A Modified Hybrid Genetic Algorithm for Solving Nonlinear Optimal Control Problems

<table class="table-group" id="tab9"><tr><td><table class="table"><tr><td class="thead-hr" colspan="5"><hr/></td></tr><tr class="thead"><td align="left">Method</td><td align="center">Mean rank</td><td align="center">Sum of ranks</td><td align="center">Test statistics</td><td align="center">Value</td></tr><tr><td class="thead-hr" colspan="5"><hr/></td></tr><tr><td align="left">LI</td><td align="center">13.54</td><td align="center">176.0</td><td align="center">Mann-Whitney <svg height="9.21094pt" id="M758" style="vertical-align:-0.2129908pt" version="1.1" viewbox="-0.0498162 -8.99795 9.98608 9.21094" width="9.98608pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M715 650H476L470 622C550 619 562 610 562 570C562 536 553 481 535 384L523 321C485 121 408 37 283 37C178 37 99 109 134 297L178 532C193 612 199 616 278 622L284 650H29L23 622C105 616 109 609 94 532L48 291C23 160 51 83 98 40C143 -1 202 -16 260 -16C321 -16 387 3 439 49C509 111 542 204 563 312L577 384C595 479 608 533 619 569C629 603 637 620 709 622L715 650Z" id="g113-86"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="727" vert-adv-y="727"></glyph.data></g></svg></td><td align="center">84</td></tr><tr><td align="left">SI</td><td align="center">13.46</td><td align="center">175.0</td><td align="center">Wilcoxon <svg height="9.21094pt" id="M759" style="vertical-align:-0.2129908pt" version="1.1" viewbox="-0.0498162 -8.99795 13.3451 9.21094" width="13.3451pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M962 650H739L732 622L760 619C812 613 819 604 798 552C760 457 671 267 606 131H604L511 638H480L237 131H233L183 554C177 607 183 614 226 619L248 622L257 650H24L17 622C88 615 92 611 103 524L173 -11H203L450 491H453L543 -11H575L839 529C882 609 886 613 953 622L962 650Z" id="g113-88"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="974" vert-adv-y="974"></glyph.data></g></svg></td><td align="center">175</td></tr><tr><td align="left">—</td><td align="center">—</td><td align="center">—</td><td align="center"><span style="width: 9.51013ptpx;"><svg height="9.38774pt" id="M760" style="vertical-align:-0.04981995pt" version="1.1" viewbox="-0.0498162 -9.33792 9.51013 9.38774" width="9.51013pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M669 636L663 650H260C230 650 221 652 208 675H186C177 621 162 548 144 481L173 479C195 534 211 560 226 578C248 604 266 615 373 615H544C460 511 114 112 23 16L31 0H561C577 34 611 135 622 170L594 182C566 125 544 88 515 65C481 38 416 35 319 35C249 35 197 37 152 41C270 184 542 501 669 636Z" id="g113-91"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="692" vert-adv-y="692"></glyph.data></g></svg></span></td><td align="center"><span style="width: 37.2928ptpx;"><svg height="9.06134pt" 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Results of Mann-Whitney test on relative errors of the pair (LI, SI) for <svg height="14.758pt" id="M757" style="vertical-align:-5.760051pt" version="1.1" viewbox="-0.0498162 -8.99795 16.6404 14.758" width="16.6404pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M743 650H503L496 622L527 618C563 613 564 603 532 573C449 495 371 431 323 392C301 374 272 355 246 346L280 522C297 609 300 614 379 622L385 650H135L129 622C209 614 215 609 198 522L124 133C106 39 99 35 23 28L17 0H271L277 28C193 35 192 39 208 133L239 316C264 328 280 325 303 288C368 183 435 90 502 0H652L659 28C602 34 584 43 543 94C495 154 403 283 347 369L574 554C634 603 659 612 735 624L743 650Z" id="g113-76"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="755" vert-adv-y="755"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,9.619,3.264)"><path d="M659 416L649 451C600 438 564 414 546 396C527 377 518 354 506 308C495 265 474 200 456 152C437 102 400 45 326 34L424 510L415 514L351 494L266 44C213 55 182 94 182 174C182 200 187 243 191 277C194 304 196 331 196 357C196 415 178 451 138 451C108 451 66 431 24 386L32 355C49 370 66 377 81 377C91 377 110 370 110 328C110 300 107 269 104 240C101 214 99 188 99 162C99 37 166 -5 257 -13L220 -202C215 -228 229 -251 236 -257L274 -218L318 -11C349 -4 378 8 400 21C444 48 489 89 518 157L586 327C601 369 619 393 659 416Z" id="g50-245"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="671" vert-adv-y="671"></glyph.data></g></svg>.

Mathematical Problems in Engineering

tab9

Table 9

Table 9: A Modified Hybrid Genetic Algorithm for Solving Nonlinear Optimal Control Problems 