Optimal Production Plan for Maximizing Revenue: Gentleman and Ladies’ Hats

Introduction

This report utilizes linear programming to identify the optimal production plan for maximizing revenue from gentleman hats and ladies’ hats, given specific constraints on cutting and sewing times.

Problem Description

We aim to maximize revenue from producing gentleman hats and ladies’ hats under the constraints of available cutting and sewing times. The specifics are as follows:

• Gentleman Hat: Price = 50 EUR, Cutting Time = 5 min, Sewing Time = 10 min.
• Ladies’ Hat: Price = 75 EUR, Cutting Time = 10 min, Sewing Time = 8 min.
• Available capacity for cutting and sewing: 480 minutes each.

The objective is to determine the optimal number of gentleman hats and ladies’ hats to produce to maximize revenue.

Methodology

The problem is approached through linear programming, utilizing the R package lpSolve to find the optimal production quantities that maximize revenue.

# Ensure the lpSolve package is available
if (!require("lpSolve")) {
  install.packages("lpSolve")
}

## Loading required package: lpSolve

library(lpSolve)

# Define the profit per unit for each product
profit <- c(50, 75) # Gentleman hat, Ladies' hat

# Define the matrix of constraints
constraints <- matrix(c(5, 10,  # Cutting times
                        10, 8),  # Sewing times
                      nrow = 2, byrow = TRUE)

# Define the available capacity for cutting and sewing
capacity <- c(480, 480)

# Define the direction of the constraints
directions <- c("<=", "<=")

# Solve the LP problem
solution <- lp("max", profit, constraints, directions, capacity, all.int = TRUE)

Results

The optimal production plan derived from the linear programming model is as follows:

if(solution$status == 0) {
  cat("Optimal production plan to maximize revenue:\n")
  cat("Gentleman Hats:", solution$solution[1], "units\n")
  cat("Ladies' Hats:", solution$solution[2], "units\n")
  cat("Maximum Revenue (EUR):", sum(solution$solution * profit), "\n")
} else {
  cat("An optimal solution could not be found.\n")
}

## Optimal production plan to maximize revenue:
## Gentleman Hats: 16 units
## Ladies' Hats: 40 units
## Maximum Revenue (EUR): 3800

Conlusion

From these results, I can interpret that the linear programming model has found that to maximize revenue within the constraints of the available cutting and sewing times, you should produce 16 gentleman hats and 40 ladies’ hats. This production plan will yield the highest possible revenue, which is calculated to be 3800 EUR.

The model takes into account the different times required for cutting and sewing each type of hat, as well as their respective selling prices. By allocating production time between the two products in the proportions suggested, you make the most efficient use of the time available to generate the greatest amount of revenue.

Discussion

Solving this optimization problem in R was a fascinating and educational experience. It showcased the power of linear programming and the efficiency of R for data analysis and operational research. The lpSolve package proved to be an invaluable tool, providing a straightforward interface to implement complex mathematical models.

One of the key learnings from this exercise was the importance of accurately defining the constraints and objective function to ensure that the solution is not only optimal but also feasible in a real-world production environment.

My experiences

In tackling this assignment, I journeyed from drawing graphs to using Excel’s Solver and finally to crafting an R Markdown document. Each step brought its own insights and challenges.

Initially, the graph made the problem clear, but it was only a starting point. It was a good way to visualize the problem, but it didn’t give me all the answers. Next, I moved to using Excel’s Solver. This was a step up—it did the math for me and felt more hands-on. It was offering a more automated way to find the solution. However, it was the move to R Markdown that brought everything together. The lpSolve package in R handled the computations, and Markdown took care of the reporting.

At first, the complexity of linear programming was overwhelming. Linear programming sounded complicated, and I wasn’t sure I could handle it, but by the end of the project, as I saw the optimal solution emerge step by step, my shock turned to joy. Completing the task brought a deep sense of accomplishment. It’s been a journey from daunting to doable, ending in a satisfying success.

Working through this problem showed me how powerful these tools can be. It was a bit scary at first, but now I see how they help us make better decisions. It’s not just about hats—it’s about learning to solve problems, and feeling good when you do.

When I first took on this task, I dove straight into using R Markdown. I thought it would be straightforward—a neat way to blend my analysis and reporting. But somehow, things didn’t go as planned. The code didn’t work on my first try, and it was frustrating.

One of the key learnings from this exercise was the importance of accurately defining the constraints and objective function to ensure that the solution is not only optimal but also feasible in a real-world production environment. The process underscored the need for careful data preparation and a deep understanding of the problem domain.

P.S. It would be interesting to utilize R Markdown for our data analyses to our thesis.