TOOLS
AND
TECHNIQUES
Quantitative Inventory
Models
15
Inventory modeling has been the subject of research in OM for many decades. The bestknown and most fundamental inventory model is the economic order quantity (EOQ)
model, which was developed in the early 1900s, and many variations of it have been developed or extended to different problem scenarios. In this section we present the development
of the EOQ and other common inventory models.
ECONOMIC ORDER QUANTITY (EOQ) MODEL
Several important assumptions are made in using the EOQ model:
°
Only a single item under continuous review—an (r, Q) system—is considered
°
The entire quantity ordered arrives in the inventory at one time
°
The demand for the item has a constant, or nearly constant, rate
The condition of constant demand rate simply means that the same number of units are taken
from inventory each period of time—such as five units every day, 25 units every week, 100
units every four-week period, and so on.This model also assumes that the lead-time is constant and that no stockouts are allowed. Further, the EOQ model is concerned with two basic
decisions: how much to order, and when to place an order.
The how-much-to-order decision involves selecting an order quantity that is a compromise between (1) keeping small inventories and ordering frequently and (2) keeping large
inventories and ordering infrequently.The first alternative would probably result in undesirably high ordering costs, while the second alternative would probably result in undesirably
high inventory-holding costs.The EOQ model minimizes the total cost equal to the sum of
the inventory-holding cost and the ordering cost.
We begin by defining Q to be the size of the order.The inventory level has a maximum
value of Q units when the order of size Q is received from the manufacturer.The inventory
is depleted, at which time another shipment of Q units will be received. With the assumption of a constant demand rate, a sketch of the inventory level from the time that Q units are
received until the inventory is depleted is shown in Figure T15-1. Note that the sketch indicates that the average inventory level for the period in question is 1/2 Q.This should appear
reasonable, since the maximum inventory level is Q, the minimum is 0, and the inventory
level declines at a constant rate over the period.
Figure T15-1 shows the inventory pattern during one order-cycle period, T.As time goes
on, this pattern will repeat.The complete inventory pattern is shown in Figure T15-2. If the
average inventory during each cycle is 1/2 Q, the average inventory level over any number
of cycles is also 1/2 Q. So, as long as the time period involved contains an integral number
of order cycles, the average inventory for the period will be 1/2 Q.
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T O O L S
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TOOLS AND TECHNIQUES 15
FIGURE T15-1
|
Quantitative Inventory Models
Sketch of Inventory Level
Maximum inventory level
Inventory Level
1 5
Q
Average inventory level
1/2 Q
Minimum inventory level
0
0
Time
T
Length of time required to deplete
an inventory of Q units
FIGURE T15-2
|
Inventory Pattern for the EOQ Model
Inventory is used at the constant demand
rate of 2,000 units per month
Inventory Level
Q
Average
inventory
level
1/2 Q
0
Time
The inventory-holding cost can be calculated by multiplying the average inventory by the
cost of carrying one unit in inventory for the stated period.The period of time selected for
the model is up to the user; it can be one week, one month, one year, or more. However,
since the inventory-carrying costs for many industries and businesses are expressed as an
annual percentage or rate, most inventory models are developed on an annual cost basis. Let
I = annual inventory–carrying charge
C = unit cost of the inventory item
The cost of storing one unit in inventory for the year, denoted by Ch, is given by Ch = IC.
Thus, the general equation for annual inventory-holding cost is
annual inventory–holding cost = (average inventory)
1
(annual holding cost per unit) = QCh
2
Quantitative Inventory Models
Thus, the total annual cost—inventory-holding cost plus ordering cost—can be expressed as
1
D
TC = QCh + Co
2
Q
The next step is to find the order quantity, Q, that does, in fact, minimize the total cost.
By using differential calculus, we can show that the quantity that minimizes the total cost,
denoted by Q*, is given by the formula
Q* =
2DCo
Ch
This formula is referred to as the economic order quantity (EOQ) formula.
