The Improved Combination Rule of D Numbers and Its Application in Radiation Source Identification

<table class="fixed-width table-group" id="tab2"><tr><td><table class="table"><colgroup><col style="width:7.53em"/><col style="width:5.79em"/><col style="width:9.94em"/></colgroup><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td class="align_left"><span style="width: 26.8887ptpx;"><svg height="11.5564pt" id="M158" style="vertical-align:-2.45076pt" version="1.1" viewbox="-0.0498162 -9.10564 26.8887 11.5564" width="26.8887pt" 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="M405 650H141L135 622C222 616 230 610 215 535L133 116C118 41 113 33 29 28L23 0H289L295 28C209 33 205 40 219 116L298 535C312 609 317 616 399 622L405 650Z" id="g113-74"></path></g><g transform="matrix(.013,0,0,-0.013,7.746,.183)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path></g><g transform="matrix(.013,0,0,-0.013,12.261,0)"><path d="M743 375C743 570 607 650 407 650H136L130 622C214 612 220 606 204 526L125 122C110 43 105 36 24 27L17 0H250C382 0 482 23 571 76C680 141 743 248 743 375ZM644 379C644 188 520 36 305 36C277 36 249 38 229 46C205 55 202 71 212 126L293 553C300 588 306 600 317 606C330 613 356 617 390 617C547 617 644 537 644 379Z" id="g113-69"></path></g><g transform="matrix(.013,0,0,-0.013,22.219,.183)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path></g></svg></span></td><td class="align_center">D numbers</td><td class="align_center">Improved D numbers</td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td class="align_left"><span style="width: 81.3787ptpx;"><svg height="14.5316pt" id="M159" style="vertical-align:-3.9384pt" version="1.1" viewbox="-0.0498162 -10.5932 81.3787 14.5316" width="81.3787pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,2.33939e-05)"><path d="M308 -259V-228H299C192 -219 190 -215 190 -109V616C190 755 193 759 299 768H308V799H127V-259H308Z" id="g119-1"></path></g><g transform="matrix(.013,0,0,-0.013,4.854,0)"><path d="M743 375C743 570 607 650 407 650H136L130 622C214 612 220 606 204 526L125 122C110 43 105 36 24 27L17 0H250C382 0 482 23 571 76C680 141 743 248 743 375ZM644 379C644 188 520 36 305 36C277 36 249 38 229 46C205 55 202 71 212 126L293 553C300 588 306 600 317 606C330 613 356 617 390 617C547 617 644 537 644 379Z" id="g113-69"></path></g><g transform="matrix(.0091,0,0,-0.0091,14.747,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path></g><g transform="matrix(.013,0,0,-0.013,22.599,0)"><path d="M573 255C573 407 448 530 294 530S15 407 15 255S140 -20 294 -20S573 103 573 255ZM520 280H323V484C426 471 508 386 520 280ZM265 280H68C80 386 162 471 265 484V280ZM265 26C162 39 80 124 68 230H265V26ZM520 230C508 124 426 39 323 26V230H520Z" id="g117-68"></path></g><g transform="matrix(.013,0,0,-0.013,33.136,0)"><path d="M743 375C743 570 607 650 407 650H136L130 622C214 612 220 606 204 526L125 122C110 43 105 36 24 27L17 0H250C382 0 482 23 571 76C680 141 743 248 743 375ZM644 379C644 188 520 36 305 36C277 36 249 38 229 46C205 55 202 71 212 126L293 553C300 588 306 600 317 606C330 613 356 617 390 617C547 617 644 537 644 379Z" id="g113-69"></path></g><g transform="matrix(.0091,0,0,-0.0091,43.029,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g><g transform="matrix(.013,0,0,-0.013,47.975,2.33939e-05)"><path d="M244 -259V799H63V768H72C178 759 181 755 181 616V-109C181 -215 179 -219 72 -228H63V-259H244Z" id="g119-2"></path></g><g transform="matrix(.013,0,0,-0.013,55.837,0)"><path d="M573 255C573 407 448 530 294 530S15 407 15 255S140 -20 294 -20S573 103 573 255ZM520 280H323V484C426 471 508 386 520 280ZM265 280H68C80 386 162 471 265 484V280ZM265 26C162 39 80 124 68 230H265V26ZM520 230C508 124 426 39 323 26V230H520Z" id="g117-68"></path></g><g transform="matrix(.013,0,0,-0.013,66.374,0)"><path d="M743 375C743 570 607 650 407 650H136L130 622C214 612 220 606 204 526L125 122C110 43 105 36 24 27L17 0H250C382 0 482 23 571 76C680 141 743 248 743 375ZM644 379C644 188 520 36 305 36C277 36 249 38 229 46C205 55 202 71 212 126L293 553C300 588 306 600 317 606C330 613 356 617 390 617C547 617 644 537 644 379Z" id="g113-69"></path></g><g transform="matrix(.0091,0,0,-0.0091,76.267,3.132)"><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" id="g50-52"></path></g></svg></span></td><td class="align_center">0.1719</td><td