Entropy-Based Weighting for Multiobjective Optimization: An Application on Vertical Turning

<table class="table-group" id="tab4"><tr><td><table class="table"><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td align="left">Life of cutting tool</td><td align="center">Productivity </td><td align="center">Cost</td></tr><tr class="thead"><td align="left">(mm)</td><td align="center">(part/h)</td><td align="center">(US$/part)</td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td align="left"><span style="width: 35.2513ptpx;"><svg height="11.6856pt" id="M65" style="vertical-align:-1.5831pt" version="1.1" viewbox="-0.0498162 -10.1025 35.2513 11.6856" width="35.2513pt" 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="M306 382C339 403 362 417 379 433C398 450 411 473 411 500C411 578 352 635 248 635H247C181 635 134 604 106 573L56 506L82 482C117 529 151 557 201 557C246 557 287 531 287 466C287 404 228 364 143 336L150 302C165 306 186 309 204 309C258 309 324 275 324 179C324 85 285 44 235 44C175 44 128 88 106 108C92 118 80 116 69 106C56 97 39 79 39 61C39 47 46 34 63 21C83 5 123 -12 174 -12C236 -12 454 58 454 222C454 297 403 362 306 380V382Z" id="g13-49"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="500" vert-adv-y="500"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.5,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z" id="g190-45"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="228" vert-adv-y="228"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,9.475,0)"><path d="M409 0V34C329 39 322 56 322 125V635C234 611 160 593 94 583V554L145 551C186 547 190 541 190 488V125C190 54 181 39 102 34V0H409Z" id="g13-47"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="500" vert-adv-y="500"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,15.975,0)"><path d="M486 174V231H396V629H324C229 512 123 368 15 215V174H269V116C269 55 264 41 179 34V0H477V34C403 41 396 51 396 112V174H486ZM269 231H90C155 335 219 427 267 493H269V231Z" id="g13-50"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="500" vert-adv-y="500"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,22.475,0)"><path d="M252 635C85 635 28 460 28 311C28 167 82 -12 250 -12C419 -12 472 166 472 312C472 457 419 635 252 635ZM248 594C314 594 335 455 335 314C335 174 318 29 252 29S165 180 165 315S181 594 248 594Z" id="g13-46"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="500" vert-adv-y="500"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,29.073,-5.741)"><path d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 290 288C243 319 222 338 189 380C167 381 142 376 125 366C125 347 133 322 145 304C198 296 225 288 275 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vert-adv-y="500"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.5,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z" id="g190-45"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="228" vert-adv-y="228"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,9.475,0)"><path d="M262 635C141 635 57 556 57 461C57 387 108 338 166 306C137 289 92 261 74 245C56 228 32 204 32 157C32 55 118 -12 249 -12C346 -12 472 51 472 178C472 266 409 313 339 351C378 376 396 392 412 406C435 428 445 453 445 480C445 569 371 635 262 635ZM246 595C286 595 338 561 338 475C338 417 318 386 301 370C224 406 172 439 172 507C172 549 199 595 246 595ZM260 28C210 28 152 72 152 169C152 221 175 265 204 287C301 235 350 209 351 137C352 71 315 28 260 28Z" id="g13-54"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="500" vert-adv-y="500"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,15.975,0)"><path d="M152 515H407L439 613L433 623H115L70 303C91 310 134 322 179 322C270 322 327 264 327 174C327 100 297 45 235 45C178 45 130 84 106 106C93 117 82 120 70 112C53 99 38 76 38 61S43 36 63 20C84 2 126 -12 166 -12C226 -11 297 15 354 56C424 105 455 164 455 226C455 312 382 394 254 394C228 394 175 380 125 362L152 515Z" id="g13-51"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="500" vert-adv-y="500"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,22.475,0)"><path d="M252 635C85 635 28 460 28 311C28 167 82 -12 250 -12C419 -12 472 166 472 312C472 457 419 635 252 635ZM248 594C314 594 335 455 335 314C335 174 318 29 252 29S165 180 165 315S181 594 248 594Z" id="g13-46"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="500" vert-adv-y="500"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,29.073,-5.741)"><path d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 290 288C243 319 222 338 189 380C167 381 142 376 125 366C125 347 133 322 145 304C198 296 225 288 275 262C225 236 198 227 145 220C133 201 125 177 126 158C141 148 167 143 189 144C222 186 243 205 290 236C288 179 282 152 262 102C272 82 289 63 306 54C322 63 340 82 350 102C330 152 324 179 321 236C368 205 390 186 422 144C444 143 470 148 486 158Z" id="g50-43"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g></svg></span></td><td