Now that we know how much to order, we want to answer the second question of when
to order, expressed most often in terms of the reorder point. For inventory systems using the
constant-demand-rate assumption and a fixed lead-time, the reorder point is the same as the
lead-time demand—that is, the demand that occurs between the time the order is placed and
when it is received.The general expression for the reorder point is
r = dm
where r is the reorder point, d is demand per day, and m is lead-time for a new order in days.We
can now compute the frequency of orders, or the time between orders.This is given by
N
NQ*
T = = D/Q*
D
where N is the number of days of operation for the year.
EXAMPLE T15.1. Computing the EOQ. A pharmaceutical distributor has determined
that the demand for a common pain reliever is relatively constant at 2,000 cases per month.The
cost is $12.00 per case, and the company estimates its cost of capital at a rate of 12 percent.
Insurance, taxes, breakage, pilferage, and warehouse overhead are estimated to be approximately
6 percent of the value of the inventory.Thus, the annual inventory-holding costs are estimated
to be 18 percent of the value of the inventory.Therefore, the cost of holding one case in inventory for one year is 0.18($12.00) = $2.16.The cost of placing an order includes the salaries of
the purchasing agents and clerical support staff, transportation costs, and miscellaneous costs
such as paper, postage, and telephone costs, which are estimated to be $38.00 regardless of the
quantity purchased. From this information, we have the data necessary to apply the EOQ
model:
D = 24,000 cases per year
Co = $38.00 per order
I = 18 percent
C = $12.00 per case
Ch = IC = $2.16
1 5
D
annual ordering cost = (number of orders per year)(cost per order) = ()Co
Q
T O O L S
To complete the total-cost model, we must now include the ordering cost.The goal is to
express this cost in terms of the order quantity, Q. Since the inventory-holding cost is
expressed on an annual basis, we need to express ordering costs as an annual cost. Letting D
denote the annual demand for the product, we know that by ordering Q units each time we
order, we have to place D/Q orders per year. If Co is the cost of placing one order, the general expression for the annual ordering cost is
3
T O O L S
4
TOOLS AND TECHNIQUES 15
Quantitative Inventory Models
The total-cost model becomes
1 5
1
24,000
TC = Q($2.16) + ($38.00)
2
Q
912,000
= 1.08Q + Q
Thus, the minimum-cost order quantity given by the EOQ formula is
Q* =
2(24,000)(38)
= 919 cases (rounded to a whole number)
2.16
Using this order quantity shows that the inventory can be handled with a total annual cost of
TC = 1.08(919) + 912,000/919 = $1,984.90
If we compare this value with the current purchasing policy, which calls for monthly orders of
the amount Q = 2,000, the total annual cost is
TC = 1.08(2000) + 912,000/2,000 = $2,616.00
Thus, the EOQ analysis has resulted in a $2,616.00 – $1,984.90 = $631.10, or 24.1 percent, cost
reduction. Notice also that the total ordering costs ($992.52) are equal to the total holding costs
($992.38), with the small difference due to rounding. In general, this will always be true for the
EOQ model.
The manufacturer of the product guarantees a three-day delivery to the distributor.
Considering weekends and holidays, the distributor operates 250 days per year. So, on a daily
basis, the annual demand of 24,000 cases corresponds to a demand of 24,000/250 = 96 cases.
Thus, we anticipate
(3 days)(96 cases per day) = 288
cases will be sold during the three days it takes a new order to reach the warehouse. Since the
three-day delivery period is the lead-time for a new order, the 288 cases of demand during this
period is the lead-time demand. Therefore, the distributor should order a new shipment from
the manufacturer when the inventory position reaches a reorder level of 288 cases. Finally, we
note that the distributor will place 24,000/919 = 26.12 orders, or approximately 26 orders, per
year.With 250 working, or operating, days per year, it should be placing an order every 250/26
= 9.6 days, or roughly every 9 to 10 days.
Figure T15-3 shows a plot of the holding, ordering, and total cost as a function of the
order quantity.You can see the tradeoffs between holding and ordering and that the optimal
quantity occurs when these costs are equal. Also, the total-cost curve is relatively flat close to
the optimal value, Q*. So we can conclude that small variations in the actual order quantity
used will have little effect on the total cost.Therefore, the EOQ model is insensitive to small
variation or errors in the cost estimates.
