At which vertex is the objective function

To solve a linear programming problem, follow these steps. Now find the coordinates of the vertices of the region formed. Evaluate the objective function at each vertex.

Subjects Near Me. Sign in via your Institution. You could not be signed in, please check and try again. Sign in with your library card Please enter your library card number. Related Content Related Overviews linear programming. Show Summary Details Overview vertex method. All rights reserved. Now, we see that every point in the feasible region satisfies all the constraints, and since there are infinitely many points, it is not evident how we should go about finding a point that gives a maximum value of the objective function.

To handle this situation, we use the following theorems which are fundamental in solving linear programming problems. The proofs of these theorems are beyond the scope of the book. When Z has an optimal value maximum or minimum , where the variables x and y are subject to constraints described by linear inequalities, this optimal value must occur at a corner point vertex of the feasible region.

If R is bounded, then the objective function Z has both a maximum and a minimum value on R and each of these occurs at a corner point vertex of R.

I searched wikipedia here to get under Optimal vertices and rays of polyhedra :. This principle underlies the simplex algorithm for solving linear programs.. But I couldn't understand it. Can someone explain? It might be helful to add that I have studied basic Calculus. Notice that, for different values of C, you get different straight lines of varying y intercepts but they will have same slope :. Only the lines that cut through the feasible region satisfy all the given constraints because you can cookup x,y values such that they fall in both feasible region and the objective function.

Consequently the vertex A gives the maximum value for the objective function. The only way you could get stuck is that for every possible improving direction, some face blocks you. If you do not already stand at a vertex, then while staying on the hyperplane, you can walk to a vertex without changing your objective value.

In the above explanation, I have used the boundedness of the feasible region: Without boundedness, it might be possible to walk in an improving direction forever, or it might be possible to walk along a face of optimal solutions without ever arriving at a vertex. I have also used convexity: If I do not stand at an optimal solution, then there will always exist an improving direction.

Suppose you are not at a vertex of the convex region. If you are in the strict interior of the convex region, then you can move a little bit in any direction you want.

Put all the pieces together and you get that some vertex provides the optimal solution. Of these three values, the minimum value is negative 27 and the maximum value is one. And these are not just the minimum and maximum values on the vertices of the feasible region; they are the minimum and maximum values on the entire feasible region including the interior and the edges.

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