# Monte-Carlo Simulation Technique

Simulation is the process of replicating the real world based on a set of assumptions and conceived models of reality.

Monte-Carlo simulation is required for problems involving ran­dom variables with known (assumed) probability distributions.

This method of simulation was started as an idea by Enrico Fermi in the 1930s. Stanislaw Ulam in 1946 first had the idea and later con­tacted John von Neumann to work on it, and he started to use this simulation in a secret project. After World War II this simulation was published in many papers as a simulation technique.

The Monte-Carlo simulation technique is frequently used to verify results of analytical methods. Rushedi (1984) used the Monte-Carlo simulation approach to obtain the first two statistical moments (mean, value, and standard deviation) of the failure mode expression of brittle and ductile frames and, consequently, a system safety index. Ayyub and Haider (1985) suggested advanced simula­tion methods for the estimation of system reliability.

Fellow et. al. (1993) used the Monte-Carlo simulation program (М-Star) to understand the load and resistance factor design (LRFD). Nikolaos (1995) used the Monte-Carlo simulation to study the reliability of reinforced concrete members strengthened with carbon-fiber-reinforced plastic.

This method depends on simulating the case of study by its parameters, and each parameter will be represented by its prob­ability distribution, mean, and standard deviation.

The simulation will have two parameters: a variable and uncer­tainty. For example, the length of the men in a country is a variable as it represents a normal distribution. But managing a project by time and cost is usually uncertain and is represented by a triangle distribution by knowing the minimum, maximum, and most likely.

So, the risk assessment for the cost estimate and the risk assess­ment for the project’s time through the PERT method also uses Monte-Carlo simulation. If you want to predict the cost of a large

project, you should break it into parts, define the cost of each part, and add them together. As will be discussed concerning time management in chapter four, the project’s time schedule plan is also broken up into smaller activities and based on the PERT method.

Each random variable is described by its statistical parameters: mean, standard deviation, and type of distribution. The distribu­tion type of the random variable is chosen among the different probability distributions provided by the program.

Figure (3.23) presents an overview of the Monte-Carlo simula­tion technique. The input data for the variables will be a probability distribution and, after simulation, one obtains the outputs by the graphs and statistical data.

The simulation model contains all the input data of the deterministic parameters, the random variables, and the equations. The model will run for at least 10,000 trials, as in the following flowchart. The Monte-Carlo simulation technique is simple and is presented in Figure (3.24).

The final output results in displays by software at the end of the simulation and contains the statistical parameters of the variable Z, describing the limit state equation of the cost and time.

After the whole trials are completed, this program will calcu­late and present the mean, standard deviation, and the statistical

 Figure 3.23 Monte Carlo simulation.

Figure 3.24 Flow chart for monte-carlo simulation.

parameters. Also, it can provide the frequency distribution of the value of outcomes of Z and determine the probability of increasing the cost to the limit of the budget.

As shown in Figure (3.25), the input data for all the variable parameters is selected by choosing the probability, arithmetic mean, and standard deviation or coefficient of variation. After running the simulation, the output will be a probabilities distribution curve.

Then, run the random numbers as per the "Mid-square Method" (Von Neumann and Metropolis, 1940s), which produces pseudo­random four digit numbers:

2. Square the seed and extract the center most four digits.