Inventory Policy Decisions
Inventory Policy Decisions
Inventory Policy Decisions
Auto repair shops are faced with maintaining thousands of parts to repair a
variety of automobiles from different model years. An automobile can contain
15,000 parts. To provide the fastest turnaround, repair shops carry a limited
inventory of the more popular parts such as spark plugs, fan belts, and batteries.
A second tier of inventories is maintained by the auto manufacturer in regional
warehouses from which parts can be transported by airfreight, so that the repair
shops can, in some cases, receive them the same day. A high level of parts
availability can be achieved with a minimum of on-site inventory.
Reduce Costs
Although holding inventories has a cost associated with it, it can indirectly reduce
operating costs in other activities and may more than offset the carrying cost.
First, holding inventories may encourage economies of production by allowing
larger, longer and more level production runs. Production output can be
decoupled from the variation in demand requirements when inventories exist to
act as buffers between the two.
∗∗
The material in this handout is extracted from Chapter IV, “ Business Logistics Management” by
Ronald H..Ballou, Prentice-Hall International Inc. Fourth Edition,
Second, holding inventories fosters economies in purchasing and transportation.
A purchasing department may buy in quantities beyond the firm's immediate
needs in order to realize price-quantity discounts. The cost of holding the excess
quantities until they are needed is balanced with the price reduction that can be
achieved. In a similar manner, transportation costs can often be reduced by
shipping in larger quantities that require less handling per unit. However,
increasing the shipment size results in increased inventory levels that need to be
maintained at both ends of the transportation channel. The reduction in
transportation costs justifies the carrying of an inventory.
Fourth, variability in the time that it takes to produce and transport goods
throughout the operating channel can cause uncertainties that impact on
operating costs as well as customer service levels. Inventories are frequently
used at many points in the channel to buffer the effects of this variability and,
thereby, help to smooth operations.
Fifth, unplanned and unanticipated shocks can befall the logistics system. Labour
strikes, natural disasters, surges in demand, and delays in supplies are the types
of contingencies against which inventories can afford some protection. Having
some inventory at key points throughout the logistics channel allows the system
to operate for a period of time while the effect of the shock can be diminished.
It has been claimed that management's job is much easier having the security of
inventories. Criticism for being overstocked is much more defensible than being
short of supplies. The major portion of inventory carrying costs is of an
opportunity cost nature and, therefore, goes unidentified in normal accounting
reports. To the extent that inventory levels have been too high for the
reasonable support of operations, the criticism is perhaps deserved.
Critics have challenged the holding of inventories along several lines. First,
inventories are considered wasteful. They absorb capital that might otherwise be
put to better use, such as to improve productivity or competitiveness. Also, they
do not contribute any direct value to the products of the firm, although they do
store value.
Second, they can mask quality problems. When quality problems surface, the
tendency is to work off existing inventories to protect the capital investment.
Correcting quality problems can be slow.
TYPES OF INVENTORIES
Second, some stocks may be held for speculation, but they are still part of the
total inventory base that must be managed. Raw materials such as copper, gold,
and silver are purchased as much for price speculation as they are to meet
operating requirements. Where price speculation takes place for time periods
beyond the foreseeable needs of operations, such resulting inventories are
probably more the concern of financial management than logistics management.
However, when inventories are built up in anticipation of seasonal selling or
occur due to forward buying activities, these inventories are likely to be the
responsibility of logistics.
Third, stocks may be regular or cyclical in nature. These are the inventories
necessary to meet the average demand during the time between successive
replenishments. The amount of cycle stock is highly dependent on production lot
sizes, economical shipment quantities, storage space limitations, replenishment
lead times, price-quantity discount schedules, and inventory carrying costs.
Nature of Demand
On the other hand, some products are highly seasonal or have a one-time, or
spike, demand pattern. Inventories that are held to meet such a demand pattern
usually cannot be sold off without deep price discounting. A single order for
inventories must be placed with little or no opportunity to reorder or return
goods if demand has been inaccurately projected. Fashion clothing, Christmas
trees, and political campaign buttons, are examples of this type of demand
pattern. ,
Similarly, demand may display a lumpy, or erratic, pattern. The demand may be
perpetual, but there are periods of little or no demand followed by periods of
high demand. The timing of demand is not as predictable as for seasonal
demand, which usually occurs at the same time every year. Items in inventory
are typically a mixture of lumpy and perpetual demand items. A reasonable test
to separate these is to recognize that lumpy items have a high variance around
their mean demand level. If the standard deviation of the distribution of demand,
or the forecast error, is greater than the average demand, or forecast, the item
is probably lumpy. Inventory control of such items is best handled by intuitive
procedures or by mathematical procedures.
