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What steps are involved in linear programming?

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Linear programming involves taking a series of linear inequalities, known as the constraints of the system, and determining the optimal values for a system given those constraints. Linear systems consist of at least three inequalities. Solving them in pairs provides the intersection points and the means to derive the optimal solution. The area between the constraints of the system is called the feasibility region.

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For example, take the system of y < -x +3, y < 2x - 1, and y > x +1. Arranging each of those inequalities into pairs and solving for x and y yields (1,2), (4/3, 5/3) and (2,3) as the three points at which the lines intersect. The optimal value for the linear system is always at one of these points, because it represents the edge of the constraints.

Next, assume the expression for choosing optimal values is p = 2x + 4y. Take each of the three vertices of the system, entering each x and y value into the equation to see which has the largest value. Respectively, the results are 10, 9.3 and 16. The values of (2,3) for x and y produce the largest result. Linear programming can be used to determine cost-benefit ratio in resource use and other areas.

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