Representing Uncertainty by Probability and Possibility What s the Difference?

Transcription

1 Representing Uncertainty by Probability and Possibility What s the Difference? Presentation at Amsterdam, March 29 30, 2011 Hans Schjær Jacobsen Professor, Director RD&I Ballerup, Denmark [email protected] 1

2 Agenda 1. Why do we need uncertainty management? 2. Alternative representations of uncertainty 3. Some principles of New Budgeting 4. Introducing uncertainty in the cost model 5. Numerical examples 6. Resumé and perspectives 2

3 1. Why do we need uncertainty management? 3

4 Cost overruns and demand shortfalls in urban rail Average cost escalation for urban rail projects is 45% in constant prices For 25% of urban rail projects cost escalations are at least 60% Actual ridership is on average 51% lower than forecast For 25% of urban rail projects actual ridership is at least 68% lower than forecast (Flyvbjerg 2007) 4

5 2. Alternative representations of uncertainty 5

6 Uncertainty Two worlds of risk and uncertainty World Representation and calculation Imprecision Ignorance Lack of knowledge Possibility Possibility distributions [a; ; b] Interval arithmetic Global optimisation Statistical nature Randomness Variability Probability Probability distributions {µ; σ} Linear approximation Monte Carlo simulation 6

7 Rectangular representation [a; b] and {µ; σ} Possibility distribution [a; b] 1 Alternative interpretations Probability distribution {µ; σ} h h = 1/(b-a) μ = (a+b)/2 σ 2 = (b-a) 2 / a b 7

8 Trapezoidal representation [a; c; d; b] and {µ; σ} Possibility distribution [a; c; b] 1 Alternative interpretations Probability distribution {µ; σ} h α α-cut h = 2/(b-a+d-c) μ = h((b 3 -d 3 )/(b-d)-(c 3 -a 3 )/(c-a))/6 σ 2 = (3(r+2s+t) 4 +6(r 2 +t 2 )(r+2s+t) 2 -(r 2 -t 2 ) 2 )/(12(r+2s+t)) 2 r s t 0 0 a c d b 8

9 Triangular representation [a; c; b] and {µ; σ} Possibility distribution [a; c; b] 1 Alternative interpretations Probability distribution {µ; σ} h h = 2/(b-a) μ = (a+b+c)/3 σ 2 = (a 2 +b 2 +c 2 -ab-ac-bc)/18 α α-cut 0 0 a c b 9

10 Independent stochastic variables Intervals Triple estimates {μ; σ} = {μ 1 ; σ 1 } # {μ 2 ; σ 2 } [a; b] = [a 1 ; b 1 ] # [a 2 ; b 2 ] [a; c; b] = [a 1 ; c 1 ; b 1 ] # [a 2 ; c 2 ; b 2 ] Addition μ = μ 1 + μ 2 ; σ 2 = σ σ 2 a = a 1 + a 2 ; b = b 1 + b 2 a = a 1 + a 2 ; c = c 1 + c 2 ; b = b 1 + b 2 Subtraction μ = μ 1 - μ 2 ; σ 2 = σ σ 2 a = a 1 - b 2 ; b = b 1 - a 2 a = a 1 - b 2 ; c = c 1 - c 2 ; b = b 1 - a 2 Multiplication μ = μ 1 μ 2 ; σ 2 σ 12 μ σ 22 μ 1 a = min(a 1 a 2, a 1 b 2, b 1 a 2, b 1 b 2 ); b = max(a 1 a 2, a 1 b 2, b 1 a 2, b 1 b 2 ) a = min(a 1 a 2, a 1 b 2, b 1 a 2, b 1 b 2 ); c = c 1 c 2 ; b = max(a 1 a 2, a 1 b 2, b 1 a 2, b 1 b 2 ) Division μ = μ 1 /μ 2 ; σ 2 σ 2 1 /μ σ 22 μ 2 1 /μ 4 2, if μ 2 0 a = min(a 1 /b 2, a 1 /a 2, b 1 /b 2, b 1 /a 2,); b = max(a 1 /b 2, a 1 /a 2, b 1 /b 2, b 1 /a 2 ), if 0 [a 2 ; b 2 ] a = min(a 1 /b 2, a 1 /a 2, b 1 /b 2, b 1 /a 2,); c = c 1 /c 2 ; b = max(a 1 /b 2, a 1 /a 2, b 1 /b 2, b 1 /a 2 ), if 0 [a 2 ; b 2 ] Table 1. Formulas for basic calculations with alternative representations of uncertain variables. 10

11 Modelling by possibility distributions i.e. intervals, fuzzy intervals, etc. The actual economic problem is modelled by a function Y of n uncertain variables Y = Y(X 1, X 2,, X n ). NB: Function can be arranged in different ways. In case of intervals Y is calculated by means of interval arithmetic (only applicable in the simple case) or global optimisation (applicable in the general case). In case of triple estimates Extreme values of Y are calculated as above. In case of fuzzy intervals As above, for all α cuts. 11