There are products whose demand terminates at some predictable time in the
future, which is usually longer than one year. Inventory planning here involves
maintaining inventories to just meet demand requirements, but some reordering
within the limited time horizon is allowed. Textbooks with planned revisions,
spare parts for military aircraft, and pharmaceuticals with a shelf life are
examples of products with a defined life. Because the distinction between these
products and those with a perpetual life is often blurred, they will not be treated
differently from perpetual-life products for the purposes of developing a
methodology to control them. Finally, the demand pattern for an item may be
derived from demand for some other item. The demand for packaging materials
is derived from the demand for the primary product. The inventory control of
such dependent demand items is best handled by materials requirements
planning (MRP) and distribution requirements planning (DRP) procedures.
Management Philosophy
INVENTORY OBJECTIVES
Inventory management
involves balancing product
availability, or customer
service, on the one hand with
the costs of providing a given
level of product availability on
the other. Since there may be
more than one way of meeting
the customer service target,
we seek to minimize
inventory-related costs for
each level of customer service
See Figure. Let us begin the
development of the methodology to control inventories with a way to define
product availability and an identification of the costs relevant to managing
inventory levels.
Product Availability
We will see that controlling the service level for single items is computationa1ly
convenient. However, customers frequently request more than one item at a
time. Therefore, the probability of filling the customer order completely can be of
greater concern than single-item service levels. For example, suppose that five
items are requested on an order where each item has a service level of 0.95,
that is, only a 5 percent chance of not being in stock. Filling the entire order
without any item being out of stock would be 0.95 x 0.95 x 0.95 x 0.95 x 0.95 =
0.77. The probability of filling the order completely is somewhat less than the
individual item probabilities.
A number of orders from many customers will show that a mixture of items can
appear on anyone order. The service level is then more properly expressed as a
weighted average fill rate (WAFR). The WAFR is found by multiplying the
frequency with which each combination of items appears on the order by the
probability of filling the order completely, given the number of items on the
order. If a target W AFR is specified, then the service levels for each item must
be adjusted so as to achieve this desired WAFR.
A specialty chemical company receives orders for one of its paint products. The
paint product line contains three separate items that are ordered by customers in
various combinations. From a sampling of orders over a period of time, the items
appear on orders in seven different combinations with frequencies as noted in
the Table. Also from the company's historical records, the probability of having
each item in stock is SL1=0.95; SL2=0.90; and SL3=0.80. As the calculations in
the Table show, the WAFR is 0.801. There will be about one order in five where
the company cannot supply all items at the time of the customer request.
Relevant Costs
Procurement Costs
Some of these procurement costs are fixed per order and do not vary with the
order size. Others, such as transportation, manufacturing, and materials-handling
costs vary to a degree with order size. Each requires slightly different analytical
treatment.
Carrying Costs
Inventory carrying costs result from storing, or holding, goods for a period of
time and are roughly proportional to the average quantity of goods on hand.
These costs can be collected into four classes: space costs, capital costs,
inventory service costs, and inventory risk costs.
Space Costs
Space costs are charges made for the use of the cubic footage inside the storage
building. When the space is rented, storage rates are typically charged by weight
for a period of time, for example, $/cwt/month. If the space is privately owned
or contracted, space costs are determined by allocating space-related operating
costs such as heat and light, as well as fixed costs, such as building and storage
equipment costs, on a volume-stored basis. Space costs are irrelevant when
calculating carrying costs for in-transit inventories.
Capital Costs
Capital costs refer to the cost of the money tied up in inventory. This cost may
represent more than 80 percent of total inventory cost, yet it is the most
intangible and subjective of all the carrying cost elements. There are two reasons
for this. First, inventory represents a mixture of short-term and long-term assets,
as some stocks may serve seasonal needs and others are held to meet longer-
term demand patterns. Second, the cost of capital may vary from the prime rate
of interest to the opportunity cost of capital.
The exact cost of capital for inventory purposes has been debated for some time.
Many firms use their average cost of capital, whereas others use the average
rate of return required of company investments. The hurdle rate has been
suggested as most accurately reflecting the true capital cost. The hurdle rate is
the rate of return on the most lucrative investments forgone by the firm.
