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Inventory Management

Inventory Management

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Inventory Management

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  1. Inventory Management Henry C. Co Technology and Operations Management, California Polytechnic and State University

  2. Short-range decisions about supplies, inventories, production levels, staffing patterns, schedules, and distribution -operations infrastructure • 4 R’s • Right Material • Right Amount • Right Place • Right Time. Inventory Management (Henry C. Co)

  3. Motivations Economies of Scale Uncertainties

  4. Economies of Scale • Over-investment • Ties up capacity and financial resources • Inventory carrying cost, obsolescence, etc. • Insufficient/late availability causes • Idle personnel or equipment when components ran out • Lost sales/customer-goodwill if items are out-of-stock. Inventory Management (Henry C. Co)

  5. Uncertainties • Safety stock (demand or supply uncertainty) • In-transit inventories (lead times) • Hedge inventories Inventory Management (Henry C. Co)

  6. Drivers

  7. Approximately 16 % of total assets are invested in inventories (1986) • In manufacturing firms, 25 to 35% of total assets of typical are tied in inventories • Materials’ average share in a manufacturer’s cost of goods sold • 40% in 1945 • 50% in 1960 • > 60% Today • Spare parts (service parts) inventories of a typical manufacturer • ~ $ 5 million - $ 15 million Inventory Management (Henry C. Co)

  8. Consequently … distribution and inventory (logistics) costs are quite substantial • Value of inventories in U.S. ~ $ 1 trillion (1993) Inventory Management (Henry C. Co)

  9. Basic definitions • An inventory is an accumulation of a commodity that will be used to satisfy some future demand. • Inventories of the following form: • Raw material • Components • Semi-finished goods • Spare parts • Purchased products in retailing. Inventory Management (Henry C. Co)

  10. inventory cycle stock time a cycle Functional classification • Cycle Inventories: Produce or buy in larger quantities than needed. • Economies of scale • Quantity discounts • Restrictions (technological, transportation,…) Inventory Management (Henry C. Co)

  11. inventory place order at this time time Safety stock reorder level • Safety Stock: Provides protection against irregularities and uncertainties. Inventory Management (Henry C. Co)

  12. Average Anticipation stock Low demand season • Anticipation stock: Low demand in one part of the year build up stock for the high demand season Inventory Management (Henry C. Co)

  13. Hedge inventories : expect changes in the conditions (price, strike, supply, etc.) • Pipeline (or work-in-process) inventories: goods in transit, between levels of a supply chain, between work stations. Inventory Management (Henry C. Co)

  14. Service Operations • No tangible items to purchase or inventory, materials management is of minor concern • Operations that provide repair or refurbishment services carry inventory of replacement parts and supplies • Examples: • Automobile service centers carry automotive parts • Hospitals carry inventory of food, linens, medicine, and medical supplies Inventory Management (Henry C. Co)

  15. Functions of Inventory • To meet anticipated demand • To smooth production requirements • To decouple components of the production-distribution • To protect against stock-outs • To take advantage of order cycles • To help hedge against price increases or to take advantage of quantity discounts Inventory Management (Henry C. Co)

  16. Conflicting Needs • Some Excuses for Holding Excess Inventory • “Inaccurate sales forecast” • “Poor quality” • “Unsynchronized processes” • “Poor schedules” • “Unreliable suppliers” • “Unreliable shippers” • “Poor attitudes” Inventory Management (Henry C. Co)

  17. Pressures to Cut Inventory • Interest/opportunity cost • Storage and handling • Property taxes • Insurance premiums • Shrinkage • INVENTORIES HIDE PROBLEMS! Inventory Management (Henry C. Co)

  18. Profile of Inventory Level Over Time Q Usage rate Quantity on hand Reorder point Receive order Place order Place order Receive order Receive order Lead time Inventory Management (Henry C. Co)

  19. Profile of … Frequent Orders Inventory Management (Henry C. Co)

  20. The Classic EOQ Model

  21. The Economic Order Quantity (EOQ) • The total cost curve reaches its minimum where the carrying and ordering costs are equal. • EOQ represents trade-off between fixed cost associated with production or procurement against inventory holding costs. D = Rate of demand, units/year S = Fixed cost of procurement, $/order v = Variable cost of procurement. h = Cost/unit time of holding each unit of inventory, $/unit/year Q = Quantity ordered, units Inventory Management (Henry C. Co)

  22. Annual Cost QO The Classical EOQ Model • The total cost curve reaches its minimum where the carrying and ordering costs are equal. Ordering Costs Order Quantity (Q) (optimal order quantity) Inventory Management (Henry C. Co)

  23. Using calculus, we take the derivative of the total cost function (TC) and set the derivative (slope) equal to zero and solve for Q. Inventory Management (Henry C. Co)

