LESSON 18: INVENTORY MODELS (STOCHASTIC)
Q,R SYSTEMS
OPTIMIZATION WITHOUT SERVICE
Outline
• Multi-Period Models
– Lot size-Reorder Point (Q, R) Systems
• Optimization without service
– Procedure
– Example
Procedure to find the Optimal (Q,R) Policy
Without Any Service Constraint
Goal: Given , , h, K , p find (Q,R) to minimize total cost
Step 1: Take a trial value of Q = EOQ
Step 2: Find a trial value of R =   z where  and 
are respectively mean and standard deviation of the
lead-time demand and z is the normal distribution
variate corresponding to the area on the right, 1-F(z)
= Qh / p see Table A-4, pp. 835-841
Step 3: Find the expected number of stock-outs per
cycle, n  L(z ) where L(z ) is the standardized loss
function available from Table A-4, pp. 835-841
Procedure to find the Optimal (Q,R) Policy
Without Any Service Constraint
Step 4: Find the modified
2
np  K 
Q
h
Step 5: Find the modified value of R =   z where z
is the recomputed value of the normal distribution
variate corresponding to the area on the right, 1-F(z)
= Qh / p see Table A-4, pp. 835-841
Step 6: If any of modified Q and R is different from the
previous value, go to Step 3. Else if none of Q and R
is modified significantly, stop.
Example - Optimal (Q,R) Policy
Annual demand for number 2 pencils at the campus store
is normally distributed with mean 2,000 and standard
deviation 300. The store purchases the pencils for 10
cents and sells them for 35 cents each. There is a twomonth lead time from the initiation to the receipt of an
order. The store accountant estimates that the cost in
employee time for performing the necessary paper work
to initiate and receive an order is $20, and recommends a
25 percent annual interest rate for determining holding
cost. The cost of a stock-out is the cost of lost profit plus
an additional 20 cents per pencil, which represents the
cost of loss of goodwill. Find an optimal (Q,R) policy
Example - Optimal (Q,R) Policy
Fixed ordering cost, K 
Holding cost, h  Ic 
Penalty cost, p 
Mean annual demand,  
Standard deviation of annual demand,  y 
Lead time,  
Mean lead - time demand,    
Standard deviation of lead - time demand,    y 

Example - Optimal (Q,R) Policy
Iteration 1
Step 1:
Q  EOQ 
2 K

h
Step 2:
1 F ( z) 
Qh

p
z
(Table A-4)
R    z 
Example - Optimal (Q,R) Policy
Step 3: L(z ) 
n  L(z ) 
(Table A-4)
Step 4:
2
np  K  
Q
h
Question: What are the stopping criteria?
Qh
Step 5: 1  F ( z )  p 
z
R    z 
(Table A-4)
Iteration 2 Example - Optimal (Q,R) Policy
Step 3: L(z ) 
(Table A-4)
n  L(z ) 
Step 4:
2
np  K  
Q
h
Question: Do the answers converge?
Qh
Step 5: 1  F ( z )  p 
z
R    z 
(Table A-4)
Fixed cost (K )
Holding cost (h )
Penalty cost (p )
Mean annual demand ()
Lead time (in years
Lead time demand parameters:
Mean,
Standard deviation, 
Step 1 Q =
Step 2 Area on the right=1-F (z )
z=
R=
Step 3 L(z )=
n=
Step 4 Modified Q =
Step 5 Area on the right=1-F (z )
z=
Modified R =
Note: K , h , and p
are input data
input
input data
<--- computed
input data
Iteration 1 Iteration 2
EO Q
Q h / p
Table A1/A4
  z
Table A4
 L( z)
2  np  K  / h
Q h / p
Table A1/A4
  z
READING AND EXERCISES
Lesson 18
Reading:
Section 5.4, pp. 262-264 (4th Ed.), pp. 253-255 (5th
Ed.)
Exercise:
13a, p. 271 (4th Ed.), p. 261