Distributed Vehicle Routing Approximation Presentation at IPDPS'17 May 31, 2017 Distributed Vehicle Routing Approximation Akhil Krishnan Mikhail Markov Borzoo Bonakdarpour

Outline Autonomous vehicles & Distributed VRP Our solution Presentation at IPDPS'17 May 31, 2017 Outline Autonomous vehicles & Distributed VRP Our solution Simulations and experiments Conclusion

Presentation at IPDPS'17 May 31, 2017 Autonomous Vehicles  There is growing popularity in employing autonomous vehicles Driverless cars Unmanned aerial systems (UAS)

Unmanned aerial systems Presentation at IPDPS'17 May 31, 2017 Unmanned aerial systems Agriculture Search & rescue Patrolling Surveys & inspection

Presentation at IPDPS'17 May 31, 2017 Topographic surveys Sumac Geomatics Inc. employs UASs for data/image acquisition to conduct surveys on Activities of mining and lumber companies Analysis of vegetation Climate change (Canadian northern territories) Problem: Serious energy limits of UASs Solution: Swarming!

Research problem How can we design an autonomous fleet of UASs that Presentation at IPDPS'17 May 31, 2017 Research problem How can we design an autonomous fleet of UASs that Carries out a joint mission Optimizes a set of costs: Size of the fleet Time Job (search, imaging, etc) Energy Sounds like a combinatorial optimization problem!

The vehicle routing problem (VRP) Presentation at IPDPS'17 May 31, 2017 The vehicle routing problem (VRP)  VRP is concerned with the optimal design of routes to be used by a fleet of vehicles to service a set of customers. VRP generalizes the multiple traveling salesmen problem (mTSP).

The vehicle routing problem (VRP) Presentation at IPDPS'17 May 31, 2017 The vehicle routing problem (VRP) Let A be a finite set of vehicles and be an undirected finite graph. Each vehicle is associated with a vertex called the depot. We consider two cost functions: Cost of travel: Cost of service:

The vehicle routing problem (VRP) Presentation at IPDPS'17 May 31, 2017 The vehicle routing problem (VRP) Total cost of a path taken by vehicle a: Goal: Find a set P of paths and a plan function such that: The paths cover all vertices of G for servicing. That is, And, F respects the following minimization objective

Presentation at IPDPS'17 May 31, 2017 Distributed VRP We aim at solving the problem in a distributed fashion: Each vehicle runs a local algorithm and communicates with other vehicles to compute its route. Challenge: Thinking globally and acting locally. Finding an accurate approximation bound.

Distributed VRP Model of computation Presentation at IPDPS'17 May 31, 2017 Distributed VRP Model of computation The communication network is a complete graph. Each vehicle has a unique id Each vehicle knows the structure of the graph, and cost functions of all vehicles.

Presentation at IPDPS'17 May 31, 2017 The Main challenge Designing some distributed solution to the VRP is not overly difficult. What’s difficult is computing the approximation ratio.

Outline Autonomous vehicles & Distributed VRP Our solution Presentation at IPDPS'17 May 31, 2017 Outline Autonomous vehicles & Distributed VRP Our solution Simulations and experiments Conclusion

Our strategy We design a synchronous algorithm, where in each round Presentation at IPDPS'17 May 31, 2017 Our strategy We design a synchronous algorithm, where in each round the vehicles engage in internal path computation and then send/receive broadcast messages. the error is bound and given a fixed number of rounds, one can compute the approximation ratio.

Solution sketch: ILP Representation Presentation at IPDPS'17 May 31, 2017 Solution sketch: ILP Representation First, we transform the VRP problem into an integer linear program (ILP) ILP variables Paths are simple Each node is serviced exactly once Subtour elimination If a vehicle services a node, then it must travel to that node Objective function

Solution sketch: LP Relaxation and Duality Presentation at IPDPS'17 May 31, 2017 Solution sketch: LP Relaxation and Duality Second, we apply an LP relaxation Third, we generate the dual LP: In each round of the distributed algorithm, we update primal and dual variables to compute the approximation bound.

