Maximizing Z = 716x₁ + 516x₂ A Graphical Method Solution

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Consider the objective function: Maximize Z = (700 + 16)x₁ + (500 + 16)x₂ Subject to 8x₁ + 5x₂ ≤ 400, 20x₁ + 8x₂ ≤ 250, 15x₁ + 12x₂ ≤ 500. Use the graphical method to find all optimal solutions.

Introduction to Linear Programming

In the realm of mathematics and optimization, linear programming stands as a powerful technique for determining the best possible outcome or solution from a set of mathematical requirements, which are represented as linear relationships. These relationships often take the form of inequalities, which define a feasible region within which the optimal solution must lie. The primary goal in linear programming is to maximize or minimize a linear objective function, subject to these constraints. This method is widely used in various fields, including economics, engineering, and operations research, to make informed decisions regarding resource allocation, production planning, and logistics. The beauty of linear programming lies in its ability to handle complex problems with multiple variables and constraints, providing a systematic approach to finding the optimal solution. The graphical method, a cornerstone of linear programming, provides a visual approach to solving problems with two decision variables, offering an intuitive understanding of the solution space and the location of optimal solutions.

Problem Statement: Maximizing the Objective Function

At the heart of this article is a classic linear programming problem. We aim to maximize the objective function, which is defined as: Z = 716x₁ + 516x₂. This equation represents the quantity we want to maximize, where x₁ and x₂ are the decision variables. These variables represent the quantities of two different products or activities, and our goal is to find the combination of x₁ and x₂ that yields the highest value for Z. However, we cannot simply choose any values for x₁ and x₂; our choices are constrained by a set of inequalities. These constraints represent limitations on resources, production capacity, or other factors that affect the decision variables. In this particular problem, we have three constraints:

  1. 8x₁ + 5x₂ ≤ 400
  2. 20x₁ + 8x₂ ≤ 250
  3. 15x₁ + 12x₂ ≤ 500

These inequalities define the feasible region, which is the set of all possible combinations of x₁ and x₂ that satisfy all the constraints simultaneously. Our task is to find the point within this feasible region that yields the maximum value for the objective function Z. To solve this problem, we will employ the graphical method, a powerful visual tool for linear programming with two decision variables. This method allows us to plot the constraints as lines on a graph, identify the feasible region, and then determine the optimal solution by examining the corner points of this region. The graphical method provides an intuitive understanding of the problem and its solution, making it a valuable tool for decision-making.

Step-by-Step Graphical Method

The graphical method provides a visual approach to solving linear programming problems with two decision variables. This method involves plotting the constraints as lines on a graph, identifying the feasible region, and then determining the optimal solution by examining the corner points of this region. The step-by-step process is outlined below:

1. Plotting the Constraints

Begin by converting each inequality constraint into an equation. This allows us to plot each constraint as a line on a graph. For instance, the first constraint, 8x₁ + 5x₂ ≤ 400, becomes 8x₁ + 5x₂ = 400. To plot this line, we can find two points that satisfy the equation. A simple way to do this is to set x₁ = 0 and solve for x₂, and then set x₂ = 0 and solve for x₁. This gives us two points that lie on the line. Repeat this process for all three constraints, generating a set of lines on the graph. Each line represents the boundary of the region defined by the corresponding inequality. The direction of the inequality determines which side of the line is feasible. For example, for the constraint 8x₁ + 5x₂ ≤ 400, the feasible region lies below the line, as it represents all points where the sum of 8x₁ and 5x₂ is less than or equal to 400.

2. Identifying the Feasible Region

The feasible region is the area on the graph that satisfies all the constraints simultaneously. It is the intersection of the regions defined by each individual constraint. To identify the feasible region, we need to consider the direction of each inequality. For "less than or equal to" constraints (≤), the feasible region lies on or below the line. For "greater than or equal to" constraints (≥), the feasible region lies on or above the line. The feasible region is the area where all these individual feasible regions overlap. This region is typically a polygon, and its corners are known as corner points or vertices. These corner points are crucial because the optimal solution to a linear programming problem always occurs at one of these points.

