A2 June 2024 Q5
5. Two friends, Anaira and Tommi, play a game involving two positive numbers \(x\) and \(y\)
Anaira gives Tommi the following clues to see if he can correctly determine the value of \(x\) and the value of \(y\)
- \(x\) is greater than \(y\) and the difference between the two is at least 100
- \(x\) is at most 5 times as large as \(y\)
- the sum of \(2x\) and \(3y\) is at least 350
- the sum of \(x\) and \(y\) is as small as possible
Tommi decides to solve this problem by using the big-M method.
As part of your solution, you must show
- how the constraints were made into equations using one slack variable, exactly two surplus variables and exactly two artificial variables
- how the objective function was formed
The big-M method is applied until the tableau containing the optimal solution to the problem is found. One row of this final tableau is as follows.
| b.v. | \(x\) | \(y\) | \(s_1\) | \(s_2\) | \(s_3\) | \(a_1\) | \(a_2\) | Value |
|---|---|---|---|---|---|---|---|---|
| \(x\) | 1 | 0 | \(-\dfrac{3}{5}\) | 0 | \(-\dfrac{1}{5}\) | \(\dfrac{3}{5}\) | \(\dfrac{1}{5}\) | 130 |
| Scheme | Marks | AO | |||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(x - y \geqslant 100 \Rightarrow x - y - s_1 + a_1 = 100\) \(x - 5y \leqslant 0 \Rightarrow x - 5y + s_2 = 0\) \(2x + 3y \geqslant 350 \Rightarrow 2x + 3y - s_3 + a_2 = 350\) | B3,2,1 | 3.3 3.3 3.3 | |||||||||||||||||||||||||||||||||||||||||||||
| \(P = -x - y - M(a_1 + a_2)\) and substitute expressions for \(a_1\) and \(a_2\) \(\big(P + (1 - 3M)x + (1 - 2M)y + Ms_1 + Ms_3 = -450M\big)\) | M1 | 2.1 | |||||||||||||||||||||||||||||||||||||||||||||
e.g.
| M1 A1 | 3.3 2.2a | |||||||||||||||||||||||||||||||||||||||||||||
| (6) |
Notes
(a) NOTE: if correct they must use one slack, two surplus and two artificial variables. Accept alternative letters for these as long as the artificial variables are clearly identifiable
B1: one correct equation or two correct inequalities (do not accept strict inequalities)
B1: two correct equations or three correct inequalities
B1: all three equations correct (please check suffices on \(s\) and \(a\) terms carefully – they may be in a different order)
M1: setting up the new objective which must be \(P = -x - y - M(a_1 + a_2)\) and an attempt to substitute for their \(a_1\) and \(a_2\) (accept any equivalent form, which may not be fully simplified) (accept the use of \(Q\) instead of \(P\) throughout)
M1: setting up initial tableau – all four rows complete with two correct rows (but ignore b.v. column for this mark) (Note the order of rows may be different from above) Check that the slack, surplus and artificial variables correspond to their equations
A1: CAO (any equivalent correct form, but the terms in the objective row must be simplified)
| Scheme | Marks | AO |
|---|---|---|
| (i) \(x = 130\) | B1 | 3.4 |
| (ii) When \(x = 130 \Rightarrow y \leqslant 30,\ y \geqslant 26,\ y \lt 130\) and \(y \geqslant 30\) | M1 | 3.1a |
| \(y = 30\) | A1 | 2.2a |
| (3) | ||
| (9 marks) |
Notes
(i) B1: CAO (\(x = 130\))
(ii) M1: Substitute \(x = 130\) into candidate’s inequalities from (a) (at least 3 inequalities seen or both \(y \geqslant 30\) and \(y \leqslant 30\)) (condone \(y \leqslant 130\))
A1: CAO (\(y = 30\)) we must see both \(y \geqslant 30\) and \(y \leqslant 30\) explicitly stated for this mark (must not follow from any incorrect working)