Maths for PM – TIME

Having covered in the previous article the maths applicable to SCOPE, I would like here to explore the maths for TIME and scheduling.

Some elements of the maths and formulas presented previously might have an application also when dealing with the time and schedule of a project. There is a logical connection with financial considerations as well (covered in the next article) and also an interesting application of probability.

I here covered the subject under three subtopics:

• Critical path method
• PERT (Project Evaluation and Review Technique)
• Schedule Performance.

Most of the consideration will be related to a “waterfall” approach. Some of the concepts, for example the one about “critical path”, will only be applicable to that traditional approach. Agile would probably be slightly different. I would welcome comments from those with a specific experience with Agile.

Critical path method

In a project the critical path is the sequence of critical tasks. These tasks are those that, if delayed, will delay the completion of the project. [This will clearly apply to traditional “waterfall” projects.]

The Critical Path method calculates the minimum length of time in which the project can be completed, identifying those critical tasks. Most planning tools will help with a facilitated identification of the path.

The maths is simple and it entails using expected duration, early start and finish, late start and finish, the calculation of ‘slack’ and understanding dependencies. I found the table below for the tasks to build a house very useful:

Task ID	Task name	Expected duration (weeks)	Precedence	ES	LS	EF	LF	LS-ES (slack)
A	Lay foundation, build walls	2		0	1	2	3	1
B	Have doors, windows made	7		0	0	7	7	0
C	Build interior	4	A	2	3	6	7	1
D	Install roof, finish outside	3	A	2	6	5	9	4
E	Install windows, finish interior	2	B,C	7	7	9	9	0

This article by the PMI well explains the maths. In the example above, Tasks B and E form the critical path with a total project duration of 9 weeks.

Modern scheduling tools will have algorithms and features that will work out the critical path for you. See the example below, with the critical path in red:

PERT (Project Evaluation and Review Technique)

PERT is a technique for monitoring and controlling complex projects, those consisting of a large number of individual tasks that have to be executed in a certain order. This technique works similarly to the CPM method.

Unlike CPM, the task times are not specified as a fixed length, but by three numbers as it aims to address the unpredictability of the schedule. These three times are: t_opt (optimistic time), t_mod (the most likely time or its mode) and t_pes (a pessimistic time). Using the example of the house build as presented for the CPM method we may have the following situation:

Task ID	Task name	Optimistic time	Mode	Pessimistic time
A	Lay foundation, build walls	1 week	2 weeks	3 weeks
B	Have doors, windows made	5 weeks	7 weeks	9 weeks
C	Build interior	3 weeks	4 weeks	5 weeks
D	Install roof, finish outside	2 weeks	3 weeks	4 weeks
E	Install windows, finish interior	1 week	2 weeks	3 weeks

The critical path is worked out similarly to CPM (tasks B and E set the critical path). PERT uses formulas to make statements about the approximate probability that the project can be completed in less than a given time T, based on a normal approximation. [The method calls in for some statistics].

First, the average time t for each critical task and its standard deviation s are approximated by a particular form of the beta probability distribution [I take here the formulas as presented in the book I used as reference]:

t = (t_opt + 4t_mod + t_pes) / 6

s = (t_pes + t_opt) / 6

The expected length of the critical path is given by the sum of the average time for the tasks on the path (tasks B and E) and the standard deviation as the square root of the sum of the squared standard deviations for the same tasks. According to my book, this means that the critical path for the example above has an expected time of 9 weeks and a standard deviation of 0.75 week. The probability that the house is finished within 10 weeks is about 0.9. Interesting, isn’t it?

What about Agile projects?

You may want to consult specific articles, like this link, for example. The formulas seem pretty basic. What I see that might be relevant for the Time & Schedule is the concept of Velocity (as the story points of completed items in an iteration), calculated as:

Velocity = sum of story points of completed items / iteration

Or Average Velocity, as an average value of several iterations, calculated as:

Average velocity = sum of story points of completed items from n iterations / n

Additionally, the concept of Throughput can be considered, as a measure of the number of work items completed during an iteration or another period of time. Here, I would welcome feedback from the readers who have more experience with Agile.

---

In this article I provided an overview of the maths and formulas that in my opinion would help in the project TIME and schedule. In the next article I am venturing into COST and finance (good luck to me!). Please do not hesitate to contact me for any comments and constructive feedback.

Marco Bottacini, Senior Portfolio Manager, GALVmed

---

The views and opinions expressed in this blog are those of the author and do not necessarily reflect the views and opinion of GALVmed.