fundamental-engineering-optimization-methods.pdf

# B the solution does not improve upon an available ip

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b) The solution does not improve upon an available IP solution. c) An improved IP solution is returned and is recorded as current optimal. d) A non-integer solution that is better than the current optimal is returned. 4. Fathoming. In the first three cases above the current branch is excluded from further consideration. The algorithm then backtracks to the most recently unbranched node in the tree and continues with examining the next node in a last in first out (LIFO) search strategy. Finally, the process ends when all branches have been fathomed, and an integer optimal solution to the problem, if one exists, has been found. Let NF denote the set of nodes not yet fathomed, F denote the feasible region for the original IP problem, ܨ denote the feasible region for the LP relaxation, ܨ denote the feasible region at node ݇ ³ ܵ denote the subproblem defined as: ݖ ൌ ࢉ ࢞ ǡ ࢞ א ܨ ³ and let ݖ denote the lower bound on the optimal solution. Then, the BB algorithm is given as follows:

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Download free eBooks at bookboon.com Click on the ad to read more Fundamental Engineering Optimization Methods 121 iscrete Optimization Branch-and-bound Algorithm (Sierksma, p. 219): Initialize: set ܨ ൌ ܨ ǡ ܰܨ ൌ ሼͲሽǡ ݖ ൌ െλ ² . While ܰܨ ് ׎ǡ 1. Select a label ݇ א ܰܨ ² . 2. Determine if there exists an optimal solution ሺݖ ǡ ࢞ WR ܵ ǡ HOVH VHW ݖ ൌ െλ ² 3. If ݖ ൐ ݖ ǡ WKHQ LI א ܨǡ VHW ݖ ൌ ݖ ² 4. If ݖ ൑ ݖ ǡ VHW ܰܨ ൌ ܰܨ̳ሼ݇ሽ ² 5. If ݖ ൐ ݖ DQG ב ܨǡ partition ܨ L into two or more subsets as follows: choose a variable ݔ א ࢞ with fractional value, ݔ ൌ ܫ ൅ ߜ ǡ ܫ ൌ ہݔ ۂǡ Ͳ ൏ ߜ ൏ ͳǤ Define two new subprograms: ܨ ൌ ܨ ת ሼݔ ൑ ܫሽǡ ܨ ൌ ܨ ת ሼݔ ൒ ܫ ൅ ͳሽ ² 6HW ܰܨ ൌ ܰܨ ׫ ሼ݇ ǡ ݇ ² An example is now presented to illustrate the BB algorithm.
Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 122 iscrete Optimization Example 6.3: Branch and bound algorithm We consider the following IP problem (Belegundu and Chandrupatla, p. 383): A tourist bus company having a budget of \$10M is considering acquiring a fleet with a mix of three models: a 15-seat van costing \$35,000, a 30-seat minibus costing \$60,000, and a 60-seat bus costing \$140,000. A total capacity of 2000 seats is required. At least one third of the vehicles must be the big buses. If the estimated profits per seat per month for the three models are: \$4, \$3, and \$2 respectively, determine the number of vehicles of each type to be acquired to maximize profit. Let ݔ ǡ ݔ ǡ ݔ denote the quantities to be purchased for each of the van, minibus, and big bus; then, the optimization problem is formulated as: 0D[LPL]H ݖ ൌ ͸Ͳݔ ൅ ͻͲݔ ൅ ͳʹͲݔ 6XEMHFW WR± » ͷݔ ൅ ͸Ͳݔ ൅ ͳͶͲݔ ൑ ͳͲͲͲǡ ͳͷݔ

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• Winter '17
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