Problem 27 - College Algebra

Question

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

Step-by-Step Solution

Answer

To solve a linear programming problem, first understand the problem and identify the decision variables, constraints, and the objective function. Then, formulate these into mathematical expressions. Finally, use the graphical or Simplex method to find where the objective function is optimized subject to the constraints. Interpret the results in terms of the original problem situation.

Step 1: Understand the Problem

The first thing to be done is identify the decision variables, limitations or constraints and the objective function from our problem statement. Decision variables are those quantities that we have control over and want to choose in a way that achieves our goal. Constraints are the set of limitations or conditions that are imposed on us and the decision variables. The objective function is the function that we wish to maximize or minimize.

Step 2: Formulate the Mathematical Model

After identifying the key parts, these need to be translated into mathematical expressions. Decision variables are usually denoted by symbols like \(x\), \(y\), etc. Constraints are expressed as linear inequalities in terms of the decision variables. The objective function is expressed as a linear equation in terms of the decision variables and represents the quantity to be minimized or maximized.

Step 3: Apply Linear Programming Method

Finally, once we have a proper represented mathematical model, we are now ready to solve the problem. If our problem involves two variables, we can use graphical methods: plot the feasible region which is determined by the constraints and then find the point in this region where the objective function is maximized or minimized. If our problem involves more than two variables, numerical methods such as the Simplex method can be used to find the solution.

Step 4: Interpretation of Results

After solving the problem, the obtained solution must be interpreted in terms of the original problem, checking whether it's feasible or not and it satisfies the posed conditions.