TCTO numerical with time-cost chart (five activities)

Given data and method

Indirect cost = 1.6 thousand 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. To draw the full trade-off chart, crashing is continued to the minimum possible duration.

$$\text{Cost slope}=\frac{\text{Crash cost}-\text{Normal cost}}{\text{Normal duration}-\text{Crash duration}}$$

Paths at normal duration

• A-C = 36 weeks
• A-D-E = 42 weeks
• B-E = 40 weeks

Critical activities (on the longest path or paths): A, D, E; normal duration = 42 weeks; normal direct cost = 26 thousand Rs.

Crashing, step by step (costs in thousand Rs)

(At 40 weeks A-D-E and B-E are both critical and D is fully crashed, so E, common to both, is crashed. At 36 weeks A-C also becomes critical, so C+E is the cheapest cut of all three paths. At 34 weeks D and E are at their crash limits, so A and B must be crashed together. At 28 weeks B and E are both fully crashed, so B-E cannot be shortened further.)

Result

• Optimum (minimum-cost) schedule: duration 40 weeks, direct cost 29 + indirect 64 = 93 thousand Rs.
• Activities crashed at the optimum: D by 2. Durations: A=26, B=28, C=10, D=2, E=12.
• Crashing beyond 40 weeks costs more than the Rs 1.6 thousand/week saved (E slope 1.83 > 1.6), so the total cost rises.
• Minimum possible duration: 28 weeks, with total cost 113.8 thousand Rs (direct 69 + indirect 44.8).

Time-cost trade-off chart (total cost against duration)

 Duration   Total cost (Rs '000); bar length = cost above 90
 (weeks)
  42 | ###### 93.20  (normal)
  41 | ###### 93.10
  40 | ###### 93.00  <-- minimum total cost (optimum)
  39 | ###### 93.23
  38 | ####### 93.47
  37 | ####### 93.70
  36 | ######## 93.93
  35 | ########## 95.17
  34 | ############# 96.40
  33 | ################### 99.30
  32 | ######################## 102.20
  31 | ############################## 105.10
  30 | #################################### 108.00
  29 | ########################################## 110.90
  28 | ################################################ 113.80  (crash)

On graph paper, plot direct cost (falling with duration), indirect cost (a straight line rising at 1.6 per week) and their sum; the total cost curve is lowest at 40 weeks.

Answer: optimum duration 40 weeks; minimum total cost Rs 93.0 thousand (direct 29.0 + indirect 64.0). The shortest possible duration is 28 weeks, at a total cost of Rs 113.8 thousand.