Solving Linear Programming Problems Step by Step

Let's explore how to set up a linear programming problem using a furniture factory example.First, we need to define our decision variables. These represent the quantities we want to optimize.Next, we write our objective function. This represents the total profit we want to maximize.Now we identify all constraints. These represent the limitations on our resources.Finally, we convert all constraints to standard form by adding slack variables.Now that we have our problem in standard form, we're ready to create the initial simplex tableau.Let's move on to setting up our initial simplex tableau.Now we'll construct our initial simplex tableau using the coefficients from our objective function and constraints.The bottom row contains our objective function coefficients. We look for the most negative value to identify our pivot column.This negative three is our most negative coefficient, making this our pivot column.To find our pivot row, we calculate the ratios of the right-hand side values to the corresponding pivot column entries.The smallest non-negative ratio identifies our pivot row. Here, four is smaller than twelve, making this our pivot element.We perform row operations to get our pivot element to one and eliminate all other entries in the pivot column.After this first iteration, our objective value has improved from zero to twelve, but we're not done yet since we still have negative values in the objective row.Let's continue our simplex method iterations to find the optimal solution.We'll perform pivot operations until we have no negative entries in our objective row.To check for optimality, we examine the objective row for any remaining negative entries.After our final iteration, we arrive at the optimal tableau with no negative entries in the objective row.From our optimal tableau, we can read our solution. x₁ equals 3, x₂ equals 2, giving us an optimal objective value of 13.Let's verify that our solution satisfies all original constraints.For our first constraint, two times three plus two equals eight, which satisfies our inequality.For our second constraint, three plus four equals seven, which is less than ten, satisfying our inequality.