Real Options Switching Strategies in Dynamic Transport Service Operations Qian-wen Guo a , Joseph Y.J. Chow b, *, Paul Schonfeld c a Sun Yat-Sen University, Guangzhou, China b New York University, NY, USA c University of Maryland, College Park, MD, USA INFORMS 2017, Houston, TX
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 2
Our Contribution Using (automated) last mile transit as primary application… … we developed an optimal switching algorithm (available on GitHub) for data-driven decisions minimizing transit fleet https://github.com/BUILTNYU/Optimal-Switching operating costs Slide 3
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 4
Background: Last Mile Transit Ops Increasing need for optimization of automated, shared, on-demand transit to serve last mile Qiu et al. 2014 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 Slide 5
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 No methodology to dynamically optimize switching between different Chang & Schonfeld (1991) operating states in last mile transit Slide 6
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 𝑒𝑅 = 𝜈 𝑛 − 𝑅 𝑒𝑢 + 𝜏𝑅𝑒𝑥 urban transport operations Mean Increment in Long term reversion rate Wiener Volatility mean density process Slide 7
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 of switching Slide 8
Use cases Switching between fixed route and on- demand Switching between one-module and two- module vehicles (vehicle “size”) Slide 9
Problem Definition (1) ➢ A last mile region: 𝑀 × 𝑋 region connected to a hub via line haul of length 𝐾 ➢ Demand density: spatially uniformly distributed and Line haul temporally as mean- reverting process. ➢ The fixed-route conventional mode subdivided into 𝑂 𝑑 routes of Line haul width 𝑠 and length 𝑀 ➢ The flexible service mode Kim & Schonfeld (2014) subdivided into a grid of 𝑂 𝑔 zones of area A. Slide 10
Problem Definition (2) ➢ Given: ➢ Parameters of a stochastic process for demand density (as mean-reverting) fitted to historical data Line haul ➢ Current demand density ➢ Current operating state (fixed vs flexible, or 1- veh flexible vs 2-veh flexible) ➢ Determine whether or not to Line haul switch operating state at current time to minimize Kim & Schonfeld (2014) operating cost Slide 11
“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 : where a single threshold 𝑅 ∗ exists 𝐷 0 𝑅 𝑢 , 𝐷 1 𝑅 𝑢 for deterministically choosing one mode over the other Optimal policy under infinite horizon determines threshold 𝑅 𝑀 and 𝑅 𝐼 to optimize value function (current and future expected payoffs) Slide 12
SWITCHING BETWEEN FIXED ROUTE (MODE 1) AND ON-DEMAND (MODE 0) Slide 13
