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Capacity allocation Lecture 4 Capacity allocation • How many low-fare seats (hotel rooms, rental cars) to allow to be booked while facing the possibility of future high-fare demand • Airlines, rental car companies, hotels, cruise lines, freight transportation, made-to-order manufacturing The two-class problem • The basic model: all the discount bookings occur any full-fare passengers seek to book • Maximizing revenue/taking into account incremental costs and ancilliary contribution/taking into account sunk costs • Determine the discount booking limit • Tradeoff between setting it too high or too low (dilution vs. spoilage) • ***7.1 ff(C-b=y) 1 0 10 20 30 40 60 70 80 90 100 60 70 80 90 100 y 0.8 Ff(C-b=y) 50 0.6 0.4 0.2 0 0 10 20 30 40 50 Y ff(C-b=y) 1 0 10 20 30 40 60 70 80 90 100 60 70 80 90 100 y 0.8 Ff(C-b=y) 50 0.6 0.4 0.2 0 0 10 20 30 40 50 Y ff(C-b=y) 1 0 10 20 30 40 60 70 80 90 100 60 70 80 90 100 y 0.8 Ff(C-b=y) 50 0.6 0.4 0.2 0 0 10 20 30 40 50 Y ff(C-b=y) 1 0 10 20 30 40 60 70 80 90 100 60 70 80 90 100 y 0.8 Ff(C-b=y) 50 0.6 0.4 0.2 0 0 10 20 30 40 50 Y • B=60->61 • PlaneC=100 – – – – – – – – – • • • • • • *Dd=50 *Df=45; Df=55 *86% **Dd=65 **Df=30 **14%..*6.5% ***Dd=65 ***Df=45 ***14%..*93.5% =6.5%*190+93.5%*(190-200)=3 =86%*0+14%*3=0.42 =86%*0+14%*(6.5%*190+93.5%*(190-200)) =14%*(6.5%*190+93.5%*(190-200))=>6.5%*190+93.5%*(190-200) =190—93.5%*200=> Pd—93.5%*Pf>0=>Pd/Pf>93.5%;190/200=95% 50%*10000+50%*20000= • Discount booking limit for an airplane with 150 seats is 80 seats. • The airplane is being substituted for one with 100 seats. • What is the discount booking limit now? ff(C-b=y) 100 90 80 70 60 50 40 30 20 10 60 70 80 90 0 y 1 1-Ff(C-b=y) 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 b 100 ff(C-b=y) 1 0 10 20 30 40 60 70 80 90 100 60 70 80 90 100 y 0.8 Ff(C-b=y) 50 0.6 0.4 0.2 0 0 10 20 30 40 50 Y ff(C-b=y) 100 90 80 70 60 50 40 30 20 10 60 70 80 90 0 y 1 1-Ff(C-b=y) 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 b Kui pd/pf = 0.5, siis paneme me täishinnaga müüki alati mü=66 piletit, olenemata sellest kui suur on full demand standardhälve 100 ff(C-b=y) 100 90 80 70 60 50 40 30 20 10 60 70 80 90 0 y 1 1-Ff(C-b=y) 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 b Kui diskonteeritud piletid on väga kallid, siis mida riskantsem on täishinnaga piletite müük, seda vähem me täishinnale pileteid protectime 100 ff(C-b=y) 100 90 80 70 60 50 40 30 20 10 60 70 80 90 0 y 1 1-Ff(C-b=y) 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 b Kui diskonteeritud piletid on väga odavad, siis mida riskantsem on täishinnaga piletite müük, seda rohkem me täishinnale pileteid protectime 100 Relation to the newsvendor problem, 1882 • • • • • • Overage and underage costs O=20 U=5 With respect to the high price class O=pd U=pf-pd Using Littlewood’s (1972) rule • The algorithm for setting b: – Set b to 0; set pd=190; set pf=200->pd/pf=95% – Set b=b+1 (e.g. remainder = 99, if PlaneCapacity=100) – Check whether 1-Ff(C-b) > 95%– probability, that there is too much full demand, is over 95% – If no, go back to increasing b – Otherwise stop at previous b Dynamic programming, Bellman’s principle Expected Marginal Seat Revenue (EMSR) Heuristics, the case with three price classes, heuristic EMSRa • Littlewood lookup for the distribution of P1: – P3/P1 • Littlewood lookup for the distribution of P2: – P3/P2 • Booking limit b3= PlaneC minus the two quantities computed above • The cases with more classes analogous – e.g. first calculate b4, then b3, then b2 Heuristic EMSRb, this time, for example, for four classes • First of all, we want the booking limit b4 • For the classes above it, calculate a following „weighted average“ class • AvgNew=Avg3+Avg2+Avg1 • StdevNew=sqrt(Stdev3^2+Stdev2^2+Stdev1^2) • priceNew=price3*Avg3 /AvgNew + price2*Avg2/AvgNew + price1*Avg1/AvgNew • Do a Littlewood lookup on price4/priceNew on a distribution given by AvgNew and StdevNew to determing b4 • Proceed analogously for b3, then b2 Comparison of EMSRa, EMSRb and dynamic programming, see Belobaba 1992 Demand dependence • Demand in each fare class is independent of demand in the other fare classes – E.g. no cannibalization – opening a discount class, has no effect on full-fare demand – Also, no buy-up/sell-up – closing a discount fare class does not lead to increased demand in higher fare classes Two-class capacity allocation with demand dependence • Instead of =pd/pf, we have the modified formula =1/(1-alfa)*(pd/pf-alfa) • to look up using Littlewood’s rule, • alfa being the fraction of customers, who will switch to full price, after being denied a discounted ticket Modified EMSR Heuristics, Belobaba and Weatherford 1996 • Allows buy-up to the next highest class only • Modified EMSRa – use modified formula for the calculation of the next level only, original formula for all the classes above that • Modified EMSRb – use modified formula with respect to „weighted average“ class created for EMSRb (create this class in the original way) • Though, they are „heuristics grafted on to another heuristic“ Measuring the effectiveness of revenue management • • • • Total revenue opportunity No revenue management Revenue Opportunity Metric Example: – ROM = ($45 000 – $35 000)/($50 000 - $35 000) – =67% • But, ROM for flights with very low demand will always be close to 100% • And, see Fig7.5 above, if full-fare demand is high, but uncertain, ROM will be lower, than when full-fare demand is high, but more certain (low stdev) • Thus, as these examples show, changes in ROM, might also be due to changes in the underlying factors of demand, rather than due to changes in the effectiveness of revenue management • LECTURE 4 ENDS HERE Tactical revenue management • Calculates and periodically updates the booking limits • Resources – Units of capacity (flight departure, hotel room night, rental car day) • Products – Customers are seeking to purchase those (a seat on Flight 130 from St. Louis to Cleveland on Monday, June 30 – single resource; A two-night stay at the Sheraton Cleveland, arriving on March 19 and departing on March 21 – two resources) • Fare classes – A combination of a price and a set of restrictions on who can purchase and when (e.g. group and regional pricing) • The fact that RM operates fare classes, does not change much from customers view – he still sees only the lowest available fare • Since airlines still respond to the offers made by the competition, RM supplements rather than replaces pricing Capacity allocation • How many seats (hotel rooms, rental cars) to allow low-fare customers to book – given the possible future high-fare demand • Two-class problem – Discount customers – Full-fare customers • BASIC MODEL – all discount bookings happen before full-fare bookings • We maximize expected revenue – incremental costs and ancillary contribution are zero • In reality companies should maximize expected total contribution