Simon Fraser University

Industry sponsor: Dynamex Inc.

Courier Routing Problem in Atlanta area

07/04/05 2Atlanta area

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• Same day delivery service in Atlanta area.

• Approximately 1000 requests per day (each request consists a delivery job).

Courier routing problem

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• Requests arrive offline (online requests can be updated).

• About 17 vehicles available.

• Atlanta area road-map: 125,000 intersections, 154,000 road segments, Approximated 70 miles in diameter.

Typical input

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54

1

3

11

126

2

7

8

9

1518

17

13

14

16

10

0

Courier routing (static version)

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54

1

3

11

126

2

7

8

9

1518

17

13

14

16

10

0

• new jobs arrive• old job is either cancelledor its specifications have changed.

Courier routing (dynamic version)

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minimize

cost(,s) = dsum() or tsum()

• more complex objective functions are ‘welcome’.

traveltime

traveldistance

Vehicle Routing Problem (VRP)

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• Single route optimization:– Efficient graph based TSP algorithms.

• Multiple routes inter-exchanging:– Meta-heuristic techniques using local search.– Dynamic programming.

Solving methodologies:

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Improvement algorithm

We start with a feasible solution = (1, 2, …, m). We then obtain abetter feasible solution ’ by means of exchanges of jobs within a route and between the routes.

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We start with a feasible solution = (1, 2, …, m). We then obtain abetter feasible solution ’ by means of exchanges of jobs within a route and between the routes.

x

x’

Improvement algorithm

Neighborhood of

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x

x ’’x’

Improvement algorithm

We start with a feasible solution = (1, 2, …, m). We then obtain abetter feasible solution ’ by means of exchanges of jobs within a route and between the routes.

Neighborhood of

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x

Neighborhood of

x ’’x’

Improvement algorithm

We start with a feasible solution = (1, 2, …, m). We then obtain abetter feasible solution ’ by means of exchanges of jobs within a route and between the routes.

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The concept of improvement graphallows one to identify profitable cyclic/path exchanges with certain constraints.

Each node represents a job. Two nodesare adjacent if the corresponding jobscan be exchanged.

Improvement Graph

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• Improvement graph• Local search• Distance approximation (on graphs):

• Well-Separated Sets.• Gabriel neighbours.

• TSP heuristics

References

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ReferencesWell-Separated Sets: •Paul B. Callahan and S. Rao Kosaraju, A decomposition of multi-dimensional point-sets with applications to k-nearest-neighbors and n-body potential fields, 546-556, 1992• Binay Bhattacharya et al. The travel time between an arbitrary pair of locations can be computed in logarithmic time (expected). manuscript 2005.Neighborhood Computation: • Michael Houle. SASH: A spatial approximation sample hierarchy for similarity search. Tech. Report RT-0517, IBM Tokyo Research Laboratory, 2003.• Kaustav Mukherjee. Application of the gabriel graph to instance-based learning. M.sc. project, School of Computing Science, SFU, 2004. (Supervised by Binay Bhattacharya) • Binay Bhattacharya, Kaustav Mukherjee and Godfried Toussaint, Geometric decision rules for high dimensions, 2005 (invited paper). Local Search:• T. Ibaraki, S. Imahori, M. Kubo, T. Masuda, T. Uno and M. Yagiura, Effective Local Search Algorithms for Routing and Scheduling Problems with General Time-Window Constraints, 2005.TSP heuristics:• S. Lin and B.W. Kernighan, An effective heuristic algorithm for the TSP, 1973.Improvement Graphs:• R.K. Ahuja, J.B. Orlin, and D. Sharma. Very large-scale neighborhood search, 2000.• R.K. Ahuja, O. Ergun, J.B. Orlin, A.P. Punnen. A survey of very large-scale neighborhood search techniques, 2002.

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Three types of algorithms:

• Inner route optimization (opt #1):• Inter-routes exchange - load balanced (opt #2):• Inter-routes exchange – route deletion (opt #3):

SFU improved algorithms

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Three types of algorithms:

• Inner route optimization (opt #1):optimizes only the jobs order within each route (using realistic graph base distances)

• Inter-routes exchange - load balanced (opt #2):• Inter-routes exchange – route deletion (opt #3):

SFU improved algorithms

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Three types of algorithms:

• Inner route optimization (opt #1):• Inter-routes exchange - load balanced (opt #2):allows job swaps between routes – keep the original number of routes (balanced).

