What is true about feasible solutions in a linear programming problem?

Feasible solutions in a linear programming problem refer to any potential solution that satisfies all of the constraints imposed on the problem. When considering the options, the statement that they may exist anywhere within the feasible region is accurate. The feasible region is defined as the area (or space) that is bounded by the constraints, typically illustrated graphically as a polygon or polytope in the case of two or more variables.

Every point in this feasible region represents a viable solution to the linear programming problem, where all constraints are satisfied. These solutions can include both extreme points (vertices of the polygon) as well as any point within the region itself. Thus, it’s correct that feasible solutions may exist anywhere within the entire feasible region, including points that are not extreme points.

This perspective aligns with the nature of linear programming, where the objective is often to find the best solution among these feasible solutions. It is important to note that while extreme points are of particular interest in optimization problems (since the optimal solution will occur at a vertex in a convex feasible region), feasible solutions can indeed exist at any point in the feasible region.