Let's break down and solve the problem step by step based on the questions and the data provided in Table Q.1.

Part (a): Draw a precedence diagram and determine the project duration under normal conditions

1. Activities and Precedence:

   • START → A, B, C
   • A → D, E
   • D and C → F
   • F and E → FINISH

2. Precedence Diagram: I'll create a precedence diagram (Activity-on-Node format).

   • Nodes:
     • START → A, B, C
     • A → D, E
     • C + D → F
     • F + E → FINISH

3. Duration Under Normal Conditions: To determine the time for project completion, compute the critical path.

   • Paths and their durations:
     • START → A → D → F → FINISH: $5 + 4 + 4 + 3 = 16$
     • START → A → E → FINISH: $5 + 2 + 3 = 10$
     • START → B → FINISH: $3 + 3 = 6$
     • START → C → F → FINISH: $7 + 4 + 3 = 14$

   The critical path is START → A → D → F → FINISH, with a total duration of 16 days.

Part (b): Minimum cost of completing the project in 15 days

• To reduce the duration to 15 days, we need to crash activities on the critical path.
• Critical Path (Original): START → A → D → F → FINISH
  • Current duration: $5 + 4 + 4 + 3 = 16$
  • Target duration: $15$

Step 1: Identify crashable activities and costs

• Only activities on the critical path need crashing.
• From the table:
  • $A: 5 \rightarrow 3$ days, cost = $7200 - 6000 = 1200$
  • $D: 4 \rightarrow 2$ days, cost = $6800 - 6400 = 400$
  • $F: 4 \rightarrow 2$ days, cost = $3700 - 3200 = 500$

Step 2: Crash activities (reduce by 1 day at a time)

1. Crash $D$ by 1 day (cost = £200/day):
   • New duration: $16 - 1 = 15$
   • New critical path duration = $15$
   • Additional cost: £200

Step 3: Total costs

• Direct Costs:

  • Normal cost: $2000 + 6000 + 2400 + 5600 + 6400 + 2400 + 3200 = £28,000$
  • Crashing $D$: $400$
  • Total Direct Cost = $28,400$

• Indirect Costs:

  • $15 \text{ days} \times £500 = £7500$

• Combined Cost: $28,400 + 7500 = £35,900$

The minimum cost of completing the project in 15 days is £35,900.

Part (c): Cost/Project Time Relationship

1. Direct Costs:

   • Direct costs increase with crashing as we spend more to reduce activity durations.

2. Indirect Costs:

   • Indirect costs decrease as the project duration reduces (less overhead time).

3. Combined Costs:

   • Total cost is the sum of direct and indirect costs.
   • There's an optimal duration where the combined costs are minimized (trade-off between direct and indirect costs).

Would you like me to visualize the precedence diagram or any of these relationships (e.g., cost vs. time)? Let me know! Here are some further questions to guide your understanding:

1. What is the impact of crashing beyond the target duration (e.g., 14 days)?
2. How would changing the indirect cost rate affect the combined cost?
3. Can you identify other non-critical paths that might become critical after crashing?
4. What happens to the project cost if all activities are completed at their crash durations?
5. How would adding parallel tasks impact the overall schedule?

Tip: Always re-evaluate the critical path after crashing to ensure you’re targeting the correct activities!