| Exam Board | OCR |
|---|---|
| Module | Further Discrete AS (Further Discrete AS) |
| Year | 2019 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Formulation from word problem |
| Difficulty | Moderate -0.3 This is a straightforward linear programming formulation with three variables and simple constraints that translate directly from the word problem. Part (a) requires standard variable definition and constraint writing. Parts (b)(i-iii) involve routine graphical LP solution with g=8 fixed, reducing to 2D, plus basic interpretation questions. While it's a multi-part question worth several marks, it requires no novel insight—just systematic application of standard LP techniques taught in Decision Maths modules. Slightly easier than average A-level due to the direct constraint translation and standard graphical method. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| Answer | Marks | Guidance |
|---|---|---|
| \(g, r, s =\) time spent on gym work, running, swimming, respectively, in hours | B1 [1.1] | Define variables; Or equivalent with other letters |
| Maximise \(3g + r + 1.5s\) | B1 [3.3] | Objective (maximise and positive coefficients in ratio 6:2:3); Max \(g + \frac{1}{3}r + \frac{1}{2}s\); NOT Max \(g + 3r + 2s\) |
| \(g + r + s \leq 18\) | B1 [1.1] | |
| \(g \leq 8\) | B1 [3.3] | Ignore extra constraints provided they are not inconsistent |
| \(r \geq 4\) | ||
| \(g + s \leq r\) | B1 [1.1] | |
| \(g \geq 0\) and \(s \geq 0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Set \(g = 8\): Maximise \(r + 1.5s\) (+24) subject to \(r + s \leq 10,\ s + 8 \leq r\) and \(s \geq 0\) | B1 [3.1b] | Reformulate as a 2-variable problem with an objective function and at least two non-trivial constraints; A linear expression or linear equation and at least two non-trivial constraints in their \(r\) and \(s\) |
| A finite FR identified on graph with variables their \(r\) and \(s\) | M1 [1.1] | |
| Correct FR identified | A1 [3.4] | |
| Optimal at \(r = 9, s = 1\) | M1 [1.1] | A vertex where 2 of their lines (with variables their \(r\) and \(s\)) cross; M1 may be implied from correct solution in context |
| Spend 9 hours running and 1 hour swimming | A1 [3.4] | Interpret their solution (with non-negative times) in context |
| Answer | Marks | Guidance |
|---|---|---|
| e.g. would need to rest/eat/sleep; e.g. need transition time between activities | B1 [3.5b] | A reason why their solution is not practical; e.g. should not train for 18 hours just before a race |
| Answer | Marks | Guidance |
|---|---|---|
| e.g. build in recovery time; e.g. for each hour allocated to each type of training only 50 minutes is spent on the training | B1 [3.5c] | Describing how model can be adapted to make their solution more realistic; e.g. restrict the preparation time to something that is less than 18 hours |
# Question 5:
## Part (a)
$g, r, s =$ time spent on gym work, running, swimming, respectively, in hours | **B1** [1.1] | Define variables; Or equivalent with other letters
Maximise $3g + r + 1.5s$ | **B1** [3.3] | Objective (maximise and positive coefficients in ratio 6:2:3); Max $g + \frac{1}{3}r + \frac{1}{2}s$; NOT Max $g + 3r + 2s$
$g + r + s \leq 18$ | **B1** [1.1] |
$g \leq 8$ | **B1** [3.3] | Ignore extra constraints provided they are not inconsistent
$r \geq 4$ | | |
$g + s \leq r$ | **B1** [1.1] |
$g \geq 0$ and $s \geq 0$ | | |
## Part (b)(i)
Set $g = 8$: Maximise $r + 1.5s$ (+24) subject to $r + s \leq 10,\ s + 8 \leq r$ and $s \geq 0$ | **B1** [3.1b] | Reformulate as a 2-variable problem with an objective function and at least two non-trivial constraints; A linear expression or linear equation and at least two non-trivial constraints in their $r$ and $s$
A finite FR identified on graph with variables their $r$ and $s$ | **M1** [1.1] |
Correct FR identified | **A1** [3.4] |
Optimal at $r = 9, s = 1$ | **M1** [1.1] | A vertex where 2 of their lines (with variables their $r$ and $s$) cross; M1 may be implied from correct solution in context
Spend 9 hours running and 1 hour swimming | **A1** [3.4] | Interpret their solution (with non-negative times) in context
## Part (b)(ii)
e.g. would need to rest/eat/sleep; e.g. need transition time between activities | **B1** [3.5b] | A reason why their solution is not practical; e.g. should not train for 18 hours just before a race
## Part (b)(iii)
e.g. build in recovery time; e.g. for each hour allocated to each type of training only 50 minutes is spent on the training | **B1** [3.5c] | Describing how model can be adapted to make their solution more realistic; e.g. restrict the preparation time to something that is less than 18 hours
---5 Corey is training for a race that starts in 18 hours time. He splits his training between gym work, running and swimming.
\begin{itemize}
\item At most 8 hours can be spent on gym work.
\item At least 4 hours must be spent running.
\item The total time spent on gym work and swimming must not exceed the time spent running.
\end{itemize}
Corey thinks that time spent on gym work is worth 3 times the same time spent running or 2 times the same time spent swimming. Corey wants to maximise the worth of the training using this model.
\begin{enumerate}[label=(\alph*)]
\item Formulate a linear programming problem to represent Corey's problem.
Your formulation must include defining the variables that you are using.
Suppose that Corey spends the maximum of 8 hours on gym work.
\item \begin{enumerate}[label=(\roman*)]
\item Use a graphical method to determine how long Corey should spend running and how long he should spend swimming.
\item Describe why this solution is not practical.
\item Describe how Corey could refine the LP model to make the solution more realistic.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete AS 2019 Q5 [12]}}