NFSCI 310 Computation in Information ScienceHomework 5

Linear programming

Consider the linear programming problem below both by R and hand.

1. Draw the feasible solution space.
2. Solve for the optimal solution with the simplex method.
3. Solve for the optimal solution using R.

Maximize

                    Z = 12 X1 + 15X2      

Subject to

                   4X1 +  3X2  ≤  12
                   
                   2X1  + 5X2  ≤  10
                   
                   X1  ≥ 0   
                   
                   X2 ≥  0 
                   

library(lpSolve)

# Define the objective function coefficients
objective.coef <- c(12, 15)

# Define the constraints
const.mat <- matrix(c(4, 2, 3, 5), nrow = 2, byrow = TRUE)
const.rhs <- c(12, 10)
const.dir <- c("<=", "<=")

# Solve the problem
lp.solution <- lp(direction = "max", objective.in = objective.coef, 
                  const.mat = const.mat, const.dir = const.dir, const.rhs = const.rhs, 
                  all.int = FALSE)

# Check if the solution is found
if (lp.solution$status == 0) {
    # Output the optimal values of X1 and X2
    optimal.values <- lp.solution$solution
    cat("Optimal values:\n")
    cat("X1 =", optimal.values[1], "\n")
    cat("X2 =", optimal.values[2], "\n")
    
    # Output the maximum value of Z
    max.Z <- sum(optimal.values * objective.coef)
    cat("Maximum value of Z =", max.Z, "\n")
} else {
    cat("Solution not found or infeasible problem.")
}

Optimal values:
X1 = 2.857143 
X2 = 0.2857143 
Maximum value of Z = 38.57143