INVENTORY CONTROL
Inventories are carried mainly to meet forecasted demand, to protect against stockouts and to decouple (provide some independence between) the successive components of
the manufacturing-distribution systems. The main inventory types are raw material
inventories, work-in-process (WIP) inventories and finished goods inventories. This chapter
is on inventory planning for independent demand items. Independent demand items are
finished products or service parts sold to customers, whereas dependent demand items are
components that are used in the production of a finished product or other parent items. This
chapter focuses on inventory planning for independent demand items.
It is important for the firms to know how much inventory they should carry. If they
carry too much inventory, it will be costly. There is holding or carrying cost for inventories
due to the cost of capital(dividends or interest on funds invested in inventory or the
opportunity cost if the company does not borrow money), cost of storage(warehouse related
costs), obsolescence cost(when the items in inventory become outdated they lose value),
insurance and taxes on inventory, and so on. On the other hand, if the firms carry too little,
they may encounter stock-outs and lost sales.
There is also an order cost associated with inventories. Preparing the paperwork for
purchase orders, communication costs(telephone, fax, etc.), handling the delivery of the
shipment, unpacking the ordered items after the shipment arrives, inspecting, and moving
items into storage, etc. are the cost elements involved in placing and processing an order.
Therefore, it is important to know how much to order (economic order quantity) by
taking into account both the inventory holding costs and ordering costs. In order to
determine the economic order quantity, we must first determine the total cost of carrying
inventories and ordering inventories.
Total Cost Equation
When the order quantity (Q) is received, the maximum inventory level is equal to Q.
When it is depleted, the inventory level is 0 (zero). Therefore, the average inventory is (Q +
0)/2 = Q/2. If Q/2 units are carried on the average, and if the annual carrying cost per unit is
H, then the annual inventory holding cost is:
(Q/2)H
On the other hand, the number of orders annually is D/Q. For example, if the annual
demand (D) is 1,000 and if we order a quantity (Q) of 100 units each time, then we would
have to order 10 times to meet the demand of 1,000, i.e. 1,000(D)
100(Q)
If the number of orders is D/Q and cost per order is Co, then the annual ordering cost is:
(D/Q)Co
Hence, the total of inventory and ordering costs would be:
TC = (Q/2)H + (D/Q)Co
As we order larger quantities the inventory costs will increase but the ordering costs
will decrease. If we order smaller quantities the inventory costs will decrease but this time
we will have to order more frequently and the ordering costs will increase as illustrated in
the following situation. Suppose that Annual Demand = 1,200, Inventory Holding Cost =
$3 per unit per year, Order Cost = $6 per order. Let's analyze the case of Order Quantity =
100 vs. Order Quantity = 10. The following table depicts this situation.
Number of
Orders During
A Year
Annual
Inventory
Holding Cost
Annual
Order
Cost
Order
Quantity
Average
Inventory
100
50
12
150
72
10
5
120
15
720
(decreasing)
(increasing)
As can be seen , as we try to decrease one cost, the other increases. Therefore, it is
impossible to minimize both costs separately and at the same time; but we can minimize the
total of both costs by determining the economic order quantity(EOQ) that would generate
the minimum total cost. EOQ finds a balance between inventory carrying and ordering
costs.
Economic Order Quantity (EOQ) Model
The EOQ model determines the order size which minimizes the total cost (total of
annual ordering and holding inventory) on the assumption of constant demand. It occurs at
the point where inventory holding costs are equal to ordering cost (this point is also the
minimum total cost point). It is derived by differentiating the total cost equation with
respect to quantity with the following result:
EOQ = 2 DCo /H
where
D = annual demand
Co = order cost (per order)
H = annual holding cost per unit
EOQ is depicted graphically below:
Total Cost Curve
Cost
Inventory Holding Cost Curve
Ordering Cost Curve
EOQ
Quantity
Example:
JAV Plastics buys raw plastic resin in 1,000-pound cartons. The company uses
approximately 3,125 cartons per year. The cost of placing an order is $20 per order.
Storage and handling costs are $50 per carton. Assume 250 working days a year.
a.
b.
c.
d.
e.
What is the most economical order size?
Determine the average number of cartons on hand.
How many orders will be placed per year?
What is the ordering interval?
What is the total cost of ordering and holding the raw
plastic?
Solution:
D = 3,125 cartons, Co = $20, H = $50
a . EOQ = 2 DCo /H = 2(3,125) (20) / 50 = 50 cartons
b. Q/2 = 50/2 = 25 cartons
c. D/Q = 3,125/50 = 62.5 (i.e., order 62 times one year and 63
assuming all the parameters remain
constant)
times the other year
d. Q/D = 50/3,125 = 0.016 year = 0.016x250 days = 4 days
e. TC = (Q/2)H + (D/Q)C0 = (50/2)(50) + (3,125/50)20
= 1,250 + 1,250 = $2,500
Economic Production Quantity(EPQ)
If an item is purchased, the critical question is: "How much should we order each
time?" and the answer is given by Economic Order Quantity(EOQ). If an item is
manufactured the question becomes "How much should we produce each time?" and the
answer is given Economic Production Quantity(EPQ). The equation for EPQ is:
EPQ = 2 DCs / H(1 - D/P)
where
D
Cs
H
P
= annual demand
= machine set up cost for production
= holding cost
= annual production
Here, Co is substituted by Cs because we now don't purchase but we produce.
