Deterministic Inventory Management

Report
INVENTORY MANAGEMENT
Operations Management
Dr. Ron 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
U S In v e n to ry , G D P ($ B )
1 4 ,0 0 0
1 2 ,0 0 0
1 0 ,0 0 0
8 ,0 0 0
6 ,0 0 0
4 ,0 0 0
2 ,0 0 0
B u s in e s s In v e n to rie s
US G D P
04
20
02
20
00
20
98
19
96
19
94
19
92
19
90
19
88
19
86
19
19
84
-
US In v e n to r ie s a s % o f G D P
2 5 .0 %
% of GD P
2 0 .0 %
1 5 .0 %
1 0 .0 %
5 .0 %
Ye a r
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
TC ( Q )  H *
Q
2
S*
D
Q
C*D
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 =
H*
Q
S*
2
Take derivative
with respect to Q =
H
S*
2
D
C*D
Q
D
Q
2
Set equal
 0 to zero
Solve for Q:
H
2

DS
Q
2
Q 
2
2 DS
H
Q 
2 DS
H
Adding Lead Time

Use same order size
Q 


2 DS
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:
Q
2 DS
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) 
Co  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 
xm
sL
Safety stock
& Safety stock = zsL
Therefore, z =
sL
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 =
sL 
L days s day

4 5  10
R  d * L  zs

L
 d *L z
R = 40 + 3 * 10 = 70
L days s day
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 )  z s 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 #
Vo l.
206
26,000
$ 36
$936,000
71.1
A
105
200
600
120,000
9.1
A
019
2,000
55
110,000
8.4
B
144
20,000
4
80,000
6.1
B
207
7,000
10
70,000
5.3
C
1,316,000
100.0
Total
Cost
$ V o l.
%
ABC

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