| Exam Board | OCR |
|---|---|
| Module | Further Discrete (Further Discrete) |
| Year | 2018 |
| Session | September |
| Marks | 8 |
| Topic | Linear Programming |
| Type | Formulation from word problem |
| Difficulty | Standard +0.3 This is a straightforward linear programming formulation and simplex algorithm application. Part (i) requires simple comparison of profit per unit of limiting resource (a standard technique), and part (ii) is a routine simplex execution with clear constraints and objective function. While it requires careful arithmetic, it involves no novel insights and follows standard Further Maths decision mathematics procedures. |
| Spec | 7.07a Simplex tableau: initial setup in standard format7.07b Simplex iterations: pivot choice and row operations7.07c Interpret simplex: values of variables, slack, and objective7.07e Graphical interpretation: iterations as edges of convex polygon |
| Type of drink | Coffee | Foamed milk | Profit |
| w Americano | 80 | 0 | 1.20 |
| \(x\) Cappuccino | 60 | 120 | X |
| \(y\) Flat White | 60 | 100 | 1.40 |
| \(z\) Latte | 40 | 120 | 1.50 |
| Available | 900 | 1500 |
6 Kai mixes hot drinks using coffee and steamed milk.\\
The amounts ( ml ) needed and profit ( $\pounds$ ) for a standard sized cup of four different drinks are given in the table. The table also shows the amount of the ingredients available.
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
Type of drink & Coffee & Foamed milk & Profit \\
\hline
w Americano & 80 & 0 & 1.20 \\
\hline
$x$ Cappuccino & 60 & 120 & X \\
\hline
$y$ Flat White & 60 & 100 & 1.40 \\
\hline
$z$ Latte & 40 & 120 & 1.50 \\
\hline
Available & 900 & 1500 & \\
\hline
\end{tabular}
\end{center}
Kai makes the equivalent of $w$ standard sized americanos, $x$ standard sized cappuccinos, $y$ standard sized flat whites and $z$ standard sized lattes. He can make different sized drinks so $w , x , y , z$ need not be integers.
Kai wants to find the maximum profit that he can make, assuming that the customers want to buy the drinks he has made.\\
(i) What is the minimum value of X for it to be worthwhile for Kai to make cappuccinos?
Kai makes no cappuccinos.\\
(ii) Use the simplex algorithm to solve Kai's problem.
The grids in the Printed Answer Booklet should have at least enough rows and columns and there should be at least enough grids to show all the iterations needed.
Only record the output from each iteration, not any intermediate stages.\\
Interpret the solution and state the maximum profit that Kai can make.
\hfill \mbox{\textit{OCR Further Discrete 2018 Q6 [8]}}