Problem 27 - Algebra and Trigonometry

Question

In your own words, describe how to solve a linear programming problem.

Step-by-Step Solution

Answer

To solve a linear programming problem, you must first formulate a mathematical model of the problem. This model comprises the objective function (what you're trying to maximise or minimise) and the constraints within which you must work. Then, graph these constraints to identify the feasible region. Use the method of corners to find the optimal points (vertices of the feasible region) that give the maximum or minimum value of the objective function.

Step 1: Formulate the Mathematical Model

Identify all the variables, constraints, and the objective function in the problem. Write these out in mathematical form. For example, if you want to maximise a function \(Z = 2x + 3y\), subject to the constraints \(x \geq 0\), \(y \geq 0\), \(x + y \leq 4\), and \(3x + 2y \leq 6\), you would turn this into a model.

Step 2: Graph the Feasible Region

On a graph, plot all the constraints, usually as inequality regions. The area where all the constraints overlap is known as the feasible region.

Step 3: Identify the Objective Function

The objective function should already be identified in Step 1. It's the function you're trying to maximise or minimise.

Step 4: Use the Method of Corners

Examine each vertex (corner) of the feasible region to find the maximum or minimum value of the objective function. These values will be the solution to the problem.