- 2075 Bhadra · 7 marks
TCTO numerical: lowest cost schedule, Rs 400/week
Short answer
Time-cost trade-off: crash B by 2, C by 1, and E by 1 week. Lowest-cost schedule is 22 weeks, total cost Rs 28,050. Further crashing E would cost Rs 250/week vs. Rs 400 saved, so 22 weeks is optimal.
The details of a project are shown below. If the indirect cost per week is Rs. 400, find the lowest cost schedule of the project.
| Activity | Predecessor(s) | Time (weeks) Normal | Time (weeks) Crash | Cost (Rs) Normal | Cost (Rs) Crash |
|---|---|---|---|---|---|
| A | - | 8 | 5 | 2000 | 2200 |
| B | - | 10 | 8 | 3800 | 4000 |
| C | B | 6 | 5 | 2700 | 2600 |
| D | A | 9 | 6 | 4100 | 4300 |
| E | C | 10 | 9 | 3100 | 3350 |
| F | B | 12 | 12 | 3100 | 3100 |
Given data and method
Indirect cost = 400 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.
Assumption: for C the table shows crash cost (2,600) lower than normal cost (2,700). The two values are taken as interchanged, so normal cost = Rs 2,600 and crash cost = Rs 2,700. F cannot be crashed.
| Activity | Normal dur | Crash dur | Normal cost | Crash cost | Slope |
|---|---|---|---|---|---|
| A | 8 | 5 | 2,000 | 2,200 | 66.67 |
| B | 10 | 8 | 3,800 | 4,000 | 100 |
| C | 6 | 5 | 2,600 | 2,700 | 100 |
| D | 9 | 6 | 4,100 | 4,300 | 66.67 |
| E | 10 | 9 | 3,100 | 3,350 | 250 |
| F | 12 | 12 | 3,100 | 3,100 | cannot crash |
Paths at normal duration
- A-D = 17 weeks
- B-C-E = 26 weeks
- B-F = 22 weeks
Critical activities (on the longest path or paths): B, C, E; normal duration = 26 weeks; normal direct cost = 18,700 Rs.
Crashing, step by step
| Step | Crash (by 1 unit) | Slope/cost | Duration | Direct | Indirect | Total |
|---|---|---|---|---|---|---|
| 0 | - | - | 26 | 18,700 | 10,400 | 29,100 |
| 1 | B | 100 | 25 | 18,800 | 10,000 | 28,800 |
| 2 | B | 100 | 24 | 18,900 | 9,600 | 28,500 |
| 3 | C | 100 | 23 | 19,000 | 9,200 | 28,200 |
| 4 | E | 250 | 22 | 19,250 | 8,800 | 28,050 |
Result
- Optimum (minimum-cost) schedule: duration 22 weeks, direct cost 19,250 + indirect 8,800 = 28,050 Rs.
- Activities crashed at the optimum: B by 2, C by 1, E by 1. Durations: A=8, B=8, C=5, D=9, E=9, F=12.
- Minimum possible duration: 22 weeks, with total cost 28,050 Rs (direct 19,250 + indirect 8,800).
Answer: lowest-cost schedule = 22 weeks, total cost Rs 28,050.
✓ Answer checked by Claude Opus 5.5 (high effort)
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