| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Simplex algorithm execution |
| Difficulty | Moderate -0.8 This is a standard simplex algorithm question requiring mechanical application of the tableau method. Part (a) is direct reading of equations from the tableau, part (b) follows the prescribed pivot rule with routine row operations, and part (c) is reading the final solution. No problem-solving insight or novel reasoning required—purely algorithmic execution of a well-practiced technique. |
| Spec | 7.07a Simplex tableau: initial setup in standard format7.07b Simplex iterations: pivot choice and row operations |
| Basic variable | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| \(r\) | 7 | 10 | 10 | 1 | 0 | 0 | 3600 |
| \(s\) | 6 | 9 | 12 | 0 | 1 | 0 | 3600 |
| \(t\) | 2 | 3 | 4 | 0 | 0 | 1 | 2400 |
| \(P\) | \(-35\) | \(-55\) | \(-60\) | 0 | 0 | 0 | 0 |
| Answer | Marks |
|---|---|
| \[P - 35x - 55y - 60z = 0\] | B2,1,0(4) |
| (b) [Simplex tableau - initial and after pivoting operations] | m1 A1 m1 A1(4) |
| (c) \(P = 20400\); \(x = 0\); \(y = 240\); \(z = 120\); \(r = 0\); \(s = 0\); \(t = 1200\) | m1 B2,1,0(3) |
| Answer | Marks |
|---|---|
| (i) 10 in z column | m0 m1 A1 mo |
| (ii) 4 in z column | m1 A1(4) |
| Answer | Marks |
|---|---|
| (i) Choose 7 in x column | m1 A1(2) |
| (ii) Choose 10 in y column | m1 A1(4) |
| Final tableau note: \(\to\) my final tableau | m1 A1 |
| Answer | Marks |
|---|---|
| First 3 equations c.a.o - 1 each error, but penalise only 1 error per equation unequalities set \(P\) | B2 CAO(B1 for a 'list 5t.') |
| (b) m1: Correct pivot chosen inconsistent to deal with 'old' m0; A1: Pivot as correct c.a.o including b.v.; m1: Correct ro operation used (ob2)-at least 1 non-row or 1 rencorrect in each ro, while all X same; A1/non-pivot ro correct ✓ on errors, pivot chose only | (4) |
| (5) B1: RA opera correctly stated. (London last ✓ RA : 1) must be less of rest, positive s value, proceed tableau or m0); m1/correct pivot chose ✓ from prev. tableu. No negative s value > presen tableau or m0); C.A.O including b.v. but ✓ for previous table; m1 Correct ro operation used (ob 1)-at least 1 non-row el 1 rencorrect in each ro, while all X same; (4 A1) C.A.O-on non critical | (4) |
| (C) m1: 3 variable stated - must have completed b.v. and table vi. (1'snd ran) ∈ no length. Any register... Dno | m0 |
| A2/ all 7 correct | (4) |
(a)
$$7x + 10y + 10z + r = 3600$$
$$6x + 9y + 12z + s = 3600$$
$$2x + 3y + 4z + t = 2400$$
$$P - 35x - 55y - 60z = 0$$ | B2,1,0(4)
(b) [Simplex tableau - initial and after pivoting operations] | m1 A1 m1 A1(4)
(c) $P = 20400$; $x = 0$; $y = 240$; $z = 120$; $r = 0$; $s = 0$; $t = 1200$ | m1 B2,1,0(3)
---
## Question D1: June 2006 Q6(L) - Wrong pivot choice
(i) 10 in z column | m0 m1 A1 mo
(ii) 4 in z column | m1 A1(4)
---
## Question D1: June 2006 Q6(L) - Misread
(i) Choose 7 in x column | m1 A1(2)
(ii) Choose 10 in y column | m1 A1(4)
Final tableau note: $\to$ my final tableau | m1 A1
(−2 for Misread)
---
## Question 6(B2) B1
First 3 equations c.a.o - 1 each error, but penalise only 1 error per equation unequalities set $P$ | B2 CAO(B1 for a 'list 5t.')
(b) m1: Correct pivot chosen inconsistent to deal with 'old' m0; A1: Pivot as correct c.a.o including b.v.; m1: Correct ro operation used (ob2)-at least 1 non-row or 1 rencorrect in each ro, while all X same; A1/non-pivot ro correct ✓ on errors, pivot chose only | (4)
(5) B1: RA opera correctly stated. (London last ✓ RA : 1) must be less of rest, positive s value, proceed tableau or m0); m1/correct pivot chose ✓ from prev. tableu. No negative s value > presen tableau or m0); **C.A.O including b.v. but ✓ for previous table**; m1 Correct ro operation used (ob 1)-at least 1 non-row el 1 rencorrect in each ro, while all X same; (4 A1) **C.A.O-on non critical** | (4)
(C) m1: 3 variable stated - must have completed b.v. and table vi. (1'snd ran) ∈ no length. Any register... Dno | m0
**A2/** all 7 correct | (4)
**A1/' at least 4 correct**
---The tableau below is the initial tableau for a maximising linear programming problem.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
Basic variable & $x$ & $y$ & $z$ & $r$ & $s$ & $t$ & Value \\
\hline
$r$ & 7 & 10 & 10 & 1 & 0 & 0 & 3600 \\
\hline
$s$ & 6 & 9 & 12 & 0 & 1 & 0 & 3600 \\
\hline
$t$ & 2 & 3 & 4 & 0 & 0 & 1 & 2400 \\
\hline
$P$ & $-35$ & $-55$ & $-60$ & 0 & 0 & 0 & 0 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Write down the four equations represented in the initial tableau above. [4]
\item Taking the most negative number in the profit row to indicate the pivot column at each stage, solve this linear programming problem. State the row operations that you use. [9]
\item State the values of the objective function and each variable. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2006 Q6 [16]}}