heuristic_functions.py 45.6 KB
 cyts committed Dec 17, 2020 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 ``````from operator import itemgetter import random import numpy as np import networkx as nx import time from itertools import product from utils import ( graph_cost, graph_with_calc_edge_capacity, create_compl_graph_from_other, ) """ This script generates spanning trees and iterates over the heuristic network optimization algorithms """ def optimal_from_prufer_sequence(input_graph, cost_function): """ Finds global minimum-cost graph from another graph object, by iterating over all possible trees. Based on the method described in the appendices of Heijenen et al 2020: "A method for designing minimum‐cost multisource multisink network layouts" https://doi.org/10.1002/sys.21492 Args: input_graph: NetworkX Graph cost_function: a cost function of the capacity Returns: optimal networkx Graph cost of optimal graph """ graph = input_graph.copy() #Number of nodes n = len(list(graph)) nodes_list = graph.nodes() #Rename nodes as integers dict_lab = dict(zip(range(n), nodes_list)) graph = nx.convert_node_labels_to_integers(graph) cost_initial = np.inf #Generate prufer sequences of all possible trees prufer_sequences = list(product(range(n), repeat = n-2)) #Check every possible tree for a lower cost solution for tree in prufer_sequences: candidate_graph = graph.copy() candidate_graph.remove_edges_from(list(candidate_graph.edges())) candidate_graph.add_edges_from(list(nx.from_prufer_sequence(tree).edges())) #Relabel with original node labels candidate_graph = nx.relabel_nodes(candidate_graph, dict_lab) candidate_graph = graph_with_calc_edge_capacity(candidate_graph) new_cost = graph_cost(cost_function, candidate_graph) if new_cost < cost_initial: incumbent_graph = candidate_graph.copy() cost_initial = new_cost return incumbent_graph, cost_initial def local_search(input_graph, cost_function): """ Heuristic to provide minimum-cost tree from another tree, by creating and breaking cycles in the trees before calculating cost. Steepest-descent algorithm. Not guaranteed to provide global minima solutions. Based on the method described in Brimberg 2003 "An oil pipeline design problem." http://dx.doi.org/10.1287/opre.51.2.228.12786 Args: input_graph: NetworkX Graph cost_function: a cost function of the capacity Returns: optimized Networkx Graph cost of optimized graph """ complete_graph = create_compl_graph_from_other(input_graph) initial_cost = graph_cost(cost_function, input_graph) nodes_i = list(input_graph.nodes()) #This variable serves to stop the while loop if no better solution is found better_found = True candidate_graph = input_graph.copy() while better_found == True: initial_cost = graph_cost(cost_function, candidate_graph) better_found = False #Potential list of cheaper trees: results = [] #Add current best solution results.append([initial_cost, candidate_graph.copy()]) for node_iter_i in nodes_i: #Find all potential neighbours to node_i nodes_ij = list(complete_graph[node_iter_i]) #Find all existing neighbours to node_i existing_neighbours = list(candidate_graph.neighbors(node_iter_i)) #Get candiate nodes that aren't connected to node_i for node_neighbour in existing_neighbours: nodes_ij.remove(node_neighbour) #Create cycle by connecting candidate nodes for node_ij_iter in nodes_ij: candidate_graph2 = candidate_graph.copy() candidate_graph2.add_edge( node_ij_iter, node_iter_i, weight=complete_graph[node_ij_iter][node_iter_i]["weight"], ) edges_cycle = nx.find_cycle(candidate_graph2) try: edges_cycle.remove((node_ij_iter, node_iter_i)) except ValueError: edges_cycle.remove((node_iter_i, node_ij_iter)) #Test removing each edge in the new cycle and calculate graph cost for candidate_edge_remove in edges_cycle: candidate_graph3 = candidate_graph2.copy() candidate_graph3.remove_edge(*candidate_edge_remove) candidate_graph3 = graph_with_calc_edge_capacity(candidate_graph3) new_cost = graph_cost(cost_function, candidate_graph3) if new_cost < initial_cost: results.append([new_cost, candidate_graph3]) #If better solution found: if len(results) > 1: better_found = True results.sort(key=lambda x: x[0]) candidate_graph = results[0][1] #Otherwise exit whle loop (better_found is stil True) and return the incumbent solution else: candidate_graph = results[0][1] return candidate_graph, graph_cost(cost_function, candidate_graph) def delta_change(input_graph, cost_function): """ Heuristic to provide