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Bilevel Programming Approaches to Revenue Management and Price Setting Problems

Bilevel Programming Approaches to Revenue Management and Price Setting Problems. Gilles Savard, École Polytechnique de Montréal, GERAD, CRT

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Bilevel Programming Approaches to Revenue Management and Price Setting Problems

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  1. Bilevel Programming Approaches to Revenue Management and Price Setting Problems Gilles Savard, École Polytechnique de Montréal, GERAD, CRT Collaborators: P. Marcotte and C. Audet, L. Brotcorne, M. Gendreau, J. Gauvin, P. Hansen, A. Haurie, B. Jaumard, J. Judice, M. Labbé, D. Lavigne, R. Loulou, F. Semet, L. Vicente, D.J. White, D. Zhu Students: so many including J.-P. Côté, V. Rochon, A. Schoeb, É. Rancourt, F. Cirinei, M. Fortin, S. Roch, J. Guérin, S. Dewez, K. Lévy

  2. Outline • The revenue management problem • The bilevel programming problem • A price setting paradigm • … applied to toll setting • … a TSP instance • … applied to airline • Conclusion 16 janvier, 2004

  3. The revenue management problem • …the optimal revenue management of perishable assets through price segmentation (Weatherford and Bodily 92) • Fixed (or almost) capacity • Market segmentation • Perishable products • Presales • High fixed cost • Low variable cost 16 janvier, 2004

  4. The revenue management problem RM Business process • Forecasting • Schedule with capacity • Pricing • Booking limits • Seat sales 16 janvier, 2004

  5. The revenue management problem Some issues in airline industry: • How to design the booking classes? • Restriction, min stay, max stay, service, etc… • … at what price? • Willingness to pay, competition, revenue, etc… • … how many tickets? • Given the evolution of sales (perishables) • … at what time? • Given the inventory and the date of flight 16 janvier, 2004

  6. The revenue management problem Evolution of Pricing & RM • 1960’s: AA starts to use OR models for RM decisions • 1970’s: AA develops SABRE, providing automatic update of availability and prices • 1980’s: First RM software available • 1990’s: RM grows, even beyond airlines(hotel, rail, car rental, cruise, telecom,…) • 2000’s: networks 16 janvier, 2004

  7. The revenue management problem Decision Support Tools focus on bookinglimits BUT mostly ignore pricing • Complex problem: • Must take into account its own action and the competition, as well as passenger behaviour • Highly meshed network (hub-and-spoke) • OD-based vs. Leg-based approach • Data intensive 16 janvier, 2004

  8. The revenue management problem • «Pricing has been ignored» P.Belobaba (MIT) • « Interest in RRM … is rising dramatically … RRM should be one of the top IT priorities for most retailers » AMR Research • «Pricing Decision Support Systems will spur the next round of airline productivity gains» L. Michaels (SH&E) 16 janvier, 2004

  9. The revenue management problem • Until recently, capacity allocation and pricing were performed separately: capacity allocation is based on average historical prices; pricing is done without considering capacity. • However, there is a strong duality relationship between these two aspects. • A bilevel model combines both aspects while taking into account the topological structure of the network. 16 janvier, 2004

  10. The revenue management problem Maximize expected revenue by determining over time the products the prices the inventory the capacity taking into account the market response pricing seat inventory overbooking forecasting… 16 janvier, 2004

  11. Outline • The revenue management problem • The bilevel programming problem • A price setting paradigm • … applied to toll setting • … a TSP instance • … applied to airline • Conclusion 16 janvier, 2004

  12. Bilevel programming problem Leader Follower 16 janvier, 2004

  13. Bilevel programming problem … or MPEC problems IV 16 janvier, 2004

  14. Bilevel programming problem A linear instance… F2 x’ x’’ 16 janvier, 2004

  15. Bilevel programming problem • Typically non convex, disconnected and strongly NP-hard (HJS92)(even for local optimality (VSJ94)) • Optimal solution  pareto solution (HSW89, MS91) • Steepest descent: BLP linear/quadratic (SG93) • Many instances: • Linear/linear (HJS92, JF90, BM90) • Linear/quadratic (BM92) • Convex/quadratic (JJS96) • Bilinear/bilinear (BD02, LMS98, BLMS01) • Bilinear/convex • Convex/convex 16 janvier, 2004

