Feasible and Optimal Solutions A feasible solution to a linear programming problem is a set of values for the decision variables that satisfies the system o...
Feasible and Optimal Solutions
A feasible solution to a linear programming problem is a set of values for the decision variables that satisfies the system of linear inequalities and equalities. An optimal solution is a set of values for the decision variables that not only satisfies the system of linear inequalities and equalities, but also maximizes or minimizes a specified objective function.
A feasible solution must satisfy the following conditions:
Each component (i) of each inequality must be non-negative.
Each component (i) of each equality must be zero.
The decision variables must take on real values.
An optimal solution must satisfy the following conditions:
The objective function must be maximized.
Each decision variable must be non-negative.
A linear programming problem is a special case of a linear optimization problem where the objective function is linear. In other words, the objective function can be represented by a linear equation.
The graphical method for solving linear programming problems involves finding the intersection points of the constraint lines. These intersection points represent the feasible solutions to the problem. The objective function is then evaluated at each intersection point to determine the optimal solution.
Here are some examples of feasible and optimal solutions:
A farmer with 10 acres of land wants to plant corn or soybeans. He can plant a maximum of 5 acres of corn and 3 acres of soybeans. The profit per acre of corn is 15. The farmer should plant a mix of corn and soybeans that will maximize his profit.
A company produces two products, A and B, which have different demand curves. The company faces a fixed cost of 5 per unit of product A and $8 per unit of product B. The company should produce a quantity of product A that yields the highest profit.
An investor has $100 to invest in two stocks, A and B. The return on stock A is 10% per year, while the return on stock B is 15%. The investor should allocate his investment equally between stocks A and B to maximize his profit