The ( Q , r ) Model

1

The (Q, r) 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 a fixed cost associated with placing an order

Orders that cannot be fulfilled immediately from on-hand inventory are backordered

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The (Q, r) Policy

Start with an initial amount of inventory R. When inventory level reaches level r, place an order in the amount Q = R-r to bring inventory position back up to level R. Thereafter whenever inventory position drops to r, place an order of size Q.

The base-stock policy is the special case of the (Q, r) policy where Q = 1.

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Inventory versus Time

Q

Inve

nto

ry

Time

l

r

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Notation

I: inventory level, a random variableB: number of backorders, a random variableX: Leadtime demand (inventory on-order), a random variableIP: inventory positionE[I]: Expected inventory levelE[B]: Expected backorder levelE[X]: Expected leadtime demandE[D]: average demand per unit time (demand rate)

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Inventory Position

Inventory position = on-hand inventory + inventory on-order – backorder level

Under the (Q, r) policy, inventory position, IP, takes on values takes on values r+1, r+2, ..., r+Q

The time IP remains at any value is the time between consecutive demand arrivals. Since the times between consecutive arrivals are independent and identically distributed, the long run fraction of time IP remains at any value is the same for all values

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Inventory Position (Continued…)

IP is equally likely to take on values r+1, r+2, ..., r+Q, or equivalently, IP is uniformly distributed with

Pr(IP = i)=1/Q where i = r+1, ..., r+Q.

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Net Inventory

Net inventory IN=I–B increases when a delivery is made and decreases when demand occurs.

Let D(t, t+L] refer to the amount of demand that takes place in the interval (t, t+L].

All the inventory that was on order at t will be delivered in the interval (t, t+L]

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Net Inventory (Continued…)

IN(t+L)=IN(t) + IO(t) – D(t, t+L]IN(t+L)=IP(t) – D(t, t+L]

IN=IP – X

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Inventory and Backorders

I=IN+=(IP -X)+

B=IN-=(IP-X)- =(X - IP)+

E[B]=E[(X - IP)+] E[I]=E[(IP - X)+]

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Probability of Stocking Out

1

1

1

Pr(stocking out) 1 ( , ) Pr( 0)

Pr( 0 | )Pr( )

Pr( 0 | )

Pr( 0 | )

r Q

x r

r Q

x r

r Q

x r

S Q r IN

IN IP x IP x

IN IP x

Q

IP X IP x

Q

1 1

1

Pr( ) Pr( 1) =

Pr( )

r Q r Q

x r x r

r Q

x r

X x X x

Q Q

X x

Q

Since IP and X are independent, we can condition on IP to obtain

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Probability of Not Stocking Out

1

1

1

1

1

Pr( )Pr(not stocking out) ( , ) 1

1 Pr( )

Pr( )

1 ( )

1 ( 1)

r Q

x r

r Q

x r

r Q

x r

r Q

x r

r Q

x r

X xS Q r

Q

X x

Q

X x

Q

G xQ

G xQ

Probability of not stocking out for the base-stock policy

with base-stock x

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It can be shown that

[( ) ] [( ( )) ]( , ) 1

[ ( )] [ ( )] 1

E X r E X r QS Q r

Q

E B r E B r Q

Q

Expected backorder level under a base-stock policy with base-stock level r+1

Expected backorder level under a base-stock policy with base-stock level r+Q+1

1

0

( [ ( )] [ ] [1 ( )])r

x

E B r E D L G x

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Service Level Approximations

Type I Service:

Type II Service:

)(, rGr)S(Q

, 1 [ ( )]/S(Q r) E B r Q

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Expected Backorders and Inventory

To emphasize the dependency of inventory and backorder levels on the choice of Q and R, we let I(Q, r) and B(Q, r) denote respectively inventory level I and backorder level B when we use a (Q, r) policy.

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Expected Backorders and Inventory

(Continued…)

Conditioning on the value IP also leads to

1 1

1 1

[( ) ] [ ( )][ ( , )]

[( ) ] [ ( )][ ( , )]

r Q r Q

x r x r

r Q r Q

x r x r

E X x E B xE B Q r

Q Q

E x X E I xE I Q r

Q Q

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Since IN = IP – X, we have

E[IN] = E[IP] – E[X] = r +(Q+1)/2 –[D]L

E[I(Q,r)] = r +(Q+1)/2 –E[D]L + E[B(Q,r)]

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h: inventory holding cost per unit per unit time b: backorder cost per unit per unit time. A: ordering cost per order

The Expected Total Cost

( , ) [ ]/ [ ( , )] [ ( , )]Y Q r AE D Q hE I Q r bE B Q r

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The Expected Total Cost (Continued…)

( , ) [ ]/ [ ( , )] [ ( , )]

[ ]/ ( 1) / 2 [ ] [ ( , )] [ ( , )]

[ ]/ ( 1) / 2 [ ] ( ) [ ( , )]

Y Q r AE D Q hE I Q r bE B Q r

AE D Q h r Q E D L E B Q r bE B Q r

AE D Q h r Q E D L h b E B Q r

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We want to choose r and Q so that expected total cost (the sum of expected ordering cost, inventory holding cost and backorder cost per unit time) is minimized.

