Statistical Inventory Models
 Newsperson
Model:
– Single order in the face of uncertain demand
– No replenishment
 Base
Stock Model:
– Replenish one at a time
– How much inventory to carry
 (Q,
r) Model
– Order size Q
– When inventory reaches r
Issues
 How
much to order
– Newsperson problem
 When
to order
– Variability in demand during lead-time
– Variability in lead-time itself
Newsperson Problem
 Ordering
for a One-time market
– Seasonal sales
– Special Events
 How
much do we order?
– Order more to increase revenue and
reduce lost sales
– Order less to avoid additional
inventory and unsold goods.
Newsperson Problem
Order up to the point that the expected costs
and savings for the last item are equal
 Costs: Co
– cost of item less its salvage value
– inventory holding cost (usually small)
 Savings:
Cs
– revenue from the sale
– good will gained by not turning
away a customer
Newsperson Problem
 Expected
Savings:
– Cs *Prob(d < Q)
 Expected
Costs:
– Co *[1 - Prob(d < Q)]
 Find
Q so that Prob(d < Q) is
Co
Cs + Co
Example
 Savings:
– Cs = $0.25 revenue
 Costs:
– Co = $0.15 cost
 Find
Q so that Prob(d < Q) is 0.375
0.15
0.25 + 0.15
Finding Q (An Example)
Normal Distribution (Upper Tail)
z
0.0
0.1
0.2
0.3
0.00
0.50000
0.46017
0.42074
0.38209
0.01
0.49601
0.45620
0.41683
0.37828
0.02
0.49202
0.45224
0.41294
0.37448
0
0.03
0.48803
0.44828
0.40905
0.37070
0.04
0.48405
0.44433
0.40517
0.36693
z
0.05
0.48006
0.44038
0.40129
0.36317
0.06
0.47608
0.43644
0.39743
0.35942
Example Continued
the process is Normal with mean  and
std. deviation , then
 If
(X- )/  is Normal with mean 0 and std. dev. 1
 If
in our little example demand is N(100,
10) so  = 100 and .
– Find z in the N(0, 1) table: z = .32
– Transform to X: (X-100)/10 = .32
X = 103.2
Extensions
 Independent,
periodic demands
 All unfilled orders are backordered
 No setup costs
Cs = Cost of one unit of backorder one period
Co = Cost of one unit of inventory one period
Extensions
 Independent,
periodic demands
 All unfilled orders are lost
 No setup costs
Cs = Cost of lost sale (unit profit)
Co = Cost of one unit of inventory one period
Base Stock Model
 Orders
placed with each sale
– Auto dealership
 Sales
occur one-at-a-time
 Unfilled orders backordered
 Known lead time l
 No
setup cost or limit on order
frequency
Different Views
 Base
Stock Level: R
– How much stock to carry
 Re-order
point: r = R-1
– When to place an order
 Safety
Stock Level: s
– Inventory protection against variability in lead
time demand
– s = r - Expected Lead-time Demand
Different Tacks
 Find
the lowest base stock that supports a
given customer service level
 Find the customer service level a given base
stock provides
 Find the base stock that minimizes the costs
of back-ordering and carrying inventory
Finding the Best Trade-off
 As
with the newsperson
– Cost of carrying last item in inventory =
– Savings that item realizes
 Cost
of carrying last item in inventory
– h, the inventory carrying cost $/item/year
 Cost
of backordering
– b, the backorder carrying cost $/item/year
Finding Balance
 Cost
the last item represents:
– h*Fraction of time we carry inventory
– h*Probability Lead-time demand is less than R
– h*P(X < R)
 Savings
the last item represents:
– b*Fraction of time we carry backorders
– b*Probability Lead-time demand exceeds R
– b*(1-P(X < R))
 Choose
R so that P(X < R) = b/(h + b)
Customer Service Level
 What
customer service level does base
stock R provide?
 What fraction of customer orders are filled
from stock (not backordered)?
 What fraction of our orders arrive before the
demand for them?
 What’s the probability that lead time
demand is smaller than R?
 P(X < R)
Smallest Base Stock
 What’s
the smallest base stock that provides
desired customer service level? e.g. 99% fill
rate.
 What’s the smallest R so that P(X < R) >
.99?
Control Policies
 Periodic
Review
– eg, Monthly Inventory Counts
– order enough to last till next review + cushion
– orders are different sizes, but at regular
intervals
 Continuous
Review
– constant monitoring
– (Q, R) policy
– orders are the same
size but at irregular intervals
Inventory
Continuous Review
Order Quantity
Reorder Level
Safety Stock
Time
Safety Stock
 Inventory
used to protect against variability
in Lead-Time Demand
Lead-Time Demand: Demand between the
time the order to restock is placed and the
time it arrives
Reorder Point is:
R = Average Lead-Time Demand
+ Safety Stock
Order Quantity
 Trade-off
– fixed cost of placing/producing order, A
– inventory carrying cost, h
A Model
 Choose
Q and r to minimize sum of
– Setup costs
– holding costs
– backorder costs
Approximating the Costs
 Setup
Costs
– Setup D/Q times per year
 Average
Inventory is
– cycle stock: Q/2
– safety stock: s
– Total: Q/2+s