3. Finding the Corner Points

The corner points of the feasible region are the points where the constraint lines intersect. These points represent the extreme values of the feasible region and are potential candidates for the optimal solution. To find the coordinates of the corner points, we need to solve the system of equations formed by the intersecting lines. For example, if two lines intersect at a corner point, we can solve the equations of these two lines simultaneously to find the x₁ and x₂ coordinates of the intersection point. This can be done using various algebraic techniques, such as substitution or elimination. Each corner point represents a specific combination of x₁ and x₂ that satisfies the constraints, and we need to evaluate the objective function at each of these points to determine the optimal solution. The corner points are the key to finding the maximum or minimum value of the objective function within the feasible region.

4. Evaluating the Objective Function

Once we have identified all the corner points of the feasible region, the next step is to evaluate the objective function at each of these points. This involves substituting the x₁ and x₂ coordinates of each corner point into the objective function equation, Z = 716x₁ + 516x₂, and calculating the corresponding value of Z. The corner point that yields the highest value for Z is the optimal solution for a maximization problem, while the corner point that yields the lowest value for Z is the optimal solution for a minimization problem. This is based on the fundamental theorem of linear programming, which states that the optimal solution to a linear programming problem always occurs at a corner point of the feasible region. By evaluating the objective function at each corner point, we can systematically determine the optimal solution and the corresponding optimal value of the objective function.

5. Determining the Optimal Solution

After evaluating the objective function at each corner point, we can compare the values of Z to determine the optimal solution. In this case, we are looking to maximize Z, so the corner point with the highest value of Z represents the optimal solution. The x₁ and x₂ coordinates of this corner point indicate the optimal quantities of the decision variables, and the corresponding value of Z represents the maximum value of the objective function. This optimal solution satisfies all the constraints of the problem and yields the best possible outcome. It is important to note that there may be cases where multiple corner points yield the same optimal value of Z. In such cases, there are multiple optimal solutions, and any combination of x₁ and x₂ along the line segment connecting these corner points will also be optimal. The graphical method provides a clear and intuitive way to identify the optimal solution and understand the relationship between the constraints, the feasible region, and the objective function.

Applying the Graphical Method to the Problem

Now, let's apply the graphical method to our specific problem. We'll start by plotting the constraints on a graph. The constraints are:

  1. 8x₁ + 5x₂ ≤ 400
  2. 20x₁ + 8x₂ ≤ 250
  3. 15x₁ + 12x₂ ≤ 500

We'll convert each inequality into an equation to plot the lines. For the first constraint, 8x₁ + 5x₂ = 400, we can find two points: (0, 80) and (50, 0). For the second constraint, 20x₁ + 8x₂ = 250, we can find the points (0, 31.25) and (12.5, 0). And for the third constraint, 15x₁ + 12x₂ = 500, we can find the points (0, 41.67) and (33.33, 0). Plotting these lines on a graph, we can identify the feasible region. The feasible region is the area that satisfies all three constraints simultaneously. It's the polygon formed by the intersection of the regions defined by each inequality. Next, we need to find the corner points of the feasible region. These are the points where the constraint lines intersect. We can find these points by solving the equations of the intersecting lines simultaneously. The corner points will be the potential candidates for the optimal solution. Once we have the corner points, we'll evaluate the objective function, Z = 716x₁ + 516x₂, at each of these points. The corner point that yields the highest value for Z will be the optimal solution. This point represents the combination of x₁ and x₂ that maximizes our objective function while satisfying all the constraints. By following this process, we can effectively use the graphical method to solve our linear programming problem and find the optimal solution.

