Abstract
We address a family of hard benchmark instances for the Simple Plant Location Problem (also known as the Uncapacitated Facility Location Problem). The recent attempt by Fischetti et al. Manag Sci 63(7): 2146–2162 (2017) to tackle the Körkel–Ghosh instances resulted in seven new optimal solutions and 22 improved upper bounds. We use automated generation of heuristics to obtain a new algorithm for the Simple Plant Location Problem. In our experiments, our new algorithm matched all the previous best known and optimal solutions, and further improved 12 upper bounds, all within shorter time budgets compared to the previous efforts.
Our algorithm design process is split into two phases: (1) development of algorithmic components such as local search procedures and mutation operators, and (2) composition of a metaheuristic from the available components. Phase (1) requires human expertise and often can be completed by implementing several simple domain-specific routines known from the literature. Phase (2) is entirely automated by employing the Conditional Markov Chain Search (CMCS) framework. In CMCS, a metaheuristic is flexibly defined by a set of parameters, called configuration. Then the process of composition of a metaheuristic from the algorithmic components is reduced to an optimisation problem seeking the best performing CMCS configuration.
We discuss the problem of comparing configurations, and propose a new efficient technique to select the best performing configuration from a large set. To employ this method, we restrict the original CMCS to a simple deterministic case that leaves us with a finite and manageable number of meaningful configurations.
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Appendices
Appendix 1: Optimal Solutions for Instances Solved to Optimality
Instance | Obj. v. | Opened sites |
|---|---|---|
ga250a-3 | 257953 | 22 35 39 46 57 66 76 86 97 100 105 112 114 116 121 124 126 127 144 154 155 176 192 196 200 207 211 219 223 227 229 237 246 249 |
ga250a-5 | 258190 | 13 17 29 35 40 43 49 55 60 63 79 82 110 126 135 139 150 157 161 174 178 179 198 201 204 208 211 230 232 241 248 |
ga500c-5 | 621313 | 4 75 183 259 360 491 |
gs500c-3 | 621204 | 98 195 216 245 333 429 |
gs500c-5 | 623180 | 22 51 276 355 439 444 |
ga250a-1 | 257957 | 21 32 38 47 53 56 58 84 94 100 101 103 111 129 136 139 144 146 149 150 168 170 175 178 203 204 219 224 234 238 239 250 |
ga250a-2 | 257502 | 11 37 55 62 64 84 88 99 100 103 107 114 115 116 118 132 146 157 158 160 171 191 200 211 213 217 218 221 237 238 240 250 |
ga250a-4 | 257987 | 4 5 7 30 31 37 53 55 69 73 74 75 84 92 93 103 108 115 119 127 129 153 163 168 173 174 188 199 200 203 208 213 219 236 250 |
ga250b-1 | 276296 | 10 56 60 94 106 129 149 150 170 203 219 250 |
ga250b-2 | 275141 | 37 55 88 103 135 141 158 191 211 213 231 |
ga250b-3 | 276093 | 1 18 22 35 39 50 97 192 200 229 246 |
ga250b-4 | 276332 | 5 7 36 56 77 92 124 160 228 236 250 |
Instance | Obj. v. | Opened sites |
|---|---|---|
