Question 7 - A-Level Maths

Edexcel D2 — Question 7 18 marks

Exam Board	Edexcel
Module	D2 (Decision Mathematics 2)
Marks	18
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Matchings and Allocation
Type	Transportation problem: north-west corner method
Difficulty	Moderate -0.5 This is a standard textbook application of the north-west corner method and stepping-stone algorithm for transportation problems. While it requires multiple steps and careful bookkeeping across parts (a)-(e), it follows a completely algorithmic procedure taught in D2 with no problem-solving insight required—students simply execute learned algorithms on given data.
Spec	7.06a LP formulation: variables, constraints, objective function 7.06b Slack variables: converting inequalities to equations 7.06c Working with constraints: algebra and ad hoc methods 7.06d Graphical solution: feasible region, two variables

7. A distributor has six warehouses. At one point the distributor needs to move 25 lorries from warehouses \(W _ { 1 } , W _ { 2 }\) and \(W _ { 3 }\) to warehouses \(W _ { \mathrm { A } } , W _ { \mathrm { B } }\) and \(W _ { \mathrm { C } }\) for the minimum possible cost. The transportation tableau below shows the unit cost, in tens of pounds, of moving a lorry between two warehouses, and the relevant figures regarding the number of lorries available or required at each warehouse.

	\(W _ { \text {A } }\)	\(W _ { \mathrm { B } }\)	\(W _ { \mathrm { C } }\)	Available
\(W _ { 1 }\)	7	8	10	10
\(W _ { 2 }\)	9	6	5	8
\(W _ { 3 }\)	11	5	7	7
Required	5	12	8

Write down the initial solution given by the north-west corner rule.
Obtain improvement indices for the unused routes.
Use the stepping-stone method to find an improved solution and state why it is degenerate.
Placing a zero in cell \(( 2,2 )\), show that the improved solution is optimal and state the transportation pattern.

Find the total cost of the optimal solution. \section*{Please hand this sheet in for marking}

Stage	State	Destination	Cost	Total cost
\multirow[t]{3}{*}{1}	Marquee	Deluxe Cuisine
	Castle	Deluxe Castle Cuisine
	Hotel	Deluxe Cuisine Hotel
\multirow[t]{3}{*}{2}	Church	Marquee Castle Hotel
	Castle	Marquee Castle
	Registry Office	Marquee Castle Hotel
3	Home	Castle Church Registry

\section*{Please hand this sheet in for marking}

A B \(C\) D \(E\) \(F\) \(G\) \(H\)
A - 85 59 31 47 52 74 41
B 85 - 104 73 51 68 43 55
C 59 104 - 54 62 88 61 45
D 31 73 54 - 40 59 65 78
E 47 51 62 40 - 56 71 68
\(F\) 52 68 88 59 56 - 53 49
\(G\) 74 43 61 65 71 53 - 63
\(H\) 41 55 45 78 68 49 63 -
A \(B\) \(C\) D \(E\) \(F\) \(G\) \(H\)
A - 85 59 31 47 52 74 41
B 85 - 104 73 51 68 43 55
C 59 104 - 54 62 88 61 45
D 31 73 54 - 40 59 65 78
E 47 51 62 40 - 56 71 68
\(F\) 52 68 88 59 56 - 53 49
G 74 43 61 65 71 53 - 63
\(H\) 41 55 45 78 68 49 63 -

Show mark scheme Show mark scheme source

Question 7:

Part (a):

Answer	Marks	Guidance
Answer	Marks	Guidance
NW corner: \(W_1W_A=5\), \(W_1W_B=5\), \(W_2W_B=7\), \(W_2W_C=1\), \(W_3W_C=7\)	M1 A1	M1 correct method, A1 all correct

Part (b):

Answer	Marks	Guidance
Answer	Marks	Guidance
Set up shadow costs \(u_i + v_j = c_{ij}\) for used routes	M1
Solve for \(u\)'s and \(v\)'s	A1
Improvement index for each unused cell \(= c_{ij} - u_i - v_j\)	M1
All indices calculated correctly	A1 A1	One A1 for method, one for all correct values

Part (c):

