Bilevel Programming and Price Setting Problems

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Bilevel Programming and Price Setting Problems

• Martine Labbé
• Département dInformatique
• Université Libre de Bruxelles

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In collaboration with

• Luce Brotcorne, U. de Valenciennes
• Sophie Dewez, ULB
• Patrice Marcotte, CRT, Montréal
• Gilles Savard, GERAD, montréal

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Basic References

• M. Labbé, P. Marcotte, and G. Savard (1998),  A
  bilevel model of taxation and its application to
  optimal highway pricing, Management Science, Vol.
  44, 1608-1622.
• M. Labbé, M., P. Marcotte and G.Savard (2000), On
  a class of bilevel programs, in Nonlinear
  Optimization and Related Topics, G. Di Pillo and
  F. Giannessi (eds.), Kluwer,183-206.

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Bilevel Program
where
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Adequate Framework for Stackelberg Game

• Leader 1st level,
• Follower 2nd level.
• Leader takes followers optimal reaction into
  account.

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Adequate Framework for Principal/Agent Model

• Principal first level,
• Agent second level
• Principal wants to delegate task to Agent against
  reward.
• Principal optimizes his utility s.t.
• Agent accepts task
• Agent performs optimally

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Adequate framework for Price Setting Problem

• 1st level set taxes on some activites
• 2nd level selects activities among taxed and
  untaxed ones to minimize operating costs

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Price Setting Problem with linear constraints
(PSP)
nonempty and bounded
nonempty
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A 2-dimensional example
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The profit function
T6
T4
T5
T1
T2
T3
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Reformulation
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Reformulation-contd

Dual of lower level with infinite taxes
Lower level with zero taxes
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Price setting problem

• Bilinear bilevel program
• Reformulation as linear bilevel program
• Reformulation as bilinear single level program

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Toll setting problem

• Network with taxed arcs (A1) and non taxed arcs
  (A2).
• Costs on arcs
• K commodities (ok, dk, nk)
• Routing on cheapest (costtoll) path
• Maximize total profit

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Toll Setting Problem
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Additional References - Toll Optimization

• L. Brotcorne, M. Labbé, P. Marcotte, and G.
  Savard (2001), A Bilevel Model for Toll
  Optimization on a Multicommodity Transportation
  Network, Transportation Science, Vol. 35,
  345-358.
• L. Brotcorne, M. Labbé, P. Marcotte and G.
  Savard (2003), Joint Design and Pricing on a
  Network, submitted
• S. Dewez, Toll Optimization problems
  Formulations and Solutions algorithms, Ph.D.
  dissertation, June 2004.

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Example
UB on T1 T2 SPL(T ?) - SPL(T0) 22 - 6 16
T23 5, T45 10
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Flow not assigned to SP(T0)Example
T23 T45 T15 7, profit 7
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Example with negative toll
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Toll Setting 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
  (logA1)/21

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Additional References

• P. Marcotte, G. Savard, F. Semet(2004), A bilevel
  approach to the travelling salesman problem,
  Operations Research Letters, Vol. 32, 240-248.
• S. Roch, G. Savard, P. Marcotte, Design and
  analysis of an approximation Stackelberg network
  pricing, submitted.

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One toll arc - algorithm

• For each k, compute UB(k) on profit(k) if k uses
  toll arc
• UB(1) ? UB(2) ? ?UB(K)
• Ta UB(i), with

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Toll setting - reformulation
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Linearize Tak Taxak
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Solution approach

• Formulate TSP as MIP
• Tight bound M on tax, if arc used and if arc not
  used, very effective
• Add valid inequalities to strengthen LP relaxation

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The shortest path graph model

• Routing on shortest path in original graph

• Routing in SPGM

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A graph and its SPGM
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Formulation with path variables
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Computational results
60 nodes 208 arcs
20 commodities
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Computational results
60 nodes 208 arcs
30 commodities