1. 1 Modeling Extensions for a Class of Business-to-Business Revenue Management Problems Nicola Secomandi Carnegie Mellon University Tepper School of Business Phone: (412) 268-9596 E-mail: ns7@andrew.cmu.edu Joint work with Kirk Abbott, PROS Revenue Management 4-th Annual Conference of the INFORMS Revenue Management and Pricing Section, MIT June 10, 2004

2. 2 Outline Motivation A Class of Business-to-Business (B2B) Revenue Management Problems Unification and Extension of Traditional Models Possible Types of Control Policies Conclusions

3. 3 Motivation Traditional revenue management supports commercial reservation processes for perishable capacity –Airline case: itinerary bookings –Hotel case: room-night bookings –Rental-car case: car-day bookings “Airline revenue management illustrates a successful e-commerce model” (Boyd and Bilegan 2003) –Interplay of central reservation and revenue management systems

4. 4 Motivation Traditional revenue management deals mainly with business-to-consumer (B2C) transactions But B2B commerce accounts for more than 90% of all commercial transactions E-commerce enables the use of revenue management to support B2B commercial reservation processes

5. 5 Motivation The opportunity to apply revenue management to B2B environments is huge and remains essentially untapped Research questions –To what B2B domains are traditional revenue management concepts relevant? –Are traditional models adequate? –If not, how can they be extended?

6. 6 A Class of B2B Revenue Management Problems The class of B2B problems with rentable resources –Companies that sell B2B services rendered through rentable resources Examples –Companies that provide transportation services as natural-gas, telecom, freight, and cargo carriers –Companies that lease commercial and industrial equipment, physical space, data storage, and web-based computing machinery

7. 7 A Class of B2B Revenue Management Problems B2B transactions are predominantly based on long-term contracts –They support production and distribution processes –B2C transactions support consumption processes Negotiation is the main B2B transaction mechanism –Bilateral trades –Requests for proposal/quote –Auctions –Price, quantity, quality, and terms and conditions are jointly negotiated

8. 8 A Class of B2B Revenue Management Problems For this class of problems, revenue management must –Integrate pricing and inventory controls –Support the negotiation of long-term contracts involving multiple services But traditional models –Separate inventory and pricing control –Are “1-dimensional” in either resources or time

9. 9 A Class of B2B Revenue Management Problems Booking period Services are delivered hereAirline Setting Beginning of the finite horizon End of the finite horizon Booking period Service is delivered over time Hotel/Rental-Car Setting Beginning of the finite horizon End of the finite horizon

10. 10 Unification of Traditional Models Recent review papers –McGill & van Ryzin (1999) –Boyd & Bilegan (2003) –Bitran & Caldentey (2003) –Elmagraby & Keskinocak (2003) Books –Talluri & van Ryzin (2004) –Phillips (2004)

11. 11 Unification of Traditional Models Separation of inventory and pricing control in traditional revenue management is mainly an organizational issue Within the simple deterministic framework of traditional models it is not mathematically warranted Gallego and van Ryzin (GV97) deterministic network pricing model subsumes the classical demand-to-come deterministic LP model and its related control mechanisms

12. 12 Unification of Traditional Models The GV97 model reformulated as a demand-to-come model max p,x  j p j x j s.t.  j a ij x j  c i,  i; (y i ) 0  x j  E[N j (p j )],  j p j  0,  j At optimality the allocation variables x j can be eliminated

13. 13 Unification of Traditional Models GV97 KKT conditions imply that If  i a ij y* i > 0 then p* j = [1 + MU j (p* j )]  i a ij y* i MU j (p* j ): optimal mark-up factor for service j GV97 optimal prices include bid prices –When prices are set correctly “the effect of allocation schemes appears to be relatively minor” (GV97)

14. 14 Unification of Traditional Models Extending a result reported by Boyd and Bilegan (2003) the GV97 model can be decomposed into –A Lagrangian dual pricing problem max   i v i (  ) s.t.  i  ij  0,  j –And a set of Lagrangian allocation subproblems,  i v i (  )  max x  j  ij x ij s.t.  j a ij x ij  c i 0  x ij  E[N j (  j )],  j with p j =  i  ij

15. 15 Unification of Traditional Models The traditional price-proration approach combined with local optimizations can be used with the GV97 model –EMSR-based virtual nesting –DP-based bid price tables Ignoring demand uncertainty, pricing is more important than inventory control But inventory control remains important to account for demand uncertainty in between re- optimizations

16. 16 Extension of Traditional Models The following contract-type is the basic modeling entity for the class of B2B problems with rentable resources –Start time and duration –Set of services requiring the usage of multiple resources –Service prices and quantities –Take-or pay terms and conditions The firm structures its commercial agreements as contract-type instances (CTIs) The firm observes demand for CTIs

17. 17 Extension of Traditional Models CTIs have overlapping booking and service periods

