1. **Problem Statement:** We are given tasks with durations and dependencies for an event planning project. We need to analyze the project using CPM (Critical Path Method). 2. **Step 1: Draw the CPM Network Diagram** - Tasks: A, B, C, D, E, F, G - Dependencies: - A has no predecessor. - B, C, D depend on A. - E depends on B and D. - F depends on E. - G depends on B, C, and D. 3. **Step 2: Forward Pass to Calculate EST (Earliest Start Time) and EFT (Earliest Finish Time)** - Formula: $$\text{EFT} = \text{EST} + \text{Duration}$$ - Rules: EST of first task with no predecessors is 0. Calculate: - A: EST=0, EFT=0+2=2 - B: EST=EFT of A=2, EFT=2+3=5 - C: EST=EFT of A=2, EFT=2+2=4 - D: EST=EFT of A=2, EFT=2+1=3 - E: EST=max(EFT of B, EFT of D)=max(5,3)=5, EFT=5+2=7 - F: EST=EFT of E=7, EFT=7+3=10 - G: EST=max(EFT of B, C, D)=max(5,4,3)=5, EFT=5+1=6 4. **Step 3: Backward Pass to Calculate LST (Latest Start Time) and LFT (Latest Finish Time)** - Formula: $$\text{LST} = \text{LFT} - \text{Duration}$$ - Rules: LFT of last task(s) = project duration (max EFT), here max EFT=10 (from F) Calculate: - F: LFT=10, LST=10-3=7 - E: LFT=LST of F=7, LST=7-2=5 - G: LFT=10 (since G is not predecessor of any task, assume project end), LST=10-1=9 - B: LFT=min(LST of E, LST of G)=min(5,9)=5, LST=5-3=2 - C: LFT=LST of G=9, LST=9-2=7 - D: LFT=min(LST of E, LST of G)=min(5,9)=5, LST=5-1=4 - A: LFT=min(LST of B, C, D)=min(2,7,4)=2, LST=2-2=0 5. **Step 4: Identify Critical Path and Project Duration** - Critical path tasks have zero slack (EST=LST and EFT=LFT). - Slack = LST - EST Calculate slack: - A: 0-0=0 - B: 2-2=0 - C: 7-2=5 - D: 4-2=2 - E: 5-5=0 - F: 7-7=0 - G: 9-5=4 Critical path: A -> B -> E -> F Project duration = 10 days 6. **Step 5: Slack and Critical Path Tasks** - Slack for each task: - A: 0 (critical) - B: 0 (critical) - C: 5 - D: 2 - E: 0 (critical) - F: 0 (critical) - G: 4 - Tasks on critical path: A, B, E, F --- **Summary of Concepts:** 1. Slack (or float) time is the amount of time a task can be delayed without delaying the project completion. Critical path tasks have zero slack. 2. A slack of 4 days means the task can be delayed up to 4 days without affecting the overall project timeline. 3. Knowing slack helps project managers prioritize tasks and allocate resources efficiently. 4. Identifying slack helps manage limited resources by allowing flexibility in scheduling non-critical tasks.