Inventory 1.1


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  • Inventory 1.1

    1. 1. Inventory Management <ul><li>Using Analytics for Better Decisions </li></ul>
    2. 2. Why We Want to Hold Inventories <ul><li>Improve customer service </li></ul><ul><li>Reduce certain costs such as </li></ul><ul><ul><li>ordering costs </li></ul></ul><ul><ul><li>stockout costs </li></ul></ul><ul><ul><li>acquisition costs </li></ul></ul><ul><ul><li>start-up quality costs </li></ul></ul><ul><li>Contribute to the efficient and effective operation of the production system </li></ul>
    3. 3. Why We Want to Hold Inventories <ul><li>Finished Goods </li></ul><ul><ul><li>Essential in produce-to-stock positioning strategies </li></ul></ul><ul><ul><li>Necessary in level aggregate capacity plans </li></ul></ul><ul><ul><li>Products can be displayed to customers </li></ul></ul><ul><li>Work-in-Process </li></ul><ul><ul><li>Necessary in process-focused production </li></ul></ul><ul><ul><li>May reduce material-handling & production costs </li></ul></ul><ul><li>Raw Material </li></ul><ul><ul><li>Suppliers may produce/ship materials in batches </li></ul></ul><ul><ul><li>Quantity discounts and freight/handling lead to savings </li></ul></ul>
    4. 4. Why We Do Not Want to Hold Inventories <ul><li>Certain costs increase such as </li></ul><ul><ul><li>carrying costs </li></ul></ul><ul><ul><li>cost of diluted return on investment </li></ul></ul><ul><ul><li>reduced-capacity costs </li></ul></ul><ul><ul><li>large-lot quality cost </li></ul></ul><ul><ul><li>cost of production problems </li></ul></ul>
    5. 5. Why We Do Not Want to Hold Inventories <ul><li>Difficult to Control </li></ul><ul><li>Hides Production problems </li></ul>
    6. 6. Trade-offs <ul><li>Lot-size – Inventory (Bullwhip effect) </li></ul><ul><li>Inventory – Transportation (Ordering/Setup) cost </li></ul><ul><li>Lead Time – Transportation cost </li></ul><ul><li>Product variety – Inventory </li></ul><ul><li>Cost – Customer service </li></ul>
    7. 7. How do we know we have a good inventory management system?
    8. 8. Effective Inventory Management <ul><li>A system to keep track of inventory </li></ul><ul><li>A reliable forecast of demand </li></ul><ul><li>Knowledge of lead time </li></ul><ul><li>Reasonable estimate of - Holding cost - Ordering cost - Shortage cost </li></ul><ul><li>A classification system </li></ul>
    9. 9. Nature of Inventory <ul><li>Two Fundamental Inventory Decisions </li></ul><ul><li>Terminology of Inventories </li></ul><ul><li>Independent Demand Inventory Systems </li></ul><ul><li>Dependent Demand Inventory Systems </li></ul><ul><li>Inventory Costs </li></ul>
    10. 10. Two Fundamental Inventory Decisions <ul><li>How much to order of each material when orders are placed with either outside suppliers or production departments within organizations </li></ul><ul><li>When to place the orders </li></ul>
    11. 11. Independent Demand Inventory Systems <ul><li>Demand for an item carried in inventory is independent of the demand for any other item in inventory </li></ul><ul><li>Finished goods inventory is an example </li></ul><ul><li>Demands are estimated from forecasts and/or customer orders </li></ul>
    12. 12. Dependent Demand Inventory Systems <ul><li>Items whose demand depends on the demands for other items </li></ul><ul><li>For example, the demand for raw materials and components can be calculated from the demand for finished goods </li></ul><ul><li>The systems used to manage these inventories are different from those used to manage independent demand items </li></ul>
    13. 13. Inventory Costs <ul><li>Costs associated with ordering too much (represented by carrying costs) </li></ul><ul><li>Costs associated with ordering too little (represented by ordering costs) </li></ul><ul><li>These costs are opposing costs, i.e., as one increases the other decreases </li></ul><ul><li>. . . more </li></ul>
    14. 14. Carrying cost <ul><li>Obsolescence </li></ul><ul><li>Insurance </li></ul><ul><li>Extra staffing </li></ul><ul><li>Interest </li></ul><ul><li>Pilferage </li></ul><ul><li>Damage </li></ul><ul><li>Warehousing </li></ul><ul><li>Etc. </li></ul>
    15. 15. Carrying Cost (Approximate ranges) <ul><li>Category Cost as % of </li></ul><ul><li>Inventory Value </li></ul><ul><li>Investments costs 11% </li></ul><ul><li> (6 – 24%) </li></ul><ul><li>Labour cost from extra handling 3% </li></ul><ul><li> (3 – 5%) </li></ul><ul><li>Housing cost 6% </li></ul><ul><li> (3 – 10%) </li></ul><ul><li>Other costs also range between 1 – 5%. </li></ul>
    16. 16. Ordering cost <ul><li>Processing Supplies </li></ul><ul><li>Forms </li></ul><ul><li>Order processing </li></ul><ul><li>Clerical support </li></ul><ul><li>Etc. </li></ul>
    17. 17. Set-up costs <ul><li>Clean-up cost </li></ul><ul><li>Re-tooling cost </li></ul><ul><li>Adjustment cost </li></ul><ul><li>Etc. </li></ul>
