Statistical Inventory Models

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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

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Issues

• How much to order
• Newsperson problem
• When to order
• Variability in demand during lead-time
• Variability in lead-time itself

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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.

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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

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Newsperson Problem

• Expected Savings
• Cs Prob(d lt Q)
• Expected Costs
• Co 1 - Prob(d lt Q)
• Find Q so that Prob(d lt Q) is
• Co
• Cs Co

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Example

• Savings
• Cs 0.25 revenue
• Costs
• Co 0.15 cost
• Find Q so that Prob(d lt Q) is 0.375
• 0.15
• 0.25 0.15

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Finding Q (An Example)
Normal Distribution (Upper Tail)
z
0
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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

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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

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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

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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

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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

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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

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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

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Finding Balance

• Cost the last item represents
• hFraction of time we carry inventory
• hProbability Lead-time demand is less than R
• hP(X lt R)
• Savings the last item represents
• bFraction of time we carry backorders
• bProbability Lead-time demand exceeds R
• b(1-P(X lt R))
• Choose R so that P(X lt R) b/(h b)

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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?
• Whats the probability that lead time demand is
  smaller than R?
• P(X lt R)

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Smallest Base Stock

• Whats the smallest base stock that provides
  desired customer service level? e.g. 99 fill
  rate.
• Whats the smallest R so that P(X lt R) gt .99?

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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

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Continuous Review
Order Quantity
Inventory
Reorder Level
Safety Stock
Time
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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

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Order Quantity

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

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A Model

• Choose Q and r to minimize sum of
• Setup costs
• holding costs
• backorder costs

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Approximating the Costs

• Setup Costs
• Setup D/Q times per year
• Average Inventory is
• cycle stock Q/2
• safety stock s
• Total Q/2s
• Q/2 r - Expected Lead-time Demand
• Q/2 r - ?

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Estimating The Costs

• Backorder Costs
• Number of backorders in a cycle
• 0 if lead-time demand lt 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

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The Objective

• minimize Total Variable Cost
• AD/Q (Setup cost)
• h(Q/2 r - ?) (Holding cost)
• bn(r)D/Q (Backorder cost)

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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

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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...)

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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 zSqrt(Var(X))
• Choose z to provide desired level
• of protection.

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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

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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)

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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) 9210 920
• Var(X) 1.510 4(8464)
• 34,000

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Example

• Look up 1 in a million on the Normal Upper Tail
  Chart
• z 4.6
• Compute Safety Stock
• s 4.6Sqrt(34,000) 4.6184 846
• Compute Reorder Point
• r 920 846 1,766

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Other Issues

• Why Carry Inventory?
• How to Reduce Inventory?
• Where to focus Attention?

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Why Carry Inventory?

• Buffer Production Rates From
• Seasonal Demand
• Seasonal Supplies
• Anticipation Inventory

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Other Types of Inventory

• Decoupling Inventory
• Allows Processes to Operate Asynchronously
• Examples
• DCs decouple our distribution from individual
  
• customer orders
• Holding tanks decouple 20K gal. syrup mixes
• from 5gal. bag-in-box units.

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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!

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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

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Other Types of Inventory

• Safety Stock
• Buffer against Variability in
• Demand
• Production Process
• Supplies
• Avoid Stockouts or Shortages

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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

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Reducing Inventory

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

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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

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Reducing Inventory

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

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Reducing Inventory

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

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ABC Analysis

• Where to focus Attention
• Dollar Volume Unit Price Annual Demand
• Category A 20 of the Stock Keeping Units
  (SKUs) account for 80 of the Dollar Volume
• Category C 50 of the SKUs with
• lowest Dollar Volume
• Category B Remaining 30 of
• the SKUs