Answer	Marks	Guidance
Answer	Marks	Guidance
Identify most negative index; form stepping-stone loop	M1
Improved solution obtained	A1
Degenerate because number of basic variables \(< m+n-1 = 3+3-1=5\)	A1	Must state reason

Part (d):

Answer	Marks	Guidance
Answer	Marks	Guidance
Place zero in cell \((2,2)\); recalculate shadow costs	M1 A1
All improvement indices \(\geq 0\), therefore optimal	M1 A1
State transportation pattern (all allocations)	A1 A1

Part (e):

Answer	Marks	Guidance
Answer	Marks	Guidance
Calculate total cost from optimal allocation	M1
Total cost \(= \) correct value in tens of pounds	A1	ft from (d)

I can see this is a answer sheet for students to complete (Sheet for answering question 4), not a mark scheme. The page shows:

- Two distance matrices (i) and (ii) for nodes A through H

- Blank dotted lines for student working/answers

- The matrices are identical, both showing distances between 8 nodes (A, B, C, D, E, F, G, H)

This document does not contain a mark scheme. It is a blank answer/working sheet that students would fill in and hand in for marking. To provide the mark scheme content you're requesting, I would need to see the actual mark scheme document, which would be a separate document typically showing:

- Correct answers

- Method marks (M1)

- Accuracy marks (A1)

- Follow-through marks (ft)

- etc.

If you have the mark scheme document, please share that image and I can extract and format the content as requested.

# Question 7:

## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| NW corner: $W_1W_A=5$, $W_1W_B=5$, $W_2W_B=7$, $W_2W_C=1$, $W_3W_C=7$ | M1 A1 | M1 correct method, A1 all correct |

## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Set up shadow costs $u_i + v_j = c_{ij}$ for used routes | M1 | |
| Solve for $u$'s and $v$'s | A1 | |
| Improvement index for each unused cell $= c_{ij} - u_i - v_j$ | M1 | |
| All indices calculated correctly | A1 A1 | One A1 for method, one for all correct values |

## Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Identify most negative index; form stepping-stone loop | M1 | |
| Improved solution obtained | A1 | |
| Degenerate because number of basic variables $< m+n-1 = 3+3-1=5$ | A1 | Must state reason |

## Part (d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Place zero in cell $(2,2)$; recalculate shadow costs | M1 A1 | |
| All improvement indices $\geq 0$, therefore optimal | M1 A1 | |
| State transportation pattern (all allocations) | A1 A1 | |

## Part (e):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Calculate total cost from optimal allocation | M1 | |
| Total cost $= $ correct value in tens of pounds | A1 | ft from (d) |

I can see this is a answer sheet for students to complete (Sheet for answering question 4), not a mark scheme. The page shows:

- Two distance matrices (i) and (ii) for nodes A through H
- Blank dotted lines for student working/answers
- The matrices are identical, both showing distances between 8 nodes (A, B, C, D, E, F, G, H)

**This document does not contain a mark scheme.** It is a blank answer/working sheet that students would fill in and hand in for marking. To provide the mark scheme content you're requesting, I would need to see the actual mark scheme document, which would be a separate document typically showing:
- Correct answers
- Method marks (M1)
- Accuracy marks (A1)
- Follow-through marks (ft)
- etc.

If you have the mark scheme document, please share that image and I can extract and format the content as requested.

Show LaTeX source

7. A distributor has six warehouses. At one point the distributor needs to move 25 lorries from warehouses $W _ { 1 } , W _ { 2 }$ and $W _ { 3 }$ to warehouses $W _ { \mathrm { A } } , W _ { \mathrm { B } }$ and $W _ { \mathrm { C } }$ for the minimum possible cost. The transportation tableau below shows the unit cost, in tens of pounds, of moving a lorry between two warehouses, and the relevant figures regarding the number of lorries available or required at each warehouse.

\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
 & $W _ { \text {A } }$ & $W _ { \mathrm { B } }$ & $W _ { \mathrm { C } }$ & Available \\
\hline
$W _ { 1 }$ & 7 & 8 & 10 & 10 \\
\hline
$W _ { 2 }$ & 9 & 6 & 5 & 8 \\
\hline
$W _ { 3 }$ & 11 & 5 & 7 & 7 \\
\hline
Required & 5 & 12 & 8 &  \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Write down the initial solution given by the north-west corner rule.
\item Obtain improvement indices for the unused routes.
\item Use the stepping-stone method to find an improved solution and state why it is degenerate.
\item Placing a zero in cell $( 2,2 )$, show that the improved solution is optimal and state the transportation pattern.
\item Find the total cost of the optimal solution.