  16. Bilevel programming problem 16 janvier, 2004

  17. Bilevel programming problem 16 janvier, 2004

  18. Bilevel programming model • Resolution approaches • Combinatorial approaches (global solution) • Lower level structure: combinatorial structure • Descent approaches (on the bilevel model) • Sensitivity analysis (local approach) (Outrata+Zowe) • Descent approaches (on an approximated one-level model) • Model still non convex (e.g. penalization of the second level KKT conditions) (Scholtes+Stöhr) 16 janvier, 2004

  19. Bilevel programming model 1.Combinatorial approaches: convex/quadratic 16 janvier, 2004

  20. Bilevel programming model KKT 16 janvier, 2004

  21. Bilevel programming model The one level formulation: 16 janvier, 2004

  22. Bilevel programming model 16 janvier, 2004

  23. Bilevel programming model B&B: the subproblem and the relaxation 16 janvier, 2004

  24. Bilevel programming model • An efficient B&B algorithm can be developedby • Exploiting the monotonicity principle • Using two subproblems (primal and dual) to drive the selection of the constraints • Efficient separation schemes • Using degradation estimation by penalties • Using cuts • Size (exact solution): 60x60 to 300X150 • Heuristics: 600x600 (tabou, pareto) 16 janvier, 2004

  25. Bilevel programming model 2. Descent approach within a trust region approach (BC) • A good trust region model to bilevel program is a bilevel program that • is easy to solve (combinatorial lower-level structure) • is a good approximation of the original bilevel program • Such a non convex submodel (with exact algorithm) can track part of the non convexity of the original problem 16 janvier, 2004

  26. Bilevel programming model • Potential models: 16 janvier, 2004

  27. Bilevel programming model Notations real actual current predicted 16 janvier, 2004

  28. Bilevel programming model Classic steps: 16 janvier, 2004

  29. Bilevel programming model With a linesearch step (to guaranty a strong stationary point) 16 janvier, 2004

  30. Bilevel programming model b-stationary convergence 16 janvier, 2004

  31. Bilevel programming model 16 janvier, 2004

  32. Outline • The revenue management problem • The bilevel programming problem • A price setting paradigm • … applied to toll setting • … a TSP instance • … applied to airline • Conclusion 16 janvier, 2004

  33. A generic price setting model T: tax or price vector x: level of taxed activities y: level of untaxed activities 16 janvier, 2004

  34. A generic price setting model If the revenue is proportional to the activities we obtain the so-called bilinear/bilinear problem: 16 janvier, 2004

  35. A generic price setting model 16 janvier, 2004

  36. A generic price setting model 16 janvier, 2004

  37. A generic price setting model 1. The one level formulation: combinatorial approach 16 janvier, 2004

  38. A generic price setting model 16 janvier, 2004

  39. A generic price setting model 2. One level formulation: continuous approach 16 janvier, 2004

  40. A generic price setting model The combinatorial equivalent problem… The continuous equivalent problem… 16 janvier, 2004

  41. Outline • The revenue management problem • The bilevel programming problem • A price setting paradigm • … applied to toll setting • … a TSP instance • … applied to airline • Conclusion 16 janvier, 2004

  42. … on a transportation network Pricing over a network 16 janvier, 2004

  43. Free arcs Toll arcs … on a transportation network 5 1 1 1 1 2 3 4 5 10 Leader max Tx Follower min (c+T)x + dy Ax+By=b x,y >=0 T Toll vector x Toll arcs flow y Free arcs flow 16 janvier, 2004

  44. … on a transportation network A feasible solution... 5 1 +4 1 +1 1 +8 1 2 3 4 5 10 PROFIT = 4 16 janvier, 2004

  45. … on a transportation network …the optimal solution. 5 1 + 4 1 - 1 1 + 4 1 2 3 4 5 10 PROFIT = 7 16 janvier, 2004

  46. … on a transportation network The algorithms: • Branch-and-cut approach on various MIP-paths and/or arcs reformulations (LMS98, LB, SD, DMS01) • Primal-dual approaches (BLMS99, BLMS00, AF) • Gauss-Seidel approaches (BLMS03) 16 janvier, 2004

  47. … on a transportation network Replacing the lower level problem by its optimality conditions, the only nonlinear constraints are: We can linearize this term (exploiting the shortest paths): 16 janvier, 2004

  48. … on a transportation network 1. A MIP formulation 16 janvier, 2004

  49. … on a transportation network 2. Primal-dual approach (LB) 16 janvier, 2004

  50. … on a transportation network Step 1: Solve for T and λ (Frank-Wolfe) Step 2: Solve for x,y Step 3: Inverse optimisation Step 4: Update the M1 and M2 16 janvier, 2004

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