- Asked 2/19
- 2073 Magh · 6 marks
TCTO numerical: optimal crashed time, Rs 3000/week
Short answer
Optimal crashed completion time: 21 weeks, minimum total cost Rs 86,350. Achieved by crashing B (2 wks), E (1 wk), and G (1 wk) on the critical path B-E-G. Cost slopes: B Rs 150/wk, E Rs 200/wk, G Rs 350/wk.
Consider the data of a project as shown in table below. If the indirect cost per week is Rs. 3000, find the optimal crashed project completion time and corresponding minimum cost.
| Activity | Immediate Predecessor | Normal time (Week) | Normal Cost (Rs.) | Crash Duration (Week) | Crash Cost (Rs.) |
|---|---|---|---|---|---|
| A | - | 8 | 2000 | 5 | 2300 |
| B | - | 10 | 4000 | 8 | 4300 |
| C | A | 6 | 3000 | 5 | 3125 |
| D | A | 9 | 5000 | 6 | 5225 |
| E | B | 10 | 2500 | 9 | 2700 |
| F | B | 13 | 5000 | 13 | - |
| G | D, E | 5 | 1000 | 3 | 1700 |
Given data and method
Indirect cost = 3,000 Rs/week. Time-cost trade-off (TCTO) crashes the activities on the critical path, one time-unit at a time, always choosing the cheapest cost slope, as long as the saving in indirect cost is more than the extra direct cost.
F cannot be crashed (crash duration equals normal duration).
| Activity | Normal dur | Crash dur | Normal cost | Crash cost | Slope |
|---|---|---|---|---|---|
| A | 8 | 5 | 2,000 | 2,300 | 100 |
| B | 10 | 8 | 4,000 | 4,300 | 150 |
| C | 6 | 5 | 3,000 | 3,125 | 125 |
| D | 9 | 6 | 5,000 | 5,225 | 75 |
| E | 10 | 9 | 2,500 | 2,700 | 200 |
| F | 13 | 13 | 5,000 | 5,000 | cannot crash |
| G | 5 | 3 | 1,000 | 1,700 | 350 |
Paths at normal duration
- A-C = 14 weeks
- A-D-G = 22 weeks
- B-E-G = 25 weeks
- B-F = 23 weeks
Critical activities (on the longest path or paths): B, E, G; normal duration = 25 weeks; normal direct cost = 22,500 Rs.
Crashing, step by step
| Step | Crash (by 1 unit) | Slope/cost | Duration | Direct | Indirect | Total |
|---|---|---|---|---|---|---|
| 0 | - | - | 25 | 22,500 | 75,000 | 97,500 |
| 1 | B | 150 | 24 | 22,650 | 72,000 | 94,650 |
| 2 | B | 150 | 23 | 22,800 | 69,000 | 91,800 |
| 3 | E | 200 | 22 | 23,000 | 66,000 | 89,000 |
| 4 | G | 350 | 21 | 23,350 | 63,000 | 86,350 |
Result
- Optimum (minimum-cost) schedule: duration 21 weeks, direct cost 23,350 + indirect 63,000 = 86,350 Rs.
- Activities crashed at the optimum: B by 2, E by 1, G by 1. Durations: A=8, B=8, C=6, D=9, E=9, F=13, G=4.
- Minimum possible duration: 21 weeks, with total cost 86,350 Rs (direct 23,350 + indirect 63,000).
Answer: optimal crashed project completion time = 21 weeks; minimum total cost = Rs 86,350.
✓ Answer checked by Claude Opus 5.5 (high effort)
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