N EWSVENDOR & R EVENUE M ANAGEMENT P ROFESSOR D AVID G ILLEN (U NIVERSITY OF B RITISH C OLUMBIA ) & P ROFESSOR B ENNY M ANTIN (U NIVERSITY OF W ATERLOO ) Istanbul Technical University Logistic Management in Air Transport Air Transportation Management Module 7-8 M.Sc. Program 17 December 2015
THE NEWSVENDOR AND APPLICATIONS 2
T HE N EWSVENDOR M ODEL 3
O’N EILL ’ S H AMMER 3/2 W ETSUIT TIMELINE Generate forecast of demand and submit an order to TEC Spring selling season Nov Dec Jan Feb Mar Apr May Jun Jul Aug Receive order Left over from TEC at the units are end of the discounted month • Marketing’s forecast for sales is 3200 units. 4
O’N EILL ’ S H AMMER 3/2 W ETSUIT TIMELINE • Marketing’s forecast for sales is 3200 units. • O’Neill sells each suit for p = $190 • O’Neill purchases each suit from its supplier for c = $110 per suit • Discounted suits sell for v = $90 – This is also called the salvage value . • How many units shall O’Neill order?! • O’Neill is facing a “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. 5
F ORECASTING : D EMAND D ISTRIBUTION • Marketing's forecast is a single number. • However, demand is more likely to follow some distribution. • Traditional distributions from statistics include, e.g., the normal, gamma, Poisson distributions GAMMA POISSON NORMAL Probability Probability 0 1 2 3 4 5 6 7 8 9 10 0 50 100 150 200 6 Demand Demand
F ORECASTING : D EMAND D ISTRIBUTION • How can we uncover the underlying distribution? • We can look at historical forecast performance at O’Neill 7000 6000 Actual demand . 5000 4000 3000 2000 1000 0 0 1000 2000 3000 4000 5000 6000 7000 Forecast 7 Forecasts and actual demand for surf wet-suits from the previous season
F ORECASTING : D EMAND D ISTRIBUTION • The idea: use historical actual to forecast ratio (the A/F ratio) from past observations. Actual demand A/F ratio Forecast • Calculate average and standard deviation of A/F ratio. • Set the mean of the normal distribution to Expected actual demand (Expected A/F ratio) Forecast • Set the standard deviation of the normal distribution to Standard deviation of actual demand (Standard deviation of A/F ratios) Forecast N 1 2 ( ) • Recall: x i N 8 i 1
F ORECASTING : D EMAND D ISTRIBUTION Product description Forecast Actual demand Error A/F Ratio JR ZEN FL 3/2 90 140 -50 1.5556 Note: Full data EPIC 5/3 W/HD 120 83 37 0.6917 in hidden JR ZEN 3/2 140 143 -3 1.0214 slide WMS ZEN-ZIP 4/3 170 156 14 0.9176 … … … … … ZEN 3/2 3190 1195 1995 0.3746 ZEN-ZIP 4/3 3810 3289 521 0.8633 WMS HAMMER 3/2 FULL 6490 3673 2817 0.5659 Average 0.9975 Standard deviation 0.3690 • Marketing’s forecast for sales is 3200 units . Hence: . 0 9975 3200 3192 Expected actual demand 0 369 3200 1181 Standard deviation of actual demand . • O’Neill can choose a normal distribution with mean 3192 and standard deviation 1181 to represent demand for the Hammer 3/2 during the Spring season. 9
D ISTRIBUTION AND DENSITY FUNCTIONS Distribution function Density function 1.00 0.80 0.60 Probability Probability 0.40 0.20 0.00 0 50 100 150 200 0 50 100 150 200 Demand Demand The probability the outcome will The probability of a particular be a particular value or smaller outcome occurring 11
N ORMAL DISTRIBUTION • Characterized by mean ( ) and standard deviation ( ) • All normal distributions are related to the standard normal that has mean = 0 and standard deviation = 1. • For example: Q – Let Q be some quantity – N ( , ) describes the demand distribution Probability – Prob {demand is Q or lower} = Prob {the outcome of a standard normal is z or lower}, where Q 0 50 100 150 200 or z Q z Demand – Look up z in the Standard Normal Distribution Function Table to get the desired probability or use Excel: Prob {} = normsdist ( z ) 12
T HE S TANDARD N ORMAL D ISTRIBUTION Standard Normal Distribution Function Table (continued), F ( z ) z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 • Q: What is the probability the outcome of a standard normal will be z = 0.28 or smaller? • A: The answer is 0.6103, or 61.03% 13
T HE S TANDARD N ORMAL D ISTRIBUTION Standard Normal Distribution Function Table (continued), F ( z ) z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 • Q: For what z is there a 70.19% chance that the outcome of a standard normal will be that z or smaller? • A: the answer is z = 0.53 14
O’N EILL ’ S H AMMER 3/2 W ETSUIT TIMELINE • Marketing’s forecast for sales is 3200 units. – our demand model is a normal distribution with mean 3192 and standard deviation 1181 • O’Neill sells each suit for p = $190 • O’Neill purchases each suit from its supplier for c = $110 per suit • Discounted suits sell for v = $90 – This is also called the salvage value . • How many units shall O’Neill order?! • O’Neill is facing a “too much/too little problem”: – What are the costs associated with too much/too little? 15
“T OO MUCH ” AND “ TOO LITTLE ” COSTS • C o = overage cost – The consequence of ordering one more unit than what you would have ordered had you known demand. – For the Hammer 3/2 C o = Cost – Salvage value = c – v = 110 – 90 = 20 • C u = underage cost – The consequence of ordering one fewer unit than what you would have ordered had you known demand. – For the Hammer 3/2 C u = Price – Cost = p – c = 190 – 110 = 80 16
“T OO MUCH ” AND “ TOO LITTLE ” COSTS • Idea: balance the risk vs. benefit of an additional unit • Ordering one more unit increases the chance of overage … – Expected loss on the Q th unit = C o · F ( Q ) • … but the benefit/gain of ordering one more unit is the reduction in the chance of underage: – Expected gain on the Q th unit = C u · (1- F ( Q )) 90 • As more units are ordered, 80 70 Expected gain or loss . the expected benefit from Expected gain 60 ordering one unit decreases 50 40 while the expected loss of Expected loss 30 ordering one more unit increases! 20 10 0 17 0 800 1600 2400 3200 4000 4800 5600 6400 Q th unit ordered
M AXIMIZE EXPECTED PROFIT Expected profit is maximized when the expected loss on the Q th unit • equals the expected gain on the Q th unit: ( ) 1 C F Q C F Q o u • Rearrange terms in the above equation: C u ( ) F Q C C o u • The ratio C u / ( C o + C u ) is called the critical ratio . • Hence, to maximize profit, choose Q such that the probability we satisfy all demand (i.e., demand is Q or lower) equals the critical ratio. 80 C u 0 . 80 • For the Hammer 3/2 the critical ratio is 18 20 80 C C o u
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