ga250b-5 | 276404 | 40 57 79 110 157 161 183 184 208 211 241 246 |
ga250c-1 | 334135 | 100 154 175 231 |
ga250c-2 | 330728 | 45 55 88 99 |
ga250c-3 | 333662 | 22 97 127 138 |
ga250c-4 | 332423 | 5 124 143 188 |
ga250c-5 | 333538 | 74 110 157 247 |
ga500c-1 | 621360 | 127 269 378 403 430 |
ga500c-2 | 621464 | 28 107 212 315 344 456 |
ga500c-3 | 621428 | 68 187 314 326 370 474 |
ga500c-4 | 621754 | 56 97 307 350 436 |
gs250a-1 | 257964 | 4 10 12 25 27 30 47 51 63 71 119 123 126 132 137 143 145 155 161 163 169 176 177 178 203 214 232 234 236 238 245 246 249 |
gs250a-2 | 257573 | 9 24 25 40 43 46 52 74 77 86 87 88 95 96 98 100 101 113 114 120 130 139 154 160 161 165 166 184 191 241 245 250 |
gs250a-3 | 257626 | 9 20 33 34 37 38 55 60 67 69 71 72 91 110 120 121 132 139 144 148 166 172 174 177 187 189 190 199 204 209 223 229 234 |
gs250a-4 | 257961 | 3 20 31 36 46 54 101 102 104 115 118 128 139 143 144 159 160 163 168 179 188 193 195 207 208 217 221 226 233 237 |
gs250a-5 | 257896 | 18 33 36 47 49 60 76 77 89 98 104 114 118 122 124 133 137 156 161 168 172 189 204 207 209 212 213 217 223 227 228 230 235 250 |
gs250b-1 | 276761 | 8 27 47 63 71 113 137 145 170 178 229 232 |
gs250b-2 | 275675 | 25 43 52 69 77 87 120 139 149 160 221 245 |
gs250b-3 | 275710 | 32 55 57 60 67 69 82 166 172 174 210 |
gs250b-4 | 276114 | 13 19 31 35 97 106 139 144 157 177 191 247 |
gs250b-5 | 275916 | 18 36 118 122 124 137 166 172 177 204 209 230 |
gs250c-1 | 332935 | 63 170 176 232 |
gs250c-2 | 334630 | 25 52 83 144 |
gs250c-3 | 333000 | 57 60 69 166 |
gs250c-4 | 333158 | 52 144 157 191 |
gs250c-5 | 334635 | 18 84 114 186 |
gs500c-1 | 620041 | 29 102 112 242 440 |
gs500c-2 | 620434 | 70 286 424 439 495 |
gs500c-4 | 620437 | 96 247 283 316 390 |
Appendix 2: Best Known Solutions for Instances Not Yet Solved to Optimality
The objective values of these solutions can be found in Table 2, column ‘Our best’.
Instance | Opened sites |
|---|---|
ga500a-1 | 22 28 52 59 65 70 73 79 86 90 100 103 111 126 142 152 156 173 177 189 199 205 208 219 221 234 245 246 260 265 269 275 280 299 301 303 313 375 378 397 410 419 430 463 475 486 490 494 |
ga500a-2 | 34 51 54 70 84 116 120 122 126 144 155 158 169 177 188 193 195 204 212 213 218 222 223 238 255 289 313 315 321 333 338 345 360 372 391 397 399 401 413 415 437 450 466 470 476 478 485 487 500 |
ga500a-3 | 12 33 36 37 44 49 55 65 75 85 88 92 94 96 114 132 145 148 150 166 178 184 185 187 192 196 201 220 241 252 257 278 285 288 290 303 340 364 367 370 375 378 386 393 431 451 474 |
ga500a-4 | 18 19 29 35 39 42 43 49 56 65 74 78 102 119 138 140 144 155 188 197 204 214 267 273 280 281 282 293 317 329 340 346 350 360 364 371 377 388 404 411 417 419 430 436 448 456 466 484 496 |
ga500a-5 | 4 11 14 22 34 36 38 40 47 51 55 95 100 120 123 125 127 133 155 174 181 183 199 216 229 283 284 301 316 321 326 328 332 336 348 369 371 380 382 387 390 397 399 424 429 487 488 491 497 |
ga500b-1 | 34 100 127 153 156 176 184 189 199 236 277 375 378 379 410 430 470 |
ga500b-2 | 28 34 51 137 212 213 225 238 241 245 249 268 315 336 338 344 459 478 |