  24. Example A grocery store pays $100 for each delivery of milk from the dairy. They sell 200 gallons per week. Each gallon costs the store $2, and they sell it for $3. They earn a 15% return on cash that is invested in milk. They have a large refrigerator that holds up to 2,000 gallons. It costs them $10,000 per year to maintain. D ~ 200(52) = 10,400 gallons per year. h ~ 15%($2) = $.30 per gallon per year S ~ $100 Q* = 2,633  Roughly one replenishment every 9 weeks?!?! Inventory Management (Henry C. Co)

  25. Suppose that the refrigerator held only 100 gallons. How much would it be worth to expand capacity to 200 gallons? • $5,185 per year in operational savings from doubling freezer size. Inventory Management (Henry C. Co)

  26. Example A bank has determined that it costs $30 to replenish the cash in one of its suburban ATMs. Customers take cash out of the ATM at a rate of $2000 per day (365 days per year). The bank earns a 10% return on cash that is not sitting in an ATM. Let us define $1000 to be a basic unit of cash. D ~ $2K(365) = $730K per year. h ~ 10%($1K) = $100 per $K per year S ~ $30 Q* = $20.9K  Roughly one replenishment every 10 days. Inventory Management (Henry C. Co)

  27. Potential Analytical Errors • Using different time units for holding cost versus the rate of demand. • Determining the true opportunity cost associated with holding inventory. (Physical costs as well as financial.) • Dealing with operational constraints. Inventory Management (Henry C. Co)

  28. When to Reorder? • Reorder Point R – When the quantity on hand of an item drops to this amount, the item is reordered • Safety Stock SS – Stock that is held in excess of expected demand due to variable demand rate and/or lead time. • Service Level - Probability that demand will not exceed supply during lead time. Inventory Management (Henry C. Co)

  29. Continuous Review = Average Lead Time Demand SS = Safety Stock; R = + SS Inventory Management (Henry C. Co)

  30. Maximum probable demand during lead time Quantity Expected demand during lead time Reorder point R Safety stock LT Inventory Management (Henry C. Co)

  31. Reorder Point Service level = Risk of a stock-out probability of no stock-out Quantity Expected demand Safety-stock 0 z z-scale R Inventory Management (Henry C. Co)

  32. Determining Reorder Point R • The Average lead time demand is equal to the average rate of demand (D) multiplied by the length of the lead time (L). • The amount of safety stock is influenced by the variability of demand, and our preference for avoiding inventory versus satisfying all of the demand. • If distribution of demand is stable over time, and the demand in one interval of time is statistically independent from that in another interval, then the standard deviation of lead time demand is proportional to the square root of the length (L) of the lead time: Inventory Management (Henry C. Co)

  33. Standard Deviation of Lead-time Demand • If daily demand has mean 100, and standard deviation 40, and the lead time L= 9 days: Then Lead Time Demand has mean = 900, and the standard deviation of lead time demand =120 (=SQRT[9] * 40). • If annual demand has mean 1040, and standard deviation 600, and the lead time L = 2 weeks: Then Lead Time Demand has mean = 40 (= 1040*(2/52) and the standard deviation of lead time demand = 117.67 (=SQRT[2/52]*600. Inventory Management (Henry C. Co)

  34. If weekly demand has mean 20, and standard deviation 83.2, and the lead time L = 2 weeks: Then Lead Time Demand has mean = 40 and the standard deviation of lead time demand = 117.67. (Note that this example is identical to the previous one, but the demand was specified in terms of weekly demand instead of annual.) Inventory Management (Henry C. Co)

  35. Safety Stock • Safety stock determines the expected amount of unsatisfied demand • Whenever the amount of demand during the lead time exceeds R, the excess represents the amount of unsatisfied demand. The expected amount of unsatisfied demand is: where g(x) represents the distribution of Lead Time Demand. Inventory Management (Henry C. Co)

  36. A Retail Stocking Problem • Daily demand (7 days/wk) is normally distributed with mean = 60, standard deviation = 30. • Orders can be placed at any time, and will be filled in exactly 6 days. • It costs $10 to place and order, and each unit costs $5. • Annual holding costs are 10% of the value of the item. • We want to determine an efficient inventory policy that allows us to satisfy 99% of demand from inventory. Inventory Management (Henry C. Co)

  37. First, we need to determine how much to order at a time: • Thus, our average cycle stock is 936/2 =468. • In order to determine the re-order point, we need to know the length of the lead time (6 days), and the standard deviation of lead time demand. Inventory Management (Henry C. Co)

  38. Now, to satisfy 99% of demand from inventory, we need: From the table in the lecture note, we can see that E(z) = .127 implies that z = .77 (or so). We can now calculate the re-order point, which consists of expected lead time demand plus safety stock. Inventory Management (Henry C. Co)

