Which Ordered Pair Minimizes The Objective Function C 60x 85y

Which Ordered Pair Minimizes the Objective Function C = 60x + 85y? A Comprehensive Guide to Linear Programming

Finding the ordered pair that minimizes an objective function like C = 60x + 85y is a core concept in linear programming. This technique is widely used across various fields, from optimizing business operations to managing resources efficiently. This comprehensive guide will walk you through the process, explaining the underlying principles and providing practical examples.

Understanding Linear Programming and Objective Functions

Linear programming is a mathematical method used to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. A crucial part of this is the objective function, which represents the quantity you want to maximize or minimize. In our case, the objective function is C = 60x + 85y. This means we're aiming to find the values of x and y that result in the smallest possible value for C.

Key Components:

• Objective Function: The function you want to optimize (minimize or maximize). In this instance, it's C = 60x + 85y.
• Decision Variables: These are the variables you can control, which are x and y in our example. They represent quantities or choices that affect the objective function.
• Constraints: These are limitations or restrictions on the decision variables. They often represent real-world limitations, such as resource availability or production capacity. We'll explore these constraints in detail later.

The Graphical Method: Visualizing the Solution

Before diving into complex algorithms, let's understand the graphical method for solving linear programming problems. This method is particularly useful for problems with two decision variables (like our example).

1. Defining the Constraints

To minimize C = 60x + 85y, we need constraints. These constraints define the feasible region—the set of all possible combinations of x and y that satisfy all the limitations. Let's assume we have the following constraints:

• x ≥ 0: The value of x must be non-negative.
• y ≥ 0: The value of y must be non-negative.
• x + y ≤ 10: The sum of x and y cannot exceed 10.
• 2x + y ≤ 16: A second constraint limiting the combined values of x and y.

These constraints represent practical limitations; for example, they could represent limitations on available resources or production capacity.

2. Graphing the Constraints

Plot these constraints on a graph with x as the horizontal axis and y as the vertical axis. Each inequality defines a region. The feasible region is the area where all the constraint regions overlap.

• x ≥ 0: This is the region to the right of the y-axis.
• y ≥ 0: This is the region above the x-axis.
• x + y ≤ 10: This inequality can be rewritten as y ≤ -x + 10. Plot the line y = -x + 10, and shade the region below the line.
• 2x + y ≤ 16: This inequality can be rewritten as y ≤ -2x + 16. Plot the line y = -2x + 16 and shade the region below this line.

The feasible region is the area where all shaded regions overlap. It will be a polygon (in this case, a quadrilateral).

3. Identifying the Corner Points

The optimal solution (the point that minimizes or maximizes the objective function) will always lie at one of the corner points (vertices) of the feasible region. Identify the coordinates of these corner points.

In our example, you'll find that the corner points are:

• (0, 0)
• (0, 10)
• (6, 4)
• (8, 0)

4. Evaluating the Objective Function

Substitute the coordinates of each corner point into the objective function C = 60x + 85y:

• (0, 0): C = 60(0) + 85(0) = 0
• (0, 10): C = 60(0) + 85(10) = 850
• (6, 4): C = 60(6) + 85(4) = 360 + 340 = 700
• (8, 0): C = 60(8) + 85(0) = 480

5. Determining the Minimum Value

By comparing the values of C for each corner point, we find that the minimum value of the objective function is 0, which occurs at the ordered pair (0, 0).

The Simplex Method: A More Robust Approach

For problems with more than two decision variables, the graphical method becomes impractical. The simplex method is an iterative algorithm that systematically explores the feasible region to find the optimal solution. It's more complex but handles higher-dimensional problems efficiently. The detailed explanation of the simplex method is beyond the scope of this introductory article, but it's important to know that it's a powerful tool for larger linear programming problems.

Applications of Linear Programming and Optimization

The principles of linear programming and objective function minimization have extensive real-world applications:

• Production Planning: Determining the optimal production levels of different products to maximize profit while considering resource constraints (materials, labor, machine time).
• Transportation: Optimizing routes and shipment sizes to minimize transportation costs.
• Portfolio Management: Creating investment portfolios that maximize returns while managing risk.
• Resource Allocation: Distributing limited resources (budget, personnel) across different projects or departments to achieve the best overall outcome.
• Scheduling: Optimizing schedules for tasks, personnel, or machines to minimize delays or maximize efficiency.

Software Tools for Linear Programming

Several software packages are available to solve linear programming problems. These tools automate the process, especially for complex problems with numerous variables and constraints. These include specialized software packages and even add-ins for spreadsheet programs.

Beyond the Basics: Advanced Concepts

This introduction covered fundamental concepts. More advanced topics in linear programming include:

• Integer Programming: Constraints that require decision variables to be integers (whole numbers). This is essential when dealing with indivisible units, like the number of machines or employees.
• Nonlinear Programming: Problems where the objective function or constraints are nonlinear. These require more sophisticated solution techniques.
• Sensitivity Analysis: Studying how changes in the objective function coefficients or constraint parameters affect the optimal solution. This helps in understanding the robustness of the solution.
• Duality Theory: A powerful concept in linear programming that provides insights into the problem's structure and allows for alternative solution approaches.

Conclusion: Mastering the Art of Optimization

Understanding how to minimize an objective function like C = 60x + 85y is a valuable skill in various fields. Whether you use the graphical method for simpler problems or the simplex method for complex ones, the ability to optimize resource allocation and achieve the best possible outcome is crucial for efficient decision-making. Remember that the core principles of identifying constraints, defining the feasible region, and evaluating corner points remain consistent across different problem complexities. As you delve deeper into linear programming, exploring advanced techniques and software tools will enhance your ability to tackle more challenging optimization problems effectively.