minimum-cost tree from another tree, by creating and breaking cycles in the trees before calculating cost. First-descent algorithm. Not guaranteed to provide global minima solutions. Based on the method described by Rothfarb 1970 "Optimal Design of Offshore Natural-Gas Pipeline Systems" https://doi.org/10.1287/opre.18.6.992 with additions (highlighted here) from Andre 2013 "Design and dimensioning of hydrogen transmission pipeline networks" https://doi.org/10.1016/j.ejor.2013.02.036 Args: input_graph: NetworkX Graph cost_function: a cost function of the capacity Returns: optimized Networkx Graph cost of optimized graph """ complete_graph = create_compl_graph_from_other(input_graph) nodes_i = list(input_graph.nodes()) #Randomisation of node list as indicated in Andre 2013: random.shuffle(nodes_i) #This variable serves to stop the while loop if no better solution is found better_found = True candidate_graph = input_graph.copy() while better_found == True: better_found = False `````` cyts committed Mar 03, 2021 205 `````` initial_cost = graph_cost(cost_function, candidate_graph) `````` cyts committed Dec 17, 2020 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 `````` for node_iter_i in nodes_i: #The following paragraph of code would correspond to an addition of Andre 2013 #In which only the closest nodes to reconnect are considered. #The value of 5 closest nodes is set here. #We choose not to implement this addition #Here the weight property of the graph is simply distance #sorted_dist = [(x[0],x[1]['weight']) for x in dict(complete_graph[node_iter_i]).items()] #sorted_dist.sort(key=itemgetter(1)) #nodes_ij = [x[0] for x in sorted_dist][:5] #Find all potential neighbours to node_i nodes_ij = list(complete_graph[node_iter_i]) #Find all existing neighbours to node_i existing_neighbours = list(candidate_graph.neighbors(node_iter_i)) #Get candiate nodes that aren't connected to node_i for node_neighbour in existing_neighbours: nodes_ij.remove(node_neighbour) #Create cycle by connecting candidate nodes for node_ij_iter in nodes_ij: candidate_graph2 = candidate_graph.copy() candidate_graph2.add_edge( node_ij_iter, node_iter_i, weight=complete_graph[node_ij_iter][node_iter_i]["weight"], ) edges_cycle = nx.find_cycle(candidate_graph2) try: edges_cycle.remove((node_ij_iter, node_iter_i)) except ValueError: edges_cycle.remove((node_iter_i, node_ij_iter)) #Test removing each edge in the new cycle and calculate graph cost for candidate_edge_remove in edges_cycle: candidate_graph3 = candidate_graph2.copy() candidate_graph3.remove_edge(*candidate_edge_remove) candidate_graph3 = graph_with_calc_edge_capacity(candidate_graph3) new_cost = graph_cost(cost_function, candidate_graph3) #As soon as better solution is found break all loops and restart: if new_cost < initial_cost: `````` cyts committed Mar 03, 2021 255 `````` `````` cyts committed Dec 17, 2020 256 257 258 259 `````` candidate_graph = candidate_graph3.copy() initial_cost = new_cost better_found = True break `````` cyts committed Mar 03, 2021 260 `````` `````` cyts committed Dec 17, 2020 261 262 263 264 265 266 267 `````` if better_found == True: break if better_found == True: break return candidate_graph, graph_cost(cost_function, candidate_graph) `````` cyts committed Mar 03, 2021 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 `````` def delta_change_recursive_under(input_graph, cost_function,recursion_level): complete_graph = create_compl_graph_from_other(input_graph) nodes_i = list(input_graph.nodes()) candidate_graph = input_graph.copy() better_found = False initial_cost = graph_cost(cost_function, candidate_graph) for node_iter_i in nodes_i: #Find all potential neighbours to node_i nodes_ij = list(complete_graph[node_iter_i]) #Get candiate nodes that aren't connected to node_i existing_neighbours = list(candidate_graph.neighbors(node_iter_i)) for node_neighbour in existing_neighbours: nodes_ij.remove(node_neighbour) #Create cycle by connecting candidate nodes for node_ij_iter in nodes_ij: candidate_graph2 = candidate_graph.copy() candidate_graph2.add_edge( node_ij_iter, node_iter_i, weight=complete_graph[node_ij_iter][node_iter_i]["weight"], ) edges_cycle = nx.find_cycle(candidate_graph2) try: edges_cycle.remove((node_ij_iter, node_iter_i)) except ValueError: edges_cycle.remove((node_iter_i, node_ij_iter)) #Test removing each edge in the new cycle and calculate graph cost for candidate_edge_remove in edges_cycle: candidate_graph3 = candidate_graph2.copy() candidate_graph3.remove_edge(*candidate_edge_remove) candidate_graph3 = graph_with_calc_edge_capacity(candidate_graph3) new_cost = graph_cost(cost_function, candidate_graph3) #As soon as better solution is found break all loops and restart: if new_cost < initial_cost: candidate_graph = candidate_graph3.copy() initial_cost = new_cost better_found = True break if recursion_level >1: nested_candidate,nested_cost = delta_change_recursive(candidate_graph3, cost_function,recursion_level-1) if nested_cost < initial_cost: candidate_graph = nested_candidate.copy() initial_cost = nested_cost better_found = True break if better_found == True: break if better_found == True: break return candidate_graph, graph_cost(cost_function, candidate_graph) def delta_change_recursive(input_graph, cost_function,recursion_level = 2): `````` cyts committed Dec 17, 2020 346 347 348 349 350 351 `````` """ Heuristic to provide minimum-cost tree from another tree, by creating and breaking cycles in the trees before calculating cost. First-descent algorithm. Not guaranteed to provide global minima solutions. `````` cyts committed Mar 03, 2021 352 `````` This algorithm is a recursive version of a simpler algorithm "delta_change". `````` cyts committed Dec 17, 2020 353 `````` In this algorithm the delta_change is performed, `````` cyts committed Mar 03, 2021 354 355 `````` and on each intermediate candidate tree, a nested delta_change is performed, and so on... allowing jumps into the neigbourhood solution space of the recursion level in one step. `````` cyts committed Dec 17, 2020 356 357 `````` In this implementation we remove the Andre's (2013) additions the delta_change algorithm (and references to them), although including them (notably only taking closest candidate nodes) could reduce calculation time. `````` cyts committed Mar 03, 2021 358 359 360 `````` Note that calcaulation time becomes usually becomes impratical for input recusion levels above 3 and node numbers above 10. `````` cyts committed Dec 17, 2020 361 362 363 364 365 366 367 368 369 370 371 372 `````` Args: input_graph: NetworkX Graph cost_function: a cost function of the capacity Returns: optimized Networkx Graph cost of optimized graph """ complete_graph = create_compl_graph_from_other(input_graph) nodes_i = list(input_graph.nodes()) `````` cyts committed Mar 03, 2021 373 374 375 376 `````` #Randomisation of node list as indicated in Andre 2013: random.shuffle(nodes_i) `````` cyts committed Dec 17, 2020 377 378 379 380 381 382 383 384 385 386 `````` #This variable serves to stop the while loop if no better solution is found better_found = True candidate_graph = input_graph.copy() while better_found == True: better_found = False initial_cost = graph_cost(cost_function, candidate_graph) for node_iter_i in nodes_i: `````` cyts committed Mar 03, 2021 387 `````` #if node_iter `````` cyts committed Dec 17, 2020 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 `````` #Find all potential neighbours to node_i nodes_ij = list(complete_graph[node_iter_i]) #Get candiate nodes that aren't connected to node_i existing_neighbours = list(candidate_graph.neighbors(node_iter_i)) for node_neighbour in existing_neighbours: nodes_ij.remove(node_neighbour) #Create cycle by connecting candidate nodes for node_ij_iter in nodes_ij: candidate_graph2 = candidate_graph.copy() candidate_graph2.add_edge( node_ij_iter, node_iter_i, weight=complete_graph[node_ij_iter][node_iter_i]["weight"], ) edges_cycle = nx.find_cycle(candidate_graph2) try: edges_cycle.remove((node_ij_iter, node_iter_i)) except ValueError: edges_cycle.remove((node_iter_i, node_ij_iter)) #Test removing each edge in the new cycle and calculate graph cost for candidate_edge_remove in edges_cycle: candidate_graph3 = candidate_graph2.copy() candidate_graph3.remove_edge(*candidate_edge_remove) candidate_graph3 = graph_with_calc_edge_capacity(candidate_graph3) new_cost = graph_cost(cost_function, candidate_graph3) #As soon as better solution is found break all loops and restart: if new_cost < initial_cost: candidate_graph = candidate_graph3.copy() initial_cost = new_cost better_found = True break `````` cyts committed Mar 03, 2021 431 `````` if recursion_level >1: `````` cyts committed Dec 17, 2020 432 `````` `````` cyts committed Mar 03, 2021 433 434 435 436 437 438 439 440 `````` nested_candidate,nested_cost = delta_change_recursive_under(candidate_graph3, cost_function,recursion_level-1) if