𝐷 1 𝑅 𝑢 and 𝐷 0 𝑅 𝑢 For fixed route transit, 𝐷 1 = the sum of the bus operating cost 𝐷 𝑝 , user in-vehicle cost 𝐷 𝑤 , user waiting cost 𝐷 𝑥 , and user access cost 𝐷 𝑦 . 𝐷 1 𝑅 = 𝐷 𝑝 + 𝐷 𝑤 + 𝐷 𝑥 + 𝐷 𝑦 = 𝑏 1 + 𝑐 1 𝑅 + 𝑒 1 𝑅 + 𝑓 1 𝑅 𝐷 0 = sum of bus operating cost 𝐷 𝑝 , user in-vehicle cost 𝐷 𝑤 , user waiting cost 𝐷 𝑥 , and user access cost 𝐷 𝑦 4 2 5 + 𝑐 0 𝑅 3 + 𝑒 0 𝑅 𝐷 0 𝑅 = 𝑏 0 𝑅 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 𝑇 𝑑 are endogenously determined to minimize cost (from Chang and Schonfeld, 1991a) Slide 14
Cost savings function Φ 𝑅 𝑢 The immediate cost savings accrued from time 𝑢 to 𝑢 + 𝑒𝑢 when switching from flexible to fixed route mode is: Φ 𝑅 𝑢 = 𝐷 0 𝑅 𝑢 ; 𝑇 𝑔 − 𝐷 1 𝑅 𝑢 ; 𝑇 𝑑 Fixed Flexible 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 15
Policy valuation based on asset equilibrium pricing The option value of using flexible bus operating mode 𝑊 0 𝑅 1 ′′ 𝑅 + 𝜈 𝑛 − 𝑅 𝑊 ′ 𝑅 − 𝜍𝑊 2 𝜏 2 𝑅 2 𝑊 0 𝑅 = 0 0 0 The option value of using conventional bus operating mode 𝑊 1 𝑅 1 ′′ 𝑅 + 𝜈 𝑛 − 𝑅 𝑊 ′ 𝑅 − 𝜍𝑊 2 𝜏 2 𝑅 2 𝑊 1 𝑅 + Φ 𝑅 = 0 1 1 Slide 16
Asset equilibrium conditions Define 𝑅 𝑀 as demand threshold to switch from fixed to flexible, and 𝑅 𝐼 from flexible to fixed When 𝐺 + = 𝐺 − = 0 , 𝑅 𝐼 = 𝑅 𝑀 At asset equilibrium, value matching between two modes 1 𝑅 𝐼 − 𝐺 + V 0 𝑅 𝐼 = 𝑊 0 𝑅 𝑀 − 𝐺 − V 1 𝑅 𝑀 = 𝑊 Smooth pasting ′ 𝑅 𝐼 = 𝑊 ′ 𝑅 𝐼 V 0 1 ′ 𝑅 𝑀 = 𝑊 ′ 𝑅 𝑀 V 0 1 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 17
Asset equilibrium conditions General solution of 𝑊 0 𝑅 and 𝑊 1 𝑅 1−𝑥 0 2𝜈𝑛 𝑅 𝛿 0 𝑊 0 𝑅 = 𝐵 0 𝐼 −𝛿 0 , 𝑥 0 , 𝑦 + 𝐶 0 𝐼 1 − 𝛿 0 − 𝑥 0 , 2 − 𝑥 0 , 𝑦 𝜏 2 𝑅 𝑊 1 𝑅 1−𝑥 1 2𝜈𝑛 𝑅 𝛿 1 = 𝐵 1 𝐼 −𝛿 1 , 𝑥 1 , 𝑦 + 𝐶 1 𝐼 1 − 𝛿 1 − 𝑥 1 , 2 − 𝑥 1 , 𝑦 𝜏 2 𝑅 ∞ 𝑓 −𝜍 𝑡−𝑢 𝑒𝑡 + 𝐹 𝑢 න Φ 𝑅 𝑡 𝑢 𝐼 ∙ is a confluent hypergeometric (“ Kummer ”) function 𝑥 𝑦 + 𝛿 𝛿 + 1 𝑦 2 𝛿 + 2 𝑦 3 𝐼 𝛿, 𝑥, 𝑦 = 1 + 𝛿 𝑥 𝑥 + 1 2! + 𝛿 𝛿 + 1 𝑥 + 2 3! + ⋯ 𝑥 𝑥 + 1 Slide 18
Asset equilibrium conditions 𝒀 = 𝑅 𝐼 , 𝑅 𝑀 , 𝐵 0 , 𝐵 1 ′ is uniquely determined by solving ∞ 𝛿 0 + ∆ 1 𝐵 0 − 𝐵 1 𝐼 1 𝑅 𝐼 𝑅 𝐼 𝛿 1 − 𝐹 𝑢 න Φ 𝑅 𝑡 ׀ 𝑅 𝑢 = 𝑅 𝐼 𝑓 −𝜍 𝑡−𝑢 𝑒𝑡 + 𝐺 + 𝐵 0 𝐼 0 𝑅 𝐼 𝑅 𝐼 𝑢 ∞ 𝛿 0 + ∆ 1 𝐵 0 − 𝐵 1 𝐼 1 𝑅 𝑀 𝑅 𝑀 𝛿 1 − 𝐹 𝑢 න Φ 𝑅 𝑡 ׀ 𝑅 𝑢 = 𝑅 𝐼 𝑓 −𝜍 𝑡−𝑢 𝑒𝑡 − 𝐺 − 𝐵 0 𝐼 0 𝑅 𝑀 𝑅 𝑀 𝐆 𝐘 = 𝑢 ∞ Φ 𝑅 𝑡 ׀ 𝑅 𝑢 = 𝑅 𝐼 𝑓 −𝜍 𝑡−𝑢 𝑒𝑡 𝜖𝐹 𝑢 𝛿 0 + ∆ 1 𝐵 0 − 𝐵 1 𝑁 1 𝑅 𝐼 𝑅 𝐼 𝛿 1 + 𝑢 𝐵 0 𝑁 0 𝑅 𝐼 𝑅 𝐼 𝜖𝑅 ∞ Φ 𝑅 𝑡 ׀ 𝑅 𝑢 = 𝑅 𝐼 𝑓 −𝜍 𝑡−𝑢 𝑒𝑡 𝜖𝐹 𝑢 𝛿 0 + ∆ 1 𝐵 0 − 𝐵 1 𝑁 1 𝑅 𝑀 𝑅 𝑀 𝛿 1 + 𝑢 𝐵 0 𝑁 0 𝑅 𝑀 𝑅 𝑀 𝜖𝑅 Due to the complexity of the equations, we obtain the solution numerically. Slide 19
Model properties Sensitivity of switching policy to transportation system parameters 𝑹 𝟏 =32 trips/mile 2 /hr, operating as flexible Solutions service initially Headway, ℎ 𝑑 0.42 Fixed Transit Vehicle size, 𝑇 𝑑 75 Fleet size, 𝐺 5 𝑑 Route spacing, 𝑠 1.41 Total cost, 𝐷 𝑡𝑑 2881.1 Headway, ℎ 𝑔 0.08 Vehicle size, 𝑇 𝑔 7 Flexible Fleet size, 𝐺 58 𝑔 Transit Service zone, 𝐵 3.02 Total cost, 𝐷 𝑡𝑔 2883.0 Φ 𝑅 1.9 ∞ Φ 𝑅 𝑓 −𝜍 𝑡−𝑢 𝑒𝑡 -39.4 𝐹 𝑢 න 𝑢 23.5 𝑊 0 𝑅 0 17.4 𝑊 1 𝑅 0 𝑅 𝑀 28.7 𝑅 𝐼 41.8 Indifference band ( 𝑅 𝐼 − 𝑅 𝑀 ) 13.1 Slide 20
Illustration of policy in action Flexible Flexible Fixed transit Fixed transit Slide 21
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