INVENTORY MANAGEMENT
Operations Management
Dr. Ron Tibben-Lembke
Purposes of Inventory

Meet anticipated demand
 Demand
variability
 Supply variability

Decouple production & distribution
 permits



constant production quantities
Take advantage of quantity discounts
Hedge against price increases
Protect against shortages
2006
2007
13.81
1857
24.0%
446
801
58
1305
9.9
US Inventory, GDP ($B)
14,000
12,000
10,000
8,000
6,000
4,000
2,000
Business Inventories
US GDP
04
20
02
20
00
20
98
19
96
19
94
19
92
19
90
19
88
19
86
19
19
84
-
US Inventories as % of GDP
25.0%
% of GDP
20.0%
15.0%
10.0%
5.0%
Year
Source: CSCMP, Bureau of Economic Analysis
04
20
02
20
00
20
98
19
96
19
94
19
92
19
90
19
88
19
86
19
19
84
0.0%
Two Questions
Two main Inventory Questions:
 How much to buy?
 When is it time to buy?
Also:
Which products to buy?
From whom?
Types of Inventory






Raw Materials
Subcomponents
Work in progress (WIP)
Finished products
Defectives
Returns
Inventory Costs
What costs do we experience because we carry
inventory?
Inventory Costs
Costs associated with inventory:
 Cost of the products
 Cost of ordering
 Cost of hanging onto it
 Cost of having too much / disposal
 Cost of not having enough (shortage)
Shrinkage Costs

How much is stolen?
2% for discount, dept. stores, hardware, convenience,
sporting goods
 3% for toys & hobbies
 1.5% for all else


Where does the missing stuff go?
Employees: 44.5%
 Shoplifters: 32.7%
 Administrative / paperwork error: 17.5%
 Vendor fraud: 5.1%

Inventory Holding Costs
Category
Housing (building) cost
Material handling
Labor cost
Opportunity/investment
Pilferage/scrap/obsolescence
Total Holding Cost
% of Value
4%
3%
3%
9%
2%
21%
Inventory Models

Fixed order quantity models
 How
much always same, when changes
 Economic order quantity
 Production order quantity
 Quantity discount

Fixed order period models
 How
much changes, when always same
Economic Order Quantity
Assumptions
 Demand rate is known and constant
 No order lead time
 Shortages are not allowed
 Costs:
S - setup cost per order
 H - holding cost per unit time

EOQ
Inventory
Level
Q*
Optimal
Order
Quantity
Decrease Due to
Constant Demand
Time
EOQ
Inventory
Level
Q*
Optimal
Order
Quantity
Instantaneous
Receipt of Optimal
Order Quantity
Time
EOQ
Inventory
Level
Q*
Reorder
Point
(ROP)
Time
Lead Time
EOQ
Inventory
Level
Q*
Average
Inventory Q/2
Reorder
Point
(ROP)
Time
Lead Time
Total Costs






Average Inventory = Q/2
Annual Holding costs = H * Q/2
# Orders per year = D / Q
Annual Ordering Costs = S * D/Q
Cost of Goods = D * C
Annual Total Costs = Holding + Ordering + CoG
Q
D
TC (Q)  H *  S *  C * D
2
Q
How Much to Order?
Annual Cost
Holding Cost
= H * Q/2
Order Quantity
How Much to Order?
Annual Cost
Ordering Cost
= S * D/Q
Holding Cost
= H * Q/2
Order Quantity
How Much to Order?
Annual Cost
Total Cost
= Holding + Ordering
Order Quantity
How Much to Order?
Total Cost
= Holding + Ordering
Annual Cost
Optimal Q
Order Quantity
Optimal Quantity
Total Costs =
Q
D
H *  S * C*D
2
Q
Take derivative
with respect to Q =
H
D
S* 2 0
2
Q
Set equal
to zero
Solve for Q:
H DS
 2
2 Q
2 DS
Q 
H
2
2DS
Q
H
Adding Lead Time



Use same order size
2DS
Q
H
Order before inventory depleted
R = d * L where:
 d = average demand rate (per day)
L
= lead time (in days)
 both in same time period (wks, months, etc.)
A Question:

If the EOQ is based on so many horrible
assumptions that are never really true, why is it the
most commonly used ordering policy?
 Profit
function is very shallow
 Even if conditions don’t hold perfectly, profits are close
to optimal
 Estimated parameters will not throw you off very far
Quantity Discounts


How does this all change if price changes
depending on order size?
Holding cost as function of cost:
H

=I*C
Explicitly consider price:
2DS
Q
I C
Discount Example
D = 10,000
S = $20
PriceQuantity EOQ
c = 5.00
Q < 500
4.50
501-999
3.90
Q >= 1000
I = 20%
633
666
716
Discount Pricing
Total Cost
Price 1
Price 2
Price 3
X 633
X 666
X 716
500
1,000
Order Size
Discount Pricing
Total Cost
Price 1
Price 2
Price 3
X 633
X 666
X 716
500
1,000
Order Size
Discount Example
Order 666 at a time:
Hold 666/2 * 4.50 * 0.2= $299.70
Order 10,000/666 * 20 = $300.00
Mat’l 10,000*4.50 =
$45,000.00
45,599.70
Order 1,000 at a time:
Hold 1,000/2 * 3.90 * 0.2=$390.00
Order 10,000/1,000 * 20 = $200.00
Mat’l 10,000*3.90 =
$39,000.00 39,590.00
Discount Model
1. Compute EOQ for next cheapest price
2. Is EOQ feasible? (is EOQ in range?)
If EOQ is too small, use lowest possible Q to get
price.
3. Compute total cost for this quantity
4.
Repeat until EOQ is feasible or too big.
5.
Select quantity/price with lowest total cost.
INVENTORY MANAGEMENT
-- RANDOM DEMAND
Random Demand




Don’t know how many we will sell
Sales will differ by period
Average always remains the same
Standard deviation remains constant
Impact of Random Demand
How would our policies change?
 How would our order quantity change?
 How would our reorder point change?
Mac’s Decision





How many papers to buy?
Average = 90, st dev = 10
Cost = 0.20, Sales Price = 0.50
Salvage = 0.00
Cost of overestimating Demand, CO
 CO

= 0.20 - 0.00 = 0.20
Cost of Underestimating Demand, CU
 CU
= 0.50 - 0.20 = 0.30
Optimal Policy
G(x) = Probability demand <= x
Optimal quantity:
Cu
Pr(D  Q) 
C o  Cu
Mac: G(x) = 0.3 / (0.2 + 0.3) = 0.6
From standard normal table, z = 0.253
=Normsinv(0.6) = 0.253
Q* = avg + zs = 90+ 2.53*10 = 90 +2.53 = 93
Optimal Policy



If units are discrete, when in doubt, round up
If u units are on hand, order Q - u units
Model is called “newsboy problem,” newspaper
purchasing decision
 By
time realize sales are good, no time to order more
 By time realize sales are bad, too late, you’re stuck

Similar to the problem of # of Earth Day shirts to
make, lbs. of Valentine’s candy to buy, green beer,
Christmas trees, toys for Christmas, etc., etc.
Random Demand –
Fixed Order Quantity

If we want to satisfy all of the demand 95% of the
time, how many standard deviations above the
mean should the inventory level be?
Probabilistic Models
Safety stock = x m
From statistics, z 
Therefore, z =
xm
sL
Safety stock
sL
& Safety stock = zsL
From normal table z.95 = 1.65
Safety stock = zsL = 1.65*10 = 16.5
R = m + Safety Stock
=350+16.5 = 366.5 ≈ 367
Random Example

What should our reorder point be?
demand over the lead time is 50 units,
 with standard deviation of 20
 want to satisfy all demand 90% of the time
 (i.e., 90% chance we do not run out)




To satisfy 90% of the demand, z = 1.28
Safety stock = zσL= 1.28 * 20 = 25.6
R = 50 + 25.6 = 75.6
St Dev Over Lead Time

What if we only know the average daily demand,
and the standard deviation of daily demand?
 Lead
time = 4 days,
 daily demand = 10,
 standard deviation = 5,