nested_cost < initial_cost: candidate_graph = nested_candidate.copy() initial_cost = nested_cost better_found = True `````` cyts committed Dec 17, 2020 441 `````` break `````` cyts committed Mar 03, 2021 442 `````` `````` cyts committed Dec 17, 2020 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 `````` if better_found == True: break if better_found == True: break return candidate_graph, graph_cost(cost_function, candidate_graph) def edge_turn(input_graph, cost_function): """ Heuristic to provide minimum-cost tree from another tree, by breaking the tree in two, and reconnecting the two connected components elswehere in the trees before calculating cost. Steepest-descent algorithm. Not guaranteed to provide global minima solutions. Based on the edge-turn method described in Heijenen et al 2019: "A method for designing minimum‐cost multisource multisink network layouts" https://doi.org/10.1002/sys.21492 Args: input_graph: NetworkX Graph cost_function: a cost function of the capacity Returns: optimized Networkx Graph cost of optimized graph """ #This variable serves to stop the while loop if no better solution is found better_found = True candidate_graph = input_graph.copy() while better_found == True: better_found = False initial_cost = graph_cost(cost_function, candidate_graph) #Potential list of cheaper trees: results = [] #Add current best solution results.append([initial_cost, candidate_graph.copy()]) #Test removing edges and reconnecting elswehere for edge in candidate_graph.edges: candidate_graph2 = candidate_graph.copy() candidate_graph2.remove_edge(*edge) #For each removed edge, all nodes of either connected component are re-attached to the node in the oher connected compoenent from the removed edge # the following codeblock in then repeated only twice in the for loop: once for each connected component #Iterates over both connected components: for component in list(nx.connected_components(candidate_graph2)): connected = list(component) #Finding the terminal in the removed edge that is not in the connected component considered if edge[0] not in connected: term = edge[0] else: term = edge[1] for comp in connected: candidate_graph_test = candidate_graph2.copy() candidate_graph_test.add_edge(term, comp) candidate_graph_test = graph_with_calc_edge_capacity(candidate_graph_test) new_cost = graph_cost(cost_function, candidate_graph_test) if new_cost < initial_cost: results.append([new_cost, candidate_graph_test]) #If better solution found: if len(results) > 1: better_found = True results.sort(key=lambda x: x[0]) candidate_graph = results[0][1] #Otherwise exit whle loop (check is stil True) and return the incumbent solution else: candidate_graph = results[0][1] input_graph = candidate_graph.copy() return input_graph, graph_cost(cost_function, input_graph) `````` cyts committed Mar 03, 2021 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 `````` def edge_turn_recursive_under(input_graph, cost_function,recursion_level): #This variable serves to stop the while loop if no better solution is found candidate_graph = input_graph.copy() better_found = False initial_cost = graph_cost(cost_function, candidate_graph) #Potential list of cheaper trees: results = [] #Add current best solution results.append([initial_cost, candidate_graph.copy()]) #Iterates over the edges to remove for edge in candidate_graph.edges: candidate_graph2 = candidate_graph.copy() candidate_graph2 = graph_with_calc_edge_capacity(candidate_graph2) candidate_graph2.remove_edge(*edge) #For each removed edge, all nodes of either connected component are re-attached #to the node in the oher connected compoenent from the removed edge. #The following codeblock in then repeated only twice in the for loop: once for each connected component #Iterates over both connected components: for component in list(nx.connected_components(candidate_graph2)): connected = list(component) #Finding the terminal in the removed edge that is not in the connected component considered if edge[0] not in connected: term = edge[0] else: term = edge[1] for comp in connected: candidate_graph_test = candidate_graph2.copy() candidate_graph_test.add_edge(term, comp) candidate_graph_test = graph_with_calc_edge_capacity(candidate_graph_test) new_cost = graph_cost(cost_function, candidate_graph_test) if new_cost < initial_cost: results.append([new_cost, candidate_graph_test]) if recursion_level >1: nested_candidate,nested_cost = edge_turn_recursive_under(candidate_graph_test, cost_function,recursion_level-1) if nested_cost < initial_cost: results.append([nested_cost, nested_candidate]) #If better solution found: if len(results) > 1: better_found = True results.sort(key=lambda x: x[0]) candidate_graph = results[0][1] #Otherwise return the incumbent solution else: candidate_graph = results[0][1] input_graph = candidate_graph.copy() return input_graph, graph_cost(cost_function, input_graph) def edge_turn_recursive(input_graph, cost_function,recursion_level = 2): `````` cyts committed Dec 17, 2020 596 597 598 599 600 601 `````` """ Heuristic to provide minimum-cost tree from another tree, by creating and breaking cycles in the trees before calculating cost. Steepest-descent algorithm. Not guaranteed to provide global minima solutions. `````` cyts committed Mar 03, 2021 602 `````` This algorithm is a recursive version of a simpler algorithm "edge_turn". `````` cyts committed Dec 17, 2020 603 `````` In this algorithm the edge turn algorithm is performed, `````` cyts committed Mar 03, 2021 604 605 `````` and on each intermediate candidate tree, a nested turn algorithm is performed, and so on... allowing jumps into the neigbourhood solution space of the recursion level in one step `````` cyts committed Dec 17, 2020 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 `````` In our implementation, the steepest descent applies to both the initial and to the nested edge turn algorithm, and the chosen best tree is chosen over all the initial and nested edge turns. Args: input_graph: NetworkX Graph cost_function: a cost function of the capacity Returns: optimized Networkx Graph cost of optimized graph """ #This variable serves to stop the while loop if no better solution is found better_found = True candidate_graph = input_graph.copy() while better_found == True: better_found = False initial_cost = graph_cost(cost_function, candidate_graph) #Potential list of cheaper trees: results = [] #Add current best solution results.append([initial_cost, candidate_graph.copy()]) #Iterates over the edges to remove for edge in candidate_graph.edges: candidate_graph2 = candidate_graph.copy() candidate_graph2 = graph_with_calc_edge_capacity(candidate_graph2) candidate_graph2.remove_edge(*edge) #For each removed edge, all nodes of either connected component are re-attached #to the node in the oher connected compoenent from the removed edge. #The following codeblock in then repeated only twice in the for loop: once for each connected component #Iterates over both connected components: for component in list(nx.connected_components(candidate_graph2)): connected = list(component) #Finding the terminal in the removed edge that is not in the connected component considered if edge[0] not in connected: term = edge[0] else: term = edge[1] for comp in connected: candidate_graph_test = candidate_graph2.copy() candidate_graph_test.add_edge(term, comp) candidate_graph_test = graph_with_calc_edge_capacity(candidate_graph_test) new_cost = graph_cost(cost_function, candidate_graph_test) if new_cost < initial_cost: results.append([new_cost, candidate_graph_test]) `````` cyts committed Mar 03, 2021 664 665 `````` if recursion_level >1: nested_candidate,nested_cost = edge_turn_recursive_under(candidate_graph_test, cost_function,recursion_level-1) `````` cyts committed Dec 17, 2020 666 `````` `````` cyts committed Mar 03, 2021 667 668 669 `````` if nested_cost < initial_cost: results.append([nested_cost, nested_candidate]) `````` cyts committed Dec 17, 2020 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 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1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 `````` #If better solution found: if len(results) > 1: better_found = True results.sort(key=lambda x: x[0]) candidate_graph = results[0][1] #Otherwise exit whle loop (better_found is stil False) and return the incumbent solution else: candidate_graph = results[0][1] input_graph = candidate_graph.copy() return input_graph, graph_cost(cost_function, input_graph) def vns(input_graph, cost_function): """ Metaheuristic to provide minimum-cost tree from another tree, by exploring higher-cost solutions as ways out of local minima. Makes jumps to neighbourhoods by creating and breaking cycles a given number of times before performing local_search on the cheapest found tree in the new optimization space. This algorithm is based on the description for VNS given by Brimberg