Cooperative Strategies for Maximum-Flow Problem in Uncertain Decentralized Systems Using Reliability Analysis

<table class="table-group" id="tab4"><tr><td><table class="table"><tr><td class="thead-hr" colspan="5"><hr/></td></tr><tr class="thead"><td class="align_left"> </td><td class="align_center">Owner</td><td class="align_center">Shapley</td><td class="align_center"><svg height="6.1673pt" id="M127" 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" id="g113-241"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="483" vert-adv-y="483"></glyph.data></g></svg>-value</td><td class="align_center">Core center</td></tr><tr><td class="thead-hr" colspan="5"><hr/></td></tr><tr><td class="align_left" rowspan="4">Coalition</td><td class="align_center"><span style="width: 15.4188ptpx;"><svg height="11.5564pt" id="M128" style="vertical-align:-2.45076pt" version="1.1" viewbox="-0.0498162 -9.10564 15.4188 11.5564" width="15.4188pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,.183)"><path d="M293 -169V-141C218 -131 209 -90 209 -44C209 19 222 85 222 152C222 207 198 255 139 269V273C199 286 222 334 222 388C222 454 209 523 209 588C209 632 218 671 293 681V709C234 709 148 695 148 577C147 513 155 438 155 372C155 337 152 291 64 285V256C152 250 155 204 155 169C156 105 148 31 148 -41C148 -157 234 -169 293 -169Z" id="g113-124"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="347" vert-adv-y="347"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,4.528,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,10.768,.183)"><path d="M283 255V284C195 290 192 337 192 375C192 436 200 508 200 580C200 696 116 709 54 709V681C127 671 139 634 139 588C138 524 125 452 125 387C125 333 149 286 209 272V268C148 253 125 206 125 151C126 87 139 16 139 -47C139 -92 127 -131 54 -141V-169C115 -169 200 -152 200 -41C200 32 192 104 192 164C192 202 195 249 283 255Z" id="g113-126"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="347" vert-adv-y="347"></glyph.data></g></svg></span></td><td class="align_center">40</td><td class="align_center">39.877</td><td class="align_center">39.673</td></tr><tr><td class="align_center"><span style="width: 15.4188ptpx;"><svg height="11.5564pt" id="M129" style="vertical-align:-2.45076pt" version="1.1" viewbox="-0.0498162 -9.10564 15.4188 11.5564" width="15.4188pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,.183)"><path d="M293 -169V-141C218 -131 209 -90 209 -44C209 19 222 85 222 152C222 207 198 255 139 269V273C199 286 222 334 222 388C222 454 209 523 209 588C209 632 218 671 293 681V709C234 709 148 695 148 577C147 513 155 438 155 372C155 337 152 291 64 285V256C152 250 155 204 155 169C156 105 148 31 148 -41C148 -157 234 -169 293 -169Z" id="g113-124"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="347" vert-adv-y="347"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,4.528,0)"><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" id="g113-51"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,10.768,.183)"><path d="M283 255V284C195 290 192 337 192 375C192 436 200 508 200 580C200 696 116 709 54 709V681C127 671 139 634 139 588C138 524 125 452 125 387C125 333 149 286 209 272V268C148 253 125 206 125 151C126 87 139 16 139 -47C139 -92 127 -131 54 -141V-169C115 -169 200 -152 200 -41C200 32 192 104 192 164C192 202 195 249 283 255Z" id="g113-126"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="347" vert-adv-y="347"></glyph.data></g></svg></span></td><td class="align_center">84.5</td><td class="align_center">84.585</td><td class="align_center">84.688</td></tr><tr><td class="align_center"><span style="width: 15.4188ptpx;"><svg height="11.5564pt" id="M130" style="vertical-align:-2.45076pt" version="1.1" viewbox="-0.0498162 -9.10564 15.4188 11.5564" width="15.4188pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,.183)"><path d="M293 -169V-141C218 -131 209 -90 209 -44C209 19 222 85 222 152C222 207 198 255 139 269V273C199 286 222 334 222 388C222 454 209 523 209 588C209 632 218 671 293 681V709C234 709 148 695 148 577C147 513 155 438 155 372C155 337 152 291 64 285V256C152 250 155 204 155 169C156 105 148 31 148 -41C148 -157 234 -169 293 -169Z" id="g113-124"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="347" vert-adv-y="347"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,4.528,0)"><path d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z" id="g113-52"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,10.768,.183)"><path d="M283 255V284C195 290 192 337 192 375C192 436 200 508 200 580C200 696 116 709 54 709V681C127 671 139 634 139 588C138 524 125 452 125 387C125 333 149 286 209 272V268C148 253 125 206 125 151C126 87 139 16 139 -47C139 -92 127 -131 54 -141V-169C115 -169 200 -152 200 -41C200 32 192 104 192 164C192 202 195 249 283 255Z" id="g113-126"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="347" vert-adv-y="347"></glyph.data></g></svg></span></td><td class="align_center">128.5</td><td class="align_center">128.538</td><td class="align_center">128.638</td></tr><tr><td class="align_center">Stable</td><td class="align_center">Yes</td><td class="align_center">Yes</td><td class="align_center">Yes</td></tr><tr class="table-tr"><td colspan="5"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Assigning of the coalition payoff by variant methods.

Mathematical Problems in Engineering

tab4

Table 4

Table 4: Cooperative Strategies for Maximum-Flow Problem in Uncertain Decentralized Systems Using Reliability Analysis 