# Question 8:

## Part (a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $x + y + z \leq 39$ | B1 | cao |
| $\frac{2}{5}(x + y + z) \leq x \Rightarrow -3x + 2y + 2z \leq 0$ | M1 A1 | M1 for $\frac{2}{5}(x+y+z) \mathbin{\square} x$ where $\square$ is any inequality or equals; A1 cao |
| $x + z \geq 28$ | B1 | cao |
| Maximise $P = y + z$ $(\Rightarrow P - y - z = 0)$ | B1 | Correct objective function plus 'maximise' or 'max' but not 'maximum' |

**(5 marks)**

## Part (b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $x+y+z \leq 39 \Rightarrow x+y+z+s_1=39$ and $-3x+2y+2z \leq 0 \Rightarrow -3x+2y+2z+s_2=0$ | M1 A1 | M1 for one $\leq$ constraint as equation using slack variables; A1 cao (both $\leq$ constraints) |
| $x+z \geq 28 \Rightarrow x+z-s_3+a_1=28$ | B1 | $\geq$ constraint re-formulated using one surplus and one artificial variable |
| $I = -a_1 \Rightarrow I - x - z + s_3 = -28$ | M1 | Formulates second objective with $I = -a_1$ and their expression for $a_1$ |
| Initial tableau with all five rows complete (two correct rows) | M1 A1 | M1 for setting up tableau; A1 cao (any equivalent correct form) |

Example initial tableau:

| b.v. | $x$ | $y$ | $z$ | $s_1$ | $s_2$ | $s_3$ | $a_1$ | Value |
|---|---|---|---|---|---|---|---|---|
| $s_1$ | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 39 |
| $s_2$ | $-3$ | 2 | 2 | 0 | 1 | 0 | 0 | 0 |
| $a_1$ | 1 | 0 | 1 | 0 | 0 | $-1$ | 1 | 28 |
| $P$ | 0 | $-1$ | $-1$ | 0 | 0 | 0 | 0 | 0 |
| $I$ | $-1$ | 0 | $-1$ | 0 | 0 | 1 | 0 | $-28$ |

**(6 marks)**

## Part (c):

| Answer/Working | Mark | Guidance |
|---|---|---|
| The only negative in the objective row is $-1$ so the pivot is from the $z$-column | B1 | Correct reasoning that pivot is from $z$-column; condone any mention of negative value in $P$ row |
| The 5 in the $s_2$ row is the pivot because $\frac{62}{5}$ is less than $\frac{28}{1}$ | B1 | Must compare or state that 12.4 is less than 28; not sufficient to just state 12.4 is the least |

**(2 marks)**

## Part (d):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Pivot row correct including change of b.v. | B1 | |
| All values in one non-pivot row correct **or** one of the non-zero columns ($s_1$, $s_2$ or value) correct | M1 | From their choice of pivot |
| Row operations used correctly at least twice (two of $s_1$, $s_2$ or value) | A1 | |
| All values and row operations correctly stated | A1 | Do not penalise lack of correct b.v. in pivot row twice; condone blank Row Ops in first row only |

Final tableau:

| b.v. | $x$ | $y$ | $z$ | $s_1$ | $s_2$ | $s_3$ | Value | Row Ops |
|---|---|---|---|---|---|---|---|---|
| $y$ | 0 | 1 | 0 | 1 | 0 | 1 | 11 | R1 |
| $z$ | 0 | 0 | 1 | $-\frac{2}{5}$ | $\frac{1}{5}$ | $-1$ | $\frac{62}{5}$ | $\frac{1}{5}$R2 |
| $x$ | 1 | 0 | 0 | $\frac{2}{5}$ | $-\frac{1}{5}$ | 0 | $\frac{78}{5}$ | R3$-$R2 |
| $P$ | 0 | 0 | 0 | $\frac{3}{5}$ | $\frac{1}{5}$ | 0 | $\frac{117}{5}$ | R4$+$R2 |

| Spend 15.6 hours swimming, 11 hours cycling and 12.4 hours running | A1 | Must be in context, not just $x$, $y$, $z$ |

**(5 marks)**

**Total: 18 marks**