in 2003: "An oil pipeline design problem." http://dx.doi.org/10.1287/opre.51.2.228.12786 The two parameters "k_max" and "time_stopped" are fixed here at values of 5 and 10 times the time required for an initial local_search on the graph, as set by Brimberg. These parameters correspond to the kmax and stopping criterion as described by Brimberg. Args: input_graph: NetworkX Graph cost_function: a cost function of the capacity Returns: optimized Networkx Graph cost of optimized graph """ #Initiate initial solution from local_search start = time.time() candidate_graph, cost_orig = local_search(input_graph, cost_function) end = time.time() local_search_time = end - start #Set stoppping and kmax parameters k_max = 5 time_stopped = 10*local_search_time complete_graph = create_compl_graph_from_other(input_graph) nodes_i = list(candidate_graph.nodes()) #Initiate stopping timer and neighbourhood distance continue_k = False start_full = time.time() #While stopping criterion not reached while time.time() - start_full < time_stopped: #Start k at 1 and increase as long as kmax not reached if continue_k == False: k = 1 if continue_k == True: k = k + 1 candidate_graph2 = candidate_graph.copy() nodes_done = [] #Modify create and break a cycle in initial graph k times for dummy in range(k): works = True #This while loop ensures that the node pairs to connect #are not the same as old ones while works == True: nodes_i_it = nodes_i.copy() node1_rand = random.choice(nodes_i_it) existing_neighbours = list(candidate_graph.neighbors(node1_rand)) for node_neighbour in existing_neighbours: nodes_i_it.remove(node_neighbour) node2_rand = random.choice(nodes_i_it) while ( node2_rand == node1_rand or set([node1_rand, node2_rand]) in nodes_done ): node2_rand = random.choice(nodes_i_it) if node2_rand == node1_rand and [node2_rand] == nodes_i_it: works = False break if works == False: works = True continue #If new node pair is valid, connect them to create a cycle candidate_graph2.add_edge( node1_rand, node2_rand, weight=complete_graph[node1_rand][node2_rand]["weight"], ) edges_cycle = nx.find_cycle(candidate_graph2) #Dont allow the algorithm to remove the newly created edge try: edges_cycle.remove((node1_rand, node2_rand)) except ValueError: edges_cycle.remove((node2_rand, node1_rand)) pot_graphs = [] #Find and keep the cheapest graph from the new cycle for candidate_edge_remove in edges_cycle: candidate_graph3 = candidate_graph2.copy() candidate_graph3.remove_edge(*candidate_edge_remove) candidate_graph3 = graph_with_calc_edge_capacity(candidate_graph3) new_cost = graph_cost(cost_function, candidate_graph3) pot_graphs.append((new_cost, candidate_graph3)) pot_graphs.sort(key=lambda x: x[0]) candidate_graph2 = pot_graphs[0][1] nodes_done.append(set([node1_rand, node2_rand])) works = False #Do local search on the graph moved out of local minimum k times candidate_graph_end, new_cost = local_search(candidate_graph2, cost_function) #If a better solution found set as the incumbent solution and reinitialize k to 1 if new_cost < cost_orig: candidate_graph = candidate_graph_end.copy() continue_k = False cost_orig = new_cost #Else explore further away from incumbent solution if k + 1 >= k_max: continue_k = False return candidate_graph, cost_orig def tabu(input_graph, cost_function): """ Metaheuristic to provide minimum-cost tree from another tree, by exploring higher-cost solutions as ways out of local minima. Makes jumps to different solutions using by creating and breaking cycles, and pursuing potentially higher-cost solutions in hope of finding a lower minimum in the wider optimization space. Once a transformation is made the reverse transformation is not allowed and is added to the Tabu list. This algorithm is based on the description for Tabu search given by Brimberg in 2003: "An oil pipeline design problem." http://dx.doi.org/10.1287/opre.51.2.228.12786 The parameter "stopped" is fixed here at a value of 10. This parameter corresponds to the stopping criterion as described by Brimberg. Args: s_ref: NetworkX Graph func: a cost function of the capacity Returns: optimized Networkx Graph cost of optimized graph . """ start = time.time() #Initiate initial solution from local_search best_graph, best_cost = local_search(input_graph, cost_function) end = time.time() local_time = end - start #Set stopping parameter time_stopped = 10*local_time length_tabu = 7 candidate_graph = best_graph.copy() #Complete graph for edges to