Chapter 14
Inventory Models: Deterministic Demand




Economic Order Quantity (EOQ) Model
Economic Production Lot Size Model
Inventory Model with Planned Shortages
Quantity Discounts for the EOQ Model
© 2009 South-Western, a part of Cengage Learning
Slide 1
Inventory Models


The study of inventory models is concerned with
two basic questions:
• How much should be ordered each time
• When should the reordering occur
The objective is to minimize total variable cost over
a specified time period (assumed to be annual in the
following review).
© 2009 South-Western, a part of Cengage Learning
Slide 2
Inventory Costs





Ordering cost -- salaries and expenses of processing
an order, regardless of the order quantity
Holding cost -- usually a percentage of the value of
the item assessed for keeping an item in inventory
(including cost of capital, insurance, security costs,
taxes, warehouse overhead, and other related variable
expenses)
Backorder cost -- costs associated with being out of
stock when an item is demanded (including lost
goodwill)
Purchase cost -- the actual price of the items
Other costs
© 2009 South-Western, a part of Cengage Learning
Slide 3
Deterministic Models


The simplest inventory models assume demand
and the other parameters of the problem to be
deterministic and constant.
The deterministic models covered in this chapter
are:
• Economic order quantity (EOQ)
• Economic production lot size
• EOQ with planned shortages
• EOQ with quantity discounts
© 2009 South-Western, a part of Cengage Learning
Slide 4
Economic Order Quantity (EOQ)



The most basic of the deterministic inventory
models is the economic order quantity (EOQ).
The variable costs in this model are annual holding
cost and annual ordering cost.
For the EOQ, annual holding and ordering costs are
equal.
© 2009 South-Western, a part of Cengage Learning
Slide 5
Economic Order Quantity

Assumptions
• Demand D is known and occurs at a constant rate.
• The order quantity Q is the same for each order.
• The cost per order, $Co, is constant and does not
depend on the order quantity.
• The purchase cost per unit, C, is constant and does
not depend on the quantity ordered.
• The inventory holding cost per unit per time
period, $Ch, is constant.
• Shortages such as stock-outs or backorders are not
permitted.
• The lead time for an order is constant.
• The inventory position is reviewed continuously.
© 2009 South-Western, a part of Cengage Learning
Slide 6
Economic Order Quantity

Formulas
• Optimal order quantity:
Q*=
• Number of orders per year:
2DCo/Ch
D/Q *
• Time between orders (cycle time): Q */D years
• Total annual cost:
[Ch(Q*/2)] + [Co(D/Q *)]
(holding + ordering)
© 2009 South-Western, a part of Cengage Learning
Slide 7
Example: Bart’s Barometer Business

Economic Order Quantity Model
Bart's Barometer Business is a retail outlet that
deals exclusively with weather equipment. Bart is
trying to decide on an inventory and reorder policy
for home barometers.
Barometers cost Bart $50 each and demand is
about 500 per year distributed fairly evenly
throughout the year.
© 2009 South-Western, a part of Cengage Learning
Slide 8
Example: Bart’s Barometer Business

Economic Order Quantity Model
Reordering costs are $80 per order and holding
costs are figured at 20% of the cost of the item. Bart's
Barometer Business is open 300 days a year (6 days a
week and closed two weeks in August). Lead time is
60 working days.
© 2009 South-Western, a part of Cengage Learning
Slide 9
Example: Bart’s Barometer Business

Total Variable Cost Model
Total Costs = (Holding Cost) + (Ordering Cost)
TC = [Ch(Q/2)] + [Co(D/Q)]
= [.2(50)(Q/2)] + [80(500/Q)]
= 5Q + (40,000/Q)
© 2009 South-Western, a part of Cengage Learning
Slide 10
Example: Bart’s Barometer Business

Optimal Reorder Quantity
Q*=

2DCo /Ch =
2(500)(80)/10 = 89.44  90
Optimal Reorder Point
Lead time is m = 60 days and daily demand is
d = 500/300 or 1.667.
Thus the reorder point r = dm = (1.667)(60) = 100.
Bart should reorder 90 barometers when his
inventory position reaches 100 (that is 10 on hand
and one outstanding order).
© 2009 South-Western, a part of Cengage Learning
Slide 11
Example: Bart’s Barometer Business

Number of Orders Per Year
Number of reorder times per year = (500/90) = 5.56
or once every (300/5.56) = 54 working days (about
every 9 weeks).

