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Economic Production Quantity (EPQ) Model Overview

The EPQ model extends the EOQ model by considering production/delivery not being instantaneous. It involves producing and delivering units one at a time with a finite production rate. The model aims to minimize total costs while ensuring no stockouts occur. Utilizing cyclic policies and order quantities, the EPQ model provides insights into optimal order intervals and quantities for efficient inventory management. Comparative analysis with the EOQ model and implications for production facility utilization are also discussed.

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Economic Production Quantity (EPQ) Model Overview

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  1. The Economic Production Quantity (EPQ) Model 1

  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 2

  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) 3

  4. Inventory versus Time Qmax Inventory Tp TD Time 4

  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 5

  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 6

  7. The Economic Production Quantity  ( ) dQ (1 / ) dY Q h D P D A    0 2 Q 2 2  AD D P  Q * (1 / ) h   ( ) 2 (1 / ) Y Q ADh D P * 7

  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. 8

  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? 9

  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 10

  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 11

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

  13. N D   i  1 i  In order to ensure feasibility, we must have P 1 i 2  A D *   Choosing can lead to stockouts i i Q i 1 ( i h / ) D P i i 13

  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 Qiof each product i is produced and consumed; therefore, Qi= DiT 14

  15. Inventory versus Time P1 Inventory P2 P3 Time 15

  16. Order Quantities and Order Interval  2 1 ( / ) h Q D P D i i i i i     Cost for Product i: Q ( ) Y A c D i i i i Q i 16

  17. Order Quantities and Order Interval  2 1 ( / ) h Q D P D i i i i i     Cost for Product i: Q ( ) Y A c D i i i i Q i  1 ( i / ) h Q D P D N i i i i   i     Total Cost: ( , ,..., ) { } Y Q Q Q A c D 1 2 N i i i 2 Q 1 i 17

  18. Order Quantities and Order Interval  2 1 ( / ) h Q D P D i i i i i     Cost for Product i: Q ( ) Y A c D i i i i Q i  1 ( i / ) h Q D P D N i i i i   i     Total Cost: ( , ,..., ) { } Y Q Q Q A c D 1 2 N i i i 2 Q 1 i N hDT A T '      Since Qi/Di= T, ( , Y Q Q ,..., ) { } Q c D i i i 1 2 N i i 2  1 i   (1 / ) h h D P ' i i i i 18

  19. Optimal Order Interval and Order Quantities N 2 iA   i 1  * T N ' ih D   i i 1 * i Q  * D T i 19

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