Reference: Ma, YM., Hu, XB. & Zhou, H. A deterministic and nature-inspired algorithm for the fuzzy multi-objective path optimization problem. Complex Intell. Syst. (2022). https://doi.org/10.1007/s40747-022-00825-3
Each arc in the fuzzy multi-objective path optimization problem has multiple crisp and fuzzy weights. The fuzzy weights are represented by triangular or trapezoidal fuzzy numbers. The ripple-spreading algorithm can find all Pareto-optimal paths from one node to all other nodes within a single run.
| Variables | Meaning |
|---|---|
| network | Dictionary, {node1: {node2: [c1, c2, ... (crisp weights)], [f1, f2, ... (fuzzy weights)], ...}, ...} |
| 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 |
| n_cirsp | The number of crisp weights |
| n_fuzzy | The number of fuzzy weights |
| f | f = 3: triangular fuzzy number, f = 4: trapezoidal fuzzy number |
| neighbor | Dictionary, {node1: [the neighbor nodes of node1], ...} |
| v | The ripple-spreading speed (i.e., the minimum length of arcs) |
| t | The simulated time index |
| nr | The number of ripples - 1 |
| epicenter_set | List, the epicenter node of the i-th ripple is epicenter_set[i] |
| path_set | List, the path of the i-th ripple from the source node to node i is path_set[i] |
| radius_set | List, the radius of the i-th ripple is radius_set[i] |
| active_set | List, active_set contains all active ripples |
| objective_set | List, the objective value of the traveling path of the i-th ripple is objective_set[i] |
| Omega | Dictionary, Omega[n] = i denotes that ripple i is generated at node n |
| Arc | Crisp 1 | Crisp 2 | Fuzzy 1 | Fuzzy 2 |
|---|---|---|---|---|
| 0-1 | 8 | 1 | (4, 7, 15) | (12, 19, 20) |
| 0-2 | 4 | 5 | (8, 12, 17) | (6, 14, 15) |
| 0-4 | 9 | 7 | (14, 15, 19) | (2, 19, 20) |
| 1-3 | 2 | 2 | (8, 14, 16) | (2, 19, 20) |
| 1-4 | 6 | 8 | (2, 5, 13) | (2, 10, 12) |
| 2-4 | 5 | 8 | (13, 18, 19) | (5, 9, 13) |
| 2-5 | 8 | 1 | (7, 8, 13) | (2, 10, 11) |
| 3-4 | 6 | 2 | (14, 17, 20) | (7, 11, 20) |
| 3-6 | 4 | 5 | (4, 6, 17) | (2, 12, 16) |
| 4-6 | 6 | 9 | (4, 7, 11) | (7, 10, 20) |
| 4-7 | 4 | 3 | (2, 11, 12) | (17, 19, 20) |
| 4-9 | 9 | 7 | (2, 5, 16) | (6, 10, 20) |
| 5-4 | 9 | 4 | (2, 3, 17) | (2, 11, 20) |
| 5-7 | 4 | 3 | (8, 15, 20) | (5, 11, 19) |
| 6-8 | 6 | 8 | (8, 9, 17) | (5, 6, 11) |
| 6-9 | 4 | 7 | (3, 12, 15) | (6, 12, 17) |
| 7-9 | 9 | 2 | (5, 15, 19) | (11, 14, 19) |
| 7-10 | 1 | 6 | (9, 10, 14) | (7, 9, 12) |
| 8-9 | 3 | 8 | (2, 11, 18) | (5, 12, 19) |
| 8-11 | 7 | 1 | (6, 11, 20) | (4, 9, 19) |
| 9-11 | 9 | 6 | (10, 18, 19) | (2, 3, 7) |
| 10-9 | 5 | 3 | (4, 8, 16) | (9, 13, 19) |
| 10-11 | 9 | 2 | (4, 7, 19) | (5, 13, 16) |
if __name__ == '__main__':
test_network = {0: {1: [[8, 1], [[4, 7, 15], [12, 19, 20]]], 2: [[4, 5], [[8, 12, 17], [6, 14, 15]]],
4: [[9, 7], [[14, 15, 19], [2, 19, 20]]]},
1: {3: [[2, 2], [[8, 14, 16], [2, 19, 20]]], 4: [[6, 8], [[2, 5, 13], [2, 10, 12]]]},
2: {4: [[5, 8], [[13, 18, 19], [5, 9, 13]]], 5: [[8, 1], [[7, 8, 13], [2, 10, 11]]]},
3: {4: [[6, 2], [[14, 17, 20], [7, 11, 20]]], 6: [[4, 5], [[4, 6, 17], [2, 12, 16]]]},
4: {6: [[6, 9], [[4, 7, 11], [7, 10, 20]]], 7: [[4, 3], [[2, 11, 12], [17, 19, 20]]],
9: [[9, 7], [[2, 5, 16], [6, 10, 20]]]},
5: {4: [[9, 4], [[2, 3, 17], [2, 11, 20]]], 7: [[4, 3], [[8, 15, 20], [5, 11, 19]]]},
6: {8: [[6, 8], [[8, 9, 17], [5, 6, 11]]], 9: [[4, 7], [[3, 12, 15], [6, 12, 17]]]},
7: {9: [[9, 2], [[5, 15, 19], [11, 14, 19]]], 10: [[1, 6], [[9, 10, 14], [7, 9, 12]]]},
8: {9: [[3, 8], [[2, 11, 18], [5, 12, 19]]], 11: [[7, 1], [[6, 11, 20], [4, 9, 19]]]},
9: {11: [[9, 6], [[10, 18, 19], [2, 3, 7]]]},
10: {9: [[5, 3], [[4, 8, 16], [9, 13, 19]]], 11: [[9, 2], [[4, 7, 19], [5, 13, 16]]]},
11: {}}
source = 0
result_RSA = main(test_network, source)
print(result_RSA[11]) # all Pareto-optimal paths from the source node to node 11[
{'path': [0, 4, 7, 10, 11], 'objective': [23, 18, [29, 43, 64], [31, 60, 68]]},
{'path': [0, 2, 5, 7, 10, 11], 'objective': [26, 17, [36, 52, 83], [25, 57, 73]]},
{'path': [0, 4, 9, 11], 'objective': [27, 20, [26, 38, 54], [10, 32, 47]]},
{'path': [0, 1, 3, 6, 8, 11], 'objective': [27, 17, [30, 47, 85], [25, 65, 86]]},
{'path': [0, 1, 3, 4, 7, 10, 11], 'objective': [30, 16, [41, 66, 96], [50, 90, 108]]},
{'path': [0, 4, 7, 9, 11], 'objective': [31, 18, [31, 59, 69], [32, 55, 66]]},
{'path': [0, 2, 5, 7, 9, 11], 'objective': [34, 17, [38, 68, 88], [26, 52, 71]]},
{'path': [0, 1, 3, 4, 7, 9, 11], 'objective': [38, 16, [43, 82, 101], [51, 85, 106]]}
]
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