Plowing the Streets of Pittsburgh

Transcription

1 Plowing the Streets of Pittsburgh A Dynamic Route Planning System J. Kinable 1,2 W-J. van Hoeve 2 S.F. Smith 1 1 Robotics Institute - Carnegie Mellon University, USA 2 Tepper School of Business - Carnegie Mellon University, USA January 19, 2016 J. Kinable (Carnegie Mellon University) January 19, / 33

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3 Motivation & Goals "Drivers have printed routing directions laying on the passenger seat. Most of them know the route (or an interpretation thereof) by heart." "A lot of time is lost due to emergency plow requests." "People complain that primary roads in their district are full of snow, while in other districts tertiary roads have already been serviced." J. Kinable (Carnegie Mellon University) January 19, / 33

4 Motivation (2) 1 Consumable resource costs: 4,3M$ (2014/2015), 3.3M$ (2013/2014) 2 Labor costs: 3.3M$ (2014/2015) 3 Equipment costs: 800K$ (2014/2015) 4 Snow and ice control staff: 275 safety economy environment Generalizes to street sweeping, street maintenance, garbage collection, etc. J. Kinable (Carnegie Mellon University) January 19, / 33

5 Goal Design a route planning system that issues optimized turn-by-turn instructions to the vehicles as they execute routes, and dynamically revises these plans as unexpected events force changes. J. Kinable (Carnegie Mellon University) January 19, / 33

6 Problem Description - Routes Road segments: start/end # lanes priority class shape driving speed directionality J. Kinable (Carnegie Mellon University) January 19, / 33

7 Problem Description - Vehicles Vehicles: Location Salt Capacity Fuel Capacity Salt economy Fuel economy J. Kinable (Carnegie Mellon University) January 19, / 33

8 Problem Description - Depots Depots: Location Salt (y/n) Fuel (y/n) J. Kinable (Carnegie Mellon University) January 19, / 33

9 Related models Chinese postman problem Vehicle routing with resource constraints Resource-constrained project scheduling with renewable and non-renewable resources For overview of Snow plowing literature, see N. Perrier, A. Langevin, et. al: [1, 2, 3, 4] Synchronized arc routing for snow plowing operations [5] City of Dieppe: 24,000 inhabitants, 462 intersections, 1,234 road segments ( 20% Downtown Pittsburgh) J. Kinable (Carnegie Mellon University) January 19, / 33

10 Model - Objective Objectives: 1 Makespan 2 Minimize deadheading 3 Weighted objective of Completion times per priority class J. Kinable (Carnegie Mellon University) January 19, / 33

11 Model - Constraints Constraints: 1 Vehicles start/end at depot 2 Every lane on a street segment must be covered, subject to traffic direction. 3 Renewable resource constraints: fuel, salt 4 Time constraints J. Kinable (Carnegie Mellon University) January 19, / 33

12 Model Job: 1 ID 2 start location 3 end location 4 duration 5 fuel & salt requirement Job types: 1 source depot job 2 target depot job 3 plow job 4 refuel job 5 resupply salt job J. Kinable (Carnegie Mellon University) January 19, / 33

13 Mathematical models Mixed Integer Programming Constraint Programming J. Kinable (Carnegie Mellon University) January 19, / 33

14 Mixed Integer Programming Model - Outline Variables: Three-index formulation: Vehicle k performs job j after job i. Variables recording resource utilization: time, salt, fuel Constraints: All plow jobs have to be performed, refuel/resupply jobs are optional Decide on direction for bidirectional jobs Resource constraints for fuel/salt/time J. Kinable (Carnegie Mellon University) January 19, / 33

15 Constraint Programming Model - Outline Variables: Job (interval) variables. Job to vehicle assignment variables Constraints (outline): Job sequencing constraints for each vehicle Alternative constraint for job to vehicle assignment Cumulative functions to manage resources J. Kinable (Carnegie Mellon University) January 19, / 33

16 Heuristic Constructive Heuristic Late Acceptance improvement heuristic J. Kinable (Carnegie Mellon University) January 19, / 33

17 Constructive Heuristic - Phase 1 1 Order jobs based on their priority 2 Constructively build schedules - Insert jobs 1 by 1? J. Kinable (Carnegie Mellon University) January 19, / 33

18 Constructive Heuristic - Phase 2 1 Insert Refuel and Resupply salt jobs Fuel Time?? J. Kinable (Carnegie Mellon University) January 19, / 33

19 Late acceptance improvement heuristic Quality (lower is better) Compare with solution L iterations ago Candidate solution Solution L iterations ago Iteration J. Kinable (Carnegie Mellon University) January 19, / 33

