Building Ride-sharing and Routing Engine for Autonomous Vehicles: A State-space-time Network Modeling Approach

Transcription

1 Building Ride-sharing and Routing Engine for Autonomous Vehicles: A State-space-time Network Modeling Approach Xuesong Zhou (xzhou7@asu.edu) Associate professor, School of Sustainable Engineering and the Built Environment Arizona State University Presentation at University of Nevada, Reno

2 Open-source Free Software Package NEXTA: front-end Graphical User Interface GUI (C++) DTALite: Open-source computational engine (C++) Light-weight and agent-based DTA Simplified kinematic wave model (Newell) Built-in OD demand matrix estimation (ODME) program Emission prediction (light-weight MOVES interface) Simplified car follow modeling (Newell) Research areas Train routing and scheduling Vehicle routing and scheduling (new)

3 Computational Challenges Shared memory-based parallel computing for agent-based path finding and mesoscopic traffic simulation (based on OpenMP) Maryland State-wide model: 0 K nodes, 7K links,,000 zones, 8 M agents CPU time: 0 min per UE iteration on a 0-core workstation with 9 GB RAM

4 Outline. Introduction: Modeling next-generation of transportation systems. Vehicle routing pick up and delivery constraints with time windows. Methodology for traffic signal control and automated vehicle routing

5 . Challenges in modeling next-generation of transportation systems: from simulation to optimization Based on Paper titled Finding Optimal Solutions for Vehicle Routing Problem with Pickup and Delivery Services with Time Windows: A Dynamic Programming Approach Based on State-space-time Network Representations Monirehalsadat Mahmoudi, Xuesong Zhou Submitted Transportation Research Part B; 5

6 Motivation Concept of Ride Sharing: Advantages of Ride Sharing: Private SAV Providers Cloud Computing Center for traffic data storage and congestion pricing/crediting Regional Traffic Management Center Reducing driver stress and driving cost Increasing safety Increasing road capacity and reducing costs Increasing fuel efficiency and reducing pollution depot passenger 's destination passenger 's destination kindergarten passenger shopping center passenger 's destination passenger passenger office OD trip request and confirmation SAV route 6

7 Ride Sharing Companies One-to-One Matching Innovative Pricing Mechanism Accessibility for Low-income Families 7

8 Key Questions How many cars a city should use to support the overall transportation activity demand, at different levels of coordination and pre-trip scheduling? How much energy is used at the optimal state (optimal state is a condition in which 00% of travelling is supported by ride sharing)? How much emissions can be minimized? To address the first question: we propose a new mathematical model for pickup and delivery problem with time windows (PDPTW) to present a holistic optimization approach for synchronizing travel activity schedules, transportation services, and infrastructure on urban networks. 8

9 Vehicle Routing Problem (VRP) & Vehicle Routing Problem with Time Windows (VRPTW) Inputs: Passengers Location Passengers Preferred Service Time Window Output: Routing Route Depot 5 Customers 6 Route Route 9 9

10 Vehicle Routing Problem with Pick up and Delivery with Time Windows (VRPPDTW) Inputs: Passengers Origin and Destination Location Passengers Preferred Pick up and Delivery Time Windows Outputs: Vehicle Routing O Vehicle Scheduling Assigning Passengers to Vehicles (Many-to- One Relationship) Pricing D O Route O5 D O Depot Route O Route D D O6 D5 D6 0

11 Problem Statement by a Simple 6-node Transportation Network 5 6 D Two passengers Two vehicles Passenger and s origin: node Passenger and s destination: node O Origin O Vehicle and s origin depot: node Vehicle and s destination depot: node D Destination Depot

12 Problem Statement by a Simple 6-node Transportation Network D Passenger s preferred time window for departure from origin: [,7] Passenger s preferred time for arrival at destination: [9,] O Origin O 5 6 Passenger s preferred time window for departure from origin: [0,] Passenger s preferred time for arrival at destination: [5,7] D Destination Depot

