Forward and Backward Scanning Revision Notes for HSC SSCE Mathematics Standard

Forward and Backward Scanning

Introduction

In critical path analysis, we need to determine two important pieces of information for each activity in a project:

When each activity can earliest begin
When each activity must latest begin to avoid delaying the project

These are calculated using forward scanning and backward scanning respectively. Understanding both techniques helps us plan projects efficiently and identify which activities have flexibility in their scheduling.

infoNote

Both forward and backward scanning work together to give us a complete picture of project scheduling. Forward scanning identifies the minimum project duration, while backward scanning reveals which activities have scheduling flexibility.

Forward scanning

What is earliest starting time (EST)?

The earliest starting time (EST) is the earliest time any activity can begin after all prior activities have been completed.

For example, an $\text{EST} = 8$ means an activity cannot start until $8$ time units (such as hours or days) after the project begins.

Key points about EST:

Activities that start the project have an $\text{EST} = 0$ (no prior activities)
Activities that depend on others have an $\text{EST} > 0$
The EST at the project finish gives us the minimum time needed to complete the entire project

The forward scanning method

Forward scanning is the process we use to calculate the EST for each activity. We work systematically from the start of the project toward the finish, moving forward through the network diagram.

Step-by-step process:

Step 1: Draw a box split into two cells next to each vertex in your network diagram. If multiple activities begin at a vertex, draw a box for each activity.

Step 2: For activities that start the project (no predecessors), write $0$ in the left cell. This represents $\text{EST} = 0$ .

Step 3: For each subsequent activity, calculate its EST by adding:

$\text{EST of activity} = \text{EST of predecessor} + \text{duration of predecessor}$

Step 4: If an activity has multiple predecessors, calculate the EST using each predecessor separately, then choose the largest value. This ensures all prerequisite activities are complete before the activity begins.

Step 5: The EST value at the finish vertex represents the minimum completion time for the entire project.

chatImportant

Critical Rule for Multiple Predecessors:

When an activity has more than one predecessor, you must calculate the EST using each predecessor separately and then choose the largest value. This is because all predecessor activities must be completed before the activity can begin, so we need to wait for the longest path to finish.

Worked example: Forward scanning

Let's work through a complete example to see forward scanning in action.

lightbulbExample

Worked Example: Forward Scanning

Question: Find the earliest starting time (EST) for the project shown in the network diagram. Activity durations are in days.

Solution:

We follow the forward scanning steps systematically:

Draw boxes at each vertex (shown in the diagrams above)
Activities $A$ and $B$ start the project, so both have $\text{EST} = 0$
Activity $C$ follows activity $A$ (duration $8$ days): $\text{EST of } C = 0 + 8 = 8$
Continue for each activity. Notice the special cases:
- Activity $F$ $F$ has two predecessors:
  - Via activity $D$ : $\text{EST} = 6 + 2 = 8$
  - Via activity $C$ : $\text{EST} = 9 + 0 = 9$
  - We choose the larger value: $\text{EST of } F = 9$

Activity $G$ $G$ also has two predecessors:
- Via activity $F$ : $\text{EST} = 9 + 1 = 10$
- Via activity $E$ : $\text{EST} = 9 + 3 = 12$
- We choose the larger value: $\text{EST of } G = 12$

At the finish, we find $\text{EST} = 15$

Answer: The earliest starting time of the project is 15 days. This is the minimum time required to complete the entire project.

infoNote

Why choose the largest EST?

When an activity has multiple predecessors, all of them must be completed before the activity can begin. Taking the largest EST ensures we wait for the longest path to complete. If we took a smaller EST, we might try to start the activity before all its prerequisites are finished, which is impossible in practice.

Backward scanning

What is latest starting time (LST)?

The latest starting time (LST) is the latest time an activity can start without delaying the entire project, assuming all prior activities have been completed.

Certain activities have flexibility in their scheduling—they don't need to start at their earliest possible time. The LST tells us how late we can afford to start each activity while still completing the project in minimum time.

infoNote

Understanding LST is crucial for project management because it reveals which activities have "slack time" or flexibility. Some activities must start precisely at their EST (these are on the critical path), while others can be delayed without impacting the overall project completion time.

The backward scanning method

Backward scanning is the reverse of forward scanning. We calculate LSTs by working from the project finish back toward the start, moving backward through the network diagram.

Step-by-step process:

Step 1: Use your network diagram with EST values already calculated from forward scanning.

