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PROJECT PLANNING. BY. MONTE CARLO SIMULATION. Part II. Prof. Dr. Ahmed Farouk Abdul Moneim. Probability Density Functions Activities Durations :. Rand. Triangular. 0 ≤ Rand ≤ 1. a. m. t. b. Rand. Truncated Normal. t. t MIN. t e. t MAX. Rand. Truncated Exponential. t.
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PROJECT PLANNING BY MONTE CARLO SIMULATION Part II Prof. Dr. Ahmed Farouk Abdul Moneim
Probability Density Functions Activities Durations: Rand • Triangular 0 ≤ Rand ≤ 1 a m t b Rand • Truncated Normal t t MIN t e t MAX Rand • Truncated Exponential t Rand t MIN t MAX From Tables OR Excel Normal Distribution • Truncated Weibull Φ(Z) t t MIN t MAX Z Prof. Dr. Ahmed Farouk Abdul Moneim
Truncated Normal Distribution GIVEN a Probability = Rand Find the corresponding value of t t-Plane Show that the following function is the right PDF T MIN t T e T MAX f (Z) Rand Z-Plane 0 Z Z MIN Z MAX Z Take the Inverse function of Φ (Φ-1 ) Prof. Dr. Ahmed Farouk Abdul Moneim
Triangular Distribution Rand 1-Rand GIVEN a Probability = Rand Find the corresponding value of t Rand H h h Area of the Triangle = (b-a)*H/2 = 1 a m t b t Then Area of the Left part of the Triangle = (m-a)*H/2 = If Otherwise Prof. Dr. Ahmed Farouk Abdul Moneim
f(t) • Truncated Exponential Distribution Consider the following Probability Density Function μ - T Min 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 t T Max T Min t Important Notice! For TRUNCATED Exponential Distribution, The following condition SHOULD BE SATISFIED As R tends to Infinity Prof. Dr. Ahmed Farouk Abdul Moneim
f(t) • Truncated Exponential Distribution Rand t T Max T Min t Prof. Dr. Ahmed Farouk Abdul Moneim
f(t) • Truncated Weibull Distribution Consider the following Probability Density Function T Min t t Prof. Dr. Ahmed Farouk Abdul Moneim
Truncated Weibull Distribution f(t) Rand T Min t t Prof. Dr. Ahmed Farouk Abdul Moneim
** From Tables or Excel *** **** Prof. Dr. Ahmed Farouk Abdul Moneim
SUMMARY • Distribution • Simulated Time t • Truncated Normal If • Triangular Otherwise • Truncated Exponential • Truncated Weibull Prof. Dr. Ahmed Farouk Abdul Moneim
Example See Excel Sheet for solution Prof. Dr. Ahmed Farouk Abdul Moneim