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## Uncertain Demand: The Newsvendor Model

содержание презентации «Uncertain Demand: The Newsvendor Model.ppt»
 Сл Текст Сл Текст 1 Inventory Models. Uncertain Demand: 9 22. 24. 26. 28. 30. 32. 34. 36. The Newsvendor Model. Probability. .05. .10. .15. .20. .20. .15. 2 Background: expected value. A fruit .10. .05. seller example. Undamaged mango. Damaged 10 Stockout and Markdown Risks. 1. Mrs. mango. Profit. \$ 4. \$ 1. Probability. 80%. Kandell has only one chance to order until 20%. What is the expected profit for a the sales begin: no information to revise stock of 100 mangoes ? 0.8 x 100 (\$4) + the forecast; after the sales start: too 0.2 x 100 x (\$1) = 320 + 20 = \$340. random late to order more. 2. She has to decide variable: ai. probability: pi. Expected an order quantity Q now. D total demand value = a1 p1 + a2 p2 + … + ak pk = Si = before Christmas F(x) the demand 1,,k aipi. distribution, D > Q ? stockout, at a 3 Probabilistic models: Flower seller cost of: cu (D – Q)+ = cu max{D –Q, 0} D example. Wedding bouquets: Selling price: < Q ? overstock, at a cost of co (Q–D)+ \$50 (if sold on same day), \$ 0 (if not = co max{Q – D, 0}. sold on that day) Cost = \$35. number of 11 Key elements of the model. 1. bouquets. 3. 4. 5. 6. 7. 8. 9. Uncertain demand 2. One chance to order probability. 0.05. 0.12. 0.20. 0.24. 0.17. (long) before demand 3. ( order > 0.14. 0.08. How many bouquets should he demand OR order < demand) ? COST. make each morning to maximize the expected 12 Model development. Stockout cost = cu profit? max{D –Q, 0} Overstock cost = co max{Q – 4 Probabilistic models: Flower seller D, 0}. Total cost = G(Q) = cu (D – Q)+ + example.. number of bouquets. 3. 4. 5. 6. co (Q – D)+. 7. 8. 9. probability. 0.05. 0.12. 0.20. 13 Model Development: generalization. 0.24. 0.17. 0.14. 0.08. CASE 1: Make 3 Suppose Demand ? a continuous variable ++ bouquets probability( demand ? 3) = 1. good approximation when number of Exp. Profit = 3x50 – 3x35 = \$45. CASE 2: possibilities is high -- difficult to Make 4 bouquets if demand = 3, then generate probabilities, but… ++ revenue = 3x \$50 = \$150 if demand = 4 or probability distribution can be guessed. more, then revenue = 4x \$50 = \$200. prob = 14 Model solution. g(Q) is a convex 0.05. prob = 0.95. Exp. Profit = 150x0.05 function: it has a unique minimum when + 200x0.95 – 4x35 = \$57.5. g(Q) is at minimum value, F(Q) = cu/(cu + 5 Probabilistic models: Flower seller co). example. Compute expected profit for each 15 The Critical Ratio. Solution to the case ? number of bouquets. 3. 4. 5. 6. 7. Newsvendor problem: ? = cu /(co + cu ) is 8. 9. probability. 0.05. 0.12. 0.20. 0.24. called the critical ratio. b ? relative 0.17. 0.14. 0.08. Expected profit. 45. importance of stockout cost vs. markdown 57.5. 64. 60.5. 45. 21. -10. Making 5 cost. bouquets will maximize expected profit. 16 Mrs. Kandell’s Problem, solved: co = 6 Probabilistic models: definitions. 25 – 15 = \$10. cu = 55 – 25 = \$30. ? = cu Discrete random variable. Probability (sum /(co + cu ) = 30/(30 + 10) = 0.75. Past of all likelihoods = 1). Continuous random Data. ? optimum ? 31. NOTE: E(D) = 22x variable: Example, height of people in a 0.05 + 24 x 0.1 + … + 36 x 0.05 = 29. city. Probability density function (area 17 Newsvendor model: effect of critical under curve = integral over entire range = ratio. ? = cu /(co + cu ) = 30/(30 + 10) = 1). number of bouquets. 3. 4. 5. 6. 7. 8. 0.75 ? optimum: 31. ? overstock cost less 9. probability. 0.05. 0.12. 0.20. 0.24. significant ? order more. ? overstock cost 0.17. 0.14. 0.08. dominates ? order less. 7 Probabilistic models: normal 18 Summary. When demand is uncertain, we distribution function. Standard normal minimize expected costs newsvendor model: distribution curve: mean = 0, std dev. = single period, with over- and under-stock 1. P( a? x ? b) = ?ab f(x) dx. Property: costs Critical ratio determines the normally distributed random variable x, optimum order point Critical ratio affects mean = m, standard deviation = s, the direction and magnitude of order Corresponding standard random variable: z quantity. = (x – m)/ s z is normally distributed, 19 Concluding remarks on inventory with a m = 0 and s = 1. control. Inventory costs lead to 8 The Newsvendor Model. Assumptions: - success/failure of a company. Drive to Plan for single period inventory level - reduce inventory costs was main motivation Demand is unknown - p(y) = probability( for Supply Chain Management. Example: Dell demand = y), known - Zero setup (ordering) Inc. “Dell's direct model enables us to cost. keep low component inventories that enable 9 Example: Mrs. Kandell’s Christmas Tree us to give customers immediate savings Shop. Order for Christmas trees must be when component prices are reduced, ... placed in Sept. How many trees should she Because of our inventory management, Dell order? If she orders too few, the unit is able to offer some of the newest shortage cost is cu = 55 – 25 = \$30. If technologies at low prices while our she orders too many, the unit overage cost competitors struggle to sell off older is co = 25 – 15 = \$10. Past Data. Sales. products.”. next: Quality Control. Uncertain Demand: The Newsvendor Model.ppt
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## Uncertain Demand: The Newsvendor Model

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