OPIM 310-Lecture #5 Instructor: Jose Cruz Inventory Management OPIM 310-Lecture #5 Instructor: Jose Cruz

Inventory Stock of items held to meet future demand Inventory management answers two questions How much to order When to order

Types of Inventory Raw materials Purchased parts and supplies Labor In-process (partially completed) products Component parts Working capital Tools, machinery, and equipment

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 Independent Items used to produce final products Independent Items demanded by external customers

Inventory Costs Carrying Cost Ordering Cost Shortage 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 Continuous system (fixed-order-quantity) Constant amount ordered when inventory declines to predetermined level Periodic system (fixed-time-period) Order placed for variable amount after fixed passage of time

Assumptions of Basic EOQ Model Demand is known with certainty and is constant over time No shortages are allowed Lead time for the receipt of orders is constant The order quantity is received all at once

The Inventory Order Cycle Demand rate Time Lead time Order placed Order receipt Inventory Level Reorder point, R Order quantity, Q Figure 10.1

EOQ Cost Model Co - cost of placing order D - annual demand Cc - annual per-unit carrying cost Q - order quantity Annual ordering cost = CoD Q Annual carrying cost = CcQ 2 Total cost = +

EOQ Cost Model TC = + CoD Q CcQ 2 = + Q2 Cc TC Q 0 = + C0D Qopt = = + Q2 Cc TC Q 0 = + C0D Qopt = 2CoD Deriving Qopt Proving equality of costs at optimal point = CoD Q CcQ 2 Q2 = 2CoD Cc Qopt = Co - cost of placing order D - annual demand Cc - annual per-unit carrying cost Q - order quantity Annual ordering cost = CoD Q Annual carrying cost = CcQ 2 Total cost = +

EOQ Cost Model Annual cost ($) CoD Q Ordering Cost = Order Quantity, Q Figure 10.2

EOQ Cost Model Annual cost ($) CcQ 2 Carrying Cost = CoD Q Order Quantity, Q Annual cost ($) Carrying Cost = CcQ 2 Ordering Cost = CoD Q Figure 10.2

EOQ Cost Model Slope = 0 Total Cost Order Quantity, Q Annual cost ($) Minimum total cost Optimal order Qopt Carrying Cost = CcQ 2 Ordering Cost = CoD Q Figure 10.2

EOQ Example Cc = $0.75 per yard Co = $150 D = 10,000 yards Qopt = 2CoD 2(150)(10,000) (0.75) Qopt = 2,000 yards TCmin = + CoD Q CcQ 2 TCmin = + (150)(10,000) 2,000 (0.75)(2,000) TCmin = $750 + $750 = $1,500 Orders per year = D/Qopt = 10,000/2,000 = 5 orders/year Order cycle time = 311 days/(D/Qopt) = 311/5 = 62.2 store days Example 10.2

EOQ with Noninstantaneous Receipt Q(1-d/p) Inventory level (1-d/p) Q 2 Time Order receipt period Begin order receipt End Maximum inventory level Average Figure 10.3

EOQ with Noninstantaneous Receipt p = production rate d = demand rate Maximum inventory level = Q - d = Q 1 - Q p d Average inventory level = 1 - 2 TC = + 1 - CoD CcQ Qopt = 2CoD Cc 1 - d p

Production Quantity Cc = $0.75 per yard Co = $150 D = 10,000 yards d = 10,000/311 = 32.2 yards per day p = 150 yards per day Qopt = = = 2,256.8 yards 2CoD Cc 1 - d p 2(150)(10,000) 0.75 1 - 32.2 150 TC = + 1 - = $1,329 d p CoD Q CcQ 2 Production run = = = 15.05 days per order Q p 2,256.8 150 Example 10.3

