A certain assembly project requires the completion of each of 5 tasks. In the diagram, the tasks are represented by...

Owning the Dataset

Table 1: Text Analysis

Table 2: Chart Analysis

Key Insights

1. The longest (critical) path is begin → D (21 days) → E (5 days) → end, totaling 26 days, which sets the earliest possible project completion time.
2. Tasks A, B, and C can be executed in parallel to D and E, but their total duration (A+B or A+C) is less than D+E, so they don't affect project duration unless heavily delayed.
3. Increasing the duration of task C by 1 day does not delay project completion, as it is not on the critical path.
4. The main constraint is that E cannot start until D is finished, establishing \(\mathrm{D→E}\) as the pace-setter for the whole project.

Step-by-Step Solution

Question 1: Minimum Time to Complete the Project

Complete Statement:

The least number of days needed to complete the project is _______ days.

Breaking Down the Statement

• Statement Breakdown 1:
  • Key Phrase: least number of days
    • Meaning: This is asking for the shortest possible duration from project start to finish.
    • Relation to Chart: In a project network, the overall duration is determined by the longest path from start to finish (the 'critical path') because tasks on that path cannot be delayed.
    • Important Implications: We must consider all possible sequences of tasks, but the one that takes the longest time is what actually determines project completion.

• Statement Breakdown 2:
  • Key Phrase: complete the project
    • Meaning: This refers to reaching the 'end' node only after all required tasks are done.
    • Relation to Chart: Completion can only occur after all paths through the network have been followed and finished.
    • Important Implications: Parallel paths can finish at different times, but the last finishing (slowest) path sets the project's minimum time.

• What is needed: The duration of the longest path through the project network diagram.

Solution:

• Condensed Solution Implementation:
  List all possible paths from 'begin' to 'end', calculate the time for each, and identify the longest (the critical path).
• Necessary Data points:
  Possible paths and their durations are: Path 1: begin → A → B → end \(\mathrm{(8+5=13\text{ days})}\), Path 2: begin → A → C → end \(\mathrm{(8+7=15\text{ days})}\), Path 3: begin → D → E → end \(\mathrm{(21+5=26\text{ days})}\).
  • Calculations Estimations:
    Path 1: \(\mathrm{8+5=13}\) days. Path 2: \(\mathrm{8+7=15}\) days. Path 3: \(\mathrm{21+5=26}\) days. The longest path is Path 3 at 26 days.
  • Comparison to Answer Choices:
    Among the listed choices, 26 days matches our calculated critical path duration.

FINAL ANSWER Blank 1: 26

Question 2: Effect of Extending Task C

Complete Statement:

If the number of days needed to complete Task C is increased by 1 day, then the least number of days needed to complete the project _______.

Breaking Down the Statement

• Statement Breakdown 1:
  • Key Phrase: Task C is increased by 1 day
    • Meaning: Task C will now take 8 days instead of 7.
    • Relation to Chart: This changes the duration of any path including Task C, primarily Path 2 (begin → A → C → end).

• Statement Breakdown 2:
  • Key Phrase: least number of days needed to complete the project
    • Meaning: We must see if this change alters the longest duration path through the network.
    • Relation to Chart: Compare new Path 2 duration to Path 3; if Path 3 is still longest, project duration is unchanged.

• What is needed: Whether the overall project duration changes if Task C takes 1 extra day.

Solution:

• Condensed Solution Implementation:
  Increase Task C's duration by 1 day and recalculate Path 2, then compare with current critical path (Path 3).
• Necessary Data points:
  Original Path 2: \(\mathrm{8\text{ (A)} + 7\text{ (C)} = 15}\) days. New Path 2: \(\mathrm{8\text{ (A)} + 8\text{ (C)} = 16}\) days. Path 3 \(\mathrm{(D → E)}\) is still \(\mathrm{21 + 5 = 26}\) days.
  • Calculations Estimations:
    New Path 2 is 16 days, which is still less than Path 3 as \(\mathrm{16 < 26}\), so the critical path and project duration remain unchanged.
  • Comparison to Answer Choices:
    Since the 26-day path is still the longest, the project duration 'remains the same', matching that answer choice.

FINAL ANSWER Blank 2: remains the same

Summary

The minimum time to complete the project is 26 days, determined by the critical path \(\mathrm{D→E}\). Even if Task C is extended by one day, the 26-day path \(\mathrm{(D→E)}\) remains the critical path, so the project completion time stays the same.

Question Independence Analysis

The two blanks are dependent: correctly identifying the critical path in question 1 is essential for understanding why a 1-day delay in Task C does not affect the project completion time in question 2.