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The Base Stock Model

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Assumptions

Demand occurs continuously over time Times between consecutive orders are stochastic butindependent and identically distributed ( i.i.d. )Inventory is reviewed continuouslySupply leadtime is a fxed constant L

There is no fxed cost associated with placing an order

Orders that cannot be ulflled immediately rom onhand inventory are bac!ordered

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The Base-Stock Policy

Start with an initial amount o inventory R" #ach timea new demand arrives$ place a replenishment orderwith the supplier"

%n order placed with the supplier is delivered L unitso time a ter it is placed"&ecause demand is stochastic$ we can have multipleorders (inventory on order) that have been placed

but not delivered yet"

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The Base-Stock Policy

The amount o demand that arrives during thereplenishment leadtime L is called the leadtimedemand"

'nder a base stoc! policy$ leadtime demand andinventory on order are the same"

hen leadtime demand (inventory on order) exceedsR$ we have bac!orders"

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Notation

I inventory level$ a random variableB number o bac!orders$ a random variable

X *eadtime demand (inventory on order)$ a random variableIP inventory positionE+I, #xpected inventory levelE+B, #xpected bac!order levelE+ X , #xpected leadtime demandE+D, average demand per unit time (demand rate)

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Inventory Balance Equation

Inventory position - on hand inventory . inventoryon order / bac!order level

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Inventory Balance Equation

Inventory position - on hand inventory . inventory onorder / bac!order level'nder a base stoc! policy with base-stock level R$inventory position is always !ept at R ( Inventory position - R )

IP I! X - B R

E "I# + E " X # E "B # = R

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$eadtime %emand

'nder a base stoc! policy$ the leadtime demand X is independent o R and depends only on L and D with E+ X , = E +D, L (the textboo! re ers to this

0uantity as )"

The distribution o X depends on the distributiono D.

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I - max+1$ I / B,- + I / B, . B-max+1$ B I, - + B I, .

Since R I ! X & B ' (e also have

I / B - R / X I - + R / X , .

B -+ X / R, .

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E+I, - R / E+2, . E+B, - R / E+ X , . E+( X / R) . ,

E+B, - E+I, . E+2, / R - E+(R / X) . , . E+2, / R

3r(stoc!ing out) - 3r( X R)

3r(not stoc!ing out) - 3r( X R 4)

5ill rate - #(D) 3r( X R 4)6#(D) - 3r( X R 4)

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)b*ective

Choose a value for R that minimizes the sumof expe ted inventor! holdin" ost and

expe ted #a $order ost% Y(R)= hE "I# +bE "B #% &here h is the unit holdin" ost perunit time and b is the #a $order ost per unit

per unit time.

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The +ost ,unction

( ) [ ] [ ]

( [ ] [ ]) [ ]

( [ ]) ( ) [ ]

( [ ] ) ( ) ([ ] )

( [ ] ) ( ) ( ) Pr( ) x R

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 x

=

= +

= + +

= + +

= + +

= + + =

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The )ptimal Base-Stock $evel

The optimal value o R is the smallest integer that

satisfes ( 1) ( ) 0.Y R Y R+

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( )

( )

( )

( )

1

1

1

( 1) - ( ) 1 [ ] ( ) ( 1) Pr( )

[ ] ( ) ( ) Pr( )

( ) ( 1) ( ) Pr( )

( ) Pr( )

( ) Pr( 1)

( ) 1 Pr( )( ) Pr( )

x R

x R

x R

x R

Y R Y R h R E D L h b x R X x

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 Rb h b X R

= +

=

= +

= +

+ = + + + =

+ =

= + + =

= + =

= + +

= + = + +

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( 1) - ( ) 0

( ) Pr( ) 0

Pr( )

Y R Y R

b h b X R

b X R b h

+

+ +

+

7hoosing the smallest integer R that satisfes ' (R.4) /

' (R) 1 is e0uivalent to choosing the smallest integerR that satisfes

Pr( ) b

X Rb h

+

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E ample .

Demand arrives one unit at a time according to a3oisson process with mean " I D(t ) denotes theamount o demand that arrives in the interval o timeo length t $ then

*eadtime demand$ X $ can be shown in this case toalso have the 3oisson distribution with

( )

Pr( ( ) ) , 0.!

x t t e D t x x

x

= =

( )Pr( ) , [ ] , and ( ) .

!

x L L e X x E X L Var X L

x

= = = =

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The Normal Appro imation

/( )

/( )

* ( ) ( )

( *) ( ) ( ) ( )

b b h

b b h

R E D L z Var X

Y R h b Var X z

+

+

= +

= +

I X can be approximated by a normal distribution$then

In the case where X has the 3oisson distributionwith mean L

/( )

/( )

*

( *) ( ) ( )

b b h

b b h

R L z L

Y R h b L z

+

+

= +

= +

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E ample/

1

( ) (1 ).

[ ]1

Pr( )

Pr( ) 1

x

x

x

P X x

E X

X x

X x

+

= =

= =

=

I 2 has the geometric distribution with parameter $ 1 4

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E ample / 0+ontinued12

*

*

1 *

*

Pr( )

ln[ ]1 1

ln[ ]

ln[ ]

ln[ ]

R

b

X R b h

b

b b h Rb h

b

b h R

+

+

+

+

+

=

The optimal base stoc! level is the smallest integerR8 that satisfes

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+omputin3 E pectedBackorders

0[ ] ( ) Pr( )

R

x E I R x X x

== =

It is sometimes easier to frst compute ( or a givenR)$

and then obtain E+B,- E+I, . E+ X , / R"

5or the case where leadtime demand has the3oisson distribution (with mean - #( D)L)$ the

ollowing relationship ( or a fxed R) applies

E+B,- 3r( X - R).( R)+4 3r( X R),