SERVICE LEVELS AND UNCERTAINTY IN DEMAND
One of the critical assumptions of the EOQ model is that demand is constant. In many cases,
though, demand is uncertain and can be described by a probability distribution. If ordering
policies are developed based solely on average demand, the firm runs a significant risk of running out of stock during the lead-time. Variation in the lead-time and the quality of goods
received may also contribute to this risk.As a result, most firms find it necessary to carry safety
stock to ensure product availability at the end of the order cycle.The amount of safety stock
carried is a function both of the level of uncertainty present and a policy decision called the
service level—the desired probability that a stockout will not occur during a lead-time. For
example, a service level of 95 percent means that we wish to ensure that 95 percent of all
demand can be satisfied; equivalently, there is only a 5 percent chance of a stockout. With
higher safety stock, fewer orders will be placed, and less risk will be incurred.
Quantitative Inventory Models
|
T O O L S
FIGURE T15-3
Cost Curves for the EOQ Model
Cost (dollars)
1 5
Annual
total cost
3000
2000
Annual
ordering cost
1000
Annual inventoryholding cost
0
0
400
800
1200
1600
2000
2400
Q*
Order Quantity (Q)
EXAMPLE T15.2. Computing Safety Stock. Let us consider an office supply company
that sells laser printer paper.The paper is purchased in reams from a firm in Portland, Oregon.
Ordering costs are $45.00 per order, one ream of paper costs $3.80, and the annual inventorycarrying cost rate is 20 percent.Thus, the inventory-holding cost is Ch = 0.20($3.80) = $0.76.
Demand is not specifically known, but historical sales data indicate the annual demand to be
15,000 reams. As an approximation of the best order quantity, we can apply the EOQ model
with the expected annual volume substituted for the annual demand, D.This yields
Q* =
2DCo
=
Ch
5
2(15,000)(45)
= 1.333 reams of paper
0.76
It usually takes two weeks to receive a new supply of paper from the manufacturer; thus,
the average lead-time demand is (15,000 reams/52 weeks) 2 = 577 reams. If demands are
uncertain and distributed symmetrically about 577, then demand will be greater than 577 reams
and result in a shortage roughly 50 percent of the time. If we can estimate the lead-time-demand
probability distribution, we can find the probability of stockout because stockouts occur whenever the lead-time demand exceeds the reorder point.
When a normal probability distribution provides a good approximation of lead-time
demand, the general expression for the reorder point is
r = + z
where z = number of standard deviations necessary to achieve the acceptable service level
= expected lead-time demand
= standard deviation of lead-time demand
Let us assume that the lead-time-demand distribution for the printer paper is normally
distributed with a mean of 577 reams and a standard deviation of 100 reams. This distribution
is shown in Figure T15-4.
If we desire a service level of 0.95, we need only determine the critical value of the
lead-time distribution for which the upper tail area is 0.05 (the probability of a stockout).This
is shown in Figure T15-5. From a table of the normal distribution, we find that a reorder point
that is 1.645 standard deviations above the mean will allow stockouts only 5 percent of the time.
Therefore, the reorder point, r, is determined by
r = 577 + 1.645(100) = 742 reams (rounded up)
T O O L S
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TOOLS AND TECHNIQUES 15
FIGURE T15-4
Quantitative Inventory Models
|
Determining Lead-Time-Demand Distribution
1 5
= 577
277
= 100
377
477
577
677
777
877
Lead-Time Demand
In this calculation, we see that the reorder point is based on the expected lead-time
demand, plus a safety stock of 165 reams.Thus, the inventory decision is to order 1,333 reams
whenever the inventory position reaches the reorder point of 742.