Step 2: Copy the minimum completion time (the EST at finish) into the right cell of the box at the finish vertex.

Step 3: For each activity, calculate its LST by subtracting:

$\text{LST of activity} = \text{LST of following activity} - \text{duration of current activity}$

Step 4: If multiple activities share the same predecessor, calculate the LST using each following activity separately, then choose the smallest value. This ensures the activity finishes in time for all its successors to start when needed.

Step 5: Continue backward through the network until all LST values are calculated.

chatImportant

Critical Rule for Multiple Successors:

When an activity has multiple successors, you must calculate the LST using each following activity separately and then choose the smallest value. This is because the activity must finish early enough for all its successor activities to start on time, so we need to accommodate the most time-critical successor.

Worked example: Backward scanning

We'll use the same network diagram from the forward scanning example.

lightbulbExample

Worked Example: Backward Scanning

Question: Find the latest starting time (LST) for each activity. Activity durations are in days.

Solution:

Starting with our completed forward scanning:

Copy the minimum completion time ( $15$ ) into the right cell at the finish vertex: $\text{LST} = 15$
Work backwards:
- Activity $H$ : $\text{LST} = 15 - 1 = 14$
- Activity $G$ : $\text{LST} = 14 - 2 = 12$
For activities with multiple successors, calculate each path:
- Activity $E$ $E$ :
  - Path via $G$ : $\text{LST} = 12 - 3 = 9$
  - This is the only path, so $\text{LST of } E = 9$
- Activity $C$ $C$ :
  - Path via $E$ : $\text{LST} = 9 - 1 = 8$
  - Path via $F$ : $\text{LST} = 11 - 1 = 10$
  - Choose the smaller value: $\text{LST of } C = 8$
Continue backward to the start:
- Activity $A$ : Since it leads to $C$ , $\text{LST} = 8 - 8 = 0$
- Activity $B$ : Through its path, $\text{LST} = 3$

infoNote

Why choose the smallest LST?

When an activity has multiple successors, it must finish early enough for all of them to start on time. Taking the smallest LST ensures the activity completes before the most time-critical successor needs to begin. Choosing a larger LST could delay critical successors and extend the entire project duration.

Comparing EST and LST

The table below highlights the key differences between forward and backward scanning:

Earliest Starting Time (EST)	Latest Starting Time (LST)
The earliest time any activity can start after all prior activities are complete	The latest time any activity can start without delaying the project
Calculated using forward scanning	Calculated using backward scanning
Start from the beginning and work forward	Start from the end and work backward
$\text{EST} = 0$ if no prior activities	LST at finish equals the minimum completion time
With multiple predecessors: choose the largest EST	With multiple successors: choose the smallest LST
Formula: $\text{EST} = \text{EST of predecessor} + \text{duration of predecessor}$	Formula: $\text{LST} = \text{LST of successor} - \text{duration of activity}$

infoNote

The difference between an activity's LST and EST represents its float or slack time—the amount of scheduling flexibility that activity has. Activities where $\text{EST} = \text{LST}$ are on the critical path and have no flexibility.

Exam tips

Always complete forward scanning before starting backward scanning
Draw clear boxes at each vertex—label the left cell for EST and right cell for LST
When activities have multiple predecessors or successors, write out all calculations to avoid mistakes
Remember: forward scanning uses largest values, backward scanning uses smallest values
The EST at the finish vertex always equals the LST at the finish vertex (both equal minimum completion time)
Double-check your arithmetic—errors in early calculations affect all subsequent values

chatImportant

A common mistake is confusing when to choose the largest versus smallest value. Remember this simple rule:

Forward (EST): Add durations, choose LARGEST
Backward (LST): Subtract durations, choose SMALLEST

Remember!

bookmarkSummary

Key Points to Remember:

Forward scanning calculates the earliest starting times (EST) by working from start to finish through the network
Backward scanning calculates the latest starting times (LST) by working from finish to start through the network
With multiple predecessors: add durations and choose the largest EST (all predecessors must complete)
With multiple successors: subtract durations and choose the smallest LST (must finish before earliest successor)
The EST at the project finish gives the minimum completion time for the entire project
Always complete forward scanning first, then use those values for backward scanning
The difference between LST and EST for any activity represents its scheduling flexibility (float)

Forward and Backward Scanning (HSC SSCE Mathematics Standard): Revision Notes