Towards Robust Revenue Management: Capacity Control Using Limited Demand Information Michael Ball, Huina Gao, Yingjie Lan & Itir Karaesmen Robert H Smith School of Business University of Maryland June 6, 2006

Background on Competitive Analysis Algorithmalgorithminput Evil DesignerstreamAdversary i 1, i 2, i 3, … Competitive Ratio = Min input streams {(alg performance)/(best performance)} “Traditional” revenue management analysis has assumed: Demand can be forecast reasonably well Risk neutrality Are these valid??

Review of BQ Results BQ: Ball & Queyranne, “Toward Robust Revenue Management: Competitive Analysis of Online Booking” Sample Result:  Flight has 95 available seats, three fare classes: $1,000, $750, $500  Policy that guarantees at least 63% of the max possible revenue:  Protect 15 high fare seats  Protect 35 seats for two higher fare classes (i.e. sell at most 60 lowest fare seats)

2-fare Case Continuous booking problem – can accept a fraction of an order  n = number of seats  f1 = higher fare; f2 = lower fare.  r = f2/f1 = discount ratio.  Key quantity: b(r) = 1 / (2 – r) Theorem: For the continuous two-fare booking problem, the booking policy with protection level (1 – b(r)) n has competitive ratio b(r). This is best possible among all online booking policies.

Comparison to first-come first-served and partitioned protection levels first come-first served partitioned protection levels nested protection levels

More General Results Define:  = m -  {i=2,m} f i / f i-1  i = (n /  ) (i -  {j=1,i} f j+1 / f j ) Theorem: For the continuous m-fare problem, the protection level policy using protection levels  i achieves a competitive ratio of at least 1 /  and this is the best possible. Discrete problem: results generally can be extended with small loss in guaranteed performance; randomized algorithms can achieve same performance as for continuous case. Results apply to standard and theft nesting (order quantity control); results also for bid-price controls. Dynamic policies introduced.

Goals of Our Research  Empirically evaluate online policies and compare to other well-known methods  Evaluate the benefit of using limited demand information  Test sensitivity to demand parameters or models

Experiments  Average performance computed using simulation  S = 300 samples  Performance metric: Average performance gap (average of ratio of policy revenue to offline revenue)  Aggregate demand Poisson distributed  Vary: demand mix, demand factor, fares, arrival regime

Basic BQ policies w time homogeneous arrivals (stat: static BQ, stat-t: static BQ w theft, dyn: dynamic BQ, fcfs: first-come, first-served) f 1 =2000, f 2 =1000, n=100, arrivals time-homogeneous

(cnt’d) No. of seats unsold f 1 =2000, f 2 =1000, n=100, arrivals time-homogeneous

(cont’d) Performance with LBH arrivals f 1 =2000, f 2 =1000, n=100, arrivals LBH

Demand Bounds BQ policies do not seem to perform well when high fare demand is low. But these policies are behaving “as advertised”, i.e. protecting against possibility of influx of high demand orders. If such an influx of high demand orders is not possible then we need to “tell” the policy  Demand bounds: L i  class i demand  U i

Deriving Policies for Bounded Demand Case online revenue from input stream I and policy b optimal offline revenue Problem: space of input streams can get extremely large

Reducing problem size using CAST’s (category attack streams) fare class m class 1 fare class m class 1 num orders = lower bound num orders = upper bound  All input streams with the same profile are dominated by the unique LBH stream.  Any LBH stream is dominated by one of the CAST streams, there are m in total.

MIP Formulation for Choosing Best Policy competitive ratio. b i – b i gap between nested booking limits offline optimal revenue from CAST j revenue guaranteed by lower bounds from requests with fares higher than class j. Capacity remaining after allocating space for orders of class j or higher guaranteed by bounds

Closed Form Solution for Continuous Problem (LP relaxation)

Policy Optimality and Discrete Case  For the continuous m-fare problem with bounds, no policy can achieve a better competitive ratio than z*.  For the discrete problem, the seller can achieve the same competitive ratio as the continuous problem, by randomizing among a certain set of policies.

Dynamic Policies  Prior analysis assumes adversary always follows an optimal strategy.  If adversary makes a mistake, then from that point forward we can adjust the policy and possibly guarantee a better level of performance.  The analysis of the bounded case can easily be adjusted for this case since adversary mistakes  adjustments to bounds.  Dynamic policies cannot guarantee better a priori performance – only better performance given certain partial input streams

Policies evaluated  STAT: static online policy of BQ  DYN: dynamic online policy suggested by BQ  BSTAT: static online policy with demand bounds  BDYN: dynamic online policy with demand bounds  EMSR: Littlewood’s rule (optimal for m=2 and LBH)  MDP: Markov Decision Process model  VRM-30: adaptive method of van Ryzin and McGill, computed only with the first 30 runs  VRM: adaptive method of van Ryzin and McGill, computed by discarding initial runs

Benefit of demand information: time-homogeneous arrivals f 1 =2000, f 2 =1000, n=100, arrivals time-homogeneous

Benefits of demand information: LBH Arrivals f 1 =2000, f 2 =1000, n=100, LBH arrivals

Comparison to VRM (2 to 1 fare ratio) f 1 =2000, f 2 =1000, n=100, LBH arrivals, demand factor 1.5

Multiple fare classes m=3, n=100, f 1 =3000, f 2 =2000, f 3 =1000, 3 =30, LBH arrivals

Sensitivity to demand information (mean arrival rate varies – bounds remain the same) f 1 =2000, f 2 =1000, n=100, estimated mean [60,60], demand bounds 2 std.dev. away from estimated mean, time-homogeneous arrivals

Sensitivity to demand information (all arrivals in first half of booking period) f 1 =2000, f 2 =1000, n=100, arrivals estimated to be time-homogeneous, actual arrivals occur only in the first-half of the booking period

Observations based on Computations  When correct bounds are used, online policies yield maximal revenue (  EMSR)  Difference between static and dynamic policies is negligible when bounds are used when demand is stationary  Policies relatively robust to quality of aggregate demand information (  EMSR)

Summary and Conclusions  Competitive analysis of online booking  Incorporated limited demand information  Closed form solutions obtained via sequence reduction and MIP/LP formulation  Optimal online policies  Nesting structure preserved and is optimal  Computational complexity minimal

Summary and Conclusions  Demand information significantly improves the average revenues obtained by online policies  Close to maximal revenues  Policies are robust  Distribution free  Free of demand modeling errors  Work well with error in estimates  No risk-neutrality assumption