Chapter 2 Development of Methods for Logistical System Performance Analysis
2.3 Inventory Control
2.3.6 Multi-product inventory situations
To evaluate the replenishment order size Qp, which maximises the gross profit per replenishment order, it is necessary to know the selling price per unit, Ca, which produces a profit per unit of Ca-Cm (Cm being the material and labour cost of one unit of inventory) and the value of Qp is given by (Lewis, 1981):
2
2 o o
m a
p Q
C C
Q =C − (2.10)
Substituting for Qo in equation (2.10):
C A C Q C
h m a
p = −
from which it is apparent that, to maximise profit per replenishment, order sizes should be chosen that are directly proportional to the sales value rather than their square root.
Maximising profit per replenishment does not maximise profit per unit time, that is really required, and that the economic order quantity is in fact more useful in this respect. In spite of this, when deciding which method to employ in evaluating replenishment order quantities, it is useful to know that to maximise profit per replenishment order, a larger investment must be made as Qp is in practice always greater than the economic order quantity. The criterion which any particular business might choose in selecting their replenishment order sizes will depend on the type of product range they manage, their management objectives and the market within which they operate.
2.3.6.1 Optimisation of economic order sizes when a limitation is set on the capital invested in stocks
When the decision is made to limit the amount of capital invested in stocked items, and the average capital invested in stocks of all items using the unrestricted replenishment order sizes (such as the economic order quantity) exceeds the capital restriction, the calculation of the replenishment order quantities must be modified. If a capital restriction is not in effect, then V, the average total value invested in stocks, is found by:
∑
== J
j
mj jC q g V
1
(2.11) where qj is the replenishment order quantity for the jth item
Cmj is the per unit material and labour cost of the jth item g is the so-called ‘normalising factor’
This normalising factor g is introduced to allow replenishment orders for individual items to arrive at different times rather than simultaneously. Should it so happen that all replenishment orders were received at the same time, it would produce a maximum investment situation and the value of g would therefore be one. If, however, it is assumed that the receipt of replenishment orders is spread over time in such a way that capital investment is half the maximum value, then g would be equal to one-half.
Generally, it can be assumed that g lies somewhere between a value of one-half and one, the exact figure depending on the particular situation (Scarf, 1993). Assuming that V the average total value invested in stocks exceeds Vmax (the average maximum value allowed), the problem is to know in what manner each individual item’s replenishment order size should be reduced to meet effectively the overall capital restriction. Thus we have the condition
∑
==
≥ J
j mj jC q g V V
1
max
or, alternatively,
∑
=≥
− J
j m jC j q g V
1
max 0 (2.12)
An intermediate Langrange multiplier z can then be defined by:
z<0 when
∑
=
=
− J
j m jC j q g V
1
max 0
z=0 when
∑
=
>
− J
j m jC j q g V
1
max 0
As
−
∑
= J
j
mj jC q g V z
1
max is always zero by the above definition, the total annual inventory operating costs (excluding stock-out costs and letting the holding cost for the jth item be represented by Cmji) can then be found by:
−
+
+
=
∑ ∑
= j
j j
m j J
j
m j j
j
o q C i z V g q C
q A
C C max
1 2 (2.13)
To minimise this total operating cost function, C is differentiated with respect to qj and the result set equal to zero thus:
2 0
2 + − =
−
= j j mj
m j
j o j
i zgC C q
A C dq
dC
Rewriting qj now as Qo'j, the modified economic order quantity we obtain:
) 2 (
' 2
gz i C
A Q C
j j
m j o oj
= − (2.14)
At this stage, it is generally recommended that different increasing values of z should be tried until the capital restriction is just met. This procedure can become very tedious, and it should be avoided. For this particular model of multi-product inventory system it is possible to calculate z directly because the material cost Cmj is a common factor to both the cost of holding inventory and the capital invested. Qo'j the modified economic order quantity can be rewritten in terms of the unrestricted economic order quantity as:
gz i Q i Qoj oj
2
'
= − (2.15)
Now, ideally we wish that
∑
= =− J
j j m o C j Q g V
1 '
max 0
and substituting for Qo'jusing equation (2.15) this then becomes
∑
=− =
− J
j
m ojC j gz Q
i g i V
1
max 0
2 (2.16)
Thus, if the maximum capital invested,
∑
= J
j
m ojC j Q g
1
, is calculated and the average maximum capital allowed, Vmax is known, z can be evaluated directly by using:
−
−
=
∑
= 1
2
2
max 1
V C Q g g z i
J
j
m oj j
(2.17)
In practice, however, to calculate the modified replenishment order quantities required to meet the capital restriction, it is not necessary to actually calculate z as these order sizes can be evaluated from the unrestricted economic order quantity Qojand the value of