    18. 18. Inventory Costs (continued) <ul><li>The sum of the two costs is the total stocking cost (TSC) </li></ul><ul><li>When plotted against order quantity, the TSC decreases to a minimum cost and then increases </li></ul><ul><li>This cost behavior is the basis for answering the first fundamental question: how much to order </li></ul><ul><li>It is known as the economic order quantity (EOQ) </li></ul>
    19. 19. Balancing Carrying against Ordering Costs Annual Cost ($) Order Quantity Minimum Total Annual Stocking Costs Annual Carrying Costs Annual Ordering Costs Total Annual Stocking Costs Smaller Larger Lower Higher EOQ
    20. 20. Fixed Order Quantity Systems <ul><li>Behavior of Economic Order Quantity (EOQ) Systems </li></ul><ul><li>Determining Order Quantities </li></ul><ul><li>Determining Order Points </li></ul>
    21. 21. Behavior of EOQ Systems <ul><li>As demand for the inventoried item occurs, the inventory level drops </li></ul><ul><li>When the inventory level drops to a critical point, the order point, the ordering process is triggered </li></ul><ul><li>The amount ordered each time an order is placed is fixed or constant </li></ul><ul><li>When the ordered quantity is received, the inventory level increases </li></ul><ul><li>. . . more </li></ul>
    22. 22. Behavior of EOQ Systems <ul><li>An application of this type system is the two-bin system </li></ul><ul><li>Two bin system : Two containers of inventory; order when one is empty </li></ul><ul><li>A perpetual inventory accounting system is usually associated with this type of system </li></ul><ul><li>Perpetual inventory system : System that keeps track of removals from inventory continuously; thus monitoring current levels of each item. </li></ul>
    23. 23. Determining Order Quantities <ul><li>Basic EOQ </li></ul><ul><li>EOQ for Production Lots </li></ul><ul><li>EOQ with Quantity Discounts </li></ul>
    24. 24. Model I: Basic EOQ <ul><li>Typical assumptions made </li></ul><ul><ul><li>annual demand (D), carrying cost (C) and ordering cost (S) can be estimated </li></ul></ul><ul><ul><li>average inventory level is the fixed order quantity (Q) divided by 2 which implies </li></ul></ul><ul><ul><ul><li>no safety stock </li></ul></ul></ul><ul><ul><ul><li>orders are received all at once </li></ul></ul></ul><ul><ul><ul><li>demand occurs at a uniform rate </li></ul></ul></ul><ul><ul><ul><li>no inventory when an order arrives </li></ul></ul></ul><ul><ul><li>. . . more </li></ul></ul>
    25. 25. Model I: Basic EOQ <ul><li>Assumptions (continued) </li></ul><ul><ul><li>Stockout, customer responsiveness, and other costs are inconsequential </li></ul></ul><ul><ul><li>acquisition cost is fixed, i.e., no quantity discounts </li></ul></ul><ul><li>Annual carrying cost = (average inventory level) x (carrying cost) = (Q/2)C </li></ul><ul><li>Annual ordering cost = (average number of orders per year) x (ordering cost) = (D/Q)S </li></ul><ul><li>. . . more </li></ul>
    26. 26. Model I: Basic EOQ <ul><li>Total annual stocking cost (TSC) = annual carrying cost + annual ordering cost = (Q/2)C + (D/Q)S </li></ul><ul><li>The order quantity where the TSC is at a minimum (EOQ) can be found using calculus (take the first derivative, set it equal to zero and solve for Q) </li></ul>
    27. 27. Example: Basic EOQ <ul><li>Zartex Co. stocks fertilizer to sell to retailers. One item – calcium nitrate – is purchased from a nearby manufacturer at Rs. 22.50 per ton. Zartex estimates it will need 5,750,000 tons of calcium nitrate next year. </li></ul><ul><li>The annual carrying cost for this material is 40% of the acquisition cost, and the ordering cost is Rs. 595. </li></ul><ul><li>a) What is the most economical order quantity? </li></ul><ul><li>b) How many orders will be placed per year? </li></ul><ul><li>c) How much time will elapse between orders? </li></ul>
    28. 28. Example: Basic EOQ <ul><li>Economical Order Quantity (EOQ) </li></ul><ul><li>D = 5,750,000 tons/year </li></ul><ul><li>C = .40(22.50) = Rs. 9.00/ton/year </li></ul><ul><li>S = Rs. 595/order </li></ul><ul><li>= 27,573.135 tons per order </li></ul>
    29. 29. Example: Basic EOQ <ul><li>Total Annual Stocking Cost (TSC) </li></ul><ul><li>TSC = (Q/2)C + (D/Q)S </li></ul><ul><li> = (27,573.135/2)(9.00) </li></ul><ul><li> + (5,750,000/27,573.135)(595) </li></ul><ul><li> = 124,079.11 + 124,079.11 </li></ul><ul><li> = Rs.248,158.22 </li></ul>Note: Total Carrying Cost equals Total Ordering Cost
    30. 30. Example: Basic EOQ <ul><li>Number of Orders Per Year </li></ul><ul><li>= D/Q </li></ul><ul><li>= 5,750,000/27,573.135 </li></ul><ul><li>= 208.5 orders/year </li></ul><ul><li>Time Between Orders </li></ul><ul><li>= Q/D </li></ul><ul><li>= 1/208.5 </li></ul><ul><li>= .004796 years/order </li></ul><ul><li>= .004796(365 days/year) = 1.75 days/order </li></ul>Note: This is the inverse of the formula above.