class="align_center">0.4756</td></tr><tr><td class="align_left"><span style="width: 81.3787ptpx;"><svg height="14.5316pt" id="M160" style="vertical-align:-3.9384pt" version="1.1" viewbox="-0.0498162 -10.5932 81.3787 14.5316" width="81.3787pt" 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="M743 375C743 570 607 650 407 650H136L130 622C214 612 220 606 204 526L125 122C110 43 105 36 24 27L17 0H250C382 0 482 23 571 76C680 141 743 248 743 375ZM644 379C644 188 520 36 305 36C277 36 249 38 229 46C205 55 202 71 212 126L293 553C300 588 306 600 317 606C330 613 356 617 390 617C547 617 644 537 644 379Z" id="g113-69"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.893,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path></g><g transform="matrix(.013,0,0,-0.013,17.745,0)"><path d="M573 255C573 407 448 530 294 530S15 407 15 255S140 -20 294 -20S573 103 573 255ZM520 280H323V484C426 471 508 386 520 280ZM265 280H68C80 386 162 471 265 484V280ZM265 26C162 39 80 124 68 230H265V26ZM520 230C508 124 426 39 323 26V230H520Z" id="g117-68"></path></g><g transform="matrix(.013,0,0,-0.013,28.281,2.33939e-05)"><path d="M308 -259V-228H299C192 -219 190 -215 190 -109V616C190 755 193 759 299 768H308V799H127V-259H308Z" id="g119-1"></path></g><g transform="matrix(.013,0,0,-0.013,33.136,0)"><path d="M743 375C743 570 607 650 407 650H136L130 622C214 612 220 606 204 526L125 122C110 43 105 36 24 27L17 0H250C382 0 482 23 571 76C680 141 743 248 743 375ZM644 379C644 188 520 36 305 36C277 36 249 38 229 46C205 55 202 71 212 126L293 553C300 588 306 600 317 606C330 613 356 617 390 617C547 617 644 537 644 379Z" id="g113-69"></path></g><g transform="matrix(.0091,0,0,-0.0091,43.029,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g><g transform="matrix(.013,0,0,-0.013,50.88,0)"><path d="M573 255C573 407 448 530 294 530S15 407 15 255S140 -20 294 -20S573 103 573 255ZM520 280H323V484C426 471 508 386 520 280ZM265 280H68C80 386 162 471 265 484V280ZM265 26C162 39 80 124 68 230H265V26ZM520 230C508 124 426 39 323 26V230H520Z" id="g117-68"></path></g><g transform="matrix(.013,0,0,-0.013,61.417,0)"><path d="M743 375C743 570 607 650 407 650H136L130 622C214 612 220 606 204 526L125 122C110 43 105 36 24 27L17 0H250C382 0 482 23 571 76C680 141 743 248 743 375ZM644 379C644 188 520 36 305 36C277 36 249 38 229 46C205 55 202 71 212 126L293 553C300 588 306 600 317 606C330 613 356 617 390 617C547 617 644 537 644 379Z" id="g113-69"></path></g><g transform="matrix(.0091,0,0,-0.0091,71.31,3.132)"><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" id="g50-52"></path></g><g transform="matrix(.013,0,0,-0.013,76.256,2.33939e-05)"><path d="M244 -259V799H63V768H72C178 759 181 755 181 616V-109C181 -215 179 -219 72 -228H63V-259H244Z" id="g119-2"></path></g></svg></span></td><td class="align_center">0.2267</td><td class="align_center">0.4756</td></tr><tr><td class="align_left"><span style="width: 81.3787ptpx;"><svg height="14.5316pt" id="M161" style="vertical-align:-3.9384pt" version="1.1" viewbox="-0.0498162 -10.5932 81.3787 14.5316" width="81.3787pt" 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="M743 375C743 570 607 650 407 650H136L130 622C214 612 220 606 204 526L125 122C110 43 105 36 24 27L17 0H250C382 0 482 23 571 76C680 141 743 248 743 375ZM644 379C644 188 520 36 305 36C277 36 249 38 229 46C205 55 202 71 212 126L293 553C300 588 306 600 317 606C330 613 356 617 390 617C547 617 644 537 644 379Z" id="g113-69"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.893,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" id="g50-51"></path></g><g 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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" id="g50-52"></path></g><g transform="matrix(.013,0,0,-0.013,76.256,2.33939e-05)"><path d="M244 -259V799H63V768H72C178 759 181 755 181 616V-109C181 -215 179 -219 72 -228H63V-259H244Z" id="g119-2"></path></g></svg></span></td><td class="align_center">0.2302</td><td class="align_center">0.4756</td></tr><tr class="table-tr"><td colspan="3"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Combined D number’s integration representation.</div>

Mathematical Problems in Engineering

tab2

Table 2

Table 2: The Improved Combination Rule of D Numbers and Its Application in Radiation Source Identification 