gz i
i
−2 which from equation (2.16) is obtained by:
j
j m
J
j
o C
Q g
V gz
i i
∑
=− =
1 max
2 (2.18)
It is apparent from Figure 2.3 on page 49, that because the replenishment order sizes have been reduced to accommodate the capital investment restriction, the total inventory operating costs must increase as a result of the restriction being in force compared to the unrestricted case. This rise in operating costs results because the increased cost of ordering smaller replenishment orders more frequently is not offset by the reduced holding costs accrued from lower average inventory levels. It follows that the total inventory operating costs C for the unrestricted situation is obtained by:
∑
=
+
= J
j
m o o
j
o Q C i
Q A
C C j j
j j
1 2
and this expression can be further simplified by substituting from equation (2.9) on page 48 and then becomes:
∑
== J
j
m ojC j Q i C
1
(2.19) C’ the annual inventory operating cost for the restricted situation is given by:
∑
=
+
= J
j
m o o
j
o Q C i
Q A
C C j j
j j
1 2
' ' '
which by substitution from equation (2.15) on page 53 can be shown to be:
∑
=− −
= J
j
m ojC j gz Q
i gz i i C
2 1
) ( '
or more simply as:
( )
{
max}
' 1 i gzV
C = g − . (2.20)
2.3.6.2 Significance of the Lagrange multiplier z
Defining δC as the increased annual operating cost due to the imposition of the capital restriction, this can be evaluated as C’-C which when substituting from equation (2.19) and (2.20) can be defined as:
( ) ∑
=
−
− −
= J
j
m ojC j Q gz i
i gz i i C
2 1
δ
If for most practical situations it can be assumed that
gz i
i
−2 is approximately unity, then:
−
≅
∑
= J
j
m ojC j Q g z C
1
δ
Thus, z can be considered as the unit proportional increase in annual inventory operating costs resulting from the imposition of the capital restriction.
Level at which the increased inventory operating costs become prohibitive:
It is useful to know at what value of the Lagrange multiplier z the increase in inventory operating costs resulting from the capital restriction becomes prohibitive.
Earlier in this chapter it was indicated generally that below a value of Q45o (the minimum economic order quantity which subtends an angle of 45° with the horizontal) the gradient of the cost versus the order quantity curve increased at such a rate as to make the operation below this point uneconomic. Now, if it is required that no modified economic order quantity should fall below its Q45o value as a result of the capital restriction, it follows that:
j
j Q
Qo' ≥ 45o Thus
2 2 ) 2 ( 2
≥ +
− m
j o m
j o
iC A C gz
i C
A
C j
j j
which can be simplified down to:
mj
z gC1
≤
−
For the first individual item whose modified economic order quantity (which as a result of increasing z in an attempt to meet the capital restriction) falls below its particular value of Q45o, it follows that:
max
1 Cm
z≤
− (2.21)
where
mmx
C is the highest material and labour cost for any individual item in the group.
Substituting for this value of z in equation (2.16), Vmin, the minimum capital investment level below which it would not be advisable to operate because of the increased annual inventory operating costs, is given by:
∑
=≥ + J
j
m o m
m
j
jC
g Q iC
V iC
1
min 2
max
max (2.22)
Thus, if the solution of a multi-product capital restriction problem is solved for a value of z greater than
max
1
Cm , or if the capital restriction is unrealistic then it causes an excessive increase in inventory operating costs.
2.3.6.3 Optimisation of economic order sizes when a limitation is set on total storage space
Where the maximum allowable total storage space is limited to a value Smax, the average value of the space occupied by stocks would be given by S as:
∑
== J
j j jW q g S
1
(2.23)
where Wj is the storage space required per unit for the jth item
g is again a normalising factor with a similar range of values to that considered earlier for the capital restriction case.
Following through the same stages as for the capital restriction model we have:
∑
=≥
− J
j j jW q g S
1
max 0 (2.24)
and the same procedure would eventually produce the value of the modified economic order quantity ''
oj
Q under this storage space restriction as:
j m
j o
o iC gW
A Q C
j j
j 2θ
'' 2
= − (2.25)
where θ is again an intermediate Langrange multiplier.
For this particular multi-product model where a limitation is placed on storage space, no simplification is possible and successive values of the modified economic order quantities must, therefore, be evaluated for different increasing values of θ until the total space limitation expressed by the restriction equation (2.23) is met.
The value for the Lagrange multiplier can be considered as the imputed value per cubic meter of storage space. Thus, if the value of θ found to solve this space restriction model is greater than the known rental value of storage space, additional storage space should be rented to reduce the overall inventory operating costs.