align="center">0.04298</td></tr><tr><td align="left">3,067</td><td align="center">1,718</td><td align="center"><span style="width: 48.3001ptpx;"><svg height="10.335pt" id="M67" style="vertical-align:-0.2325001pt" version="1.1" viewbox="-0.0498162 -10.1025 48.3001 10.335" width="48.3001pt" 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="M252 635C85 635 28 460 28 311C28 167 82 -12 250 -12C419 -12 472 166 472 312C472 457 419 635 252 635ZM248 594C314 594 335 455 335 314C335 174 318 29 252 29S165 180 165 315S181 594 248 594Z" id="g13-46"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="500" vert-adv-y="500"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,6.5,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z" id="g190-47"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="228" vert-adv-y="228"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,9.475,0)"><path d="M252 635C85 635 28 460 28 311C28 167 82 -12 250 -12C419 -12 472 166 472 312C472 457 419 635 252 635ZM248 594C314 594 335 455 335 314C335 174 318 29 252 29S165 180 165 315S181 594 248 594Z" id="g13-46"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="500" vert-adv-y="500"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,15.975,0)"><path d="M306 382C339 403 362 417 379 433C398 450 411 473 411 500C411 578 352 635 248 635H247C181 635 134 604 106 573L56 506L82 482C117 529 151 557 201 557C246 557 287 531 287 466C287 404 228 364 143 336L150 302C165 306 186 309 204 309C258 309 324 275 324 179C324 85 285 44 235 44C175 44 128 88 106 108C92 118 80 116 69 106C56 97 39 79 39 61C39 47 46 34 63 21C83 5 123 -12 174 -12C236 -12 454 58 454 222C454 297 403 362 306 380V382Z" id="g13-49"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="500" vert-adv-y="500"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,22.475,0)"><path d="M152 515H407L439 613L433 623H115L70 303C91 310 134 322 179 322C270 322 327 264 327 174C327 100 297 45 235 45C178 45 130 84 106 106C93 117 82 120 70 112C53 99 38 76 38 61S43 36 63 20C84 2 126 -12 166 -12C226 -11 297 15 354 56C424 105 455 164 455 226C455 312 382 394 254 394C228 394 175 380 125 362L152 515Z" id="g13-51"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="500" vert-adv-y="500"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,28.975,0)"><path d="M152 515H407L439 613L433 623H115L70 303C91 310 134 322 179 322C270 322 327 264 327 174C327 100 297 45 235 45C178 45 130 84 106 106C93 117 82 120 70 112C53 99 38 76 38 61S43 36 63 20C84 2 126 -12 166 -12C226 -11 297 15 354 56C424 105 455 164 455 226C455 312 382 394 254 394C228 394 175 380 125 362L152 515Z" id="g13-51"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="500" vert-adv-y="500"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,35.475,0)"><path d="M262 635C141 635 57 556 57 461C57 387 108 338 166 306C137 289 92 261 74 245C56 228 32 204 32 157C32 55 118 -12 249 -12C346 -12 472 51 472 178C472 266 409 313 339 351C378 376 396 392 412 406C435 428 445 453 445 480C445 569 371 635 262 635ZM246 595C286 595 338 561 338 475C338 417 318 386 301 370C224 406 172 439 172 507C172 549 199 595 246 595ZM260 28C210 28 152 72 152 169C152 221 175 265 204 287C301 235 350 209 351 137C352 71 315 28 260 28Z" id="g13-54"></path><glyph.data ascent="1024" descent="-360" horiz-adv-x="500" vert-adv-y="500"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,42.121,-5.741)"><path d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 290 288C243 319 222 338 189 380C167 381 142 376 125 366C125 347 133 322 145 304C198 296 225 288 275 262C225 236 198 227 145 220C133 201 125 177 126 158C141 148 167 143 189 144C222 186 243 205 290 236C288 179 282 152 262 102C272 82 289 63 306 54C322 63 340 82 350 102C330 152 324 179 321 236C368 205 390 186 422 144C444 143 470 148 486 158Z" id="g50-43"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g></svg></span></td></tr><tr class="table-tr"><td colspan="3"><hr class="tbody-hr"/></td></tr></table></td></tr><tr class="table-fn"><td><svg height="10.1524pt" id="M68" style="vertical-align:-0.04990005pt" version="1.1" viewbox="-0.0498162 -10.1025 6.17869 10.1524" width="6.17869pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0091,0,0,-0.0091,0,-5.741)"><path d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 290 288C243 319 222 338 189 380C167 381 142 376 125 366C125 347 133 322 145 304C198 296 225 288 275 262C225 236 198 227 145 220C133 201 125 177 126 158C141 148 167 143 189 144C222 186 243 205 290 236C288 179 282 152 262 102C272 82 289 63 306 54C322 63 340 82 350 102C330 152 324 179 321 236C368 205 390 186 422 144C444 143 470 148 486 158Z" id="g50-43"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g></svg>Optimum individual.<br/></td></tr></table>

Payoff matrix for the objective functions.

Mathematical Problems in Engineering

tab4

Table 4

Table 4: Entropy-Based Weighting for Multiobjective Optimization: An Application on Vertical Turning 