Statistical Inventory Models F Newsperson Model: –Single order in the face of uncertain demand –No replenishment F Base Stock Model: –Replenish one at a time –How much inventory to carry F (Q, r) Model –Order size Q –When inventory reaches r

Issues F How much to order –Newsperson problem F When to order –Variability in demand during lead-time –Variability in lead-time itself

Newsperson Problem F Ordering for a One-time market –Seasonal sales –Special Events F 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 F Costs: C o –cost of item less its salvage value –inventory holding cost (usually small) F Savings: C s –revenue from the sale –good will gained by not turning away a customer

Newsperson Problem F Expected Savings: –C s *Prob(d < Q) F Expected Costs: –C o *[1 - Prob(d < Q)] F Find Q so that Prob(d < Q) is C o C s + C o

Example F Savings: –C s = $0.25 revenue F Costs: –C o = $0.15 cost F Find Q so that Prob(d < Q) is

Finding Q (An Example) Normal Distribution (Upper Tail) z0

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 F Independent, periodic demands F All unfilled orders are backordered F No setup costs C s = Cost of one unit of backorder one period C o = Cost of one unit of inventory one period

Extensions F Independent, periodic demands F All unfilled orders are lost F No setup costs C s = Cost of lost sale (unit profit) C o = Cost of one unit of inventory one period

Base Stock Model F Orders placed with each sale –Auto dealership F Sales occur one-at-a-time F Unfilled orders backordered  Known lead time l F No setup cost or limit on order frequency

Different Views F Base Stock Level: R –How much stock to carry F Re-order point: r = R-1 –When to place an order F Safety Stock Level: s –Inventory protection against variability in lead time demand –s = r - Expected Lead-time Demand

Different Tacks F Find the lowest base stock that supports a given customer service level F Find the customer service level a given base stock provides F Find the base stock that minimizes the costs of back-ordering and carrying inventory

Finding the Best Trade-off F As with the newsperson –Cost of carrying last item in inventory = –Savings that item realizes F Cost of carrying last item in inventory –h, the inventory carrying cost $/item/year F Cost of backordering –b, the backorder carrying cost $/item/year

Finding Balance F 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) F Savings the last item represents: –b*Fraction of time we carry backorders –b*Probability Lead-time demand exceeds R –b*(1-P(X < R)) F Choose R so that P(X < R) = b/(h + b)

Customer Service Level F What customer service level does base stock R provide? F What fraction of customer orders are filled from stock (not backordered)? F What fraction of our orders arrive before the demand for them? F What’s the probability that lead time demand is smaller than R? F P(X < R)

Smallest Base Stock F What’s the smallest base stock that provides desired customer service level? e.g. 99% fill rate. F What’s the smallest R so that P(X.99?

Control Policies F Periodic Review –eg, Monthly Inventory Counts –order enough to last till next review + cushion –orders are different sizes, but at regular intervals F Continuous Review –constant monitoring –(Q, R) policy –orders are the same size but at irregular intervals

Continuous Review Time Inventory Reorder Level Order Quantity Safety Stock

F 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 F Trade-off –fixed cost of placing/producing order, A –inventory carrying cost, h

A Model F Choose Q and r to minimize sum of –Setup costs –holding costs –backorder costs

Approximating the Costs F Setup Costs –Setup D/Q times per year F Average Inventory is –cycle stock: Q/2 –safety stock: s –Total: Q/2+s u Q/2 + r - Expected Lead-time Demand  Q/2 + r - 

Estimating The Costs F Backorder Costs –Number of backorders in a cycle u 0 if lead-time demand < r u x-r if lead-time demand x, exceeds r  n(r) =  r  (x-r)g(x)dx –Expected backorders per year u n(r)D/Q

The Objective F minimize Total Variable Cost  AD/Q (Setup cost)  h(Q/2 + r -  )(Holding cost)  bn(r)D/Q(Backorder cost)

An Answer F Q = Sqrt(2D(A + bn(r))/h) F P(XŠ r) = 1 - hQ/bD F Compute iteratively: –Initiate: With n(r) = 0, calculate Q –Repeat: u From Q, calculate r u With this r, calculate Q

Another Tack F Set the desired service level and figure the Safety Stock to Support it. F Use trade-off in Inventory and Setups to determine Q (EOQ, EPQ, POQ...)

Variability in Lead-Time Demand F Variability in Lead-Time F Variability in Demand  X =  X t : period t in lead-time) F Var(X) = Var(X t )E(LT) + Var(LT)E(X t ) 2 F s = z*Sqrt(Var(X)) F Choose z to provide desired level of protection.

Safety Stock F 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 C o =Stockouts/period* C s Stockouts/period = C o / C s

Example F Safety Stock of Raw Material X –Cost of Stocking out? u Lost sales u Unused capacity u Idle workers –Cost of Carrying Inventory u 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 F Assuming: – Average Demand is 6,000/qtr (~ 92/day) –Variance in Demand is 100 units 2 /qtr (1.5/day) –Average Lead Time is 2 weeks (10 days) –Variance in Lead-Time is 4 days 2 –Lead-Time Demand is normally distributed F E(X) = 92*10 = 920 F Var(X) = 1.5*10 + 4*(8464) ~ 34,000

Example F Look up 1 in a million on the Normal Upper Tail Chart –z ~ 4.6 F Compute Safety Stock –s = 4.6*Sqrt(34,000) = 4.6*184 = 846 F Compute Reorder Point –r = = 1,766

Other Issues F Why Carry Inventory? F How to Reduce Inventory? F Where to focus Attention?

Why Carry Inventory? F Buffer Production Rates From: – Seasonal Demand – Seasonal Supplies “Anticipation Inventory”

Other Types of Inventory “Decoupling Inventory” –Allows Processes to Operate Asynchronously –Examples: u DC’s “decouple” our distribution from individual customer orders u Holding tanks “decouple” 20K gal. syrup mixes from 5gal. bag-in-box units.

Other Types of Inventory F “Cycle Stock” –Consequence of Batch Production –Used to Reduce Change Overs: u 8 hours and 400 tons of “red stripe” to change Pulp Mill from Hardwood to Pine Pulp u 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: u Parts made in Mexico, Taurus Assembled in Atlanta

Other Types of Inventory “Safety Stock” –Buffer against Variability in u Demand u Production Process u Supplies –Avoid Stockouts or Shortages

Using Inventory F Inventory Finished Goods or Raw Materials? F Inventory at Central Facility or at DCs? F Extremes: –High Demand, Low Cost Product –Low Demand, High Cost Product

Reducing Inventory F Reducing Anticipation Inventories –Manage Demand with Promotions, etc. –Reduce overall seasonality through product mix –Expand Markets

Reducing Inventory F Reducing Cycle Stock –Reduce the length of Setups u Redesign the Products u Redesign the Process –Move Setups Offline –Fixturing, etc. –Reduce the number of Setups u Narrow Product Mix u Consolidate Production

Reducing Inventory F Reducing Pipeline Inventory –Move the Right Products, eg, Syrup not Coke –Consolidate Production Processes –Redesign Distribution System –Use Faster Modes

Reducing Inventory F Reducing Safety Stock –Reduce Lead-Time –Reduce Variability in Lead-Time –Reduce the Number of Products –Consolidate Inventory

ABC Analysis F 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