Inventory Models
1
Overview of Inventory Issues
• Proper control of inventory is crucial to the success
of an enterprise.
• Typical inventory problems include:
– Basic inventory
– Quantity discount
– Production lot size
– Planned shortage
– Periodic review
– Single period
• Inventory models are often used to develop an
optimal inventory policy, consisting of:
– An order quantity, denoted Q.
– A reorder point, denoted R.
2
Type of Costs in Inventory Models
• Inventory analyses can be thought of as costcontrol techniques.
• Categories of costs in inventory models:
– Holding (carrying costs)
– Order/ Setup costs
– Customer satisfaction costs
– Procurement/Manufacturing costs
3
Type of Costs in Inventory Models
• Holding Costs (Carrying costs):
These costs depend on the order
size
–
–
–
–
–
–
–
–
Cost of capital
Storage space rental cost
Costs of utilities
Labor
Insurance
Security
Theft and breakage
Deterioration or Obsolescence
Ch = H * C
Ch = Annual holding cost per unit
in inventory
H = Annual holding cost rate
C = Unit cost of an item
4
Type of Costs in Inventory Models
• Order/Setup Costs
These costs are independent of the order
size.
– Order costs are incurred when purchasing a good
from a supplier. They include costs such as
•
•
•
•
Telephone
Order checking
Labor
Transportation
Co = Order cost
or setup cost
– Setup costs are incurred when producing goods
for sale to others. They can include costs of
• Cleaning machines
• Calibrating equipment
• Training staff
5
Type of Costs in Inventory Models
• Customer Satisfaction Costs
– Measure the degree to which a
customer is satisfied.
Cb = Fixed administrative costs
of an out of stock item
– Unsatisfied customers may:
• Switch to the competition (lost
sales).
• Wait until an order is supplied.
– When customers are willing to
wait there are two types of
costs incurred:
($/stockout unit).
Cs = Annualized cost of a
customer awaiting an out
of stock item
($/stockout unit per
year).
6
Type of Costs in Inventory Models
• Procurement/Manufacturing
Cost
– Represents the unit purchase
cost (including transportation) in
case of a purchase.
– Unit production cost in case of inhouse manufacturing.
C = Unit purchase or
manufacturing cost.
7
Demand in Inventory Models
• Demand is a key component affecting an inventory
policy.
• Projected demand patterns determine how an
inventory problem is modeled.
• Typical demand patterns are:
– Constant over time (deterministic inventory models)
– Changing but known over time (dynamic models)
– Variable (randomly) over time (probabilistic models)
D = Demand rate (usually per year)
8
Inventory Classifications
Inventory can be classified in various ways:
By
ByProcess
Process
Raw
Rawmaterials
materials
Work
Workininprogress
progress
Finished
Finishedgoods
goods
By
ByImportance
Importance
Used typically by
accountants at
manufacturing firms.
Enables management to
track the production
process.
Items are classified by
their relative
importance
in terms of the firm’s
capital needs.
A,
A,B,
B,CC
By
ByShelf
ShelfLife
Life
Perishable
Perishable
Nonperishable
Nonperishable
Management of items
with short shelf life
and long shelf life is
very different
9
Review Systems
• Two types of review systems are used:
– Continuous review systems.
• The system is continuously monitored.
• A new order is placed when the inventory reaches a
critical point.
– Periodic review systems.
• The inventory position is investigated on a regular basis.
• An order is placed only at these times.
10
Review Systems
continuous review system
– Continuous review systems.
• The system is continuously monitored.
• A new order is placed when the inventory reaches a
critical point.
– EOQ
11
ALLEN APPLIANCE COMPANY (AAC)
• AAC wholesales small appliances.
• AAC currently orders 600 units of the Citron
brand juicer each time inventory drops to 205
units.
• Management wishes to determine an optimal
ordering policy for the Citron brand juicer
12
ALLEN APPLIANCE COMPANY (AAC)
• Data
– Co = $12 ($8 for placing an order) + (20 min. to check)($12
per hr)
– Ch = $1.40 [HC = (14%)($10)]
– C = $10.
