Doing Monte Carlo Simulation in Minitab Statistical Software
Doing Monte Carlo simulations in Minitab Statistical Software is very easy. This article illustrates how to use Minitab for Monte Carlo simulations using both a known engineering formula and a DOE equation. by Paul Sheehy and Eston Martz
Monte Carlo simulation uses repeated random sampling to simulate data for a given mathematical model and evaluate the outcome. This method was initially applied back in the 1940s, when scientists working on the atomic bomb used it to calculate the probabilities of one fissioning uranium atom causing a fission reaction in another. With uranium in short supply, there was little room for experimental trial and error. The scientists discovered that as long as they created enough simulated data, they could compute reliable probabilities—and reduce the amount of uranium needed for testing. Today, simulated data is routinely used in situations where resources are limited or gathering real data would be too expensive or impractical. By using Minitab’s ability to easily create random data, you can use Monte Carlo simulation to:
Simulate the range of possible outcomes to aid in decision-making Forecast financial results or estimate project timelines Understand the variability in a process or system Find problems within a process or system
Manage risk by understanding cost/benefit relationships
Steps in the Monte Carlo Approach
Depending on the number of factors involved, simulations can be very complex. But
at a basic level, all Monte Carlo simulations have four simple steps: 1. Identify the Transfer Equation
To do a Monte Carlo simulation, you need a quantitative model of the business activity, plan, or process you wish to explore. The mathematical expression of your process is called the “transfer equation.” This may be a known engineering or business formula, or it may be based on a model created from a designed experiment (DOE) or regression analysis. 2. Define the Input Parameters For each factor in your transfer equation, determine how its data are distributed. Some inputs may follow the normal distribution, while others follow a triangular or uniform distribution. You then need to determine distribution parameters for each input. For instance, you would need to specify the mean and standard deviation for inputs that follow a normal distribution. 3. Create Random Data To do valid simulation, you must create a very large, random data set for each input—something on the order 100,000 instances. These random data points simulate the values that would be seen over a long period for each input. Minitab can easily create random data that follow almost any distribution you are likely to encounter. 4. Simulate and Analyze Process Output With the simulated data in place, you can use your transfer equation to calculate simulated outcomes. Running a large enough quantity of simulated input data through your model will give you a reliable indication of what the process will output over time, given the anticipated variation in the inputs.
Those are the steps any Monte Carlo simulation needs to follow. Here’s how to apply them in Minitab.
Monte Carlo Using a Known Engineering Formula
A manufacturing company needs to evaluate the design of a proposed product: a small piston pump that must pump 12 ml of fluid per minute. You want to estimate the probable performance over thousands of pumps, given natural variation in piston diameter (D), stroke length (L), and strokes per minute (RPM). Ideally, the pump flow across thousands of pumps will have a standard deviation no greater than 0.2 ml. Step 1: Identify the Transfer Equation The first step in doing a Monte Carlo simulation is to determine the transfer equation. In this case, you can simply use an established engineering formula that measures pump flow: Flow (in ml) = π(D/2)2 ? L ? RPM Step 2: Define the Input Parameters Now you must define the distribution and parameters of each input used in the transfer equation. The pump’s piston diameter and stroke length are known, but you must calculate the strokes-per-minute (RPM) needed to attain the desired 12 ml/minute flow rate. Volume pumped per stroke is given by this equation: π(D/2)2 * L Given D = 0.8 and L = 2.5, each stroke displaces 1.256 ml. So to achieve a flow of 12 ml/minute the RPM is 9.549.
Based on the performance of other pumps your facility has manufactured, you can say that piston diameter is normally distributed with a mean of 0.8 cm and a standard deviation of 0.003 cm. Stroke length is normally distributed with a mean of 2.5 cm and a standard deviation of 0.15 cm. Finally, strokes per minute is normally distributed with a mean of 9.549 RPM and a standard deviation of 0.17 RPM.
利用MINIB做蒙特卡洛模拟