The anticipated annual cost for this system is as follows:
15,000
D
Co = 45 = $506
Ordering cost
1,333
Q
Q
1,333
Holding cost, normal inventory
Cn = (.76) = $507
2
2
Holding cost, safety stock
(165)Ch = (165)(.76) = $125
Total cost
$1,138
Note that the safety stock of 165 units costs an additional $125 per year.
FIGURE T15-5
|
Determining the Safety Stock Level
Service level = .95
r
No stockout
(demand r )
95%
277
377
477
577
677
Lead-Time Demand
Stockout
(demand r )
5%
777
877
Quantitative Inventory Models
1. With S backorders existing when a new shipment of size Q arrives, the S backorders
will be shipped to the appropriate customers immediately, and the remaining (Q – S)
units will be placed in inventory.
2. Q – S will be the maximum inventory level.
3. The inventory cycle of T days will be divided into two distinct phases: t1 days, when
inventory is on hand and orders are filled as they occur, and t2 days, when there is a
stockout and all orders are placed on backorder.
The inventory pattern for this model, where negative inventory represents the number of
back orders, is shown in Figure T15-6.
FIGURE T15-6
|
Inventory Pattern for Backorder Situation
t1 = length of inventory period
t2 = length of stockout, or back-order, period
Maximum
inventory
Inventory Level
Q-S
0
–S
t1
t2
T
Time
Backordering costs usually involve labor and special-delivery costs directly associated with
the handling of backorders. Another portion of the backorder cost can be expressed as a loss
of goodwill with customers due to their having to wait for their orders. Since the goodwill
cost depends on how long the customer has to wait, it is customary to adopt the convention
of expressing all backorder costs in terms of how much it costs to have a unit on backorder
for a stated period of time.This method of computing cost is similar to the method we used
to compute the inventory-holding cost.
Admittedly, the backorder cost rate (especially the goodwill cost) is difficult to determine
in practice. However, noting that EOQ models are rather insensitive to the cost estimates, we
should feel confident that reasonable estimates of the backorder cost will lead to a good
approximation of the overall minimum-cost inventory decision.
1 5
In some cases, it may be desirable—from an economic point of view—to plan for and allow
shortages. This situation is most common when the value per unit of the inventory is very
high, and hence the inventory-holding cost is high. An example is a new-car dealer’s inventory. Most customers do not find the specific car they want in stock, but are willing to backorder it. We present an extension to the EOQ model that allows for backorders. If we let S
indicate the number of backorders that have accumulated when an order of size Q is received,
the inventory system has these characteristics:
T O O L S
INVENTORY MODELS WITH BACKORDERS
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T O O L S
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TOOLS AND TECHNIQUES 15
Quantitative Inventory Models
Letting Cb be the cost to maintain one unit on back order for one year, the three sources
of cost in this planned-shortage inventory model can be expressed as in the following
equations:
1 5
(Q – S)2
Inventory-holding cost = Ch
2Q
D
Ordering cost = Co
Q
S2
Backordering cost = Cb
2Q
Thus, our total-annual-cost expression (TC) becomes
D
S2
(Q – S)2
TC = Ch + Co + Cb
Q
2Q
2Q
The minimum-cost values for Q and S can be found using calculus and are
Q* =
and
2DC o Ch + Cb
Cb
Ch
Ch
S* = Q* Ch + Cb
EXAMPLE T15.3. Planned Shortage Model. To illustrate the use of this model, consider
an electronics company that is concerned about an expensive part used in television repair.The
cost of the part is $125, and the inventory-holding rate is 20 percent.The cost to place an order
is estimated to be $40.The annual demand, which occurs at a constant rate throughout the year,
is 800 parts. Currently, the inventory policy is based on the EOQ model, with
Q* =
2DCo
=
Ch
2(800)(40)
= 51 parts
.20(125)
The total annual cost of inventory holding and ordering has been
1
D
1
800
TC = QCh + Co = (51)(.20)(125) + (40) = $637.50 + $627.50 = $1,265
2
Q
2
51
Because of the relatively high inventory investment, backordering is being considered.
