Least Expected Time Paths in Stochastic Schedule-Based Transit Networks

<div>The relationship between nondominated and LET and the fastest <svg height="6.1673pt" id="M478" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 6.20645 6.1673" width="6.20645pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M445 282C445 390 384 448 297 448C266 448 231 437 189 418C96 376 23 279 23 151C23 65 66 -12 169 -12C233 -12 294 18 337 53C411 113 445 201 445 282ZM355 280C355 161 302 62 243 36C231 31 219 30 207 30C147 30 111 78 111 154C111 302 180 380 220 399C231 404 246 407 264 407C311 407 355 367 355 280Z" id="g113-112"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="468" vert-adv-y="468"></glyph.data></g></svg>-<svg height="6.02377pt" id="M479" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -5.81737 5.98461 6.02377" width="5.98461pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M219 86C216 168 211 250 206 337C201 410 189 448 163 448C131 448 79 396 43 344L60 322C91 359 110 375 118 375S132 358 136 298C141 238 152 81 155 -12H182C242 62 331 177 390 258C435 321 451 360 451 391C450 424 432 448 408 448C390 448 372 435 366 419C362 410 362 401 366 394C373 383 376 367 376 350C376 283 262 138 221 86H219Z" id="g185-40"></path><glyph.data ascent="989" descent="-303" horiz-adv-x="447" vert-adv-y="447"></glyph.data></g></svg> paths for departure time <svg height="8.02022pt" id="M480" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -7.81382 4.54925 8.02022" width="4.54925pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z" id="g113-117"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="341" vert-adv-y="341"></glyph.data></g></svg> over the universal scenario set <svg height="9.46864pt" id="M481" style="vertical-align:-0.54566pt" version="1.1" viewbox="-0.0498162 -8.92298 9.12938 9.46864" width="9.12938pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M663 549C663 621 607 680 521 680C283 680 82 444 82 188C82 80 142 -15 258 -15C311 -15 361 -1 408 23C428 -8 451 -38 500 -38C535 -38 584 -16 633 44L618 58C582 23 543 10 516 10C485 10 469 28 454 52C564 133 637 268 637 396C637 502 561 542 497 542C414 542 336 469 336 362C336 331 339 284 371 232L393 242C369 289 366 322 366 347C366 448 430 506 494 506C539 506 586 470 586 396C586 266 508 157 469 118C458 106 446 97 435 86C418 115 396 142 349 142C313 142 284 123 241 84L254 67C283 90 305 96 330 96C359 96 375 77 390 53C350 29 307 15 256 15C179 15 142 78 142 150C142 331 347 645 517 645C590 645 624 603 643 547L663 549Z" id="g198-18"></path><glyph.data ascent="687" descent="-226" horiz-adv-x="692" vert-adv-y="692"></glyph.data></g></svg> and its subset <svg height="8.68572pt" id="M482" style="vertical-align:-0.0498209pt" version="1.1" viewbox="-0.0498162 -8.6359 9.97754 8.68572" width="9.97754pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M719 146L691 147C666 77 657 72 612 72H528C572 125 688 224 688 364C688 546 558 665 384 665C212 665 77 541 77 361C77 221 188 124 239 72H150C110 72 97 80 73 147L46 146L63 0H307L314 18C284 71 178 203 178 360C178 529 256 631 383 631C509 631 588 523 588 363C588 197 485 67 453 18L458 0H701L719 146Z" id="g113-153"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="757" vert-adv-y="757"></glyph.data></g></svg>.</div>

Mathematical Problems in Engineering

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Figure 3

Figure 3: Least Expected Time Paths in Stochastic Schedule-Based Transit Networks 