Q/2 + r - Expected Lead-time Demand
Q/2 + r - 
Estimating The Costs
 Backorder
Costs
– Number of backorders in a cycle
0
if lead-time demand < r
 x-r if lead-time demand x, exceeds r
 n(r) = r(x-r)g(x)dx
– Expected backorders per year
 n(r)D/Q
The Objective
 minimize
Total Variable Cost
AD/Q
h(Q/2 + r - )
bn(r)D/Q
(Setup cost)
(Holding cost)
(Backorder cost)
An Answer
Q
= Sqrt(2D(A + bn(r))/h)
 P(XŠ r) = 1 - hQ/bD
 Compute iteratively:
– Initiate: With n(r) = 0, calculate Q
– Repeat:
 From
Q, calculate r
 With this r, calculate Q
Another Tack
 Set
the desired service level and figure the
Safety Stock to Support it.
 Use trade-off in Inventory and Setups to
determine Q (EOQ, EPQ, POQ...)
Variability in Lead-Time
Demand
 Variability
in Lead-Time
 Variability in Demand
 X =  Xt: period t in lead-time)
 Var(X) = Var(Xt)E(LT) + Var(LT)E(Xt)2
 s = z*Sqrt(Var(X))
 Choose z to provide desired level
of protection.
Safety Stock
 Analysis
similar to Newsperson problem
sets number of stockouts:
– Savings of Inventory carrying cost
– Cost of One more item short each time we
stocked out
Co =Stockouts/period* Cs
Stockouts/period = Co / Cs
Example
 Safety
Stock of Raw Material X
– Cost of Stocking out?
 Lost
sales
 Unused capacity
 Idle workers
– Cost of Carrying Inventory
 Say,
10% of value or $2.50/unit/year
– Number of times to stock out:
2.50/2,500,000 or 1 in a million (exaggerated)
Example
 Assuming:
–
–
–
–
–
Average Demand is 6,000/qtr (~ 92/day)
Variance in Demand is 100 units2/qtr (1.5/day)
Average Lead Time is 2 weeks (10 days)
Variance in Lead-Time is 4 days2
Lead-Time Demand is normally distributed
 E(X)
= 92*10 = 920
 Var(X) = 1.5*10 + 4*(8464)
~ 34,000
Example
 Look
up 1 in a million on the Normal Upper
Tail Chart
– z ~ 4.6
 Compute
Safety Stock
– s = 4.6*Sqrt(34,000) = 4.6*184 = 846
 Compute
Reorder Point
– r = 920 + 846 = 1,766
Other Issues
 Why
Carry Inventory?
 How to Reduce Inventory?
 Where to focus Attention?
Why Carry Inventory?
 Buffer
Production Rates From:
– Seasonal Demand
– Seasonal Supplies
“Anticipation Inventory”
Other Types of Inventory
“Decoupling Inventory”
– Allows Processes to Operate Asynchronously
– Examples:


DC’s “decouple” our distribution from individual
customer orders
Holding tanks “decouple” 20K gal. syrup mixes
from 5gal. bag-in-box units.
Other Types of Inventory
 “Cycle
Stock”
– Consequence of Batch Production
– Used to Reduce Change Overs:
8 hours and 400 tons of “red stripe” to change Pulp
Mill from Hardwood to Pine Pulp
 4 hours to change part feeders on a
Chip Shooter

Reduce Setup Time!
Other Types of Inventory
“Pipeline Inventory”
–
–
–
–
Goods in Transit
Work in Process or WIP
Allows Processes to be in Different Places
Example:
 Parts
made in Mexico, Taurus
Assembled in Atlanta
Other Types of Inventory
“Safety Stock”
– Buffer against Variability in
 Demand
 Production
Process
 Supplies
– Avoid Stockouts or Shortages
Using Inventory
 Inventory
Finished Goods or Raw Materials?
 Inventory at Central Facility or at DCs?
 Extremes:
– High Demand, Low Cost Product
– Low Demand, High Cost Product
Reducing Inventory
 Reducing Anticipation
Inventories
– Manage Demand with Promotions, etc.
– Reduce overall seasonality through product mix
– Expand Markets
Reducing Inventory
 Reducing
Cycle Stock
– Reduce the length of Setups
 Redesign
the Products
 Redesign the Process
– Move Setups Offline
– Fixturing, etc.
– Reduce the number of Setups
 Narrow
Product Mix
 Consolidate Production
Reducing Inventory
 Reducing
–
–
–
–
Pipeline Inventory
Move the Right Products, eg, Syrup not Coke
Consolidate Production Processes
Redesign Distribution System
Use Faster Modes
Reducing Inventory
 Reducing
–
–
–
–
Safety Stock
Reduce Lead-Time
Reduce Variability in Lead-Time
Reduce the Number of Products
Consolidate Inventory
ABC Analysis
 Where
to focus Attention:
Dollar Volume = Unit Price * Annual Demand
– Category A: 20% of the Stock Keeping Units
(SKU’s) account for 80% of the Dollar Volume
– Category C: 50% of the SKU’s with
lowest Dollar Volume
– Category B: Remaining 30% of
the SKU’s