12 Modelling by probability distributions The actual economic problem is modelled by a function Y of n independent uncertain variables Y = Y(X 1, X 2,, X n ). Linear approximation Y is approximated by means of a Taylor series Y Y(μ 1,, μ n ) + Y/ X 1 (X 1 μ 1 ) + Y/ X 2 (X 2 μ 2 ) + + Y/ X n (X n μ n ), where Y/ X i is the partial derivative of Y with respect to X i calculated at (μ 1,, μ n ). The expected value is given by E(Y) = μ = Y(μ 1,, μ n ). The variance is approximated by VAR(Y) = σ 2 ( Y/ X 1 ) 2 σ ( Y/ X n ) 2 σ n2. Monte Carlo simulation 12

13 Y = X(1-X), X = [0; 1] pdf Global optimisation X = [0; 1] Y = [0; 0,25] (Normalised as pdf) Monte Carlo simulation X = RiskUniform(0; 1) Y = {0,167; 0,075} 5 0 0,00 0,05 0,10 0,15 0,20 0,25 Y = X(1-X) 13

14 Sum of 10 identical rectangular cost elements [7; 15] 0,06 0,05 pdf 0,04 0,03 Independent variables Monte Carlo simulation N{110; 7,3} Fuzzy variables Fuzzy arithmetic [70; 150] (normalized as pdf) 0,02 0,01 0, Sum 14

15 Sum of 10 identical trapezoidal cost elements [7; 9; 11; 15] 0,08 0,07 0,06 0,05 Independent variables Monte Carlo simulation N{106; 5,4} pdf 0,04 0,03 0,02 Fuzzy variables Fuzzy arithmetic [70; 90; 110; 150] (normalized as pdf) 0,01 0, Sum 15

16 Sum of 10 identical triangular cost elements [7; 10; 15] 0,08 0,07 0,06 Independent variables Monte Carlo simulation N{107; 5,2} 0,05 pdf 0,04 0,03 Fuzzy variables Fuzzy arithmetic [70; 100; 150] (normalized as pdf) 0,02 0,01 0, Sum 16

17 Sum of 10 identical triangular cost elements [7; 10; 15] 17

18 Sum of 10 identical triangular cost elements [7; 10; 15] 18

19 Sum of 10 identical triangular cost elements [7; 10; 15] 19

20 Sum of 10 identical triangular cost elements [7; 10; 15] 20

21 Sum of 10 identical triangular cost elements [7; 10; 15] 21

22 Y = X i, X i = [8; 10; 16], i = 1,,10 0,08 0,07 0,06 Monte Carlo simulation X = RiskTriangular(8; 10; 16) (Uncorrelated variables) Y = {113,3; 5,35} pdf 0,05 0,04 0,03 Fuzzy arithmetic X = [8; 10; 16] Y = [80; 100; 160] (Normalised as pdf) Monte Carlo simulation X = RiskTriangular(8; 10; 16) (100% correlated variables) Y = {113,3; 17,00} 0,02 0,01 0, Y = sum of X i, i = 1,,10 22

23 n Y n = X i, X i = [90/n;100/n;140/n] i =1 cdf n = 1 n = 5 n = 10 n = 15 23

24 Some observations of probability vs. possibility With numerically identical input variables probability results in less numerical output uncertainty than does possibility Uniform probability representation is different from interval possibility representation Probability uncertainty decreases with increasing analytical complexity whereas possibility uncertainty is independent Possibility uncertainty corresponds to fully correlated input probability variables Monte Carlo simulation does not generally produce possibility results 24

25 3. Some principles of New Budgeting 25

26 A Danish governmental initiative Best realistic budget based on available knowledge Budget control is done by standardised budgets and logging of follow up results Risk and uncertainty management is conducted during entire project Estimates of unit prices, quantities and particular risks Experience based supplementary budget of one third of 50% of rough budget is allocated Likelihood of event multiplied by impact is not accepted Acceptable to incur additional cost to reduce risk and uncertainty 26

27 The Anchor Budget Project with a number of activities A Each activity: unit price p and quantity q Total cost C of activities at time t = 0 Subsequently, additional activities and costs may be introduced 27

28 The Event Impact Matrix Risk events E are identified at any time t = Additional activities may be initiated Impacts of Risk Events on all p and q are estimated We keep track of accumulated cost impacts for all individual risk events Impacts from interacting (co acting) Risk Events are pooled 28

29 The Risk Budget All identified Risk Events are assumed to occur Resulting p, q and cost for each activity is calculated Total cost for project is calculated Deviations from Anchor Budget is calculated 29

