Revenue-Maximizing Dynamic Tolls

Revenue-Maximizing Dynamic Tolls INFORMS National Conference San Francisco, CA November 9, 2014

Robert Phillips Columbia Business School Nomis Solutions

Agenda

• Introduction • Queueing Model Structural Results • Stochastic Simulation Results

Columbia Business School

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Managed Lane Concept

Free Lanes (GPL)

Tolled Lanes (ML) Controlled Minimum Speed

Free Lanes (GPL)

“Express Tollway within an Existing Highway”

• •

Solution to congestion on existing (urban) corridors by adding capacity

•

The managed lanes are physically separated from the free lanes, and have a controlled access (on-off ramps at selected locations).

•

All electronic, open access (no toll booths, no barriers).

•

Key Question: How should tolls be set and updated in order to maximize expected profitability?

The additional lanes are operated under a dynamic tolling regime


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The LBJ and the NTE

DFW Airport

Fort-Worth

• • • •

2 Managed Lane Projects in Dallas/FW Area Built by Cintra US as a Public/Private Partnership Cintra builds managed lanes Cintra sets dynamic tolls and keeps revenue Columbia Business School

Dallas

LBJ Express North Tarrant Express NTE Seg 3A – 3B

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Research Questions

●

For a managed lane project: – What is the nature of the dynamic revenue-maximizing tolling policy? – How do various heuristics perform? – What is the importance of dynamic adjustment? – How important is frequent adjustment?

●

Research performed with Caner Göçmen of Nomis Solutions and Garrett van Ryzin of Columbia Business School.


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Model Basics Two sets of parallel lanes: – Managed Lanes for which a toll may be charged – Unmanaged Lanes that are always free to use Arriving vehicles choose which set of lanes to take based on: – The current toll – The travel time savings from taking the managed lanes A controller is able to set and update the tolls dynamically in response to current conditions. (Every 5 minutes in the case of the LBJ). The goal of the controller is to maximize expected revenue.


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Model Dynamics

Managed Lanes Traffic 𝑥𝑚 (t)

Toll p(t)

Time Saving Δ𝑇(𝑥𝑚 t , 𝑥𝑢 (t)) Vehicle Arrivals 𝜆𝑡

Unmanaged Lanes Traffic 𝑥𝑢 (t) Columbia Business School

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Agenda

• Introduction • Queueing Model • Stochastic Simulation


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A Queuing Model • Each set of lanes is an M/M/1 queue • 𝜇𝑢 , 𝜇𝑚 = service rates for unmanaged and managed lanes respectively. • 𝒙 𝑡 = (𝑥𝑢 𝑡 , 𝑥𝑚 (𝑡)) = number of vehicles in the unmanaged and managed lanes respectively. • 𝜆 = total vehicle arrival rate. • 𝑝 𝑡 = toll at time t. • V = consumer value of travel-time saved (a random variable). Assumed bounded above. • F(V) = C.D.F on V.


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Queuing and Traffic Flow

Speed

Speed

Free-flow Speed

Capacity

Typical Traffic Model Columbia Business School

Traffic Volume

Traffic Volume

Queuing Model 10

Possible Policies ● Static

Policies:

– Single Toll. A single toll is set and does not change over time. – Multiple Toll. Pre-set tolls vary with time-of-day but do not change in response to current conditions.

● Adaptive

Policies:

– Myopic. Tolls are set to maximize the expected revenue from

every entering vehicle given the current congestion levels. – Optimal. Tolls are set to maximize expected total revenue.


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Dynamic Program Homogenous arrivals – infinite horizon. We assume that time is divided into intervals sufficiently small such that only of the following events can occur in each time period: • An arrival (probability 𝜆) • A departure from the managed lanes (probability 𝜇𝑚 ) • A departure from the unmanaged lanes (probability 𝜇𝑢 ) Maximize expected discounted revenue:


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Bellman Equation

Renormalize the problem using

Which gives the Bellman equation:


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Structural Properties of the Queueing Model ● 𝐽∗ (𝒙)

● The

is convex in 𝑥𝑢 and concave in 𝑥𝑚

optimal toll 𝑝∗ (𝒙) is

– Increasing in 𝑥𝑢 – Decreasing in 𝑥𝑚 ● The

myopic toll 𝑝𝑚 (𝒙) satisfies 𝑝∗ 𝒙 = 𝑘∆𝑇(𝒙)

● 𝑝∗

𝒙 ≥ 𝑝𝑚 𝒙 : The revenue-maximizing toll is greater than the myopic toll.


