fundamental-engineering-optimization-methods.pdf

Ͳݔ ? ͳݔ ? ʹͳͳͳ? ݔ ଵ ݔ ? ʹݔ ? ͳǣ

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൅ ͵Ͳݔ ൅ ͸Ͳݔ ൒ ʹͲͲͲǡ ݔ ൅ ݔ െ ʹݔ ൑ ͲǢ ݔ ǡ ݔ ǡ ݔ ൒ Ͳ DQG LQWHJHU Following steps are taken to solve the problem. The progress is also shown in a decision tree in Fig. 6.2: 1. ܵ ǣ the LP relaxation problem ሺܨ ൌ ܨ is first solved and produces an optimum solution: ݔ כ Ͳǡ ݔ כ ൌ ͹Ǥ͸ͻǡ ݔ כ ൌ ͵Ǥͺͷǡ ݂ כ ൌ ͳͳͷ͵Ǥͺ ³ which serves as an upper bound for IP solution. 2. ܵ ǣ ܨ ׫ ሼݔ ൑ ͵ሽ is solved and produces an integer solution: ݔ כ ൌ Ͳǡ ݔ כ ൌ ͸ǡ ݔ כ ൌ ͵ǡ ݂ כ ൌ ͻͲͲ ² This is recorded as current optimum. 3. ܵ ǣ ܨ ׫ ሼݔ ൒ Ͷሽ produces a non-integer solution: ݔ כ ൌ ͳǤ͸ǡ ݔ כ ൌ ͸ǤͶǡ ݔ כ ൌ Ͷǡ ݂ כ ൌ ͳͳͷʹ ² . 4. ܵ ǣ ܨ ׫ ሼݔ ൑ ͸ሽ produces a non-integer solution: ݔ כ ൌ ʹǤͳǡ ݔ כ ൌ ͸ǡ ݔ כ ൌ ͶǤͲͷǡ ݂ כ ൌ ͳͳͷͳǤͶ ² 5. ܵ ǣ ܨ ׫ ሼݔ ൑ Ͷሽ produces an integer solution: ݔ כ ൌ ʹǡ ݔ כ ൌ ͸ǡ ݔ כ ൌ Ͷǡ ݂ כ ൌ ͳͳͶͲ ² This is recorded as the new optimum and the branch is fathomed. 6. ܵ ǣ ܨ ׫ ሼݔ ൒ ͷሽ produces a non-integer solution: ݔ כ ൌ ͺǤͷ͹ǡ ݔ כ ൌ Ͳǡ ݔ כ ൌ ͷǡ ݂ כ ൌ ͳͳͳͶǤ͵ ³ which is lower than the current optimum, so the branch is fathomed. 7. ܵ ǣ ܨ ׫ ሼݔ ൒ ͹ሽ produces a non-integer solution: ݔ כ ൌ ͲǤͷ͹ǡ ݔ כ ൌ ͹ǡ ݔ כ ൌ Ͷǡ ݂ כ ൌ ͳͳͶͶǤ͵ 8. ܵ ǣ ܨ ׫ ሼݔ ൑ Ͳሽ produces a non-integer solution: ݔ כ ൌ Ͳǡ ݔ כ ൌ ͹Ǥ͵͵ǡ ݔ כ ൌ Ͷǡ ݂ כ ൌ ͳͳͶͲ ³ which does not improve upon the current optimum. The branch is fathomed. 9. ܵ ǣ ܨ ׫ ሼݔ ൒ ͳሽ has no feasible solution. The branch is fathomed. 10. All branches having been fathomed, the optimal solution is: ݔ כ ൌ ሺʹǡ͸ǡͶሻǡ ݂ כ ൌ ͳͳͶͲ ²
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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 123 iscrete Optimization ܵ ǣ ܨ ൌ ܨ כ ൌ ሺͲǡ ͹Ǥ͸ͻǡ ͵Ǥͺͷሻǡ ݂ כ ൌ ͳͳͷ͵Ǥͺ ܵ ǣ ܨ ׫ ሼݔ ൑ ͵ሽ כ ൌ ሺͲǡ ͸ǡ͵ሻǡ ݂ כ ൌ ͻͲͲ ܵ ǣ ܨ ׫ ሼݔ ൒ Ͷሽ כ ൌ ሺͳǤ͸ǡ ͸ǤͶǡ Ͷሻǡ ݂ כ ൌ ͳͳͷʹ ܵ ǣ ܨ ׫ ሼݔ ൑ ͸ሽ כ ൌ ሺʹǤͳǡ ͸ǡ ͶǤͲͷሻǡ ݂ כ ൌ ͳͳͷͳǤͶ ܵ ǣ ܨ ׫ ሼݔ ൒ ͹ሽ כ ൌ ሺͲǤͷ͹ǡ ͹ǡ Ͷሻǡ ݂ כ ൌ ͳͳͶͶǤ͵ ܵ ǣ ܨ ׫ ሼݔ ൑ Ͷሽ כ ൌ ሺʹǡ ͸ǡ Ͷሻǡ ݂ כ ൌ ͳͳͶͲ ܵ ǣ ܨ ׫ ሼݔ ൒ ͷሽ כ ൌ ሺͺǤͷ͹ǡ Ͳǡ ͷሻǡ ݂ כ ൌ ͳͳͳͶǤ͵ ܵ ǣ ܨ ׫ ሼݔ ൑ Ͳሽ כ ൌ ሺͲǡ ͹Ǥ͵͵ǡ Ͷሻǡ ݂ כ ൌ ͳͳͶͲ ܵ ǣ ܨ ׫ ሼݔ ൒ ͳሽ ܰܨܵ Fig. 6.2: The decision tree for Example 6.3.
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