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1. 1
The Base Stock Model

2. 2
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

3. 3
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.

4. 4
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.

5. 5
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)

6. 6
Inventory Balance Equation
 Inventory position = on-hand inventory + inventory on-order
– backorder level

7. 7
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 = I+X - B = R
E[I] + E[X] – E[B] = R

8. 8
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.

9. 9
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]+

10. 10
 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 ³ R)
 Pr(not stocking out) = Pr(X £ R-1)
 Fill rate = E(D) Pr(X £ R-1)/E(D) = Pr(X £ R-1)

11. 11
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.

12. 12
The Cost Function
Y R hE I bE B
= +
= - + +
= - + +
= - + + -
= - + + å - =
( ) [ ] [ ]
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 x
( [ ] [ ]) [ ]
( [ ]) ( ) [ ]
( [ ] ) ( ) ([ ] -
)
( [ ] ) ( ) ¥
( )Pr( ) x =
R

13. 13
The Optimal Base-Stock Level
The optimal value of R is the smallest integer that satisfies
Y (R + 1) - Y (R) ³ 0.

14. Y R + Y R = h R + - E D L + h + b x - R - X =
x
( 1) - ( ) 1 [ ] ( ) ( 1)Pr( )
14
( )
( )
h R E D L h b x R X x
h h b x R x R X x
h h b X x
h h b X R
h h b X R
b h b X R
- - - + - =
= + + - - - - =
= - + =
= - + ³ +
= - + - £
= - + + £
[ ] ( ) ( )Pr( )
( )
( ) ( 1) ( ) Pr( )
( ) Pr( )
( )Pr( 1)
( ) ( 1 Pr( )
)
( )Pr( )
1
1
1
x R
x R
x R
x R
¥
= +
¥
=
¥
= +
¥
= +
å
å
å
å

15. 15
Y R + Y R
³
b h b X R
( 1) - ( ) 0
( )Pr( ) 0
Pr( )
Û - + + £ ³
X R b
b h
Û £ ³
+
Choosing the smallest integer R that satisfies Y(R+1) – Y(R) ³ 0
is equivalent to choosing the smallest integer R that satisfies
Pr(X R) b
b h
£ ³
+

16. X x l L x e l L E X l L Var X l
L
16
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
D t x l t x e -l
t x
Pr( ( ) = ) = ( ) , ³
0.
!
x
 Leadtime demand, X, can be shown in this case to also have
the Poisson distribution with
Pr( ) ( ) , [ ] , and ( ) .
!
x
-
= = = =

17.  If X can be approximated by a normal distribution, then:
17
The Normal Approximation
R E D L z Var X
Y R h b Var X f z
* ( ) ( )
/( )
( *) ( ) ( ) ( )
/( )
b b h
b b h
+
+
= +
= +
 In the case where X has the Poisson distribution with
mean lL
*
/( )
( *) ( ) ( )
/( )
b b h
b b h
R l L z l
L
+
Y R h b l L f
z
+
= +
= +

18. If X has the geometric distribution with parameter r , 0 £ r £ 1
18
Example 2
x
= = -
r r
r
( ) (1 ).
1
[ ]
1
Pr( )
Pr( ) 1
x
x
P X x
E X
X x
X x
r
r
r +
=
-
³ =
£ = -

19. The optimal base-stock level is the smallest integer R* that
satisfies
19
Example 2 (Continued…)
*
*
Pr( )
1 *
*
ln[ ]
1 1
ln[ ]
ln[ ]
ln[ ]
R
X R b
b h
b
b R b h
b h
b
R b h
r
r
r
+
£ ³
+
Þ - ³ Þ ³ + -
+
ê ú
ê + ú Þ = ê ú
ê ú
ë û

20. 20
Computing Expected Backorders
 It is sometimes easier to first compute (for a given R),
E I =åx= R -x X = x
0 [ ] ( )Pr( ) 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)]