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Three Alternative Scenarios We compared three different policies: – Multiple-Price, static – Myopic – Optimal

On three different demand scenarios. Each scenario had the same total arrival rate:


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Policy Comparison by Scenario

Case

Revenue

Revenue Gap*

Optimal

Myopic

Static

Myopic

Static

No Peak

9.43

8.11

7.34

14.0%

22.2%

Low Peak

9.39

8.08

7.31

14.0%

22.2%

High Peak

9.26

7.98

7.19

13.8%

22.4%

* Revenue Gap = 100 x (Policy Revenue – Optimal)/Optimal


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Optimal Price Tracks

Low Peak Case

High Peak Case

In both cases, it is optimal to increase tolls substantially before the peak.


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“Jam and Harvest”

Repeated simulations show that entering a peak traffic period, it is optimal to set tolls high ahead of the peak. This encourages drivers to take the unmanaged lanes which, in turn creates congestion in those lanes. Once the unmanaged lanes begin to get jammed, drivers will pay higher tolls for the use of the much faster managed lanes.

This general pattern has appeared in all of our simulations – both deterministic and stochastic – and accounts for much of the benefit of the optimal policy relative to the myopic policy.


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Time Savings and Optimal Toll An important implication is that the optimal toll is not simply a function of the current state, but also of the expected future traffic. Optimal Toll versus Time Savings Simulation “Small Peak” Case

Toll

$8.00 $7.00 $6.00 $5.00 $4.00 $3.00 $2.00

$1.00 $0.00 0.00

1.00

2.00

3.00

4.00

5.00

Time Savings Columbia Business School

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Agenda

• Introduction • Queueing Model • Stochastic Simulation


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Modeling We compared policies using a simulation based on realistic consumerchoice and traffic models. Travel time Difference

Demand Generation

Total Traffic

Consumer Choice

Traffic by Lane

Traffic Simulation

The model was calibrated using a 10-mile stretch of the SR-91 in Orange County, CA*: •

Weekdays for July, 2011

•

Traffic and tolls available at 5 minute intervals

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17% of data discarded due to lane-closures and accidents

•

We simulated both Eastbound and Westbound traffic * Data available at http://pems.dot.ca.gov.


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Demand Generation Travel time Difference

Demand Generation

• • •

Total Traffic

Consumer Choice

Traffic by Lane

Traffic Simulation

Hourly traffic generated by 𝑌𝑡 = 𝛽𝑡 + 𝛼𝑌𝑡−1 + 𝛾𝑌𝑡−2 + 𝜗𝑌𝑡−3 + 𝜖𝑡 Calibrated to SR-91 Eastbound (above) and Westbound Allocated to 5 minute intervals based on historical fractions


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Consumer Choice Model Travel time Difference

Demand Generation

Total Traffic

Consumer Choice

Traffic by Lane

Traffic Simulation

• Logistic regression used to model lane choice of arriving demand • Multiple models considered, chosen model used: • Toll by time-of-day • (Time Saving)^2 • Fit shown above Columbia Business School

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Traffic Simulation Travel time Difference

Demand Generation

Total Traffic

Consumer Choice

Traffic by Lane

Traffic Simulation

• Mesoscopic simulation similar to DYNASMART (Jayakrishnan, et al., 1994) • Calibrated to SR-91 observed speed/density relationship (above) Columbia Business School

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Policies Tested

1.

Myopic: Set the toll that maximizes expected revenue from each arriving vehicle based on current conditions

2.

Certain Equivalent Optimal: Set optimal tolls based on the expected demand pattern over the day.

3.

Linearly Adjusted Tolls: Start with the Certain Equivalent Optimal tolls then apply a linear adjustment based on the deviation of the actual travel time difference (TTD) from the expected TTD.

𝑝 𝑡, ∆𝑇 𝑡

= 𝑝 𝑡 + 𝛼 + (𝑡)(∆𝑇 𝑡 − ∆𝑇 𝑡 )+ + 𝛼 − (𝑡)(∆𝑇 𝑡 − ∆𝑇(𝑡))+

What did we learn?


