Managing a Dual-Channel Supply Chain with Fairness and Channel Preference

<table class="table-group" id="tab1"><tr><td><table class="table"><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: 17.5807ptpx;"><svg height="11.927pt" id="M16" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 17.5807 11.927" width="17.5807pt" 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="M703 650H36V622C118 616 126 607 126 526V124C126 45 117 34 40 28V0H297V28C219 34 211 44 211 124V581C211 609 214 613 245 613H494C521 613 527 610 527 583V124C527 45 519 34 434 28V0H700V28C620 34 612 44 612 124V526C612 605 620 616 703 622V650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.594,3.132)"><path d="M781 91L765 118C733 86 702 70 693 70C686 70 685 77 692 106L738 297C772 439 736 451 712 451S675 445 647 431C605 410 526 356 452 257H450L459 294C491 426 459 451 429 451C403 451 386 445 361 431C312 404 239 343 170 253H168L188 333C208 411 205 451 175 451C148 451 85 417 24 346L39 318C57 338 100 374 112 374C120 374 122 372 115 339C91 226 62 112 28 -6L37 -12C59 -4 87 3 114 6C125 68 142 127 154 168C184 227 316 383 372 383C397 383 396 366 381 289C362 192 338 92 312 -6L317 -12C339 -5 367 3 396 6L434 170C468 234 598 383 650 383C668 383 676 369 664 319L609 90C593 23 600 -12 629 -12C654 -12 722 23 781 91Z"></path></g></svg></span></td><td class="align_center">Manufacturer’s profit</td></tr><tr><td class="align_left"><span style="width: 14.128ptpx;"><svg height="11.927pt" id="M17" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 14.128 11.927" width="14.128pt" 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="M703 650H36V622C118 616 126 607 126 526V124C126 45 117 34 40 28V0H297V28C219 34 211 44 211 124V581C211 609 214 613 245 613H494C521 613 527 610 527 583V124C527 45 519 34 434 28V0H700V28C620 34 612 44 612 124V526C612 605 620 616 703 622V650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.594,3.132)"><path d="M397 380C406 395 404 411 396 425S369 451 350 451C302 451 239 372 192 294H189L199 338C214 407 207 451 180 451C152 451 83 405 30 345L48 318C87 354 117 372 125 372S135 362 127 324L55 -5L61 -12C87 -5 117 3 139 5C154 87 168 162 179 207C198 250 240 310 260 332C281 355 297 366 307 366C321 366 333 360 347 346C351 342 360 342 369 348C379 355 389 366 397 380Z"></path></g></svg></span></td><td class="align_center">Retailer’s profit</td></tr><tr><td class="align_left"><span style="width: 16.589ptpx;"><svg height="11.927pt" id="M18" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 16.589 11.927" width="16.589pt" 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="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"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.606,3.132)"><path d="M781 91L765 118C733 86 702 70 693 70C686 70 685 77 692 106L738 297C772 439 736 451 712 451S675 445 647 431C605 410 526 356 452 257H450L459 294C491 426 459 451 429 451C403 451 386 445 361 431C312 404 239 343 170 253H168L188 333C208 411 205 451 175 451C148 451 85 417 24 346L39 318C57 338 100 374 112 374C120 374 122 372 115 339C91 226 62 112 28 -6L37 -12C59 -4 87 3 114 6C125 68 142 127 154 168C184 227 316 383 372 383C397 383 396 366 381 289C362 192 338 92 312 -6L317 -12C339 -5 367 3 396 6L434 170C468 234 598 383 650 383C668 383 676 369 664 319L609 90C593 23 600 -12 629 -12C654 -12 722 23 781 91Z"></path></g></svg></span></td><td class="align_center">Manufacturer’s utility</td></tr><tr><td class="align_left"><span style="width: 13.1363ptpx;"><svg height="11.927pt" id="M19" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 13.1363 11.927" width="13.1363pt" 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="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"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.606,3.132)"><path d="M397 380C406 395 404 411 396 425S369 451 350 451C302 451 239 372 192 294H189L199 338C214 407 207 451 180 451C152 451 83 405 30 345L48 318C87 354 117 372 125 372S135 362 127 324L55 -5L61 -12C87 -5 117 3 139 5C154 87 168 162 179 207C198 250 240 310 260 332C281 355 297 366 307 366C321 366 333 360 347 