Insurance and taxes are also a part of inventory carrying costs because their
level roughly depends on the amount of inventory on hand. Insurance coverage
is carried as a protection against losses from fire, storm, or theft. Inventory taxes
are levied on the inventory levels found on the day of assessment. Although the
inventory at the point in time of the tax assessment only crudely reflects the
average inventory level experienced throughout the year, taxes typically
represent only a small portion of total carrying cost. Tax rates are readily
available from accounting or public records.
Out-of-stock costs are incurred when an order is placed but cannot be filled from
the inventory to which the order is normally assigned. There are two kinds of
out-of- stock costs: lost sales costs and back order costs. Each presupposes
certain actions on the part of the customer, and, because of their intangible
nature, they are difficult to measure accurately. A lost sales cost occurs when the
customer, faced with an out-of-stock situation, chooses to withdraw his or her
request for the product. The cost is the profit that would have been made on this
particular sale and may also include an additional cost for the negative effect
that the stock out may have on future sales. Products for which the customer is
very willing to substitute competing brands, such as bread, gasoline, or soft
drinks, are those that are most likely to incur lost sales. A back order cost occurs
when a customer will wait for his or her order to be filled so that the sale is not
lost, only delayed. Back orders can create additional clerical and sales costs for
order processing, and additional transportation and handling costs when such
orders are not filled through the normal distribution channel. These costs are
fairly tangible, so measurement of them is not too difficult. There also may be
the in- tangible cost of lost future sales. This cost is very difficult to measure.
Products that can be differentiated in the mind of the consumer (automobiles
and major appliances) are more likely to be back ordered than substituted.
Let us begin to develop methods for controlling inventory levels with the push
philosophy. Recall that this method is appropriate where production or purchase
quantities exceed the short-term requirements of the inventories into which the
quantities are to be shipped. If these quantities cannot be stored at the
production site for lack of space or other reasons, then they must be allocated to
the stocking points, hopefully in some way that makes economic sense. We need
to address the following questions: How much inventory should be maintained at
each stocking point? For a particular production run or purchase, how much
should be allocated to each stocking point? How should the excess supply over
requirements be apportioned among the stocking points?
A method for pushing quantities into stocking points involves the following steps:
For the upcoming month, the needs of each warehouse were forecasted, the
current sock levels checked, and desired stock availability level noted for each
warehouse. The findings are tabulated in the following Table
From the normal distribution curve z= 1.28. Hence, the total requirements for
each warehouse would be 12,560 = 10,000 + (1.28 ×2,000). Other warehouse
total requirements are computed similarly.
Net requirements are found as the difference between total requirements and
the quantity on hand in the warehouse. Summing the net requirements
(110,635) shows that 125,000 -110,635 =14,365 which is the excess production
that needs to e prorated to the warehouses.
Warehouse (1)Total (2) On Hand (3) =(1)-(2) (4) Proration (5) =(3)
Requirements Net of Excess +(4)
Requirements Allocation
1 12,560 lb. 5,000 7,560 lb. 1,105 lb. 8,665 lb
2 52,475 15,000 37,475 5,525 43,000
3 95,600 30,000 65,600 7,735 73,335
160,635 110,635 14,365 125,000
Single-Order Quantity
Many practical Inventory Problems exist where the products involved are
perishable or the demand for them is a one-time event. Products such as fruits
and vegetables or the cut flowers, newspapers and some pharmaceuticals have a
short and defined shelf life and they are not available for subsequent selling
periods. Others such as toys and fashion clothes for the immediate selling
season, posters for political campaign etc. have a one-time demand level that
usually cannot be estimated with certainty. Only one order can be placed for
these products to meet such demand. The objective is to determine how large
the single order should be.
To find the most economic order size (Q*), we can use the marginal economic
analysis. That is Q* is found at the point where the marginal profit on the next
unit sold equals the marginal loss of not selling the next unit. The marginal profit
per unit obtained by selling a unit is
Considering the probability of a given number of units being sold, the expected
profits and the losses are balanced at this point. That is,
CPn (Loss) = (1 - CPn) (Profit)
Where CPn represents the cumulative frequency of selling at least n units of the
product.
Solving the above expression for CPn, we have
This says that we should continue to increase the order quantity until the
cumulative probability of selling additional units just equals the ratio of the Profit
/ (Profit + Loss).
A grocery store estimates that it will sell 100 pounds of its specially prepared
potato salad in the next week. The distribution of demand is normally distributed
with a standard deviation of 20 pounds. The supermarket can sell the salad for $
5.99 per pound for the ingredients. Because no preservatives are used, any
unsold salad is given to charity at no cost.