– H = 14% (10% ann. interest rate) + (4% miscellaneous)
– D = demand information of the last 10 weeks was collected:
Sales of Juicers over the last 10 weeks
Week
1
2
3
4
Sales
105
115
125
120
Week
6
7
8
9
Sales
120
135
115
110
5
125
10
130
13
ALLEN APPLIANCE COMPANY (AAC)
• Data
– The constant demand rate seems to be a good
assumption.
– Annual demand = (120/week)(52weeks) = 6240
juicers.
14
AAC – Solution:
EOQ and Total Variable Cost
• Current ordering policy calls for Q = 600 juicers.
TV( 600) = (600 / 2)($1.40) + (6240 / 600)($12) = $544.80
EOQ = sqrt((2*6240*12)/1,4)
= 327,065
= 327
Savings
of 16%
Savings
of 16%
TV(327) = (327 / 2)($1.40) + (6240 / 327) ( $12) = $457.89
15
AAC – Solution:
Reorder Point and Total Cost
• Under the current ordering policy AAC holds 13 units
safety stock (how come? observe):
•
AAC is open 5 day a week.
–
–
–
–
–
The average daily demand = 120/week/5 = 24 juicers.
Lead time is 8 days. Lead time demand is (8)(24) = 192 juicers.
Reorder point without Safety stock = LD = 192.
Current policy: R = 205.
Safety stock = 205 – 192 = 13.
• For safety stock of 13 juicers the total cost is
TC(327) = 457.89 + 6240($10) + (13)($1.40) = $62,876.09
TV(327) + Procurement
cost
+ Safety stock
holding cost
16
AAC – Solution:
Sensitivity of the EOQ Results
• Changing the order size
– Suppose juicers must be ordered in increments of 100 (order 300 or 400)
– AAC will order Q = 300 juicers in each order.
– There will be a total variable cost increase of $1.71.
– This is less than 0.5% increase in variable costs.
• Changes in input parameters
–
–
–
–
Suppose there is a 20% increase in demand. D=7500 juicers.
The new optimal order quantity is Q* = 359.
Only 0.4%
The new variable total cost = TV(359) = $502
increase
If AAC still orders Q = 327, its total variable costs becomes
TV(327) = (327/2)($1.40) + (7500/327)($12) = $504.13
17
AAC – Solution:
Cycle Time
• For an order size of 327 juicers we have:
– T = (327/ 6240) = 0.0524 year.
= 0.0524(52)(5) = 14 days.

working days per week
– This is useful information because:
• Shelf life may be a problem.
• Coordinating orders with other items might be
desirable.
18
AAC – Excel Spreadsheet
=1/E11
Copy to cell H12
=$B$15*$B$10+$B$16INT(($B$15*$B$10+$B$16)/E10)*E10
Copy to cell H13
=(E10/2)*$B$13+($B$10/E10)*$B$14
Copy to cell H14
=SQRT(2*$B$10*$B
$14/$B$13)
=E10/B10
Copy to cell H11
=$B$10*$B$11+E14+$B$13*B16
Copy to Cell H15
19
Service Levels and
Safety Stocks
20
Determining Safety Stock Levels
• Businesses incorporate safety stock
requirements when determining reorder
points.
• A possible approach to determining safety stock
levels is by specifying desired service level .
21
Two Types of Service Level
Service levels can be viewed in two
ways.
• The cycle service level
• The unit service level
– The probability of not
incurring a stockout during
an inventory cycle.
– The percentage of demands
that are filled without
incurring any delay.
– Applied when the likelihood
of a stockout, and not its
magnitude, is important for
the firm.
– Applied when the
percentage of unsatisfied
demand should be under
control.
22
The Cycle Service Level Approach
• In many cases short run demand is variable
even though long run demand is assumed
constant.
• Therefore, stockout events during lead time
may occur unexpectedly in each cycle.
• Stockouts occur only if demand during lead
time is greater than the reorder point.
23
The Cycle Service Level Approach
• To determine the reorder point we need to know:
– The lead time demand distribution.
– The required service level.
• In many cases lead time demand is approximately
normally distributed. For the normal distribution
case the reorder point is calculated by
R = mL + zasL
1 –a = service level
24
The Cycle Service Level Approach
Service level =
P(DL<R) =
1– a
P(DL>R) = a
m=192
R
P(DL> R) = P(Z > (R – mL)/sL) = a. Since
P(Z > Za) = a, we have Za = (R – mL)/sL,
which gives…
R = mL + zasL
25
AAC Cycle Service Level Approach
• Assume that lead time demand is normally
distributed.