346C351 342 360 342 369 348C379 355 389 366 397 380Z"></path></g></svg></span></td><td class="align_center">Retailer’s utility</td></tr><tr><td class="align_left"><span style="width: 9.39034ptpx;"><svg height="6.1673pt" id="M20" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 9.39034 6.1673" width="9.39034pt" 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="M689 332C689 394 670 448 646 448C620 448 597 421 597 396C597 386 600 381 608 372C619 359 620 334 620 315C620 150 538 45 454 45C414 45 386 67 386 122C386 138 388 158 394 180L457 426L452 432L377 416L315 156C302 100 259 45 216 45C176 45 148 67 148 122C148 133 152 158 156 180C162 212 173 259 194 332C201 357 206 384 206 405C206 430 198 448 174 448C125 448 66 406 23 342L43 319C84 368 110 383 121 383C126 383 128 382 128 377C128 370 127 359 122 343C99 268 84 204 77 156C74 137 70 111 70 104C70 25 125 -12 180 -12C228 -12 276 12 319 50C338 8 378 -12 418 -12C549 -12 689 166 689 332Z"></path></g></svg></span></td><td class="align_center">Wholesale price</td></tr><tr><td class="align_left"><span style="width: 11.864ptpx;"><svg height="10.2124pt" id="M21" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -6.78297 11.864 10.2124" width="11.864pt" 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,-0.0091,7.384,3.132)"><path d="M398 365C398 411 356 451 297 451C250 451 198 430 151 397C65 336 24 230 24 139C24 14 97 -12 148 -12C198 -12 284 7 373 104L355 130C306 84 246 53 198 53C132 53 111 109 111 163V189C209 211 398 261 398 365ZM317 351C317 308 273 262 117 225C134 338 189 383 221 399C230 404 249 407 263 407C295 407 317 389 317 351Z"></path></g></svg></span></td><td class="align_center">Online channel price</td></tr><tr><td class="align_left"><span style="width: 11.9097ptpx;"><svg height="10.2124pt" id="M22" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -6.78297 11.9097 10.2124" width="11.9097pt" 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,-0.0091,7.384,3.132)"><path d="M397 380C406 395 404 411 396 425S369 451 350 451C302 451 239 372 192 294H189L199 338C214 407 207 451 180 451C152 451 83 405 30 345L48 318C87 354 117 372 125 372S135 362 127 324L55 -5L61 -12C87 -5 117 3 139 5C154 87 168 162 179 207C198 250 240 310 260 332C281 355 297 366 307 366C321 366 333 360 347 346C351 342 360 342 369 348C379 355 389 366 397 380Z"></path></g></svg></span></td><td class="align_center">Offline channel price</td></tr><tr><td class="align_left"><i>c</i></td><td class="align_center">Manufacturer’s cost</td></tr><tr><td class="align_left"><span style="width: 6.59789ptpx;"><svg height="9.49473pt" id="M23" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 6.59789 9.49473" width="6.59789pt" 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="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z"></path></g></svg></span></td><td class="align_center">Consumer’s online channel preference</td></tr><tr><td class="align_left"><i>a</i></td><td class="align_center">Total market demand</td></tr><tr><td class="align_left"><i>b</i></td><td class="align_center">Price elasticity</td></tr><tr><td class="align_left"><span style="width: 7.30254ptpx;"><svg height="9.49473pt" id="M24" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 7.30254 9.49473" width="7.30254pt" 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="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z"></path></g></svg></span></td><td class="align_center">Cross-price elasticity</td></tr><tr><td class="align_left"><span style="width: 7.51131ptpx;"><svg height="6.1673pt" id="M25" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 7.51131 6.1673" width="7.51131pt" 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="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"></path></g></svg></span></td><td class="align_center">Coefficient of retailer fairness preference</td></tr><tr class="table-tr"><td colspan="2"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Mathematical Problems in Engineering

tab1

Table 1

Table 1: Managing a Dual-Channel Supply Chain with Fairness and Channel Preference 