Hence, there is no order cost but there is a setup cost because we must set up the machines
after each completion of the lot or batch size represented by EPQ. In case of purchasing we
dealt with the number of orders whereas in case of manufacturing we deal with the number
of setups.
Also, EPQ involves the component 1-D/P, whereas EOQ does not. This is due to
the fact that manufactured inventory accumulates gradually, whereas purchased (ordered)
inventory is received all in bulk. An example showing the computation of EPQ follows.
Example:
ABC Petroleum produces 500 barrels of oil per day. Demand for oil is 200 barrels
per day. Preparation costs are $150, and storage and handling costs are $5 per barrel. ABC
operates 250 days per year.
a.
b.
c.
d.
Compute the EPQ.
What is the maximum inventory?
What is the approximate length of a production run in days?
Determine the approximate number of workdays in a production
cycle.
Solution:
d = 200 barrels, D = 50,000 barrels, Cs = $150, H = $5
p = 500, P = 125,000
a . EPQ = 2 DCs / H(1 - d/p) =
2(50,000)150 / 5(1 - 200 / 500) 
2,236 barrels
b. Imax = Q(1-D/P) = Q(1-d/p) = 2,236(1-200/500)=1,341.6 barrels
c. Run Time = Q/p = 2,236 / 500 = 4.47 days
d. Time Between Production Runs = Q/d = 2,236 / 200 = 11.18 days
Reorder Point (ROP) Model
It is not enough to know how much to order; it is also important to know at what
point to order. The ROP model determines when to order. When the quantity on hand
drops to a predetermined amount, it is time to reorder. This amount includes expected
demand during lead time and usually some safety stock to reduce the probability of a stockout.
Suppose the daily demand (d) is 20 units and the lead time (LT) is 3 days. This
basically means if we order now, our order will arrive after 3 days. Hence, we must order 3
days in advance. Since the daily demand is 20, the demand or consumption during these 3
days will be:
20 x 3 = 60
(d) (LT)= dLT
Hence, we should place our order 3 days in advance or as soon as our inventory level drops
to 60. This is the reorder point. Therefore,
ROP = 20 x 3 = 60
ROP = dLT
We must have 60 units in stock to meet the demand during the lead time of 3 days.
Hence, we can call 60(dLT) the lead-time demand. We must at least have enough inventory
to cover the lead-time demand. Since the lead-time demand may vary, it is advisable to
carry some extra inventories (safety stock) on top of the lead-time demand.
The amount of safety stock depends on  which is the standard deviation of the leadtime demand. The safety stock is a multiple of  or z where z is the multiple determined
from the standard normal table. For example, if the objective is to meet the demand during
90% of the lead-time cycles, the z value would be approximately 1.28 and the safety stock
would be 1.28; 90% here is called the service level. Hence, ROP is established above the
average lead-time demand (dLT) by incorporating safety stocks. Actually, the complete
ROP equation becomes
ROP = dLT + z
Example:
The demand during the lead time of six weeks is 150. The standard deviation of the
demand during the lead time is 5, and the desired service level is 98.9%. What is the reorder
point?
Solution:
dLT = 150,  = 5, z = 2.29
ROP = dLT + z
= 150 + 2.29(5)
= 161.45 units
Note: Here, the lead-time demand is variable as there is a standard deviation associated
with it. This variability may either be due to the variability of the demand per period
(units/day, units/week, etc.) or the variability of the lead time (days, weeks, etc.) or both. In
this example we do not know whether the demand or the lead time is variable or both are
variable, therefore, we use the general case formula of ROP = dLT + z, where dLT =
average demand during the lead time and  = standard deviation of the demand during the
lead time.
Example:
A fast food restaurant uses an average rate of 30 ten-pound bags of frozen french
fries per day. Daily demand is normal and has a standard deviation of 4 bags per day. Lead
time is 6 days, and the restaurant wants a service level of 99.5%. What is the reorder point?
Solution:
d = 30 bags per day, d = 4 bags per day, LT = 6 days, z = 2.575
ROP = dLT + z LT  d (variable demand, constant lead time)
= 30(6) + (2.575) 6 (4)
= 205.23bags
Note: In this example the lead time is constant, but the demand per period(daily demand) is
variable causing the lead-time demand to be variable. We can also use the general case
formula,
ROP = dLT + z by computing  = LT  d and substituting into the
general ROP formula.
Example:
Everything is the same as in the previous example, except now daily demand is
constant and the lead time is variable with a standard deviation of 2 days. Determine the
ROP.
Solution:
d = 30 bags/day , LT = 6 days, LT = 2 days, z = 2.575
ROP = dLT + zdLT (constant demand, variable lead time case)
= 30(6) + 2.575 (30)(2)
= 334.5 bags
Note: In this example, the demand per period(daily demand), d, is constant, but the demand
during the lead time,  is variable due to the variability of the lead time. We can also use
the general case formula, ROP = dLT + z by computing  = dLT and substituting it into
the general case ROP formula.