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In linear programming, the optimal solution represents the best possible outcome that either maximizes or minimizes the objective function.

The primary goal of linear programming is to optimize a linear function while adhering to a set of linear constraints. The optimal solution consists of values for the decision variables that either maximize or minimize the objective function, all while ensuring that every constraint is satisfied.

There are two distinct types of optimal solutions: maximum and minimum. A maximum optimal solution corresponds to the set of values that maximizes the objective function, whereas a minimum optimal solution corresponds to the set of values that minimizes the objective function.

To determine the optimal solution, we employ the simplex method, which is an iterative algorithm that transitions from one feasible solution to another until the optimal solution is identified. The simplex method involves constructing a tableau—a matrix that represents the linear program in standard form. For a more comprehensive understanding, please refer to our additional resources.

The optimal solution is achieved when all coefficients in the objective row of the tableau are non-negative. If this condition is not satisfied, we utilize the simplex method to perform pivot operations on the tableau until the necessary condition is met. Once the optimal solution is identified, the optimal value of the objective function can be calculated by evaluating it at the optimal solution.

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