Objective

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Solution Approach

Y(Q, r) is jointly convex Q and in r. Therefore, an efficient computational search can be implemented to solve for Q* and r*, the optimal values of Q and r, respectively.

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An Approximate Solution Approach

1. Approximate E[B(Q,r)] by E[B(r)]

2. Assume demand is continuous

3. Treat Q and r as continuous variables

( , ) [ ]/ ( 1) / 2 [ ] ( ) [ ( )]Y Q r AE D Q h r Q E D L h b E B r

2 [ ]*

AE DQ

h hb

brG

*)(

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An Approximate Solution Approach

If the distribution of leadtime demand is approximated by a normal distribution, then the optimal reorder point can be approximated by

*/( )

/( )

/( )

[ ] ( )

[ ] ( )

[ ]

b b h

b b h

b b h D

r E D L z Var X

E D L z Var D L

E D L z L

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Model with Backorder Costs per Occurrence

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Y(Q, r) = AE[D]/Q + hE[I(Q,r)]+ kE[D](1 - S(Q,r))

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Using the approximation

leads to

Y(Q,r) AE[D]/Q+h(r+(Q + 1)/2 – E[D]L+ E[B(r)])+ kE[D]E[B(r)]/Q

, 1 [ ( )]/S(Q r) E B r Q

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Using the approximation

leads to

Y(Q,r) AE[D]/Q + h(r + (Q + 1)/2 – E[D]L+ E[B(r)])+ kE[D]E[B(r)]/Q

An approximate solution is then given by

, 1 [ ( )]/S(Q r) E B r Q

2 ( )*

AE DQ

h

( )( *)

( ) *

kE DG r

kE D hQ

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Insights from the (Q, r) Model

Main Insights: Increasing the reorder point r increases safety stock but provides a greater buffer against stockouts

Increasing the order quantity increases cycle stock but reduces ordering (or setup) costs

Increasing the order quantity leads to a decrease in the reorder point

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Insights from the (Q, r) Model

Main Insights: Increasing the reorder point r increases safety stock but provides a greater buffer against stockouts

Increasing the order quantity increases cycle stock but reduces ordering (or setup) costs

Increasing the order quantity leads to a decrease in the reorder point

Other Insights: Increasing Ltends to increase the optimal reorder point Increasing the variability of the demand process tends to increase the optimal reorder point

Increasing the holding cost tends to decrease the optimal order quantity and reorder point

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The (R, r) Model

This is usually called the (S, s) model Each demand order can be for multiple units Demand orders are stochastic A replenishment order is placed to bring inventory position back to R

Decision variables are R (instead of Q) and r

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Dealing with Lead Time Variability

L: replenishment lead time (a random variable)E(L): expected replenishment leadtimeVar(L): variance of lead timeD: demand per periodE(D) = expected demand per period Var(D): variance demand

E(X)=E(L)E(D)Var(X)=E(L)Var(D) + E(D)2Var(L)

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Under the Normal approximation, the optimal reorder can be obtained as

Dealing with Lead Time Variability (Continued…)

*/( )

2/( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

b b h

b b h

r E D E L z Var X

E D E L z Var D E L E D Var L

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1 1

1 1

[ ] [ ] [ [ | ]]

Since [ | ] [ ] [ ],

[ ] [ [ ]] [ ] [ ]

L L

i ii i

L n

i ii i

E X E D E E D L n

E D L n E D nE D

E X E nE D E L E D

Expected Leadtime Demand

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22

1 1

2 2

1 1

Var[ ]

| ,

L L

i ii i

L L

i ii i

X E D E D

E D E E D L n

Variance of Leadtime Demand

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2 2

1 1

22 2

1 1

22 2

1

22 2

1

|

Var Var [ ]

| [ ]Var [ ] [ ]

[ ]Var [ ] [ ]

L n

i ii i

n n

i ii i

L

ii

L

ii

E D L n E D

D E D n D n E D

E E D L n E L D E L E D

E D E L D E L E D

Variance of Leadtime Demand

36

22

1 1

22 2

2 2 2

2

Var( )

= [ ]Var( ) [ ] [ ] [ ] [ ]

[ ]Var( ) [ ] [ ] [ ]

= [ ]Var( ) [ ] ( )

L L

i ii iX E D E D

E L D E L E D E L E D

E L D E D E L E L

E L D E D Var L

Variance of Leadtime Demand (Continued..)

The ( Q , r ) Model

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