Differentiated Pricing of Urban Transportation Networks. Networks with Vehicle-Tracking Technologies

Transcription

1 of Urban Transportation Networks with Vehicle-Tracking Technologies Mahmood Zangui Yafeng Yin* Siriphong Lawphongpanich Shigang Chen Department of Civil and Coastal Engineering University of Florida July 18, 2013 of Urban Transportation Networks

2 Price Differentiation Economic concept Defined by Dupuit (1894) Identical products sold at different prices Examples in transit fare One two-way ticket is cheaper than two one-way tickets Senior citizens pay lower bus fare Literature of congestion pricing Differentiation with respect to vehicle type Differentiation with respect to value of time We investigate a new differentiated congestion pricing scheme that differentiates travelers with respect to their travel characteristics. of Urban Transportation Networks

6 How does a differentiated pricing scheme work? of Urban Transportation Networks

7 Definitions Introduction Basic Idea Examples Anonymous scheme: Everyone pays the same toll for the same link. Differentiated scheme: Toll depends on travel characteristics. Level of differentiation: L0: None anonymous scheme L1: Origin origin-specific scheme L2: Origin and destination OD-specific scheme L3: Path path-based scheme of Urban Transportation Networks

8 Definitions Introduction Basic Idea Examples Anonymous scheme: Everyone pays the same toll for the same link. Differentiated scheme: Toll depends on travel characteristics. Level of differentiation: L0: None anonymous scheme L1: Origin origin-specific scheme L2: Origin and destination OD-specific scheme L3: Path path-based scheme of Urban Transportation Networks

9 Level of Differentiation Basic Idea Examples Path L0 L1 L2 L π 9 π 2 9 π 2,3 9 π 4,8,9, π 9 π 1 9 π 1,3 9 π 2,8,9, π 9 π 1 9 π 1,4 9 π 2,8,9, π 9 π 1 9 π 1,4 9 π 1,6,9,12 of Urban Transportation Networks

10 Benefits of Differentiation Basic Idea Examples Price differentiation π 1 a π 2 a Relaxation of anonymous pricing First-best condition: reduce toll revenue Second-best condition: reduce travel time Potential for better results of Urban Transportation Networks

11 Example: First-best Condition Basic Idea Examples All links are tollable System optimum is achievable with anonymous tolling x a = x a a A Benefits of price differentiation can only be reflected on a secondary objective Toll revenue is a financial burden on travelers Higher toll revenue implies less public acceptance Choose the tolls with minimum revenue min w W p P w π pf p of Urban Transportation Networks

12 Example: First-best Condition Basic Idea Examples 1 (6, 18) (3, 35) (5, 12) 5 (8, 26) (2, 11) 9 (8, 30) (7, 32) (4, 26) 7 (6, 24) (8, 39) (3, 25) 3 2 (9, 35) 6 (6, 33) 8 (6, 43) 4 Table : First-best pricing for nine-node network Tolling Scheme Toll Revenue OD Generalized Travel Cost Amount Reduction [1, 3] [1, 4] [2, 3] [2, 4] Anonymous % Origin-specific % OD-specific % Path-based % of Urban Transportation Networks

13 Basic Idea Examples Example: Second-best Condition Suppose links a Ψ are untollable Origin-specific γa o(w) = 0 w W, a Ψ OD-specific γa w = 0 w W, a Ψ Travel time as the performance measure Choose tolls that minimize total system travel time min w W p P w t p(f )f p of Urban Transportation Networks

14 Basic Idea Examples Example: Second-best Condition 1 (6, 18) (3, 35) (5, 12) 5 (8, 26) (2, 11) 9 (8, 30) (7, 32) (4, 26) 7 (6, 24) (8, 39) (3, 25) 3 2 (9, 35) 6 (6, 33) 8 (6, 43) 4 Table : Second-best pricing for nine-node network Tolling Scheme Total Travel Time OD Generalized Travel Cost Amount Saving [1, 3] [1, 4] [2, 3] [2, 4] UE % Anonymous % Origin-specific % OD-specific % SO % of Urban Transportation Networks

15 Basic Idea Examples How to design optimal differentiated pricing schemes? of Urban Transportation Networks

16 Models Algorithms Finding Optimal Differentiated Schemes Feasible region: p P w f p = d w f p (t p (f ) + π p λ w ) = 0 t p (f ) + π p λ w 0 f p 0 π p 0 x a = δ ap f p w W p P w w W p P w, w W p P w, w W p P w, w W p P w, w W a A of Urban Transportation Networks

