Real Options Switching Strategies in Dynamic Transport Service - - PowerPoint PPT Presentation

Qian-wen Guoa, Joseph Y.J. Chowb,*, Paul Schonfeldc

Real Options Switching Strategies in Dynamic Transport Service Operations

INFORMS 2017, Houston, TX

a Sun Yat-Sen University, Guangzhou, China b New York University, NY, USA c University of Maryland, College Park, MD, USA

Slide 2

Premise

 Need for policies to inform “when to switch” between

two operational regimes:

 Automation

 Shared autonomous vehicle operations  Dynamic tolls/pricing (transit, freight, roads, parking, etc.)  Dynamic infrastructure use (traffic control, lanes, parking,

etc.)

 Dynamic fleet operations (dispatch, rebalancing, customer

incentives, etc.)

 Highly uncertain sequential decisions

 New technology adoption  Rapidly growing/changing community

Slide 3

Our Contribution

Using (automated) last mile transit as primary application… … we developed an

• ptimal switching algorithm

(available on GitHub) for data-driven decisions minimizing transit fleet

• perating costs

https://github.com/BUILTNYU/Optimal-Switching

Slide 4

Outline

 Background and literature review  Problem definition

 Problem illustration  Formal definition

 Proposed model

 Dynamic switching between fixed and flexible transit  Model variation: modular vehicle sizes

 Model properties  Computational evaluations

Slide 5

Dual-mode transit fleet operating strategies tend to be static

➢ Fixed route vs flexible service e.g. Kim and Schonfeld, 2013 ➢ Vehicle size e.g. Fu and Ishkanov, 2004 ➢ Headway control e.g. Thomas, 2007 ➢ Idle vehicle relocation. e.g. Yuan et al., 2011, Sayarshad and Chow, 2017 ➢ Ridesharing options M to 1, M to Few, M to M. Daganzo, 1978; Chang and

Schonfeld, 1991a,b; Quadrifoglio and Li, 2009

Background: Last Mile Transit Ops

Qiu et al. 2014

Increasing need for optimization of automated, shared,

• n-demand transit to serve last mile

Slide 6

Research Gap

 Chang & Schonfeld

(1991): optimize fixed route or flexible service for a last mile (M-to-1) region – threshold exists

 Kim & Schonfeld (2012):

extended to multiple deterministic periods Chang & Schonfeld (1991) No methodology to dynamically optimize switching between different

• perating states in last mile

transit

Slide 7

Research Gap (2)

 Dixit (1989): Analytical expression for optimal

switching timing for GBM stochastic process

 Sødal et al. (2009): applied to shipping wih

stochastic freight rates

 Tsekrekos (2010): using an infinite series

(Kummer series) to derive a solution for stationary mean-reverting (Ornstein- Uhlenbeck) process

Not yet applied to urban transport

• perations

𝑒𝑅 = 𝜈 𝑛 − 𝑅 𝑒𝑢 + 𝜏𝑅𝑒𝑥

Mean reversion rate Long term mean density Volatility Increment in Wiener process

Slide 8

Why not deterministic threshold?

 Hysteresis effect: presence of inertia

Sødal et al. Review of Financial Economics, 2008

• Using deterministic threshold is a myopic

policy

• Buffer can be designed to account for

characteristics of stochastic process and cost

• f switching

Slide 9

Use cases

 Switching between fixed route and on-

demand

 Switching between one-module and two-

module vehicles (vehicle “size”)

Slide 10

Problem Definition (1)

Line haul

Kim & Schonfeld (2014)

➢ A last mile region: 𝑀 × 𝑋 region connected to a hub via line haul of length 𝐾 ➢ Demand density: spatially uniformly distributed and temporally as mean- reverting process.

Line haul

➢ The fixed-route conventional mode subdivided into 𝑂𝑑 routes of width 𝑠 and length 𝑀 ➢ The flexible service mode subdivided into a grid

• f

𝑂

𝑔 zones of area A.

Slide 11

Problem Definition (2)

Line haul

Kim & Schonfeld (2014)

➢ Given: ➢ Parameters of a stochastic process for demand density (as mean-reverting) fitted to historical data ➢ Current demand density ➢ Current operating state (fixed vs flexible, or 1- veh flexible vs 2-veh flexible) ➢ Determine whether or not to switch operating state at current time to minimize

• perating cost

Line haul

Slide 12

“Market entry-exit” switching option with OU process

 Two operating modes (e.g. “in market” vs “out of

market”)

 Both modes governed by single OU stochastic

process, e.g. demand 𝑅 𝑢

 Each mode’s incremental cost or payoff function at 𝑢:

𝐷0 𝑅 𝑢 , 𝐷1 𝑅 𝑢 where a single threshold 𝑅∗ exists for deterministically choosing one mode over the

