A Supply Chain-Logistics Super-Network Equilibrium Model for Urban Logistics Facility Network Optimization

<table class="fixed-width table-group" id="tab4"><tr><td><table class="table"><colgroup><col style="width:4.86em"/><col style="width:37.29em"/></colgroup><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr class="thead"><td class="align_left">Notation </td><td class="align_center">Definition </td></tr><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr><td class="align_left"><span style="width: 40.1376ptpx;"><svg height="18.3157pt" id="M87" style="vertical-align:-5.8304pt" version="1.1" viewbox="-0.0498162 -12.4853 40.1376 18.3157" width="40.1376pt" 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="M383 397C383 424 344 448 299 448C244 448 172 409 132 375C66 319 23 227 23 146C23 42 74 -12 146 -12C208 -12 298 30 359 103L343 124C315 95 248 48 192 48C145 48 111 85 111 163C111 228 129 294 151 330C171 363 201 401 241 401C275 401 302 384 325 356C332 347 339 344 348 348C373 360 383 381 383 397Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.282,-5.741)"><path d="M509 90L493 118C465 91 427 70 418 70S409 77 415 103L469 327C494 429 466 451 436 451C412 451 387 444 356 427C307 400 222 342 160 264H158L246 667C253 701 252 710 243 710C228 710 169 680 87 673L83 644H117C155 644 160 640 149 588L24 -4L32 -12C57 -4 86 4 108 9L143 187C198 275 324 386 375 386C394 386 399 367 385 311L331 87C315 22 322 -12 351 -12C387 -12 452 24 509 90Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,4.173,3.784)"><path d="M485 418C485 434 469 451 441 451C389 451 312 387 255 335C221 304 194 278 159 242H157L259 678C264 698 264 710 255 710C242 710 183 681 96 673L90 643L128 642C165 641 171 640 162 602L24 -5L31 -12C51 -4 81 4 110 9L145 181C154 195 180 220 205 242C233 170 263 106 291 53C318 1 335 -12 360 -12C383 -12 425 0 483 82L463 105C437 79 411 57 400 57C388 57 378 68 355 110C329 156 288 244 268 298C299 333 355 376 405 376C426 376 440 373 448 368C453 365 465 368 468 373C477 386 485 405 485 418Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.924,3.784)"><path d="M238 675C244 702 241 710 231 710C220 710 155 682 74 673L70 643H101C147 643 151 635 142 594L39 111C18 11 36 -12 55 -12C89 -12 170 38 223 103L207 129C176 98 130 68 118 68C113 68 108 72 114 102L238 675Z"></path></g><g transform="matrix(.013,0,0,-0.013,14.094,2.0537e-05)"><path d="M378 -385C239 -253 158 0 158 271S238 795 378 927L356 953C165 798 79 550 79 272V271C79 -10 165 -259 356 -411L378 -385Z"></path></g><g transform="matrix(.013,0,0,-0.013,19.874,0)"><path d="M699 368C699 549 574 666 407 666C186 666 23 488 23 277C23 113 129 -3 288 -13L307 -26C431 -111 501 -139 533 -147C559 -154 613 -163 658 -164L666 -141C597 -111 507 -66 430 -11L416 -1C580 42 699 190 699 368ZM601 371C601 227 518 54 381 22L354 40L278 24C175 47 120 145 120 269C120 451 235 631 398 631C540 631 601 521 601 371Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,29.26,-5.741)"><path d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"></path></g><g transform="matrix(.013,0,0,-0.013,34.207,2.0537e-05)"><path d="M364 271V272C364 550 278 798 87 953L65 927C205 795 285 542 285 271S204 -253 65 -385L87 -411C278 -259 364 -10 364 271Z"></path></g></svg></span></td><td class="align_center">transaction cost for customer <i>l</i> buying product<i> h</i> from retailer <i>k</i></td></tr><tr><td class="align_left"><span style="width: 7.83752ptpx;"><svg height="10.2124pt" id="M88" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -6.78297 7.83752 10.2124" width="7.83752pt" 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="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"></path></g></svg></span></td><td class="align_center"><i>lh</i>-dimensional column vector of demand market prices with component <i>lh </i>denoted by <svg height="16.2017pt" id="M89" style="vertical-align:-3.943501pt" version="1.1" viewbox="-0.0498162 -12.2582 13.1493 16.2017" width="13.1493pt" 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="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.71,-5.741)"><path d="M509 90L493 118C465 91 427 70 418 70S409 77 415 103L469 327C494 429 466 451 436 451C412 451 387 444 356 427C307 400 222 342 160 264H158L246 667C253 701 252 710 243 710C228 710 169 680 87 673L83 644H117C155 644 160 640 149 588L24 -4L32 -12C57 -4 86 4 108 9L143 187C198 275 324 386 375 386C394 386 399 367 385 311L331 87C315 22 322 -12 351 -12C387 -12 452 24 509 90Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.384,3.784)"><path d="M238 675C244 702 241 710 231 710C220 710 155 682 74 673L70 643H101C147 643 151 635 142 594L39 111C18 11 36 -12 55 -12C89 -12 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-12C57 -4 86 4 108 9L143 187C198 275 324 386 375 386C394 386 399 367 385 311L331 87C315 22 322 -12 351 -12C387 -12 452 24 509 90Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.46,3.784)"><path d="M238 675C244 702 241 710 231 710C220 710 155 682 74 673L70 643H101C147 643 151 635 142 594L39 111C18 11 36 -12 55 -12C89 -12 170 38 223 103L207 129C176 98 130 68 118 68C113 68 108 72 114 102L238 675Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.951,2.33939e-05)"><path d="M332 -241C217 -138 149 60 149 271C149 483 216 680 332 783L312 808C148 686 74 491 74 272V271C74 50 148 -146 312 -266L332 -241Z"></path></g><g transform="matrix(.013,0,0,-0.013,17.975,0)"><path d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"></path></g><g transform="matrix(.013,0,0,-0.013,25.685,2.33939e-05)"><path d="M311 271V272C311 491 237 686 73 808L53 783C169 680 236 483 236 271C236 60 168 -138 53 -241L73 -266C237 -146 311 50 311 271Z"></path></g></svg></span></td><td class="align_center">demand function of customer<i> l</i> for product <i>h</i></td></tr><tr class="table-tr"><td colspan="2"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Mathematical Problems in Engineering

tab4

Table 4

Table 4: A Supply Chain-Logistics Super-Network Equilibrium Model for Urban Logistics Facility Network Optimization 