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

2. 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

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

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

5. 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!

6. 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!

7. 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).

8. 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:

9. 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

10. 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.

11. 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.

12. 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.

13. 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

14. 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.

15. 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

16. 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.

17. 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

18. 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

19. 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

20. 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

21. 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.

22. Presentation at IPDPS'17
May 31, 2017
Proof

23. 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

24. 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

25. 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

26. 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.

27. 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

28. Presentation at IPDPS'17
May 31, 2017
Video Clip

29. 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

30. 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.

31. 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

32. 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

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