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PROJECT PLANNING MONTE CARLO SIMULATION Prof. Dr. Ahmed Farouk Abdul Moneim BY Part II

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Probability Density Functions Activities Durations : Triangular Truncated Normal Truncated Exponential Truncated Weibull amb t e t MIN t MAX t MIN t MAX t MIN t MAX t t t t Rand 0 ≤ Rand ≤ 1 Φ(Z) Z From Tables OR Excel Normal Distribution Prof. Dr. Ahmed Farouk Abdul Moneim

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T e T MIN T MAX Truncated Normal Distribution GIVEN a Probability = Rand Find the corresponding value of t t Rand Show that the following function is the right PDF t-Plane Z MIN Z MAX Z Z-Plane 0 f (Z) Z Take the Inverse function of Φ (Φ -1 ) Prof. Dr. Ahmed Farouk Abdul Moneim

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GIVEN a Probability = Rand Find the corresponding value of t Triangular Distribution amb t Rand If Otherwise H Area of the Triangle = (b-a)*H/2 = 1 Then h Area of the Left part of the Triangle = (m-a)*H/2 = t Rand h 1-Rand Prof. Dr. Ahmed Farouk Abdul Moneim

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Truncated Exponential Distribution T Min T Max t Consider the following Probability Density Function To show that this is a PDF, the integral over the whole Range R should equal to one Now, find an expression for the mean μ μ - T Min f(t) t Important Notice! For TRUNCATED Exponential Distribution, The following condition SHOULD BE SATISFIED As R tends to Infinity Prof. Dr. Ahmed Farouk Abdul Moneim

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Truncated Exponential Distribution T Min T Max t f(t) t Rand Prof. Dr. Ahmed Farouk Abdul Moneim

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Truncated Weibull Distribution Consider the following Probability Density Function T Min t f(t) t Prof. Dr. Ahmed Farouk Abdul Moneim

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Truncated Weibull Distribution T Min t f(t) t Rand Prof. Dr. Ahmed Farouk Abdul Moneim

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DistributionGiven DataParametersFormulas Triangulara, m, b m-a, b-a, b-m Truncated Normal μ, σ, T min, T max ** Truncated Exponential T min, T max, μ λ *** Truncated WeibullT min, μ, σ β, η**** *** **** ** From Tables or Excel Prof. Dr. Ahmed Farouk Abdul Moneim

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SUMMARY Distribution Simulated Time t Truncated Normal Triangular If Otherwise Truncated Exponential Truncated Weibull Prof. Dr. Ahmed Farouk Abdul Moneim

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Example Activity Predecesso r(s) Distribution G I V E N D A T APARAMETERS ANone Triangular a10m12b14 m-a2 b-a 4 BNone Truncated Normal T min10T max12 μ 6 σ 3 CNone Triangular a6m8b10 D A Truncated Weibull Tmin8 μ 15 σ 4 β 1.812 η 7.874 EB,D Truncated Normal Tmin7Tmax9 μ 8 σ 4 FE Truncated Normal Tmin7Tmax13 μ 10 σ 6 GB,D Truncated Exponential Tmin5Tmax20 μ 6 λ 0.999 85 HF,G Triangular a5m8b11 See Excel Sheet for solution Prof. Dr. Ahmed Farouk Abdul Moneim

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Probability Density Functions (PDF). The total area under a probability density function is equal to 1 Particular probabilities are found by finding.

Probability Density Functions (PDF). The total area under a probability density function is equal to 1 Particular probabilities are found by finding.

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