Let us suppose t_{ij} be duration of activity (i, j)

**Earliest Start Time (ES):**

It is considered as the earliest time (occurrence time) of any event with which different activity gets started.

For any beginning event, ES_{1} = 0

Suppose ES_{i} is equal to ES such that all the activities start from node I, then we can consider,

ES_{j} = Max { ES_{i} + t_{ij})

**Earliest Finish (EF) or Completion Time:**

It can be formulated as-

EF_{1} = ES_{i} + t_{ij}

For this, consider an example of a part of network-

ES_{j} = Max {ES_{1} + t_{1j}, ES_{2} + t_{2j}} = Max (3 + 2 + 1} = 5

EF_{1} = 3 + 2 = 5

EF_{2} = 1 + 3 = 4

**Latest Finish (LF) orCompletion Time:**

LF is the latest occurrence time of different events when the activities are ready to terminate or finish.

LF_{i} = Min_{j} {LF_{1} – t_{ij}}

For this, consider the below example as a part of network-

3 ① LF_{1} = 8

i

4 ② LF_{2} = 7

Then,

LF_{i} = Min {LF_{1} – t_{i1}, LF2 – t_{i2}}

= Min {8 – 3, 7 – 4} = 3

**Latest Start Time or LS:**

It is the last time when any of the events occur without any delay in the completion of the project.

**Total Floats or TF:**

TF is considered as the time duration when any activity can get delayed without affecting project’s completion time.

TF_{ij} = LF_{j} – ES_{i} – t_{ij}

= LF_{j} – (ES_{i} + t_{ij})

= LF_{j} – EF_{ij}

We know that,

TF_{ij} = LF_{ij} – ES_{i}

= (LF_{j}– t_{ij}) – ES_{i}

**Free Floats or FF:**

It is the time duration that can be taken without affecting the actual completion time form the earliest start time of its immediate successor in the network.

EF_{ij} = EF_{j} – ES_{i} – t_{ij}

= EF_{j} – (ES_{i} + t_{ij})

= EF_{j} – EF_{ij}

Here, any activity will be said as critical if it satisfied the following:

EF_{ij} = LF_{ij}, ES_{i}

= LF_{j} ES_{i} – ES_{i}

= LF_{j}–LF_{j}

= t_{ij}

It is found that any critical activity should have zero TF (total float and zero FF (free float)

**Independent Floats:**

The difference between free float and tail stack is called independent float.

**Note:**

Generally, slack is considered with PERT as it is reference to any event whereas float is considered with CPM as it is reference to any activity. Moreover, it is possible to be used interchangeably.

**Links of Previous Main Topic:-**

- Introduction to statistics
- Knowledge of central tendency or location
- Definition of dispersion
- Moments
- Bivariate distribution
- Theorem of total probability addition theorem
- Random variable
- Binomial distribution
- What is sampling
- Estimation
- Statistical hypothesis and related terms
- Analysis of variance introduction
- Definition of stochastic process
- Introduction operations research
- Introduction and mathematical formulation in transportation problems
- Introduction and mathematical formulation
- Queuing theory introduction
- Inventory control introduction
- Simulation introduction

**Links of Next Statistics Topics:-**