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PERT (PROGRAM EVALUATION AND REVIEW TECHNIQUE) ASSUMPTIONS: • Activities are statistically independent (no corrective actions) • The critical path is significantly longer than any near critical path • The critical path can analysed on the basis of the central limit theorem (each activity gives little contribution to the total variance) LIUC - Master in Project Management Enrico Cagno © • The duration of the single activity is described by independent statistical variables • Every single activity is described by a Beta 2 distribution with mean t and variance σ t • The estimate of the distribution parameters considers three values a, m, b, corresponding to optimistic, the most likely and the pessimistic duration of the activity considered (5% percentile; mode; 95% percentile) LIUC - Master in Project Management Enrico Cagno © • The estimate of the variance is given by: 2 σ 2 t (b − a ) = 6 N.B.: as a rule of a thumb for unimodal distributions standard deviation corresponds to a 1/6 of the variation range of the considered variable. This rule stems from the fact that at least 89% of every distribution can be found within 3 standard deviations from the mean; for the normal distribution the percentage grows over 99,7% LIUC - Master in Project Management Enrico Cagno © • The value t can be approximated by: t= ( a + 4m + b ) LIUC - Master in Project Management 6 Enrico Cagno © BETA DISTRIBUTION The distribution of the duration of any activity is estimated by means of a beta distribution defined by the function: 0 α γ f ( t ) = k ⋅ ( t − a ) ⋅ (b − t ) 0 −∞ < t < a a ≤ t ≤ b; a , b ≥ 0 α , β > −1 As a matter of fact, the actual distribution is not known so that the formulae proposed are approximations. LIUC - Master in Project Management Enrico Cagno © Nevertheless this distribution has some good properties: • two non-negative intercepts; • continuous; • unimodal; • low probabilities of occurrence given to the pessimistic (a) and optimistic (b) durations; • high flexibility; But…why not triangular distribution? LIUC - Master in Project Management Enrico Cagno © The project expected duration is given by: N TM = ∑t i i=1 and the corresponding variance by: N 2 T σ = ∑σ i =1 2 ti ⇒ Normal distribution and central limit theorem (n>15; al least n>6) LIUC - Master in Project Management Enrico Cagno © • The probability that the project completion time will be shorter than a fixed one (Ts) will be Ts − TM P { T ≤ Ts } = P z ≤ σT where z dove z represents the statistical variable from a Standardized Normal Distribution. LIUC - Master in Project Management Enrico Cagno © EXAMPLE (AOA): Act. Pred. a m b t σ t2 A - 2 4 12 5 2.78 B - 3 6 9 6 1.00 C A 1 2 9 3 1.78 D A 3 3 9 4 1.00 E B 1 2 3 2 0.11 F B 2 8 8 7 1.00 G C 1 2 9 3 1.78 H D, E 4 5 12 6 1.78 I F 1 3 5 3 0.44 LIUC - Master in Project Management Enrico Cagno © C (3) 2 5 A (5) G (3) D (4) 1 4 H (6) 7 E (2) I (3) B (6) 3 F (7) C (3) 6 G (3) A (5) D (4) H (6) E (2) B (6) I (3) F (7) LIUC - Master in Project Management Enrico Cagno © B-F-I (critical path) T1 = 1 6 σ T2 = 2 .4 4 1 σ T = 1.5 6 1 P { B − F − I ≤ 1 5} z = 15 − T1 15 − 16 = = − 0 . 64 σ T1 1 . 56 Φ ( 0 .6 4 ) = 0 .2 4 Φ ( − 0 . 6 4 ) = 0 .5 − 0 . 2 4 = 0 . 2 6 LIUC - Master in Project Management Enrico Cagno © A-D-H (near critical path) T2 = 1 5 σ T2 = 5.5 6 2 σ T = 2 .3 6 2 P { A − D − H ≤ 1 5} z = 15 − T 2 15 − 15 = = 0 σ T2 2 . 