On an annual basis, a unit back-order cost of $60 was assigned. Using the formulas
above, the optimal order quantity, Q*, and the optimal number of back orders, S*, become
Q* =
and
2(800)(40) .20(125) + 60
= 60 parts
.20(125)
60
.20(125)
S* = 60 = 18 parts
.20(125) + 60
Both Q* and S* have been rounded to whole numbers to simplify the remaining calculations.
We find the following costs associated with the inventory policy.
Quantitative Inventory Models
The single-period inventory model applies to inventory situations in which one order is
placed for the product in anticipation of a future selling season where demand is uncertain.
At the end of the period, the product has either sold out, or there is a surplus of unsold items
to sell for a salvage value. Single-period models are used in situations involving seasonal or
perishable items that cannot be carried in inventory and sold in future periods. One example would be ordering dough for a pizza restaurant, which stays fresh for only three and onehalf days. Other examples include daily newspapers and seasonal clothing such as bathing suits
and winter coats. In such a single-period inventory situation, the only inventory decision is
how much of the product to order at the start of the period. Because newspaper sales are a
typical example of the single-period situation, the single-period inventory problem is sometimes referred to as the newsboy problem.
The newsboy problem can be solved using a technique called marginal analysis, which
compares the cost or loss of ordering one additional unit with the cost or loss of not ordering one additional unit.The costs involved are defined as follows:
co = the cost per unit of overestimating demand; this cost represents the loss of ordering one additional unit and finding that it cannot be sold.
cu = the cost per unit of underestimating demand; this cost represents the opportunity
loss of not ordering one additional unit and finding that it could have been sold.
The optimal order quantity is the value of Q* that satisfies the expression
cu
P(demand Q*) = cu + co
EXAMPLE T15.4. Single-Period Inventory Model. To illustrate this model, let us consider a buyer for a department store who is ordering fashion swimwear.The purchase must be
made in the winter, and the store plans to hold an August clearance sale to sell whatever has not
been sold by July 31. Each swimsuit costs $40 and sells for $60. At the sale price of $30 per
swimsuit, the store expects it can sell any remaining stock during the August sale.We will assume
that a uniform probability distribution ranging from 350 to 650 units, shown in Figure T15-7,
describes the demand.The expected demand is 500.
1 5
SINGLE-PERIOD INVENTORY MODEL
T O O L S
(Q – S)2
(60 – 18)2
Inventory-holding cost = Ch = (.20)(125) = $367.50
2Q
2(60)
D
800
Ordering cost = Co = (40) = $533.33
Q
600
S2
(18)2
Backordering cost = Cb = (60) = $162.00
2Q
2(60)
The total cost is $1,062.83, and hence the backordering policy provides a $1,265 – $1,062.83
= $202.17, or 16 percent, cost reduction when compared to the EOQ model. Note that the
daily demand for the part is (800 parts)/(250 days) = 3.2 parts per day. Since the maximum
number of backorders is 18, we see that the length of the backorder period will be 18/3.2 =
5.6 days.
9
T O O L S
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TOOLS AND TECHNIQUES 15
FIGURE T15-7
Quantitative Inventory Models
|
Probability Distribution for Single-Period Model
1 5
Expected demand = 500
350
500
650
Q
Demand
The retailer will incur the cost of overestimating demand whenever it orders too much
and has to sell the extra units remaining after July. Thus, the cost per unit of overestimating
demand is equal to the purchase cost per unit minus the August sale price per unit; that is, co =
$40 – $30 = $10. In other words, the retailer will lose $10 for each unit that it orders over the
quantity demanded.The cost of underestimating demand is the lost profit (opportunity loss) due
to the fact that a unit could have been sold but was not available in inventory. Thus, the perunit cost of underestimating demand is the difference between the regular selling price per unit
and the purchase cost per unit; that is, cu = $60 – $40 = $20.The optimal order size must satisfy this condition:
cu
20
20
2
P(demand Q*) = = = = cu + co
20 + 10
30
3
Because the demand distribution is uniform, the value of Q* is two-thirds of the way from 350
to 650.Thus, Q* = 550.
