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

Inve

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ry

TimeTp 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

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

Costs

7

The Economic Production Quantity

2

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dY Q h D P A

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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

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?

Production Facility Utilization

10

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

EOQ vs. EPQ

11

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

Systems with Multiple Products

12

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

Objective

13

In order to ensure feasibility, we must have

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14

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

A Cyclic Policy

15

Inventory versus Time

P1

P2

P3

Time

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ry

16

Order Quantities and Order Interval

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17

Order Quantities and Order Interval

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Total Cost: }2

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18

Order Quantities and Order Interval

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19

Optimal Order Interval and Order Quantities

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12

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