Production Quantity Cc = $0.75 per yard Co = $150 D = 10,000 yards d = 10,000/311 = 32.2 yards per day p = 150 yards per day Number of production runs = = = 4.43 runs/year D Q 10,000 2,256.8 Maximum inventory level = Q 1 - = 2,256.8 1 - = 1,772 yards d p 32.2 150 Qopt = = = 2,256.8 yards 2CoD Cc 1 - d p 2(150)(10,000) 0.75 1 - 32.2 150 TC = + 1 - = $1,329 CoD Q CcQ 2 Production run = = = 15.05 days per order 2,256.8 Example 10.3

P = per unit price of the item Quantity Discounts Price per unit decreases as order quantity increases TC = + + PD CoD Q CcQ 2 where P = per unit price of the item D = annual demand

P = per unit price of the item Quantity Discounts Price per unit decreases as order quantity increases TC = + + PD CoD Q CcQ 2 where P = per unit price of the item D = annual demand ORDER SIZE PRICE 0 - 99 $10 100 - 199 8 (d1) 200+ 6 (d2)

Quantity Discount Model Qopt Carrying cost Ordering cost Inventory cost ($) Q(d1 ) = 100 Q(d2 ) = 200 TC (d2 = $6 ) TC (d1 = $8 ) TC = ($10 ) Figure 10.4

Quantity Discount Model Qopt Carrying cost Ordering cost Inventory cost ($) Q(d1 ) = 100 Q(d2 ) = 200 TC (d2 = $6 ) TC (d1 = $8 ) TC = ($10 ) Figure 10.4

Quantity Discount QUANTITY PRICE 1 - 49 $1,400 50 - 89 1,100 90+ 900 1 - 49 $1,400 50 - 89 1,100 90+ 900 Co = $2,500 Cc = $190 per computer D = 200 Qopt = = = 72.5 PCs 2CoD Cc 2(2500)(200) 190 TC = + + PD = $233,784 CoD Qopt CcQopt 2 For Q = 72.5 TC = + + PD = $194,105 CoD Q CcQ 2 For Q = 90 Example 10.4

When to Order Reorder Point is the level of inventory at which a new order is placed R = dL where d = demand rate per period L = lead time

Reorder Point Example Demand = 10,000 yards/year Store open 311 days/year Daily demand = 10,000 / 311 = 32.154 yards/day Lead time = L = 10 days R = dL = (32.154)(10) = 321.54 yards Example 10.5

Safety Stocks Safety stock Stockout Service level 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

Variable Demand with a Reorder Point point, R Q LT Time Inventory level Figure 10.5

Reorder Point with a Safety Stock point, R Q LT Time Inventory level Safety Stock Figure 10.6

Reorder Point With Variable Demand R = dL + zd L where d = average daily demand L = lead time d = the standard deviation of daily demand z = number of standard deviations corresponding to the service level probability zd L = safety stock

Reorder Point for a Service Level Probability of meeting demand during lead time = service level a stockout R Safety stock dL Demand zd L Figure 10.7

Reorder Point for Variable Demand The carpet store wants a reorder point with a 95% service level and a 5% stockout probability d = 30 yards per day L = 10 days d = 5 yards per day For a 95% service level, z = 1.65 R = dL + z d L = 30(10) + (1.65)(5)( 10) = 326.1 yards Safety stock = z d L = (1.65)(5)( 10) = 26.1 yards Example 10.6

Order Quantity for a Periodic Inventory System Q = d(tb + L) + zd tb + L - I where d = average demand rate tb = the fixed time between orders L = lead time sd = standard deviation of demand zd tb + L = safety stock I = inventory level

Fixed-Period Model with Variable Demand d = 6 bottles per day sd = 1.2 bottles tb = 60 days L = 5 days I = 8 bottles z = 1.65 (for a 95% service level) Q = d(tb + L) + zd tb + L - I = (6)(60 + 5) + (1.65)(1.2) 60 + 5 - 8 = 397.96 bottles