    31. 31. Model II: EOQ for Production Lots <ul><li>Used to determine the order size, production lot, if an item is produced at one stage of production, stored in inventory, and then sent to the next stage or the customer </li></ul><ul><li>Differs from Model I because orders are assumed to be supplied or produced at a uniform rate (p) rate rather than the order being received all at once </li></ul><ul><li>. . . more </li></ul>
    32. 32. Model II: EOQ for Production Lots <ul><li>It is also assumed that the supply rate, p, is greater than the demand rate, d </li></ul><ul><li>The change in maximum inventory level requires modification of the TSC equation </li></ul><ul><li>TSC = (Q/2)[(p-d)/p]C + (D/Q)S </li></ul><ul><li>The optimization results in </li></ul>
    33. 33. Example: EOQ for Production Lots <ul><li>Highland Electric Co. buys coal from Cedar Creek Coal Co. to generate electricity. CCCC can supply coal at the rate of 3,500 tons per day for $10.50 per ton. HEC uses the coal at a rate of 800 tons per day and operates 365 days per year. </li></ul><ul><li>HEC’s annual carrying cost for coal is 20% of the acquisition cost, and the ordering cost is $5,000. </li></ul><ul><li>a) What is the economical production lot size? </li></ul><ul><li>b) What is HEC’s maximum inventory level for coal? </li></ul>
    34. 34. Example: EOQ for Production Lots <ul><li>Economical Production Lot Size </li></ul><ul><li>d = 800 tons/day; D = 365(800) = 292,000 tons/year </li></ul><ul><li>p = 3,500 tons/day </li></ul><ul><li>S = $5,000/order C = .20(10.50) = $2.10/ton/year </li></ul><ul><li> = 42,455.5 tons per order </li></ul>
    35. 35. Example: EOQ for Production Lots <ul><li>Total Annual Stocking Cost (TSC) </li></ul><ul><li>TSC = (Q/2)((p-d)/p)C + (D/Q)S </li></ul><ul><li> = (42,455.5/2)((3,500-800)/3,500)(2.10) </li></ul><ul><li> + (292,000/42,455.5)(5,000) </li></ul><ul><li> = 34,388.95 + 34,388.95 </li></ul><ul><li> = $68,777.90 </li></ul>Note: Total Carrying Cost equals Total Ordering Cost
    36. 36. Example: EOQ for Production Lots <ul><li>Maximum Inventory Level </li></ul><ul><li>= Q(p-d)/p </li></ul><ul><li>= 42,455.5(3,500 – 800)/3,500 </li></ul><ul><li>= 42,455.5(.771429) </li></ul><ul><li>= 32,751.4 tons </li></ul>Note: HEC will use 23% of the production lot by the time it receives the full lot.
    37. 37. Key Points from EOQ Model <ul><li>In deciding the optimal lot size, the tradeoff is between setup (order) cost and holding cost. </li></ul><ul><li>If demand increases by a factor of 4, it is optimal to increase batch size by a factor of 2 and produce (order) twice as often. Cycle inventory (in days of demand) should decrease as demand increases . </li></ul><ul><li>If lot size is to be reduced, one has to reduce fixed order cost. To reduce lot size by a factor of 2, order cost has to be reduced by a factor of 4. </li></ul>
    38. 38. Model III: EOQ with Quantity Discounts <ul><li>Under quantity discounts, a supplier offers a lower unit price if larger quantities are ordered at one time </li></ul><ul><li>This is presented as a price or discount schedule, i.e., a certain unit price over a certain order quantity range </li></ul><ul><li>This means this model differs from Model I because the acquisition cost (ac) may vary with the quantity ordered, i.e., it is not necessarily constant </li></ul><ul><li>. . . more </li></ul>
    39. 39. All-Unit Quantity Discounts <ul><li>Pricing schedule has specified quantity break points q 0 , q 1 , …, q r , where q 0 = 0 </li></ul><ul><li>If an order is placed that is at least as large as q i but smaller than q i+1 , then each unit has an average unit cost of C i </li></ul><ul><li>The unit cost generally decreases as the quantity increases, i.e., C 0 >C 1 >…>C r </li></ul><ul><li>The objective for the company (a retailer in our example) is to decide on a lot size that will minimize the sum of material, order, and holding costs </li></ul>
    40. 40. Model III: EOQ with Quantity Discounts <ul><li>Under this condition, acquisition cost becomes an incremental cost and must be considered in the determination of the EOQ </li></ul><ul><li>The total annual material costs (TMC) = Total annual stocking costs (TSC) + annual acquisition cost </li></ul><ul><li> TSC = (Q/2)C + (D/Q)S + (D)ac </li></ul><ul><li>. . . more </li></ul>
    41. 41. All-Unit Quantity Discounts: Example Cost/Unit Rs. 3 Rs. 2.96 Rs.2.92 Order Quantity 5,000 10,000 Order Quantity 5,000 10,000 Total Material Cost