Total Annual Variable Cost
TC = 5(90) + (40,000/90) = 450 + 444 = $894
© 2009 South-Western, a part of Cengage Learning
Slide 12
Sensitivity Analysis for the EOQ Model

Optimal Order Quantities for Several Costs
Possible
Inventory
Possible
Cost Per
Holding Cost
Order
18%
Optimal
Order
Projected Total
Annual Cost
Qnty. (Q*) Using Q* Using Q = 90
$75
91 units
$822
$822
18
85
97
875
877
22
75
83
908
912
22
85
88
967
967
© 2009 South-Western, a part of Cengage Learning
Slide 13
Example: Bart’s Barometer Business
We’ll now use a spreadsheet to implement
the Economic Order Quantity model. We’ll confirm
our earlier calculations for Bart’s problem and
perform some sensitivity analysis.
This spreadsheet can be modified to accommodate
other inventory models presented in this chapter.
© 2009 South-Western, a part of Cengage Learning
Slide 14
Example: Bart’s Barometer Business

Partial Spreadsheet with Input Data
A
B
1 BART'S ECONOMIC ORDER QUANTITY
2
3
Annual Demand
500
4
Ordering Cost
$80.00
5
Annual Holding Rate %
20
6
Cost Per Unit
$50.00
7
Working Days Per Year
300
8
Lead Time (Days)
60
© 2009 South-Western, a part of Cengage Learning
Slide 15
Example: Bart’s Barometer Business

Partial Spreadsheet Showing Formulas for Output
A
10 Econ. Order Qnty.
11 Request. Order Qnty
12 % Change from EOQ
13
14 Annual Holding Cost
15 Annual Order. Cost
16 Tot. Ann. Cost (TAC)
17 % Over Minimum TAC
18
19 Max. Inventory Level
20 Avg. Inventory Level
21 Reorder Point
22
23 No. of Orders/Year
24 Cycle Time (Days)
B
C
=SQRT(2*B3*B4/(B5*B6/100))
=(C11/B10-1)*100
=B5/100*B6*B10/2
=B4*B3/B10
=B14+B15
=B5/100*B6*C11/2
=B4*B3/C11
=C14+C15
=(C16/B16-1)*100
=B10
=B10/2
=B3/B7*B8
=C11
=C11/2
=B3/B7*B8
=B3/B10
=B10/B3*B7
=B3/C11
=C11/B3*B7
© 2009 South-Western, a part of Cengage Learning
Slide 16
Example: Bart’s Barometer Business

Partial Spreadsheet Showing Output
A
10 Econ. Order Qnty.
11 Request. Order Qnty.
12 % Change from EOQ
13
14 Annual Holding Cost
15 Annual Order. Cost
16 Tot. Ann. Cost (TAC)
17 % Over Minimum TAC
18
19 Max. Inventory Level
20 Avg. Inventory Level
21 Reorder Point
22
23 No. of Orders/Year
24 Cycle Time (Days)
B
C
89.44
75.00
-16.15
$447.21
$447.21
$894.43
$375.00
$533.33
$908.33
1.55
89.44
44.72
100
75
37.5
100
5.59
53.67
6.67
45.00
© 2009 South-Western, a part of Cengage Learning
Slide 17
Example: Bart’s Barometer Business

Summary of Spreadsheet Results
• A 16.15% negative deviation from the EOQ
resulted in only a 1.55% increase in the Total
Annual Cost.
• Annual Holding Cost and Annual Ordering Cost
are no longer equal.
• The Reorder Point is not affected, in this model, by
a change in the Order Quantity.
© 2009 South-Western, a part of Cengage Learning
Slide 18
Economic Production Lot Size