13 Problem Statement by a Simple 6-node Transportation Network D 5 6 O O Origin D Destination Depot

14 Problem Statement by a Simple 6-node Transportation Network D 5 6 O O Origin D Destination Depot

15 Existing Network representation Pick up node Delivery node Depot 0 0 5

16 Optimization-based Approaches for Solving VRPPDTW Reference Method (Algorithm) Type of problem Objective Function Psaraftis (980) Psaraftis (98) Sexton and Bodin (985) Desrosiers, Dumas, and Soumis (986) Dumas, Desrosiers, and Soumis (99) Exact backward dynamic programming Forward dynamic programming Benders decomposition Exact forward dynamic programming Column Generation Single vehicle VRPPD Single vehicle VRPPDTW Single vehicle VRPPD with one sided windows single vehicle VRPPDTW Multiple vehicle VRPPDTW Weighted combination of the total service time and the total customer inconvenience Sum of waiting and riding times Total customer inconvenience Total distance traveled Total travel cost 6

17 Optimization-based Approaches for Solving VRPPDTW (contd.) Ruland (995, 997) Savelsbergh and Sol (998) Lu and Dessouky (00) Ropke, Cordeau, Laporte (007) Ropke and Cordeau (009) Polyhedral approach Branch-and-price Branch-and-cut Branch-and-cut-andprice Branch-and-cut-andprice Single Vehicle VRPPD without capacity constraints Multiple vehicle VRPPDTW Multiple vehicle VRPPDTW Multiple vehicle VRPPDTW Multiple vehicle VRPPDTW Total travel cost Primary objective function: Total number of vehicles, secondary: Total distance traveled Total travel cost and the fixed vehicle cost Total traveled distance Total traveled distance Baldacci, Bartolini, Mingozzi (0) Set-partitioning formulation improved by additional cuts Multiple vehicle VRPPDTW Primary objective function: Route costs, secondary: Sum of vehicle fixed costs and then sum of route costs 7

18 Current Mathematical Model for VRPPDTW[Cordeau (006)] Min v V i N j N c ij v x ij v objective function: minimizing the total routing cost v V j N j N x ij v x ij v = j N v x n+i,j i P = 0 i P, v V guarantees that each passenger is definitely picked up ensure that each passenger s origin and destination are visited exactly once by the same vehicle j N x v 0j = v V each vehicle v starts its route from the origin depot x v ji x v ij = 0 j N j N i P D, v V flow balance on each node i N v x i,n+ = v V each vehicle v ends its route to the destination depot 8

19 Current Mathematical Model for VRPPDTW[Cordeau (006)] (contd.) x ij v B i v + d i + t ij B j v Non-linear Constraint i N, j N, v V validity of the time variables x ij v Q i v + q j Q j v Non-linear Constraint i N, j N, v V validity of the load variables L v v i = B n+i B i v + d i i P, v V defines each passenger s ride time v B n+ B v 0 T v v V impose maximal duration of each route e i B i v l i i N, v V impose time windows constraints t i,n+i L i v L i P, v V max 0, q i Q i v min Q v, Q v + q i i N, v V impose the ride time of each passenger constraints impose capacity constraints x ij v 0, i N, j N, v V 9

20 Current Theoretical and Computational Challenges Single vs multiple vehicles The focus of most research was on solving the PDPTW for a single vehicle (simpler case) Single vs multiple depots Limited number of transportation requests (passengers) The most Current Algorithm: Baldacci et al. (0) based on a set-partitioning formulation solved instances of approximately requests with tight time windows. Time windows Some preprocessing steps to find feasible transportation requests are needed Research only focus on tight time windows to prevent some fluctuation in results 0

21 Current Theoretical and Computational Challenges (contd.) Fixed routing cost (travel time) over time Existing network for PDPTW: an offline network in which each link has a fixed routing cost (travel time) Existence of sub-tour in the optimal solution Some additional constraints are needed to avoid the existence of any sub-tour in the optimal solution

22 Opening Statement about Our Method (contd.) Description of the PDPTW in Space-Time Transportation Network [Mahmoudi, M. & Zhou, X. (0)] d [,] d [,6] d Pickup node Delivery node Depot Transportation node [.,.] Time windows o (a) [,7] o o [8,0] (b)

23 Space Physical Vehicle s Space-time Transportation Network 6 5 o d d Dummy pick up node Dummy delivery node Dummy depot Transportation node Transportation arcs Waiting Arc Service arc corresponding pick up Service arc corresponding drop off o o d Time window for vehicle v at starting and ending depots Passenger p s preferred departure time window from origin Passenger p s preferred arrival time window to destination Time

