Benchmark problems

We should pose some simple benchmark problems to further test our hypotheses. Some thoughts on what they could look like:

a) y = (u*v+w)^2

- u characterized by tfn(1,2,3,4)
- v characterized by pbox with upper cdf U(1,2) and lower cdf U(3,4)
- w characterized by DSS { ([1,3], 1/4), ([2,4], 1/2), ([2,2.5], 1/4) }
- all variables are thought to be strictly independent
- to compute
  - y <= 10 (failure)
  - 10 <= y <= 300 (ok)
  - y >= 300 (suboptimal)
  - E[y]

b) same as a) but with unknown interaction, i.e. the Frechet case

c) same as a) but with perfect dependence between u and v

d) z = x(T, p, x0) + q for T=1s, dx/dt = -(1+p)*x and x(0) = x0

- p is time-invariant, but positive with mean 1, i.e. a Markov distribution
- q has mean 0 and variance 1, i.e. a Chebychev distribution
- x0 is [100m, 110m]
- p, q and x0 are thought to be strictly independent
- x(T, p, x0) = exp( -(1+p)*T ) * x0
- to compute
  - z <= 20 (good)
  - 20 <= z <= 80 (ok)
  - z >= 80 (bad)
  - E[z]

e) same as d) but with unknown interaction, i.e. the Frechet case

f) same as d) but with perfect dependence between p and x0

The different arguments would have to be 'cast' into the respective frameworks (pba, possibility) and evaluated accordingly.

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Problem 1.

Plot of variables

Original (u, v, w) PBOX (u, v, w) Fuzzy (u, v, w)

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y = (u*v+w)^2

failure: y <= 10

ok: y in [10, 300]

suboptimal: y >= 300

red is independence, blue is perfect , green is frechet

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Pbox calculation

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failure Probabilities

Independent | [0, 0.085]

Perfect | [0, 0.22]

Frechet | [0, 0.495]

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ok Probabilities

Independent | [0.844999, 1]

Perfect | [0.57, 1]

Frechet | [0, 1]

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suboptimal Probabilities

Independent | [0, 0.0700001]

Perfect | [0, 0.21]

Frechet | [0, 0.775001]

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Expectation

Independent | [15.96, 243.111]

Perfect | [16.6056, 246.004]

Frechet | [11.0681, 300.745]

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Fuzzy calculation

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failure Probabilities

Independent | [0, 0.435]

Perfect | [0, 0.25]

Frechet | [0, 0.5]

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ok Probabilities

Independent | [0.0799999, 1]

Perfect | [0.534999, 1]

Frechet | [0, 1]

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suboptimal Probabilities

Independent | [0, 0.920001]

Perfect | [0, 0.465001]

Frechet | [0, 1]

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Expectation

Independent | [8.92575, 352.959]

Perfect | [14.3019, 286.075]

Frechet | [7.61867, 371.798]

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Fuzzy Independent Revised

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Fuzzy calculation

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failure Probabilities

Independent | [0, 0.390001]

Perfect | [0, 0.25]

Frechet | [0, 0.5]

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ok Probabilities

Independent | [0.574999, 1]

Perfect | [0.534999, 1]

Frechet | [0, 1]

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suboptimal Probabilities

Independent | [0, 0.425]

Perfect | [0, 0.465001]

Frechet | [0, 1]

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Expectation

Independent | [11.5462, 297.522]

Perfect | [14.3019, 286.075]

Frechet | [7.61867, 371.798]

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