Modeling Extensions for a Class of Business-to-Business Revenue Management Problems

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. 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. 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. 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. 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. A Class of B2B Revenue Management Problems • The class of B2B problems with rentableresources • 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. 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. 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. Airline Setting Services are delivered here Beginning of the finite horizon End of the finite horizon Booking period A Class of B2B Revenue Management Problems Service is delivered over time Hotel/Rental-Car Setting Beginning of the finite horizon End of the finite horizon Booking period

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. 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. Unification of Traditional Models • The GV97 model reformulated as a demand-to-come model maxp,xSj pjxj s.t.Sjaijxj£ ci, "i; (yi) 0 £ xj£ E[Nj(pj)],"j pj³0,"j • At optimality the allocation variables xj can be eliminated

13. Unification of Traditional Models • GV97 KKT conditionsimply that If Si aijy*i > 0 then p*j = [1 + MUj(p*j)]Si aijy*i MUj(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. 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 maxpSi vi(p) s.t. Sipij³0, "j • And a set of Lagrangian allocation subproblems,"i vi(p) := maxxSjpijxij s.t.Sjaijxij£ ci 0 £ xij£ E[Nj (pj)], "j with pj = Si pij

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

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. 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 variablecost 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. Extension of Traditional Models • Index sets: CTI set J, service set M, resource set I, time-period set K • Parameters aj: Length of CTI j service period Njk(pjk): r.v. # of CTI j requests received in bookingperiodk as a function of the price vector pjk= (pjkm, m Î M) Sjkm: r.v. size of service m asked for by one CTI j request in period k aik’m: consumption of resource i by service m in serviceperiodk’ cik’: resource i available capacity in period k’ fcik: resource i convex variable-cost function in period k’ • Decisions variables xjk: number of accepted CTI j requests in period k pjkm: price of service m for CTI j in period k

21. Extension of Traditional Models • Deterministic network pricing model maxp,xSj,k,majpjkmE[Sjkm]xjk –Si,k’ fcik’(Aik’) s.t.Sj,k,maik’mE[Sjkm]xjk£ cik’, "i, k’; (yik’) 0 £ xjk£ E[Njk(pjk)], "j, k pjkm³0, "j, k, m with Aik’:=Sj,k,maik’mE[Sjkm]xjk • At optimality variables xjk can be eliminated

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. Extension of Traditional Models • KKT conditions imply that the optimal prices are mark-ups over the sum of marginal and per-unitopportunity costs p*jkm= aj–1[1 + MUjkm(p*jk)]Si,k’ [fc’ik’(A*ik’)+ y*ik’]aik’m withMUjkm(p*jk) the optimal mark-up factor (³ 0) for service m of CTI j in period k

24. Extension of Traditional Models • Lagrangian dual pricing problem maxpSi,k’ vik’(pik’) s.t. Si,k’pijkk’m³0, "j, k, m • Lagrangian allocation subproblems, "i, k’ vik’(pik’) := maxxSj,k,mpijkk’mE[Sjkm]xijkk’ – fcik’(Bik’) s.t.Sj,k,maik’mE[Sjkm]xijkk’£ cik’ 0 £ xijkk’£ E[Njk(pijkk’)], "j, k withajpjkm = Si,k’pijkk’m and Bik’ :=Sj,k,maik’mE[Sjkm]xijkk’

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. 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 unitprices according to the optimization model then negotiate sizes • Set size-dependent prices to maximize the expected profitability of the transaction

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

28. Possible Types of Control Policies IC(sk) :=Si,k’ [VCik’(sk) + OCik’(sk)] VCik’(sk): Variable cost of resource i in period k’ with cbik’ already booked capacity VCik’(sk) :=fcik’ (cbik’ + Sj,k,maik’mskm) – fcik’ (cbik’) OCik’(sk): Approximate opportunity cost of resource i in period k’, e.g. OCik’(sk) := Smaik’mskmy*ik’

29. Possible Types of Control Policies • Given the incremental cost, compute prices to maximize expected profit maxP [aSm Pkmskm –IC(sk)] × Pr(Çm{Wm ³ Pkm}) With Wm the random variable buyer’s willingness to pay for service m

30. Possible Types of Control Policies • Given the incremental cost, compute total revenue to maximize expected profit maxR [R–IC(sk)] × Pr(V³ 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. 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