The economic production lot size model is a variation
of the basic EOQ model.
A replenishment order is not received in one lump
sum as it is in the basic EOQ model.
Inventory is replenished gradually as the order is
produced (which requires the production rate to be
greater than the demand rate).
This model's variable costs are annual holding cost
and annual set-up cost (equivalent to ordering cost).
For the optimal lot size, annual holding and set-up
costs are equal.
© 2009 South-Western, a part of Cengage Learning
Slide 19
Economic Production Lot Size

Assumptions
• Demand occurs at a constant rate of D items per
year or d items per day.
• Production rate is P items per year or p items per
day (and P > D, p > d ).
• Set-up cost: $Co per run.
• Holding cost: $Ch per item in inventory per year.
• Purchase cost per unit is constant (no quantity
discount).
• Set-up time (lead time) is constant.
• Planned shortages are not permitted.
© 2009 South-Western, a part of Cengage Learning
Slide 20
Economic Production Lot Size

Formulas
• Optimal production lot-size:
Q*=
2DCo /[(1-D/P )Ch]
• Number of production runs per year:
• Time between set-ups (cycle time):
• Total annual cost:
D/Q *
Q */D years
[Ch(Q*/2)(1-D/P )] + [Co/(D/Q *)]
(holding + ordering)
© 2009 South-Western, a part of Cengage Learning
Slide 21
Example: Beauty Bar Soap

Economic Production Lot Size Model
Beauty Bar Soap is produced on a production line
that has an annual capacity of 60,000 cases. The annual
demand is estimated at 26,000 cases, with the demand
rate essentially constant throughout the year. The
cleaning, preparation, and setup of the production line
cost approximately $135. The manufacturing cost per
case is $4.50, and the annual holding cost is figured at a
24% rate. Other relevant data include a five-day lead
time to schedule and set up a production run and 250
working days per year.
© 2009 South-Western, a part of Cengage Learning
Slide 22
Example: Beauty Bar Soap

Total Annual Variable Cost Model
This is an economic production lot size problem with
D = 26,000, P = 60,000, Ch = 1.08, Co = 135
TC = (Holding Costs) + (Set-Up Costs)
= [Ch(Q/2)(1 - D/P )] + [Co(D/Q)]
= [1.08(Q/2)(1 – 26,000/60,000)] + [135(26,000/Q)]
= .306Q + 3,510,000/Q
© 2009 South-Western, a part of Cengage Learning
Slide 23
Example: Beauty Bar Soap

Optimal Production Lot Size
Q*=

2DCo/[(1 -D/P )Ch]
=
2(26,000)(135) /[(1.08)(1 – 26,000/60,000)]
=
3,387
Number of Production Runs (Cycles) Per Year
D/Q * = 26,000/3,387
= 7.6764
times per year
© 2009 South-Western, a part of Cengage Learning
Slide 24
Example: Beauty Bar Soap

Total Annual Variable Cost
Optimal TC = .306(3,387) + 3,510,000/3,387
= 1,036.42 + 1,036.32
= $2,073
© 2009 South-Western, a part of Cengage Learning
Slide 25
Example: Beauty Bar Soap

Idle Time Between Production Runs
There are 7.6764 cycles per year.
Thus, each cycle lasts (250/7.6764) = 32.567 days.
The time to produce 3,387 per run = (3,387/60,000)250
= 14.1125 days.
Thus, the production line is idle for:
32.567 – 14.1125 =
18.4545
days between runs.
The production line is utilized:
14.1125/32.567(100) = 43.33%
© 2009 South-Western, a part of Cengage Learning
Slide 26
Example: Beauty Bar Soap


Maximum Inventory
Maximum inventory = (1-D/P )Q *
= (1-26,000/60,000)3,387  1,919.3
Machine Utilization
Machine is producing D/P = 26,000/60,000
= .4333 of the time.
© 2009 South-Western, a part of Cengage Learning
Slide 27
EOQ with Planned Shortages