ABC Classification System Demand volume and value of items vary Classify inventory into 3 categories, typically on the basis of the dollar value to the firm PERCENTAGE PERCENTAGE CLASS OF UNITS OF DOLLARS A 5 - 15 70 - 80 B 30 15 C 50 - 60 5 - 10

ABC Classification PART UNIT COST ANNUAL USAGE 1 $ 60 90 2 350 40 1 $ 60 90 2 350 40 3 30 130 4 80 60 5 30 100 6 20 180 7 10 170 8 320 50 9 510 60 10 20 120 PART UNIT COST ANNUAL USAGE Example 10.1

ABC Classification PART UNIT COST ANNUAL USAGE 1 $ 60 90 2 350 40 1 $ 60 90 2 350 40 3 30 130 4 80 60 5 30 100 6 20 180 7 10 170 8 320 50 9 510 60 10 20 120 PART UNIT COST ANNUAL USAGE TOTAL % OF TOTAL % OF TOTAL PART VALUE VALUE QUANTITY % CUMMULATIVE 9 $30,600 35.9 6.0 6.0 8 16,000 18.7 5.0 11.0 2 14,000 16.4 4.0 15.0 1 5,400 6.3 9.0 24.0 4 4,800 5.6 6.0 30.0 3 3,900 4.6 10.0 40.0 6 3,600 4.2 18.0 58.0 5 3,000 3.5 13.0 71.0 10 2,400 2.8 12.0 83.0 7 1,700 2.0 17.0 100.0 $85,400 Example 10.1

ABC Classification A B C PART UNIT COST ANNUAL USAGE 1 $ 60 90 1 $ 60 90 2 350 40 3 30 130 4 80 60 5 30 100 6 20 180 7 10 170 8 320 50 9 510 60 10 20 120 PART UNIT COST ANNUAL USAGE TOTAL % OF TOTAL % OF TOTAL PART VALUE VALUE QUANTITY % CUMMULATIVE 9 $30,600 35.9 6.0 6.0 8 16,000 18.7 5.0 11.0 2 14,000 16.4 4.0 15.0 1 5,400 6.3 9.0 24.0 4 4,800 5.6 6.0 30.0 3 3,900 4.6 10.0 40.0 6 3,600 4.2 18.0 58.0 5 3,000 3.5 13.0 71.0 10 2,400 2.8 12.0 83.0 7 1,700 2.0 17.0 100.0 $85,400 A B C Example 10.1

ABC Classification A B C PART UNIT COST ANNUAL USAGE 1 $ 60 90 1 $ 60 90 2 350 40 3 30 130 4 80 60 5 30 100 6 20 180 7 10 170 8 320 50 9 510 60 10 20 120 PART UNIT COST ANNUAL USAGE TOTAL % OF TOTAL % OF TOTAL PART VALUE VALUE QUANTITY % CUMMULATIVE 9 $30,600 35.9 6.0 6.0 8 16,000 18.7 5.0 11.0 2 14,000 16.4 4.0 15.0 1 5,400 6.3 9.0 24.0 4 4,800 5.6 6.0 30.0 3 3,900 4.6 10.0 40.0 6 3,600 4.2 18.0 58.0 5 3,000 3.5 13.0 71.0 10 2,400 2.8 12.0 83.0 7 1,700 2.0 17.0 100.0 $85,400 A B C % OF TOTAL % OF TOTAL CLASS ITEMS VALUE QUANTITY A 9, 8, 2 71.0 15.0 B 1, 4, 3 16.5 25.0 C 6, 5, 10, 7 12.5 60.0 Example 10.1

ABC Classification C B A % of Value | | | | | | 0 20 40 60 80 100 100 – 80 – 60 – 40 – 20 – 0 – | | | | | | 0 20 40 60 80 100 % of Quantity % of Value A B C

Assumptions of Basic EOQ Model Demand is known with certainty and is constant over time No shortages are allowed Lead time for the receipt of orders is constant The order quantity is received all at once