Number of Watches Constraint

Oct 13, 2025 by windailys.com 85 views

Introduction

John is a skilled watch repairman who has to fix at least 12 watches within a time frame of 40 hours. His task involves repairing both digital and analog watches, each requiring a different amount of time to fix. In this article, we will explore the inequalities that represent this situation and discuss how John can optimize his time management to complete the task efficiently.

Digital Watches

Digital watches take John 2 hours to fix. Let's assume that John has to fix $x$ digital watches. The total time spent on fixing digital watches can be represented as $2x$ hours.

Analog Watches

Analog watches take John $\delta$ hours to fix. Let's assume that John has to fix $y$ analog watches. The total time spent on fixing analog watches can be represented as $\delta y$ hours.

Total Time Constraint

John has to complete the task within 40 hours. The total time spent on fixing both digital and analog watches cannot exceed 40 hours. This can be represented as an inequality:

2x + \delta y \leq 40

Number of Watches Constraint

John has to fix at least 12 watches. The total number of digital and analog watches cannot be less than 12. This can be represented as an inequality:

x + y \geq 12

Non-Negativity Constraints

Since John cannot fix a negative number of watches, both $x$ and $y$ must be non-negative. This can be represented as:

x \geq 0

y \geq 0

Optimization

John wants to optimize his time management to complete the task efficiently. He wants to minimize the total time spent on fixing the watches while ensuring that he fixes at least 12 watches. This can be represented as a linear programming problem:

Minimize: $2x + \delta y$ Subject to:

2x + \delta y \leq 40

x + y \geq 12

x \geq 0

y \geq 0

Solution

To solve this linear programming problem, we can use various methods such as the graphical method, simplex method, or interior-point method. For simplicity, let's assume that we use the graphical method.

We can plot the constraints on a graph and find the feasible region. The feasible region is the area where all the constraints are satisfied.

The objective function is to minimize the total time spent on fixing the watches. We can plot the objective function as a line on the graph.

The optimal solution is the point where the objective function is minimized within the feasible region.

Conclusion

In this article, we have discussed the inequalities that represent John's situation of fixing at least 12 watches within 40 hours. We have also discussed how John can optimize his time management to complete the task efficiently using linear programming.

The solution to the linear programming problem involves finding the feasible region and minimizing the objective function within that region. The optimal solution is the point where the objective function is minimized within the feasible region.

References

[1] Chvátal, V. (1983). Linear Programming. W.H. Freeman and Company.
[2] Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research. McGraw-Hill Education.
[3] Winston, W. L. (2018). Operations Research: Applications and Algorithms. Cengage Learning.

Q: What is the main goal of John's task?

A: John's main goal is to fix at least 12 watches within a time frame of 40 hours.

Q: What is the time constraint for fixing digital watches?

A: Digital watches take John 2 hours to fix.

Q: What is the time constraint for fixing analog watches?

A: Analog watches take John $\delta$ hours to fix.

Q: What is the total time constraint for fixing both digital and analog watches?

A: The total time spent on fixing both digital and analog watches cannot exceed 40 hours, which can be represented as the inequality $2x + \delta y \leq 40$ .

Q: What is the number of watches constraint?

A: John has to fix at least 12 watches, which can be represented as the inequality $x + y \geq 12$ .

Q: What are the non-negativity constraints?

A: Since John cannot fix a negative number of watches, both $x$ and $y$ must be non-negative, which can be represented as $x \geq 0$ and $y \geq 0$ .

Q: What is the objective function in the linear programming problem?

A: The objective function is to minimize the total time spent on fixing the watches, which can be represented as $2x + \delta y$ .

Q: How can John optimize his time management to complete the task efficiently?

A: John can use linear programming to optimize his time management. He can find the feasible region and minimize the objective function within that region to find the optimal solution.

Q: What is the optimal solution to the linear programming problem?

A: The optimal solution is the point where the objective function is minimized within the feasible region.

Q: How can John use the graphical method to solve the linear programming problem?

A: John can plot the constraints on a graph and find the feasible region. He can then plot the objective function as a line on the graph and find the point where the objective function is minimized within the feasible region.

Q: What are some common methods for solving linear programming problems?

A: Some common methods for solving linear programming problems include the graphical method, simplex method, and interior-point method.

Q: What are some resources for learning more about linear programming and operations research?

A: Some resources for learning more about linear programming and operations research include textbooks such as "Linear Programming" by V. Chvátal and "Introduction to Operations Research" by F.S. Hillier and G.J. Lieberman, as well as online resources such as Stanford University's Linear Programming tutorial and MIT OpenCourseWare's Operations Research course.

Q: What are some real-world applications of linear programming and operations research?

A: Some real-world applications of linear programming and operations research include scheduling, resource allocation, and supply chain management.