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Problem 14

# Describe the three steps in solving a linear programming problem.

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The three steps in solving a linear programming problem are as follows: Step 1: Formulate the Linear Programming Problem, which involves identifying the objective function, decision variables, and linear constraints. The objective function represents what we aim to maximize or minimize, the decision variables are the unknowns that influence the objective function, and the linear constraints represent the restrictions of the problem. Step 2: Graph the Feasible Region by plotting the linear constraints on a coordinate plane. The feasible region is the area where all constraints are satisfied simultaneously. This region holds all potential solutions. Step 3: Identify the Optimal Solution within the feasible region. For maximization problems, find the point(s) in the region where the objective function has the highest value. For minimization problems, seek the point(s) where the objective function has the lowest value. These points are the optimal solutions. Even though, sometimes, there can exist multiple optimal solutions or no solution at all based on the specifics of the feasible region and the problem.
See the step by step solution

## Step 1: Formulate the Linear Programming Problem

To solve a linear programming problem, the first step is to formulate the problem by identifying the objective function, decision variables, and the linear constraints. The objective function represents the quantity (such as profit or cost) that needs to be maximized or minimized. The decision variables are the unknowns that we need to find that will influence the objective function, and the linear constraints represent the limitations of the problem.

## Step 2: Graph the Feasible Region

Once the problem is formulated, plot the feasible region on a coordinate plane by graphing the linear constraints as equations. The feasible region is the area where all the constraints are satisfied simultaneously, often in the form of a polygon. It contains all the possible solutions to the linear programming problem.

## Step 3: Identify the Optimal Solution

The last step is to identify the optimal solution within the feasible region. If the problem is to maximize the objective function, find the vertex or vertices in the feasible region where the objective function has the highest value. If the problem is to minimize the objective function, find the vertex or vertices in the feasible region where the objective function has the lowest value. These points represent the optimal solution of the linear programming problem. Sometimes, there may be multiple optimal solutions or no solution at all, depending on the feasible region and the nature of the problem.

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