With the EOQ with planned shortages model, a
replenishment order does not arrive at or before the
inventory position drops to zero.
Shortages occur until a predetermined backorder
quantity is reached, at which time the replenishment
order arrives.
The variable costs in this model are annual holding,
backorder, and ordering.
For the optimal order and backorder quantity
combination, the sum of the annual holding and
backordering costs equals the annual ordering cost.
© 2009 South-Western, a part of Cengage Learning
Slide 28
EOQ with Planned Shortages

Assumptions
• Demand occurs at a constant rate of D items/year.
• Ordering cost: $Co per order.
• Holding cost: $Ch per item in inventory per year.
• Backorder cost: $Cb per item backordered per year.
• Purchase cost per unit is constant (no qnty. discount).
• Set-up time (lead time) is constant.
• Planned shortages are permitted (backordered
demand units are withdrawn from a replenishment
order when it is delivered).
© 2009 South-Western, a part of Cengage Learning
Slide 29
EOQ with Planned Shortages

Formulas
• Optimal order quantity:
Q * = 2DCo/Ch (Ch+Cb )/Cb
• Maximum number of backorders:
S * = Q *(Ch/(Ch+Cb))
• Number of orders per year: D/Q *
• Time between orders (cycle time): Q */D years
• Total annual cost:
[Ch(Q *-S *)2/2Q *] + [Co(D/Q *)] + [S *2Cb/2Q *]
(holding + ordering + backordering)
© 2009 South-Western, a part of Cengage Learning
Slide 30
Example: Higley Radio Components Co.

EOQ with Planned Shortages Model
Higley has a product for which the assumptions of
the inventory model with backorder are valid. Demand
for the product is 2,000 units per year. The inventory
holding cost rate is 20% per year. The product costs
Higley $50 to purchase. The ordering cost is $35 per
order. The annual backorder cost is estimated to be $30
per unit per year.
© 2009 South-Western, a part of Cengage Learning
Slide 31
Example: Higley Radio Components Co.

Optimal Order Policy
D = 2,000; Co = $25; Ch = .20(50) = $10; Cb = $30
Q*=
=
2DCo/Ch
(Ch + Cb)/Cb
2(2000)(25)/10 x
(10+30)/30
= 115.47  115
S * = Q *(Ch/(Ch+Cb))
= 115(10/(10+30)) = 28.87
© 2009 South-Western, a part of Cengage Learning
Slide 32
Example: Higley Radio Components Co.

Maximum Inventory
Q – S = 115.47 – 28.87 = 86.6 units

Cycle Time
T = Q/D(250) = 115.47/2000(250) = 14.43 working days
© 2009 South-Western, a part of Cengage Learning
Slide 33
Example: Higley Radio Components Co.

Total Annual Cost
Holding Cost:
Ch(Q –S)2/(2Q) = 10(115.47 – 28.87)2/(2(115.47))
= $324.74
Ordering Cost:
Co(D/Q) = 25(2000/115.47) = $433.01
Backorder Cost:
Cb(S2/(2Q) = 30(28.87)2/(2(115.47)) = $108.27
Total Cost:
324.74 + 433.01 + 108.27 = $866.02
© 2009 South-Western, a part of Cengage Learning
Slide 34
Example: Higley Radio Components Co.

Stockout: When and How Long
Question:
How many days after receiving an order does
Higley run out of inventory? How long is Higley
without inventory per cycle?
© 2009 South-Western, a part of Cengage Learning
Slide 35
Example: Higley Radio Components Co.

