Inventory Models Production Lot Size Models

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 I MAX that is reached when production is ceased

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) I MAX = Q - D(Q/P) = (1-D/P)Q Average inventory = I MAX /2 = ((1-D/P)/2)Q

PRODUCTION LOT SIZE -- TOTAL ANNUAL COST Q = The production lot size C O = 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) = C O (D/Q) + C h ((1-D/P)/2)Q + CD

OPTIMAL PRODUCTION LOT SIZE, Q* TC(Q) = C O (D/Q) + C h ((1-D/P)/2)Q + CD

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 (C O = 150) Holding Cost rate: 40% (C h =.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

OPTIMAL PRODUCTION LOT SIZE

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

OTHER QUANTITES Length of a Production runLength of a Production run = Q*/P = 31,449/8,760,000 =.00359yrs. =.00359(365) 1.31 days = 1.31 days Length of a Production cycleLength of a Production cycle = Q*/D = days 31,449/613,200 = yrs. = (365) = days # of Production runs/yr.19.5# of Production runs/yr. = D/Q* = 19.5 I MAX 29,248I MAX = (1-(613,200/8,760,000))(31,449) = 29,248

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

Using the Template Production Lot Size Worksheet Enter Parameters Optimal Values

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 I MAX = 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