The Shaded Area In The Following Graph Represents The Feasib
The Shaded Area In The Following Graph Represents The Feasible Region
The shaded area in the following graph represents the feasible region of a linear programming problem whose objective function is to be maximized, where x1 and x2 represent the level of the two activities. Label each of the following statements as True or False, and then justify your answer based on the graphical method. In each case, give an example of an objective function that illustrates your answer.
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The problem presented involves analyzing the properties of a feasible region in a linear programming (LP) problem, depicted graphically by a shaded area. The key aspects involve understanding the implications of the location of points within this region relative to optimal solutions, as well as the characteristics of specific points such as (3, 3), (0, 2), (6, 3), and (0, 0). These insights are essential for effectively applying linear programming techniques to optimize a given objective function under certain constraints.
a. If (3, 3) produces a larger value of the objective function than (0, 2) and (6, 3), then (3, 3) must be an optimal solution.
False. The fact that (3, 3) yields a higher value of the objective function compared to (0, 2) and (6, 3) does not necessarily imply it is an optimal solution. In linear programming, an optimal solution is generally a vertex (corner point) of the feasible region. If (3, 3) is within the feasible region and produces a higher value, it may indeed be optimal; however, if it is not a vertex or boundary point of the feasible region, then it cannot be considered optimal, despite its higher objective value compared to some other points. For example, suppose the objective function is Z = 2x1 + 2x2. If (3, 3) results in Z = 12, (0, 2) results in Z= 4, and (6, 3) results in Z= 18, then (6, 3) is actually the optimal point, not (3, 3). Therefore, higher objective function value at some point does not guarantee optimality unless that point lies at a boundary vertex where the maximum occurs.
b. If (3, 3) is an optimal solution and multiple optimal solutions exist, then either (0, 2) or (6, 3) must also be an optimal solution.
False. Multiple optimal solutions arise when the objective function is maximized along a whole edge or face of the feasible region, not necessarily at the specific points (0, 2) or (6, 3). The presence of multiple optima suggests that the objective function is parallel to a boundary edge of the feasible region. While (0, 2) and (6, 3) could be optimal points if they lie on the same boundary line where the maximum occurs, it is not obligatory that these particular points are optimal. Other points along a boundary segment could also serve as optimal solutions. For example, if the objective function is Z = 3x1 + 3x2, and the feasible region's boundary includes the line segment connecting (0, 2) and (6, 3), any point along that segment is optimal, not just the endpoints. Thus, the statement is false because optimality could involve a collection of points, not necessarily (0, 2) or (6, 3).
c. The point (0, 0) cannot be an optimal solution.
True. The point (0, 0) is often the intersection of the axes and typically represents the origin. In many LP problems, the origin is feasible only if it satisfies all constraints. However, if the constraints exclude the origin, either because of non-negativity restrictions or other constraint boundaries, then (0, 0) cannot be an optimal solution. Moreover, in the context of maximizing an objective function such as Z = 2x1 + 2x2, the point (0, 0) would yield Z = 0, which is unlikely to be the maximum if the feasible region contains points with higher Z-values. Example: If the feasible region is constrained by x1 ≥ 1 or x2 ≥ 1, then (0, 0) is infeasible and cannot be optimal. Therefore, unless the constraints include the origin, (0, 0) cannot be the optimal point.
References
- Winston, W. L. (2004). Operations Research: Applications and Algorithms. Duxbury Press.