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)

z0

z 0.00 0.01 0.02 0.03 0.04 0.05 0.060.0 0.50000 0.49601 0.49202 0.48803 0.48405 0.48006 0.476080.1 0.46017 0.45620 0.45224 0.44828 0.44433 0.44038 0.436440.2 0.42074 0.41683 0.41294 0.40905 0.40517 0.40129 0.397430.3 0.38209 0.37828 0.37448 0.37070 0.36693 0.36317 0.35942

Example Continued

If the process is Normal with mean and std. deviation , then (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

Continuous Review

Time

Inve

ntor

y

Reorder Level

Order Quantity

Safety Stock

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 CostAD/Q (Setup cost)h(Q/2 + r - ) (Holding cost)bn(r)D/Q (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