Stockout: When and How Long
Answer
Inventory exists for Cb/(Cb+Ch) = 30/(30+10) = .75
of the order cycle. (Note, (Q *-S *)/Q * = .75 also,
before Q * and S * are rounded.)
An order cycle is Q */D = .057735 years = 14.434
days. Thus, Higley runs out of inventory .75(14.434)
= 10.823 days after receiving an order.
Higley is out of stock for approximately 14.434 –
10.823 = 3.611 days.
© 2009 South-Western, a part of Cengage Learning
Slide 36
EOQ with Quantity Discounts



The EOQ with quantity discounts model is applicable
where a supplier offers a lower purchase cost when an
item is ordered in larger quantities.
This model's variable costs are annual holding,
ordering and purchase costs.
For the optimal order quantity, the annual holding
and ordering costs are not necessarily equal.
© 2009 South-Western, a part of Cengage Learning
Slide 37
EOQ with Quantity Discounts

Assumptions
• Demand occurs at a constant rate of D items/year.
• Ordering Cost is $Co per order.
• Holding Cost is $Ch = $CiI per item in inventory per
year (note holding cost is based on the cost of the
item, Ci).
• Purchase Cost is $C1 per item if the quantity
ordered is between 0 and x1, $C2 if the order
quantity is between x1 and x2 , etc.
• Delivery time (lead time) is constant.
• Planned shortages are not permitted.
© 2009 South-Western, a part of Cengage Learning
Slide 38
EOQ with Quantity Discounts

Formulas
• Optimal order quantity:
the procedure for
determining Q * will be demonstrated
• Number of orders per year: D/Q *
• Time between orders (cycle time): Q */D years
• Total annual cost: [Ch(Q*/2)] + [Co(D/Q *)] + DC
(holding + ordering + purchase)
© 2009 South-Western, a part of Cengage Learning
Slide 39
Example: Nick's Camera Shop

EOQ with Quantity Discounts Model
Nick's Camera Shop carries Zodiac instant print
film. The film normally costs Nick $3.20 per roll, and
he sells it for $5.25. Zodiac film has a shelf life of 18
months. Nick's average sales are 21 rolls per week.
His annual inventory holding cost rate is 25% and it
costs Nick $20 to place an order with Zodiac.
© 2009 South-Western, a part of Cengage Learning
Slide 40
Example: Nick's Camera Shop

EOQ with Quantity Discounts Model
If Zodiac offers a 7% discount on orders of 400
rolls or more, a 10% discount for 900 rolls or more,
and a 15% discount for 2000 rolls or more, determine
Nick's optimal order quantity.
-------------------D = 21(52) = 1092; Ch = .25(Ci); Co = 20
© 2009 South-Western, a part of Cengage Learning
Slide 41
Example: Nick's Camera Shop

Unit-Prices’ Economical Order Quantities
• For C4 = .85(3.20) = $2.72
To receive a 15% discount Nick must order
at least 2,000 rolls. Unfortunately, the film's shelf
life is 18 months. The demand in 18 months (78
weeks) is 78 x 21 = 1638 rolls of film.
If he ordered 2,000 rolls he would have to
scrap 372 of them. This would cost more than the
15% discount would save.
© 2009 South-Western, a part of Cengage Learning
Slide 42
Example: Nick's Camera Shop

Unit-Prices’ Economical Order Quantities
• For C3 = .90(3.20) = $2.88
Q3* =
2DCo/Ch =
2(1092)(20)/[.25(2.88)] = 246.31
(not feasible)
The most economical, feasible quantity for C3 is 900.
• For C2 = .93(3.20) = $2.976
Q2* =
2DCo/Ch = 2(1092)(20)/[.25(2.976)] = 242.30
(not feasible)
The most economical, feasible quantity for C2 is 400.
© 2009 South-Western, a part of Cengage Learning
Slide 43
Example: Nick's Camera Shop

Unit-Prices’ Economical Order Quantities
• For C1 = 1.00(3.20) = $3.20
Q1* =
2DCo/Ch = 2(1092)(20)/.25(3.20) = 233.67
(feasible)
When we reach a computed Q that is feasible we
stop computing Q's. (In this problem we have no more
to compute anyway.)
© 2009 South-Western, a part of Cengage Learning
Slide 44
Example: Nick's Camera Shop

