fundamental-engineering-optimization-methods.pdf

B the solution does not improve upon an available ip

Info icon This preview shows pages 120–123. Sign up to view the full content.

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:
Image of page 120

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

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.
Image of page 121
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
Image of page 122

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Image of page 123
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern