| Exam Board | OCR |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2011 |
| Session | January |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Critical Path Analysis |
| Type | Schedule with limited workers - determine minimum time |
| Difficulty | Moderate -0.8 This is a straightforward linear programming problem with simple constraint formation from a table and basic graphical solution. The question guides students through each step methodically, requires only standard D1 techniques (forming constraints, plotting feasible regions, finding optimal vertices), and involves no complex reasoning or novel problem-solving approaches—easier than average A-level maths. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06c Working with constraints: algebra and ad hoc methods |
| Check contents | Check postage | Check address | |
| New | 3 | 4 | 3 |
| Occasional | 5 | 3 | 4 |
| Regular | 2 | 3 | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x=\) number of new parcels checked per hour, \(y=\) occasional, \(z=\) regular | B1 | [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3x+5y+2z \leq 60\) (contents check) | B1 | |
| \(4x+3y+3z \leq 60\) (postage check) | B1 | |
| \(3x+4y+3z \leq 60\) (address check) | B1 | |
| \(x,y,z \geq 0\) | B1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(z\) removed from objective: Maximise \(P=8x+7y\) | B1 | Constraints simplified by setting \(z=0\) or removing \(z\) terms |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct feasible region drawn | M1 A1 | |
| Objective line drawn with correct gradient | M1 | |
| Optimal point identified | A1 | |
| Optimal values of \(x,y\) stated | A1 A1 | [6] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Constraints become equalities or \(\leq 60\) with integer solutions sought | M1 | |
| Correct optimal integer solution found | A1 A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| The number of parcels must be a whole number / integer values required but LP gives non-integer solution | B1 | [1] |
# Question 5:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x=$ number of new parcels checked per hour, $y=$ occasional, $z=$ regular | B1 | **[1]** |
## Part (ii) - Constraints
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3x+5y+2z \leq 60$ (contents check) | B1 | |
| $4x+3y+3z \leq 60$ (postage check) | B1 | |
| $3x+4y+3z \leq 60$ (address check) | B1 | |
| $x,y,z \geq 0$ | B1 | **[4]** |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $z$ removed from objective: Maximise $P=8x+7y$ | B1 | Constraints simplified by setting $z=0$ or removing $z$ terms | B1 | **[2]** |
## Part (iv) - Graphical Method
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct feasible region drawn | M1 A1 | |
| Objective line drawn with correct gradient | M1 | |
| Optimal point identified | A1 | |
| Optimal values of $x,y$ stated | A1 A1 | **[6]** |
## Part (v) - Exactly One Hour
| Answer | Marks | Guidance |
|--------|-------|----------|
| Constraints become equalities or $\leq 60$ with integer solutions sought | M1 | |
| Correct optimal integer solution found | A1 A1 | **[3]** |
## Part (vi)
| Answer | Marks | Guidance |
|--------|-------|----------|
| The number of parcels must be a whole number / integer values required but LP gives non-integer solution | B1 | **[1]** |
---5 An online shopping company selects some of its parcels to be checked before posting them. Each selected parcel must pass through three checks, which may be carried out in any order. One person must check the contents, another must check the postage and a third person must check the address.
The parcels are classified according to the type of customer as 'new', 'occasional' or 'regular'. The table shows the time taken, in minutes, for each check on each type of parcel.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\hline
& Check contents & Check postage & Check address \\
\hline
New & 3 & 4 & 3 \\
\hline
Occasional & 5 & 3 & 4 \\
\hline
Regular & 2 & 3 & 3 \\
\hline
\end{tabular}
\end{center}
The manager in charge of checking at the company has allocated each type of parcel a 'value' to represent how useful it is for generating additional income. In suitable units, these values are as follows.
$$\text { new } = 8 \text { points } \quad \text { occasional } = 7 \text { points } \quad \text { regular } = 4 \text { points }$$
The manager wants to find out how many parcels of each type her department should check each hour, on average, to maximise the total value. She models this objective as
$$\text { Maximise } P = 8 x + 7 y + 4 z .$$
(i) What do the variables $x , y$ and $z$ represent?\\
(ii) Write down the constraints on the values of $x , y$ and $z$.
The manager changes the value of parcels for regular customers to 0 points.\\
(iii) Explain what effect this has on the objective and simplify the constraints.\\
(iv) Use a graphical method to represent the feasible region for the manager's new problem. You should choose scales so that the feasible region can be clearly seen. Hence determine the optimal strategy.
Now suppose that there is exactly one hour available for checking and the manager wants to find out how many parcels of each type her department should check in that hour to maximise the total value. The value of parcels for regular customers is still 0 points.\\
(v) Find the optimal strategy in this situation.\\
(vi) Give a reason why, even if all the timings and values are correct, the total value may be less than this maximum.
\section*{Question 6 is printed overleaf.}
\hfill \mbox{\textit{OCR D1 2011 Q5 [17]}}