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Chapter 3: Problem 52

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. An optimal solution of a linear programming problem is a feasible solution, but a feasible solution of a linear programming problem need not be an optimal solution.

Short Answer

Expert verified
The statement "An optimal solution of a linear programming problem is a feasible solution, but a feasible solution of a linear programming problem need not be an optimal solution" is true. An optimal solution satisfies all constraints and minimizes or maximizes the objective function, making it a feasible solution. However, a feasible solution only satisfies the constraints without necessarily optimizing the objective function, as shown in the example with solutions (2,2) and (3,2).

Step by step solution

01

Part 1: An optimal solution is a feasible solution

An optimal solution, by definition, minimizes or maximizes the objective function while also satisfying all constraints of the linear programming problem. Since the optimal solution satisfies all constraints, it is also considered a feasible solution. Therefore, this part of the statement is true.
02

Part 2: A feasible solution need not be an optimal solution

A feasible solution, by definition, satisfies all constraints of the linear programming problem, but it may not necessarily minimize or maximize the objective function. Let's consider an example to illustrate this: Objective function: Minimize \( z = x + y \) Constraints: 1. \( x + y \leq 5 \) 2. \( x \geq 0 \) 3. \( y \geq 0 \) Let's take two feasible solutions: a. (2, 2) which gives a z-value of 4 b. (3, 2) which gives a z-value of 5 Both of these solutions satisfy the constraints but do not minimize the objective function. The optimal solution for this problem is (0, 0) which gives a z-value of 0. It satisfies all constraints and minimizes the objective function. In this example, we showed that some feasible solutions are not optimal solutions since they do not minimize (or maximize) the objective function. Therefore, this part of the statement is also true.
03

Conclusion

Since both parts of the statement are true, we can conclude that the statement "An optimal solution of a linear programming problem is a feasible solution, but a feasible solution of a linear programming problem need not be an optimal solution" is true.

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Most popular questions from this chapter

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