1. © Nuffield Foundation 2011 Nuffield Free-Standing Mathematics Activity Refurbishing a room © Nuffield Foundation 2011

2. Refurbishing a room You can answer such questions by using critical path analysis. This activity will show you how to do this. Where shall I start? What order shall I do things in? How soon can it be ready?’

3. The techniques of critical path analysis were developed in the late 1950s by two American companies, Dupont and Remington Rand. They were used to plan large-scale projects such as the development of the Polaris Missile. The algorithm determines the minimum time for a project to be completed and optimises manpower and resources to achieve this minimum completion time. Critical Path Analysis

4. Activity Time (hours) Preceding activities Decorating and furnishing a spare bedroom 1 Remove old furniture Remove carpet Take down curtains and rail Remove wallpaper 0.5 2 A BC A B C D Prepare walls Prepare woodwork Paint walls & ceiling 1 st coat (& dry) Paint woodwork (and dry) Lay new carpet Put up curtain rail and hang curtains Arrange new furniture Paint walls & ceiling 2 nd coat (& dry) E F G H I J K L M 0.5 1.5 5 8 5 2 1 1 0.5 Put up posters D D E F G HI HI J I Think about… What else needs to be done? Which jobs need to be done before others? How long each job will take?

5. K 1 End 0 M 0.5 L 1 I 5 G 5 F 1.5 E 0.5 D 2 C B 1 AStart 0 00 01 1 1.5 0 3.54 6 1.5 3.5 4 9 9 14 J 2 16 14 17 1417 16 17 Draw an activity network Carry out a forward pass to show earliest possible start times Carry out a reverse pass to show latest possible finish times Show the time needed for each activity 14 Find a critical path Start A B D E G I J L End 8 H 5 14 Think about… What will the rest of the network look like?

6. Critical path analysis Step 1List the activities with a time estimate for each. Step 2 Note which activities must precede others. Step 3 Draw an activity network, including the time for each activity. Step 4 Carry out a forward pass to find the earliest possible start times. Step 5 Carry out a reverse pass to find the latest possible finish times. Step 6 Identify critical activities and find a critical path.

7. Critical activities must start on time if the project is not to be delayed. They are those for which: AFurniture removal B Carpet removal D Wallpaper removal E Wall preparation G 1 st coat on walls I 2 nd coat on walls J Carpet laying L Arranging new furniture must start at 0 hours must start at 1 hours must start at 1.5 hours must start at 3.5 hours must start at 4 hours must start at 9 hours must start at 14 hours must start at 16 hours Latest finish time = earliest start time + duration Think about… What comes next?

8. Activity Earliest start Latest finish DurationFloat The other activities have some flexibility in their start time. C Remove curtain & rail F Prepare woodwork H Paint woodwork K Put up curtain & rail M Put up posters 0 1.5 1 h 3.5 6 1 h 5 14 1 h 14 17 2 h 14 17 2.5 h Float = latest finish time – (earliest start time + duration) 0.5 1.5 8 1 0.5 Think about… Which activities have float?

9. Reflect on your work Summarise the steps in working out a critical path. Describe what is meant by ‘float’. What effect will the number of helpers involved have on the minimum completion time? When on your finished network will there be the need for at least one helper to allow some of the activities to take place simultaneously? What practical considerations need to be taken into account when working out the minimum completion time? Refurbishing a room