Use Graphical Methods To Solve The Linear Programming Problem

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However, unfortunately one is never as lucky to find a real world solution that involves two or less decision variables.

As we saw in the article on formulation, there can indeed be quite a few decision variables in even a simple problem.

We will more or less follow the same principles — the only problem being — we are not exactly sure about solving linear inequalities which are what, in most cases, will represent our constraints.

In this article, we discuss an algebraic method to solve linear programs.

The maximum value of the objective function is 33, and it corresponds to the values x = 3 and y = 12 (G-vertex coordinates).

In Graphical method is necessary to calculate the value of the objective function at each vertex of feasible region, while the Simplex method ends when the optimum value is found. For example, in the last low, s1 takes the value -20 which violates our sign restriction s1 = 0.The rest of the rows represent solutions that do not violate any of our constraints and hence are feasible.The input base variable in the Simplex method determines towards what new vertex is performed the displacement.In this example, as P1 (corresponding to 'x') enters, the displacement is carried out by the OF-edge to reach the F-vertex, where the Z-function value is calculated.Out of these, we select the best solution to be presented in row 8 which is colored in a shade of green.Please note that the selections for which we arrive at valid solutions above are called Basic Feasible Solutions of the LP.What we will strive to do is to keep the geometric intuition that we built in the last chapter alive and relate our algebraic method with it at every step.In this interest, we shall solve the same problem that we solved in the last article.Hence if we have any hope of finding out the vertices of the feasible region with the tools of algebra, we must convert the inequalities that currently represent our constraints into equations.It must be worthwhile to pause here and think about how one can do this instead of reading ahead to the solution.


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