add complete_graph = create_compl_graph_from_other(candidate_graph) #Initial Tabu list and stopping timer tabu_list = [] start_full = time.time() while time.time() - start_full < time_stopped: sets_complete = [set(x) for x in complete_graph.edges()] sets_candidate = [set(x) for x in candidate_graph.edges()] #Potential edges to add edges_not_there = [tuple(x) for x in sets_complete if x not in sets_candidate] #Initiate potential solutions list results = [] #Iterate over neighbourhood for edge_test in edges_not_there: candidate_graph2 = candidate_graph.copy() candidate_graph2.add_edge( edge_test[0], edge_test[1], weight=complete_graph[edge_test[0]][edge_test[1]]["weight"],) edges_cycle = nx.find_cycle(candidate_graph2) #Dont allow the algorithm to remove the newly created edge try: edges_cycle.remove((edge_test[0], edge_test[1])) except ValueError: edges_cycle.remove((edge_test[1], edge_test[0])) #Break cycle and add solutions for edge_rem in edges_cycle: #Don't add solutions from Tabu list if (set(edge_test),set(edge_rem)) in tabu_list or (set(edge_rem),set(edge_test)) in tabu_list: continue candidate_graph3 = candidate_graph2.copy() candidate_graph3.remove_edge(*edge_rem) candidate_graph3 = graph_with_calc_edge_capacity(candidate_graph3) new_cost = graph_cost(cost_function, candidate_graph3) results.append([new_cost, candidate_graph3, edge_test, edge_rem]) #Choose current "exploratory" solution as best from neighbourhood results.sort(key=lambda x: x[0]) candidate_graph = results[0][1].copy() #Add the transformation to Tabu list if len(tabu_list) 2] #Iterating over initial high valency nodes for cand in candidates_high_valency: sorted_dist = [ (x[0], x[1]["weight"]) for x in dict(complete_graph[cand[0]]).items() ] #Find closest neighbours to transfer edges to (here closest 4) sorted_dist.sort(key=itemgetter(1)) nodes_ij = [x[0] for x in sorted_dist][:4] #Iterating over nodes to transfer edges to for node_ij in nodes_ij: candidate_graph2 = candidate_graph.copy() #Edges to transfer edges = list(candidate_graph2[cand[0]]) for to_reconnect in edges: if to_reconnect != node_ij: candidate_graph2.add_edge(node_ij, to_reconnect) candidate_graph2.remove_edge(cand[0], to_reconnect) #After edges removal, if node is not connected, reconnect to candidate if len(list(candidate_graph2[cand[0]])) == 0: candidate_graph2.add_edge(nodes_ij[0], cand[0]) #Break cycle (if found) and add solutions try: edges_cycle = nx.find_cycle(candidate_graph2) for candidate_edge_remove in edges_cycle: candidate_graph3 = candidate_graph2.copy() candidate_graph3.remove_edge(*candidate_edge_remove) candidate_graph3 = graph_with_calc_edge_capacity(candidate_graph3) #Local Search is the lower-level heuristic candidate_graph4, new_cost = local_search(candidate_graph3, cost_function) results.append([new_cost, candidate_graph4]) except nx.exception.NetworkXNoCycle: candidate_graph2 = graph_with_calc_edge_capacity(candidate_graph2) candidate_graph3, new_cost = local_search(candidate_graph2, cost_function) results.append([new_cost, candidate_graph3]) results.sort(key=lambda x: x[0]) if results[0][0] < orig_cost: orig_cost = results[0][0] candidate_graph = results[0][1].copy() better_found = True return candidate_graph, orig_cost def high_valency_shuffle_edge_turn(input_graph, cost_function): """ Metaheuristic to provide minimum-cost tree from another tree, by switching high valency nodes with neighbours. After a local minimum is achieved with a lower-level heuristic, high valency nodes are identified and their edes are transferered to a nearby node. Cycles are broken in case any are created. The lower-level heuristic used here is the edge-turn algorithm. This metaheuristic algorithm is original work, currently under review for publication. Args: input_graph: NetworkX Graph cost_function: a cost function of the capacity Returns: optimized Networkx Graph cost of optimized graph . """ #Initiate initial solution from edge_turn candidate_graph, orig_cost = edge_turn(input_graph, cost_function) #Complete graph is used to identify closest nodes for node shuffle complete_graph = create_compl_graph_from_other(input_graph) better_found = True while better_found == True: better_found = False #Initiate results list results = [] results.append([orig_cost, candidate_graph]) #Find high valency (here chosen as <2) nodes candidates = [(x[0], x[1]) for x in list(candidate_graph.degree) if x[1] > 2] #Iterating over initial high valency nodes for cand