Question 2 - A-Level Maths

Edexcel D2 2011 June — Question 2 9 marks

Exam Board	Edexcel
Module	D2 (Decision Mathematics 2)
Year	2011
Session	June
Marks	9
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Matchings and Allocation
Type	Transportation problem: stepping-stone method
Difficulty	Moderate -0.3 This is a standard textbook transportation problem using well-defined algorithms (north-west corner, shadow costs, stepping-stone method). While it requires multiple steps and careful arithmetic, it involves direct application of learned procedures with no novel problem-solving or insight required. Slightly easier than average due to its algorithmic nature.
Spec	7.07a Simplex tableau: initial setup in standard format

2. The table below shows the cost of transporting one unit of stock from each of four supply points, 1 , 2, 3 and 4, to each of three demand points, \(\mathrm { A } , \mathrm { B }\) and C . It also shows the stock held at each supply point and the stock required at each demand point. A minimal cost solution is required.

	A	B	C	Supply
1	31	29	32	20
2	22	33	27	22
3	25	27	32	20
4	23	26	38	38
Demand	35	25	30

Add a dummy demand point and appropriate values to Table 1 in the answer book. Table 2 shows an initial solution given by the north-west corner method.
Table 3 shows some of the improvement indices for this solution. \begin{table}[h]
A B C D
1 20
2 15 7
3 18 2
4 28 10
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table} \begin{table}[h]
A B C D
1 - 13 - 9
2 - 11
3
4 1 - 7
\captionsetup{labelformat=empty} \caption{Table 3}
\end{table}
Calculate the shadow costs and the missing improvement indices and enter them into Table 3 in the answer book.
Taking the most negative improvement index to indicate the entering square, use the steppingstone method once to obtain an improved solution. You must make your route clear and state your entering cell and exiting cell.

Show mark scheme Show mark scheme source

Question 2:

Part (a) – Dummy demand point

Answer	Marks	Guidance
Answer	Marks	Guidance
Total supply \(= 100\), total demand \(= 90\), so dummy demand \(= 10\) with all costs \(= 0\)	B1

Part (b) – Shadow costs and improvement indices

Answer	Marks	Guidance
Answer	Marks	Guidance
Set \(u_1 = 0\); use basic cells to find all \(u_i, v_j\)	M1	Correct method
\(v_A = 31, v_B = 29\); \(u_2 = -7, u_3 = -2, u_4 = -3\); \(v_C = 29, v_D = 13\)	A1
Missing indices calculated as \(c_{ij} - u_i - v_j\) for non-basic cells	M1
All correct entries in Table 3	A1

Part (c) – Stepping stone method

Answer	Marks	Guidance
Answer	Marks	Guidance
Most negative index identified (e.g. \(-13\) at 1B)	B1	Entering cell stated
Loop traced through basic cells	M1	Valid loop shown
Correct reallocation with exiting cell identified	A1
Improved solution table correct	A1	cao

# Question 2:

## Part (a) – Dummy demand point

| Answer | Marks | Guidance |
|--------|-------|----------|
| Total supply $= 100$, total demand $= 90$, so dummy demand $= 10$ with all costs $= 0$ | B1 | |

## Part (b) – Shadow costs and improvement indices

| Answer | Marks | Guidance |
|--------|-------|----------|
| Set $u_1 = 0$; use basic cells to find all $u_i, v_j$ | M1 | Correct method |
| $v_A = 31, v_B = 29$; $u_2 = -7, u_3 = -2, u_4 = -3$; $v_C = 29, v_D = 13$ | A1 | |
| Missing indices calculated as $c_{ij} - u_i - v_j$ for non-basic cells | M1 | |
| All correct entries in Table 3 | A1 | |

## Part (c) – Stepping stone method

| Answer | Marks | Guidance |
|--------|-------|----------|
| Most negative index identified (e.g. $-13$ at 1B) | B1 | Entering cell stated |
| Loop traced through basic cells | M1 | Valid loop shown |
| Correct reallocation with exiting cell identified | A1 | |
| Improved solution table correct | A1 | cao |

---

Show LaTeX source

2. The table below shows the cost of transporting one unit of stock from each of four supply points, 1 , 2, 3 and 4, to each of three demand points, $\mathrm { A } , \mathrm { B }$ and C . It also shows the stock held at each supply point and the stock required at each demand point. A minimal cost solution is required.

\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
 & A & B & C & Supply \\
\hline
1 & 31 & 29 & 32 & 20 \\
\hline
2 & 22 & 33 & 27 & 22 \\
\hline
3 & 25 & 27 & 32 & 20 \\
\hline
4 & 23 & 26 & 38 & 38 \\
\hline
Demand & 35 & 25 & 30 &  \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Add a dummy demand point and appropriate values to Table 1 in the answer book.

Table 2 shows an initial solution given by the north-west corner method.\\
Table 3 shows some of the improvement indices for this solution.

\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
 & A & B & C & D \\
\hline
1 & 20 &  &  &  \\
\hline
2 & 15 & 7 &  &  \\
\hline
3 &  & 18 & 2 &  \\
\hline
4 &  &  & 28 & 10 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 2}
\end{center}
\end{table}

\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
 & A & B & C & D \\
\hline
1 &  & - 13 &  & - 9 \\
\hline
2 &  &  & - 11 &  \\
\hline
3 &  &  &  &  \\
\hline
4 & 1 & - 7 &  &  \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 3}
\end{center}
\end{table}
\item Calculate the shadow costs and the missing improvement indices and enter them into Table 3 in the answer book.
\item Taking the most negative improvement index to indicate the entering square, use the steppingstone method once to obtain an improved solution. You must make your route clear and state your entering cell and exiting cell.
\end{enumerate}

\hfill \mbox{\textit{Edexcel D2 2011 Q2 [9]}}

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