24 Binary representation and equivalent character-based representation for passenger carrying states Binary Equivalent character-based representation representation [0,0,0] [ _] [,0,0] [p ] [0,,] [_ p p ] o d o d [p p _ ] [ _ p _ ] o o d d o o d d Depot Passenger p s origin Passenger p s destination o d o d (a) Shared ride (b) Serving single passenger once a time

25 All possible combinations of passenger carrying states w w [ _] [p ] [_ p _] [ p ] [p p _] [p _ p ] [_ p p ] [ _] no change pickup pickup pickup [p ] drop-off no change pickup pickup [_ p _] drop-off no change pickup pickup [ p ] drop-off no change pickup pickup [p p _] drop-off drop-off no change [p _ p ] drop-off drop-off no change [_ p p ] drop-off drop-off no change 5

26 Finite states graph showing all possible passenger carrying state transition (pickup or drop-off) [ _ ] [ _ p p ] [ p ] [ _ p _ ] [ p _ p ] [ p p _ ] Pick up Drop off [ p ] 6

27 Projection on state-space network representation for ridesharing path (pick up passenger p and then p ). [ p p ] d 5 6 o [ p _ ] [ p _ ] d Dummy pick up node Dummy delivery node Dummy depot o Transportation node Service link corresponding passenger p s pick up [ ] Service link corresponding passenger p s drop off 5 6 7

28 Problem Definition Min Z = v (V V ) i,j,t,s,w,w B v c v, i, j, t, s, w, w y v, i, j, t, s, w, w s.t. Flow balance constraints at vehicle v s origin vertex y v, i, j, t, s, w, w = i = o v, t = e v, w = w = w 0, v (V V ) i,j,t,s,w,w B v Flow balance constraint at vehicle v s destination vertex y v, i, j, t, s, w, w = j = d v, s = l v, w = w = w 0, v (V V ) i,j,t,s,w,w B v Flow balance constraint at intermediate vertex j,s,w y v, i, j, t, s, w, w y v, j, i, s, t, w, w = 0 i, t, w o v, e v, w 0, d v, l v, w 0, v (V V ) j,s,w 8

29 Problem Definition (contd.) Passenger p s pick-up request constraint v (V V ) i,j,t,s,w,w Ψ p,v y v, i, j, t, s, w, w = p P Passenger p s drop-off request constraint v (V V ) i,j,t,s,w,w Φ p,v y v, i, j, t, s, w, w = p P Binary definitional constraint y v, i, j, t, s, w, w {0, } i, j, t, s, w, w B v, v (V V ) 9

30 Lagrangian Relaxation-based Solution Approach o Pickup and drop-off constraints : each passenger is picked up and dropped off exactly once by a vehicle (either physical or virtual). o Flow balance constraints on intermediate nodes force the vehicle to end its route at the destination depot with the empty passenger carrying state. o If vehicle v picks up passenger p from his origin, to maintain the flow balance constraints on intermediate nodes, the vehicle must drop-off the passenger at his destination node so that the vehicle comes back to its ending depot with the empty passenger carrying state. o As a result, constraint (6) is redundant L = c v, i, j, t, s, w, w y v, i, j, t, s, w, w v (V V ) i,j,t,s,w,w B v + p P λ p y v, i, j, t, s, w, w v (V V ) i,j,t,s,w,w Ψ p,v 0

31 New Relaxed Problem Min L s.t. i,j,t,s,w,w B v y v, i, j, t, s, w, w = i = o v, t = e v, w = w = w 0, v (V V ) i,j,t,s,w,w B v y v, i, j, t, s, w, w = j = d v, s = l v, w = w = w 0, v (V V ) j,s,w y v, i, j, t, s, w, w j,s,w y v, j, i, s, t, w, w = 0 i, t, w o v, e v, w 0, d v, l v, w 0, v (V V ) y v, i, j, t, s, w, w {0, } i, j, t, s, w, w B v, v (V V ) The simplified Lagrangian function L: L = ξ v, i, j, t, s, w, w y v, i, j, t, s, w, w λ p v (V V ) i,j,t,s,w,w B v p P

