Managing for Quality

Inventory Management Outline Basic Definitions and Ideas •Reasons to Hold Inventory •Inventory Costs Inventory Control Systems •Continuous Review Models •Basic EOQ Model •Quantity Discounts •Safety Stock •Special Case: The News Vendor Problem •Discrete Probability Example •Continuous Probability Example •Periodic Review Model What is Inventory? • Inventory is a stock of items held to meet future demand. • Inventory management answers two questions: – How much to order – When to order Basic Concepts of Inventory Management can be expanded to apply to a broad array of types of “inventory”: – – – – – – – – Raw materials Purchased parts and supplies Labor In-process (partially completed) products Component parts Working capital Tools, machinery, and equipment Finished goods Reasons to Hold Inventory • Meet unexpected demand • Smooth seasonal or cyclical demand • Meet variations in customer demand • Take advantage of price discounts • Hedge against price increases • Quantity discounts Two Forms of Demand • Dependent – items used to produce final products • Independent – items demanded by external customers Inventory Costs • Carrying Cost – cost of holding an item in inventory • Ordering Cost – cost of replenishing inventory • Shortage Cost – temporary or permanent loss of sales when demand cannot be met Inventory Control Systems • Fixed-order-quantity system (Continuous) – constant amount ordered when inventory declines to predetermined level • Fixed-time-period system (Periodic) – order placed for variable amount after fixed passage of time Continuous Review Models • Basic EOQ Model • Quantity Discounts • Safety Stock The Basic EOQ Model (Economic Order Quantity) Assumptions of the Basic EOQ Model: Demand is known with certainty Demand is relatively constant over time No shortages are allowed Lead time for the receipt of orders is constant – The order quantity is received all at once – – – – Inventory Order Cycle Basic EOQ Model 750 500 Inventory 250 0 -250 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 Days EOQ Model Costs S = cost of placing order H = annual per-unit carrying cost Annual ordering cost = SD/Q Total cost = SD/Q + HQ/2 D = annual demand Q = order quantity Annual carrying cost = HQ/2 Q* = Economic Order Quantity EOQ Cost Curves EOQ Example 1000 750 Total Annual Cost 500 Ordering Holding Total 250 0 0 100 200 300 400 500 600 700 800 900 1000 Order Quantity EOQ Example If D = 1,000 per year, S = $62.50 per order, and H = $0.50 per unit per year, what is the economic order quantity? Q* 2DS H 2 * 1000 * 62.5  0.5  500  Quantity Discounts Price per unit decreases as order quantity increases: Quantity 1-49 50-74 75-149 150-299 300-499 500+ Price $35.00 $34.75 $33.55 $32.35 $31.15 $30.75 Quantity Discount Costs DS QH TC    CD Q 2 C  per unit price D = annual demand Quantity Discount Cost Curves EOQ with Quantity Discounts $85,000 $82,500 Total Annual Cost $80,000 $77,500 0-49 50-74 75-149 150-299 300-499 500+ $75,000 $72,500 $70,000 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 Order Quantity Quantity Discount Algorithm Step 1. Calculate a value for Q*. Step 2: For any discount, if the order quantity is too low to qualify for the discount, adjust Q upward to the lowest feasible quantity. Step 3: Calculate the total annual cost for each Q*. Quantity Discount Algorithm Step 1. Calculate a value for Q*. Q*   2DS H 2 * 2 , 400 * 10 3.33  120 Quantity Discount Algorithm Step 2: For any discount, if the order quantity is too low to qualify for the discount, adjust Q* upward to the lowest feasible quantity. Quantity 1-49 50-74 75-149 150-299 300-499 500+ Price $35.00 $34.75 $33.55 $32.35 $31.15 $30.75 Min 1 50 75 150 300 500 Q* 120 120 120 120 120 120 Adj. Q* 120 120 120 150 300 500 EOQ with Quantity Discounts $85,000 $82,500 Total Annual Cost $80,000 $77,500 0-49 50-74 75-149 150-299 300-499 500+ $75,000 $72,500 $70,000 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 Order Quantity Quantity Discount Algorithm Step 3: Calculate the total annual cost for each Q*. Quantity 1-49 50-74 75-149 150-299 300-499 500+ Price Min Q* Adj. Q* Holding Cost Ordering Cost Purchasing Cost Total Cost $35.00 1 120 120 $ 199.90 $ 199.90 $ 84,000.00 $84,399.80 $34.75 50 120 120 $ 199.90 $ 199.90 $ 83,400.00 $83,799.80 $33.55 75 120 120 $ 199.90 $ 199.90 $ 80,520.00 $80,919.80 $32.35 150 120 150 $ 249.75 $ 160.00 $ 77,640.00 $78,049.75 $31.15 300 120 300 $ 499.50 $ 80.00 $ 74,760.00 $75,339.50 $30.75 500 120 500 $ 832.50 $ 48.00 $ 73,800.00 $74,680.50 EOQ with Quantity Discounts $85,000 $82,500 Total Annual Cost $80,000 $77,500 $75,000 $72,500 $70,000 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 Order Quantity When to Order Reorder Point = level of inventory at which to place a new order (a.k.a. ROP, R) R = dL Where d = demand rate per period L = lead time 750 500 Inventory 250 0 -250 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 Days Lead time for one of your fastest-moving products is 21 days. Demand during this period averages 100 units per day. What would be an appropriate reorder point? R  dL  100 * 21  2,100 What About Random Demand? (Or Random Lead Time?) 4000 3500 3000 2500 Inventory 2000 1500 1000 500 0 -500 -1000 0 50 100 150 200 250 300 350 Days •Safety stock – buffer added to on-hand inventory during lead time •Stockout – an inventory shortage •Service level – probability that the inventory available during lead time will meet demand Reorder Point with Variable Demand (Leadtime is Constant) R  d L  z , where d = averagedaily demand L = lead time  d  standarddeviationof daily demand   standarddeviationof demand during lead time z = number of standarddeviationsfor desiredservice level z  safety stock A carpet store wants a reorder point with a 95% service level and a 5% stockout probability during the leadtime. d = 30 yards per day L = 10 days  d  5 yards per day Determining the z-value for Service Level Variance of Lead Time Demand=(daily variances)x (number of days of lead time) 2 = d L Standard deviation  d L = 2 = d L d = 30 yards per day L = 10 days  d  5 yards per day R  d L  z  ( 30)(10)  ( 1.65)(5)( 10 )  300  26.1  326.1 yards Safety stock  z  ( 1.65)(5)( 10 )  26.1 yards Determining the Safety Stock from the z-value What If Leadtime is Random? Random Variable L = Leadtime d = Demand dL = Demand during Leadtime Mean L d dL Standard Deviation L d 2 2 L d  d  L 2 Special Case: The Newsboy Problem The News Vendor Problem is a special “single period” version of the EOQ model, where the product drops in value after a relatively brief selling period. The name comes from newspapers, which are much less valuable after the day they are originally published. This model may be useful for any product with a short product life cycle, such as •Time-sensitive Materials (newspapers, magazines) •Fashion Goods (some kinds of apparel) •Perishable Goods (some food products) Two new assumptions: •There are two distinct selling periods: •an initial period in which the product is sold at a regular price •a subsequent period in which the item is sold at a lower “salvage” price. •Two revenue values: •a regular price P, at which the product can be sold during the initial selling period •a salvage value V, at which the product can be sold after the initial selling period. The salvage value is frequently less than the cost of production C, and in general we wish to avoid selling units at the salvage price. “Damned if you do; damned if you don’t”: • If we order too many, there will be extra units left over to be sold at the disadvantageous salvage price. • If we order too few, some customer demand will not be satisfied, and we will forego the profits that could have been made from selling to the customer. Discrete Probability Example Demand 300 400 500 600 700 800 Probability 0.05 0.10 0.40 0.30 0.10 0.05 C = $8.00 P = $20.00 V = $4.00 Newsboy Solution In this case, it is useful to examine the marginal benefit from each unit purchased. The expected profit from any unit purchased is: Expected Profit = P(Selling at Regular Price)*(Profit if Sold at Regular Price) + P(Selling at Salvage Price)*(Profit if Sold at Salvage Price) = P(Selling at Regular Price)*(P - C) + P(Selling at Salvage Price)*(V - C) Demand 300 400 500 600 700 800 Probability 0.05 0.10 0.40 0.30 0.10 0.05 Demand 300 400 500 600 700 800 P(Sell) 1.00 0.95 0.85 0.45 0.15 0.05 P(Not Sell) 0.00 0.05 0.15 0.55 0.85 0.95 Demand 300 400 500 600 700 800 Prob. 0.05 0.10 0.40 0.30 0.10 0.05 P(Sell) 1.00 0.95 0.85 0.45 0.15 0.05 Profit if Sold $ 12.00 $ 12.00 $ 12.00 $ 12.00 $ 12.00 $ 12.00 P(Not Sell) 0.00 0.05 0.15 0.55 0.85 0.95 Profit if Not Sold $ (4.00) $ (4.00) $ (4.00) $ (4.00) $ (4.00) $ (4.00) Weighted Average Profit $ 12.00 $ 11.20 $ 9.60 $ 3.20 $ (1.60) $ (3.20) Marginal Expected Profit $14 $12 $10 $5,000 $8 $6 $4 $2 $2,000 $$(2) $(4) 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 Total Expected Profit $7,000 $6,000 $4,000 $3,000 $1,000 $0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 Quantity Ordered Quantity Ordered Based on this analysis, we would order 600 units. Continuous Probability Example Using the same mean and standard deviation as in the previous case (545.0 and 111.7), what would be optimal if demand were normally distributed? Define CO and CU to be the “costs” of over-ordering and under-ordering, respectively. In this case: CO  V  C   4  8   $4.00 CU  P C  20  8  $12.00 It can be shown that the optimal order quantity is the value in the demand distribution that corresponds to the “critical probability”: Critical probability  CU CU  CO 12  12  4 12  16  0.75 From the standard normal table, the z-value corresponding to a 0.75 probability is 0.6745. Q    0.6745  545.0  0.6745111.7   620.3 Periodic Review Models Sometimes a continuous review system doesn’t make sense, as when the item is not very expensive to carry, and/or when the customers don’t mind waiting for a backorder. A periodic review system only checks inventory and places orders at fixed intervals of time. A basic periodic review system might work as follows: Every T time periods, check the inventory level I, and order enough to bring inventory back up to some predetermined level. This “order-up-to” level should be enough to cover expected demand during the lead time, plus the time that will elapse before the next periodic review. Q  dT  L  I We might also build some safety stock in to the “order-up-to” quantity. Q  d T  L   z T  L  I Summary Basic Definitions and Ideas •Reasons to Hold Inventory •Inventory Costs Inventory Control Systems •Continuous Review Models •Basic EOQ Model •Quantity Discounts •Safety Stock •Special Case: The News Vendor Problem •Discrete Probability Example •Continuous Probability Example •Periodic Review Model