ga500b-3 | 36 66 92 94 166 185 187 189 241 290 300 303 326 340 370 393 431 474 |
ga500b-4 | 24 138 204 282 293 329 330 343 360 388 396 436 448 451 456 466 484 496 |
ga500b-5 | 2 14 55 123 124 135 142 147 181 183 231 258 259 349 360 382 399 414 |
gs500a-1 | 6 22 42 53 55 58 94 95 115 116 120 121 126 127 129 144 149 154 164 171 173 212 239 252 270 285 294 300 320 321 327 335 336 343 352 377 379 384 385 389 399 420 426 429 434 442 464 490 |
gs500a-2 | 9 50 53 62 70 99 109 110 115 124 168 169 175 185 198 202 204 218 229 233 241 247 269 276 289 290 294 295 301 316 333 335 336 356 358 376 383 394 400 422 426 437 439 453 457 459 460 463 464 470 |
gs500a-3 | 7 17 28 38 41 55 65 67 74 84 86 110 117 147 152 153 162 173 212 219 223 244 256 259 269 271 273 287 300 301 308 310 365 369 371 377 381 385 394 401 413 417 421 437 453 456 493 494 |
gs500a-4 | 9 10 14 18 56 67 68 84 87 93 95 123 124 136 137 161 165 173 180 189 194 196 202 217 229 231 258 273 277 281 290 350 356 359 363 371 378 380 390 391 435 438 453 458 464 484 490 491 495 |
gs500a-5 | 3 4 7 13 15 30 40 47 60 69 78 86 122 123 136 153 156 159 160 171 174 185 192 231 233 234 235 250 251 257 268 281 304 312 316 322 331 338 384 391 411 424 431 444 459 460 481 498 |
gs500b-1 | 22 29 45 82 102 112 116 193 215 257 258 313 385 410 440 468 470 471 |
gs500b-2 | 24 70 85 95 115 168 233 247 329 356 358 382 383 408 437 439 457 488 |
gs500b-3 | 7 41 116 216 235 245 255 269 273 279 287 299 308 333 347 371 392 429 |
gs500b-4 | 76 91 103 120 132 142 165 171 173 212 230 351 380 406 437 438 447 491 |
Instance | Opened sites |
|---|---|
gs500b-5 | 3 7 13 40 65 105 153 185 226 235 319 322 372 384 431 444 460 486 |
ga750a-1 | 2 18 35 50 52 67 69 71 74 105 110 111 117 118 127 152 155 209 219 233 234 242 263 280 296 305 309 316 330 335 346 381 430 431 435 446 449 457 478 484 494 512 538 540 548 559 564 587 616 640 644 647 650 667 680 689 711 713 716 729 738 745 |
ga750a-2 | 1 18 40 45 71 102 104 109 110 114 120 126 131 144 150 154 168 170 180 183 211 214 227 234 235 237 239 259 283 288 289 296 301 316 351 352 357 359 362 369 375 382 387 436 447 455 491 511 528 530 539 550 581 582 615 642 644 645 673 684 710 722 |
ga750a-3 | 49 68 71 75 88 95 101 109 113 127 183 202 206 212 254 266 295 298 334 339 349 356 361 379 389 400 415 419 428 433 436 439 446 447 450 464 465 483 506 560 561 565 570 575 580 581 585 590 606 626 627 645 657 666 669 679 682 694 698 731 |
ga750a-4 | 25 54 79 87 99 100 104 108 112 149 154 160 163 169 176 195 224 226 258 266 271 279 291 292 303 305 319 324 326 363 386 400 413 416 418 420 437 454 468 487 496 534 536 551 561 568 628 635 636 652 656 669 672 684 693 695 703 718 733 737 748 |
ga750a-5 | 18 34 35 67 75 77 80 87 93 117 119 146 161 167 168 187 189 195 224 228 235 246 247 271 315 320 325 329 359 365 367 373 389 411 421 424 426 429 452 456 475 476 507 522 523 524 532 553 562 565 578 588 655 678 702 706 709 713 734 736 747 749 |
ga750b-1 | 58 67 71 100 117 184 214 335 346 386 478 484 512 559 589 593 616 647 662 711 720 745 746 |