30 Risk events modify Anchor Budget For the i th activity A i, we get the modified estimated cost C i at time C i = = (p i + p i1 + p i2 + + p ij + + p im ) (q i + q i1 + q i2 + + q ij + + q im ) = C i + p i ( q i1 + q i2 + + q ij + + q im ) + ( p i1 + p i2 + + p ij + + p im ) q i + p i1 ( q i1 + q i2 + + q ij + + q im ) + p i2 ( q i1 + q i2 + + q ij + + q im ) + + p ij ( q i1 + q i2 + + q ij + + q im ) + + p im ( q i1 + q i2 + + q ij + + q im ) 30

31 Convenient set up for calculations Event Impact Matrix at t = Anchor Budget Risk Budget at t = 0 Interaction at t = E 1 E 2 E 3 Sum p q Cost p q Cost p q Cost p q Cost Cost Cost p q Cost Activity A 1 p 1 q 1 C 1 p 11 q 11 C 11 A 2 p 2 q 2 C 2 p 21 q 21 C 21 A 3 p 3 q 3 C 3 p 31 q 31 C 31 A 4 p 4 q 4 C 4 p 41 q 41 C 41 A 5 p 5 q 5 C 5 p 51 q 51 C 51 Sum C C 1 p 12 q 12 C 12 p 22 q 22 C 22 p 32 q 32 C 32 p 42 q 42 C 42 p 52 q 52 C 52 C 2 p 13 q 13 C 13 p 23 q 23 C 23 p 33 q 33 C 33 p 43 q 43 C 43 p 53 q 53 C 53 C 3 c 1 c 2 c 3 c 4 c 5 C 1 C 2 C 3 C 4 C 5 p 1 p 2 p 3 p 4 p 5 q 1 q 2 q 3 q 4 q 5 C 1 C 2 C 3 C 4 C 5 c C C Table 1. Convenient set-up for calculations, n = 5, m = 3. Anchor Budget Event Impact Matrix Risk Budget 31

32 Event Impact Matrix at t = Anchor Budget Risk Budget at t = 0 Interaction at t = E 1 E 2 E 3 Sum p q Cost p q Cost p q Cost p q Cost Cost Cost p q Cost Activity A , , , , , , ,000 A , , ,000 10, , ,000 A , ,000 15, ,000 A 4 1, , ,000 10,000 1, ,000 A ,000 45, ,000 Sum 850,000 47,000 12,500 35, , ,000 Table 2. Numerical test example without event and impact uncertainty. A 1 E 1 : Cost = (p+ p) (q+ q) - Cost = p q + p q + p q = , = 32,000 A 1 E 2 : Cost = (p+ p) (q+ q) - Cost = p q + p q + p q = , = 2,500 A 1 Interaction: Cost = =

33 4. Introducing uncertainty in the cost model 33

34 Uncertain impacts and likelihoods Uncertain impact of Risk Events Uncertainty for all p and q from Anchor Budget Additional activities may become necessary How to estimate and represent uncertainty? Calculate uncertain Risk Budget Likelihood of Risk Events occurring Estimate probabilities of Risk Events occurring Construct probability distribution of total project cost 34

35 Triangular representation [a; c; b] Uncertain impact on unit price p = 100: p = [18; 20; 25] Uncertain impact on quantity q = 1000: q = [95; 100; 125] Uncertain impact on cost p q Cost = (p+ p) (q+ q) p q = p q + p q + p q Cost = [9,500; 10,000; 12,500] + [18,000; 20,000; 25,000] + [1,710; 2,000; 3,125] = [29,210; 32,000; 40,625] 35

36 Triangular representation {µ; σ} Uncertain impact on unit price p = 100: p = {21.0; 1.47} Uncertain impact on quantity q = 1000: q = {106.7; 6.56} Uncertain impact on cost p q Cost = (p+ p) (q+ q) p q = p q + p q + p q Cost = {33,907; 1,802} (by Monte Carlo simulation) min = 29,551, max = 39,966 36

37 5. Numerical examples 37

38 Activity Event Impact Matrix at t = E 1 E 2 E 3 Interaction Sum p q Cost p q Cost p q Cost Cost Cost A 1 [18; 25] [95; 125] [29,210; 40,625] [21; 31] [2,100; 3,100] [378; 775] [31,688; 44,500] A 2 [-210; -175] [-10,500; -8,750] [-10,500, -8,750] A 3 [25; 45] [12,500; 22,500] [12,500; 22,500] A 4 [7; 16] [7,000; 16,000] [7,000; 16,000] A 5 [145; 165] [280; 350] [40,600; 57,750] [40,600; 57,750] Sum [41,710; 63,125] [9,100; 19,100] [30,100; 49,000] [378; 775] [81,288; 132,000] Table 3. Event Impact Matrix using interval uncertainty representation [a; b]. (Anchor Budget of Table 2). 38

