Let { X $}$ Be The Number Of Chairs And { Y $}$ The Number Of Sofas. A Furniture Manufacturer Produces Chairs And Sofas. Each Chair Requires 10 Yards Of Fabric, And Each Sofa Requires 20 Yards Of Fabric. The Manufacturer Has 300
Introduction
In this article, we will explore a classic problem in linear programming, which is a method to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships. The problem is as follows: Let { x $}$ be the number of chairs and { y $}$ the number of sofas. A furniture manufacturer produces chairs and sofas. Each chair requires 10 yards of fabric, and each sofa requires 20 yards of fabric. The manufacturer has 300 yards of fabric available. The goal is to determine the optimal production levels of chairs and sofas to maximize profit.
Problem Formulation
To solve this problem, we need to formulate it as a linear programming problem. Let's define the variables:
- { x $}$: number of chairs produced
- { y $}$: number of sofas produced
- { z $}$: total profit (in dollars)
The objective function is to maximize the total profit, which is given by:
{ z = 50x + 100y $}$
The constraints are:
- { 10x + 20y \leq 300 $}$: the total amount of fabric used cannot exceed 300 yards
- { x \geq 0 $}$: the number of chairs produced cannot be negative
- { y \geq 0 $}$: the number of sofas produced cannot be negative
Graphical Method
To solve this problem, we can use the graphical method, which involves graphing the constraints on a coordinate plane and finding the feasible region. The feasible region is the area where all the constraints are satisfied.
The first constraint, { 10x + 20y \leq 300 $}$, can be graphed as a line with a slope of -1/2 and a y-intercept of 15. The second and third constraints, { x \geq 0 $}$ and { y \geq 0 $}$, are the x-axis and y-axis, respectively.
The feasible region is the area below the line { 10x + 20y = 300 $}$ and above the x-axis and y-axis.
Optimal Solution
To find the optimal solution, we need to find the point in the feasible region that maximizes the objective function. This point is called the optimal solution.
The optimal solution can be found by drawing a line with a slope of -50/100 = -1/2 and a y-intercept of 5, which is the negative reciprocal of the slope of the constraint line. The point where this line intersects the feasible region is the optimal solution.
The optimal solution is { x = 10, y = 10 $}$, which means that the manufacturer should produce 10 chairs and 10 sofas to maximize profit.
Sensitivity Analysis
Sensitivity analysis is a technique used to analyze how changes in the coefficients of the objective function and constraints affect the optimal solution. In this problem, we can perform sensitivity analysis by changing the coefficients of the objective function and constraints and re-solving the problem.
For example, if the price of a chair increases by 10%, the new objective function becomes { z = 55x + 100y $}$. The optimal solution remains the same, { x = 10, y = 10 $}$.
However, if the price of a sofa increases by 10%, the new objective function becomes { z = 50x + 110y $}$. The optimal solution changes to { x = 10, y = 8.18 $}$.
Conclusion
In this article, we have solved a linear programming problem using the graphical method. We have found the optimal solution, which is the point in the feasible region that maximizes the objective function. We have also performed sensitivity analysis to analyze how changes in the coefficients of the objective function and constraints affect the optimal solution.
The problem is a classic example of a linear programming problem, and it has many real-world applications in fields such as business, economics, and engineering.
References
- Chvátal, V. (1983). Linear Programming. W.H. Freeman and Company.
- Dantzig, G.B. (1963). Linear Programming and Extensions. Princeton University Press.
- Hillier, F.S., & Lieberman, G.J. (2015). Introduction to Operations Research. McGraw-Hill Education.
Further Reading
- Linear Programming: A Tutorial by Stanford University
- Linear Programming: A Guide by MIT OpenCourseWare
- Linear Programming: A Tutorial by University of California, Berkeley
Glossary
- Linear Programming: a method to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships.
- Objective Function: a mathematical function that represents the goal of the problem.
- Constraints: limitations or restrictions on the variables in the problem.
- Feasible Region: the area where all the constraints are satisfied.
- Optimal Solution: the point in the feasible region that maximizes the objective function.
- Sensitivity Analysis: a technique used to analyze how changes in the coefficients of the objective function and constraints affect the optimal solution.
Introduction
In our previous article, we explored a classic problem in linear programming, which is a method to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships. The problem is as follows: Let { x $}$ be the number of chairs and { y $}$ the number of sofas. A furniture manufacturer produces chairs and sofas. Each chair requires 10 yards of fabric, and each sofa requires 20 yards of fabric. The manufacturer has 300 yards of fabric available. The goal is to determine the optimal production levels of chairs and sofas to maximize profit.
Q&A
Q: What is linear programming?
A: Linear programming is a method to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships.
Q: What is the objective function in linear programming?
A: The objective function is a mathematical function that represents the goal of the problem. In our example, the objective function is { z = 50x + 100y $}$, which represents the total profit.
Q: What are the constraints in linear programming?
A: The constraints are limitations or restrictions on the variables in the problem. In our example, the constraints are { 10x + 20y \leq 300 $}$, { x \geq 0 $}$, and { y \geq 0 $}$.
Q: What is the feasible region in linear programming?
A: The feasible region is the area where all the constraints are satisfied. In our example, the feasible region is the area below the line { 10x + 20y = 300 $}$ and above the x-axis and y-axis.
Q: How do I find the optimal solution in linear programming?
A: To find the optimal solution, you need to find the point in the feasible region that maximizes the objective function. This point is called the optimal solution.
Q: What is sensitivity analysis in linear programming?
A: Sensitivity analysis is a technique used to analyze how changes in the coefficients of the objective function and constraints affect the optimal solution.
Q: How do I perform sensitivity analysis in linear programming?
A: To perform sensitivity analysis, you need to change the coefficients of the objective function and constraints and re-solve the problem.
Q: What are some real-world applications of linear programming?
A: Linear programming has many real-world applications in fields such as business, economics, and engineering.
Q: What are some common mistakes to avoid in linear programming?
A: Some common mistakes to avoid in linear programming include:
- Not defining the objective function clearly
- Not identifying all the constraints
- Not checking for feasibility
- Not performing sensitivity analysis
Q: How do I choose the right method for solving a linear programming problem?
A: To choose the right method for solving a linear programming problem, you need to consider the size and complexity of the problem, as well as the availability of computational resources.
Conclusion
In this article, we have answered some common questions about linear programming, including what it is, how to find the optimal solution, and how to perform sensitivity analysis. We have also discussed some real-world applications of linear programming and some common mistakes to avoid.
References
- Chvátal, V. (1983). Linear Programming. W.H. Freeman and Company.
- Dantzig, G.B. (1963). Linear Programming and Extensions. Princeton University Press.
- Hillier, F.S., & Lieberman, G.J. (2015). Introduction to Operations Research. McGraw-Hill Education.
Further Reading
- Linear Programming: A Tutorial by Stanford University
- Linear Programming: A Guide by MIT OpenCourseWare
- Linear Programming: A Tutorial by University of California, Berkeley
Glossary
- Linear Programming: a method to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships.
- Objective Function: a mathematical function that represents the goal of the problem.
- Constraints: limitations or restrictions on the variables in the problem.
- Feasible Region: the area where all the constraints are satisfied.
- Optimal Solution: the point in the feasible region that maximizes the objective function.
- Sensitivity Analysis: a technique used to analyze how changes in the coefficients of the objective function and constraints affect the optimal solution.