Heuristics for the Buffer Allocation Problem with Collision Probability Using Computer Simulation

<div>(a) The relationship between <svg height="10.9007pt" id="M340" style="vertical-align:-3.16517pt" version="1.1" viewbox="-0.0498162 -7.73553 18.9077 10.9007" width="18.9077pt" 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="M329 433H203L239 587L230 596L147 534L123 433H57L30 395L34 388H115L61 129C37 16 59 -12 85 -12C147 -12 222 58 260 98L241 125C212 95 160 62 144 62C132 62 127 71 138 126L192 386L305 394L329 433Z" id="g50-117"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="347" vert-adv-y="347"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,4.446,3.001)"><path d="M303 37L293 66C280 59 256 49 231 49S173 63 173 143V396H281C294 404 298 428 287 439H173V575L158 577L92 509V439H46L17 408L22 396H92V107C92 27 128 -12 192 -12C203 -12 218 -9 237 2L303 37Z" id="g58-114"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="312" vert-adv-y="312"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,7.34,3.001)"><path d="M442 39L431 68C420 63 406 57 394 57C377 57 359 71 359 118V301C359 353 349 393 314 424C291 443 261 451 230 451C171 438 107 401 79 381C59 366 46 355 46 340C46 316 72 295 91 295C106 295 117 303 122 319C131 350 139 371 152 384C163 397 178 404 197 404C246 404 281 365 281 293V278C260 262 186 232 124 213C69 194 41 161 41 112C41 47 92 -12 163 -12C195 -12 243 25 283 51C288 35 293 22 308 7C319 -4 339 -12 354 -12L442 39ZM281 87C262 68 226 51 199 51C169 51 129 74 129 126C129 162 151 181 190 198C211 208 259 229 281 240V87Z" id="g58-95"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="449" vert-adv-y="449"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,11.398,3.001)"><path d="M396 114C349 71 317 59 274 59C216 59 123 104 123 241C123 342 180 400 246 400C283 400 317 386 348 359C355 352 361 348 366 348C378 348 401 371 402 394C402 405 398 413 385 424C369 437 337 451 294 451H293C254 451 193 434 141 394C73 342 39 277 39 198C39 90 114 -12 243 -12C303 -12 371 31 415 90L396 114Z" id="g58-97"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="432" vert-adv-y="432"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,15.403,3.001)"><path d="M303 37L293 66C280 59 256 49 231 49S173 63 173 143V396H281C294 404 298 428 287 439H173V575L158 577L92 509V439H46L17 408L22 396H92V107C92 27 128 -12 192 -12C203 -12 218 -9 237 2L303 37Z" id="g58-114"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="312" vert-adv-y="312"></glyph.data></g></svg> and <svg height="8.64656pt" id="M341" style="vertical-align:-0.04981995pt" version="1.1" viewbox="-0.0498162 -8.59674 8.03326 8.64656" width="8.03326pt" 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="M598 511C598 627 497 650 401 650H141L131 618C219 611 222 606 210 538L130 119C117 43 108 39 25 32L18 0H236C324 0 396 5 456 30C531 59 585 119 585 193C585 285 520 331 428 351V353C529 373 598 423 598 511ZM502 508C502 418 431 368 330 368H263L298 559C303 584 308 595 317 600C329 608 345 612 369 612C428 612 502 595 502 508ZM483 197C483 95 404 39 309 39C225 39 206 58 219 139L256 330H317C395 330 483 301 483 197Z" id="g50-67"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="622" vert-adv-y="622"></glyph.data></g></svg> when <svg height="8.45736pt" id="M342" style="vertical-align:-0.2129202pt" version="1.1" viewbox="-0.0498162 -8.24444 29.3722 8.45736" width="29.3722pt" 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="M558 106L533 129C501 84 476 67 464 67C446 67 435 116 414 239C457 297 508 367 550 440L539 451L484 438C458 390 429 338 404 300H402C375 407 351 451 285 451C168 451 24 312 24 153C24 56 66 -12 132 -12C206 -12 284 64 345 156H347C367 24 386 -12 422 -12C457 -12 503 7 558 106ZM338 206C272 99 215 56 173 56C140 56 116 96 116 170C116 303 194 405 255 405C304 405 324 299 338 206Z" id="g50-223"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="582" vert-adv-y="582"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,11.242,0)"><path d="M556 331V384H55V331H556ZM556 140V193H55V140H556Z" id="g54-34"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,22.861,0)"><path