39 Activity Risk Budget at t = p q Cost A 1 [118; 125] [1,116; 1,156] [131,688; 144,500] A 2 50 [9,790; 9,825] [489,500; 491,250] A 3 [225; 245] 500 [112,500; 122,500] A 4 1,000 [157; 166] [157,000; 166,000] A 5 [145; 165] [280; 350] [40,600; 57,750] Sum [931,288; 982,000] Table 4. Risk Budget using interval uncertainty representation [a; b]. (Anchor Budget of Table 2 and Event Impact Matrix of Table 3). 39

40 Activity Risk Budget at t = p q Cost A 1 [118; 120; 125] [1,116; 1,125; 1,156] [131,688; 135,000; 144,500] A 2 50 [9,790; 9,800; 9,825] [489,500; 490,000; 491,250] A 3 [225; 230; 245] 500 [112,500; 115,000; 122;500] A 4 1,000 [157; 160; 166] [157,000; 160,000; 166,000] A 5 [145; 150; 165] [280; 200; 350] [40,600; 54,000; 57,750] Sum [931,288; 945,000; 982,000] Table 5. Risk Budget using triple estimate uncertainty representation [a; c; b]. (Combination of Table 2 and 4). 40

41 Activity Event Impact Matrix at t = E 1 E 2 E 3 Interaction Sum p q Cost p q Cost p q Cost Cost Cost A 1 {21.0; 1.47} {106.7; 6.56} {33,907; 1,802} {25.7; 2.05} {2,567; 205} {539; 57.9} {37,012; 1,851} A 2 {195; 7.36} {-9,750; 368} {-9,750; 368} A 3 {33.3; 4.25} {16,667; 2,125} {16,667; 2,125} A 4 {11.0; 1.87} {11,000; 1,871} {11,000; 1,871} A 5 {153.3; 4.25} {310; 14.7} {47,534; 2,619} {47,534; 2,619} Sum {50,573; 2,807} {13,567; 1,884} {37,784; 2,643} {539; 57.9} {102,462; 4,305} Table 6. Event Impact Matrix using triangular probability input distributions {µ; σ}. (Anchor Budget of Table 2). 41

42 Activity Risk Budget at t = p q Cost A 1 {121.0; 1.47} {1,132; 6.87} {137,012; 1,851} A 2 50 {9,805; 7.36} {490,250; 368} A 3 {233.3; 4.25} 500 {116,667; 2,125} A 4 1,000 {161.0; 1.87} {161,000; 1,871} A 5 {153.3; 4.25} {310.0; 14.7} {47,534; 2,619} Sum {952,462; 4,305} Table 7. Risk Budget using triangular probability input distributions {µ; σ}. (Anchor Budget of Table 2 and Event Impact Matrix of Table 6). 42

43 E 1 E 2 E 3 Probabilities of combinations Cost C Cost C pr 1 =0.6 pr 1 =0.3 pr 3 =0.2 pdf cdf [a; c; b] {µ; σ} no no no , ,000 no yes no [859,100; 862,500; 869,100] {863,567; 1,882} no no yes ,376 [880,100; 885,000; 899,000] {887,783; 2,642} yes no no [891,710; 897,000; 913,125] {900,573; 2,810} no yes yes [889,200; 897,500; 918,100] {901,350; 3,251} yes yes no [901,188; 910,000; 933,000] {914,679; 3,380} yes no yes [921,810; 932,000; 962,125] {938,356; 3,853} yes yes yes [931,288; 945,000; 982,000] {952,462; 4,305} Table 8. Distributions of C for representations by triple estimates and probabilities. 43

44 Distribution of cost C (triple estimates, Table 8) 1.0 E 1, E 2, E 3 E 1, E 2 E 1, E 3 cdf E 1 E 2, E 3 E 2 E , ,000 Cost 1,000,000 44

45 Triangular probability Monte Carlo simulation of Risk Budget. (Anchor Budget of Table 2, Event Impact Matrix of Table 6). Uncorrelated input parameters Uncorrelated input parameters Most poss.: 945, % correlated input parameters Min: 931,288 Max: 982,000 45

46 6. Resumé and perspectives 46

47 Resumé and perspectives Anchor Budget, unit prices, quantities Impacts of individual Risk Events Impacts of co acting Risk Events Impacts on individual Activities Uncertainty by probabilities and possibilities Under or overestimating uncertain impacts Research into practical applications End of presentation 47

48 Back to Table 6 48

49 Back to Table 6 49

50 Back to Table 6 50

51 Back to Table 6 51

52 Back to Table 6 52

53 Back to Table 7 53

54 Back to Table 7 54

55 Back to Table 7 55

56 Back to Table 7 56

57 Thank You! 57