  39. Batch Production

  40. Suppose a machine produces a product at a production rate = p; e.g., p = 200 units/day. • Suppose the demand rate of this product is d; e.g., d = 80 units/day. • Since p > d, inventory will increase at (p-d) or (200-80 = 120) units/day. • Suppose the current inventory is 0. • In 10 days, the inventory level would be 10 days * 120 unit/day = 1,200 units. • In 20 days, the inventory level would be 20 days * 60 unit/day = 2,400 units. • etc. Inventory Management (Henry C. Co)

  41. Suppose the machine produces a batch of this product, then stop, then resumes production at some later time when the inventory of this item is low. This is call batch production. • Batch production is very common in industry. • When a machine is used to produce two or more products, one product at a time. • One decision the production manager has to make is when to start producing each product, and when to stop. • The run time is the amount of time the machine is producing a batch. • For example, producing at 200 units/day, if we want to produce 2,000 units per batch, the run time is 2,000/200 = 10 days. Here, batch size Q = 2,000 units, the run time t = 10 days. Inventory Management (Henry C. Co)

  42. Maximum Inventory Level • If the current inventory level is 0, what is the inventory at the end of the run time? • Since inventory will be rising at (200- 80 =120) units/day, in 10 days, the inventory level will be 10 days * 120 unit/day = 1,200 units. • The inventory at the end of the run time is the maximum inventory. It is equal to (p-d)*t = (200 units/day - 80 units/day)*10 days = 1,200 units. • Why is it that the machine produced 2,000 units in 10 days, and the maximum inventory level is only 1,200? • Answer: We consumed d*t = 80 units/day * 10 days = 800 units. • After completing a batch, how long will it take to deplete the inventory? • Answer: It will take (p-d)*t/d = 1,200/80 = 15 days to deplete the inventory. This is the off-time. Inventory Management (Henry C. Co)

  43. Number of Runs Per Year • If the annual demand of this product is D = 24,000 units, how many runs of this item do we produce each year? • Answer: Since we are producing Q = 2,000 units per batch, there will be D/Q = 24,000/2,000 = 12 batches per year. • In other words, there are D/Q = 12 cycles per year. In each cycle, there is a period of time the machine is producing the product (the run time), and a period to allow the inventory to deplete (the off time). Inventory Management (Henry C. Co)

  44. Average Inventory • What is the average inventory level? • During the run time, the inventory level rises from 0 to the maximum level of (p-d)*t = 1,200 units (see page 4). • During the off time, the inventory level drops from a maximum of (p-d)*t units to 0. • The average inventory level therefore = [0 + (p-d)*t ]/2 = (0 + 1,200)/2 = 600 units. • Since p*t = Q, then t = Q/p. We can rewrite the expression for the average inventory as (p-d)*t/2 = (p-d)*(Q/p)/2 = (1-d/p)Q/2. Inventory Management (Henry C. Co)

  45. Tradeoff • Batch size = production rate * run time. Large batch size means long run time, and high average inventory. • On the contrary, if the batch size is small, the run time is short, and we need to run many batches per year. • What is the average inventory if the batch size equals the annual demand D = 24,000 units? How many batches do we have to run per year? • Answer: The average inventory = (1-d/p)Q/2 = (1-80/200) (24,000)/2 = 7,200 units. We need to run one batch per year. • What is the average inventory if the batch size equals the weekly demand of 480 units (assuming 50 weeks/year)? How many batches do we have to run per year? • Answer: 144 units; Run 50 batches per year. Inventory Management (Henry C. Co)

  46. Optimal Tradeoff • Suppose the cost to carry one unit of inventory for one year is H. Since the average inventory level is (1-d/p)Q/2, the annual inventory-carrying cost is H*(1-d/p) Q/2. • Suppose the cost to set-up the machine to produce a batch is S. Since we need to run D/Q batches per year (see page 5), the annual set-up cost is S*D/Q. • Adding the two costs, we have H*(1-d/p) Q/2 + S*D/Q. Using calculus, the optimal batch size is Inventory Management (Henry C. Co)

  47. Illustration • Annual demand D = 24,000 units. • Production rate p = 200 units/day. • Demand rate d = 80 units/day (25 days/month). • Set-up cost S = $100. • Inventory-carrying cost H = $2/unit/year. • The optimal batch size =2,000 units. Inventory Management (Henry C. Co)

  48. The machine will produce D/Q = 12 batches a year. • Run time t = Q/p = 2,000/200 = 10 days. • Maximum inventory = (p-d)*t = (100-40)*20 = 1,200 units. • Off time = 1,200 units/ 80 units/day = 15 days. • In other words, the machine will run this product for 10 days, stop (to do something else) for 15 days, before running this item again. Inventory Management (Henry C. Co)

  49. The New Boy Problem Johnson & Pike, 1999

  50. % of total annual $ usage 80 80 100 20 40 60 % of total number of SKUs A-B-C Classification A specific unit of stock to be controlled is called a Stock Keeping Unit (SKU) Inventory Management (Henry C. Co)