Stochastic Inventory Theory Professor Stephen R. Lawrence Leeds School of Business University of Colorado Boulder, CO 80309-0419

1
Stochastic Inventory TheoryProfessor Stephen
R. LawrenceLeeds School of BusinessUniversity
of ColoradoBoulder, CO 80309-0419
2
Stochastic Inventory Theory

• Single Period Stochastic Inventory Model
• Newsvendor model
• Multi-Period Stochastic Inventory Models
• Safety Stock Calculations
• Expected Demand Std Dev Calculations
• Continuous Review (CR) models
• Periodic Review (PR) models

3
Single Period Stochastic Inventory

• Newsvendor Model

4
Single-Period Independent Demand

• Newsvendor Model One-time buys of seasonal
  goods, style goods, or perishable items
• Examples
• Newspapers, Christmas trees
• Supermarket produce
• Fad toys, novelties
• Fashion garments
• Blood bank stocks.

5
Newsvendor Assumptions

• Relatively short selling season
• Well defined beginning and end
• Commit to purchase before season starts
• Distribution of demand known or estimated
• Significant lost sales costs (e.g. profit)
• Significant excess inventory costs.

6
Single-Period Inventory Example
A T-shirt silk-screening firm is planning to
produce a number of custom T-shirts for the next
Bolder Boulder running event. The cost of
producing a T-shirt is 6.00, with a selling
price of 12.00. After BB concludes, demand for
T-shirts falls off, and the manufacturer can only
sell remaining shirts for 3.00 each. Based on
historical data, the expected demand distribution
for BB T-shirts is

Quantity Probability Cumulative
1000 0.00 0.00
2000 0.05 0.05
3000 0.15 0.20
4000 0.40 0.60
5000 0.30 0.90
6000 0.10 1.00
How many T-shirts should the firm produce to
maximize profits?
7
Opportunity Cost of Unmet Demand
Define U opportunity cost of unmet demand
(underproduce - understock)
Example U sales price - cost of
production 12 - 6
6 lost profit / unit
8
Cost of Excess Inventory
Define O cost of excess inventory
(overproduce - overstock) Example O
cost of production - salvage price
6 - 3 3 loss/unit
9
Solving Single-Period Problems
Example U cost of unmet demand
(understock) U 12 - 6 6 profit O
cost of excess inventory (overstock) O 6
- 3 3 loss
Optimal Solution
Where Pr(xQ) is the critical fractile of the
demand distribution.
Produce/purchase quantity Q that satisfies the
ratio
10
Translation to Textbook Notation
Lawrence Textbook
Understock cost U Cus
Overstock cost O Cos
Probability of understocking Pr(x?Q) Pus
Critical fractile Pr(x?Q) Critical fillrate (cfr)
11
Alternate Solution
Some textbooks use an alternative representation
of the critical fractile
Where Pr(xgtQ) is the critical fractile that
represents the probability of a stockout when
starting with an inventory of Q units.
NOTE to use a standard normal Z-table, you
will need Pr(xQ), NOT Pr(xgtQ)
Produce/purchase quantity Q that satisfies the
ratio
12
Solving Single-Period Problems
Example U cost of unmet demand
(underage) U 12 - 6 6 profit O cost
of excess inventory (overage) O 6 - 3
3 loss Example Pr(x Q) 6 / ( 3 6 )
0.667
13
Solving Single-Period Problems
0.667
4,222
D
14
Inventory Spreadsheet
15
Multi-Period Stochastic Inventory Models

• Continuous Review (CR) models
• Periodic Review (PR) models

16
Key Assumptions

• Demand is probabilistic
• Average demand changes slowly
• Forecast errors are normally distributed with no
  bias
• Lead times are deterministic

17
Key Questions

• How often should inventory status be determined?
• When should a replenishment order be placed?
• How large should the replenishment be?

18
Types of Multi-Period Models

• (CR) continuous review
• Reorder when inventory falls to R (fixed)
• Order quantity Q (fixed)
• Interval between orders is variable
• (PR) periodic review
• Order periodically every T periods (fixed)
• Order quantity q (variable)
• Inventory position I at time of reorder is
  variable
• Many others

19
Notation

• B stockout cost per item
• TAC total annual cost of inv.
• L order leadtime
• D annual demand
• d(L) demand during leadtime
• h holding cost percentage
• H holding cost per item
• I current inventory position

• Q order quantity (fixed)
• Q optimal order quantity
• q order quantity (variable)
• T time between orders
• R reorder point (ROP)
• S setup or order cost
• SS safety stock
• C per item cost or value.

