Exam February 2022 solutions

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Titolo presentazione sottotitolo • Milano, XX mese 20XX Exam February 2022 Exercise 1 The manager of a company is responsible for the following project. Using the PERT algorithm identify the critical path. Compute the probability that the critical path(s) is (are) concluded after month 15. Consider the possibility to crash the activities as shown in the following table. Identify the optimal solution according to a crashing programme. A penalty of 15 000€ for each month of delay is applied in case the project is not concluded before month 14. At the end of each step of the crashing algorithm write which is the margin gained and the critical path/s. Activity Immediate Predecessor Time Optimistic (months) Average Time (months) Time Pessimistic (months) Std. deviation Duration Reduction (months) Total Crashing Cost (€) A - 3 4 5 1 1 35 000€ B A 3 5 13 5 3 30 000€ C A 1 2 9 4 2 60 000€ D B 6 7 8 2 2 10 000€ E B 4 5 6 1 - - F C 5 9 19 7 5 9 000€ G C 5 5 5 0 3 21 000€ H D,E,F,G 2 3 4 1 - - Solution Activity Immediate Predecessor Time Optimistic (months) Average Time (months) Time Pessimistic (months) Expected Duration Std. deviation Duration Reduction (months) Total Crashing Cost (€) A - 3 4 5 4 1 1 35 000€ B A 3 5 13 6 5 3 39 000€ C A 1 2 9 3 4 2 60 000€ D B 6 7 8 7 2 2 10 000€ E B 4 5 6 5 1 - - F C 5 9 19 10 7 5 9 000€ G C 5 5 5 5 0 3 21 000€ H D,E,F,G 2 3 4 3 1 - - 4 73 C 4 70 4 106 B 4 100 10 177 D 10 170 10 155 E 12 172 7 1710 F 7 170 The critical paths are A-B-D-H and A-C-F-H. The project duration is 20 months. Solution 0 44 A 0 40 7 125 G 12 175 17 203 H 17 200 Solution ➢ Path A-B-D-H: P(Duration > 15 ) = P(z > 15−20 1+25+4+1) = 𝑃 𝑧 ≻ −0.90 = 1 − 𝑃(𝑧 > 0.90) = 1-(1-P(z<0.90)) = P(z<0.90) = 81.59% ➢ Path A-C-F-H: P(Duration > 15 ) = P(z < 15−20 1+16+49+1) = 𝑃 𝑧 > −0.61…