17 Models Algorithms Finding Optimal Differentiated Schemes Additional constraints: Origin-specific pricing π p = a A δ ap γ o(w) a p P w, w W OD-specific pricing γ o(w) a 0 a A, w W π p = a A δ ap γ w a p P w, w W γ w a 0 a A, w W of Urban Transportation Networks

18 Models Algorithms Finding Optimal Differentiated Schemes Objective functions: First-best condition min w W p P w π p f p Second-best condition min w W p P w t p (f )f p of Urban Transportation Networks

19 Models Algorithms Example: Path-based Pricing Scheme (First-best Condition) min s.t. w W p P w π p f p f p = d w p P w f p (t p (f ) + π p λ w ) = 0 t p (f ) + π p λ w 0 f p 0 π p 0 x a = δ ap f p w W p P w w W p P w, w W p P w, w W p P w, w W p P w, w W a A of Urban Transportation Networks

20 Solution Algorithms Introduction Models Algorithms The formulations presented above all belong to the class of mathematical programs with complementarity constraints (MPCC) These problems are non-convex and standard stationary conditions, i.e., KKT conditions, may not hold for them because they do not satisfy Mangasarian-Fromovitz constraint qualification (MFCQ) Efficient algorithms may be developed to solve the above formulations by exploring special properties or structures that they may possess. of Urban Transportation Networks

21 Models Algorithms Example: Path-based Pricing Scheme (First-best Condition) min s.t. λ w d w w W f p = d w p P w f p ( t p + π p λ w ) = 0 t p + π p λ w 0 f p 0 π p 0 x a = δ ap f p w W p P w w W p P w, w W p P w, w W p P w, w W p P w, w W a A of Urban Transportation Networks

22 Reformulation 1: MILP-1 Models Algorithms min s.t. λ T d f F f p y p d w p P w, w W t p + π p λ w (1 y p )M p P w, w W t p + π p λ w 0 p P w, w W π p 0, y p {0, 1} p P w, w W where F = { p P w f p = d w, w W ; f p 0, p P w, w W ; x a = w W p P w δ ap f p, a A} One binary variable for each path, which is equal to 1 if the path is utilized of Urban Transportation Networks

23 Reformulation 2: MILP-2 Models Algorithms min s.t. t p y p d w w W p P w f F y p = 1 p P w f p d w k P w & t k t p y k w W p P w, w W π p 0, y p {0, 1} p P w, w W One binary variable for each path, equal to 1 if it is the longest utilized path of the OD pair. The longest utilized path is toll free Linear mixed integer model, can be solved by CPLEX. of Urban Transportation Networks

24 Computational Experiments Models Algorithms Sioux Falls Network 24 nodes 76 links 528 OD pairs Anaheim Network 416 nodes 914 links 1406 OD pairs of Urban Transportation Networks

25 Introduction Models Algorithms Table : Comparisons of toll revenues Toll Revenue Anonymous Path-based Reduction Anaheim Original Data % Rand. Data % Rand. Data % Rand. Data % Rand. Data % Sioux Falls Original Data % Rand. Data % Rand. Data % Rand. Data % Rand. Data % of Urban Transportation Networks

26 Models Algorithms Is there any issue associated with these schemes? of Urban Transportation Networks

27 Introduction Modeling Privacy Incentive Program Location privacy: the ability to prevent other parties from learning one s current or past location For anonymous tolling, it is possible to design a privacy-preserving electronic toll collection system. It is difficult, if not impossible, to design a privacy-preserving differentiated pricing system because differentiated pricing requires the location information to determine tolls of Urban Transportation Networks

28 Value of Privacy Introduction Modeling Privacy Incentive Program Individuals value their privacy differently They can be grouped into categories of privacy unconcerneds, privacy pragmatists, and privacy fundamentalists Mathematically, we can use a distribution to represent different individual valuations of privacy across the population ξ(θ) If travelers value their privacy highly, the savings of travel cost that they may enjoy from differentiated schemes will be offset by the loss of their privacy θ of Urban Transportation Networks

29 Modeling Privacy Incentive Program Who benefits from differentiated schemes? We now use origin-specific pricing as an example for modeling privacy. For each OD-pair w: Those who value their privacy less will more likely benefit from price differentiation Travel cost saving: λ w,0 λ w,1 The percentage of motorists who will be better off under origin-specific pricing: λ w,0 λ w,1 0 ξ(z)dz ξ(θ) λ w,0 λ w,1 θ of Urban Transportation Networks

