The MOEA/D for the Multi-Objective Shortest Path Problem

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Reference: Zhang Q, Li H. MOEA/D: A multiobjective evolutionary algorithm based on decomposition[J]. IEEE Transactions on Evolutionary Computation, 2007, 11(6): 712-731.

Reference: Ahn C W, Ramakrishna R S. A genetic algorithm for shortest path routing problem and the sizing of populations[J]. IEEE Transactions on Evolutionary Computation, 2002, 6(6): 566-579.

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The multi-objective aims to find a set of paths with minimized costs.

Variables	Meaning
network	Dictionary, {node 1: {node 2: [weight 1, weight 2, ...], ...}, ...}
source	The source node
destination	The destination node
h	The uniformly distributed number on each objective
lambda_list	List, the uniformly distributed vectors
gen	The maximum number of generations (iterations)
pop	Population size = len(lambda_list)
neighbor_size	Neighbor size
B	List, B[i] stores the indexes of the neighbor_size nearest neighbors of population[i]
p_mutation	The probability of mutation
neighbor	List, [[the neighbor nodes of node 1], [the neighbor nodes of node 2], ...]
nw	The number of objectives
population	List, all individuals ({'pareto rank': the Pareto rank (integer), 'chromosome': the chromosome (path), 'objective': the objective value on each objective (list)})
z	List, reference point, i.e., the best ever value of each objective
EP	List, external population to store non-dominated solutions
ep_path	List, a set to record all non-dominated paths that have been found

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Example

if __name__ == '__main__':
    test_network = {
        0: {1: [62, 50], 2: [44, 90], 3: [67, 10]},
        1: {0: [62, 50], 2: [33, 25], 4: [52, 90]},
        2: {0: [44, 90], 1: [33, 25], 3: [32, 10], 4: [52, 40]},
        3: {0: [67, 10], 2: [32, 10], 4: [54, 100]},
        4: {1: [52, 90], 2: [52, 40], 3: [54, 100]},
    }
    source_node = 0
    destination_node = 4
    print(main(test_network, source_node, destination_node, 20))

Output:

[
    {'path': [0, 3, 4], 'objective': [121, 110]}, 
    {'path': [0, 2, 4], 'objective': [96, 130]}, 
    {'path': [0, 3, 2, 4], 'objective': [151, 60]},
]