• ther

 Optimal policy under infinite horizon determines

threshold 𝑅𝑀 and 𝑅𝐼 to optimize value function (current and future expected payoffs)

Slide 13

SWITCHING BETWEEN FIXED ROUTE (MODE 1) AND ON-DEMAND (MODE 0)

Slide 14

𝐷1 𝑅 𝑢 and 𝐷0 𝑅 𝑢

 For fixed route transit, 𝐷1= the sum of the bus operating

cost 𝐷𝑝, user in-vehicle cost 𝐷𝑤, user waiting cost 𝐷𝑥, and user access cost 𝐷𝑦. 𝐷1 𝑅 = 𝐷𝑝 + 𝐷𝑤 + 𝐷𝑥 + 𝐷𝑦 = 𝑏1 + 𝑐1𝑅 + 𝑒1𝑅 + 𝑓1𝑅

 𝐷0= sum of bus operating cost 𝐷𝑝, user in-vehicle cost 𝐷𝑤,

user waiting cost 𝐷𝑥, and user access cost 𝐷𝑦 𝐷0 𝑅 = 𝑏0𝑅

4 5 + 𝑐0𝑅 2 3 + 𝑒0𝑅

where 𝑏, 𝑐, 𝑒, 𝑓 are functions of region size, fleet size, and operating speeds – route spacing 𝑠 (for fixed route), zone size 𝐵 (for flexible), and vehicle size 𝑇𝑑 are endogenously determined to minimize cost (from Chang and Schonfeld, 1991a)

Slide 15

 The immediate cost savings accrued from time 𝑢 to 𝑢 + 𝑒𝑢

when switching from flexible to fixed route mode is: Φ 𝑅 𝑢 = 𝐷0 𝑅 𝑢 ; 𝑇𝑔 − 𝐷1 𝑅 𝑢 ; 𝑇𝑑

Cost savings function Φ 𝑅 𝑢

Flexible Fixed If Φ > 0, there is cost savings switching from flexible to fixed route mode If Φ < 0, there is cost savings switching from fixed route to flexible service

Slide 16

Policy valuation based on asset equilibrium pricing

 The option value of using flexible bus operating

mode 𝑊

0 𝑅 1 2 𝜏2𝑅2𝑊 ′′ 𝑅 + 𝜈 𝑛 − 𝑅 𝑊 ′ 𝑅 − 𝜍𝑊 0 𝑅 = 0  The option value of using conventional bus operating

mode 𝑊

1 𝑅

1 2 𝜏2𝑅2𝑊

1 ′′ 𝑅 + 𝜈 𝑛 − 𝑅 𝑊 1 ′ 𝑅 − 𝜍𝑊 1 𝑅 + Φ 𝑅 = 0

Slide 17

Asset equilibrium conditions

 Define 𝑅𝑀 as demand threshold to switch from fixed to

flexible, and 𝑅𝐼 from flexible to fixed When 𝐺+ = 𝐺− = 0, 𝑅𝐼 = 𝑅𝑀

 At asset equilibrium, value matching between two modes

V0 𝑅𝐼 = 𝑊

1 𝑅𝐼 − 𝐺+

V

1 𝑅𝑀 = 𝑊 0 𝑅𝑀 − 𝐺−  Smooth pasting

V0

′ 𝑅𝐼 = 𝑊 1 ′ 𝑅𝐼

V0

′ 𝑅𝑀 = 𝑊 1 ′ 𝑅𝑀

𝐺+ is the cost assumed for switching from flexible bus service to conventional bus service 𝐺− is the cost of switching from conventional bus service to flexible bus service; .

Slide 18

Asset equilibrium conditions

 General solution of 𝑊 0 𝑅 and 𝑊 1 𝑅

𝑊

0 𝑅 = 𝐵0𝐼 −𝛿0, 𝑥0, 𝑦 + 𝐶0

2𝜈𝑛 𝜏2𝑅

1−𝑥0

𝐼 1 − 𝛿0 − 𝑥0, 2 − 𝑥0, 𝑦 𝑅𝛿0 𝑊

1 𝑅

= 𝐵1𝐼 −𝛿1, 𝑥1, 𝑦 + 𝐶1 2𝜈𝑛 𝜏2𝑅

1−𝑥1

𝐼 1 − 𝛿1 − 𝑥1, 2 − 𝑥1, 𝑦 𝑅𝛿1 + 𝐹𝑢 න

𝑢 ∞

Φ 𝑅 𝑡 𝑓−𝜍 𝑡−𝑢 𝑒𝑡 𝐼 ∙ is a confluent hypergeometric (“Kummer”) function 𝐼 𝛿, 𝑥, 𝑦 = 1 + 𝛿 𝑥 𝑦 + 𝛿 𝛿 + 1 𝑦2 𝑥 𝑥 + 1 2! + 𝛿 𝛿 + 1 𝛿 + 2 𝑦3 𝑥 𝑥 + 1 𝑥 + 2 3! + ⋯