30 Addressing Privacy Concerns Modeling Privacy Incentive Program We propose an incentive program that allows each traveler to opt in to differentiated schemes Self-selection mechanism: travelers who choose to reveal their location information will pay differentiated tolls while those who remain anonymous will pay uniform tolls Anonymous scheme preserves location privacy Incentives, such as subsidies or credits, can be provided to encourage travelers to participate in differentiated scheme In the simplest setting, the travel cost difference between differentiated and anonymous schemes can be viewed as incentive of Urban Transportation Networks

31 Design of Incentive Program Modeling Privacy Incentive Program Demand split constraints: d w,0 + d w,1 = d w w W f p,c = d w,c w W, c {0, 1} p P w d w,1 = Ξ(λ w,0 λ w,1 )d w w W d w,0, d w,1 0 w W of Urban Transportation Networks

32 Tolled User Equilibrium Modeling Privacy Incentive Program Tolled user equilibrium f p,c (t p (f ) + π p,c λ w,c ) = 0 p P w, w W, c {0, 1} t p (f ) + π p,c λ w,c 0 p P w, w W, c {0, 1} π p,c 0 p P w, w W, c {0, 1} π p,0 = δ ap γ a a A p P w, w W γ a 0 π p,1 = a A δ ap γ o(w) a a A p P w, w W γ o(w) a 0 a A, w W of Urban Transportation Networks

33 Objective functions Introduction Modeling Privacy Incentive Program First-best network conditions (all links are tollable) λw,0 λ w,1 min d w ξ(z)zdz + (π p,0 f p,0 + π p,1 f p,1 ) w W 0 p P w Second-best network conditions (some of the links are untollable) λw,0 λ w,1 min d w ξ(z)zdz + t p (f )f p 0 p P w w W of Urban Transportation Networks

34 Modeling Privacy Incentive Program Implementation on Nine-node Network Table : Incentive program under first-best conditions Pricing Scheme Distribution of β E(β) Toll Rev. Privacy Cost Total User Cost Anonymous Origin-specific U(0, 4) U(0, 8) Hybrid U(0, 16) EXP(0.500) EXP(0.250) EXP(0.125) Anonymous scheme yields highest toll revenue, and origin-specific leads to highest privacy cost. The hybrid scheme offers an option for travelers of high value of privacy to remain anonymous. Such a self-selection mechanism leads to much less loss of privacy and subsequently lower total user cost. of Urban Transportation Networks

35 Modeling Privacy Incentive Program Implementation on Nine-node Network Table : Incentive program under second-best conditions Pricing Scheme Distribution of β E(β) Travel Time Privacy Cost Total System Cost Anonymous Origin-specific U(0, 4) U(0, 8) Hybrid U(0, 16) EXP(0.500) EXP(0.250) EXP(0.125) Anonymous scheme yields highest travel time, and origin-specific leads to highest privacy cost. The hybrid scheme is able to reduce total system cost. of Urban Transportation Networks

36 Modeling Privacy Incentive Program Implementation on Sioux Falls Network Table : Incentive program under second-best conditions Pricing Scheme Distribution of β E(β) Travel Time Privacy Cost Total System Cost Anonymous Origin-specific U(0, 0.04) U(0, 0.08) Hybrid U(0, 0.16) EXP(50.0) EXP(25.0) EXP(12.5) Similar observations can be made. Hybrid schemes perform better than anonymous or differentiated pricing schemes of Urban Transportation Networks

37 Summary Introduction Contribution: Explored a new class of tolling schemes that charge different amount of toll for users with different origins, destinations, or paths. Developed an approach for modeling location privacy of travelers Proposed an incentive program that allows the tolling agency to take advantage of the potentials of differentiated pricing without doing harm to privacy rights of travelers. Conclusion: Differentiated pricing shows great promise in optimizing system performance Differentiated pricing may not be appealing for everyone Distribution of value of privacy has a significant effect on the acceptability of differentiated schemes Incentive program may create a win-win situation for all travelers of Urban Transportation Networks

38 Summary Introduction Contribution: Explored a new class of tolling schemes that charge different amount of toll for users with different origins, destinations, or paths. Developed an approach for modeling location privacy of travelers Proposed an incentive program that allows the tolling agency to take advantage of the potentials of differentiated pricing without doing harm to privacy rights of travelers. Conclusion: Differentiated pricing shows great promise in optimizing system performance Differentiated pricing may not be appealing for everyone Distribution of value of privacy has a significant effect on the acceptability of differentiated schemes Incentive program may create a win-win situation for all travelers of Urban Transportation Networks

39 Questions? of Urban Transportation Networks