An example of the cutting plane method is presented

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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
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Download free eBooks at 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.
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