1. Production Lot Size Models
Inventory Models
Production Lot Size Models

2. PRODUCTION LOT SIZE MODELS
In a production lot size model, we are a manufacturer, trying to determine how much to produce (the production lot size) during each production run.
Like the EOQ model with Q = the production lot size except:
We are producing at a rate P/yr. that is greater than the demand rate of D/yr.
Otherwise run process continuously and sell items as fast as they are produced
Inventory does not “jump” to Q but builds up to a value IMAX that is reached when production is ceased

3. What is the maximum inventory and the average inventory per cycle?
Length of a production run = Q/P
During a production run
Amount Produced = Q
Amount Demanded = D(Q/P)
IMAX = Q - D(Q/P) = (1-D/P)Q
Average inventory = IMAX/2 = ((1-D/P)/2)Q

4. Production Lot Size Model – Understanding the inventory profile
Demand accumulation
during production run = DT1
= D(Q/P) = Q(D/P)
The production increases the
inventory at a rate of P.
Length of Production Run = Q/P T1
Maximum inventory: IMAX
IMAX= Q – Q(D/P) = Q(1 – D/P)
The inventory increases at a net rate of P - D
Production
Lot Size = Q = PT1
The demand decreases the
inventory at a rate of D.
Demand accumulation
during production run

5. PRODUCTION LOT SIZE -- TOTAL ANNUAL COST
Q = The production lot size
CO = Set-up cost rather than order cost =$/setup
Number of Set-ups per year = D/Q
Average Inventory = ((1-D/P)/2)Q
Instantaneous set-up time/infinite time horizon
TC(Q) = CO(D/Q) + Ch((1-D/P)/2)Q + CD

6. OPTIMAL PRODUCTION LOT SIZE, Q*
TC(Q) = CO(D/Q) + Ch((1-D/P)/2)Q + CD

7. EXAMPLE-- Farah Cosmetics
Production Capacity 1000 tubes/hr.
Daily Demand 1680 tubes
Production cost $0.50/tube (C = 0.50)
Set-up cost $150 per set-up (CO = 150)
Holding Cost rate: 40% (Ch = .4(.50) = .20)
Since demand is 1680 per day and the production rate is 1000 per hour:
D = 1680(365) = 613,200
P = 1000(24)(365) = 8,760,000

8. OPTIMAL PRODUCTION LOT SIZE

9. TOTAL ANNUAL COST TOTAL ANNUAL COST = TC(Q) = TV(Q) + CD
TV(Q) = CO(D/Q) + Ch((1-D/P)/2)Q =
(150)(613,200/31,449) + .2((1-(613,200/8,760,000))(31,449)/2)
= $5,850
TC(Q) = TV(Q) + CD =
5, (613,200) = $312,450

10. OTHER QUANTITES Length of a Production run = Q*/P =
31,449/8,760,000 = yrs. = (365)
= 1.31 days
Length of a Production cycle = Q*/D =
31,449/613,200 = yrs. = (365) = days
# of Production runs/yr. = D/Q* = 19.5
IMAX = (1-(613,200/8,760,000))(31,449) = 29,248

11. REORDER (SETUP) POINT ANALYSIS
The reorder point (actually the setup point) and safety stock determination are not affected by the calculation of Q*.
It is found in the same way as before:
r* = LD + SS if demand is constant over lead time
r* is found using service levels if demand varies during lead time

12. Using the Template Optimal Values Enter Parameters Production Lot Size
Worksheet

13. Review
Production Lot Size Models find the amount to produce per production run
P > D, else optimal solution is to run machine continuously
Same as EOQ except IMAX = Q(1-D/P)
Length of a production run = Q*/P
Length of the production cycle = Q*/D
Reorder (setup) point analysis is not affected.
Use of template