Download the Critical Path Method Practice sheet

**How do you schedule activities in Large Project?**

Ever imagined how Managers schedule activities in a complex project. A typical project may involve several activities each of which may be dependent on one another. Determining the optimal path for a project will require you to take into account the dependencies. Thanks to Morgan Walker and James Kelly, We have the critical path method which is used by most organizations in determining the optimal duration for a Project.

In this blog I will take you through the steps involved in determining the critical path for a project. A critical path is path which covers all essential activities such that the time taken for the project is minimum.

**Scenario Overview**

I was working as a project manager for a software company. I was leading a project for developing the payroll system for a client. Typically, in such a big project you have a lot of activities. I had to overcome the challenge of scheduling the activities in such a manner that the time taken for the project is minimum. While doing so, I also had to take all the dependencies among the various activities into account. While going through the internet, I found that most Companies follow the critical path method to determine the optimal project duration.

The various major activities in this project and their dependencies are listed in the table given below. To keep the example simple I have listed only the broad activities of a software development process. Remember, there are other activities also in a software development project.

Activity | Activity Description | Duration in weeks | Preceding Activity |

A | Requirements Gathering | 6 | |

B | Feasibility Study | 5 | A |

C | Analysis | 8 | B |

D | Design | 9 | C |

E | Infrastructure and Other approval process | 4 | B |

F | Coding | 2 | D |

G | Testing | 2 | F |

H | Preparing the production Environment | 2 | E |

I | Deployment | 3 | E,G |

The various steps involved in determining a critical path for a project are

- Draw the precedence diagram
- Determine the early event time (ET) and late event time (LT)
- Determine the total float and free float
- Determine the critical path

Let us now try to draw the precedence diagram for the above software project

**Determining the early event time and late event time**

ET (1) = 0

ET (2) = ET (1) + 6=0+6=6

ET (3) = ET (2) + 5=6+5= 11

ET (4) = ET (3) + 8=11+8=19

ET (5) = ET (4) + 9=19+9=28

ET (7) = ET (2) + 4=6+4=10

ET (6) = ET (5) + 2=28+2=30

ET (8) = ET (7) + 2=10+2=12

ET (9) = max( ET (6)+2, ET (8) )= max(12,32)=32

ET (10) = ET (9) + 3=32+3=35

LT (10) = 35

LT (9) = LT (10) -3=35-3=32

LT (6) = LT (9) -2=32-2= 30

LT (8) = LT (9) -0 =32-0=32

LT (7) = LT (8) -2=32-2=30

LT (5) = LT (6) -2 =30-2=28

LT (4) = LT (5) -9=28-9=19

LT (3) = LT (4) -8 =19-8=11

LT (2) = min( LT (3)-5, LT (7) -4 )= min(6,26)=6

LT (1) = LT (2) -6=6-6=0

**Determining total float**

In project Management, float or slack is defined as the amount of time as task can be delayed in a project without causing a delay to

- Subsequent tasks ( free float)

- Project completion date ( Total Float)

Total float for any activity which connects node i, j= LT(J)- ET(i)- T(i,j)

T(i,j) is time for the activity

Activity |
T(i,j) |
Total float= LT(J)- ET(I)- T(i,j) |

A | 1,2 | 0 |

B | 2,3 | 0 |

C | 3,4 | 0 |

D | 4,5 | 0 |

E | 2,7 | 20 |

F | 5,6 | 0 |

G | 6,9 | 0 |

H | 7,8 | 20 |

I | 9,10 | 0 |

A critical path of any project is the path from the starting node to the end node. A critical path generally includes all path for which total float is zero

Thus the critical path is A-B-C-D-F-G-I

The above problem can also be solved by linear programming using MS Excel

**Solving for Critical Path using Linear Programming**

Objective function

Min X10-X1

Subject to the condition

X2 >= X1+6

X3>= X2+5

X4>=X3+8

X5>=X4+9

X6>X5+2

X9>=X6+2

X7>=X2+4

X8>=X7+2

X9>=X8

X10>=X9+3

How to use excel for solving the above problem

The above equation can be solved using excel solver. Let me now take you through how to use excel solver.

- Go to data tab of the excel sheet

- Set the target cells as cell which computes X10-X1

- Set the cells to be changed as the cells which have values X1 to X10

- Set the constraints as given by the equation above

- In the options tab of the solver check the Assume linear model option

I have also attached the excel sheet for detailed calculation

**Project Scheduling when uncertainty is there**

Many times, there can be scenarios in which determining he exact duration of an activity may not be possible. What do we do then? Don’t worry, we still have a solution. In such a scenario we determine for each activity three estimates of the activity duration i.e. Optimistic, Pessimistic and most likely.

Te=( o+4m+p)/4

Te= Time estimate

O= optimistic estimate

M= Most likely estimate

P=Pessimistic estimate

Variance of the activity= [(p-o)/6] ^2

**Using variation in time estimates for calculating the probability**

Suppose critical path for a project comes out to be 35 weeks

Sum of the variances of each activity given by [(p-o)/6]^2 is 2.5

What is the probability of the activity to be completed in 40 weeks?

Z=D-Te/Standard deviation

Z= No of standard deviation D is from Critical Path time

D= Our estimate of the project time

Te= Critical Path time

Z= 35-40/2.5=-2

I know look for the probability value corresponding to Z value of -2 which is 97.72%

Now suppose I want to be 95% sure that I will hit the target critical path. What should be my estimate of the project duration?

Using the same formula Z=D-Te/Standard deviation

We calculate D = Te+Z* Standard deviation

=40+1.65*2.5

=40+4.125

=44.125

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