4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 An Asymptotically-Optimal Dynamic Admission Policy for a Revenue Management Problem Marty Reiman and Qiong Wang Bell Labs, Lucent Technologies

4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 Problem Description Given –A set of resources (l=1,...,L) with fixed capacities C l –A set of customers classes (j=1,...,J) associated with a lj : usage of resource l (l=1,...,L) by a class j customer p j : revenue per class j customer served –Customers arrive randomly in period (0,T] without loss of generality, scale T=1 Find –An admission policy to maximize total expected revenue Example: –airline seat control –revenue management in communication networks –component allocation in a supply chain

4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 Overview of the Problem Hindsight optimum However, we may only know distributional information of  j (t) e.g, E[  j (t)] Uncertainty about arrivals implies a tradeoff between leaving capacity idle and blocking high value customers Use hindsight optimum,  opt  as a benchmark to evaluate how well an admission policy performs  j (t): exact number of arrivals in (t,1]

4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 Previous Work Review by McGill and van Ryzin (1998) –Booking Limit –Nesting –Bid Price More recent studies –Cooper (2002): LP-based, static booking limit –Li and Yao (2003): fixed-point

4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 Numerical Example Problem setup 3 resources and 11 customer classes –resource 1 used by classes 1, 4, 5, 7, 8 –resource 2 used by classes 2, 3, 6, 9, 10 –resource 3 used by classes 5, 6, 8, 10, 11 –each resource has 100 units of capacity Poisson arrival All three resources are critically loaded –(  1 =90%,  2 =104%,  3 =101%)

4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 Numerical Example Optimality gap under different policies

4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 Numerical Example Revenue change with scaling of arrivals and capacities scale both arrival rates and capacities by a factor of k –k=2, 4, 8, 16, 32, 64, 128 in all cases, the expected revenue is proportional to k

4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 Numerical Example Scaling the differences in expected revenues

4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 Asymptotic Optimality on Fluid Scale (Cooper 2002) Assumptions –the overall arrival is a simple point process (SPP) –only mean arrivals, E[  j (0)], are known Policy: accept class j customers up to x j * Optimality –scale capacity by k, C l (k) =kC l –overall arrival scaled to be a SPP.  j (k) (0)/k converges in distribution to E[  j (0)] –let  (k) be the expected revenue under the above policy,  opt (k) be the expected revenue of hindsight policy

4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 Asymptotic Optimum on Diffusion Scale Assumptions –the overall arrival process meets a second moment condition that yields a Functional Central Limit Theorem –mean arrivals in any period (t,1], E[  j (t)], are known –for the simplicity of presentation, assume here arrivals are stationary, i.e., E[  j (t)]= j (1-t) Optimality Condition –let  (k) be the expected revenue under our policy Talluri and van Ryzin(1998) studied a related but different problem and showed a bid-price control policy can achieve

4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 Outline of Our Policy and apply the revised acceptance rule beginning with the next arrival At t=0, optimize the fluid model Use x j * to formulate an acceptance rule Develop trigger function(s) –compare the expected acceptance (by fluid model) with the actual acceptance –a trigger function is a function of the differences At time  when a trigger violates a given threshold, re-optimize the fluid model

4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 “Thinning and Trigger (T 2 )” Policy Acceptance Rule accept class j customers with probability x * j / j accept all class j customers if x j * = j, reject all class j customers if x j * =0 Trigger Function let Z j (t) be the number of class j customers accepted by t Threshold Condition

4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 Asymptotic Properties of T 2 Policy scale both arrivals and capacities by k let y j (k) :hindsight-optimal numbers of customers to accept in (0,1]  (k) :the first time when the trigger is “pulled” z j (k) : the number of customers accepted before  (k)  (k) (  (k) ): the hindsight-optimal expected revenue in (  (k),1], i.e., recall that  j (k) (  (k) ) are the exact numbers of arrivals in (  (k),1]

4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 Optimality of T 2 Policy Let  (k) (  (k) ) be the expected revenue in (  (k),1] under re-optimized acceptance rule only one re-optimization is needed to get to optimality on diffusion scale Repeat Apply Fluid Model to 2)

4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 “Open and Trigger” Policy easy to implement (no probabilistic acceptance rule) may need to re-optimize many times Acceptance Rule admit all class j customers if x j * >0 Trigger Function and Threshold case 1:either x j * = j or x j * =0, no difference from T 2 policy case 2: 0< x j * < j for some j trigger functions thresholds

4th Annual INFORMS Revenue Management and Pricing Section Conference, June 2004 Summary A classical revenue management problem in a general setting –few restrictions on arrival process –only need to know mean arrivals Asymptotic optimality on diffusion scale Two proposed policies: –thinning and trigger policy: re-optimize only once –open and trigger policy: easy to implement, may re-optimize many times –in both cases, the calculation does not go beyond linear programming Numerical example shows better performance