in candidates: sorted_dist = [ (x[0], x[1]["weight"]) for x in dict(complete_graph[cand[0]]).items() ] #Find closest neighbours to transfer edges to (here closest 4) sorted_dist.sort(key=itemgetter(1)) nodes_ij = [x[0] for x in sorted_dist][:4] #Iterating over nodes to transfer edges to for node_ij in nodes_ij: candidate_graph2 = candidate_graph.copy() #Edges to transfer edges = list(candidate_graph2[cand[0]]) for to_reconnect in edges: if to_reconnect != node_ij: candidate_graph2.add_edge(node_ij, to_reconnect) candidate_graph2.remove_edge(cand[0], to_reconnect) #After edges removal, if node is not connected, reconnect to candidate if len(list(candidate_graph2[cand[0]])) == 0: candidate_graph2.add_edge(nodes_ij[0], cand[0]) #Break cycle (if found) and add solutions try: edges_cycle = nx.find_cycle(candidate_graph2) for candidate_edge_remove in edges_cycle: candidate_graph3 = candidate_graph2.copy() candidate_graph3.remove_edge(*candidate_edge_remove) candidate_graph3 = graph_with_calc_edge_capacity(candidate_graph3) #The Edge turn algorithm is the lower-level heuristic candidate_graph4, new_cost = edge_turn(candidate_graph3, cost_function) results.append([new_cost, candidate_graph4]) except nx.exception.NetworkXNoCycle: candidate_graph2 = graph_with_calc_edge_capacity(candidate_graph2) candidate_graph3, new_cost = edge_turn(candidate_graph2, cost_function) results.append([new_cost, candidate_graph3]) results.sort(key=lambda x: x[0]) if results[0][0] < orig_cost: orig_cost = results[0][0] candidate_graph = results[0][1].copy() better_found = True return candidate_graph, orig_cost def high_valency_shuffle_delta_change(input_graph, cost_function): """ Metaheuristic to provide minimum-cost tree from another tree, by switching high valency nodes with neighbours. After a local minimum is achieved with a lower-level heuristic, high valency nodes are identified and their edes are transferered to a nearby node. Cycles are broken in case any are created. The lower-level heuristic used here is the delta change algorithm. This metaheuristic algorithm is original work, currently under review for publication. Args: input_graph: NetworkX Graph cost_function: a cost function of the capacity Returns: optimized Networkx Graph cost of optimized graph . """ #Initiate initial solution from delta_change candidate_graph, orig_cost = delta_change(input_graph, cost_function) #Complete graph is used to identify closest nodes for node shuffle complete_graph = create_compl_graph_from_other(input_graph) better_found = True while better_found == True: better_found = False #Initiate results list results = [] results.append([orig_cost, candidate_graph]) #Find high valency (here chosen as >2) nodes candidates = [(x[0], x[1]) for x in list(candidate_graph.degree) if x[1] > 2] #Iterating over initial high valency nodes for cand in candidates: sorted_dist = [ (x[0], x[1]["weight"]) for x in dict(complete_graph[cand[0]]).items() ] #Find closest neighbours to transfer edges to (here closest 4) sorted_dist.sort(key=itemgetter(1)) nodes_ij = [x[0] for x in sorted_dist][:4] #Iterating over nodes to transfer edges to for node_ij in nodes_ij: candidate_graph2 = candidate_graph.copy() #Edges to transfer edges = list(candidate_graph2[cand[0]]) for to_reconnect in edges: if to_reconnect != node_ij: candidate_graph2.add_edge(node_ij, to_reconnect) candidate_graph2.remove_edge(cand[0], to_reconnect) #After edges removal, if node is not connected, reconnect to candidate if len(list(candidate_graph2[cand[0]])) == 0: candidate_graph2.add_edge(nodes_ij[0], cand[0]) #Break cycle (if found) and add solutions try: edges_cycle = nx.find_cycle(candidate_graph2) for candidate_edge_remove in edges_cycle: candidate_graph3 = candidate_graph2.copy() candidate_graph3.remove_edge(*candidate_edge_remove) candidate_graph3 = graph_with_calc_edge_capacity(candidate_graph3) #The delta change algorithm is the lower-level heuristic candidate_graph4, new_cost = delta_change(candidate_graph3, cost_function) results.append([new_cost, candidate_graph4]) except nx.exception.NetworkXNoCycle: candidate_graph2 = graph_with_calc_edge_capacity(candidate_graph2) candidate_graph3, new_cost = delta_change(candidate_graph2, cost_function) results.append([new_cost, candidate_graph3]) results.sort(key=lambda x: x[0]) if results[0][0] < orig_cost: orig_cost = results[0][0] candidate_graph = results[0][1].copy() better_found = True return candidate_graph, orig_cost``````