Finding the quantity to prepare that will maximize profit requires that we first
compute CPn. That is
From the normal distribution curve, the optimum Q* is at the point of 58.3
percent of the area under the curve (see the figure). This is a point where z
=0.21 The preparation quantity should be
When Demand is discrete, the order quantity may be between whole values. In
such cases, we will round up Q to the next higher unit to assure at least CPn is
met.
An equipment repair firm whishes to order enough spare parts to keep a
machine tool running throughout a trade show. The repairman prices the parts at
$95 each if needed for a repair. He pays $70 for each part. If all the parts are
not needed they may be returned to the supplier for a credit of $50 each. The
demand for the part is estimated according to the following distribution:
The basic EOQ formula is developed from a total cost equation involving
procurement cost and inventory carrying cost. It is expressed as
TC=-(D/Q)S+ ICQ/2
where
TC = total annual relevant inventory cost, dollars
Q = size of each order to replenish inventory, units
D = annual demand for the item in inventory, units
S = procurement cost, dollars/order
C= value of the item carried in inventory, dollars/unit I = carrying
cost as a percent of item value, %/year
The term D/Q represents the number of times per year a replenishment order is
placed on its supply source. The term Q/2 is the average amount of inventory on
hand.
As Q varies, one cost goes up as the other goes down. It can be shown
mathematically that an optimal order quantity (Q*) exists where the two costs
are in balance and the minimal total cost results. The formula for this EOQ is
2D C
Q* =
IC
N* = D
Q *
2DS 2(750)(50)
Q* = = = 92.58 or 93units
IC (0.25)(35)
Using this formula as part of a basic inventory control procedure, we see that a
sawtooth pattern of inventory depletion and replenishment occurs, as illustrated
in the Figure.
We can now introduce the idea of a reorder point, which is the quantity to which
inventory is allowed to drop before a replacement order is placed. Because there
is generally a time lapse between when the order is placed and when the items
are available in inventory, the demand that occurs over this lead time must be
anticipated. The reorder point (ROP) is
ROP = d x LT
where
The demand rate (d) and the average lead time (LT) must be expressed in the
same time dimension.
Continuing the previous example for the machine replacement part, suppose that
it takes 1.5 weeks to set up production and make the parts. The demand rate is
d=750 (units per year) /52 (weeks per year)=14.42 units per week. Therefore,
ROP=14.42 × 1.5 = 21.6 or 22 units. We can now state the inventory policy:
When the inventory level drops to 22 units, place a replenishment
order for 93 units.
Sensitivity to Data Inaccuracies
Demand and costs cannot always be known for sure. However, our computation,
the economic order quantity is not very sensitive to misestimations of the data.
For example, if demand is in fact 10 percent higher than anticipated, Q* should
only be increased by 11 . 0 =4.88 percent. If the carrying cost is 20 percent
lower than summed, Q* should be increased by only
1 = 1 1.8 p e r c e n t . These percentage changes are inserted into the
(1 - 0 .2 0 )
EOQ formula without changing the remaining cost and/or demand factors since
they remain constant. Notice the stability in values for Q*. If the incorrect order
quantity were used in these two cases, total would have been in error by only
0.11 percent and 0.62 percent, respectively.
Noninstantaneous Resupply
A built-in assumption to Ford Harris's original EOQ formula was that resupply
would be made instantaneously in a single batch of size Q*. In some
manufacturing and resupply processes, output is continuous for a period of time,
and it may take place simultaneously with demand. The basic sawtooth pattern
of on-hand inventory is modified, as shown in the Figure.
The order quantity now becomes the production run, or production order,
quantity (POQ), and we will label it Q p*. To find this, the basic order quantity
formula is modified as follows:
2D S p
Q p* = IC p - d
where p is the output rate. Computing Q p only makes sense when the output
rate p exceeds the demand rate d.
Again, for the previous parts replacement problem, suppose that the production
rate for these parts is 50 units per week. The production run quantity is
2(70)( 50) 50
Qp * =
(0.25)( 35) 50 − 14.42
= 92.58 ×1.185 = 109.74, or 110 units
The ROP quantity remains unchanged.
Two inventory control methods form the foundation for most pull-type
management philosophies with perpetual demand patterns. These are (1) the
reorder point method and (2) the period review method. Practical Control
systems may be based on either of these methods or on a combination of them.