Slide 19

Asset equilibrium conditions

𝐆 𝐘 = 𝐵0𝐼0 𝑅𝐼 𝑅𝐼

𝛿0 + ∆1𝐵0 − 𝐵1 𝐼1 𝑅𝐼 𝑅𝐼 𝛿1 − 𝐹𝑢 න 𝑢 ∞

Φ 𝑅 𝑡׀𝑅 𝑢 = 𝑅𝐼 𝑓−𝜍 𝑡−𝑢 𝑒𝑡 + 𝐺+ 𝐵0𝐼0 𝑅𝑀 𝑅𝑀

𝛿0 + ∆1𝐵0 − 𝐵1 𝐼1 𝑅𝑀 𝑅𝑀 𝛿1 − 𝐹𝑢 න 𝑢 ∞

Φ 𝑅 𝑡׀𝑅 𝑢 = 𝑅𝐼 𝑓−𝜍 𝑡−𝑢 𝑒𝑡 − 𝐺− 𝐵0𝑁0 𝑅𝐼 𝑅𝐼

𝛿0 + ∆1𝐵0 − 𝐵1 𝑁1 𝑅𝐼 𝑅𝐼 𝛿1 +

𝜖𝐹𝑢 ׬

𝑢 ∞ Φ 𝑅 𝑡׀𝑅 𝑢 = 𝑅𝐼 𝑓−𝜍 𝑡−𝑢 𝑒𝑡

𝜖𝑅 𝐵0𝑁0 𝑅𝑀 𝑅𝑀

𝛿0 + ∆1𝐵0 − 𝐵1 𝑁1 𝑅𝑀 𝑅𝑀 𝛿1 +

𝜖𝐹𝑢 ׬

𝑢 ∞ Φ 𝑅 𝑡׀𝑅 𝑢 = 𝑅𝐼 𝑓−𝜍 𝑡−𝑢 𝑒𝑡

𝜖𝑅

Due to the complexity of the equations, we obtain the solution numerically. 𝒀 = 𝑅𝐼, 𝑅𝑀, 𝐵0, 𝐵1 ′ is uniquely determined by solving

Slide 20

Model properties

 Sensitivity of switching policy to transportation system

parameters

Solutions 𝑹 𝟏 =32 trips/mile2/hr, operating as flexible service initially Fixed Transit Headway, ℎ𝑑 0.42 Vehicle size, 𝑇𝑑 75 Fleet size, 𝐺

𝑑

5 Route spacing, 𝑠 1.41 Total cost, 𝐷𝑡𝑑 2881.1 Flexible Transit Headway, ℎ𝑔 0.08 Vehicle size, 𝑇𝑔 7 Fleet size, 𝐺

𝑔

58 Service zone, 𝐵 3.02 Total cost, 𝐷𝑡𝑔 2883.0

Φ 𝑅 1.9 𝐹𝑢 න

𝑢 ∞

Φ 𝑅 𝑓−𝜍 𝑡−𝑢 𝑒𝑡

• 39.4

𝑊

0 𝑅 0

23.5 𝑊

1 𝑅 0

17.4 𝑅𝑀 28.7 𝑅𝐼 41.8 Indifference band (𝑅𝐼 − 𝑅𝑀) 13.1

Slide 21

Illustration of policy in action

Flexible Fixed transit Fixed transit Flexible

Slide 22

Verification of policy

Experimental design

For the same simulated demand density trajectory of 96

• bservations, outcome decisions (and accumulated costs) are

made for the following policies:

 Perfect information (oracle) scenario: determine the optimal

switching points deterministically as if the operator knew the demand outcome beforehand;

 Myopic policy scenario: determine the optimal switching points

whenever the incremental cost threshold (Φ=0) is crossed；

 Proposed policy scenario based on market entry-exit switching

• ption: switch to fixed transit whenever 𝑅 𝑢 > 𝑅𝐼or switch to

flexible transit whenever 𝑅 𝑢 < 𝑅𝑀.

Slide 23

Experimental design

𝜜(𝜌) = 𝑆𝑛𝑧 − 𝜌 𝑆𝑛𝑧 − 𝑆𝑞ℎ 𝑇𝑔 = 8，𝑇𝑑 = 80

Fixed vehicle sizes

𝑅𝑜+1 = 𝑅𝑜 + 𝜈 𝑛 − 𝑅𝑜 ∆𝑢 + 𝜏𝑅𝑜∆𝑥𝑜

➢ The performance of the proposed policy 𝑆𝑞ℎ perfect information scenario 𝑆𝑛𝑧 myopic scenario seats/vehicle Average demand density of 40 trips/mile2/hr

Slide 24

Experimental Results

➢ When switching cost is 0, there is just one threshold 𝑅∗ = 35.6. ➢ When there is a switching cost 𝐺+ = 𝐺− = 10, then the optimal thresholds are 𝑅𝑀 = 28.7 𝑅𝐼 = 41.8.

