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
E[I]: Expected inventory level
E[B]: Expected backorder level
E[X]: Expected leadtime demand
E[D]: average demand per unit time (demand rate)
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Inventory Balance Equation
 Inventory position = on-hand inventory + inventory onorder – backorder level
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Inventory Balance Equation
 Inventory position = on-hand inventory + inventory onorder – backorder level
 Under a base-stock policy with base-stock level R, inventory
position is always kept at R (Inventory position = R )
IP = I+X - B = R
E[I] + E[X] – E[B] = 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
E[X]= E[D]L (the textbook refers to this quantity as q).
 The distribution of X depends on the distribution of D.
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I = max[0, I – B]= [I – B]+
B=max[0, 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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 E[I] = R – E[X] + E[B] = R – E[X] + E[(X – R)+]
 E[B] = E[I] + E[X] – R = E[(R – X)+] + E[X] – 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)= hE[I] + bE[B], 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
Y (R )  hE[ I ]  bE[ B ]
 h( R  E[ X ]  E[ B ])  bE[ B ]
 h( R  E[ X ])  ( h  b) E[ B ]
 h( R  E[ D ]L)  (h  b) E ([ X  R ] )
 h( R  E[ D ]L)  (h  b)  x  R ( x  R ) Pr( X  x )

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The Optimal Base-Stock Level
The optimal value of R is the smallest integer that satisfies
Y (R  1)  Y ( R)  0.
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Y ( R  1) - Y ( R )  h  R  1  E[ D ]L   ( h  b)  x  R 1 ( x  R  1) Pr( X  x )

h  R  E [ D ]L   ( h  b) x  R ( x  R ) Pr( X  x )

 h  ( h  b) x  R 1  ( x  R  1)  ( x  R )  Pr( X  x )

 h  (h  b) x  R 1 Pr( X  x )

 h  (h  b) Pr( X  R  1)
 h  (h  b) 1  Pr( X  R ) 
 b  (h  b) Pr( X  R )
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Y ( R  1) - Y ( R )  0
 b  (h  b) Pr( X  R )  0
b
 Pr( X  R ) 
bh
Choosing the smallest integer R that satisfies Y(R+1) – Y(R)  0
is equivalent to choosing the smallest integer R that satisfies
b
Pr( X  R) 
bh
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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
(l t ) x e  l t
Pr( D(t )  x ) 
, x  0.
x!
 Leadtime demand, X, can be shown in this case to also have
the Poisson distribution with
( l L) x e  l L
Pr( X  x ) 
, E[ X ]  l L, and Var ( X )  l L.
x!
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The Normal Approximation
 If X can be approximated by a normal distribution, then:
R*  E ( D ) L  zb /( b  h ) Var ( X )
Y ( R*)  ( h  b) Var ( X ) ( zb /( b  h ) )
 In the case where X has the Poisson distribution with
mean lL
R*  l L  zb /( b  h ) l L
Y ( R*)  ( h  b) l L ( zb /( b  h ) )
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Example 2
If X has the geometric distribution with parameter  , 0    1
P ( X  x )   x (1   ).
E[ X ] 

1 
Pr( X  x )   x
Pr( X  x )  1   x 1
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Example 2 (Continued…)
The optimal base-stock level is the smallest integer R* that
satisfies
Pr( X  R* ) 
 1 
R* 1
b
bh
b

 R* 
bh
b
]
b  h 1
ln[  ]
ln[
b 

ln[
]

bh
 R*  

ln[

]




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Computing Expected Backorders
 It is sometimes easier to first compute (for a given R),
E[ I ]   x 0 ( R  x ) Pr( X  x )
R
and then obtain E[B]=E[I] + E[X] – 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
E[B]= qPr(X=R)+(q-R)[1-Pr(X R)]
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