Edexcel FD2 2021 June — Question 3 11 marks

Exam BoardEdexcel
ModuleFD2 (Further Decision 2)
Year2021
SessionJune
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicNetwork Flows
TypeTransportation problem: stepping-stone method
DifficultyStandard +0.8 This is a standard Further Maths transportation problem requiring multiple algorithmic steps (north-west corner, stepping-stone method with shadow costs and improvement indices). While methodical and time-consuming with a 4×4 table, it follows a well-defined procedure taught in FD2 with no novel insight required—just careful execution of the algorithm across multiple iterations.
Spec7.07f Algebraic interpretation: explain simplex calculations
3. The table below shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , to four sales points, \(\mathrm { P } , \mathrm { Q } , \mathrm { R }\) and S . It also shows the number of units held at each supply point and the number of units required at each sales point. A minimum cost solution is required.
PQRSSupply
A1819171328
B1615141943
C2117222329
D1620192136
Demand25414030
  1. Use the north-west corner method to obtain an initial solution.
  2. Taking AS as the entering cell, use the stepping-stone method to find an improved solution. Make your method clear.
  3. Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by showing the route and stating the
Question 3:
Part (a):
AnswerMarks Guidance
AnswerMark Guidance
Correct completed table: A row: P=25; B row: Q=38, R=5; C row: R=29; D row: R=6, S=30B1 cao
Part (b):
AnswerMarks Guidance
AnswerMark Guidance
Valid route shown with \(\theta\) values: A: Q=\(3-\theta\); B: Q=\(38+\theta\), R=\(5-\theta\); D: R=\(6+\theta\), S=\(30-\theta\)M1 A valid route shown, only one empty square used, \(\theta\)'s balance
Final table: A: P=25, S=3; B: Q=41, R=2; C: R=29; D: R=9, S=27A1 cao
Part (c):
AnswerMarks Guidance
AnswerMark Guidance
All 8 shadow costs and 9 improvement indices found for correct entriesM1 Finding all 8 shadow costs and the 9 improvement indices for the correct 9 entries
Shadow costs (rows 0,3,11,8) and IIs: A:X; B:−5,X,X; C:−8,−6,X; D:−10,0,X with columns 18,12,11,13A1 Shadow costs and II cao
Valid route shown with \(\theta\): A: P=\(25-\theta\), S=\(3+\theta\); D: P=\(\theta\), S=\(27-\theta\)M1 Valid route from most negative II, only one empty square used, \(\theta\)'s balance
Final table: A: S=28; B: Q=41, R=2; C: R=29; D: P=25, R=9, S=2; entering cell DP, exiting cell APA1 cao including the deduction (and stating) of all entering and exiting cells
Part (d):
AnswerMarks Guidance
AnswerMark Guidance
(£)2258B1 cao
Part (e):
AnswerMarks Guidance
AnswerMark Guidance
Shadow costs (rows 0,3,11,8) and IIs with columns 8,12,11,13: A:10,7,6,X; B:5,X,X,3; C:2,−6,X,−1; D:X,0,X,XM1 Finding all 8 shadow costs and all 9 negative improvement indices or sufficient number for at least 1 negative II
Correct negative II from workingA1 cao negative II from correct working
A negative II so solution is not optimalA1 cso including correct reasoning that solution is not optimal because there is a negative II 
# Question 3:

## Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Correct completed table: A row: P=25; B row: Q=38, R=5; C row: R=29; D row: R=6, S=30 | B1 | cao |

## Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Valid route shown with $\theta$ values: A: Q=$3-\theta$; B: Q=$38+\theta$, R=$5-\theta$; D: R=$6+\theta$, S=$30-\theta$ | M1 | A valid route shown, only one empty square used, $\theta$'s balance |
| Final table: A: P=25, S=3; B: Q=41, R=2; C: R=29; D: R=9, S=27 | A1 | cao |

## Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| All 8 shadow costs and 9 improvement indices found for correct entries | M1 | Finding all 8 shadow costs and the 9 improvement indices for the correct 9 entries |
| Shadow costs (rows 0,3,11,8) and IIs: A:X; B:−5,X,X; C:−8,−6,X; D:−10,0,X with columns 18,12,11,13 | A1 | Shadow costs and II cao |
| Valid route shown with $\theta$: A: P=$25-\theta$, S=$3+\theta$; D: P=$\theta$, S=$27-\theta$ | M1 | Valid route from most negative II, only one empty square used, $\theta$'s balance |
| Final table: A: S=28; B: Q=41, R=2; C: R=29; D: P=25, R=9, S=2; entering cell DP, exiting cell AP | A1 | cao including the deduction (and stating) of all entering and exiting cells |

## Part (d):
| Answer | Mark | Guidance |
|--------|------|----------|
| (£)2258 | B1 | cao |

## Part (e):
| Answer | Mark | Guidance |
|--------|------|----------|
| Shadow costs (rows 0,3,11,8) and IIs with columns 8,12,11,13: A:10,7,6,X; B:5,X,X,3; C:2,−6,X,−1; D:X,0,X,X | M1 | Finding all 8 shadow costs and all 9 negative improvement indices or sufficient number for at least 1 negative II |
| Correct negative II from working | A1 | cao negative II from correct working |
| A negative II so solution is not optimal | A1 | cso including correct reasoning that solution is not optimal because there is a negative II |

---
3. The table below shows the cost, in pounds, of transporting one unit of stock from each of four supply points, $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D , to four sales points, $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ and S . It also shows the number of units held at each supply point and the number of units required at each sales point.

A minimum cost solution is required.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
 & P & Q & R & S & Supply \\
\hline
A & 18 & 19 & 17 & 13 & 28 \\
\hline
B & 16 & 15 & 14 & 19 & 43 \\
\hline
C & 21 & 17 & 22 & 23 & 29 \\
\hline
D & 16 & 20 & 19 & 21 & 36 \\
\hline
Demand & 25 & 41 & 40 & 30 &  \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Use the north-west corner method to obtain an initial solution.
\item Taking AS as the entering cell, use the stepping-stone method to find an improved solution. Make your method clear.
\item Perform one further iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by showing the route and stating the

\begin{itemize}
  \item shadow costs
  \item improvement indices
  \item entering cell and exiting cell
\item State the cost of the solution found in (c).
\item Determine whether the solution obtained in (c) is optimal, giving a reason for your answer.
\end{itemize}
\end{enumerate}

\hfill \mbox{\textit{Edexcel FD2 2021 Q3 [11]}}
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