    42. 42. Model III: EOQ with Quantity Discounts <ul><li>To find the EOQ, the following procedure is used: </li></ul><ul><li>1. Compute the EOQ using the lowest acquisition cost. </li></ul><ul><ul><li>If the resulting EOQ is feasible (the quantity can be purchased at the acquisition cost used), this quantity is optimal and you are finished. </li></ul></ul><ul><ul><li>If the resulting EOQ is not feasible, go to Step 2 </li></ul></ul><ul><li>2. Identify the next higher acquisition cost. </li></ul>
    43. 43. Model III: EOQ with Quantity Discounts <ul><li>3. Compute the EOQ using the acquisition cost from Step 2. </li></ul><ul><ul><li>If the resulting EOQ is feasible, go to Step 4. </li></ul></ul><ul><ul><li>Otherwise, go to Step 2. </li></ul></ul><ul><li>4. Compute the TMC for the feasible EOQ (just found in Step 3) and its corresponding acquisition cost. </li></ul><ul><li>5. Compute the TMC for each of the lower acquisition costs using the minimum allowed order quantity for each cost. </li></ul><ul><li>6. The quantity with the lowest TMC is optimal. </li></ul>
    44. 44. All-Unit Quantity Discount: Example <ul><li>Order quantity Unit Price </li></ul><ul><li>0-5000 Rs. 3.00 </li></ul><ul><li>5001-10000 Rs. 2.96 </li></ul><ul><li>Over 10000 Rs. 2.92 </li></ul><ul><li>q0 = 0, q1 = 5000, q2 = 10000 </li></ul><ul><li>C0 = Rs. 3.00, C1 = Rs. 2.96, C2 = Rs. 2.92 </li></ul><ul><li>D = 120000 units/year, S = Rs. 100/lot, </li></ul><ul><li>h = 0.2 </li></ul>
    45. 45. All-Unit Quantity Discount: Example <ul><li>Step 1: Calculate Q2* = Sqrt[(2DS)/hC2] </li></ul><ul><li>= Sqrt[(2)(120000)(100)/(0.2)(2.92)] = 6410 </li></ul><ul><li>Not feasible (6410 < 10001) </li></ul><ul><li>Calculate TC2 using C2 = Rs. 2.92 and q2 = 10001 </li></ul><ul><li>TC2 = (120000/10001)(100)+(10001/2)(0.2)(2.92)+(120000)(2.92) </li></ul><ul><li>= Rs. 354,520 </li></ul><ul><li>Step 2: Calculate Q1* = Sqrt[(2DS)/hC1] </li></ul><ul><li>=Sqrt[(2)(120000)(100)/(0.2)(2.96)] = 6367 </li></ul><ul><li>Feasible (5000<6367 < 10000)  Stop </li></ul><ul><li>TC1 = (120000/6367)(100)+(6367/2)(0.2)(2.96)+(120000)(2.96) </li></ul><ul><li>= Rs. 358,969 </li></ul><ul><li>TC2 < TC1  The optimal order quantity Q* is q2 = 10001 </li></ul>
    46. 46. Example: EOQ with Quantity Discounts <ul><li> A-1 Auto Parts has a regional tire warehouse in Atlanta. One popular tire, the XRX75, has estimated demand of 25,000 next year. It costs A-1 $100 to place an order for the tires, and the annual carrying cost is 30% of the acquisition cost. The supplier quotes these prices for the tire: </li></ul><ul><li>Q ac </li></ul><ul><li>1 – 499 $21.60 </li></ul><ul><li>500 – 999 20.95 </li></ul><ul><li>1,000 + 20.90 </li></ul>
    47. 47. Example: EOQ with Quantity Discounts <ul><li>Economical Order Quantity </li></ul><ul><li>This quantity is not feasible, so try ac = $20.95 </li></ul><ul><li>This quantity is feasible, so there is no reason to try ac = $21.60 </li></ul>
    48. 48. Example: EOQ with Quantity Discounts <ul><li>Compare Total Annual Material Costs (TMCs) </li></ul><ul><li> TMC = (Q/2)C + (D/Q)S + (D)ac </li></ul><ul><li>Compute TMC for Q = 891.93 and ac = $20.95 </li></ul><ul><li>TMC 2 = (891.93/2)(.3)(20.95) + (25,000/891.93)100 </li></ul><ul><li>+ (25,000)20.95 </li></ul><ul><li> = 2,802.89 + 2,802.91 + 523,750 </li></ul><ul><li> = $529,355.80 </li></ul><ul><li>… more </li></ul>
    49. 49. Example: EOQ with Quantity Discounts <ul><li>Compute TMC for Q = 1,000 and ac = $20.90 </li></ul><ul><li>TMC 3 = (1,000/2)(.3)(20.90) + (25,000/1,000)100 </li></ul><ul><li>+ (25,000)20.90 </li></ul><ul><li> = 3,135.00 + 2,500.00 + 522,500 </li></ul><ul><li> = $528,135.00 (lower than TMC 2 ) </li></ul><ul><li> The EOQ is 1,000 tires </li></ul><ul><li>at an acquisition cost of $20.90. </li></ul>
    50. 50. Dynamics of Inventory Planning <ul><li>Continually review ordering practices and decisions </li></ul><ul><li>Modify to fit the firm’s demand and supply patterns </li></ul><ul><li>Constraints, such as storage capacity and available funds, can impact inventory planning </li></ul><ul><li>Computers and information technology are used extensively in inventory planning </li></ul>
    51. 51. <ul><li>Inventory cycle is the central focus of independent demand inventory systems </li></ul><ul><li>Production planning and control systems are changing to support lean inventory strategies </li></ul><ul><li>Information systems electronically link supply chain </li></ul>
    52. 52. Planning Supply Chain Activities Anticipatory - allocate supply to each warehouse based on the forecast Response-based - replenish inventory with order sizes based on specific needs of each warehouse