ga750b-2 | 1 45 109 110 144 168 182 214 235 237 239 283 288 296 308 329 375 505 637 644 645 712 |
ga750b-3 | 53 68 101 202 206 215 254 298 334 356 404 408 464 560 570 575 596 604 669 673 726 728 |
ga750b-4 | 47 52 55 104 115 128 149 154 202 232 266 303 358 434 468 551 635 639 672 704 733 |
ga750b-5 | 29 42 49 87 99 182 193 194 218 224 351 363 373 376 380 456 473 523 667 685 700 746 |
ga750c-1 | 214 418 476 587 593 644 711 |
ga750c-2 | 1 170 182 235 237 564 590 |
ga750c-3 | 68 101 173 215 439 548 616 |
ga750c-4 | 128 144 154 279 413 456 704 |
ga750c-5 | 14 344 376 456 577 659 685 |
gs750a-1 | 2 20 52 64 75 89 93 124 128 130 143 147 177 199 205 218 232 236 237 240 264 279 281 289 305 317 320 335 336 374 393 403 404 425 453 458 466 468 482 487 496 518 538 541 545 554 555 564 602 609 637 651 658 659 663 669 672 681 682 706 713 729 743 |
gs750a-2 | 1 19 24 62 64 70 92 93 94 100 108 113 134 137 146 157 178 180 186 213 225 265 268 278 281 325 334 336 341 348 362 393 397 398 411 444 451 460 462 493 494 498 504 554 574 575 594 595 620 625 628 636 639 646 650 661 721 724 735 |
Instance | Opened sites |
|---|---|
gs750a-3 | 6 22 35 38 46 66 96 110 117 119 133 135 171 182 192 230 250 287 288 291 317 354 357 360 367 374 385 392 396 409 412 425 437 450 458 463 465 487 499 510 528 554 561 562 583 596 609 612 624 640 677 688 702 710 715 721 728 732 746 |
gs750a-4 | 2 4 24 26 34 56 78 80 81 95 107 115 117 123 136 139 145 146 151 172 174 190 241 243 258 266 269 294 305 323 332 346 377 388 399 412 434 443 459 473 477 498 500 522 530 536 537 551 558 573 603 617 640 641 656 666 669 673 721 722 740 749 |
gs750a-5 | 3 12 31 46 54 65 74 88 91 96 102 104 112 114 119 135 150 182 198 202 207 231 234 288 302 317 356 359 386 394 397 410 411 417 421 425 483 579 582 587 607 612 638 644 654 662 663 669 672 680 693 707 716 720 721 725 731 744 |
gs750b-1 | 41 45 67 75 128 130 213 232 237 242 279 281 313 428 487 538 573 658 698 699 725 743 |
gs750b-2 | 1 10 108 126 191 213 214 257 265 314 336 341 348 367 444 450 494 508 617 636 639 650 661 |
gs750b-3 | 22 26 64 135 171 192 281 291 304 317 446 462 484 553 561 583 702 706 715 727 728 |
gs750b-4 | 4 59 81 95 107 113 165 174 179 190 238 248 261 294 305 355 431 459 616 641 654 721 |
gs750b-5 | 3 31 74 90 104 200 281 302 359 392 394 427 487 540 549 568 593 627 635 693 707 709 |
gs750c-1 | 67 112 128 428 479 603 639 |
gs750c-2 | 29 92 108 265 336 494 639 |
gs750c-3 | 6 44 304 583 624 680 698 |
gs750c-4 | 26 139 151 287 522 628 721 |
gs750c-5 | 104 198 302 557 607 635 707 |
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Karapetyan, D., Goldengorin, B. (2018). Conditional Markov Chain Search for the Simple Plant Location Problem Improves Upper Bounds on Twelve Körkel–Ghosh Instances. In: Goldengorin, B. (eds) Optimization Problems in Graph Theory. Springer Optimization and Its Applications, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-319-94830-0_7
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