\section*{Please hand this sheet in for marking}
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
Stage & State & Destination & Cost & Total cost \\
\hline
\multirow[t]{3}{*}{1} & Marquee & Deluxe Cuisine &  &  \\
\hline
 & Castle & Deluxe Castle Cuisine &  &  \\
\hline
 & Hotel & Deluxe Cuisine Hotel &  &  \\
\hline
\multirow[t]{3}{*}{2} & Church & Marquee Castle Hotel &  &  \\
\hline
 & Castle & Marquee Castle &  &  \\
\hline
 & Registry Office & Marquee Castle Hotel &  &  \\
\hline
3 & Home & Castle Church Registry &  &  \\
\hline
\end{tabular}
\end{center}

\section*{Please hand this sheet in for marking}
\begin{enumerate}[label=(\roman*)]
\item \begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
 & A & B & $C$ & D & $E$ & $F$ & $G$ & $H$ \\
\hline
A & - & 85 & 59 & 31 & 47 & 52 & 74 & 41 \\
\hline
B & 85 & - & 104 & 73 & 51 & 68 & 43 & 55 \\
\hline
C & 59 & 104 & - & 54 & 62 & 88 & 61 & 45 \\
\hline
D & 31 & 73 & 54 & - & 40 & 59 & 65 & 78 \\
\hline
E & 47 & 51 & 62 & 40 & - & 56 & 71 & 68 \\
\hline
$F$ & 52 & 68 & 88 & 59 & 56 & - & 53 & 49 \\
\hline
$G$ & 74 & 43 & 61 & 65 & 71 & 53 & - & 63 \\
\hline
$H$ & 41 & 55 & 45 & 78 & 68 & 49 & 63 & - \\
\hline
\end{tabular}
\end{center}
\item \begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
 & A & $B$ & $C$ & D & $E$ & $F$ & $G$ & $H$ \\
\hline
A & - & 85 & 59 & 31 & 47 & 52 & 74 & 41 \\
\hline
B & 85 & - & 104 & 73 & 51 & 68 & 43 & 55 \\
\hline
C & 59 & 104 & - & 54 & 62 & 88 & 61 & 45 \\
\hline
D & 31 & 73 & 54 & - & 40 & 59 & 65 & 78 \\
\hline
E & 47 & 51 & 62 & 40 & - & 56 & 71 & 68 \\
\hline
$F$ & 52 & 68 & 88 & 59 & 56 & - & 53 & 49 \\
\hline
G & 74 & 43 & 61 & 65 & 71 & 53 & - & 63 \\
\hline
$H$ & 41 & 55 & 45 & 78 & 68 & 49 & 63 & - \\
\hline
\end{tabular}
\end{center}
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{Edexcel D2  Q7 [18]}}

This paper (7 questions)

View full paper

Q1 5 Q2 6 Q3 9 Q4 11 Q5 13 Q6 13 Q7 18

	A	B	\(C\)	D	\(E\)	\(F\)	\(G\)	\(H\)
A	-	85	59	31	47	52	74	41
B	85	-	104	73	51	68	43	55
C	59	104	-	54	62	88	61	45
D	31	73	54	-	40	59	65	78
E	47	51	62	40	-	56	71	68
\(F\)	52	68	88	59	56	-	53	49
\(G\)	74	43	61	65	71	53	-	63
\(H\)	41	55	45	78	68	49	63	-

	A	\(B\)	\(C\)	D	\(E\)	\(F\)	\(G\)	\(H\)
A	-	85	59	31	47	52	74	41
B	85	-	104	73	51	68	43	55
C	59	104	-	54	62	88	61	45
D	31	73	54	-	40	59	65	78
E	47	51	62	40	-	56	71	68
\(F\)	52	68	88	59	56	-	53	49
G	74	43	61	65	71	53	-	63
\(H\)	41	55	45	78	68	49	63	-