    53. 53. Integrated planning at Shell <ul><li>“ The most successful e-supply initiatives so far have been in industries where the components converge to create a product and where prices are not volatile. Energy is different. The supply chain is divergent; there are more products than raw materials and prices are highly volatile. Shell understands this and is aiming to create a reliable, real time, multi-point-optimized, an overview of entire supply chain….. </li></ul><ul><li>Across the globe this initiative has the potential to generate new value and drive savings to the tune of multiple of million US dollars a day.” </li></ul><ul><li>- A vice president of Shell Oil Products </li></ul>
    54. 54. Integrated planning at Shell <ul><li>Requirements for IT toolsets: </li></ul><ul><li>Complete horizontal supply chain integration </li></ul><ul><li>Convergence of strategy, planning and scheduling </li></ul><ul><li>Modularity to enable phased implementation and customization </li></ul><ul><li>Scalability </li></ul><ul><li>Interactive </li></ul><ul><li>Convenient User-interfacing </li></ul><ul><li>Real time results </li></ul><ul><li>Direct links to online refinery / plant optimization </li></ul>
    55. 56. Determining Order Points <ul><li>Basis for Setting the Order Point </li></ul><ul><li>DDLT Distributions </li></ul><ul><li>Setting Order Points </li></ul>
    56. 57. Basis for Setting the Order Point <ul><li>In the fixed order quantity system, the ordering process is triggered when the inventory level drops to a critical point, the order point </li></ul><ul><li>This starts the lead time for the item. </li></ul><ul><li>Lead time is the time to complete all activities associated with placing, filling and receiving the order. </li></ul><ul><li>. . . more </li></ul>
    57. 58. Basis for Setting the Order Point <ul><li>During the lead time, customers continue to draw down the inventory </li></ul><ul><li>It is during this period that the inventory is vulnerable to stockout (run out of inventory) </li></ul><ul><li>Customer service level is the probability that a stockout will not occur during the lead time </li></ul><ul><li>. . . more </li></ul>
    58. 59. Basis for Setting the Order Point <ul><li>The order point is set based on </li></ul><ul><ul><li>the demand during lead time (DDLT) and </li></ul></ul><ul><ul><li>the desired customer service level </li></ul></ul><ul><li>Order point (OP) = Expected demand during lead time (EDDLT) + Safety stock (SS) </li></ul><ul><li>The amount of safety stock needed is based on the degree of uncertainty in the DDLT and the customer service level desired </li></ul>
    59. 60. DDLT Distributions <ul><li>If there is variability in the DDLT, the DDLT is expressed as a distribution </li></ul><ul><ul><li>discrete </li></ul></ul><ul><ul><li>continuous </li></ul></ul><ul><li>In a discrete DDLT distribution, values (demands) can only be integers </li></ul><ul><li>A continuous DDLT distribution is appropriate when the demand is very high </li></ul>
    60. 61. Setting Order Point for a Discrete DDLT Distribution <ul><li>Assume a probability distribution of actual DDLTs is given or can be developed from a frequency distribution </li></ul><ul><li>Starting with the lowest DDLT, accumulate the probabilities. These are the service levels for DDLTs </li></ul><ul><li>Select the DDLT that will provide the desired customer level as the order point </li></ul>
    61. 62. Example: OP for Discrete DDLT Distribution <ul><li>One of Sharp Retailer’s inventory items is now being analyzed to determine an appropriate level of safety stock. The manager wants an 80% service level during lead time. The item’s historical DDLT is: </li></ul><ul><li>DDLT (cases) Occurrences </li></ul><ul><li>3 8 </li></ul><ul><li>4 6 </li></ul><ul><li>5 4 </li></ul><ul><li>6 2 </li></ul>
    62. 63. OP for Discrete DDLT Distribution <ul><li>Construct a Cumulative DDLT Distribution </li></ul><ul><li>Probability Probability of </li></ul><ul><li>DDLT (cases) of DDLT DDLT or Less </li></ul><ul><li>2 0 0 </li></ul><ul><li>3 .4 .4 </li></ul><ul><li>4 .3 .7 </li></ul><ul><li>5 .2 .9 </li></ul><ul><li>6 .1 1.0 </li></ul><ul><li> To provide 80% service level, OP = 5 cases </li></ul>.8
    63. 64. OP for Discrete DDLT Distribution <ul><li>Safety Stock (SS) </li></ul><ul><li>OP = EDDLT + SS </li></ul><ul><li>SS = OP  EDDLT </li></ul><ul><li>EDDLT = .4(3) + .3(4) + .2(5) + .1(6) = 4.0 </li></ul><ul><li>SS = 5 – 4 = 1 </li></ul>