• Estimation of the normal distribution
parameters:
– Estimation of the mean weekly demand =
ten weeks average demand = 120 juicers per week.
– Estimation of the variance of the weekly demand =
Sample variance = 83.33 juicers2.
26
AAC Cycle Service Level Approach
• To find mLand sL the parameters m (per week)
and s (per week) must be adjusted since the
lead time is longer than one week.
– Lead time is 8 days =(8/5) weeks = 1.6 weeks.
• Estimates for the lead time mean demand and
variance of demand
mL  (1.6)(120) = 192;
133.33
s2L  (1.6)(83.33) =
27
AAC Service Level for a given Reorder Point
• Let us use the current reorder point of 205 juicers.
205 =
192 + z (11.55)

z = 1.13
133.33
• From the normal distribution table we have that a
reorder
point of 205 juicers results in an 87% cycle service level.
28
AAC –
Reorder Point for a given Service Level
• Management wants to improve the cycle
service level to 99%.
• The z value corresponding to 1% right hand
tail is 2.33.
R = 192 + 2.33(11.55) = 219 juicers.
29
AAC –
Acceptable Number of Stockouts per Year
• AAC is willing to run out of stock an average
of at most one cycle per year with an order
quantity of 327 juicers.
• What is the equivalent service level for this
strategy?
30
AAC –
Acceptable Number of Stockouts per Year
• There will be an average of
6240/327 = 19.08 lead times per year.
• The likelihood of stockouts = 1/19 = 0.0524.
• This translates into a service level of 94.76%
31
The Unit Service Level Approach
• When lead time demand follows a normal
distribution
service level can be calculated as follows:
– Determine the value of z that satisfy the equation
L(z) = aQ* / sL
– Solve for R using the equation
R = mL + zsL
32
AAC –
Cycle Service Level (Excel spreadsheet)
=NORMINV(B7,B5,B
6)
=NORMDIST(B8,B5,B6,TRUE)
33
EOQ Models with Quantity Discounts
• Quantity Discounts are Common Practice in Business
– By offering discounts buyers are encouraged to increase
their order sizes, thus reducing the seller’s holding costs.
– Quantity discounts reflect the savings inherent in large
orders.
– With quantity discounts sellers can reward their biggest
customers without violating the Robinson - Patman Act.
34
EOQ Models with Quantity Discounts
• Quantity Discount Schedule
– This is a list of per unit discounts and their corresponding
purchase volumes.
– Normally, the price per unit declines as the order
quantity increases.
– The order quantity at which the unit price changes is
called a break point.
– There are two main discount plans:
• All unit schedules - the price paid for all the units purchased is
based on the total purchase.
• Incremental schedules - The price discount is based only on the
additional units ordered beyond each break point.
35
All Units Discount Schedule
• To determine the optimal order quantity,
the total purchase cost must be included
TC(Q) = (Q/2)Ch + (D/Q)Co + DCi + ChSS
Ci represents the unit cost at the ith pricing level.
36
AAC - All Units Quantity Discounts
• AAC is offering all units quantity discounts to its
customers.
• Data
Quantity
Quantity Discount
Discount
Schedule
Schedule
1-299
1-299
300-599
300-599
600-999
600-999
1000-4999
1000-4999
 5000
5000
$10.00
$10.00
$9.75
$9.75
$9.40
$9.40
$9.50
$9.50
$9.00
$9.00
37
Should AAC increase its regular order of
327 juicers, to take advantage of the discount?
38
AAC – All units discount procedure
– Step 1: Find the optimal order Qi* for each discount
level “i”.
Use the formula Q *  ( 2DC o ) / C h
– Step 2: For each discount level “i” modify Q i* as
follows
• If Qi* is lower than the smallest quantity that qualifies for the i
th discount, increase Q * to that level.
i
• If Qi* is greater than the largest quantity that qualifies for the
ith discount, eliminate this level from further consideration.
– Step 3: Substitute the modified Q*i value in the total
cost formula TC(Q*i ).
