Question

A manufacturer is making products P and Q. Product P requires 2 kilograms of material and 1 hour of labour to be made and product Q requires 1 kilogram of material and 2 hours of labour time. Each day there is 6 kilograms of raw material and 8 hours of labour time available. The profits for selling each unit of products P and Q are £200 and £300, respectively. The manufacturer wishes to maximise its profit.

If xPand xQ represent the amount of products P and Q to be made, respectively, then the linear programme for meeting the manufacturers objective is
max z = 200xP+300xQ subject to 2xP+xQ ≤ 6 xP+2xQ ≤ 8 xP,xQ ≥ 0. This problem is solved using R as follows
> ObjectiveCoefficients<-c(200,300)
> ConstraintMatrix<-matrix(c(2,1,1,2),nrow=2,byrow=TRUE)
> Directions<-c("<=","<=")
> RHS<-c(6,8)
> lp("max",ObjectiveCoefficients,ConstraintMatrix,Directions,RHS)
Success: the objective function is 1266.667
> lp("max",ObjectiveCoefficients,ConstraintMatrix,Directions,RHS)$solution
[1] 1.333333 3.333333
which tells us the the optimal value of the objective function is £1266.667. This is reached when xP=1.333333 and xQ=3.333333.
The manufacturer's advisors use R as follows:
lp("max",ObjectiveCoefficients,ConstraintMatrix,Directions,RHS,compute.sens=TRUE)$duals.from
[1] 4e+00 3e+00 -1e+30 -1e+30
lp("max",ObjectiveCoefficients,ConstraintMatrix,Directions,RHS,compute.sens=TRUE)$duals.to
[1] 1.6e+01 1.2e+01 1.0e+30 1.0e+30
in no more than 50 words, give a meaningful interpretation of the results (you should ignore the numbers -1e+30 and 1e+30).

Answer 1

A linear programming problem in R has been solved to identify the optimal production levels of products P and Q, with the maximum profit calculated at £1266.67. The sensitivity analysis provides shadow prices for material and labor constraints, indicating their value to the profit maximization.

Step-by-step explanation:

The R programming language has been utilized to solve a linear programming problem to determine the production levels of two products, P and Q, to maximize profits given certain constraints on materials and labor. The optimal solution indicates producing approximately 1.33 units of product P and 3.33 units of product Q, resulting in a maximum profit of £1266.67.

The accompanying sensitivity analysis, represented by 'duals.from' and 'duals.to', gives the ranges for the shadow prices of the resources. These ranges indicate how much the profit contribution of each resource would change if the availability of the material or labor varied within certain limits. The results suggest that the constraint on material has a shadow price of £4 per kilogram, while the constraint on labor has a shadow price of £3 per hour, within the specified ranges.