Finding Optimal Solutions and the Feasible Region

To find the optimal solutions, we must first accurately identify the feasible region defined by our constraints. Let's revisit the constraints:

  1. 8x₁ + 5x₂ ≤ 400
  2. 20x₁ + 8x₂ ≤ 250
  3. 15x₁ + 12x₂ ≤ 500

And our objective function: Z = 716x₁ + 516x₂. The feasible region is the area on the graph where all constraints are satisfied simultaneously. This area is bounded by the lines corresponding to each constraint and the axes (since x₁ and x₂ are typically non-negative). To accurately plot the feasible region, we first convert each inequality into an equation and plot the corresponding line. Then, we determine which side of each line represents the feasible region based on the inequality sign. For example, for the constraint 8x₁ + 5x₂ ≤ 400, we plot the line 8x₁ + 5x₂ = 400. Since the inequality is "less than or equal to," the feasible region lies on or below the line. We repeat this process for all three constraints. The feasible region is the area where all the individual feasible regions overlap. This region is typically a polygon, and its corners are known as corner points or vertices. These corner points are crucial because the optimal solution to a linear programming problem always occurs at one of these points. The corner points are the intersections of the constraint lines. To find the coordinates of these points, we solve the systems of equations formed by the intersecting lines. For instance, to find the intersection of the lines 8x₁ + 5x₂ = 400 and 20x₁ + 8x₂ = 250, we can solve these two equations simultaneously using methods like substitution or elimination. Once we have identified all the corner points, we evaluate the objective function at each of these points. The corner point that yields the highest value for Z is the optimal solution for our maximization problem. By carefully plotting the constraints, identifying the feasible region, and finding the corner points, we can determine the optimal solutions to our linear programming problem.

Detailed Evaluation and Analysis of Results

After determining the feasible region and its corner points, the crucial step is to evaluate the objective function, Z = 716x₁ + 516x₂, at each corner point. This process involves substituting the x₁ and x₂ coordinates of each corner point into the objective function and calculating the corresponding value of Z. The corner point that yields the highest value of Z is the optimal solution for our maximization problem. Let's assume, for the sake of illustration, that we have identified the following corner points:

  • Point A: (0, 0)
  • Point B: (12.5, 0)
  • Point C: (8.82, 16.47) (Intersection of 8x₁ + 5x₂ = 400 and 20x₁ + 8x₂ = 250)
  • Point D: (0, 41.67)

Now, we evaluate the objective function at each of these points:

  • At Point A (0, 0): Z = 716(0) + 516(0) = 0
  • At Point B (12.5, 0): Z = 716(12.5) + 516(0) = 8950
  • At Point C (8.82, 16.47): Z = 716(8.82) + 516(16.47) ≈ 15396.12
  • At Point D (0, 41.67): Z = 716(0) + 516(41.67) ≈ 21507.72

Comparing these values, we can see that Point C yields the highest value for Z (approximately 15396.12). Therefore, the optimal solution is x₁ = 8.82 and x₂ = 16.47, and the maximum value of the objective function is approximately 15396.12. This analysis provides us with a clear understanding of the optimal values for our decision variables and the maximum achievable value of our objective function. It's important to note that the accuracy of the solution depends on the precision with which we plot the constraints and identify the corner points. Graphical methods are best suited for problems with two decision variables, as visualizing feasible regions becomes challenging with more variables. In real-world applications, computer software is often used to solve linear programming problems with many variables and constraints.

Conclusion: Significance of the Graphical Method

In conclusion, the graphical method is a powerful and intuitive tool for solving linear programming problems with two decision variables. It provides a visual representation of the constraints, the feasible region, and the optimal solution, making it easier to understand the problem and its solution. By plotting the constraints as lines on a graph, identifying the feasible region, and evaluating the objective function at the corner points, we can systematically determine the optimal values for the decision variables and the maximum or minimum value of the objective function. The graphical method is particularly useful for teaching and learning linear programming concepts, as it provides a clear and intuitive understanding of the solution space and the location of optimal solutions. While the graphical method is limited to problems with two decision variables, the fundamental concepts and principles it illustrates are applicable to more complex linear programming problems with many variables and constraints. For larger problems, computer software and other mathematical techniques are used to find the optimal solutions. However, the graphical method remains a valuable tool for visualizing and understanding the basic principles of linear programming. By mastering the graphical method, we can gain a solid foundation for tackling more complex optimization problems in various fields, including economics, engineering, and operations research. The ability to formulate and solve linear programming problems is a valuable skill for decision-making and resource allocation, and the graphical method is an excellent starting point for developing this skill.