1. PERSEDIAAN PROBABILISTIK: PERIODIC REVIEW

2. Replenishment Policies
When and how much to order
Continuous review with (Q,R) policy:
Inv. is continuously monitored and when it drops to R, an order of size Q is placed
Periodic review with (R,S) policy:
Inv. is reviewed at regular periodic intervals (R), and an order is placed to raise the inv. to a specified level (order-up-to level, S)

3. Periodic Review System: Order-Up-To-Level (R, S) System

4. Periodic Review Systems
Continuous Review Systems
Always knew level of on-hand inventory
Could place an order at any time
Often, we are constrained by WHEN we can order -- and it may be periodically
Train dispatched once a week
Delivery truck arrives each morning
Thus, we do not need to continuously review inventory, just check periodically

5. Periodic Review Systems
If demand were known and constant, we would just resort to our EOQ solution, possibly modifying it to meet shipping date
Now: demand is random variable
Setting:
Place an order every T periods
Policy: Order up to S
The value of Q (order quantity) will now change periodically
Previous concern: demand exceeding supply during the lead time
Now: demand exceeding supply during the period and lead time, or T + 

6. Periodic Review I(t) Order arrives. Cycle continues. S Q
Order up to S
every T periods of time.
I(t)
Order arrives. Cycle continues. S
Demand Q
Lead Time passes… t t
Time T

7. Expected cost function
Include expected holding, setup, penalty and ordering (per unit) costs
Average Inventory Level: S
At level R*,on average, order Q = S-R* units.
τ periods later, units arrive.
Inventory level? R* τ T

8. Expected cost function
Include expected: holding, setup, penalty and ordering (per unit) costs
Average Inventory Level: S
S-lt units present when
Q arrive (expected) as lt units consumed over
leading time. T

9. Expected cost function
Include expected: holding, setup, penalty and ordering (per unit) costs
Average Inventory Level: S
S - lt
lT units removed
(expected) from
inventory over
time T. T

10. Expected cost function
Include expected: holding, setup, penalty and ordering (per unit) costs
Average Inventory Level: S
S-lt lT
S-lt-lT T
Average Inventory Level =

11. Expected cost function
Include expected:
holding, setup, penalty and ordering (per unit) costs
Average Holding Cost:
Average Set-up Cost:

12. Expected cost function
Expected Shortage per Cycle:
f(x)dx = P(demand in T +  is between x and x + dx)
Expected Penalty Cost :

13. Cost Minimization Expected Cost Function: Derivative:
Recall that T and  are given:

14. Cost Minimization
Derivative:
or:

15. Example
A special control board is used in a version of a product on the production line
The board cost is $122.50
The holding cost rate is 30% per year
Reorders are placed at the start of each week, and the supplier delivers these parts in one week
The shortage cost is $100 per board due to worker downtime
Weekly demand is N (μ=125, δ2=104.17)
Set up cost (K) is $120
Find S

16. Solution Holding cost is:
h = Ic = .30 (122.50) = / 52 = $.7067 per week
Compute:
Demand Distribution is Normal
mean = variance =
Z = from Normal table
S = 125+(2.455)(104.17)1/2 =

17. Solution If penalty cost drops to $10 per unit: Compute:

18. References “Production & Operations Analysis” by S.Nahmias
“Factory Physics” by W.J.Hopp, M.L.Spearman
“Inventory Management and Production Planning and Scheduling” by E.A. Silver, D.F. Pyke, R. Peterson