LINEAR PROGRAMMING

Linear Programming Problem

One which we are to find the maximum or minimum value of a linear expression.

Objective function is the subject to a number of linear constraints of the form

Optimal Value – the largest or smallest value of the objective function and a collection of values x,y,z that gives optimal value constitute an optimal solution.

X,Y,Z are called decision variables.

Formulas.

Linear programming problem - Ax + by + cz ...

Objective function – Ax+ By+ Cz... >N

Sketching the solution set of a linear inequality

To sketch the region represented by a linear equality in two variables.A. Sketch the straight line obtained by replacing the inequality with an equalityB. Choose a test point on the line (0,0) is a good choice if the line does not pass through the origin and if the line does pass on the origin a point on one of the axes would be a good choice.

Sketching the solution set of a linear inequality

C. If the test point satisfies the inequality then the set of solutions is the entire region on the same side of the line as the test point. Otherwise it is the region on the other side of the line. In either case, shade out the side that does not contain the solutions, leaving the solution region showing.

Feasible Region

The graphical method for solving linear programming problems.

A. Graph the feasible regionB. Compute the Coordinates of the corner pointC. Substitute the coordinates of the corner points into the objective function to see which gives the optimal valueD. If the feasible region is not bounded, this method can be misleading.( Optimal solutions always exist when the feasible region is bounded but may or may not exist when the feasible region is unbounded)

Feasible Region

if the feasible region is unbounded we are minimizing the objective function and its coefficients are non-negative and s solution exists so this method yields the solution.

Feasible Region

To determine if a solution exists in the general unbound cases:

1. bound the feasible region by adding a vertical line to the right of the rightmost corner point and a horizontal line above the highest corner point

2. Calculate the coordinates of the new corner points you obtain.

3. Find