The Base Stock Model

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The Base Stock Model
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Assumptions

• Demand occurs continuously over time
• Times between consecutive orders are stochastic
  but independent and identically distributed
  (i.i.d.)
• Inventory is reviewed continuously
• Supply leadtime is a fixed constant L
• There is no fixed cost associated with placing an
  order
• Orders that cannot be fulfilled immediately from
  on-hand inventory are backordered

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The Base-Stock Policy

• Start with an initial amount of inventory R. Each
  time a new demand arrives, place a replenishment
  order with the supplier.
• An order placed with the supplier is delivered L
  units of time after it is placed.
• Because demand is stochastic, we can have
  multiple orders (inventory on-order) that have
  been placed but not delivered yet.

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The Base-Stock Policy

• The amount of demand that arrives during the
  replenishment leadtime L is called the leadtime
  demand.
• Under a base-stock policy, leadtime demand and
  inventory on order are the same.
• When leadtime demand (inventory on-order) exceeds
  R, we have backorders.

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Notation

• I inventory level, a random variable
• B number of backorders, a random variable
• X Leadtime demand (inventory on-order), a random
  variable
• IP inventory position
• EI Expected inventory level
• EB Expected backorder level
• EX Expected leadtime demand
• ED average demand per unit time (demand rate)

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Inventory Balance Equation

• Inventory position on-hand inventory
  inventory on-order backorder level

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Inventory Balance Equation

• Inventory position on-hand inventory
  inventory on-order backorder level
• Under a base-stock policy with base-stock level
  R, inventory position is always kept at R
  (Inventory position R )
• IP IX - B R
• EI EX EB R

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Leadtime Demand

• Under a base-stock policy, the leadtime demand X
  is independent of R and depends only on L and D
  with EX EDL (the textbook refers to this
  quantity as q).
• The distribution of X depends on the distribution
  of D.

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• I max0, I B I B
• Bmax0, B-I B - I
• Since R I X B, we also have
• I B R X
• I R X
• B X R

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• EI R EX EB R EX E(X R)
• EB EI EX R E(R X) EX R
• Pr(stocking out) Pr(X ? R)
• Pr(not stocking out) Pr(X ? R-1)
• Fill rate E(D) Pr(X ? R-1)/E(D) Pr(X ? R-1)

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Objective

• Choose a value for R that minimizes the sum of
  expected inventory holding cost and expected
  backorder cost, Y(R) hEI bEB, where h is
  the unit holding cost per unit time and b is the
  backorder cost per unit per unit time.

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The Cost Function
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The Optimal Base-Stock Level
The optimal value of R is the smallest integer
that satisfies
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(No Transcript)
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Choosing the smallest integer R that satisfies
Y(R1) Y(R) ? 0 is equivalent to choosing the
smallest integer R that satisfies
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Example 1

• Demand arrives one unit at a time according to a
  Poisson process with mean l. If D(t) denotes the
  amount of demand that arrives in the interval of
  time of length t, then
• Leadtime demand, X, can be shown in this case to
  also have the Poisson distribution with

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The Normal Approximation

• If X can be approximated by a normal
  distribution, then
• In the case where X has the Poisson distribution
  with mean lL

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Example 2
If X has the geometric distribution with
parameter r , 0 ? r ? 1
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Example 2 (Continued)
The optimal base-stock level is the smallest
integer R that satisfies
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Computing Expected Backorders

• It is sometimes easier to first compute (for a
  given R),
• and then obtain EBEI EX R.
• For the case where leadtime demand has the
  Poisson distribution (with mean q E(D)L), the
  following relationship (for a fixed R) applies
• EB qPr(XR)(q-R)1-Pr(X? R)