Lecture 24 Order Quantities (Continued) Books Introduction to Materials Management, Sixth Edition, J. R. Tony Arnold, P.E., CFPIM, CIRM, Fleming College, Emeritus, Stephen N. Chapman, Ph.D., CFPIM, North Carolina State University, Lloyd M. Clive, P.E., CFPIM, Fleming College Operations Management for Competitive Advantage, 11th Edition, by Chase, Jacobs, and Aquilano, 2005, N.Y.: McGraw-Hill/Irwin. Operations Management, 11/E, Jay Heizer, Texas Lutheran University, Barry Render, Graduate School of Business, Rollins College, Prentice Hall

Objectives Probabilistic Models and Safety Stock Probabilistic Demand Other probabilistic models Fixed period system EOQ consequences Period order quantity model

Probabilistic Models and Safety Stock Used when demand is not constant or certain Use safety stock to achieve a desired service level and avoid stockouts ROP = d x L + ss Annual stockout costs = the sum of the units short x the probability x the stockout cost/unit x the number of orders per year

Safety Stock Example Number of Units Probability 30 .2 40 ROP  50 .3 ROP = 50 units Stockout cost = $40 per frame Orders per year = 6 Carrying cost = $5 per frame per year Number of Units Probability 30 .2 40 ROP  50 .3 60 70 .1 1.0

Additional Holding Cost Safety Stock Example ROP = 50 units Stockout cost = $40 per frame Orders per year = 6 Carrying cost = $5 per frame per year Safety Stock Additional Holding Cost Stockout Cost Total Cost 20 (20)($5) = $100 $0 $100 10 (10)($5) = $ 50 (10)(.1)($40)(6) = $240 $290 $ 0 (10)(.2)($40)(6) + (20)(.1)($40)(6) = $960 $960 A safety stock of 20 frames gives the lowest total cost ROP = 50 + 20 = 70 frames

Probabilistic Demand Inventory level Time Safety stock 16.5 units ROP  Place order Inventory level Time Minimum demand during lead time Maximum demand during lead time Mean demand during lead time ROP = 350 + safety stock of 16.5 = 366.5 Receive order Lead time Normal distribution probability of demand during lead time Expected demand during lead time (350 kits)

Probabilistic Demand Probability of no stockout 95% of the time Mean demand 350 Risk of a stockout (5% of area of normal curve) ROP = ? kits Quantity Safety stock Number of standard deviations z

Probabilistic Demand Use prescribed service levels to set safety stock when the cost of stockouts cannot be determined ROP = demand during lead time + ZsdLT where Z = number of standard deviations sdLT = standard deviation of demand during lead time

Probabilistic Example Average demand = m = 350 kits Standard deviation of demand during lead time = sdLT = 10 kits 5% stockout policy (service level = 95%) Using Appendix I, for an area under the curve of 95%, the Z = 1.65 Safety stock = ZsdLT = 1.65(10) = 16.5 kits Reorder point = expected demand during lead time + safety stock = 350 kits + 16.5 kits of safety stock = 366.5 or 367 kits

Other Probabilistic Models When data on demand during lead time is not available, there are other models available When demand is variable and lead time is constant When lead time is variable and demand is constant When both demand and lead time are variable

Other Probabilistic Models Demand is variable and lead time is constant ROP = (average daily demand x lead time in days) + ZsdLT where sd = standard deviation of demand per day sdLT = sd lead time

Probabilistic Example Average daily demand (normally distributed) = 15 Standard deviation = 5 Lead time is constant at 2 days 90% service level desired Z for 90% = 1.28 From Appendix I ROP = (15 units x 2 days) + Zsdlt = 30 + 1.28(5)( 2) = 30 + 9.02 = 39.02 ≈ 39 Safety stock is about 9 iPods

Other Probabilistic Models Lead time is variable and demand is constant ROP = (daily demand x average lead time in days) = Z x (daily demand) x sLT where sLT = standard deviation of lead time in days

Probabilistic Example Z for 98% = 2.055 From Appendix I Daily demand (constant) = 10 Average lead time = 6 days Standard deviation of lead time = sLT = 3 98% service level desired ROP = (10 units x 6 days) + 2.055(10 units)(3) = 60 + 61.65 = 121.65 Reorder point is about 122 cameras

