1. The Economic Production Quantity (EPQ) Model

2. Similar assumptions to the EOQ model, except that production/delivery is not instantaneous
Units are produced and delivered one unit at a time
Production capacity is finite with a finite production rate P

3. Notation P: production rate (number of units/time period)
TP: production cycle (time facility is producing per order cycle)
TD: withdrawal cycle (time facility is idle per order cycle)
T: total inventory cycle (time between setups)
Qmax: maximum inventory level (units)

4. Inventory versus Time
Qmax
Inventory
Time Tp TD

5. TP = Q/P
TD = Qmax/D
Qmax = TP(P - D) = Q(1 - D/P)
Average inventory = Qmax/2
Number of orders per unit time = D/Q

6. Costs Total holding cost = hQmax/2=hQ(1-D/P)/2
Total ordering/setup cost = AD/Q
Total production/purchasing cost = cD
Total cost = AD/Q + hQ(1 - D/P)/2 + cD
Unit cost = A/Q + hQ(1 - D/P)/2D + c

7. The Economic Production Quantity

8. The EPQ is equivalent to an EOQ model with holding cost h’=h(1-D/P).
Consequently, the optimal cost under the EPQ model is lower than the optimal cost under the EOQ model with holding cost h.

9. Production Facility Utilization
U = D/P (capacity utilization)
We must not operate above capacity (i.e., always keep U  1)
What happens when D > P?
What happens when D = P?

10. EOQ vs. EPQ Q*(EPQ)  Q*(EOQ) when U  1 Q*(EPQ) = Q*(EOQ) when U  0
Q*(EPQ)  infinity when U  1 (continuous production)
Y(Q*(EPQ))  cD when U  1

11. Systems with Multiple Products
N: number of products
Di: demand rate for product i
Pi: production rate for product i
hi: holding cost per unit per unit time for product i
Ai: Ordering/setup cost for product i
ci: production cost for product i

12. Objective
Minimize total cost while guaranteeing that no stockouts occur for any product.

13. In order to ensure feasibility, we must have
Choosing can lead to stockouts

14. A Cyclic Policy
A strictly cyclic policy is used (in each cycle, there is exactly one setup per product)
Cycle time, T, is the time between two consecutive setups for any given product
During T, a quantity Qi of each product i is produced and consumed; therefore, Qi = DiT

15. Inventory versus Time P1 P2 P3
Inventory
Time

16. Order Quantities and Order Interval
Cost for Product i:

17. Order Quantities and Order Interval
Cost for Product i:
Total Cost:

18. Order Quantities and Order Interval
Cost for Product i:
Total Cost:
Since Qi/Di = T,

19. Optimal Order Interval and
Order Quantities