I'm working on Linear Programming Models.
How do I find the corner points when they fall between two numbers? Is there some formula to use?
When working with linear programming models, finding the corner points typically involves solving a system of linear equations or inequalities. These equations or inequalities represent the constraints of the problem, such as resource limitations or production requirements.
To explain the process of finding corner points, let's consider a simple example. Suppose we have the following system of inequalities:
1) 2x + 3y ≤ 10
2) x + y ≥ 4
3) x ≥ 0
4) y ≥ 0
Here's a step-by-step explanation of how to find the corner points:
1. Rewrite each inequality in the form of an equation by changing the inequality sign to an equality sign. This allows us to identify the boundary lines of the feasible region. In our example, the boundary lines are:
1') 2x + 3y = 10
2') x + y = 4
2. Plot the boundary lines on a coordinate plane, using the equations from step 1. For simplicity, graph each equation by finding the x and y-intercepts (where they cross the x and y-axes).
3. Shade the feasible region. Determine which region of the plot satisfies the given inequalities. In our example, the shaded region would be above line 2' and below line 1'.
4. Identify the corner points. Where the boundary lines intersect, you will find the corner points of the feasible region. In our example, the corner points are the intersections of lines 1' and 2'. Determine the values of (x, y) at these intersection points.
5. Check the objective function. Finally, evaluate the objective function (if given) at each corner point to determine the optimal solution. The corner point that gives the highest or lowest value, depending on the objective, would be the optimal solution.
It's important to note that not all linear programming problems have corner points falling exactly on integer coordinates. Sometimes the corner points might have fractional or decimal values. In these cases, you should still identify and evaluate those points to determine the optimal solution.
By following these steps, you should be able to find the corner points of a linear programming problem even if they fall between two numbers.