Question $3(25$ marks$)$

The sensitivity report of a particular linear programming problem $($profit maximisation$)$ is given as follows:

Adjustable cells

$\backslash$table$[[$Cell$,$Name,$\backslash$table$[[$Final$],[$value$]],\backslash$table$[[$Reduced$],[$cost$]],\backslash$table$[[$Objective$],[$coefficient$]],\backslash$table$[[$Allowable$],[$increase$]],\backslash$table$[[$Allowable$],[$decrease$]]],[$$C$$13,$Paint A$,4,0,5,1,2],[$$D$$13,$Paint B$,2,0,4,6,0\mathrm{.}6666667]]$

Constraints

$\backslash$table$[[$Cell$,$Name,$\backslash$table$[[$Final$],[$value$]],\backslash$table$[[$Shadow$],[$price$]],\backslash$table$[[$Constraint$],[$R$.$H$.$S$.]],\backslash$table$[[$Allowable$],[$increase$]],\backslash$table$[[$Allowable$],[$decrease$]]],[$$E$$7,$Raw Material X$,24,0\mathrm{.}75,24,12,4],[$$E$$8,$Raw Material Y$,6,0\mathrm{.}5,6,0\mathrm{.}6666667,2],[$$E$$9,$Demand S$,-1\mathrm{.}5,0,1,1E+30,2\mathrm{.}5],[$$E$$10,$Demand T$,1\mathrm{.}5,0,2,1E+30,0\mathrm{.}5]]$

a Regarding the linear programming problem, what is the optimal solution?

$(7$ marks$)$

b Suppose that the right$-$hand$-$side value of 'Raw Material X$\text{'}$ changes to $30,$ what can be said about the effect on the total profit? $(8$ marks$)$

c Suppose that the unit profit of 'Paint A$\text{'}$ and 'Paint B$\text{'}$ both decrease by $0\mathrm{.}6,$ would the optimal solution change? If not, what is the revised optimal profit that can be earned?

$(10$ marks$)$