Other Probabilistic Models Both demand and lead time are variable ROP = (average daily demand x average lead time) + ZsdLT where sd = standard deviation of demand per day sLT = standard deviation of lead time in days sdLT = (average lead time x sd2) + (average daily demand)2 x sLT2

Probabilistic Example Average daily demand (normally distributed) = 150 Standard deviation = sd = 16 Average lead time 5 days (normally distributed) Standard deviation = sLT = 1 day 95% service level desired Z for 95% = 1.65 From Appendix I ROP = (150 packs x 5 days) + 1.65sdLT = (150 x 5) + 1.65 (5 days x 162) + (1502 x 12) = 750 + 1.65(154) = 1,004 packs

Fixed-Period (P) Systems Orders placed at the end of a fixed period Inventory counted only at end of period Order brings inventory up to target level Only relevant costs are ordering and holding Lead times are known and constant Items are independent from one another

Fixed-Period (P) Systems Target quantity (T) On-hand inventory Time Q1 Q2 Q3 Q4 P P

Fixed-Period (P) Example 3 jackets are back ordered No jackets are in stock It is time to place an order Target value = 50 Order amount (Q) = Target (T) - On-hand inventory - Earlier orders not yet received + Back orders Q = 50 - 0 - 0 + 3 = 53 jackets

Fixed-Period Systems Inventory is only counted at each review period May be scheduled at convenient times Appropriate in routine situations May result in stockouts between periods May require increased safety stock

EOQ Assumptions Demand is relatively constant and is known. The item is produced or purchased in lots or batches and not continuously. Order prep costs & inventory-carrying costs are constant and known. Replacement occurs all at once.

EOQ Consequences Average inventory = EOQ lot size / 2 # of orders per year = Annual demand / lot size

Basic EOQ Model Basic EOQ: Demand is constant over time Inventory drops at a uniform rate over time When the inventory reaches 0, the new order is placed and received, and the inventory level again jumps to Q units The optimal order quantity will occur at a point where the total setup cost is equal to the total holding cost Basic EOQ:

Basic EOQ Model (cont.) Benefit of EOQ model: Reorder Points: It is a robust model, meaning that it gives satisfactory answers even with substantial variation in the parameters. Reorder Points: Lead Time - the time between the placement and receipt of an order. The when-to-order decision is expressed in terms of a reorder point, the inventory level at which an order should be placed.

Inventory Level Over Time (Basic EOQ Model) Maximum Inventory Level Average

Production Order Quantity Model Production Order Quantity Model is useful when: Inventory continuously flows or builds up over a period of time after an order has been placed or when units are produced and sold simultaneously. Takes into account the daily production (or inventory flow) rate and the daily demand rate. All other EOQ assumptions are valid. Production Order Quantity

Inventory Level Over Time (Production Model) Production Portion of Cycle Maximum Inventory Level Demand Portion

Period Order Quantities Calculate or determine EOQ Determine avg. weekly usage Divide EOQ by avg. weekly usage to determine period Order the amount needed during the next period to satisfy demand during that period

Practice Question 1. Sarah’s Silk Screening  Sarah’s Silk Screening sells souvenir shirts. Sarah is trying to decide how many to produce for the upcoming naming of the College of Management. The University will allow her to sell the shirts only on one day, the day that the school naming is announced. Sarah will sell the T-shirts for $20 each. When the event day is over, she will be allowed to sell the remaining stock to the Bookstore for $4 each. It costs Sarah $8 to make the specialty shirt. She estimates mean demand to be 545, with a standard deviation is 115. How many shirts should she make?

Practice Question 2. The Great Southern Automotive Co.  The Great Southern Automotive Co. buys steering wheels from a supplier. One particular steering wheel has a known and constant demand rate of 2,000 units per year. The fixed cost of ordering is $100 and the inventory holding cost is $2 per unit per year. It takes 2 weeks for an order to arrive. Compute   The optimal EOQ The reorder point The average inventory level The time between successive orders The total annual cost If demand was variable with a standard deviation of 4 units per week, and the firm aims for 98% customer satisfaction, what would the reorder point be?

End of Lecture 24