    64. 65. The Role of Safety Inventory in a Supply Chain <ul><li>Forecasts are rarely completely accurate </li></ul><ul><li>If average demand is 1000 units per week, then half the time actual demand will be greater than 1000, and half the time actual demand will be less than 1000; what happens when actual demand is greater than 1000? </li></ul><ul><li>If you kept only enough inventory in stock to satisfy average demand, half the time you would run out </li></ul><ul><li>Safety inventory: Inventory carried for the purpose of satisfying demand that exceeds the amount forecasted in a given period </li></ul>
    65. 66. Role of Safety Inventory <ul><li>Average inventory is therefore cycle inventory plus safety inventory </li></ul><ul><li>There is a fundamental tradeoff: </li></ul><ul><ul><li>Raising the level of safety inventory provides higher levels of product availability and customer service </li></ul></ul><ul><ul><li>Raising the level of safety inventory also raises the level of average inventory and therefore increases holding costs </li></ul></ul><ul><ul><ul><li>Very important in high-tech or other industries where obsolescence is a significant risk (where the value of inventory, such as PCs, can drop in value) </li></ul></ul></ul><ul><ul><ul><li>Compaq and Dell in PCs </li></ul></ul></ul>
    66. 67. Two Questions to Answer in Planning Safety Inventory <ul><li>What is the appropriate level of safety inventory to carry? </li></ul><ul><li>What actions can be taken to improve product availability while reducing safety inventory? </li></ul>
    67. 68. Determining the Appropriate Level of Safety Inventory <ul><li>Measuring demand uncertainty </li></ul><ul><li>Measuring product availability </li></ul><ul><li>Replenishment policies </li></ul><ul><li>Evaluating cycle service level and fill rate </li></ul><ul><li>Evaluating safety level given desired cycle service level or fill rate </li></ul><ul><li>Impact of required product availability and uncertainty on safety inventory </li></ul>
    68. 69. Determining the Appropriate Level of Demand Uncertainty <ul><li>Appropriate level of safety inventory determined by: </li></ul><ul><ul><li>supply or demand uncertainty </li></ul></ul><ul><ul><li>desired level of product availability </li></ul></ul><ul><li>Higher levels of uncertainty require higher levels of safety inventory given a particular desired level of product availability </li></ul><ul><li>Higher levels of desired product availability require higher levels of safety inventory given a particular level of uncertainty </li></ul>
    69. 70. Measuring Demand Uncertainty <ul><li>Demand has a systematic component and a random component </li></ul><ul><li>The estimate of the random component is the measure of demand uncertainty </li></ul><ul><li>Random component is usually estimated by the standard deviation of demand </li></ul><ul><li>Notation: </li></ul><ul><ul><li>D = Average demand per period </li></ul></ul><ul><ul><li> D = standard deviation of demand per period </li></ul></ul><ul><ul><li>L = lead time = time between when an order is placed and when it is received </li></ul></ul><ul><li>Uncertainty of demand during lead time is what is important </li></ul>
    70. 71. Measuring Demand Uncertainty <ul><li>P = demand during k periods = kD </li></ul><ul><li> = std dev of demand during k periods =  R Sqrt(k) </li></ul><ul><li>Coefficient of variation = cv =  = mean/(std dev) = size of uncertainty relative to demand </li></ul>
    71. 72. Measuring Product Availability <ul><li>Product availability: a firm’s ability to fill a customer’s order out of available inventory </li></ul><ul><li>Stockout: a customer order arrives when product is not available </li></ul><ul><li>Product fill rate (fr): fraction of demand that is satisfied from product in inventory </li></ul><ul><li>Order fill rate: fraction of orders that are filled from available inventory </li></ul><ul><li>Cycle service level: fraction of replenishment cycles that end with all customer demand met </li></ul>
    72. 73. Replenishment Policies <ul><li>Replenishment policy: decisions regarding when to reorder and how much to reorder </li></ul><ul><li>Continuous review: inventory is continuously monitored and an order of size Q is placed when the inventory level reaches the reorder point ROP </li></ul><ul><li>Periodic review: inventory is checked at regular (periodic) intervals and an order is placed to raise the inventory to a specified threshold (the “order-up-to” level) </li></ul>