Note that whenever cu < co, the formula leads to the choice of an order quantity more
likely to be less than demand; hence, a higher risk of a stockout is present. However, when cu
> co, as in the example, the optimal order quantity leads to a higher risk of a surplus.
If the demand distribution were other than uniform, the same process applies. To illustrate, suppose demand is normal with a mean of 500 and a standard deviation of 100. This
probability distribution is shown in Figure T15-8. With cu = $20 and co = $10 as previously
computed, the optimal order quantity, Q*, must still satisfy the requirement that P(demand
Q*) = 2/3.We simply use the table of areas under the normal curve (see the appendix) to
find the Q* where this condition is satisfied.This is shown in Figure T15-9. In Figure T159, the area between the mean, 500, and Q* is 0.1667. This area occurs at z = 0.43 standard
deviation above the mean.Therefore,
Q* = + .1667 = 500 + .43(100) = 543
Quantitative Inventory Models
|
Normal Probability Distribution of Demand
T O O L S
FIGURE T15-8
11
1 5
= 100
Expected demand
= 500
500
Demand
FIGURE T15-9
|
Optimal Order Quantity for Normally Distributed Demand Case
P (Demand
Q *) = 2/3
Note: Since 50% of the area is less
than 500, the area or probability
of a demand between 500
and Q * is 1/6.
500 Q*
Demand
Problems and Exercises
1. The table below gives the daily demand for a certain oil filter at an auto supply store.
Illustrate the operation for a continuous-review inventory system by graphing the inventory level versus time if Q = 40, r = 15, and the lead-time is three days. Assume that
orders are placed at the end of the day (after demand is known) and that they arrive at
the beginning of the day (before any demand is satisfied). Thus, if an order is placed at
the end of day 5, it will arrive at the beginning of day 9. Assume that 30 units are on
hand at the start of day 1.
T O O L S
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TOOLS AND TECHNIQUES 15
1 5
Day
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Quantitative Inventory Models
Demand
6
8
5
4
5
6
1
1
3
8
8
6
7
0
2
4
7
3
5
9
3
6
1
9
1
2. Kilroy’s Pizza purchases its pizza delivery boxes from a printing supplier. Kilroy’s delivers,
on average, 200 pizzas each month. Boxes cost 20 cents each, and each order costs $10 to
process. Because of limited storage space, the manager wants to charge inventory holding at 30 percent of the cost.The lead-time is one week, and the restaurant is open 360
days per year. Determine the EOQ, reorder point, number of orders per year, and total
annual cost. If the supplier raises the cost of each box to 25 cents, how would these results
change?
3. An automotive engine plant purchases a component used in the manufacture of automobile generators directly from the supplier. The generator production, which is operated at a constant rate, will require 1,200 components per month throughout the year.
Assume ordering costs are $25 per order, unit cost is $2 per component, and annual
inventory holding costs are charged at 20 percent. The company operates 250 days per
year, and the lead-time is five days. Compute the EOQ, total annual inventory-holding
and ordering costs, and the reorder point.
4. Floyd Distributors, Inc. provides a variety of auto parts to small local garages. It purchases
parts from manufacturers according to the EOQ model and then ships the parts from a
regional warehouse directly to its customers. For a particular type of muffler, Floyd’s
EOQ analysis recommends orders with Q* = 5 to satisfy an annual demand of 200 mufflers.There are 250 working days per year, and the lead-time averages 15 days.
Quantitative Inventory Models
7.
8.
9.
1 5
6.
T O O L S
5.
a. What is the reorder point if Floyd assumes a constant demand rate?
b. Suppose an analysis of Floyd’s muffler demand shows that the lead-time demand
follows a normal distribution, with = 12 and = 2.5. If Floyd’s managers can
tolerate one stockout per year, what is the revised reorder point?
c. What is the safety stock for part (b)? If Ch = $5/unit/year, what is the extra cost
due to the uncertainty of demand?
