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Inventory: Stable Demand

Inventory: Stable Demand. Dr. Ron Lembke. Economic Order Quantity. Assumptions Demand rate is known and constant No order lead time Shortages are not allowed Costs: S - setup cost per order H - holding cost per unit time. EOQ. Decrease Due to Constant Demand. Inventory Level. Q*

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Inventory: Stable Demand

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  1. Inventory:Stable Demand Dr. Ron Lembke

  2. Economic Order Quantity Assumptions • Demand rate is known and constant • No order lead time • Shortages are not allowed • Costs: • S - setup cost per order • H - holding cost per unit time

  3. EOQ Decrease Due to Constant Demand Inventory Level Q* Optimal Order Quantity Instantaneous Receipt of Optimal Order Quantity Average Inventory Q/2 Time

  4. Total Costs • Average Inventory = Q/2 • Annual Holding costs = H * Q/2 • # Orders per year = D / Q • Annual Ordering Costs = S * D/Q • Annual Total Costs = Holding + Ordering

  5. How Much to Order? Total Cost = Holding + Ordering Annual Cost Ordering Cost = S * D/Q Holding Cost = H * Q/2 Optimal Q Order Quantity

  6. Optimal Quantity Total Costs = Take derivative with respect to Q = Set equal to zero Solve for Q:

  7. EOQ Inventory Level Q* Reorder Point (R) Time Lead Time

  8. Adding Lead Time • Use same order size • Order before inventory depleted • Where: • = demand rate (per day) • L = lead time (in days) • both in same time period (wks, months, etc.)

  9. A Question: • If the EOQ is based on so many horrible assumptions that are never really true, why is it the most commonly used ordering policy? • Cost curve very flat around optimal Q, so a small change in Q means small increase in Total Costs • If overestimate D by 10%, and S by 10%, and H by 20%, they pretty much cancel each other out • Have to overestimate all in the wrong direction before Q affected

  10. Sensitivity • Suppose we do not order optimal Q*, but order Q instead. • Percentage profit loss given by: • Should order 100, order 150 (50% over): 0.5*(0.66 + 1.5) =1.08 an 8%cost increase

  11. Quantity Discounts

  12. Quantity Discounts- Price Break • How does this all change if price changes depending on order size? • Holding cost as function of cost: • H = i * C • Explicitly consider price:

  13. Discount Example D = 10,000 S = $20 i = 20% Price Quantity EOQ C = 5.00 Q < 500 633 4.50 500-999 667 3.90 Q >= 1000 716 Must Include Cost of Goods:

  14. Discount Pricing Total Cost C=$5 Price 1 Price 2 Price 3 C=$4.5 C=$3.9 X 633 X667 X 716 Order Size 500 1,000

  15. Discount Example Order 667 at a time: Hold 667/2 * 4.50 * 0.2= $300.15 Order 10,000/667 * 20 = $299.85 Mat’l 10,000*4.50 = $45,000.00 45,600.00 Order 1,000 at a time: Hold 1,000/2 * 3.90 * 0.2= $390.00 Order 10,000/1,000 * 20 = $200.00 Mat’l 10,000*3.90 = $39,000.00 39,590.00

  16. Discount Model 1. Compute EOQ for next cheapest price 2. Is EOQ feasible? (is EOQ in range?) If EOQ is too small, use lowest possible Q to get price. 3. Compute total cost for this quantity • Repeat until EOQ is feasible or too big. • Select quantity/price with lowest total cost.

  17. Summary • Economic Order Quantity • Perfectly balances ordering and holding costs • Very robust, errors in input quantities have small impact on correctness of results • Discount Model • Start with EOQ calculations, using H= iC • Compute EOQ for each price, • Determine feasible quantity • Compute Total Costs: • Holding, Ordering, and Cost of Goods.

  18. Random Demand:Fixed Order Quantity Dr. Ron Lembke

  19. Random Demand • Don’t know how many we will sell • Sales will differ by period • Average always remains the same • Standard deviation remains constant How would our policies change? • How would our order quantity change? • EOQ balances ordering vs holding, and is unchanged • How would our reorder point change? • That’s a good question

  20. Constant Demand vs Random • Steady demand • Always buy Q • Reorder at R=dL • Sell dL during LT • Inv = Q after arrival • Random demand • Always buy Q • Reorder at R=dL + ? • Sell ? during LT • Inv = ? after arrival Inv Q Q Q Q R R L L

  21. Random Demand • Reorder when on-hand inventory is equal to the amount you expect to sell during LT, plus an extra amount of safety stock • Assume daily demand has a normal distribution • If we want to satisfy all of the demand 95% of the time, how many standard deviations above the mean should the inventory level be? • Just considers a probability of running out, not the number of units we’ll be short.

  22. Demand over the Lead Time R = Expected Demand over LT + Safety Stock = Average demand per day = Lead Time in days = st deviation of demand over Lead Time z from normal table, e.g. z.95 = 1.65

  23. Random Example • What should our reorder point be? • Demand averages 50 units per day, L = 5 days • Total demand over LT has standard deviation of 100 • want to satisfy all demand 90% of the time • To satisfy 90% of the demand, z = 1.28

  24. Random Demand • Sometimes use SS • Sometimes don’t • On average use 0 SS • Random demand • Always buy Q • Reorder at R=dL + SS Inv R

  25. St Dev of Daily Demand • What if we only know the average daily demand, and the standard deviation of daily demand? • Lead time = 4 days, • daily demand = 10, • Daily demand has standard deviation = 5, • What should our reorder point be, if z = 3?

  26. St Dev Over LT If the average each day is 10, and the lead time is 4 days, then the average demand over the lead time must be 40. What is the standard deviation of demand over the lead time? Std. Dev. ≠ 5 * 4

  27. St Dev Over Lead Time We can add up variances, not standard deviations Standard deviation of demand over LT =

  28. Demand Per Day Or, same thing: = Lead time in days = average demand per day = st deviation of demand per day z from normal table, e.g. z.95 = 1.65

  29. Random DemandFixed Order Quantity • Demand per day averages 40 with standard deviation 15, lead time is 5 days, service level of 90% = 5 days = 40 = 15 = 1.30,

  30. Fixed-Time Period Model • Place an order every, say, week. • Time period is fixed, order quantity will vary • Order enough so amount on hand plus on order gets up to a target amount • Q = S – Inv • Order “up to” policies

  31. Service Level Criteria • Type I: specify probability that you do not run out during the lead time • Chance that 100% of customers go home happy • Type II: (Fill Rate) proportion of demands met from stock • 100% chance that this many go home happy, on average

  32. Two Types of Service CycleDemand Stock-Outs 1 180 0 2 75 0 3 235 45 4 140 0 5 180 0 6 200 10 7 150 0 8 90 0 9 160 0 10 40 0 Sum 1,450 55 Type I: 8 of 10 periods 80% service Type II: 1,395 / 1,450 = 96%

  33. Summary • Fixed Order Quantity – always order same • Random demand – reorder point needs to change • Standard Deviation over the LT is given • Standard Deviation per day is given • Fixed Time Period • Always order once a month, e.g. • Amount on hand plus on order will add up to S • Different service metrics

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