32 Time-dependent Forward Dynamic Programming for each vehicle v (V V ) do L.,.,. = + ; node pred of vertex.,.,. = ; time pred of vertex.,.,. = ; state pred of vertex.,.,. = ; L o v, e v, w 0 = 0; for each time t e v, l v do for each link (i, j) do for each state w do derive downstream state w based on the possible state transition on link (i, j); derive arrival time s = t + TT(i, j, t); if (L(i, t, w) + ξ v, i, j, t, s, w, w < L(j, s, w )) then L(j, s, w ) = L(i, t, w) + ξ v, i, j, t, s, w, w ; //label update node pred of vertex j, s, w = i; time pred of vertex j, s, w = t; state pred of vertex j, s, w = w;

33 Computational Results for Testing Different Cases p p p Scenario I. Two passengers are served by one vehicle through ride-sharing mode. v p Scenario II. Two passengers are served by one vehicle (not through ride-sharing mode). v v p p p 5 Scenario III. Two passengers and one vehicle; one passenger remains unserved. 6 v v p 5 Scenario IV. Two passengers and two vehicles; each vehicle is assigned to a passenger 6 p v p 5 p Scenario V. Three passengers are served by one vehicle through ride-sharing mode Transportation link v Representative of passenger p s origin-destination pair p v Scenario VI. Two vehicles compete for serving a passenger Representative of vehicle routing

34 Lagrangian multiplier value Lagrangian multipliers along 0 iterations in test case for the six-node transportation network: System Marginal Cost! Iteration number Passenger Passenger Passenger Passenger Passenger 5 Passenger 6

35 Results for the Phoenix network with,777 transportation nodes and,879 links Test case number Number of iterations Number of passengers Number of vehicles LB UB Gap (%) Number of passengers not served CPU running time (sec) % % % % %

36 Short Summary We propose a new mathematical formulation in state-space-time network for PDPTW Based on time-dependent forward dynamic programming approach in the Lagrangian reformulation framework, the main problem is transformed to easy sub-problems (Time-dependent least cost path sub-problems) which is solved independently without much effort Unlike former proposed models for PDPTW, this model is now able to solve PDPTW in large scale transportation networks. 6

37 High-dimension vs. low-dimension representation: The blind men and the elephant 7

38 : Extensions of state-space-time modeling framework: Traffic signal control for automated vehicles 8

39 Phase-time network for Signal Optimization Traffic signal phasing sequence representation NEMA ring-structure signal phase (North America) P. Li., P. Mirchandani, X. Zhou, Solving Simultaneous Route Guidance and Traffic Signal Optimization Problem Using Coupled Space-time and Phase-time Networks. Transportation Research Part B. 8, 0-0. Movement-based signal phase (same control flexibility, fewer variables) Source: Traffic Signal Timing Manual: FHWA Phase Phase Phase Phase Signal Phases Flexible Phasing Sequence Cyclic Phasing Sequence

40 Phase-time network for Signal Optimization A signal timing plan is composed of: Phasing sequence Phase duration Signal timings can also be represented with a series of phase nodes in the -D phase-time network Similar structure with a space-time network! Solution is provided like: w m, k,, h, if signal phase m is green at and then turns green to signal phase k 0, otherwise at h phase starts green at t=, then turn over green to phase at t=

41 Optimize traffic signal in phase-time network Each green phase will generate cost on other phases (e.g., delays) Find a least-cost path from origin (starting phase) to destination (end of horizon) Z Z Time Phase-Time Network: All Possible Transition Arcs from Phase Time Phase-Time Network: All Possible Transition Arcs from Phase Z Z Phase Phase Phase Phase Signal Phases Flexible Phasing Sequence Phase-Time Network: All Possible Transition Arcs from Phase Time Phase-Time Network: All Possible Transition Arcs from Phase Time

42 Intersection control constraints Mutual exclusiveness of signal phases Any signal phase has one and only one predecessor phase and successor phase at one time. (m i, τ ) (m j, h) (m i`, l ) Time

43 Conclusions Present a state-space-time (SST) based modeling framework By adding additional state dimensions, we prebuild many complex constraints into a multidimensional network Computationally efficient solution algorithm: forward dynamic programming + Lagrangian relaxation Wide range of applications (i) how to estimate macroscopic and microscopic freeway traffic states from heterogeneous measurements, (ii) how to optimize transportation systems and ride-sharing services involving vehicular routing decisions with pickup and delivery time windows (VRPPDTW). Challenges Selecting state is an art How to overcome curse of dimensionality Dynamics and uncertainty (demand/supply) Smart search space reduction and metaheuristics algorithms for real-time applications