1. **Problem Statement:** We have a project with tasks A to I, each with durations and dependencies. We need to: a. Construct the network diagram and find the critical path. b. Calculate the planned project duration in weeks. c. Identify non-critical tasks and their float (free slack). 2. **Key Concepts:** - The **critical path** is the longest path through the network, determining the shortest project duration. - **Float (free slack)** is the amount of time a task can be delayed without delaying the project. - We calculate **earliest start (ES)**, **earliest finish (EF)**, **latest start (LS)**, and **latest finish (LF)** for each task. 3. **Step 1: List tasks with durations and predecessors:** - A: 5 days, no predecessor - B: 15 days, predecessor A - C: 25 days, predecessor B - D: 15 days, predecessor B - E: 30 days, predecessor B - F: 10 days, predecessors C and D - G: 10 days, predecessors E and F - H: 5 days, predecessor G - I: 5 days, predecessor H 4. **Step 2: Calculate Earliest Start (ES) and Earliest Finish (EF):** - A: ES=0, EF=0+5=5 - B: ES=EF of A=5, EF=5+15=20 - C: ES=EF of B=20, EF=20+25=45 - D: ES=EF of B=20, EF=20+15=35 - E: ES=EF of B=20, EF=20+30=50 - F: ES=max(EF of C,D)=max(45,35)=45, EF=45+10=55 - G: ES=max(EF of E,F)=max(50,55)=55, EF=55+10=65 - H: ES=EF of G=65, EF=65+5=70 - I: ES=EF of H=70, EF=70+5=75 5. **Step 3: Calculate Latest Finish (LF) and Latest Start (LS) starting from project end:** - I: LF=EF=75, LS=LF-duration=75-5=70 - H: LF=LS of I=70, LS=70-5=65 - G: LF=LS of H=65, LS=65-10=55 - E: LF=LS of G=55, LS=55-30=25 - F: LF=LS of G=55, LS=55-10=45 - C: LF=LS of F=45, LS=45-25=20 - D: LF=LS of F=45, LS=45-15=30 - B: LF=min(LS of C,D,E)=min(20,30,25)=20, LS=20-15=5 - A: LF=LS of B=5, LS=5-5=0 6. **Step 4: Calculate Float (Free Slack) = LS - ES for each task:** - A: 0-0=0 - B: 5-5=0 - C: 20-20=0 - D: 30-20=10 - E: 25-20=5 - F: 45-45=0 - G: 55-55=0 - H: 65-65=0 - I: 70-70=0 7. **Step 5: Identify Critical Path:** Tasks with zero float form the critical path: A -> B -> C -> F -> G -> H -> I 8. **Step 6: Calculate planned duration in weeks:** Total duration in days = EF of I = 75 days Working days per week = 5 Planned duration in weeks = $\frac{75}{5} = 15$ weeks 9. **Step 7: Non-critical tasks and their floats:** - D: float 10 days - E: float 5 days **Final answers:** - Critical path: A-B-C-F-G-H-I - Planned duration: 15 weeks - Non-critical tasks: D (float 10 days), E (float 5 days)