Brauch’s Pharmacy has an expected annual demand for a leading pain reliever of 800
boxes, which sell for $6.50 each. Each order costs $6.00, and the inventory carrying
charge is 20 percent.The expected demand during the lead-time is normal, with a mean
of 25 and a standard deviation of 3.Assuming 52 weeks per year, what reorder point provides a 95 percent service level? How much safety stock will be carried? If the carrying
charge were 25 percent instead, what would be the total annual inventory-related cost?
The A&M Hobby Shop carries a line of radio-controlled model racing cars. Demand for
the cars is assumed to be constant at a rate of 30 cars per month.The cars cost $70 each,
and ordering costs are approximately $15 per order regardless of the order size.
Inventory-carrying costs are 20 percent annually.
a. Determine the EOQ and total annual cost under the assumption that no backorders are permitted.
b. Using a $45 per unit per year backorder cost, determine the minimum-cost inventory policy and total annual cost for the model racing cars.
c. What is the maximum number of days a customer would have to wait for a backorder under the policy in part (b)? Assume that the Hobby Shop is open for
business 300 days per year.
d. Would you recommend an inventory policy that allows backorders for this product? Explain.
e. If the lead-time is eight days, what is the reorder point for both a no-backorder and
a backorder inventory policy?
Marilyn’s Interiors sells silk floral arrangements and other home furnishings. Because
space is limited and she does not want to tie up a lot of money in inventory, Marilyn uses
a backorder policy for most items.A popular silk arrangement costs $40, and Marilyn sells
an average of 15 per month. Ordering costs are $30, and she values her inventory holding cost at 25 percent. Marilyn figures the backorder cost to be $40 annually.What is the
optimal order quantity and planned backorder level? What if customer cancellations and
other loss of goodwill increase the backorder cost to $100 annually?
The J&B Card Shop sells calendars featuring a different colonial picture for each month.
The once-a-year order for each year’s calendar arrives in September. From past experience, the September-to-July demand for the calendars can be approximated by a normal
distribution with = 500 and = 120. The calendars cost $3.50 each, and J&B sells
them for $7.00 each.
a. If J&B throws out all unsold calendars at the end of July (that is, salvage value is
zero), how many calendars should be ordered?
b. If J&B reduces the calendar price to $1 at the end of July and can sell all surplus
calendars at this price, how many calendars should be ordered?
The Gilbert Air-Conditioning Company is considering the purchase of a special shipment of portable air-conditioners manufactured in Japan. Each unit will cost Gilbert $80,
and it will be sold for $125. Gilbert does not want to carry surplus air-conditioners over
until the following year.Thus, all supplies will be sold to a wholesaler, who has agreed to
13
T O O L S
14
TOOLS AND TECHNIQUES 15
Quantitative Inventory Models
1 5
take all surplus units for $50 per unit.Assume that the air-conditioner demand has a normal distribution with = 20 and = 6.
a. What is the recommended order quantity?
b. What is the probability that Gilbert will sell all units it orders?
10. A popular newsstand in a large metropolitan area is attempting to determine how many
copies of the Sunday paper it should purchase each week. Demand for the newspaper on
Sundays can be approximated by a normal distribution with = 700 and = 100.The
newspaper costs the newsstand 40 cents a copy and sells for 75 cents.The newsstand does
not receive any value from surplus papers and thus absorbs a 100 percent loss on all
unsold papers.
a. How many copies of the Sunday paper should be purchased each week?
b. What is the probability that the newsstand will have a stockout?
c. The manager of the newsstand is concerned about the newsstand’s image if the
probability of stockout is high. Customers often purchase other items after coming
to the newsstand for the Sunday paper, and frequent stockouts would cause customers to go to another newsstand.The manager agrees that a $1 loss of goodwill
cost should be assigned to any stockout. What are the new recommended order
quantity and the new probability of a stockout?
APPENDIX: AREAS FOR THE STANDARD NORMAL DISTRIBUTION
Entries in the table give the area under the curve between the mean and z standard deviations above the mean. For example, for z = 1.25 the area under the curve between the mean
and z is 0.3944.
