Complexity Analysis of Pricing in a Multichannel Supply Chain with Spillovers from Online to Offline Sales

<div>Stable region for adjustment speeds and <span class="nowrap"><svg height="6.1673pt" id="M177" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 6.40217 6.1673" width="6.40217pt" 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="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z"></path></g></svg>.</span> (a) 3D stable region for <span class="nowrap"><svg height="9.25202pt" id="M178" style="vertical-align:-3.29111pt" version="1.1" viewbox="-0.0498162 -5.96091 13.4821 9.25202" width="13.4821pt" 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="M484 406C484 429 465 448 438 448C400 448 352 405 302 356L156 213L191 373C195 391 197 404 197 416C197 436 191 448 178 448C151 448 83 406 23 350L37 328C67 350 101 369 109 369C114 369 115 362 111 344L29 -5L34 -12C63 -5 86 -1 111 3C124 73 137 138 147 161C165 180 189 202 206 217C215 194 249 111 275 57C304 -1 319 -12 342 -12C368 -12 405 -5 471 92L454 115C427 80 395 55 384 55C374 55 364 58 341 106L262 274C282 296 367 378 414 378C427 378 440 370 444 367C449 363 454 362 462 368C476 379 484 395 484 406Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.266,3.132)"><path d="M698 334C698 397 680 451 655 451C627 451 602 424 602 397C602 389 605 383 614 371C622 360 624 340 624 317C624 161 546 47 458 47C419 47 389 69 389 125C389 142 391 162 397 185L457 430L451 435L380 421L315 164C301 107 266 47 222 47C182 47 153 69 153 126C153 137 157 162 161 185C166 214 179 265 197 334C203 358 208 385 208 407C208 433 201 451 176 451C124 451 64 404 24 348L44 320C86 366 113 382 121 382C126 382 128 381 128 374C128 368 126 356 122 341C102 266 88 204 81 164C77 143 73 121 73 107C73 23 131 -12 187 -12C234 -12 281 10 322 45C343 5 383 -12 422 -12C559 -12 698 174 698 334Z"></path></g></svg>,</span> <span class="nowrap"><svg height="11.4351pt" id="M179" style="vertical-align:-5.47419pt" version="1.1" viewbox="-0.0498162 -5.96091 12.3403 11.4351" width="12.3403pt" 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="M484 406C484 429 465 448 438 448C400 448 352 405 302 356L156 213L191 373C195 391 197 404 197 416C197 436 191 448 178 448C151 448 83 406 23 350L37 328C67 350 101 369 109 369C114 369 115 362 111 344L29 -5L34 -12C63 -5 86 -1 111 3C124 73 137 138 147 161C165 180 189 202 206 217C215 194 249 111 275 57C304 -1 319 -12 342 -12C368 -12 405 -5 471 92L454 115C427 80 395 55 384 55C374 55 364 58 341 106L262 274C282 296 367 378 414 378C427 378 440 370 444 367C449 363 454 362 462 368C476 379 484 395 484 406Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.266,3.132)"><path d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z"></path></g></svg>,</span> and <span class="nowrap"><svg height="6.1673pt" id="M180" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 6.40217 6.1673" width="6.40217pt" 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="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z"></path></g></svg>,</span> when <span class="nowrap"><svg height="9.49473pt" id="M181" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 37.271 9.49473" width="37.271pt" 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="M494 514C482 587 419 712 303 712C238 712 174 667 174 603C174 561 205 514 249 449C219 438 187 422 162 407C93 366 23 283 23 177C23 69 87 -12 190 -12C244 -12 288 5 328 33C406 87 444 170 444 249C444 329 404 391 331 475C265 550 222 605 222 627C222 647 238 657 267 657C355 657 421 585 484 499L494 514ZM359 234C359 143 319 30 219 30C172 30 114 75 114 178C114 275 163 343 195 378C212 397 241 415 269 425C305 382 359 313 359 234Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.354,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,21.617,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"></path></g><g transform="matrix(.013,0,0,-0.013,27.857,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"></path></g><g transform="matrix(.013,0,0,-0.013,30.821,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"></path></g></svg>.