Challenging +1.2 This is a multi-part linear programming question requiring formulation, solving a 2-variable LP graphically, and then analyzing how a parametric change affects optimality. Parts (a)-(c) are standard Further Maths Decision content. Part (d) requires understanding that the optimal solution changes when leftover beads have value, necessitating comparison of corner points with modified objective functions—this adds moderate complexity. Part (e) is a straightforward interpretation question. Overall, this is above-average difficulty due to the parametric analysis in part (d), but the LP itself is routine and the question provides significant scaffolding.
6 Sasha is making three identical bead bracelets using amber, brown and red coloured beads. Sasha has 20 amber beads, 12 brown beads and 10 red beads.
Each bracelet must use exactly 12 beads.
The profit from selling a bracelet is 6 pence for each amber bead used plus 2 pence for each brown bead used plus 3 pence for each red bead used.
Sasha wants to maximise the total profit from selling the three bracelets.
Express Sasha's problem as a linear programming formulation in two variables \(a\) and \(b\), where \(a\) represents the number of amber beads in each bracelet and \(b\) represents the number of brown beads in each bracelet.
Determine how many beads of each colour will be used in each bracelet.
By listing all the feasible solutions, identify an aspect of the optimal solution, other than the profit, that is different from all the other feasible solutions.
The beads that are not used in making the bracelets can be sold.
The profit from selling each amber bead is \(k\) pence, where \(k\) is an integer, but nothing for each brown or red bead sold.
All the previous constraints still apply.
Instead of maximising the profit from the bracelets, Sasha wants to maximise the total profit from selling the bracelets and any left over beads.
You are given that the optimal solution to the earlier problem does not maximise the total profit from selling the bracelets and any left over beads.
Determine the least possible value of Sasha's maximum total profit.
6 Sasha is making three identical bead bracelets using amber, brown and red coloured beads. Sasha has 20 amber beads, 12 brown beads and 10 red beads.
Each bracelet must use exactly 12 beads.\\
The profit from selling a bracelet is 6 pence for each amber bead used plus 2 pence for each brown bead used plus 3 pence for each red bead used.
Sasha wants to maximise the total profit from selling the three bracelets.
\begin{enumerate}[label=(\alph*)]
\item Express Sasha's problem as a linear programming formulation in two variables $a$ and $b$, where $a$ represents the number of amber beads in each bracelet and $b$ represents the number of brown beads in each bracelet.
\item Determine how many beads of each colour will be used in each bracelet.
\item By listing all the feasible solutions, identify an aspect of the optimal solution, other than the profit, that is different from all the other feasible solutions.
The beads that are not used in making the bracelets can be sold.
The profit from selling each amber bead is $k$ pence, where $k$ is an integer, but nothing for each brown or red bead sold.
All the previous constraints still apply.
Instead of maximising the profit from the bracelets, Sasha wants to maximise the total profit from selling the bracelets and any left over beads.
You are given that the optimal solution to the earlier problem does not maximise the total profit from selling the bracelets and any left over beads.
\item Determine the least possible value of Sasha's maximum total profit.
\item Why might Sasha not achieve this maximum profit?
\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete 2024 Q6 [16]}}