Total Cost Comparison
Compute the total cost for the most economical,
feasible order quantity in each price category for which
a Q * was computed.
TCi = (Ch)(Qi*/2) + (Co)(D/Qi*) + DCi
TC3 = (.72)(900/2) + (20)(1092/900) + (1092)(2.88) = 3,493
TC2 = (.744)(400/2) + (20)(1092/400) + (1092)(2.976) = 3,453
TC1 = (.80)(234/2) + (20)(1092/234) + (1092)(3.20) = 3,681
Comparing the total costs for order quantities of 234,
400 and 900, the lowest total annual cost is $3,453. Nick
should order 400 rolls at a time.
© 2009 South-Western, a part of Cengage Learning
Slide 45
Chapter 14
Inventory Models: Probabilistic Demand



Single-Period Inventory Model with Probabilistic
Demand
Order-Quantity, Reorder-Point Model with
Probabilistic Demand
Periodic-Review Model with Probabilistic Demand
© 2009 South-Western, a part of Cengage Learning
Slide 46
Probabilistic Models



In many cases demand (or some other factor) is not
known with a high degree of certainty and a
probabilistic inventory model should actually be used.
These models tend to be more complex than
deterministic models.
The probabilistic models covered in this chapter are:
• single-period order quantity
• reorder-point quantity
• periodic-review order quantity
© 2009 South-Western, a part of Cengage Learning
Slide 47
Single-Period Order Quantity



A single-period order quantity model (sometimes
called the newsboy problem) deals with a situation in
which only one order is placed for the item and the
demand is probabilistic.
If the period's demand exceeds the order quantity, the
demand is not backordered and revenue (profit) will
be lost.
If demand is less than the order quantity, the surplus
stock is sold at the end of the period (usually for less
than the original purchase price).
© 2009 South-Western, a part of Cengage Learning
Slide 48
Single-Period Order Quantity

Assumptions
• Period demand follows a known probability
distribution:
• normal: mean is µ, standard deviation is 
• uniform: minimum is a, maximum is b
• Cost of overestimating demand: $co
• Cost of underestimating demand: $cu
• Shortages are not backordered.
• Period-end stock is sold for salvage (not held in
inventory).
© 2009 South-Western, a part of Cengage Learning
Slide 49
Single-Period Order Quantity

Formulas
Optimal probability of no shortage:
P(demand < Q *) = cu/(cu+co)
Optimal probability of shortage:
P(demand > Q *) = 1 - cu/(cu+co)
Optimal order quantity, based on demand distribution:
normal:
Q * = µ + z
uniform:
Q * = a + P(demand < Q *)(b-a)
© 2009 South-Western, a part of Cengage Learning
Slide 50
Example: McHardee Press

Single-Period Order Quantity
McHardee Press publishes the Fast Food Menu
Book and wishes to determine how many copies to
print. There is a fixed cost of $5,000 to produce the
book and the incremental profit per copy is $0.45. Any
unsold copies of the the book can be sold at salvage at
a $.55 loss.
© 2009 South-Western, a part of Cengage Learning
Slide 51
Example: McHardee Press

Single-Period Order Quantity
Sales for this edition are estimated to be normally
distributed. The most likely sales volume is 12,000
copies and they believe there is a 5% chance that sales
will exceed 20,000.
How many copies should be printed?
© 2009 South-Western, a part of Cengage Learning
Slide 52
Example: McHardee Press

Single-Period Order Quantity
m = 12,000. To find  note that z = 1.65
corresponds to a 5% tail probability. Therefore,
(20,000 - 12,000) = 1.65 or  = 4848
Using incremental analysis with Co = .55 and Cu = .45,
(Cu/(Cu+Co)) = .45/(.45+.55) = .45
Find Q * such that P(D < Q *) = .45. The probability
of 0.45 corresponds to z = -.12. Thus,
Q * = 12,000 - .12(4848) =
© 2009 South-Western, a part of Cengage Learning
11,418 books
Slide 53
Example: McHardee Press

