• The Question:
  • How long will it take to complete a specific (design or implementation) project (or part of a project)?
• A Review of the Issues:
  • Units Matter
  • Task Details Matter
  • Task Dependencies Matter
  • Personnel Capabilities Matter

• The 1900s:
  • Gantt Charts
• The 1950s:
  • PERT - used time estimates
  • CPM - used time and cost estimates
• Since Then:
  • A number of techniques have been developed, all of which use the term "critical path"

• Apparently:
  • It's possible to put all of the necessary information in a table
• Unfortunately:
  • The table can be confusing
• Allaying the Confusion:
  • Use some discrete math (i.e., a graph)

• Earliest Initiation Time:
  • $I^'_i$ is the initiation time for task $i$ assuming the preceding tasks are completed as early as possible
• Earliest Completion Time:
  • $C^'_i = I^'_i + T_i$

• For the Initial Task:
  • 0
• For Other Tasks:
  • $I^'_j = \max_{i \in {\cal P}_j} C^'_i$ where ${\cal P}_j$ is the set of prerequisite tasks for task $j$

• Latest Completion Time:
  • $C^"_i$ is the last time task $i$ can be completed without delaying the project beyond its earliest time
• Latest Initiation Time:
  • $I^"_i = C^"_i - T_i$

• For the Last Task:
  • The earliest completion time
• For Other Tasks:
  • $C^"_i = \min_{j \in {\cal S}_i} I^"_j$ where ${\cal S}_i$ is the set of tasks that are subsequent to task $i$

Visualization for Tasks 7 to 14

Task 6

${\cal S}_6 = \{10, 13\}$
$C^"_6 = \min\{ I^"_{10}, I^"_{13} \} = \min\{15,21\} = 15$
$I^"_6 = C^"_6 - T_6 = 15 - 1 = 14$

• Intuition:
  • The difference between the latest completion time and the earliest completion time
• Formally:
  • $S_i = C^"_i - C^'_i$

• Definition:
  • The tasks that can't be delayed without delaying the project
• Process:
  • Find the tasks that have a slack of $0$