In the Figure, the operation of the reorder point system is illustrated for a single
item where the demand during the lead time is known only to the extent of a
normal probability distribution. This demand during lead time (DDLT) distribution
has a mean of X' and a standard deviation of s'd. The values for X' and s'd are
usually not known directly, but they can be easily estimated by summing a
single-period demand distribution over the length of the lead time. For example,
suppose weekly demand for an item is normally distributed with a mean d = 100
units and a standard deviation of sd = 10 units. Lead time is 3 weeks. We wish to
roll up the weekly demand distribution into one 3-week DDLT distribution of
demand (see Figure below).
The mean of the DDLT distribution is simply the demand rate d times the lead
time LT; or X' = d x LT = 100 x 3 = 300. The variance of DDLT distribution is
found by adding the variances of the weekly demand distributions (see Figure
10-11). That is, s'd2= LT(sd2). The standard deviation is the square root of s'd2
which is .
ROP =d x LT +z(s'd)
The term z is the number of standard deviations from the mean of the DDLT
distribution to give us the desired probability of being in stock during the lead
time period (P). The value for z is found in a normal distribution table for the
fraction of the area P under the DDLT distribution. (See Figure)
2D S 2 ( 1 11
, 0 7 ) (1 0 )
Q* = = = 1 1 , 0 0 8 u n i t s
IC ( 0 .2 0 / 1 2 ) ( 0 .1 1)
The reorder point is ROP =d × LT +z (s′ d)
where s′ d = sd √ LT = 3,099 √ 1.5 = 3,795 units. The value for z is 0.67 from
Normal Tables where the fraction of the area under the normal distribution curve
is 0.75. Thus,
ROP = 11,107 × 1.5 + 0.67 × 3,795 =19,203 units. So, when the effective
inventory level drops to 19,203 units, place a reorder for 11,008 units.
It is not unusual for the reorder point quantity to exceed the order quantity, as
was the case in the example shown above. This frequently happens when lead
times are long and/or demand rates are high. To make the reorder point control
system work properly, we simply must make sure that in deciding when to
trigger a replenishment order, we base the decision on the effective inventory
level. Recall that the effective inventory level requires that we add all stock on
order to the current quantity on hand when deciding whether the reorder point
has been penetrated. When ROP > Q*, the result of this procedure is that a
second order will be placed before the first arrives in stock.
The average inventory level (AIL) for this item is the total of the regular stock
plus safety stock. That is,
AI L = (Q/2) +z(s'd )
For the previous Tie Bar problem, the average inventory would be
AIL = 11,008 / 2 + 0.67 × 3,795 = 8047 units
The total relevant cost is useful for comparing alternative inventory policies or
determining the impact of deviations from optimum policies. We add two new
terms to the total cost formula stated in the Equation that account for
uncertainty. These are safety stock and out-of-stock terms. Total cost can now
be expressed as
Total cost = Order cost + Carrying cost (Regular Stock) + Carrying cost(Safety
Stock)+ Stock out cost
D Q D
TC = S + IC + I C z s' d + k s' d E(z )
Q 2 S
where k is the out-of-stock cost per unit. The stock out cost term requires some
explanation. First, the combined term of s'd E(z) represents the expected number
of units out of stock during an order cycle. E(z) is called the unit normal loss
integral whose values are tabled as a function of the normal deviate z. Second,
the term D/Q is the number of order cycles per period of time, usually a year.
Hence, the number of order cycles times the expected number of units out of
stock during each order cycle gives the total expected number of units out of
stock for the entire period. Then, multiplying by the out-of-stock cost yields the
total period cost.
Continuing the Tie Bar example, suppose the stockout cost is estimated at $0.01
per unit. The total annual cost for the item would be
1 1 , 1 0 7 ( 1 2 )( 1 0 ) 1 1,0 0 8 1 1,1 0 7 ( 2 )
TC = + 0 . 2 0 ( 0 .1 1) ( ) + ( 0. 2 0 ) ( 0 .1 1) (0 . 6 7 ) ( 3 7 9 5 ) + ( 0 .0 1) ( 3 7 9 5 ) (0 .1 5 0 )
1 1 , 0 0 8 2 1 1,0 0 8
Note: The value of 0.150 for E (z) =E (0.67) is from the Normal Loss
Table for z=0.67.