Area or
probability
0
z
Quantitative Inventory Models
0.01
0.0040
0.0438
0.0832
0.1217
0.1591
0.1950
0.2291
0.2612
0.2910
0.3186
0.3438
0.3665
0.3869
0.4049
0.4207
0.4345
0.4463
0.4564
0.4649
0.4719
0.4778
0.4826
0.4864
0.4896
0.4920
0.4940
0.4955
0.4966
0.4975
0.4982
0.4987
0.02
0.0080
0.0478
0.0871
0.1255
0.1628
0.1985
0.2324
0.2642
0.2939
0.3212
0.3461
0.3686
0.3888
0.4066
0.4222
0.4357
0.4474
0.4573
0.4656
0.4726
0.4783
0.4830
0.4868
0.4898
0.4922
0.4941
0.4956
0.4967
0.4976
0.4982
0.4987
0.03
0.0120
0.0517
0.0910
0.1293
0.1664
0.2019
0.2357
0.2673
0.2967
0.3238
0.3485
0.3708
0.3907
0.4082
0.4236
0.4370
0.4484
0.4582
0.4664
0.4732
0.4788
0.4834
0.4871
0.4901
0.4925
0.4943
0.4957
0.4968
0.4977
0.4983
0.4988
0.04
0.0160
0.0557
0.0948
0.1331
0.1700
0.2054
0.2389
0.2704
0.2995
0.3264
0.3508
0.3729
0.3925
0.4099
0.4251
0.4382
0.4495
0.4591
0.4671
0.4738
0.4793
0.4838
0.4875
0.4904
0.4927
0.4945
0.4959
0.4969
0.4977
0.4984
0.4988
0.05
0.0199
0.0596
0.0987
0.1368
0.1736
0.2088
0.2422
0.2734
0.3023
0.3289
0.3531
0.3749
0.3944
0.4115
0.4265
0.4394
0.4505
0.4599
0.4678
0.4744
0.4798
0.4842
0.4878
0.4906
0.4929
0.4946
0.4960
0.4970
0.4978
0.4984
0.4989
0.06
0.0239
0.0636
0.1026
0.1406
0.1772
0.2123
0.2454
0.2764
0.3051
0.3315
0.3554
0.3770
0.3962
0.4131
0.4279
0.4406
0.4515
0.4608
0.4686
0.4750
0.4803
0.4846
0.4881
0.4909
0.4931
0.4948
0.4961
0.4971
0.4979
0.4985
0.4989
0.07
0.0279
0.0675
0.1064
0.1443
0.1808
0.2157
0.2486
0.2794
0.3078
0.3340
0.3577
0.3790
0.3980
0.4147
0.4292
0.4418
0.4525
0.4616
0.4693
0.4756
0.4808
0.4850
0.4884
0.4911
0.4932
0.4949
0.4962
0.4972
0.4979
0.4985
0.4989
0.08
0.0319
0.0714
0.1103
0.1480
0.1844
0.2190
0.2518
0.2823
0.3106
0.3365
0.3599
0.3810
0.3997
0.4162
0.4306
0.4429
0.4535
0.4625
0.4699
0.4761
0.4812
0.4854
0.4887
0.4913
0.4934
0.4951
0.4963
0.4973
0.4980
0.4986
0.4990
0.09
0.0359
0.0753
0.1141
0.1517
0.1879
0.2224
0.2549
0.2852
0.3133
0.3389
0.3621
0.3830
0.4015
0.4177
0.4319
0.4441
0.4545
0.4633
0.4706
0.4767
0.4817
0.4857
0.4890
0.4916
0.4936
0.4952
0.4964
0.4974
0.4981
0.4986
0.4990
1 5
0.00
0.0000
0.0398
0.0793
0.1179
0.1554
0.1915
0.2257
0.2580
0.2881
0.3159
0.3413
0.3643
0.3849
0.4032
0.4192
0.4332
0.4452
0.4554
0.4641
0.4713
0.4772
0.4821
0.4861
0.4893
0.4918
0.4938
0.4953
0.4965
0.4974
0.4981
0.4986
T O O L S
z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
15