[M]-Linear-Optimisation.md

Linear Optimisation

Sample Task 1

Maximise profits with the given constraints.

Maximise total profit using decision variables X1 = volume of apple juice, X2 = volume of orange juice	Profit = 11X1 + 12X2
Budget Constraint	0.1X1 + 0.2X2 ≤ 12
Space Constraint	1.1X1 + 1.2X2 ≤ 34
Contract Constraint 1	X1 ≥ 5
Contract Constraint 2	X2 ≥ 5
Non-negativity Constraint 1	X1 ≥ 0
Non-negativity Constraint 2	X2 ≥ 0

Preparation Code

# Functions
library(lpSolve)

Actual Code

1. Run the linear optimisation model.

# Define parameters
objective.fn <- c(11, 22)

const.mat <- (matrix(
  c(0.1, 0.2,
    1.1, 1.2,
    1, 0,
    0, 1), # R assumes non-negativity, so the non-negativity constraint does not need to be specified
  ncol = 2, # Number of decision variables
  byrow = TRUE))

const.dir <- c("<=","<=",">=",">=")

const.rhs <- c(12,34,5,5)

# Run the model
lp.solution <- lp("max", objective.fn, const.mat, const.dir, const.rhs,
                  compute.sens = TRUE)

lp.solution
lp.solution$solution

2. If required, obtain shadow prices.

lp.solution$duals

3. If required, conduct sensitivity analysis.

lp.solution$sens.coef.to # Upper bounds to the coefficients
lp.solution$sens.coef.from # Lower bounds to the coefficients

Sample Task 2

Minimise costs with the given constraints.

Minimise total cost using decision variables X1 = volume of apple juice, X2 = volume of orange juice	Cost = 11X1 + 12X2
Budget Constraint	0.1X1 + 0.2X2 ≤ 12
Space Constraint	1.1X1 + 1.2X2 ≤ 34
Contract Constraint 1	X1 ≥ 5
Contract Constraint 2	X2 ≥ 5
Non-negativity Constraint 1	X1 ≥ 0
Non-negativity Constraint 2	X2 ≥ 0

Preparation Code

# Functions
library(lpSolve)

Actual Code

1. Run the linear optimisation model.

objective.fn <- c(11, 22)

const.mat <- (matrix(
  c(0.1, 0.2,
    1.1, 1.2,
    1, 0,
    0, 1), # R assumes non-negativity, so the non-negativity constraint does not need to be specified
  ncol = 2, # Number of decision variables
  byrow = TRUE))

const.dir <- c("<=","<=",">=",">=")

const.rhs <- c(12,34,5,5)

lp.solution <- lp("min", objective.fn, const.mat, const.dir, const.rhs,
                  compute.sens = TRUE)

lp.solution
lp.solution$solution

2. If required, obtain shadow prices.

lp.solution$duals

3. If required, conduct sensitivity analysis.

lp.solution$sens.coef.to # Upper bounds to the coefficients
lp.solution$sens.coef.from # Lower bounds to the coefficients