Chapter 5 Allocation Performance Analysis
5.3. Verification of the Ideal Traffic Distribution for Dynamic Allocation
There are two methods of Model derivation Approaches Graphical method and Calculus approach.
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3.2.1 Graphical method
The purpose of inventory graph is to present the inventory control problems in graphical terms. It plots the relationship between quantity of stock held (Q) and time (t).
Figure 1 presents a general inventory graph with various features. It shows an initial inventory of 100 items, replenished by a further 100 items continuously over a given time period. Observe as indicated that for the next time period, there was no activity, but at time period 2, 100 items were demanded, followed, over the next two periods, by a continuous demand which used up the last 100 items. This stock out position led to the delivery of additional 150 items.
Figure 1: General Inventory Graph
Total Cost
Min. TC Holding cost
Ordering cost
Purchasing cost
0 EOQ Q
Figure 2: The General Inventory Graph
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Observations about graph
TC function shape is not influenced by the purchase cost i.e it is irrelevant for EOQ determination where there are no discounts.
Total cost is minimum where holding cost = ordering cost.
3.3.2 Calculus approach
TC = Purchase Cost + Holding Cost + Ordering Cost (recall there are no shortage costs) In symbolic form:
TC = DCp + Co
Q D QCh
2
The objective now is to find Q which minimizes TC. We apply fist order and second order conditions as follows:
FOC: 0
2
2
C DC Q dQ
dTC
o h
/ 2
2 DCo Q
Ch 0
TC = Purchase Cost + Holding Cost + Ordering Cost (recall there are no shortage costs) In symbolic form:
TC = DCp + Co Q D QCh
2
The objective now is to find Q which minimizes TC. We apply fist order and second order conditions as follows:
FOC: 0
2
2
C DC Q dQ
dTC
o h
/ 2
2 DCo Q
Ch 0
Ch/2 = DCo/Q2 Re-arranging:
Q2 = 2DCo/Ch Hence
Ch
Q 2DCo
To confirm the turning point is a minimum, we apply SOC as follows;
SOC: d2TC/dQ2 = 2DCoQ-3 = 2DC0/Q3> 0 i.e +ve,
since D, Q, Co are all positive values. Hence turning point is minimum.
Note: 2
pi
EOQ DCo
C Or 2
h
EOQ DCo
C
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Example
Star Logistics Ltd has established that annual quantity for a given item is 4000 units. The cost of placing an order is ₦5000 and the price per unit is ₦ 2000. Inventory holding cost percentage is 20% of purchase cost.
Required:
a) Formulate the best (optimal) entry policy for this item i.e.
- Quantity to order (EOQ)
- Frequency for ordering and when to order
- Re-order level/point; For ROP take lead-time to be 15 days while one year has 300 working days
- Total cost associated with the policy.
b) Suppose it actually turns out that c) Ordering cost per order = ₦6000 and
d) Inventory hold cost percentage I = 15% and yet the policy formulated in (a) above is implemented for a year determine the cost of prediction error.
a) From the illustration we can see that; annual demand, D = 4000 units; cost of ordering, Co = ₦5000; carrying cost percentage, i =20% of unit cost; unit purchase cost, Cp = ₦200. Then;
2
pi
EOQ DCo
C =
2 . 0 200
5000 4000
2
= 1000 units
Frequency of ordering
This is related to the annual number of orders, N which is given as;
N = Q
D =
1000
4000 = 4 orders
Given that one year is 12 months, therefore make an order after every, 12/4 = 3 months or quarterly.
Alternatively
If one year is 300 working days make an order every 300/4 = 75 days or 2.5 months Re-order level /re-order point (ROP)
ROP- represents the quantity remaining when an order is being made
= Usage during lead time period
= daily usage x lead time
= annual usage/No of days in the year * Lead time
=
300
4000 x 15 = 200 units
TC = purchase cost + Holding cost + Ordering cost
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= Co
DQ Q Cpi
DCp 2
= 4000 x 200 +
2000.2 2
1000 x 5000
1000 +4000
= 800,000 + 20,000 + 20,000 = ₦840,000 Optimal policy
Given that some parameter estimates changed from predicted ones, there is need to calculate EOQ afresh using all correct parameters.
Cpi
EOQ DCo =
15 . 0 200
6000 4000
2
= 1265 units
Relevant total cost for Optimal policy = 1265/2 * 200* 0.15 + 4000/1265 * 6000 =
₦37947
Cost of prediction error = 39,000 – 37,947 = ₦ 1053 3.3 The Inventory Control Systems
3.3.1 Re-order Level System
This is the most commonly used control system. It generally results in lower stocks.
The system also enables items to be ordered in more economic quantities and is more responsive to fluctuations in demand than the second system discussed below.
The system sets the value of three important levels of stock as warning t or action triggers for management:
Re-order Level: This is an action level of stock which leads to the replenishment order, normally the Economic Order Quantity (EOQ). For a particular time period, the re-order level is computed as follows:
(i) Lro = maximum usage per period x maximum lead time (in periods)
(ii) Minimum Level: This is a warning level set such that only in extreme cases (above average demand or late replenishment) should it be breached. It is computed as follows:
Lmin = Re-Order Level – (normal Usage x Average lead time)
(iii) Maximum Level: This is another warning level set such that only in extreme cases (low levels of demand) should it be breached. It is computed by:
Lmax = Re-order Level + EOQ – (Minimum usage x Minimum lead time)
Example:
Suppose for a particular inventory, there exists:
(a) the weekly minimum, normal and maximum usage of 600, 1000, and 1400 respectively;
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(b) the lead time which vary between 4 and 8 weeks (average = 6 weeks); and, (c) the normal ordering quantity (EOQ) of 20,000.
It follows that:
The Re-order Level (Lro) = 1400 x 8 = 11,200 units
Minimum Stock Level (Lmin) = 11,200 – 1000 x 6 = 5,200 units Maximum Stock Level (Lmax) = 11,200 + 20,000 – 600 x 4 = 28,800
3.3.2 The Periodic Review System
i. It enables stock positions to be reviewed periodically so that the chances of obsolete stock items are minimised.
ii. Economies of scale are possible when many items are ordered at the same time or in the same sequence.