## **Solution to the Network Scheduling & Crashing Problem** ### **a. ES, EF, LS, LF, Slack, Critical Path, and Project Completion Time** #### **Step 1: Draw the Network and List Activities with Durations** | Activity | Predecessor(s) | Duration | |----------|----------------|----------| | A | - | 5 | | B | A | 7 | | C | A | 6 | | D | B | 6 | | E | C | 7 | | F | D, E | 8 | | G | F | 3 | #### **Step 2: Forward Pass (ES, EF)** - **A:** ES=, EF=5 - **B:** ES=5, EF=12 - **C:** ES=5, EF=11 - **D:** ES=12, EF=18 - **E:** ES=11, EF=18 - **F:** ES=max(18,18)=18, EF=26 - **G:** ES=26, EF=29 #### **Step 3: Backward Pass (LS, LF)** - **G:** LF=29, LS=26 - **F:** LF=26, LS=18 - **D:** LF=18, LS=12 - **E:** LF=18, LS=11 - **B:** LF=12, LS=5 - **C:** LF=11, LS=5 - **A:** LF=min(5,5)=5, LS= #### **Step 4: Slack Calculation** - **Slack = LS - ES (or LF - EF)** - **A:** - **B:** - **C:** - **D:** - **E:** - **F:** - **G:** #### **Step 5: Critical Path** All activities have zero slack, so the critical path is **A → B → D → F → G** and **A → C → E → F → G**. Since both B and C must be completed before D and E, and both D and E must be completed before F, both chains are critical. #### **Step 6: Project Completion Time** Project completion time = **29 days** --- ### **b. Project Crashing to Reduce by 3 Days** #### **Direct Cost Table** | Activity | Normal Cost | Normal Time | Crash Cost | Crash Time | |----------|-------------|-------------|------------|------------| | A | $7,000 | 5 | $8,000 | 3 | | B | $5,000 | 7 | $6,000 | 5 | | C | $4,000 | 6 | $5,200 | 4 | | D | $8,000 | 6 | $10,000 | 3 | | E | $7,000 | 7 | $8,400 | 5 | | F | $12,000 | 8 | $14,000 | 6 | | G | $7,000 | 3 | $8,000 | 2 | #### **Step 1: Crash Cost per Week Calculation** \[ \text{Crash cost per week} = \frac{\text{Crash Cost} - \text{Normal Cost}}{\text{Normal Time} - \text{Crash Time}} \] | Activity | Crash Cost/Week | |----------|-----------------| | A | ($8,000-$7,000)/(5-3) = $500 | | B | ($6,000-$5,000)/(7-5) = $500 | | C | ($5,200-$4,000)/(6-4) = $600 | | D | ($10,000-$8,000)/(6-3) = $667 | | E | ($8,400-$7,000)/(7-5) = $700 | | F | ($14,000-$12,000)/(8-6) = $1,000 | | G | ($8,000-$7,000)/(3-2) = $1,000 | #### **Step 2: Identify Critical Path Activities** - **A, B, D, F, G** and **A, C, E, F, G** (as above). #### **Step 3: Crash the Cheapest Critical Path Activities** Reduce the project by 3 days, one day at a time, always choosing the lowest cost per week on the critical path. ##### **First Crash (1st Day Reduction)** - Cheapest on critical path: **A or B ($500)**. - Crash **A** by 1 day (from 5 to 4). - Cost = $500 - Recalculate project duration: Now 28 days. ##### **Second Crash (2nd Day Reduction)** - Now, **A** can still be further crashed by 1 day, or **B** by 2 days, both at $500. Crash **B** by 1 day (from 7 to 6). - Cost = $500 - Now 27 days. ##### **Third Crash (3rd Day Reduction)** - Crash **B** again by 1 day (from 6 to 5). - Cost = $500 - Now 26 days. #### **Step 4: New Total Direct Cost** - **A:** $7,000 + $1,000 = $8,000 (crashed 2 days) - **B:** $5,000 + $1,000 = $6,000 (crashed 2 days) - **Others:** Normal cost Total direct cost = $8,000 (A) + $6,000 (B) + $4,000 (C) + $8,000 (D) + $7,000 (E) + $12,000 (F) + $7,000 (G) = **$52,000** #### **Step 5: Indirect Cost for 26 Days** - Indirect cost for 26 days = **$4,500** #### **Step 6: Total Project Cost** - Total cost = Direct cost + Indirect cost = $52,000 + $4,500 = **$56,500** #### **Step 7: New Critical Path** - Still includes A or B, as crashed, but duration reduced. --- ## **Final Summary** - **Critical Path:** A → B → D → F → G and A → C → E → F → G - **Project Completion Time:** 29 days (normal) - **After crashing by 3 days:** - **New duration:** 26 days - **Total direct cost:** $52,000 - **Indirect cost:** $4,500 - **Total project cost:** **$56,500** - **New critical path:** Path remains the same, with A and B shortened. **This step-by-step approach ensures the minimum cost for reducing the project duration by three days.**

# Summary of Network Scheduling and Crashing Problem ## **a. Network Analysis** The critical path consists of activities A, B, D, F, and G, with a total project completion time of 29 days. Each activity's early start (ES), early finish (EF), late start (LS), late finish (LF), and slack values were calculated, identifying the critical path as having zero slack. ## **b. Project Crashing** To reduce the project duration by 3 days, activities A and B were crashed, resulting in a new total project cost of $56,500, including direct and indirect costs. The new critical path remains unchanged, reflecting the adjustments made to shorten the schedule effectively.