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A Parallel Architecture for the Generalized Traveling Salesman Problem. Max Scharrenbroich AMSC 663 Mid-Year Progress Report Advisor: Dr. Bruce L. Golden R. H. Smith School of Business. Presentation Overview. Background and Review GTSP Review Applications - PowerPoint PPT Presentation
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A Parallel Architecture for the Generalized Traveling Salesman Problem
Max ScharrenbroichAMSC 663 Mid-Year Progress Report
Advisor: Dr. Bruce L. GoldenR. H. Smith School of Business
1
Presentation Overview
• Background and Review– GTSP Review– Applications– Problem Formulation and Algorithms– Review of Project Objectives
• Review of mrOX GA• Parallelism in the mrOX GA• Status Summary and Future Work
2Review >
Review of the GTSP
• The Generalized Traveling Salesman Problem (GTSP):– Variation of the well-known traveling salesman
problem.– A set of nodes is partitioned into a number of
clusters.– Objective: Find a minimum-cost tour visiting
exactly one node in each cluster.– Example on the following slides…
3
GTSP Example
• Start with a set of nodes or locations to visit.
4
GTSP Example (continued)
• Partition the nodes into clusters.
1
2
3
4
5
6
5
GTSP Example (continued)
• Find the minimum tour visiting each cluster.
1
2
3
4
5
6
6Applications >
Applications
• The GTSP has many real-world applications in the field of routing:– Mailbox collection and stochastic vehicle routing.– Warehouse order picking with multiple stock
locations.– Airport selection and routing for courier planes.
7Formulation and Algorithms >
Problem Formulation and Algorithms
• The GTSP can be formulated as a 0-1 Integer Linear Program (ILP).
• The GTSP is an NP-hard problem.• There are algorithms for solving the GTSP to
optimality, but these algorithms eventually suffer from combinatorial explosion (>90 clusters).
• There are several successful heuristic approaches to the GTSP.
• In this project I will focus on the mrOX Genetic Algorithm (J. Silberholz and B.L. Golden, 2007).
8Review of Project Objectives>
Review of Project Objectives
• Develop a generic software architecture and framework for parallelizing serial heuristics for combinatorial optimization.
• Extend the framework to host the serial mrOX GA and the GTSP problem class.
• Investigate the performance of the parallel implementation of the mrOX GA on large instances of the GTSP (> 90 clusters).
9Overview >
Presentation Overview
• Background and Review• Review of mrOX GA– mrOX Crossover– Local Search: 2-opt and 1-swap– Mutation
• Parallelism in the mrOX GA• Status Summary and Future Work
10Diagram of mrOX GA >
Review of mrOX GA
11
Pop 1
Merge
…
Start
End
Pop 2
Pop 7
Post-Merge
Initialization Phase (Light-Weight):• Sequentially generates and evolves 7 isolated populations of 50
individuals each.• Only use the rOX for crossover.• Apply one cycle of 2-opt followed by 1-swap improvement heuristic
only to new best solution.• 5% chance of mutation.• Terminate each population after no new best solution is found for
10 consecutive generations.
Post-Merge Phase:• Use the full mrOX for crossover.• Apply full cycles of 2-opt followed by 1-swap improvements until no
improvements are found to:1. Child solutions that are better than both parents.2. Random 5% of population (preserve diversity).
• 5% chance of mutation.• Terminate after no new best solution is found for 150 consecutive generations.
Select the best 50 individuals from the union of the isolated populations to continue on.
Crossover and Local Search >
Review of mrOX GA
12
rOXCrossover
2-opt 1-swap
New Best?
no
yes
Initialization Phase: rOX Crossover and Local Search Improvements
mrOXCrossover
2-opt 1-swap
Better than
Parents?
no
yes
until no improvements
Post-Merge Phase: mrOX Crossover and Local Search Improvements
mrOX Crossover >
Review of mrOX Crossover
13
1 2 3 4X Z YP1
1 43 2X Y ZP2
Permutations of Rotations and
Reversals
Combinations of Cluster Nodes at
End-Points
1 2 3 4
Randomly Generate Cut Points
1 43 2X Y ZP2’
1 2 3 4X Z YP1’ xx
1 2 3 4 Z b X Y d
Combined Tour
Example >
Example of mrOX Crossover
14
YX
Z
YX
Z
1
2 3
4x x
P1Y
XZ
1
2 3
4
P2
YdX
Zb
Check rotations and reversals with node combinations at the
end points.1
2 3
4
YdX
Zb
Child
Insert the best path.
Complexity >
Complexity of mrOX Crossover
15
1
0222 )),mod()1((),(2)(
m
ippp SminSinSN
),( 2pSin A function that returns the number of nodes in the cluster at index i in the ordered set Sp2.
