Bilevel Programming and Price Se1ing Problems. Martine Labbé Département d Informatique Université Libre de Bruxelles

Transcription

1 Bilevel Programming and Price Se1ing Problems Martine Labbé Département d Informatique Université Libre de Bruxelles

2 Bilevel Program LAAS - CNRS, Toulouse

3 Adequate Framework for Stackelberg Game Leader: 1st level, Follower: 2nd level. Leader takes follower s optimal reaction into account.

4

5 Basic References L.N. Vincente and P.H. Calamai (1994), Bilevel and multilevel programming : a bibiliography review, J. Global Optim. 5, M. Labbé, P. Marco1e, and G. Savard (1998), A bilevel model of taxation and its application to optimal highway pricing, Management Sci., Vol. 44, S. Dempe (2002), Foundations of bilevel programming. In Nonconvex optimization and its applications, volume 61. Kluwer Academic Publishers. B. Colson, P. Marco1e, and G. Savard (2005), Bilevel programming: A survey. 4OR, 3: M. Labbé and A. Violin (2013), Bilevel programming and price se1ing problems, 4OR, Vol 11, 1-30.

6 Adequate framework for Price Se1ing Problem 1st level sets taxes (prices) on some activities 2nd level selects activities among taxed and untaxed ones to minimize operating costs

7 Applications Toll optimization on highways (France, Spain, ) Truck toll systems (Germany) Express mail delivery Passenger transportation systems (train, airlines, ) Hotel room pricing Car rental pricing Travel and tourism package pricing Telecommunication package pricing LAAS - CNRS, Toulouse

8 Price Se1ing Problem with linear constraints (PSP) nonempty and bounded

9 Basic References M. Labbé, P. Marco1e, and G. Savard (1998), A bilevel model of taxation and its application to optimal highway pricing, Management Sci., Vol. 44, M. Labbé, M., P. Marco1e and G.Savard (2000), On a class of bilevel programs, in Nonlinear Optimization and Related Topics, G. Di Pillo and F. Giannessi (eds.), Kluwer,

10 Example with 2 variables in second level Max Ty 1 s.t. T 0 Min (T + c 1 )y 1 + c 2 y 2 s.t. (y 1,y 2 ) Π

11 Example: the second level LAAS - CNRS, Toulouse

12 Example: the profit function T 1 T 2 T 3 T 4 T 5 T 6 LAAS - CNRS, Toulouse

13 Price se1ing problem Bilinear bilevel program Reformulation as linear bilevel program Reformulation as bilinear single level program

14 Network pricing problem Network with taxed arcs (A 1 ) and non taxed arcs (A 2 ). Costs on arcs K commodities: (o k, d k, n k ) Routing on cheapest (cost+toll) path Maximize total profit

15 Example UB on T 1 + T 2 = SPL(T= ) - SPL(T=0) = 22-6 =16 T 23 = 5, T 45 = 10 LAAS - CNRS, Toulouse

16 Example with negative toll T12 = 4 T23 = -2 T34 = LAAS - CNRS, Toulouse

17 Flow not assigned to SP(T=0) 6 Example T 23 = T 45 = T 15 = 7, profit = 7 LAAS - CNRS, Toulouse

18 Network pricing problem Strongly NP- hard even for only one commodity. Polynomial for one commodity if lower level path is known Polynomial for one commodity if toll arcs with positive flows are known Polynomial if one single toll arc. Polynomial algorithm with worst- case guarantee of (log A 1 )/2+1

19 One toll arc - algorithm For each k, compute UB(k) on profit(k) if k uses toll arc UB(1) UB(2) UB(K) T a = UB(i * ), with i * argmax UB(i) i k i n k

20 Network Pricing Problem

21 Joint design and pricing LAAS - CNRS, Toulouse

22 Particular case: highway pricing toll arcs commodities NPP : to détermine taxes s.t à Prices high enough to generate revenue low enough to a1ract customers maximizing company s revenue customers reaction : shortest path LAAS - CNRS, Toulouse

23 Particular case: highway pricing ok Set of entry and exit nodes à complete graph ok dk dk Hyp: 1) toll edges all connected, path(= highway) 2) taxes non additives à 1 taxe for each subpath of the highway

24 Particular case: highway pricing ok Set of entry and exit nodes à complete graph dk «Clique pricing problem»

25 Particular case: highway pricing «Clique pricing problem» Possible additional constraints Monotonicity constraints Triangle constraints

26 Arc pricing versus path pricing

27 Clique pricing: formulations bilevel formulation : 1 st level (company) 2 d level (commodities) remplaced by optimality conditions

28 Clique pricing: formulations à reformulation (MIP) linearisation :

29 Clique pricing: formulations à (HP3) can be strenghtened Single commodity : FDI and complete description

30 Clique pricing: formulations The shortest path constraint can be strenghtened as

31 Numerical results Distance analysis of the Canadian highway 10 (autoroute des Cantons de l Est, Québec) à Randomly generated instances v cities à v(v- 1) commodities n entry and exit nodes à n(n- 1) toll arcs for each commodity, presolve on the toll arcs 6 instances of each size

32 Numerical results à Network pricing instances (Heilporn et al. 2010, 2011)

33 Numerical results à Network pricing instances (Heilporn et al. 2010,2011)

34 Extension: Product Pricing Toll arcs = subpaths of the original highway à Arcs can be seen as independent products à Model valid for any product pricing à Reference : R. Shioda, L. Tunçel, and T.G.J. Myklebust (2011)

35 Extension: Product Pricing Products to be priced consumers p1 r1 Seller p2 p3 r2

36 Extension: Product Pricing k p1 p2 p3 k rk

37 Numerical results à Product pricing instances (Shioda et al. 2011)

38 Conclusions Bilevel model: rich framework for pricing in network- based industries. Models: theoretically and computationally challenging. Need to exploit problem s inner structure. Analysis of basic model: relevant and useful for a1acking real applications (h1p:// Integration of real- life features (congestion, market segmentation, dynamics, randomness...). Investigate variants of product pricing.