d="M245 635C92 635 37 457 37 312C37 148 92 -12 244 -12C395 -12 449 165 449 312C449 469 394 635 245 635ZM243 598C332 598 358 454 358 312C358 172 334 26 245 26C158 26 128 174 128 313S151 598 243 598Z" id="g58-46"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg>. (b) The relationship between <svg height="10.9007pt" id="M343" style="vertical-align:-3.16517pt" version="1.1" viewbox="-0.0498162 -7.73553 18.9077 10.9007" width="18.9077pt" 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="M329 433H203L239 587L230 596L147 534L123 433H57L30 395L34 388H115L61 129C37 16 59 -12 85 -12C147 -12 222 58 260 98L241 125C212 95 160 62 144 62C132 62 127 71 138 126L192 386L305 394L329 433Z" id="g50-117"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="347" vert-adv-y="347"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,4.446,3.001)"><path d="M303 37L293 66C280 59 256 49 231 49S173 63 173 143V396H281C294 404 298 428 287 439H173V575L158 577L92 509V439H46L17 408L22 396H92V107C92 27 128 -12 192 -12C203 -12 218 -9 237 2L303 37Z" id="g58-114"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="312" vert-adv-y="312"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,7.34,3.001)"><path d="M442 39L431 68C420 63 406 57 394 57C377 57 359 71 359 118V301C359 353 349 393 314 424C291 443 261 451 230 451C171 438 107 401 79 381C59 366 46 355 46 340C46 316 72 295 91 295C106 295 117 303 122 319C131 350 139 371 152 384C163 397 178 404 197 404C246 404 281 365 281 293V278C260 262 186 232 124 213C69 194 41 161 41 112C41 47 92 -12 163 -12C195 -12 243 25 283 51C288 35 293 22 308 7C319 -4 339 -12 354 -12L442 39ZM281 87C262 68 226 51 199 51C169 51 129 74 129 126C129 162 151 181 190 198C211 208 259 229 281 240V87Z" id="g58-95"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="449" vert-adv-y="449"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,11.398,3.001)"><path d="M396 114C349 71 317 59 274 59C216 59 123 104 123 241C123 342 180 400 246 400C283 400 317 386 348 359C355 352 361 348 366 348C378 348 401 371 402 394C402 405 398 413 385 424C369 437 337 451 294 451H293C254 451 193 434 141 394C73 342 39 277 39 198C39 90 114 -12 243 -12C303 -12 371 31 415 90L396 114Z" id="g58-97"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="432" vert-adv-y="432"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,15.403,3.001)"><path d="M303 37L293 66C280 59 256 49 231 49S173 63 173 143V396H281C294 404 298 428 287 439H173V575L158 577L92 509V439H46L17 408L22 396H92V107C92 27 128 -12 192 -12C203 -12 218 -9 237 2L303 37Z" id="g58-114"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="312" vert-adv-y="312"></glyph.data></g></svg> and the average makespan when <svg height="8.45736pt" id="M344" style="vertical-align:-0.2129202pt" version="1.1" viewbox="-0.0498162 -8.24444 29.3722 8.45736" width="29.3722pt" 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="M558 106L533 129C501 84 476 67 464 67C446 67 435 116 414 239C457 297 508 367 550 440L539 451L484 438C458 390 429 338 404 300H402C375 407 351 451 285 451C168 451 24 312 24 153C24 56 66 -12 132 -12C206 -12 284 64 345 156H347C367 24 386 -12 422 -12C457 -12 503 7 558 106ZM338 206C272 99 215 56 173 56C140 56 116 96 116 170C116 303 194 405 255 405C304 405 324 299 338 206Z" id="g50-223"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="582" vert-adv-y="582"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,11.242,0)"><path d="M556 331V384H55V331H556ZM556 140V193H55V140H556Z" id="g54-34"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,22.861,0)"><path d="M245 635C92 635 37 457 37 312C37 148 92 -12 244 -12C395 -12 449 165 449 312C449 469 394 635 245 635ZM243 598C332 598 358 454 358 312C358 172 334 26 245 26C158 26 128 174 128 313S151 598 243 598Z" id="g58-46"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g></svg>. Each marker corresponds to the result obtained by the heuristics.</div>

Mathematical Problems in Engineering

fig6

Figure 6

Figure 6: Heuristics for the Buffer Allocation Problem with Collision Probability Using Computer Simulation 