Part II: When to Order? Inventory Management Under Uncertainty

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Part II When to Order? Inventory Management
Under Uncertainty

• Demand or Lead Time or both uncertain
• Even good managers are likely to run out once
  in a while (a firm must start by choosing a
  service level/fill rate)
• When can you run out?
• Only during the Lead Time if you monitor the
  system.
• Solution build a standard ROP system based on
  the probability distribution on demand during the
  lead time (DDLT), which is a r.v. (collecting
  statistics on lead times is a good starting
  point!)

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The Typical ROP System
Average Demand
ROP set as demand that accumulates during
lead time
ROP ReOrder Point
Lead Time
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The Self-Correcting Effect- A Benign Demand
Rate after ROP
Hypothetical Demand
Average Demand
ROP
Lead Time
Lead Time
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What if Demand is brisk after hitting the ROP?
Hypothetical Demand
Average Demand
ROP EDDLT SS
ROP gt
EDDLT
Safety Stock
Lead Time
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When to Order

• The basic EOQ models address how much to order Q
• Now, we address when to order.
• Re-Order point (ROP) occurs when the inventory
  level drops to a predetermined amount, which
  includes expected demand during lead time (EDDLT)
  and a safety stock (SS)
• ROP EDDLT SS.

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When to Order

• SS is additional inventory carried to reduce the
  risk of a stockout during the lead time interval
  (think of it as slush fund that we dip into when
  demand after ROP (DDLT) is more brisk than
  average)
• ROP depends on
• Demand rate (forecast based).
• Length of the lead time.
• Demand and lead time variability.
• Degree of stockout risk acceptable to management
  (fill rate, order cycle Service Level)

DDLT, EDDLT Std. Dev.
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The Order Cycle Service Level,(SL)

• The percent of the demand during the lead time (
  of DDLT) the firm wishes to satisfy. This is a
  probability.
• This is not the same as the annual service level,
  since that averages over all time periods and
  will be a larger number than SL.
• SL should not be 100 for most firms. (90?
  95? 98?)
• SL rises with the Safety Stock to a point.

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Safety Stock
Quantity
Maximum probable demand during lead time (in
excess of EDDLT) defines SS
Expected demand during lead time (EDDLT)
ROP
Safety stock (SS)
Time
LT
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Variability in DDLT and SS

• Variability in demand during lead time (DDLT)
  means that stockouts can occur.
• Variations in demand rates can result in a
  temporary surge in demand, which can drain
  inventory more quickly than expected.
• Variations in delivery times can lengthen the
  time a given supply must cover.
• We will emphasize Normal (continuous)
  distributions to model variable DDLT, but
  discrete distributions are common as well.
• SS buffers against stockout during lead time.

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Service Level and Stockout Risk

• Target service level (SL) determines how much SS
  should be held.
• Remember, holding stock costs money.
• SL probability that demand will not exceed
  supply during lead time (i.e. there is no
  stockout then).
• Service level stockout risk 100.

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Computing SS from SL for Normal DDLT

• Example 10.5 on p. 374 of Gaither Frazier.
• DDLT is normally distributed a mean of 693. and a
  standard deviation of 139.
• EDDLT 693.
• s.d. (std dev) of DDLT ? 139..
• As computational aid, we need to relate this to Z
  standard Normal with mean0, s.d. 1
• Z (DDLT - EDDLT) / ?

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Reorder Point (ROP)
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Area under standard Normal pdf from - ? to z
Z standard Normal with mean0, s.d. 1Z (X
- ? ) / ?See GF Appendix ASee Stevenson,
second from last page
P(Z ltz)
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Computing SS from SL for Normal DDLT to provide
SL 95.

• ROP EDDLT SS EDDLT z (?).
• z is the number of standard deviations SS is
  set above EDDLT, which is the mean of DDLT.
• z is read from Appendix B Table B2. Of Stevenson
  -OR- Appendix A (p. 768) of Gaither Frazier
• Locate .95 (area to the left of ROP) inside the
  table (or as close as you can get), and read off
  the z value from the margins z 1.64.
• Example ROP 693 1.64(139) 921
• SS ROP - EDDLT 921 - 693. 1.64(139) 228
• If we double the s.d. to about 278, SS would
  double!
• Lead time variability reduction can same a lot of
  inventory and (perhaps more than lead time
  itself!)

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Summary View

• Holding Cost C Q/2 SS
• Order trigger by crossing ROP
• Order quantity up to (SS Q)

QSS Target
Not full due to brisk Demand after trigger
ROP EDDLT SS
ROP gt
EDDLT
Safety Stock
Lead Time
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Part III Single-Period Model Newsvendor

• Used to order perishables or other items with
  limited useful lives.
• Fruits and vegetables, Seafood, Cut flowers.
• Blood (certain blood products in a blood bank)
• Newspapers, magazines,
• Unsold or unused goods are not typically carried
  over from one period to the next rather they are
  salvaged or disposed of.
• Model can be used to allocate time-perishable
  service capacity.
• Two costs shortage (short) and excess (long).

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Single-Period Model

• Shortage or stockout cost may be a charge for
  loss of customer goodwill, or the opportunity
  cost of lost sales (or customer!)
• Cs Revenue per unit - Cost per unit.
• Excess (Long) cost applies to the items left over
  at end of the period, which need salvaging
• Ce Original cost per unit - Salvage value per
  unit.
• (insert smoke, mirrors, and the magic of
  Leibnitzs Rule here)

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The Single-Period Model Newsvendor

• How do I know what service level is the best one,
  based upon my costs?
• Answer Assuming my goal is to maximize profit
  (at least for the purposes of this analysis!) I
  should satisfy SL fraction of demand during the
  next period (DDLT)
• If Cs is shortage cost/unit, and Ce is excess
  cost/unit, then

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Single-Period Model for Normally Distributed
Demand

• Computing the optimal stocking level differs
  slightly depending on whether demand is
  continuous (e.g. normal) or discrete. We begin
  with continuous case.
• Suppose demand for apple cider at a downtown
  street stand varies continuously according to a
  normal distribution with a mean of 200 liters per
  week and a standard deviation of 100 liters per
  week
• Revenue per unit 1 per liter
• Cost per unit 0.40 per liter
• Salvage value 0.20 per liter.

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Single-Period Model for Normally Distributed
Demand

• Cs 60 cents per liter
• Ce 20 cents per liter.
• SL Cs/(Cs Ce) 60/(60 20) 0.75
• To maximize profit, we should stock enough
  product to satisfy 75 of the demand (on
  average!), while we intentionally plan NOT to
  serve 25 of the demand.
• The folks in marketing could get worried! If
  this is a business where stockouts lose long-term
  customers, then we must increase Cs to reflect
  the actual cost of lost customer due to stockout.

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Single-Period Model for Continuous Demand

• demand is Normal(200 liters per week, variance
  10,000 liters2/wk) so ? 100 liters per week
• Continuous example continued
• 75 of the area under the normal curve must be to
  the left of the stocking level.
• Appendix shows a z of 0.67 corresponds to a
  left area of 0.749
• Optimal stocking level mean z (?) 200
  (0.67)(100) 267. liters.

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Single-Period Discrete Demand Lively Lobsters

• Lively Lobsters (L.L.) receives a supply of
  fresh, live lobsters from Maine every day. Lively
  earns a profit of 7.50 for every lobster sold,
  but a day-old lobster is worth only 8.50. Each
  lobster costs L.L. 14.50.
• (a) what is the unit cost of a L.L. stockout?
• Cs 7.50 lost profit
• (b) unit cost of having a left-over lobster?
• Ce 14.50 - 8.50 cost salvage value 6.
• (c) What should the L.L. service level be?
• SL Cs/(Cs Ce) 7.5 / (7.5 6) .56
  (larger Cs leads to SL gt .50)
• Demand follows a discrete (relative frequency)
  distribution as given on next page.

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Lively Lobsters SL Cs/(Cs Ce) .56

• Demand follows a discrete (relative frequency)
  distribution
• Result order 25 Lobsters, because that is the
  smallest amount that will serve at least 56 of
  the demand on a given night.