The Shaded Area In The Following Graph Represents The 303616
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 levels of two activities. Label each of the following statements as True or False, and justify your answer based on the graphical method. In each case, provide an example of an objective function that illustrates your answer.
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.
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.
c. The point (0, 0) cannot be an optimal solution.
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Linear programming is a mathematical method used to determine the best possible outcome in a given mathematical model, often to maximize or minimize a linear objective function subjected to a set of linear constraints. The feasible region, typically depicted graphically, is the area where all constraints intersect, representing all feasible solutions. The characteristics of this region significantly influence the optimal solutions to the problem. In the graphical approach, the vertices (corner points) of the feasible region are examined to identify the maximum or minimum value of the objective function. Understanding the properties of these vertices and the shape of the feasible region is crucial for solving linear programming problems effectively.
Part (a) posits that if the point (3, 3) produces a larger value of the objective function compared to points (0, 2) and (6, 3), then (3, 3) must be the optimal solution. This statement hinges on the assumption that the objective function increases in the direction from the compared points toward (3, 3). In the graphical method, the optimal solution is generally found at a vertex of the feasible region unless the objective function is parallel to a constraint boundary along a line segment of optimal solutions. Therefore, if (3, 3) yields a higher value than other points within the feasible region, it indicates that (3, 3) is likely a vertex at which the objective function reaches its maximum, provided the point is within or on the boundary of the feasible region. For example, consider an objective function z = 2x₁ + 3x₂. If at (0, 2), z = 6; at (6, 3), z = 21; and at (3, 3), z = 15, then (6, 3) produces the highest value, indicating an optimal solution. This aligns with the statement when (3, 3) indeed produces the largest value, making it the optimal solution.
Part (b) discusses the existence of multiple optimal solutions. If (3, 3) is an optimal point, and multiple solutions exist, then they lie along the boundary line where the objective function's level curves are parallel to a constraint boundary. In such cases, entire line segments between vertices can be optimal solutions, including points such as (0, 2) or (6, 3). The statement correctly suggests that when multiple optimal solutions exist, at least the endpoints of the optimal line segment will be solutions as well. For example, with the objective function z = 2x₁ + 3x₂, if the constraints produce a feasible region where the line segments between (0, 2) and (6, 3) all give the same maximum value, then these points are also optimal solutions. Therefore, the statement accurately describes the property of multiple optima along a constraint boundary, and the assertion that (0, 2) or (6, 3) must also be solutions holds true under these circumstances.
Part (c) considers whether the point (0, 0) can be an optimal solution. Typically, in linear programming problems, (0, 0) can be an optimal solution if it lies within or on the boundary of the feasible region and yields the highest value of the objective function. However, often the feasible region does not include the origin, especially when constraints restrict x₁ and x₂ to be positive or above certain values, or when the objective function increases with larger values of x₁ and x₂, leading to an optimal point at an interior vertex or along a boundary away from the origin. For example, if the constraints are x₁ ≥ 1 and x₂ ≥ 1, then (0, 0) is outside the feasible region, hence cannot be an optimal solution. Conversely, if the feasible region includes the origin and the objective function is decreasing in both variables, then (0, 0) could be optimal. Therefore, whether (0, 0) can be optimal depends on the specific constraints; generally, it is often not the optimal point when constraints exclude the origin or when the objective seeks maximization at higher variable levels.