Proposed policy

Flexible Fixed transit Fixed transit Flexible

Slide 25

Experimental Results

• Fig. 8. (a) Myopic switching policy, and (b) perfect information switching policy.

Perfect information Proposed policy Myopic policy Total discounted cost 46093.77 46150.62 46293.73 𝜜 𝜌 1.0000 0.7157 0.0000

The proposed policy can reduce the excess cost by 72% relative to myopic policy

Flexible Fixed transit

𝜜(𝜌) = 𝑆𝑛𝑧 − 𝜌 𝑆𝑛𝑧 − 𝑆𝑞ℎ

Slide 26

Model properties

 Switching policy sensitivity to demand density

• Fig. 4. (a) Incremental operational cost from fixed to flexible transit

(b) option value of conventional and flexible bus service with respect to demand density.

32 Flexible Fixed transit 𝟑𝟗. 𝟖 𝟓𝟐. 𝟗 35.6 .6>32

• ption value accounts for foresight

Slide 27

Model variation: modular vehicle size

 Two different vehicle sizes 𝑇1 and 𝑇2, 𝑇2 = 2𝑇1  𝐷𝑡𝑔1 𝑅; 𝑇1 = the cost of operating flexible service with vehicle size 𝑇1



𝐷𝑡𝑔2 𝑅; 𝑇2 for operating flexible service with vehicle size 𝑇2 = 2𝑇1.

 The incremental cost savings function variation Φv

Φv 𝑅 𝑢 = 𝐷𝑡𝑔1

∗

𝑅 𝑢 ; 𝑇1 − 𝐷𝑡𝑔2

∗

𝑅 𝑢 ; 𝑇2 This general structure can manage autonomous fleets such as the Next Future Mobility system

Shared autonomous fleets with modular vehicle size

Slide 28

Application to vehicle modularity

The experiment is designed to compute the option premium for the added flexibility to switch between two vehicle sizes: 𝑇1 = 10 and 𝑇2 = 20.

• Scenario 1: flexible service with only one fixed vehicle size

𝑇0 (static policy),

• Scenario 2: flexible service with two vehicle sizes in which

the proposed policy is used to determine optimal switching, assuming the system initiates at 𝑇1 and having symmetric switching costs 𝐺

𝑇 = 10.

Slide 29

Application to vehicle modularity

 𝐺 𝑇 =10

𝑅𝐼= 33.5, 𝑇1 → 𝑇2 𝑅𝑀= 22.8, 𝑇2 → 𝑇1

 𝐺 𝑇= 0

𝑅∗ = 28.2.

Fig 10. Proposed switching policy for vehicle modularity.

Proposed policy Static policy Cumulative total cost ($) 45946.64 46320.09

The flexibility to switch vehicle size in this case leads to an improvement over a static policy of $373.45 over the 24 hrs.

Slide 30

Conclusion

➢Optimize dynamic switching of transit service as a market entry-exit real options model with mean-reverting demand density; ➢Model features a quantifiable hysteresis effect ➢Relative to a myopic policy, the performance

• f the proposed policy can eliminate up to

72% of the excess cost; ➢An option premium exists for having the flexibility to switch between two vehicle sizes.

Slide 31

Further research

➢Also applicable to selecting/staging different transportation technologies (LRT vs heavy rail, electric vehicle infrastructure, autonomous vehicle infrastructure, etc.) ➢Adding significant switching duration (e.g. Li et al, 2015) and decision-dependent stochastic process ➢An empirical study on a transit operation with real data

Questions?

Qian-wen Guo: guoqw3@mail.sysu.edu.cn Joseph Y.J. Chow*: joseph.chow@nyu.edu Paul Schonfeld: pschon@umd.edu

Funding support

• Guangdong Provincial Natural Science Foundation (2016A030310223)
• National Science Foundation for Young Scientists of China (71601192)
• China Postdoctoral Science Foundation (2017T100655)
• National Science Foundation (CMMI-1634973)

Citation Guo, Q.W., Chow, J.Y.J., Schonfeld, P., 2017. Stochastic dynamic switching in fixed and flexible transit services as market entry-exit real options. Transportation Research Part C, in press, doi: 10.1016/j.trc.2017.08.008. Code and data https://github.com/BUILTNYU/Optimal-Switching