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

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

Critical Path Analysis 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.

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

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

Critical path analysis Step 1 List 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.

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

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

Refurbishing a room 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?