```| Task ID | Task Name                                                  |
|---------|------------------------------------------------------------|
| a       | Background study on e-payment implementation               |
| b       | Collect detailed requirements                              |
| c       | Develop prototype to show payment procedure                |
| d       | Design overall software architecture                       |
| e       | Design database                                            |
| f       | Design end-user interfaces on mobile phones                |
| g       | Design admin interfaces on desktop computers               |
| h       | Design signal connection with bike locks                   |
| i       | Design security scheme for internet transactions           |
| j       | Develop interface templates for sponsor review             |
| k       | Create site map/hierarchy chart                            |
| l       | Create individual web pages                                |
| m       | Create database                                            |
| n       | Source necessary library modules for client-side coding    |
| o       | Purchase bike locks for testing                            |
| p       | Code server-side modules #40;system admin and operations#41;         |
| q       | Code server-side modules #40;end-user services#41;                  |
| r       | Code client-side modules #40;system admin and operations#41;         |
| s       | Code client-side modules #40;end-user services#41;                  |
| t       | Integrate server-side modules and web pages                |
| u       | Test connection to bike locks with shared-bike app         |
| v       | Performance stress tests on the system                     |
| w       | Conduct UAT with sponsor and end-user representatives      |
| x       | Install website on government's web server                 |
| y       | Create technical training materials                        |
| z       | Train staff in operating and maintaining the system        |

## 3.2 Critical Path Analysis

Critical Path Analysis (CPA) identifies the sequence of tasks that determines the minimum project duration. Tasks on the critical path have zero slack.

### 3.2.1 Forward Pass (Calculating Early Start and Early Finish)

| Task | Duration (Days) | Predecessors | Early Start (ES) | Early Finish (EF) |
|---|---|---|---|---|
| a | 15 | - | 0 | 15 |
| b | 20 | a | 15 | 35 |
| c | 20 | b | 35 | 55 |
| d | 20 | b | 35 | 55 |
| e | 10 | d | 55 | 65 |
| f | 20 | d | 55 | 75 |
| g | 10 | f | 75 | 85 |
| h | 10 | d | 55 | 65 |
| i | 15 | d | 55 | 70 |
| j | 10 | f, g | 85 | 95 |
| k | 10 | j | 95|105|
| l | 30 | k | 105 | 135 |
| m | 15 | e | 65 | 80 |
| n | 15 | f | 75 | 90 |
| o | 10 | h | 65 | 75 |
| p | 20 | i, g | 85 | 105 |
| q | 25 | i | 70 | 95 |
| r | 20 | n | 90 | 110 |
| s | 20 | n | 90 | 110 |
| t | 15 | l, m, p, q, r, s | 135 | 150 |
| u | 5 | t, o | 150 | 155 |
| v | 5 | u | 155 | 160 |
| w | 10 | v | 160 | 170 |
| x | 10 | w | 170 | 180 |
| y | 20 | x | 180 | 200 |
| z | 15 | y | 200 | 215|

### 3.2.2 Backward Pass (Calculating Late Start and Late Finish)

Assuming the project's finish time is 244 days.

Here's the completed backward pass table:


| Task | Duration | Successors          | Late Start (LS) | Late Finish (LF) | Slack (LS - ES) |
|------|----------|---------------------|-----------------|------------------|-----------------|
| z    | 15       | -                   | 229             | 244              | 0               |
| y    | 20       | z                   | 209             | 229              | 0               |
| x    | 10       | y                   | 197             | 209              | 2               |
| w    | 10       | x                   |                 |                  | 0               |
| v    | 5        | w                   |                 |                  | 0               |
| u    | 5        | v                   |                 |                  | 0               |
| t    | 15       | u                   |                 |                  | 3               |
| l    | 30       | t                   |                 |                  | 2               |
| m    | 15       | t                   |                 |                  | 2               |
| p    | 20       | t                   |                 |                  | 2               |
| q    | 25       | t                   |                 |                  | 2               |
| r    | 20       | t                   |                 |                  | 2               |
| s    | 20       | t                   |                 |                  | 2               |
| o    | 10       | u                   |                 |                  | 0               |
| n    | 15       | r, s                |                 |                  | 2               |
| j    | 10       | k                   |                 |                  | 2               |
| k    | 10       | l                   |                 |                  | 0               |
| i    | 15       | p, q                |                 |                  | 0               |
| h    | 10       | o                   |                 |                  | 0               |
| g    | 10       | j, p                |                 |                  | 0               |
| f    | 20       | n,j, g              |                 |                  | 0               |
| e    | 10       | m                   |                 |                  | 0               |
| d    | 20       | e, f, h, i          |                 |                  | 3               |
| c    | 20       | -                   |                 |                  | 2               |
| b    | 20       | c, d                |                 |                  | 3               |
| a    | 15       | b                   |                 |                  | 0               |
```
Help identify what is not working in Slack and why, then make improvements. Total Slack time must be 29 days. Complete the back-transfer form at the end.