36 Φ ( 0 ) = 0 .5 LIUC - Master in Project Management Enrico Cagno © ex 1 (AON) Activity Predecessor a m b A - 12 14 16 B - 1 3 5 C A, B 2 5 5 D B 5 7 15 E C, D 2 4 6 F E 6 10 14 a) Determine the critical path. b) Determine the duration of the project with a probability of 90%. c) Determine the probability of completing the project within 31 months and 33 months. LIUC - Master in Project Management Enrico Cagno © a) Activity Predecessor a m b t σt σ2t A - 12 14 16 14 2/3 0.44 B - 1 3 5 3 2/3 0.44 C A, B 2 5 5 4.5 1/2 0.25 D B 5 7 15 8 5/3 E C, D 2 4 6 4 2/3 0.44 F E 6 10 14 10 4/3 2.8 1.8 A C B E F D LIUC - Master in Project Management Enrico Cagno © There are three paths in this tree: A-C-E-F, B-C-E-F e B-D-E-F. Path A-C-E-F B-C-E-F B-D-E-F T σ2T 31.2 2.93 21.2 2.93 26 5.48 σT 1.7 1.7 2.3 The critical path in terms of expected duration is A-CE-F. Nevertheless it’s clear that also path B-D-E-F should be investigated as, even if its expected duration is 26, its variance is twice bigger than the one of path A-CE-F. LIUC - Master in Project Management Enrico Cagno © b) Duration of the project with a probability of 90%: Path D90 A-C-E-F 33.4 B-C-E-F 23.4 B-D-E-F 29 Path A-C-E-F is critical e there is a probability of 90% to complete the project within 33,4 months. c) The completion probability with 31 months is near 50% (within 33 months is lower than 90%) LIUC - Master in Project Management Enrico Cagno © ex 2 (AON; succ. starts same day pred.) Duration [weeks] Activity Predecessor Min Mode Max A - 10 22 28 B A 4 4 10 C B 4 6 14 D B 1 2 3 E C 1 5 9 F D 7 8 9 G E,F 2 2 2 H G 5 7 11 I G 3 6 9 J H,I 5 7 15 LIUC - Master in Project Management Enrico Cagno © 1) Calculate the probability to complete the project within 50 weeks; 2) Starting from the 55th week, for each week of delay - for 4 weeks span – a fee of 2,5% of the contract value will be paid. Estimate the risk to pay a fee. LIUC - Master in Project Management Enrico Cagno © 1) Activity t σ2 (min+4mode+max) / 6 [(max - min) / 6]2 A 21 324/36 B 5 1 C 7 100/36 D 2 4/36 E 5 64/36 F 8 4/36 G 2 0 H 7,33 1 I 6 1 J 8 100/36 The critical path is: A,B,C,E,G,H,J N.B. The successor activity stars the same day the predecessor one ends. LIUC - Master in Project Management Enrico Cagno © 0 21 21 21 26 5 A 0 26 21 C 26 33 7 B 21 33 26 38 5 E 33 33 38 38 40 40 47,33 2 7,33 G 38 H 40 40 47,33 55,33 47,33 8 J 26 28 28 2 D 28 36 40 8 30 47,33 55,33 6 I F 30 46 38 41,33 47,33 Considering only critical path activities: Tm = Σ ti (critical) = 55,33 σ2 = Σ σi2 (critical) = 660/36 = 18,33 σ = 4,28 P ( T ≤ 50 ) = P [ z ≤ (50 – Tm) / σ] = P ( z ≤ 1,245 ) = 10,66% LIUC - Master in Project Management Enrico Cagno © 2) Calculating the probability that the project is completed beyond 55th, 56th, 57th and 58th week: P ( T > 55 ) = P ( z ≤ 0,07 ) = 52,8 % P ( T > 56 ) = 1 - P ( z ≤ 0,16 ) = 43,6 % P ( T > 57 ) = 1 - P ( z ≤0,39 ) = 34,8 % P ( T > 58 ) = 1 - P ( z ≤0,62 ) = 26,8 % P (55 < T ≤ 56) = P (T ≤ 56) - P (T ≤ 55) = = P (T > 55) - P (T > 56) = = 52,8 - 43,6 = 9,2 % P (56 < T ≤ 57) = 43,6 - 34,8 = 8,8 % P (57 < T ≤ 58) = 34,8 - 26,8 = 8 % P (T > 58 ) = 26,8 % R = 0,092 ⋅ 2,5% + 0,088 ⋅ 5% + 0,080 ⋅ 7,5% + 0,268 ⋅ 10% = 3,95 % LIUC - Master in Project Management Enrico Cagno © PERT FLAWS: • overlooks near critical paths; • assumes independent activities; • undervalues merging points; • underestimates project duration. For example: Activity 1 10, 12, 14 Milestone A Activity 2 10, 12, 14 Milestone B Activity 3 10, 12, 14 ⇒ little used (except PERT-like estimations) LIUC - Master in Project Management Enrico Cagno ©