• Inter-routes exchange – route deletion (opt #3):

SFU improved algorithms

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Three types of algorithms:

• Inner route optimization (opt #1):• Inter-routes exchange - load balanced (opt #2):

• Inter-routes exchange – route deletion (opt #3):allows any job swaps which can reduce the number of routes.

SFU improved algorithms

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We have tested 2 sets of daily delivery:• April 26, 200?: 902 jobs• April 27, 200?: 702 jobs

Each data set was tested using all three types of optimizations (#1, #2, #3).

• Distances are in miles, time: h:m:s

Dynamex versus SFU: experimental results

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Courier Statistics (27/4)

702 - jobs dynamex #1 #2 #3

Number of jobs: 702 702 702 702

Number of stops: 644 642 643 648

Driving time (all): 68:31:24 66:10:28 61:23:59 57:43:35

Driving distance (all): 2146.0 2066.9 1896.5 1735.5

Driving time per job: 0:05:51 0:05:39 0:05:14 0:04:56

Driving distance per job: 3.06 2.94 2.7 2.47

Number of vehicles: 17 17 17 11

Max. driving time per car: 6:07:02 6:02:27 5:44:59 6:48:25

Max. driving distance per car: 189.5 187.9 172.7 216.0

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Courier Statistics (26/4)

902 - jobs dynamex #1 #2 #3

Number of jobs: 902 902 902 902

Number of stops: 846 844 843 848

Driving time (all): 79:50:05 75:37:23 72:11:37 70:02:05

Driving distance (all): 2447.3 2316.3 2187.0 2118.9

Driving time per job: 0:05:18 0:05:01 0:04:48 0:04:39

Driving distance per job: 2.71 2.57 2.42 2.35

Number of vehicles: 17 17 17 13

Max. driving time per car: 5:51:08 5:37:57 7:01:25 7:05:03

Max. driving distance per car: 190.1 182.3 220.8 224.3

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Courier Statistics (27/4)

702 jobs Dynamex #1 #2 #3

Driving time (all): 100.0% 96.6% 89.6% 84.2%

Driving distance (all): 100.0% 96.3% 88.4% 80.9%

Driving time per job: 100.0% 96.6% 89.5% 84.3%

Driving distance per job: 100.0% 96.1% 88.2% 80.7%

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Dynamex versus SFU (#1)Cost (in seconds) of each route: 26.4, 902 jobs, dynamex versus #1 optimization

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Dynamex: single route example

Several self intersecting paths,Usually implies that the route can be improved.

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SFU - #1 improvement

The jobs of the route where reordered so there are less self intersecting paths - usually implies an improvement.

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SFU versus Dynamex distance comp.

Dynamex: L2 - free space TSP

SFU: graph distance TSP

SFU versus Dynamex distance comp.

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Dynamex: free space SFU: graph distances

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Dynamex – start / end points

Long distance to/from the depot to the start/end point of the route.

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SFU – #1 start / end points

shorter distances from the depot to/from the start/end points of the route.

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SFU #2 - routes examples

By allowing job swaps between routes an improved routes can be computed, while keeping ‘fairness’.

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Dynamex all routes

All 17 routes:• long distance to/from the depot to the start/end point.

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SFU #3 - all routes

Covering all jobs with 14 routes (instead of 17)

Shorter distance to/from the depot to the start/end point.

Less cars.

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Advantages of the suggested system:• improved results 4-16%.• changes can be updated dynamically.• efficient run time.• robustness & scalability:

• more jobs. • complex objective functions.• pickup & delivery with time windows.

Conclusion

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• Fine tuning the suggested methods.• Dynamic updates on road conditions.• Pick-up and delivery with multiple constrains.

Future work

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Binay Bhattacharya* , SFU

Posenjit Bose, Carleton U

Daya Ram Gaur, ULethbridge

Mark Keil, USask

David Kirkpatrick*, UBC

Ramesh Krishnamurti, SFU

Godfried Toussaint, McGill U

Collaborators | students & othersRober Benkoczi

Boaz Benmoshe

David Breton

Stephane Durocher

Yuzhuang Hu

Benjamin Lewis

Rizwan Merchant