Service Level
The customer service level, or item fill rate, achieved by a particular inventory
policy was previously defined. Restating it in the symbols now being used, we
have
(D / Q ) ( s ' d E (z )) s ' d E ( z ))
S L = 1- = 1-
D Q
3 7 9 5 ( 0 .1 5 0 )
S e r v i c e L e v e l S L = 1- = 0 .9 4 8
1 1,0 0 8
That is, the demand for Tie Bars can be met 94.8 percent of the time. Note that
this is somewhat higher than the probability of a stock out during the lead-time
of P = 0.75.
When the stock out costs are known, it is not necessary to assign a customer
service level. The optimum balance between service and cost may be calculated.
An iterative computational procedure is outlined as follows;
1. Approximate the order quantity from the basic EOQ formula; that is,
2D S
Q =
IC
2. Compute the probability of being in stock during the lead time from
QIC
P = 1-
D k
Find s'd. Find the z value that corresponds to P in the normal distribution table.
Find E(z) from the unit normal loss integral table.
2 D [ S + k s' d E ( z ) ]
Q =
IC
Repeating the Tie Bar example, with the known stockout cost of $0.01 per unit
Estimate Q
2D S 2 (1 1,1 0 7 ) ( 1 0 )
Q = = = 1 1 , 0 0 8 u n i t s
IC ( 0 . 2 0 / 1 2 ) ( 0 .1 1)
Estimate P
QIC 11,008(0.2 0)(0.11)
P = 1- = 1 - = 0 . 8 2
D k 1 1 , 1 0 7 ( 1 2 )( 0 . 0 1 )
From Normal Table z@0.82 = 0.91 from Normal Loss Table E(0.91) =0.0968.
Revise Q
The Standard deviation of DDLT was calculated previously to be s´ =3,795 units.
Now
2 D[S + k s' d E( z ) ] 2 ( 1 1 , 1 0 7 ) ( 1 2 ) [ 1 0 + 0 . 0 1 ( 3 7 9 5 )( 0 . 0 9 6 8 ) ]
Q = = = 1 2 , 8 7 2 u n it s
IC 0 . 2 0 ( 0 . 1 1 )
Revise P
12,872(0.2 0)(0.11)
P = 1 - = 0 . 7 9
1 1 , 1 0 7 ( 1 2 )( 0 . 0 1 )
Revise Q
2 ( 1 1 , 1 0 7 ) ( 1 2 ) [ 1 0 + 0 . 0 1 ( 3 7 9 5 )( 0 . 1 1 8 1 ) ]
Q = = 1 3 , 2 4 6 u n it s
0 . 2 0 ( 0 . 1 1 )
We continue this revision process until the changes in P and Q are sufficiently
small that further calculation is not practical. The final results are P=0.78:
Q*=13,395 units; ROP =19,583 units with a total relevant cost of TC =$15,019
and an actual service level of SL =96%.
The Reorder Point Method with Demand and Lead Time Uncertainty
Accounting for uncertainty in the lead time can extend the realism of the reorder
porn' model. What we wish to do is find the standard deviation (sd) of the DDLT
distribution based on uncertainty in both demand and lead time. This is found by
adding the variance of demand to the variance of lead time, giving us a revised
formula for s'd of
2 2 2
s' d = L T s + d s
d L T
In the Tie Bar problem, SLT is 0.5 months. The value for s′ d would
now be
2 2 2
s' = 1. 5 ( 3 0 9 9 ) + 1 1 1 0 7 ( 0 .5 ) = 6 ,7 2 7 u n i t s
d
Combining demand and lead-time variability in this way can greatly increase s'd
and the resulting safety stock. Brown warns that demand and lead-time
distributions may not be independent of each other. Rather, when a
replenishment order is placed, a fair idea is known as to the lead-time for that
order. Therefore, application of Equation (10-18) may lead to an overstatement
of s'd and the resulting amount of safety stock. If lead times do vary
unpredictably, Brown suggests the following precise procedure for determining
the standard deviation of demand during lead-time:
Alternately and less precisely, the longest lead time may be used as the average
lead time with s LT set at zero (0). The standard deviation is then computed as s'd
= sd√ LT.
The reorder point n i ventory control method applies. However, determining the
statistics of the demand-during-lead-time distribution requires taking the lead-
time for the entire channel into account.
Recall:
2 2 2
s' d = L T s + d s
d L T
Where
2 2 2 2
s = s + s + s
L T p i o
LT =8 p + 8 i + 8 o
=1 +4 +2= 7 days
' 2 2
s = 7 x1 0 + 1 0 0 x 1. 3 5 = 1 4 , 2 0 0 = 1 1 9 .1 6 u n i t s
d
and