Define the concept of optimal solution in linear programming.

The optimal solution in linear programming is the best possible solution that maximizes or minimizes the objective function.

In linear programming, the objective is to optimize a linear function subject to a set of linear constraints. The optimal solution is the set of values for the decision variables that maximize or minimize the objective function while satisfying all the constraints.

There are two types of optimal solutions: maximum and minimum. A maximum optimal solution is the set of values that maximize the objective function, while a minimum optimal solution is the set of values that minimize the objective function.

To find the optimal solution, we use the simplex method, which is an iterative algorithm that moves from one feasible solution to another until the optimal solution is reached. The simplex method involves constructing a tableau, which is a matrix that represents the linear program in a standard form. For a more detailed understanding, see our explanation of the simplex method in linear programming.

The optimal solution is found when all the coefficients in the objective row are non-negative. If this condition is not met, we use the simplex method to pivot the tableau until the condition is satisfied. Once the optimal solution is found, we can determine the optimal value of the objective function by evaluating it at the optimal solution.

Practical applications of these concepts can be explored in depth in the context of real-world applications of linear programming.

A-Level Maths Tutor Summary: In linear programming, the best outcome, called the optimal solution, either maximises or minimises a goal while sticking to certain rules. There are two kinds: maximum and minimum. We find it using the simplex method, which iteratively reaches the best solution. The method stops when it achieves a condition indicating the best result, showing the highest or lowest possible value for our goal. For further reading on the impact of linear programming in various fields, check our section on related rates in linear programming.