SQA Higher Applications of Mathematics Planning and Decision Making: critical path, Gantt charts and expected value

What Planning and Decision Making actually demands

This area applies mathematics to managing projects and making choices under uncertainty. The examiners test whether you can schedule a project to finish as quickly as possible, find where there is slack, represent the plan over time, and choose rationally between risky options. It draws directly on the probability and expected-value work from the statistics area. This guide ties together the three dot-point pages of the module.

Project networks and the critical path

A project is a set of activities with durations and precedence rules. Drawn as an activity network, the minimum completion time is the length of the longest path, called the critical path, because every path must finish before the project does. A forward pass finds earliest start times and a backward pass finds latest start times; an activity is critical when these are equal. The float of a non-critical activity, latest start minus earliest start, is how long it can slip without delaying the finish; critical activities have float $0$.

Gantt charts and scheduling

A Gantt chart shows each activity as a horizontal bar on a time axis, so starts, durations, finishes and overlaps are read directly, and parallel activities appear as overlapping bars. PERT focuses on the activity network and timing, often using optimistic, most likely and pessimistic estimates to allow for uncertainty. Activities run in parallel only when resources allow; a limited resource such as one team forces parallel tasks into sequence, lengthening the schedule, which the chart makes visible. A Gantt chart is also a progress-tracking tool against the plan and its deadlines.

Expected value and decision making

A decision under risk has options leading to uncertain outcomes with known probabilities. The expected value of an option, $E = \sum (\text{value} \times \text{probability})$, is its long-run average payoff; you choose the option with the best expected value, often laid out in a decision table or decision tree and folded back from outcomes to the decision. Expected value ignores risk (the spread of outcomes), so for one-off decisions or widely varying payoffs you should also comment on risk, as a cautious decision maker may prefer a safer option.

How Planning and Decision Making is examined

A typical SQA profile for this area:

• Critical path. Finding the minimum completion time and the critical path from durations and precedence.
• Float. Calculating the slack of non-critical activities and interpreting it.
• Scheduling. Reading and constructing Gantt charts, and handling resource constraints.
• Decisions. Comparing options by expected value with tables or trees, and commenting on risk.

Check your knowledge

A mix of recall and method questions covering the module. Attempt them, then check against the solutions.

1. A project's two routes take $11$ and $14$ days. State the minimum completion time. (1 mark)
2. An activity's earliest start is day $5$ and latest start is day $9$. Find its float. (1 mark)
3. On a Gantt chart, how do two activities that run in parallel appear? (1 mark)
4. An option gives $\pounds 200$ (probability $0.4$) and $\pounds 700$ (probability $0.6$). Find its expected value. (2 marks)
5. Two options have equal expected value. Why might one still be preferred? (1 mark)