20
Demand Calculations
21
Demand over Leadtime

• Multiply known demand rate D by leadtime L
• Be sure that both are in the same units!
• Example
• Mean demand is D 20 per day
• Leadtime is L 40 days
• d(L) D x L 20 x 40 800 units

22
Demand Std Deviation over Leadtime

• Multiply demand variance s 2 by leadtime L
• Example
• Standard deviation of demand s 4 units per day
• Calculate variance of demand s 2 16
• Variance of demand over leadtime L40 days is
  sL2 Ls 2 4016640
• Standard deviation of demand over leadtime L is
  sL Ls 2½ 640½ 25.3 units
• Remember
• Variances add, standard deviations dont!

23
Safety Stock Calculations
24
Safety Stock Analysis

• The world is uncertain, not deterministic
• demand rates and levels have a random component
• delivery times from vendors/production can vary
• quality problems can affect delivery quantities
• Murphy lives

Inventory Level
Q
L
R
SS
safety stock
0
stockout!
time
25
Inventory / Stockout Trade-offs
Small safety stocks Frequent stockouts Low
inventory costs
Large safety stocks Few stockouts High
inventory costs
Balanced safety stock, stockout
frequency, inventory costs
26
Safety Stock Example

• Service policies are often set by management
  judgment (e.g., 95 or 99 service level)
• Monthly demand is 100 units with a standard
  deviation of 25 units. If inventory is
  replenished every month, how much safety stock is
  need to provide a 95 service level? Assume that
  demand is normally distributed.

• Alternatively, optimal service level can be
  calculated using Newsvendor analysis

27
Continuous Review (CR) Stochastic Inventory
Models
28
(CR) Continuous Review System

• Always order the same quantity Q
• Replenish inventory whenever inventory level
  falls below reorder quantity R
• Time between orders varies
• Replenish level R depends on order lead-time L
• Requires continuous review of inventory levels

29
Safety Stock and Reorder Levels
DL
Reorder Level Safety Stock Mean Demand over
Leadtime
R SS DL
30
(CR) Order-Point, Order-Quantity

• Continuous review system
• Useful for class A, B, and C inventories
• Replenish when inventory falls to R
• Reorder quantity Q.
• Easy to understand, implement
• Two-bin variation

31
(CR) Implementation

• Implementation
• Determine Q using EOQ-type model
• Determine R using appropriate safety-stock model
• Practice
• Reserve quantity R in second bin (i.e. a baggy)
• Put order card with second bin
• Submit card to purchasing when second bin is
  opened
• Restock second bin to R upon order arrival

32
(CR) Example

• Consider a the following product
• D 2,400 units per year
• C 100 cost per unit
• h 0.24 holding fraction per year (H hC
  24/yr)
• L 1 month leadtime
• S 200 cost per setup
• B 500 cost for each backorder/stockout
• sL 125 units per month variation
• Management desires to maintain a 95 in-stock
  service level.

33
(CR) Example
Whenever inventory falls below 406, place another
order for 200 units
34
Total Inventory Costs for CR Policies

• TAC Total Annual Costs
• TAC Ordering Holding Expected Stockout Costs

• TAC 10,044 per year (CR policy)

35
Periodic Review (PR) Stochastic Inventory Models
36
Multi-Period Fixed-Interval Systems

• Requires periodic review of inventory levels
• Replenish inventories every T time units
• Order quantity q (q varies with each order)

37
Periodic Review Details

• Order quantity q must be large enough to cover
  expected demand over lead time L plus reorder
  period T (less current inventory position I )
• Exposed to demand variation over TL periods

38
(PR) Periodic-Review System

• Periodic review (often Class B,C inventories)
• Review inventory level every T time units
• Determine current inventory level I
• Order variable quantity q every T periods
• Allows coordinated replenishment of items
• Higher inventory levels than continuous review
  policies

39
(PR) Implementation

• Implementation
• Determine Q using EOQ-type model
• Set TQ/D (if possible --T often not in our
  control)
• Calculate q as sum of required safety stock,
  demand over leadtime and reorder interval, less
  current inventory level
• Practice
• Interval T is often set by outside constraints
• E.g., truck delivery schedules, inventory cycles,
  

40
(PR) Policy Example

• Consider a product with the following parameters
  
• D 2,400 units per year
• C 100
• h 0.24 per year (H hC 24/yr)
• T 2 months between replenishments
• L 1 month
• S 200
• B 500 cost for each backorders/stockouts
• I 100 units currently in inventory
• sL 125 units per month variation
• Management desires to maintain a 95 in-stock
  service level.

41
(PR) Policy Example
Suppose that this is given by circumstances
42
Total Inventory Costs for PR Policies

• TAC Total Annual Costs
• TAC Ordering Holding Expected Stockout Costs

• TAC 14,718 per year (PR policy)

43
Further Information

• American Production and Inventory Control Society
  (APICS)
• www.APICS.org
• Professional organization of production,
  inventory, and resource managers
• Offers professional certifications in production,
  inventory, and resource management

44
Further Information

• Institute for Supply Management
• (www.ISM.ws)
• Previously the National Association of Purchasing
  Managers (NAPM)
• Professional organization of supply chain
  managers
• Offers certifications in supply chain management