18. 18 Extension of Traditional Models CTI demand functions are multidimensional –They depend on the prices of all the services that belong to a CTI

19. 19 Extension of Traditional Models A time-and-space network optimization model is needed –Airline case: spatial network problem –Hotel/Rental-car case: temporal network problem In most applications the variable cost of providing a B2B service can be substantial –The marginal cost may also depend on the remaining available capacity B2B request size is not unitary

20. 20 Extension of Traditional Models Index sets: CTI set J, service set M, resource set I, time- period set K Parameters  j : Length of CTI j service period N jk (p jk ): r.v. # of CTI j requests received in booking period k as a function of the price vector p jk  (p jkm, m  M) S jkm : r.v. size of service m asked for by one CTI j request in period k a ik’m : consumption of resource i by service m in service period k’ c ik’ : resource i available capacity in period k’ f c ik : resource i convex variable-cost function in period k’ Decisions variables x jk : number of accepted CTI j requests in period k p jkm : price of service m for CTI j in period k

21. 21 Extension of Traditional Models Deterministic network pricing model max p,x  j,k,m  j p jkm E[S jkm ]x jk –  i,k’ f c ik’ (A ik’ ) s.t.  j,k,m a ik’m E[S jkm ]x jk  c ik’,  i, k’; (y ik’ ) 0  x jk  E[N jk (p jk )],  j, k p jkm  0,  j, k, m with A ik’   j,k,m a ik’m E[S jkm ]x jk At optimality variables x jk can be eliminated

22. 22 Extension of Traditional Models Main differences with respect to GV97 –Capacity constraints are “2-dimensional” (i and k’) –Objective includes a cost function –Objective and capacity constraints include expected service request size

23. 23 Extension of Traditional Models KKT conditions imply that the optimal prices are mark-ups over the sum of marginal and per-unit opportunity costs p* jkm =  j –1 [1 + MU jkm (p* jk )]  i,k’ [f c’ ik’ (A* ik’ ) + y* ik’ ]a ik’m with MU jkm (p* jk ) the optimal mark-up factor (  0) for service m of CTI j in period k

24. 24 Extension of Traditional Models Lagrangian dual pricing problem max   i,k’ v ik’ (  ik’ ) s.t.  i,k’  ijkk’m  0,  j, k, m Lagrangian allocation subproblems,  i, k’ v ik’ (  ik’ )  max x  j,k,m  ijkk’m E[S jkm ]x ijkk’ – f c ik’ (B ik’ ) s.t.  j,k,m a ik’m E[S jkm ]x ijkk’  c ik’ 0  x ijkk’  E[N jk (  ijkk’ )],  j, k with  j p jkm =  i,k’  ijkk’m and B ik’    j,k,m a ik’m E[S jkm ]x ijkk’

25. 25 Extension of Traditional Models The following traditional scheme applies –Solve the network (pricing) model to compute optimal prices and capacity duals –Compute resource/time-period combinations capacity value functions –Use these parameters to instantiate a control policy

26. 26 Possible Types of Control Policies Auction-based one-to-many negotiations –Set a minimum price for a block of capacity-to- auction to cover variable and opportunity costs Bilateral negotiations –First set unit prices according to the optimization model then negotiate sizes –Set size-dependent prices to maximize the expected profitability of the transaction

27. 27 Possible Types of Control Policies Bilateral negotiation with size-dependent prices –Consider a request for services {m} with sizes {s km } and service time-periods {k’} of length  received in time period k < k’ –Assume feasible sizes {s km } The quoted prices {P km } should –Cover the incremental cost IC(s k ) of accommodating the request –Account for the buyer’s willingness to pay or valuation

28. 28 Possible Types of Control Policies IC(s k )   i,k’ [VC ik’ (s k ) + OC ik’ (s k )] VC ik’ (s k ): Variable cost of resource i in period k’ with c b ik’ already booked capacity VC ik’ (s k )  f c ik’ (c b ik’ +  j,k,m a ik’m s km ) – f c ik’ (c b ik’ ) OC ik’ (s k ): Approximate opportunity cost of resource i in period k’, e.g. OC ik’ (s k )  m a ik’m s km y* ik’

29. 29 Possible Types of Control Policies Given the incremental cost, compute prices to maximize expected profit max P [  m P km s km – IC(s k )]  Pr(  m {W m  P km }) With W m the random variable buyer’s willingness to pay for service m

30. 30 Possible Types of Control Policies Given the incremental cost, compute total revenue to maximize expected profit max R [R – IC(s k )]  Pr(V  R) R: the total revenue from the transaction V: the buyer’s valuation or budget random variable Prices can be recovered by splitting R according to some business rule

31. 31 Conclusions Contributions –Introduced the class of B2B problems with rentable resources –Unified and extended traditional models Additional research –CTI demand modeling/forecasting and interplay with optimization/control-policies –Numerical, experimental or empirical testing