20 Late acceptance heuristic Moves: 1 Swap 2 Remove/insert Termination: 1 Time limit 2 Max number of non-improvements J. Kinable (Carnegie Mellon University) January 19, / 33

21 Experiments - Setup 1 Take 2 snapshots from OSM 2 Extract graph, pre-process and ensure Strong Connectivity 3 Execute routing algorithm J. Kinable (Carnegie Mellon University) January 19, / 33

22 Routing and Plowing Graph J. Kinable (Carnegie Mellon University) January 19, / 33

23 Experiments (4/22 instances) Kamin St: 1 28 intersections 2 45 plow jobs 3 plowing: 4mi 4 bidirectional: 38% mt Washington: intersections plow jobs 3 plowing: 52mi 4 bidirectional: 81% Residential: intersections plow jobs 3 plowing: 316mi 4 bidirectional: 64% Downtown: intersections plow jobs 3 plowing: 38mi 4 bidirectional: 38% J. Kinable (Carnegie Mellon University) January 19, / 33

24 Experiments J. Kinable (Carnegie Mellon University) January 19, / 33

25 Experiments - Kamin Coverage(%) :00:00 00:02:00 00:04:00 00:06:00 00:08:00 Time LA range LA average Constructive Heuristic CP 00:10:00 00:12:00 00:14:00 00:16:00 J. Kinable (Carnegie Mellon University) January 19, / 33

26 Experiments - Downtown Coverage(%) :00:00 00:10:00 00:20:00 00:30:00 00:40:00 00:50:00 Time LA range LA average Constructive Heuristic CP 01:00:00 01:10:00 01:20:00 01:30:00 J. Kinable (Carnegie Mellon University) January 19, / 33

27 Experiments - Mnt Washington Coverage(%) :00:00 00:10:00 00:20:00 00:30:00 00:40:00 00:50:00 01:00:00 Time 01:10:00 LA range LA average Constructive Heuristic CP 01:20:00 01:30:00 01:40:00 01:50:00 02:00:00 J. Kinable (Carnegie Mellon University) January 19, / 33

28 Experiments - Residential Coverage(%) :00:00 01:00:00 02:00:00 03:00:00 04:00:00 05:00:00 Time LA range LA average Constructive Heuristic 06:00:00 07:00:00 08:00:00 09:00:00 10:00:00 J. Kinable (Carnegie Mellon University) January 19, / 33

29 Experiments - Residential Pittsburgh solution Schedule completion time: 07:38:33 05:30:00 05:00:00 04:30:00 04:00:00 Pittsburgh Residential - Makespan (runtime 2h) Plowing Deadheading Pittsburgh Residential - Makespan (runtime 2h) Refuels Resupply salt U-turns Time 03:30:00 03:00:00 02:30:00 02:00:00 Value :30: Vehicle Vehicle J. Kinable (Carnegie Mellon University) January 19, / 33

30 Summary of preliminary results MIP not suitable: bounds are too weak CP works well for small instances Constructive heuristic finds reasonable solutions in ms. LA heuristic produces solutions of consistent quality Idea: Improve LA solutions with a CP-Large Neighborhood Search J. Kinable (Carnegie Mellon University) January 19, / 33

31 Future requirements 1 Road priorities 2 U-turns 3 Vehicle weight/width restrictions on roads 4 Efficient construction/updating of distance matrix 5 Adaptive system J. Kinable (Carnegie Mellon University) January 19, / 33

32 Bibliography I [1] N. Perrier, A. Langevin, and J. F. Campbell, A survey of models and algorithms for winter road maintenance. part i: system design for spreading and plowing. Computers & Operations Research, vol. 33, pp , [2], A survey of models and algorithms for winter road maintenance. part ii: system design for snow disposal, Computers & Operations Research, vol. 33, no. 1, pp , [3], A survey of models and algorithms for winter road maintenance. part iii: Vehicle routing and depot location for spreading, Computers & Operations Research, vol. 34, no. 1, pp , J. Kinable (Carnegie Mellon University) January 19, / 33

33 Bibliography II [4], A survey of models and algorithms for winter road maintenance. part iv: Vehicle routing and fleet sizing for plowing and snow disposal, Computers & Operations Research, vol. 34, no. 1, pp , [5] M. A. Salazar-Aguilar, A. Langevin, and G. Laporte, Synchronized arc routing for snow plowing operations. Computers & Operations Research, vol. 39, no. 7, pp , J. Kinable (Carnegie Mellon University) January 19, / 33