Single-Period Order Quantity (revised)
If any unsold copies can be sold at salvage at a
$.65 loss, how many copies should be printed?
Co = .65, (Cu/(Cu + Co)) = .45/(.45 + .65) = .4091
Find Q * such that P(D < Q *) = .4091. z = -.23
gives this probability. Thus,
Q * = 12,000 - .23(4848) = 10,885 books
However, since this is less than the breakeven
volume of 11,111 books (= 5000/.45), no copies
should be printed because if the company produced
only 10,885 copies it will not recoup its $5,000 fixed
cost.
© 2009 South-Western, a part of Cengage Learning
Slide 54
Reorder Point Quantity




A firm's inventory position consists of the on-hand
inventory plus on-order inventory (all amounts
previously ordered but not yet received).
An inventory item is reordered when the item's
inventory position reaches a predetermined value,
referred to as the reorder point.
The reorder point represents the quantity available to
meet demand during lead time.
Lead time is the time span starting when the
replenishment order is placed and ending when the
order arrives.
© 2009 South-Western, a part of Cengage Learning
Slide 55
Reorder Point Quantity



Under deterministic conditions, when both demand
and lead time are constant, the reorder point
associated with EOQ-based models is set equal to lead
time demand.
Under probabilistic conditions, when demand and/or
lead time varies, the reorder point often includes
safety stock.
Safety stock is the amount by which the reorder point
exceeds the expected (average) lead time demand.
© 2009 South-Western, a part of Cengage Learning
Slide 56
Safety Stock and Service Level




The amount of safety stock in a reorder point
determines the chance of a stock-out during lead time.
The complement of this chance is called the service
level.
Service level, in this context, is defined as the
probability of not incurring a stock-out during any
one lead time.
Service level, in this context, also is the long-run
proportion of lead times in which no stock-outs occur.
© 2009 South-Western, a part of Cengage Learning
Slide 57
Reorder Point

Assumptions
• Lead-time demand is normally distributed
with mean µ and standard deviation .
• Approximate optimal order quantity: EOQ
• Service level is defined in terms of the probability of
no stock-outs during lead time and is reflected in z.
• Shortages are not backordered.
• Inventory position is reviewed continuously.
© 2009 South-Western, a part of Cengage Learning
Slide 58
Reorder Point

Formulas
Reorder point:
r = µ + z
Safety stock:
z
Average inventory:
Q*/2 + z
Total annual cost:
[Ch(Q */2)] + [Ch z] + [Co(D/Q *)]
(hold.(normal) + hold.(safety)
+ ordering)
© 2009 South-Western, a part of Cengage Learning
Slide 59
Example: Robert’s Drug

Reorder Point Model
Robert's Drugs is a drug wholesaler supplying
55 independent drug stores. Roberts wishes to
determine an optimal inventory policy for Comfort
brand headache remedy. Sales of Comfort are relatively
constant as the past 10 weeks of data (on next slide)
indicate.
© 2009 South-Western, a part of Cengage Learning
Slide 60
Example: Robert’s Drug

Reorder Point Model
Week
1
2
3
4
5
Sales (cases)
110
115
125
120
125
Week
6
7
8
9
10
© 2009 South-Western, a part of Cengage Learning
Sales (cases)
120
130
115
110
130
Slide 61
Example: Robert’s Drug
Each case of Comfort costs Roberts $10 and
Roberts uses a 14% annual holding cost rate for its
inventory. The cost to prepare a purchase order for
Comfort is $12. What is Roberts’ optimal order
quantity?
© 2009 South-Western, a part of Cengage Learning
Slide 62
Example: Robert’s Drug

Optimal Order Quantity
The average weekly sales over the 10 week period
is 120 cases. Hence D = 120 X 52 = 6,240 cases per
year;
Ch = (.14)(10) = 1.40; Co = 12.
Q*  2DCo /C h  (2(6240)(12))/1.40  327
© 2009 South-Western, a part of Cengage Learning
Slide 63
Example: Robert’s Drug
The lead time for a delivery of Comfort has
averaged four working days. Lead time has therefore
been estimated as having a normal distribution with a
mean of 80 cases and a standard deviation of 10 cases.
Roberts wants at most a 2% probability of selling out
of Comfort during this lead time. What reorder point
should Roberts use?
© 2009 South-Western, a part of Cengage Learning
Slide 64
Example: Robert’s Drug