The algorithm Step 1 – Computing the neighborhood Presentation at IPDPS'17 May 31, 2017 The algorithm  Step 1 – Computing the neighborhood The neighborhood of a vehicle is the set of all simple paths of length k from the vehicle’s current location, where all the vertices are not yet serviced and traveled. 20 8 7 4 22 v1 v2 3 v3 3 v4 a1 6 10 v8 v7 v6 v5 3 11 20 15 10 10 8 a2 15

The algorithm Step 2 – Computing the best local path Presentation at IPDPS'17 May 31, 2017 The algorithm  Step 2 – Computing the best local path Step 2-1: Minimum cost of servicing and traveling across the nodes in a path normalized by the number of communication rounds taken. Step 2-2: Verify if a rival vehicle may enter that path in rounds of communication. 20 8 7 4 22 v1 v2 3 v3 3 v4 a1 6 10 v8 v7 v6 v5 3 11 20 15 10 10 8 a2 15

The algorithm Step 3 – Deciding to travel or service Presentation at IPDPS'17 May 31, 2017 The algorithm  Step 3 – Deciding to travel or service Decision: A vehicle services a path if no rival vehicle is likely to interfere, and no rival vehicle can do it more efficiently. Otherwise, it travels to the minimum cost travel + service unserviced vertex. Communication: After this decision, each vehicle sends a message to all other vehicles to update their global view of the graph. 20 8 7 4 22 v1 v2 3 v3 3 v4 a1 6 10 v8 v7 v6 v5 3 11 20 15 10 10 8 a2 15

They fly in different altitudes Presentation at IPDPS'17 May 31, 2017 They fly in different altitudes Example a1 a2 20 8 7 4 20 5 7 4 22 v1 v2 3 v3 3 v4 11 v1 v2 6 v3 8 v4 6 10 7 10 v8 v7 v6 v5 v8 v7 v6 v5 3 11 20 4 13 20 15 10 10 8 4 8 10 10 15

Approximation ratio The approximation ratio of our algorithm is Presentation at IPDPS'17 May 31, 2017 Approximation ratio The approximation ratio of our algorithm is where   is the maximum cost of travel or service in the input VRP instance, n is the size of the graph, and m is the number of vehicles.

Presentation at IPDPS'17 May 31, 2017 Proof

Proof sketch The Algorithm forms a feasible solution for the LP. Presentation at IPDPS'17 May 31, 2017 Proof sketch The Algorithm forms a feasible solution for the LP. At the end of each communication round, the value of the primal and dual objectives are equal. In each round, we have . In each round, we have (the sum of all 1-step costs). The approximation ratio of our algorithm is .

Presentation at IPDPS'17 May 31, 2017 Back to Integer values Naïve way: our algorithm allows a simple rounding scheme Randomized rounding: The approximation ratio of our algorithm with randomized rounding is

Outline Autonomous vehicles & Distributed VRP Our solution Presentation at IPDPS'17 May 31, 2017 Outline Autonomous vehicles & Distributed VRP Our solution Simulations and experiments Conclusion

Simulations and experiments Presentation at IPDPS'17 May 31, 2017 Simulations and experiments  Simulations compare our algorithms with the optimal solution and a simple centralized greedy algorithm.

Implementation and real experiments Presentation at IPDPS'17 May 31, 2017 Implementation and real experiments 3DR & Home-built UASs Rasberry Pi Long-range WiFi module Alfa WiFi Access point Implementation in python using MAVlink protocol for Pixhawk Structure of the graph is given in terms of GPS coordinates

Presentation at IPDPS'17 May 31, 2017 Video Clip

Outline Autonomous vehicles & Distributed VRP Our solution Presentation at IPDPS'17 May 31, 2017 Outline Autonomous vehicles & Distributed VRP Our solution Simulations and experiments Conclusion

Presentation at IPDPS'17 May 31, 2017 Summary Vehicle routing problem (VRP) is a combinatorial optimization problem for optimal design of routes by a fleet of vehicles to service a set of customers. We introduced an on-the-fly distributed synchronous algorithm and identified the approximation ratio. Our simulations show the algorithm scales well. Implementation shows that it is practical.

Future work Fault-tolerant multi-UAS networks Presentation at IPDPS'17 May 31, 2017 Future work Fault-tolerant multi-UAS networks Software defined multi-UAS networks (SDN) Adaptive multi-UAS networks (online algorithm identification of obstacles, etc) Multi-UAS networks in physically harsh environments Imprecise graphs Energy constraints in a 3D graph model Covering large areas with charging stations

Future work Other application-specific projects Agriculture Presentation at IPDPS'17 May 31, 2017 Future work Other application-specific projects Agriculture Hazard analysis Movie industry Disease diagnosis

Presentation at IPDPS'17 May 31, 2017 Thank you!