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Betting on Uncertain Demand: Newsvendor Model

Betting on Uncertain Demand: Newsvendor Model . Optional reading: Cachon’s book (reference textbook) – Ch. 11. . The Newsboy Model: an Example .

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Betting on Uncertain Demand: Newsvendor Model

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  1. Betting on Uncertain Demand: Newsvendor Model Optional reading: Cachon’s book (reference textbook) – Ch. 11.

  2. The Newsboy Model: an Example Mr. Tan, a retiree, sells the local newspaper at a Bus terminal. At 6:00 am, he meets the news truck and buys # of the paper at $4.0 and then sells at $8.0. At noon he throws the unsold and goes home for a nap. If average daily demand is 50 and he buys just 50 copies daily, then is the average daily profit =50*4 =$200? NO!

  3. Betting on Uncertain Demand • You must take a firm bet (how much stock to order) before some random event occurs (demand) and then you learn that you either bet too much or too little • More examples: Products for the Christmas season; Nokia’s new set, winter coats, New-Year Flowers, …

  4. Bossini -- Winter Clothes • Season: Dec. – Jan./Feb. • Purchase of key materials (fabrics, dyeing/printing, …) takes long times (upto 90 days) • Into the selling season, it is too late!

  5. Seattle Hong Kong Denver Case: Sport Obermeyer

  6. The SO Supply Chain Shell Fabric Subcontractors Lining Fabric Insulation mat. Cut/Sew Distr Ctr Retailers Snaps Zippers Others Textile Suppliers Obersport Obermeyer Retailers

  7. O’Neill’s Hammer 3/2 wetsuit 11-13

  8. Hammer 3/2 timeline and economics • Economics: • Each suit sells for p = $180 • TEC charges c = $110 per suit • Discounted suits sell for v = $90 • The “too much/too little problem”: • Order too much and inventory is left over at the end of the season • Order too little and sales are lost. • Marketing’s forecast for sales is 3200 units. 11-14

  9. Newsvendor model implementation steps • Gather economic inputs: • Selling price, production/procurement cost, salvage value of inventory • Generate a demand model: • Use empirical demand distribution or choose a standard distribution function to represent demand, e.g. the normal distribution, the Poisson distribution. • Choose an objective: • e.g. maximize expected profit or satisfy a fill rate constraint. • Choose a quantity to order. 11-15

  10. The Newsvendor Model: Develop a Forecast Just one approach 11-16

  11. Historical forecast performance at O’Neill Forecasts and actual demand for surf wet-suits from the previous season 11-17

  12. Empirical distribution of forecast accuracy How do we know “actual d’d” if it exceeded forecast? 11-18

  13. Normal distribution tutorial • All normal distributions are characterized by two parameters, mean = m and standard deviation = s • All normal distributions are related to the standard normal that has mean = 0 and standard deviation = 1. • For example: • Let Q be the order quantity, and (m, s) the parameters of the normal demand forecast. • Prob{demand is Q or lower} = Prob{the outcome of a standard normal is z or lower}, where • (The above are two ways to write the same equation, the first allows you to calculate z from Q and the second lets you calculate Q from z.) • Look up Prob{the outcome of a standard normal is z or lower} in the Standard Normal Distribution Function Table. 11-19

  14. Converting between Normal distributions • Start with • = 100, • = 25, Q = 125 Center the distribution over 0 by subtracting the mean Rescale the x and y axes by dividing by the standard deviation 11-20

  15. Using historical A/F ratios to choose a Normal distribution for the demand forecast • Start with an initial forecast generated from hunches, guesses, etc. • O’Neill’s initial forecast for the Hammer 3/2 = 3200 units. • Evaluate the A/F ratios of the historical data: • Set the mean of the normal distribution to • Set the standard deviation of the normal distribution to • Why not just order/buy 3200 units? It is the most likely outcome! • Forecasts always are biased, so order less than 3200 • Gross margin is 40%, should order more, if is a hit 11-21

  16. Empirical distribution of forecast accuracy 11-22

  17. Table 11.2

  18. If the coming year is a similar to the last year, i.e., the forecasting errors are similar, then, • There is a 3% chance that demand will be 800 units or fewer (0.25*3200) • There is a 90.9% chance demand is 150% of the forecast or lower (or 1.5*3200 = 4,800)

  19. O’Neill’s Hammer 3/2 normal distribution forecast • O’Neill should choose a normal distribution with mean 3192 and standard deviation 1181 to represent demand for the Hammer 3/2 during the Spring season. 11-25

  20. Empirical vs normal demand distribution Empirical distribution function (diamonds) and normal distribution function with mean 3192 and standard deviation 1181 (solid line) 11-26

  21. The Newsvendor Model: The order quantity that maximizes expected profit 11-27

  22. “Too much” and “too little” costs • Co = overage cost • The cost of ordering one more unit than what you would have ordered had you known demand. • In other words, suppose you had left over inventory (i.e., you over ordered). Co is the increase in profit you would have enjoyed had you ordered one fewer unit. • For the Hammer 3/2 Co = Cost – Salvage value = c – v = 110 – 90 = 20 • Cu = underage cost • The cost of ordering one fewer unit than what you would have ordered had you known demand. • In other words, suppose you had lost sales (i.e., you under ordered). Cu is the increase in profit you would have enjoyed had you ordered one more unit. • For the Hammer 3/2 Cu = Price – Cost = p – c = 180 – 110 = 70 11-28

  23. Balancing the risk and benefit of ordering a unit • Ordering one more unit increases the chance of overage … • Expected loss on the Qth(+1) unit = Co x F(Q) • F(Q) = Distribution function of demand = Prob{Demand <= Q) • … but the benefit/gain of ordering one more unit is the reduction in the chance of underage: • Expected gain on the Qth(+1) unit = Cu x (1-F(Q)) • As more units are ordered, the expected benefit from ordering one unit decreases while the expected loss of ordering one more unit increases. As we deal with large numbers, we omit +1 11-29

  24. Newsvendor expected profit maximizing order quantity • To maximize expected profit order Q units so that the expected loss on the Qth unit equals the expected gain on the Qth unit: • Rearrange terms in the above equation -> • The ratio Cu / (Co + Cu) is called the critical ratio. • Hence, to maximize profit, choose Q such that we don’t have lost sales (i.e., demand is Q or lower) with a probability that equals the critical ratio 11-30

  25. Finding the Hammer 3/2’s expected profit maximizing order quantity with the empirical distribution function • Inputs: • Empirical distribution function table; p = 180; c = 110; v = 90; Cu = 180-110 = 70; Co = 110-90 =20 • Evaluate the critical ratio: • Lookup 0.7778 in the empirical distribution function table • If the critical ratio falls between two values in the table, choose the one that leads to the greater order quantity (choose 0.788 which corresponds to A/F ratio 1.3) • Convert A/F ratio into the order quantity A round-up rule! See p235. 11-31

  26. Hammer 3/2’s expected profit maximizing order quantity using the normal distribution • Inputs: p = 180; c = 110; v = 90; Cu = 180-110 = 70; Co = 110-90 =20; critical ratio = 0.7778; mean = m = 3192; standard deviation = s = 1181 • Look up critical ratio in the Standard Normal Distribution Function Table: • If the critical ratio falls between two values in the table, choose the greater z-statistic • Choose z = 0.77 • Convert the z-statistic into an order quantity: 11-32

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