Optimal Reorder Point
Lead time demand is normally distributed with
m = 80,  = 10.
Since Roberts wants at most a 2% probability of
selling out of Comfort, the corresponding z value is
2.06. That is, P (z > 2.06) = .0197 (about .02).
Hence Roberts should reorder Comfort when
supply reaches r = m + z = 80 + 2.06(10) = 101 cases.
The safety stock is z = 21 cases.
© 2009 South-Western, a part of Cengage Learning
Slide 65
Example: Robert’s Drug

Total Annual Inventory Cost
Ordering: Co(D/Q *) = 12(6240/327)
= $229
Holding-Normal: Ch(Q*/2) = 1.40(327/2) = 229
Holding-Safety Stock: Ch(21) = (1.40)(21) =
29
TOTAL = $487
© 2009 South-Western, a part of Cengage Learning
Slide 66
Periodic Review System



A periodic review system is one in which the
inventory level is checked and reordering is done only
at specified points in time (at fixed intervals usually).
Assuming the demand rate varies, the order quantity
will vary from one review period to another.
At the time the order quantity is being decided, the
concern is that the on-hand inventory and the
quantity being ordered is enough to satisfy demand
from the time the order is placed until the next order
is received (not placed).
© 2009 South-Western, a part of Cengage Learning
Slide 67
Periodic Review Order Quantity

Assumptions
• Inventory position is reviewed at constant intervals.
• Demand during review period plus lead time period
is normally distributed with mean µ and standard
deviation .
• Service level is defined in terms of the probability of
no stockouts during a review period plus lead time
period and is reflected in z.
• On-hand inventory at ordering time: H
• Shortages are not backordered.
• Lead time is less than the review period length.
© 2009 South-Western, a part of Cengage Learning
Slide 68
Periodic Review Order Quantity

Formulas
Replenishment level:
M = µ + z
Order quantity:
Q=M-H
© 2009 South-Western, a part of Cengage Learning
Slide 69
Example: Ace Brush

Periodic Review Order Quantity Model
Joe Walsh is a salesman for the Ace Brush
Company. Every three weeks he contacts Dollar
Department Store so that they may place an order to
replenish their stock. Weekly demand for Ace
brushes at Dollar approximately follows a normal
distribution with a mean of 60 brushes and a
standard deviation of 9 brushes.
© 2009 South-Western, a part of Cengage Learning
Slide 70
Example: Ace Brush

Periodic Review Order Quantity Model
Once Joe submits an order, the lead time until
Dollar receives the brushes is one week. Dollar
would like at most a 2% chance of running out of
stock during any replenishment period. If Dollar
has 75 brushes in stock when Joe contacts them,
how many should they order?
© 2009 South-Western, a part of Cengage Learning
Slide 71
Example: Ace Brush

Demand During Uncertainty Period
The review period plus the following lead time
totals 4 weeks. This is the amount of time that will
elapse before the next shipment of brushes will arrive.
Weekly demand is normally distributed with:
Mean weekly demand, µ
= 60
Weekly standard deviation,  = 9
Weekly variance,  2
= 81
Demand for 4 weeks is normally distributed with:
Mean demand over 4 weeks, µ
= 4 x 60 = 240
Variance of demand over 4 weeks,  2 = 4 x 81 = 324
Standard deviation over 4 weeks,  = (324)1/2 = 18
© 2009 South-Western, a part of Cengage Learning
Slide 72
Example: Ace Brush

Replenishment Level
M = µ + z where z is determined by the desired
stock-out probability. For a 2% stock-out probability
(2% tail area), z = 2.05. Thus,
M = 240 + 2.05(18) = 277 brushes
As the store currently has 75 brushes in stock,
Dollar should order:
277 - 75 = 202 brushes
The safety stock is:
z = (2.05)(18) = 37 brushes
© 2009 South-Western, a part of Cengage Learning
Slide 73