What should our reorder point be, if z = 3?
St Dev Over LT

If the average each day is 10, and the lead time is
4 days, then the average demand over the lead
time must be 40.
d * L  10 * 4  40


What is the standard deviation of demand over the
lead time?
Std. Dev. ≠ 5 * 4
St Dev Over Lead Time

Standard deviation of demand =
s L  Ldays s day
 45  10
R  d * L  zs L  d * L  z Ldays s day

R = 40 + 3 * 10 = 70
Service Level Criteria

Type I: specify probability that you do not run out
during the lead time
 Probability

that 100% of customers go home happy
Type II: proportion of demands met from stock
 Percentage
that go home happy, on average
 Fill Rate: easier to observe, is commonly used
 G(z)= expected value of shortage, given z. Not
frequently listed in tables
G( z) 
Q
sL
1  Fill Rate 
Two Types of Service
Cycle Demand
1
180
2
75
3
235
4
140
5
180
6
200
7
150
8
90
9
160
10
40
Sum
1,450
Stock-Outs
0
0
45
0
0
10
0
0
0
0
55
Type I:
8 of 10 periods
80% service
Type II:
1,395 / 1,450 =
96%
FIXED-TIME PERIOD
MODELS
Fixed-Time Period Model



Every T periods, we look at inventory on hand and
place an order
Lead time still is L.
Order quantity will be different, depending on
demand
Fixed-Time Period Model:
When to Order?
Inventory Level
Period
Target maximum
Time
Fixed-Time Period Model: :
When to Order?
Inventory Level
Period Period
Target maximum
Time
Fixed-Time Period Model:
When to Order?
Inventory Level
Period Period
Target maximum
Time
Fixed-Time Period Model:
When to Order?
Inventory Level
Target maximum
Period Period Period
Time
Fixed-Time Period Model:
When to Order?
Inventory Level
Target maximum
Period Period Period
Time
Fixed-Time Period Model:
When to Order?
Inventory Level
Target maximum
Period Period Period
Time
Fixed Order Period

Standard deviation of demand over T+L =
s T  L  T  Ls



T = Review period length (in days)
σ = std dev per day
Order quantity (12.11) =
q  d (T  L)  zs T  L  I
Inventory Recordkeeping
Two ways to order inventory:
 Keep track of how many delivered, sold
 Go out and count it every so often
If keeping records, still need to double-check
 Annual physical inventory, or
 Cycle Counting
Cycle Counting


Physically counting a sample of total inventory on
a regular basis
Used often with ABC classification
A

items counted most often (e.g., daily)
Advantages
 Eliminates
annual shut-down for physical inventory
count
 Improves inventory accuracy
 Allows causes of errors to be identified
Fixed-Period Model


Answers how much to order
Orders placed at fixed intervals
 Inventory
brought up to target amount
 Amount ordered varies

No continuous inventory count
 Possibility

of stockout between intervals
Useful when vendors visit routinely
 Example:
P&G rep. calls every 2 weeks
ABC Analysis

Divides on-hand inventory into 3 classes
A

Basis is usually annual $ volume
$

class, B class, C class
volume = Annual demand x Unit cost
Policies based on ABC analysis
 Develop
class A suppliers more
 Give tighter physical control of A items
 Forecast A items more carefully
Classifying Items
as ABC
% Annual $ Volume
100
80
Items
A
B
C
%$Vol %Items
80
15
15
30
5
55
60
A
40
20
B
0
0
50
C
100
% of Inventory Items
150
ABC Classification Solution
Stock #
Vol.
206
105
019
144
207
26,000
200
2,000
20,000
7,000
Total
Cost
$ Vol.
$ 36 $936,000
600 120,000
55 110,000
4
80,000
10
70,000
%
71.1
9.1
8.4
6.1
5.3
1,316,000 100.0
ABC
ABC Classification Solution
Stock #
Vol.
206
105
019
144
207
26,000
200
2,000
20,000
7,000
Total
Cost
$ Vol.
$ 36 $936,000
600 120,000
55 110,000
4
80,000
10
70,000
%
71.1
9.1
8.4
6.1
5.3
1,316,000 100.0
ABC
A
A
B
B
C