t1 - insights into integer programming that have led to...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
9.8 CUTTING PLANES The cutting-plane algorithm solves integer programs by modifying linear-programming solutions until the integer solution is obtained. It does not partition the feasible region into subdivisions, as in branch-and-bound approaches, but instead works with a single linear program, which it refines by adding new constraints. The new constraints successively reduce the feasible region until an integer optimal solution is found. In practice, the branch-and-bound procedures almost always outperform the cutting-plane algorithm. Nevertheless, the algorithm has been important to the evolution of integer programming. Historically, it was the first algorithm developed for integer programming that could be proved to converge in a finite number of steps. In addition, even though the algorithm generally is considered to be very inefficient, it has provided
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: insights into integer programming that have led to other, more efficient, algorithms. Again, we shall discuss the method by considering the sample problem of the previous sections: z# = max 5x1 + 8x2, subject to: x1 + x2 + s1 = 6, 5x1 + 9x2 + s2 = 45, x1, x2, s1, s2 # 0. (11) s1 and s2 are, respectively, slack variables for the first and second constraints. Solving the problem by the simplex method produces the following optimal tableau: (-z) -5 4 s1 -3 4 s2 = -411 4 , x1 +9 4 s1 -1 4 s2 = 9 4 , x2 -5 4 s1 +1 4 s2 = 15 4 , x1, x2, s1, s2, s3 # 0. Let us rewrite these equations in an equivalent but somewhat altered form: (-z) -2s1 -s2 +42 = 3 4 -3 4 s1 -1 4 s2, x1 +2s1 -s2 - 2 = 1 4 -1 4 s1 -3 4 s2, x2 -2s1 - 3 = 3 4 -3 4 s1 -1 4 s2, x1, x2, s1, s2 # 0....
View Full Document

Ask a homework question - tutors are online