Many real-world problems involve connections, routes, and resource flows. Network models make these problems visual and tractable. General Mathematics focuses on applying network algorithms to genuine scenarios.
| Context | Network model | Algorithm |
|---|---|---|
| Road navigation | Weighted directed graph | Shortest path (Dijkstra) |
| Utility infrastructure | Undirected weighted graph | Minimal spanning tree |
| Project planning | Activity-on-edge diagram | Critical path analysis |
| Delivery routing | Complete weighted graph | Hamiltonian circuit |
| Cable/pipe layout | Undirected graph | Minimal spanning tree |
| Traffic flow | Directed graph with capacities | Maximum flow |
A telecommunications company needs to lay fibre cable connecting 5 suburbs (A, B, C, D, E). The cost (\$ thousands) of connecting each pair:
| Route | Cost |
|---|---|
| A–B | 8 |
| A–C | 5 |
| B–D | 6 |
| C–D | 3 |
| C–E | 7 |
| D–E | 4 |
Find the minimum cost to connect all suburbs (MST):
Sort by cost: C–D=3, D–E=4, A–C=5, B–D=6, C–E=7, A–B=8.
Apply Kruskal’s: Add C–D, Add D–E, Add A–C, Add B–D. Now all 5 vertices connected with 4 edges.
MST cost = 3+4+5+6 = \$18,000.
A building project has activities with durations (days):
| Activity | Predecessors | Duration |
|---|---|---|
| A | — | 3 |
| B | — | 5 |
| C | A | 4 |
| D | B | 2 |
| E | C, D | 6 |
Draw the network, find EST and LST at each vertex.
Critical path: B → D → E (5+2+6=13 days) is the longest path, so minimum project duration = 13 days.
Activities A and C have float (can be delayed without affecting completion).
Always express answers in the units of the problem:
- “The minimum time to complete the project is 13 days.”
- “The minimum cable cost is \$18,000.”
- “The shortest route from depot to delivery point is 6 km, via C and D.”
VCAA FOCUS: VCAA assessment emphasises interpretation. A correct algorithm answer that is not interpreted in context (units, meaning) will not receive full marks.
STUDY HINT: Practise drawing network diagrams neatly and labelling all edges. Messy diagrams lead to missed edges and algorithmic errors under exam conditions.