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Jam and Harvest is Robust

Expected Myopic Tolls and Hourly Demand

Expected Hourly Revenues Harvest Period

Jam Period

For all of the cases studied, the optimal policy is to price higher than myopic entering a peak, creating a larger travel-time difference and the ability to charge more during the peak. Columbia Business School

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All Dynamic Policies Beat Myopic Expected Revenue (% of Potential) Adjusted T.O.U.

T.O.U

Myopic

Eastbound

Westbound

Maximizing revenue based only on the current state of the system is consistently suboptimal


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Managing Peaks is the Key to Profitability

Optimal Time-of-Use Tolling Revenue Improvement over Myopic Tolling Stylized Deterministic Case

60.00% 50.00% 40.00%

30.00% 20.00%

3

10.00%

2

0.00% 7000 8000 9000

Length of Peak

1 10000

Peak Hourly Demand

The benefits of time-of-use tolling are highly sensitive to the size and length of the peak.


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Dynamic Toll-Setting System Toll Schedule Real-time

Intelligent Transportation System

Speed and Flow by Lane

Toll Setting Module

Weather, Incidents

Toll Updating Algorithm

Pricing Analyst Off-line Data Warehouse

Daily

Daily

Traffic Flow Simulator Columbia Business School

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Questions?


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Previous Research

Four related streams of previous research: – Toll-Setting. Setting tolls to maximize social-welfare or maintaining a minimum speed. • Liu et al. (2007), Yin and Lou (2009), Chung and Recker (2011) – Traffic Flow Modeling. • Ben-Akiva et al. (2002), Roelefson (2012) – Traffic Equilibrium. • Dafermos (1982), Nagurney (2000), Marcotte & Nguyen (2013) – Queues with Pricing and/or Quality Differentiation. • Naor (1969), Low (1974), Afeche (2010)


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The LBJ Express Project

How to set and update tolls to maximize revenue?


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LBJ Bird’s Eye View at Josey Lane 28 lanes !!!!!


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Test Cases for Constant Expected Arrival Case We generated 12 test cases by varying arrival rates and lane capacities. For each test case, we calculated expected revenue for the optimal dynamic policy, the myopic policy and the static (single price policy). Specific parameters used for the test cases were:

• 𝜆 = 2, 2.5 • 𝜇𝑢 = 3, 4

• 𝜇𝑢 = 2, 3, 4


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Optimality Loss from Alternative Policies Myopic Loss from Optimal 𝝁𝒖 = 𝟑

Static Loss from Optimal 𝝁𝒖 = 𝟑

50%

50%

40%

40%

30%

30%

20%

20%

10%

2.5

0% 2

𝜆

𝜇𝑚

2

3

10%

2.5

𝜇𝑚

0% 2

𝜆

4

𝝁𝒖 = 𝟒

2

3 4

𝝁𝒖 = 𝟒

50%

50%

40%

40%

30%

30%

20%

20%

10%

2.5

𝜇𝑚

0% 2

𝜆

2

3


4

10%

2.5

𝜇𝑚

0% 2

𝜆

2

3

4

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The LBJ Express Project

LBM Express Project Video Columbia Business School

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Update Frequency

Expected Revenue Adjusted T.O.U.

Myopic

1

5

10

15

20

30

60

Update Interval (Minutes)

For the cases we studied, update frequency did not have a significant influence on expected revenue for any policy. Columbia Business School

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The LBJ and NTE in Dallas/Ft. Worth

DFW Airport

Fort-Worth

• • •

Two projects being built by Cintra US Cintra will manage tolls dynamically for 35 years How to set the tolls to maximize expected profit?

Dallas

LBJ Express North Tarrant Express NTE Seg 3A – 3B


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Tolling Constraints Avg Speed previous 15 5th Lowest min Speed of previous 10

55mph 50mph

INCIDENT REQUIRING MANUAL INTERVENTION

LOST CONNECTIVITY WITH TRDMS

ML Speed

Travel time saved

Time Posted Tolls TR Cap Toll Control Toll Dynamic Normal

Default

Dynamic Normal

Dynamic PAM

Dynamic Mandato ry

Manual

Dynamic Mandato ry

Dynamic Normal

1. Normal Mode: Set rates under the Toll Rate Cap based on traffic conditions (every 5 minutes)

2. Penalty Avoidance Mode: If speeds on the ML approach 55mph, tolls will be raised by 50% of the difference between the previous and the cap 3. Mandatory Mode: Contract defines rules to update toll rates if ML speeed < 50 mph, with changes between 5% and 25% depending on speed and volume levels on the ML. Columbia Business School