Which one of the following is not true?
(A) The set of all feasible solutions of a linear programming problem is a closed convex set.
(B) The set of all feasible solutions of a linear programming problem is a convex set which is either closed or open.
(C) The objective function of a linear programming problem assumes its optimal value at an extreme point of the convex set of feasible solutions.
(D) A basic feasible solution to a linear programming problem corresponds to an extreme point of the convex set of feasible solutions and conversely.
Answer (2)
The statement that is not true is (B), which claims the set of feasible solutions can be either closed or open. In fact, the feasible region in linear programming is a closed convex set. Other statements concerning feasible solutions and their properties are true.
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To determine which statement is not true, let's analyze each option carefully based on the properties of linear programming.
(A) The set of all feasible solutions of a linear programming problem is a closed convex set.
In a linear programming problem, the set of all feasible solutions is represented by a polyhedron, which is a convex set that is also closed if the problem is bounded. Thus, this statement is generally true for a bounded problem.
(B) The set of all feasible solutions of a linear programming problem is a convex set which is either closed or open.
The feasible region of a linear programming problem is always convex because it is formed by the intersection of linear inequalities, which are half-spaces. However, for practical linear programming problems, the feasible region is usually closed rather than open, unless there are no constraints, which is uncommon. Therefore, this statement suggesting it could be open is misleading in typical scenarios, making it not true.
(C) The objective function of a linear programming problem assumes its optimal value at an extreme point of the convex set of feasible solutions.
This is a fundamental property of linear programming problems. The optimal value of a linear objective function, if it exists, occurs at an extreme point (or vertex) of the convex polyhedron that represents the feasible region. This statement is true.
(D) A basic feasible solution to a linear programming problem corresponds to an extreme point of the convex set of feasible solutions and conversely.
A basic feasible solution is a solution found at a vertex of the feasible region. Therefore, this statement is true as basic feasible solutions correspond to extreme points and vice versa.
Therefore, the statement that is not true is (B) : "The set of all feasible solutions of a linear programming problem is a convex set which is either closed or open." In a typical linear programming problem, the feasible set is convex and closed, not open.
Thus, option (B) is the correct choice for the statement that is not true.