</span> (b) 2D stable region for <svg height="9.25202pt" id="M182" style="vertical-align:-3.29111pt" version="1.1" viewbox="-0.0498162 -5.96091 13.4821 9.25202" width="13.4821pt" 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="M484 406C484 429 465 448 438 448C400 448 352 405 302 356L156 213L191 373C195 391 197 404 197 416C197 436 191 448 178 448C151 448 83 406 23 350L37 328C67 350 101 369 109 369C114 369 115 362 111 344L29 -5L34 -12C63 -5 86 -1 111 3C124 73 137 138 147 161C165 180 189 202 206 217C215 194 249 111 275 57C304 -1 319 -12 342 -12C368 -12 405 -5 471 92L454 115C427 80 395 55 384 55C374 55 364 58 341 106L262 274C282 296 367 378 414 378C427 378 440 370 444 367C449 363 454 362 462 368C476 379 484 395 484 406Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.266,3.132)"><path d="M698 334C698 397 680 451 655 451C627 451 602 424 602 397C602 389 605 383 614 371C622 360 624 340 624 317C624 161 546 47 458 47C419 47 389 69 389 125C389 142 391 162 397 185L457 430L451 435L380 421L315 164C301 107 266 47 222 47C182 47 153 69 153 126C153 137 157 162 161 185C166 214 179 265 197 334C203 358 208 385 208 407C208 433 201 451 176 451C124 451 64 404 24 348L44 320C86 366 113 382 121 382C126 382 128 381 128 374C128 368 126 356 122 341C102 266 88 204 81 164C77 143 73 121 73 107C73 23 131 -12 187 -12C234 -12 281 10 322 45C343 5 383 -12 422 -12C559 -12 698 174 698 334Z"></path></g></svg> and <span class="nowrap"><svg height="11.4351pt" id="M183" style="vertical-align:-5.47419pt" version="1.1" viewbox="-0.0498162 -5.96091 12.3403 11.4351" width="12.3403pt" 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="M484 406C484 429 465 448 438 448C400 448 352 405 302 356L156 213L191 373C195 391 197 404 197 416C197 436 191 448 178 448C151 448 83 406 23 350L37 328C67 350 101 369 109 369C114 369 115 362 111 344L29 -5L34 -12C63 -5 86 -1 111 3C124 73 137 138 147 161C165 180 189 202 206 217C215 194 249 111 275 57C304 -1 319 -12 342 -12C368 -12 405 -5 471 92L454 115C427 80 395 55 384 55C374 55 364 58 341 106L262 274C282 296 367 378 414 378C427 378 440 370 444 367C449 363 454 362 462 368C476 379 484 395 484 406Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.266,3.132)"><path d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z"></path></g></svg>,</span> when <span class="nowrap"><svg height="9.49473pt" id="M184" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 37.271 9.49473" width="37.271pt" 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="M494 514C482 587 419 712 303 712C238 712 174 667 174 603C174 561 205 514 249 449C219 438 187 422 162 407C93 366 23 283 23 177C23 69 87 -12 190 -12C244 -12 288 5 328 33C406 87 444 170 444 249C444 329 404 391 331 475C265 550 222 605 222 627C222 647 238 657 267 657C355 657 421 585 484 499L494 514ZM359 234C359 143 319 30 219 30C172 30 114 75 114 178C114 275 163 343 195 378C212 397 241 415 269 425C305 382 359 313 359 234Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.354,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,21.617,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"></path></g><g transform="matrix(.013,0,0,-0.013,27.857,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 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Complexity

fig1

Figure 1

Figure 1: Complexity Analysis of Pricing in a Multichannel Supply Chain with Spillovers from Online to Offline Sales 