| Exam Board | OCR MEI |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 20 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Simplex tableau interpretation |
| Difficulty | Standard +0.3 This is a standard simplex algorithm question with straightforward setup and interpretation. While it requires multiple steps (formulation explanation, simplex iterations, big-M/two-phase setup), these are routine procedures for D2 students with no novel problem-solving required. The verification in part (v) is simple arithmetic, making this slightly easier than average. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06d Graphical solution: feasible region, two variables7.07a Simplex tableau: initial setup in standard format7.07b Simplex iterations: pivot choice and row operations7.07c Interpret simplex: values of variables, slack, and objective |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a\) is the number of aardvarks, etc. defined | B1 | |
| First inequality models the furry material constraint | M1 | |
| Second inequality models the woolly material constraint | A1 | |
| Third inequality models the glass eyes constraint | ||
| Models a "pairs of glass eyes" constraint | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Problem is an IP so number of eyes used will be integer anyway | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct initial simplex tableau | M1 A1 | |
| First pivot choice and correct pivot operation | M1 A1 | |
| Second pivot choice and correct pivot operation | M1 A1 | |
| Make 6 aardvarks and 8 bears giving £58 profit, 2 eyes left over | B1 B1 B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| New constraint row added correctly | B1 | new constraint |
| Correct objective row with \(M\) | M1 A1 | objective |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(8 \times 0.5 + 2 \times 1 + 5 \times 1 = 11\) | B1 | |
| \(8 \times 2 + 2 \times 1.5 + 5 \times 1 = 24\) | ||
| \(8 \times 2 + 2 \times 2 + 5 \times 2 = 30\) | ||
| \(3 \times 8 + 5 \times 2 + 2 \times 5 = 44\) but \(3 \times 6 + 5 \times 6 + 2 \times 2 = 52\) | B1 | |
| \(1\ \text{m}^2\) of woolly material and 2 eyes left | B1 |
# Question 4:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a$ is the number of aardvarks, etc. defined | B1 | |
| First inequality models the furry material constraint | M1 | |
| Second inequality models the woolly material constraint | A1 | |
| Third inequality models the glass eyes constraint | | |
| Models a "pairs of glass eyes" constraint | B1 | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Problem is an IP so number of eyes used will be integer anyway | B1 | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct initial simplex tableau | M1 A1 | |
| First pivot choice and correct pivot operation | M1 A1 | |
| Second pivot choice and correct pivot operation | M1 A1 | |
| Make 6 aardvarks and 8 bears giving £58 profit, 2 eyes left over | B1 B1 B1 | |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| New constraint row added correctly | B1 | new constraint |
| Correct objective row with $M$ | M1 A1 | objective |
## Part (v)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $8 \times 0.5 + 2 \times 1 + 5 \times 1 = 11$ | B1 | |
| $8 \times 2 + 2 \times 1.5 + 5 \times 1 = 24$ | | |
| $8 \times 2 + 2 \times 2 + 5 \times 2 = 30$ | | |
| $3 \times 8 + 5 \times 2 + 2 \times 5 = 44$ but $3 \times 6 + 5 \times 6 + 2 \times 2 = 52$ | B1 | |
| $1\ \text{m}^2$ of woolly material and 2 eyes left | B1 | |4 The "Cuddly Friends Company" produces soft toys. For one day's production run it has available $11 \mathrm {~m} ^ { 2 }$ of furry material, $24 \mathrm {~m} ^ { 2 }$ of woolly material and 30 glass eyes. It has three soft toys which it can produce:
The "Cuddly Aardvark", each of which requires $0.5 \mathrm {~m} ^ { 2 }$ of furry material, $2 \mathrm {~m} ^ { 2 }$ of woolly material and two eyes. Each sells at a profit of $\pounds 3$.
The "Cuddly Bear", each of which requires $1 \mathrm {~m} ^ { 2 }$ of furry material, $1.5 \mathrm {~m} ^ { 2 }$ of woolly material and two eyes. Each sells at a profit of $\pounds 5$.
The "Cuddly Cat", each of which requires $1 \mathrm {~m} ^ { 2 }$ of furry material, $1 \mathrm {~m} ^ { 2 }$ of woolly material and two eyes. Each sells at a profit of $\pounds 2$.
An analyst formulates the following LP to find the production plan which maximises profit.
$$\begin{array} { l l }
\text { Maximise } & 3 a + 5 b + 2 c \\
\text { subject to } & 0.5 a + b + c \leqslant 11 , \\
& 2 a + 1.5 b + c \leqslant 24 , \\
& 2 a + 2 b + 2 c \leqslant 30 .
\end{array}$$
(i) Explain how this formulation models the problem, and say why the analyst has not simplified the last inequality to $a + b + c \leqslant 15$.\\
(ii) The final constraint is different from the others in that the resource is integer valued. Explain why that does not impose an additional difficulty for this problem.\\
(iii) Solve this problem using the simplex algorithm.
Interpret your solution and say what resources are left over.
On a particular day an order is received for two Cuddly Cats, and the extra constraint $c \geqslant 2$ is added to the formulation.\\
(iv) Set up an initial simplex tableau to deal with the modified problem using either the big-M approach or two-phase simplex. Do not perform any iterations on your tableau.\\
(v) Show that the solution given by $a = 8 , b = 2$ and $c = 5$ uses all of the resources, but that $a = 6 , b = 6$ and $c = 2$ gives more profit.
What resources are left over from the latter solution?
\hfill \mbox{\textit{OCR MEI D2 2006 Q4 [20]}}