fundamental-engineering-optimization-methods.pdf

An example of the cutting plane method is presented

Info icon This preview shows pages 125–127. Sign up to view the full content.

An example of the cutting plane method is presented below. Example 6.4: Cutting Plane method We consider the IP problem in Example 6.3 above where the LP relaxation solution was found as: ݔ כ Ͳǡ ݔ כ ൌ ͹Ǥ͸ͻǡ ݔ כ ൌ ͵Ǥͺͷǡ ݂ כ ൌ ͳͳͷ͵Ǥͺ ² The final tableau for the LP relaxation solution is given as: %DVLF 5KV º²¿º¿ ¸ º º²½»Á º²º»Á º À²¼Á ¶º²ºÁ¼ º ¸ ¶º²¹»¸ º²º¸Á º »²¿½ ܛ º²¸À» º º º²¸¸½ º²¸¸½ ¸ ¸¹»²¸ െܢ º²¸¸½ º º ¹²ºÀÀ º²½ÀÀ º ¸¸½»²¿ The following series of cuts then produces an integer optimum solution: 1R² &XW 2SWLPDO VROXWLRQ ¸² ͲǤͺͲͺݔ ൅ ͲǤͷ͵ͻݏ ൅ ͲǤͲ͵ͻݏ െ ݏ ൌ ͲǤ͸ͻʹ ݔ כ ൌ ͲǤͺͷ͹ǡ ݔ כ ൌ ͹ǡ ݔ כ ൌ ͵Ǥͻʹͻǡ ݂ כ ൌ ͳͳͷʹǤͻ ¹² ͲǤͺ͵͵ݏ ൅ ͲǤͲʹͶݏ ൅ ͲǤͺͺͳݏ െ ݏ ൌ ͲǤͻʹͻ ݔ כ ൌ ʹǤͳ͸ʹǡ ݔ כ ൌ ͷǤͻͶ͸ǡ ݔ כ ൌ ͶǤͲͷͶǡ ݂ כ ൌ ͳͳͷͳǤ͵ »² ͲǤͲͷͶݏ ൅ ͲǤͻ͹͵ݏ ൅ ͲǤͳ͵ͷݏ െ ݏ ൌ ͲǤͻͶ͸ ݔ כ ൌ ʹǤͲͺ͵ǡ ݔ כ ൌ ͷǤͻ͹ʹǡ ݔ כ ൌ ͶǤͲʹͺǡ ݂ כ ൌ ͳͳͶͷǤͺ ¾² ͲǤͲͷ͸ݏ ൅ ͲǤͳ͵ͻݏ ൅ ͲǤͻ͹ʹݏ െ ݏ ൌ ͲǤͻ͹ʹ ݔ כ ൌ ʹǡ ݔ כ ൌ ͸ǡ ݔ כ ൌ Ͷǡ ݂ כ ൌ ͳͳͶͲ
Image of page 125

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 126 ±umerical Optimization Methods 7 Numerical Optimization Methods This chapter describes the numerical methods used for solving both unconstrained and constrained optimization problems. These methods have been used to develop computational algorithms that form the basis of commercially available optimization software. The process of computationally solving the optimization problem is termed as mathematical programming and includes both linear and nonlinear programming. The basic numerical method to solve the nonlinear problem is the iterative solution method that starts from an initial guess, and iteratively refines it in an effort to reach the minimum (or maximum) of a multi-variable objective function. The iterative scheme is essentially a two-step process that seeks to determine: a) a search direction that does not violate the constraints and along which the objective function value decreases; and b) a step size that minimizes the function value along the chosen search direction. Normally, the algorithm terminates when either a minimum has been found, indicated by the function derivative being approximately zero, or when a certain maximum number of iterations has been exceeded indicating that there is no feasible solution to the problem.
Image of page 126
Image of page 127
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