    73. 74. Continuous Review Policy: Safety Inventory and Cycle Service Level <ul><li>L : Lead time for replenishment </li></ul><ul><li>D: Average demand per unit time </li></ul><ul><li> D: Standard deviation of demand per period </li></ul><ul><li>D L : Mean demand during lead time </li></ul><ul><li> L : Standard deviation of demand during lead time </li></ul><ul><li>CSL : Cycle service level </li></ul><ul><li>ss : Safety inventory </li></ul><ul><li>ROP : Reorder point </li></ul>Average Inventory = Q/2 + ss
    74. 75. <ul><li>Auto Zone sells auto parts and supplies including a popular multi-grade motor oil. When the stock of this oil drops to 20 gallons, a replenishment order is placed. The store manager is concerned that sales are being lost due to stockouts while waiting for an order. It has been determined that lead time demand is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons. </li></ul><ul><li>The manager would like to know the probability of a stockout during lead time. </li></ul>Example: OP - Continuous DDLT Distribution
    75. 76. Example: OP - Continuous DDLT Distribution <ul><li>EDDLT = 15 gallons </li></ul><ul><li> DDLT = 6 gallons </li></ul><ul><li>OP = EDDLT + Z(  DDLT ) </li></ul><ul><li>20 = 15 + Z(6) </li></ul><ul><li> 5 = Z(6) </li></ul><ul><li> Z = 5/6 </li></ul><ul><li> Z = .833 </li></ul>
    76. 77. Example: OP - Continuous DDLT Distribution <ul><li>Standard Normal Distribution </li></ul>0 .833 Area = .2967 Area = .5 Area = .2033 z
    77. 78. Example: OP - Continuous DDLT Distribution <ul><li>The Standard Normal table shows an area of </li></ul><ul><li>.2967 for the region between the z = 0 line and the </li></ul><ul><li>z = .833 line. The shaded tail area is .5 - .2967 =.2033. </li></ul><ul><li>The probability of a stock-out during lead time is .2033 </li></ul>
    78. 79. Setting Order Point for a Continuous DDLT Distribution <ul><li>The resulting DDLT distribution is a normal distribution with the following parameters: </li></ul><ul><li>EDDLT = LT(d) </li></ul><ul><li>  DDLT = </li></ul>
    79. 80. Setting Order Point for a Continuous DDLT Distribution <ul><li>The customer service level is converted into a Z value using the normal distribution table </li></ul><ul><li>The safety stock is computed by multiplying the Z value by  DDLT . </li></ul><ul><li>The order point is set using OP = EDDLT + SS, or by substitution </li></ul>
    80. 81. Example <ul><li>Q = 5; σ d = 1.5; SL = 95% </li></ul><ul><li>R = d + Z σ d = 5 + 1.645*1.5 = 5 + 2.5 </li></ul><ul><li>= 7.5 </li></ul><ul><li>Order 5 (Q) whenever the inventory level is below 7.5 (8). </li></ul><ul><li>So, what does this mean? </li></ul>
    81. 82. Example A: Estimating Safety Inventory (Continuous Review Policy) <ul><li>D = 2,500/week;  D = 500 </li></ul><ul><li>L = 2 weeks; Q = 10,000; ROP = 6,000 </li></ul><ul><li>D L = D L = (2500)(2) = 5000 </li></ul><ul><li>ss = ROP - R L = 6000 - 5000 = 1000 </li></ul><ul><li>Cycle inventory = Q/2 = 10000/2 = 5000 </li></ul><ul><li>Average Inventory = cycle inventory + ss = 5000 + 1000 = 6000 </li></ul><ul><li>Average Flow Time = Avg inventory / throughput = 6000/2500 = 2.4 weeks </li></ul>
    82. 83. Example B: Estimating Cycle Service Level (Continuous Review Policy) <ul><li>D = 2,500/week;  D = 500 </li></ul><ul><li>L = 2 weeks; Q = 10,000; ROP = 6,000 </li></ul><ul><li>Cycle service level, CSL = F(D L + ss, D L ,  L ) = </li></ul><ul><li>= NORMDIST (D L + ss, D L ,  L ) = NORMDIST(6000,5000,707,1) </li></ul><ul><li>= 0.92 (This value can also be determined from a Normal probability distribution table) </li></ul>
    83. 84. Impact of Supply Uncertainty <ul><li>D: Average demand per period </li></ul><ul><li> D: Standard deviation of demand per period </li></ul><ul><li>L : Average lead time </li></ul><ul><li> s L : Standard deviation of lead time </li></ul>
    84. 85. Impact of Supply Uncertainty <ul><li>D = 2,500/day;  D = 500 </li></ul><ul><li>L = 7 days; Q = 10,000; CSL = 0.90; s L = 7 days </li></ul><ul><li>D L = DL = (2500)(7) = 17500 </li></ul><ul><li>ss = F -1 s (CSL)  L = NORMSINV(0.90) x 17550 </li></ul><ul><li>= 22,491 </li></ul>
    85. 86. Impact of Supply Uncertainty <ul><li>Safety inventory when s L = 0 is 1,695 </li></ul><ul><li>Safety inventory when s L = 1 is 3,625 </li></ul><ul><li>Safety inventory when s L = 2 is 6,628 </li></ul><ul><li>Safety inventory when s L = 3 is 9,760 </li></ul><ul><li>Safety inventory when s L = 4 is 12,927 </li></ul><ul><li>Safety inventory when s L = 5 is 16,109 </li></ul><ul><li>Safety inventory when s L = 6 is 19,298 </li></ul>
    86. 87. Information Centralization <ul><li>Virtual aggregation </li></ul><ul><li>Information system that allows access to current inventory records in all warehouses from each warehouse </li></ul><ul><li>Most orders are filled from closest warehouse </li></ul><ul><li>In case of a stockout, another warehouse can fill the order </li></ul><ul><li>Better responsiveness, lower transportation cost, higher product availability, but reduced safety inventory </li></ul><ul><li>Examples: McMaster-Carr, Gap, Wal-Mart </li></ul>
    87. 88. Postponement <ul><li>The ability of a supply chain to delay product differentiation or customization until closer to the time the product is sold </li></ul><ul><li>Goal is to have common components in the supply chain for most of the push phase and move product differentiation as close to the pull phase as possible </li></ul><ul><li>Examples: Dell, Benetton </li></ul>
    88. 89. Impact of Replenishment Policies on Safety Inventory <ul><li>Continuous review policies </li></ul><ul><li>Periodic review policies </li></ul>
    89. 90. Estimating and Managing Safety Inventory in Practice <ul><li>Account for the fact that supply chain demand is lumpy </li></ul><ul><li>Adjust inventory policies if demand is seasonal </li></ul><ul><li>Use simulation to test inventory policies </li></ul><ul><li>Start with a pilot </li></ul><ul><li>Monitor service levels </li></ul><ul><li>Focus on reducing safety inventories </li></ul>
    90. 91. Setting Order Point for a Continuous DDLT Distribution <ul><li>Assume that the lead time (LT) is constant </li></ul><ul><li>Assume that the demand per day is normally distributed with the mean (d ) and the standard deviation (  d ) </li></ul><ul><li>The DDLT distribution is developed by “adding” together the daily demand distributions across the lead time </li></ul><ul><li>. . . more </li></ul>
    91. 92. Customer Service Criterion <ul><li>The number of units short in one year (time period) is equal to the percentage short times the annual demand. </li></ul><ul><li>(1 – SL) * D </li></ul><ul><li>This is equal to the number of units short per order ( σ d E(Z)) times the number of orders per year (time period). </li></ul><ul><li> σ d E(Z) [D/Q] </li></ul><ul><li>SL = σ d E(Z)/Q </li></ul><ul><li>E(Z) = Q(1- SL)/ σ d </li></ul>
    92. 93. Example Text <ul><li>Q = 5; σ d = 1.5 SL = 95% </li></ul><ul><li>E(Z) = Q(1 - .05)/ σ d = 5*.05/1.5 </li></ul><ul><li>= .167 </li></ul><ul><li>From the tables Z = 0.6 </li></ul><ul><li>R = d + Z σ d = 5 + 0.6*1.5 = 5 + 0.9 = 5.9 </li></ul><ul><li>Order 5 when the inventory level reaches 5.9 </li></ul>
    93. 94. Example <ul><li>A service station is located right across campus. His gas sales have been going down. To improve his sales he is considering utilizing some available space to place some soda vending machines. When he orders, he usually orders 10 cases (240 cans). He estimates that the daily demand can be approximated by a Normal distribution with a mean of 75 cans and a standard deviation of 10 cans. He also feels that an 85% (very sophisticated gas station owner) service level would be adequate. His soda supplier promises that his lead time will be exactly 4 days. </li></ul><ul><li>What should his reorder point be? </li></ul><ul><li>What is the safety stock? </li></ul>
    94. 95. Solution in terms of probability of stock-out <ul><li>a) N(75, 10) Time period correction factor </li></ul><ul><li>N(75*4, 10√4) </li></ul><ul><li>R = d + Z σ d </li></ul><ul><li>= 4*75 + (1.04)*(10 √4) </li></ul><ul><li>= 300 + 20.8 = 321 </li></ul><ul><li>b) Safety Stock </li></ul><ul><li>SS = 20.8 </li></ul>
    95. 96. Another Example <ul><li>D = annual demand 1000 units LT=15days </li></ul><ul><li>Q = 200 units Service Level = 95 % (.95) </li></ul><ul><li>Working days/yr = 250 σ d =50 units </li></ul><ul><li>Average demand/day=1000/250 = 4 units/day </li></ul><ul><li>R = d + Z σ d = 4*15 + Z(50) </li></ul><ul><li>E(Z)=(1-.95)200/50 = 0.2 from tables </li></ul><ul><li>Z = 0.49 </li></ul><ul><li>R = 4(15) + 0.49*50 = 84.5 </li></ul><ul><li>R = 4(15) + 1.645*50 = 142.25 </li></ul>
    96. 97. Example (Cont.) <ul><li>Policy: </li></ul><ul><li>When inventory level gets to 85 or less then order 200. </li></ul><ul><li>What is the expected number of units short per order? </li></ul><ul><li>E(Z) σ d = 0.2 * 50 = 10 </li></ul><ul><li>How many orders per year? </li></ul><ul><li>(1000/200)= 5 </li></ul><ul><li>Total number of units short? </li></ul><ul><li>10*5 = 50 (Service level is 95%; 950/1000) </li></ul>