)( 2pSN The number of comparisons required in a mrOX crossover operation.
2pSm The cardinality of the set Sp2.
),...,,( )1(102 mp cccS The ordered set of cluster/node pairs added from the 2nd parent.
2max2 2)( mnSN p maxnBounded by: where
is the maximum number of nodes in the clusters.
Local Search >
Local Search: 2-opt & 1-swap
16
YX
Z
12 3
4
2-optY
XZ
12 3
4
1 2 3 4X Y Z
YX
Z
12 3
4
1-swapY
XZ
12 3
4
1 2 3 4X Y Z
Mutation >
Mutation
17
YX
Z
12 3
4
(1) Randomly generate cut points.
12 3
4
(2) Isolate the path between the cut points.
12 3
4
(3) Reverse the order of the
nodes.YX
Z
12 3
4
(4) Reinsert the path.
Overview >
Presentation Overview
• Background and Review• Review of mrOX GA• Parallelism in the mrOX GA– Low Level Parallelism– Concurrent Exploration– Cellular GA Inspired Parallel Cooperation
• Status Summary and Future Work
18Low Level Parallelism >
Low-Level Parallelism in the mrOX GA
• For rOX and mrOX computational loading can be estimated (slide 13) and therefore crossover of individuals could be load balanced over a number of processors.
• The local search improvement phase (multiple cycles of 2-opt followed by 1-swap) in the post-merge phase is not deterministic and would be difficult to load balance.
• While improving execution time, load balancing the crossover and local search would introduce inefficiencies.
• Tuning the load balancing would be problem dependent and time consuming .
19Type 3 Parallelism >
Type 3 Parallelism in the mrOX GA• Genetic algorithms are amenable to parallelism via
concurrent exploration.• Cooperation between processes can be implemented
to ensure diversity while maintaining intensification.
20
P1
P2
Pn
…
start
end
P1
P2
Pn…
P1
P2
Pn
start
end
P1
P2
Pn
P1
P2
Pn
T
Parallel Cooperation w/ Mesh >
Parallel Cooperation with Mesh Topology
21
• Inspired by cellular genetic algorithms (cGAs), where individuals in a population only interact with nearest neighbors.
• Processes cooperate over a toroidal mesh topology.• Ensures diversity while maintaining intensification.
Each process has four neighbors.
Processes periodically exchange the best solutions with neighbors.
High-quality solutions diffuse through the population.
Overview >
Presentation Overview
• Background and Review• Review of mrOX GA• Parallelism in the mrOX GA• Status Summary and Future Work
22Status Summary >
Status Summary
• Investigated different ways of using parallelism in the mrOX GA.
• Learned about cellular genetic algorithms (cGAs) as a motivation for parallel cooperation schemes.
• Completed a preliminary software design and began coding.
• Coded and ran an intermediate test application to validate mesh communication pattern.
23Future Work >
Future Work
• Obtain a user account on the Deepthought cluster.
• Continue with design and coding.• Performance testing of parallel
implementation.
24End >
References• Crainic, T.G. and Toulouse, M. Parallel Strategies for Meta-Heuristics. Fleet Management and
Logistics, 205-251. • L. Davis. Applying Adaptive Algorithms to Epistatic Domains. Proceeding of the International
Joint Conference on Artificial Intelligence, 162-164, 1985.• M. Fischetti, J.J. Salazar-Gonzalez, P. Toth. A branch-and-cut algorithm for the symmetric
generalized traveling salesman problem. Operations Research 45 (3): 378–394, 1997.• G. Laporte, A. Asef-Vaziri, C. Sriskandarajah. Some Applications of the Generalized Traveling
Salesman Problem. Journal of the Operational Research Society 47: 1461-1467, 1996.• C.E. Noon. The generalized traveling salesman problem. Ph. D. Dissertation, University of
Michigan, 1988.• C.E. Noon. A Lagrangian based approach for the asymmetric generalized traveling salesman
problem. Operations Research 39 (4): 623-632, 1990.• J.P. Saksena. Mathematical model of scheduling clients through welfare agencies. CORS
Journal 8: 185-200, 1970.• J. Silberholz and B.L. Golden. The Generalized Traveling Salesman Problem: A New Genetic
Algorithm Approach. Operations Research/Computer Science Interfaces Series 37: 165-181, 2007.
• L. Snyder and M. Daskin. A random-key genetic algorithm for the generalized traveling salesman problem. European Journal of Operational Research 17 (1): 38-53, 2006.
25
Acknowledgements
Chris Groer, University of MarylandWilliam Mennell, University of MarylandJohn Silberholz, University of Maryland
26