– Step 4: Select the Q i * that minimizes TC(Q i*)
39
AAC – All units discount procedure
Step 1: Find the optimal order quantity Qi* for
each discount level “i” based on the EOQ formula
Lowest cost order size per discount level
Discount Qualifying
Price
level
order
per unit
Q*
0
1-299
10.00
327
1
300-599
9.75
331
2
600-999
9.50
336
3
1000-4999
9.40
337
 5000
4
9.00
345
40
AAC – All Units Discount Procedure
– Step 2 : Modify Q i *
$10/unit
$9.75/unit
$9.50
* *
Q1*Q2Q
3
336
1
Qualified
Qualified
Urder
Urder
1-299
1-299
300-599
300-599
600-999
600-999
1000-4999
1000-4999
 5000
5000
299 300 331
Modified
ModifiedQ*
Q*and
andtotal
totalCost
Cost
Price
Modified
Price
Modified
per
Q*
Q*
perUnit
Unit
Q*
Q*
10.00
300
****
10.00
300
****
9.75
331
331
9.75
331
331
9.50
336
600
9.50
336
600
9.40
337
1000
9.40
337
1000
9.00
345
5000
9.00
345
5000
599 600 999
Total
Total
Cost
Cost
****
****
61,292.13
61,292.13
59,803.80
59,803.80
59,388.88
59,388.88
59,324.98
59,324.98
41
AAC – All Units Discount Procedure
– Step 2 : Modify Q i *
$10/unit
1
Qualified
Qualified
Urder
Urder
1-299
1-299
300-599
300-599
600-999
600-999
1000-4999
1000-4999
5000
5000
*
Q
*
3
*Q3
*
*
*
Q
Q1 Q2Q
3 3
336
Q3* Q3* *
Q3 $9.50
Q3*
299 300 331
Modified
ModifiedQ*
Q*and
andtotal
totalCost
Cost
Price
Modified
Price
Modified
per
Q*
Q*
perUnit
Unit
Q*
Q*
10.00
300
****
10.00
300
****
9.75
331
331
9.75
331
331
9.50
336
600
9.50
336
600
9.40
337
1000
9.40
337
1000
9.00
345
5000
9.00
345
5000
600 999
Total
Total
Cost
Cost
****
****
61,292.13
61,292.13
59,803.80
59,803.80
59,388.88
59,388.88
59,324.98
59,324.98
42
AAC – All Units Discount Procedure
– Step 3: Substitute Q I * in the total cost
function Modified
ModifiedQ*
Q*and
andtotal
totalCost
Cost
Qualified
Qualified Price
Price
Urder
Urder per
perUnit
Unit
1-299
10.00
1-299
10.00
300-599
9.75
300-599
9.75
600-999
9.50
600-999
9.50
1000-4999
1000-4999 9.40
9.40
 5000
9.00
5000
9.00
– Step 4
Q*
Q*
300
300
331
331
336
336
337
337
345
345
Modified
Modified
Q*
Q*
****
****
331
331
600
600
1000
1000
5000
5000
Total
Total
Cost
Cost
****
****
61,292.13
61,292.13
59,803.80
59,803.80
59,388.88
59,388.88
59,324.98
59,324.98
AAC should order 5000 juicers
43
AAC – All Units Discount Excel
Worksheet
Calculation of Optimal Inventory Policy Under All-Units Quantity Discounts
INPUTS
Annual Demand, D =
Per Unit Cost, C =
Annual Holding Cost Rate, H =
Annual Holding Cost Per Unit, Ch =
Order Cost, Co =
Lead Time (in years), L =
Safety Stock, SS =
OPTIMAL
OUTPUTS
Values
6240.00
10.00
0.14
1.40
12.00
0.03077
13.00
Order quantity, Q* =
Cycle Time (in years), T =
# of Cycles Per Year, N =
Reorder Point, R =
Total Annual Cost, TC(Q*) =
Values
5000
0.801282051
1.248
205.0000
59341.36
DISCOUNTS
Level
0
1
2
3
4
5
6
7
8
Break point
1
300
600
1000
5000
Discount Price
10.00
9.75
9.50
9.40
9.00
Q*
327
331
336
337
345
TC(Q*)
62876.09
61309.88
59821.09
59405.99
59341.36
Modified Q*
327
331
600
1000
5000
44