| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2018 |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Travelling Salesman |
| Type | Improve Upper Bound with Shortcuts |
| Difficulty | Moderate -0.8 This is a standard D1 travelling salesman problem requiring routine application of three algorithmic methods (shortcut method, nearest neighbour, lower bound via deletion). Each part follows textbook procedures with no novel problem-solving required, making it easier than average A-level maths questions which typically involve more conceptual understanding or multi-step reasoning. |
| Spec | 7.04b Minimum spanning tree: Prim's and Kruskal's algorithms7.04c Travelling salesman upper bound: nearest neighbour method7.04d Travelling salesman lower bound: using minimum spanning tree |
| A | B | C | D | E | F | |
| A | -- | 122 | 217 | 137 | 109 | 82 |
| B | 122 | -- | 110 | 130 | 128 | 204 |
| C | 217 | 110 | -- | 204 | 238 | 135 |
| D | 137 | 130 | 204 | -- | 98 | 211 |
| E | 109 | 128 | 238 | 98 | -- | 113 |
| F | 82 | 204 | 135 | 211 | 113 | -- |
The table shows the least distances, in km, between six towns, A, B, C, D, E and F.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& A & B & C & D & E & F \\
\hline
A & -- & 122 & 217 & 137 & 109 & 82 \\
\hline
B & 122 & -- & 110 & 130 & 128 & 204 \\
\hline
C & 217 & 110 & -- & 204 & 238 & 135 \\
\hline
D & 137 & 130 & 204 & -- & 98 & 211 \\
\hline
E & 109 & 128 & 238 & 98 & -- & 113 \\
\hline
F & 82 & 204 & 135 & 211 & 113 & -- \\
\hline
\end{tabular}
\end{center}
Liz must visit each town at least once. She will start and finish at A and wishes to minimise the total distance she will travel.
\begin{enumerate}[label=(\alph*)]
\item Starting with the minimum spanning tree given in your answer book, use the shortcut method to find an upper bound below 810 km for Liz's route. You must state the shortcut(s) you use and the length of your upper bound. \hfill [2]
\item Use the nearest neighbour algorithm, starting at A, to find another upper bound for the length of Liz's route. \hfill [2]
\item Starting by deleting F, and all of its arcs, find a lower bound for the length of Liz's route. \hfill [3]
\item Use your results to write down the smallest interval which you are confident contains the optimal length of the route. \hfill [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2018 Q1 [8]}}