOQ with a price discount system can
be balanced by the JIT purchasing system by including bF
α
2 as a component of
H
.
180

This again suggests that Schniederjans and Cao (2000) overlooked that it was possible for
an inventory facility to hold the EOQ-JIT cost indifference point’s amount of inventory
when the floor area of an inventory facility reached
Edind
N . Hence, another expression of

this finding is that an EOQ based system can be more cost effective than a JIT purchasing
system when a) the size of the inventory facility is above
Edind
N , and b) the magnitude of
the annual demand is above the EOQ-JIT cost indifference point
2indrd
D , and c) the
optimal economic order quantity is below
max
Q . As suggested earlier,
Edind
N is the
minimum area of the inventory facility that can accommodate the EOQ-JIT cost
indifference point’s amount of inventory under the revised EOQ with a price discount
system. This conclusion is in line with what was reached in Chapter 4.

6.4.6.2
ξ
is greater than zero
When

the additional costs of JIT purchasing are greater than the benefits of JIT
purchasing, Eqs. (6.22) to (6.23) have three implications. First, Eq. (6.22) suggests that
the JIT purchasing approach may not be cost effective when the annual demand is low.
Second, Eqs (6.23) suggests that the upper EOQ-JIT cost indifference point (
2indrd
D
)
shifts to a lower value that is less than
(

)
kPP
kH
E
EJ
π
4
2
2
0
+−
, where
min
0
min
QPP
E
Q
EE
π
+= .
Third, the EOQ approach can always be preferred to the JIT purchasing approach if the
additional costs of JIT purchasing are substantially high. The analysis of the third
implication is presented below.

181

Comparing Eq. (6.22) and Eq. (6.23), it can be found that the condition, under which the
lower cost indifference point (
1indrd

D ) and the upper cost indifference point (
2indrd
D ) are
real, is that the sum of the additional costs and benefits resulting from JIT purchasing (
ξ
)
is below a specific value. This value is given by:
( )
(
)
(
)
k
QPPkHHkQPPHk
E
E
Q
EJEE
Q
EJ
it
π
πππ
ξξ
4
4
min
23
2
min

22
lim
minmin
−−−+−−
=≤ (6.24)
When
ξ
the sum of the additional costs and benefits resulting from JIT purchasing are
greater than
(
)
itlim
ξ
,
1indrd
D and
2indrd
D are not feasible cost indifference points.
In such a scenario, an EOQ system is always preferred to a JIT system.

6.4.6.3
ξ
is less than zero
Setting
rd
Z
in Eq. (6.21) to zero, Eq. (6.21) can be rewritten as:
( )
DH
kD

DH
kD
DH
kDDQPP
E
EE
E
Q
EJ
π
ππ
ξπ
2
2
2
2
2
2
min
min
−
−
+
−
=+−−
for
maxmin
QQQ ≤≤ (6.25)
When
ξ

is less than zero and the price discount rate
E
π
is small, it can be proved that
the left hand side of Eq. (6.25) is less than zero and the right hand side of Eq. (6.25) is
greater than zero if the annual demand is equal to
1indrd
D , where
1indrd
D is given in Eq.
(6.22).
E
π
is usually very small; this has been discussed in Section 4.4.3. Hence,
1indrd
D
is not a feasible EOQ-JIT cost indifference point when
ξ
is less than zero and if the
price discount rate
E
π
is small. Nevertheless,
2indrd
D is still a feasible EOQ-JIT cost
indifference point. Furthermore, it can be proved that
2indrd
D is greater than
182

(
)
kPP
kH
E
EJ
π
4
2
2
0
+−
when
ξ
is less than zero and where
min
0
min
QPP
E
Q
EE
π
+= . This further
suggests that JIT purchasing can probably be adopted in a much wider range than that
stipulated by
(
)
kPP
kH

E
EJ
π
4
2
2
0
+−
, if the benefits reaped from JIT purchasing, such as
improvement in quality and flexibility in production, are greater than the increased out-
of-stock costs. This conclusion is in line with what was reached from the scenario where
a price discount is not available, which was discussed in the earlier section.

6.5 Summary
This chapter developed the JPTV models for the RMC industry. The JPTV models
suggest that the local market conditions determine whether the EOQ or JIT purchasing
method is preferred.

The JPTV models can have three implications. First, when the additional costs of JIT
purchasing cannot be ignored and the annual demand is low, JIT purchasing may be an
expensive alternative. Second, when the additional costs of JIT purchasing is
substantially high, the EOQ purchasing approach may always be more cost effective than
the JIT purchasing approach. Third, when a material can be alternatively purchased in a
JIT fashion with a higher price and an EOQ fashion with a lower price, the EOQ
approach can be more cost effective than the JIT approach, if the sum of the additional
costs and benefits resulting from JIT purchasing is insignificant, the order quantity under
the EOQ system cannot be economically split, and the annual demand is high enough.
The additional costs of JIT purchasing include mainly the increased out-of-stock costs.