Linear Optimization Problem
A company wants to maximize the profit for two products A and B which are sold at $ 25 and $ 20 respectively. There are 1800 resource units available every day and product A requires 20 units while B requires 12 units. Both of these products require a production time of 4 minutes and total available working hours are 8 in a day. What should be the production quantity for each of the products to maximize profits.
The objective function in the above problem will be:
max(Sales) = max(25 x1 + 20 x2)
where,
x1 is the units of Product A produced
x2 is the units of Product B produced
x1 and x2 are also called the decision variables
The constraints (resource and time) in the problem:
20x1 + 12 x2 <= 1800 (Resource Constraint)
4x1 + 4x2 <= 8*60 (Time constraint)
## Load the package lpsolve
library(lpSolve)
## Set the coefficients of the decision variables
objective.in <- c(25, 20)
## Create constraint martix
const.mat <- matrix(c(20, 12, 4, 4), nrow=2, byrow=TRUE)
## define constraints
time_constraint <- (8*60)
resource_constraint <- 1800
## RHS for the constraints
const.rhs <- c(resource_constraint, time_constraint)
## Constraints direction
const.dir <- c("<=", "<=")
## Find the optimal solution
optimum <- lp(direction="max", objective.in, const.mat, const.dir, const.rhs)
## Display the optimum values for x1 and x2
optimum$solution
## [1] 45 75
## Check the value of objective function at optimal point
optimum$objval
## [1] 2625
From the above output, we can see that the company should produce 45 units of Product A and 75 units of Product B to get sales of $2625, which is the maximum sales that company can get given the constraints.