Programme Arithmetic: Durations, Float, Calendars and Learning

A duration is a quantity divided by an output

Every activity in a construction programme starts as the same division: the quantity of work over the output of one crew in a working day gives crew-days, and crew-days over the number of crews working at once gives working days. Solved the other way, a fixed number of working days gives the crews needed, rounded up to whole crews — which is why meeting a deadline usually leaves a little time spare.

The division is exact and the output rate is not. A crew's output varies with the size of the job, the access, the weather and how long the crew has been at the task, often by a factor of two, and a crew only divides the time if it has room to work. So the rate on every page here is the reader's own, from their records or their crew leader, and the pages carry none of their own.

The critical path is the longest chain, not the busiest

Once each activity has a duration and the activities it must follow, the critical path method (Kelley and Walker, 1959) finds the programme's length in two passes. The forward pass gives each activity its early start — the latest early finish of everything before it — and its early finish. The backward pass, from the end, gives each its late finish — the earliest late start of everything after it — and its late start.

Total float is the difference between an activity's late and early start: how far it can slip without moving the end. The activities with none form the critical path, the longest chain through the network, and a day lost on any of them is a day on the finish. Float belongs to a path rather than to one activity, so two activities in a row with four days of float have four days between them, and spending it on the first takes it from the second.

Working days are not calendar days

A programme is worked in working days and delivered on a date, and the difference between them is where programmes quietly overrun. Twenty working days on a Monday-to-Friday job is four calendar weeks, not twenty days. The count starts on the first day, passes over every day that is not in the working week, and ends on the day the required working days have been worked; a public holiday on a working day adds a whole working day to the end, and one on a Saturday adds nothing to a weekday job.

Weather is the allowance programmes most often leave out. Outdoor work in winter loses days to rain, frost and wind, and how many depends on the season, the region and the work — so the pages count them as holidays the reader adds, rather than assuming a figure.

Repetition gets faster by doublings

A crew repeating the same task gets faster, and the learning curve describes how: each time the number of units doubles, the time falls to a fixed fraction of what it was — the learning rate. At 90%, the fourth unit takes 90% of the second and the eighth 90% of the fourth. Because the saving comes per doubling, the early units learn fastest and a long run flattens out, until the crew reaches a floor the work itself sets.

There are two models, and they are not interchangeable. Wright's (1936) applies the rate to the average time of all the units so far; Crawford's applies it to each unit's own time. The same data gives a different rate under each, and the same rate gives a shorter total under Wright's, so a rate has to be used in the model it was measured under. The exponent is the same in both: the logarithm of the rate over the logarithm of two.

What none of these pages decides

Each page applies a definition: a division, the two passes of the critical path method, a count of days, a learning-curve formula. None supplies the measured inputs — the output rate, the learning rate, the weather — because those belong to a crew, a site and a season, and a published figure for any of them would be a guess presented as a fact. And none replaces scheduling software for a real job, where hundreds of activities, overlapping links, lags and resource limits interact.