The Label Correcting Algorithm for the Multi-Objective Shortest Path Problem

Reference: Paixão J M, Santos J L. Labeling methods for the general case of the multi-objective shortest path problem–a computational study[M]//Computational Intelligence and Decision Making. Springer, Dordrecht, 2013: 489-502.

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, ...], ...}, ...}
s_network	The network described by a crisp weight on which we conduct the ripple relay race
source	The source node
destination	The destination node
nn	The number of nodes
nw	The number of objectives
neighbor	Dictionary, {node1: [the neighbor nodes of node1], ...}
ind	The index of each label added to the priority queue
ni	The index of each label that is assessed as the Pareto-optimal subpath
omega	List, omega[n] contains all the indexes of labels that are assessed as the Pareto-optimal subpath from the source node to node n (Pareto-optimal labels)
obj_list	List, the objective value of the each Pareto-optimal label
path_list	List, the path of each Pareto-optimal label
queue	The priority queue, which outputs the label that has the minimum value of the summation of objectives at each iteration

Example

The red number associated with each arc is the first weight, and the green number is the second weight.

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))

Output:

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