Introduction

This report utilizes linear programming to identify the optimal production plan for maximizing revenue from standard and premium backpacks, given specific constraints on stitching and quality control times.

Problem Description

We aim to maximize revenue from producing standard and premium backpacks under the constraints of available stitching and quality control times. The specifics are as follows:

The objective is to determine the optimal number of standard and premium backpacks to produce to maximize revenue.

Methodology

The first step is to define the objective function, which represents the quantity to be optimized (e.g., maximize revenue, minimize cost). In my case, my aim to maximize revenue, so the objective function is defined as a linear combination of the profits associated with each product. Next, we specify the constraints that limit the feasible solutions. These constraints are typically linear inequalities or equalities representing limitations on resources, capacities, or other factors. Once the objective function and constraints are defined, we use the lpSolve function to solve the linear programming problem.

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(30, 50) # Standard Backpack, Premium Backpack

# Define the matrix of constraints
constraints <- matrix(c(6, 8,  # Stitching times
                        5, 10),  # Quality Control times
                      nrow = 2, byrow = TRUE)

# Define the available capacity for stitching and quality control
capacity <- c(700, 600)

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

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

Results

After solving the linear programming problem, I obtain the optimal solution, which consists of the values that maximize (or we can minimize) the objective function while satisfying all constraints. These results provide insights into the optimal production quantities for each product and the maximum achievable revenue.

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

if(solution$status == 0) {
  cat("Optimal production plan to maximize revenue:\n")
  cat("Standard Backpacks:", solution$solution[1], "units\n")
  cat("Premium Backpacks:", 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:
## Standard Backpacks: 110 units
## Premium Backpacks: 5 units
## Maximum Revenue (EUR): 3550

Conclusion

This optimal production plan suggests producing 110 units of Standard Backpacks and 5 units of Premium Backpacks to maximize revenue, resulting in a total revenue of 3550 EUR.

In linear programming, such solutions are derived by mathematically optimizing the objective function—in this case, maximizing revenue—subject to given constraints, such as stitching and quality control capacities. The model identifies the most efficient allocation of resources (in this case, stitching and quality control time) to achieve the highest possible outcome (revenue).

Discussion

Although I faced more challenges in the coding aspect compared to the previous task due to inaccuracies in providing the data, ultimately, I managed to identify the error. Each line of code, every parameter, and every function call plays a crucial role in arriving at the correct solution.

Personal experiences with linear programming and R have been enlightening. Working with R’s lpSolve package, I’ve gained a deeper understanding of optimization techniques and their practical applications.