Monte Carlo Solutions For Reliability Problems

Monte Carlo techniques solve difficult reliability problems using random numbers. Monte Carlo methods are non-deterministic, and they fall into the category of statistical calculations. Monte Carlo techniques involve the use of reliability equations and random numbers. Monte Carlo techniques require heavy use of computers for repetitively solving the problems as each solution is different from the others. However, through the noise of the Monte Carlo computer simulations comes a signal and the answer to difficult reliability problems.

Where does the random number method called Monte Carlo originate? During the Manhattan Project to construct the atomic bomb, code words were used extensively for security purposes. The code word for the problem solving technique using random numbers was called Monte Carlo. On the Manhattan Project, university professors wrote complicated equations (partial differential equations) for the nuclear systems that could not be solved by hand—remember the era was without computers! The equations were rearranged so a random number (a number between 0 and 1) could be introduced. Then random number tables developed during the depression era WPA projects were employed along with many mathematicians and statisticians to solve the problems. John von Neumann is generally considered to be the father of the methodology as he described it in 1947 but the method seemed to have been used by all the fathers of the bomb. The earliest computer ENIAC was used with a crude random number generator to solve early problems because it could draw the crude random numbers so quickly.

Using random numbers with a reliability problem results in a wide range of answers. The law of large numbers says to expect convergence to the correct answer with 1/ÖN where N is the number of sampled solutions. The 1/N^0.5 equation says that quadrupling the number of solutions, cuts the error in ½. You need to solve the problems many times to get an honest answer that approximates the true answer.

One method to arrive at a pragmatic answer for how many N simulations you need to perform is by use the jack-knife technique. The jack-knife technique says run “many simulations” (let’s start with 1,000 simulations as a single example—of course the number is too small!). Do the simulation twice to get two answers. Compare the answers; and if the results are too far apart, double the simulations to 2*N. Keep repeating the number of simulations until the error is small enough to live with. Then finally, double the number of simulation for obtaining the final answer. Err on the side of too many iterations rather than too few.

You want the Monte Carlo technique to generate random numbers without bias; however a notable bias situation did occur with the Monte Carlo roulette wheel that allowed Joseph Jagger to break the bank at Monte Carlo Casino when he discovered a bias in the late 1800s. Today the Monte Carlo roulette wheel, like other gaming devices, is studied by statistics to show it is free from bias as is done in the State of Texas with the Lotto gambling system. However, other gambling systems are allowed to be biased slightly “in favor of the house” and that’s why owning a Casino is a money maker. You hear the term “biggest payoff” in slot machines, etc.—If you believe you’re going to win, then see me as I have some nice revenue producing bridges in New York City that I can sell you at a nice price!

Microsoft’s Excel 2003 and 2007 has an improved random number simulation (it’s really a pseudo-random number generator) which is described in Knowledge Base 828795. The Excel 2003-2007 random number will repeat itself every ~10¹³ number using the Wichman and Hill methods where as earlier Excel versions repeated the string of number every ~10⁶. The Wichman and Hill method used by Excel is explained by Carl Tarum below in a Monte Carlo simulation.

Excel^® Monte Carlo Simulations (no charge downloads for instructional purposes)
1.	Random Numbers - The spreadsheet contains details about: 1) Wichman and Hill’s random number details, 2) Coin flip simulation which you can bias away from a fair coin (i.e. 50%/50%) 3) Random numbers for confidence intervals
2.	Fix When Broken Life Cycle Cost Simulation – The spreadsheet contains Weibull failure details for a single component on a fix when broken basis that generates life cycle costs details and generates NPV calculations.
3.	Life Cycle Cost Simple Model With Many Items – The spreadsheet follows a fix when broken strategy to calculate life cycle cost for many items including capability for annual electrical costs, etc. and to produce a NPV calculation.
4.	Planned Replacement Life Cycle Cost Simulation – The spreadsheet allows selection of a timed replacement strategy for a single cost item to calculate life cycle cost details and generate NPV calculations.
5.	Key Performance Production Criteria – The Monte Carlo spreadsheet shows how typical key performance indicator (KPI) statistics relate to Weibull process reliability plot characteristic value for the best 5 consecutives day average, best 7 consecutive day average, best 20 consecutive day average, etc. with errors shown as % when measured to the Weibull process plots characteristic values. See the May ’98 Problem Of The Month.
6.	Inspection Test – a) Select the items to be randomly inspected from a population of things based on a % size for the sample. b) How effective are you at performing a 100% inspection using items size given in three decimal points with a low limit tolerance and a high limit tolerance or if you’re a word person the same 100% inspection test is provided for words with 5 letters. Answers to each 100% inspection are underneath the yellow text box after you have completed your own personal 100% inspection test. You can select new random results by press of the F9 key to recalculate. (You’ll probably find you’re about 80% effective when performing a 100% inspection—much worse if you’re management and slightly better if you’re an inspector.) ALERT: Perform the 100% inspection with a strict time limit of 5 minutes—no inspection in industry can last forever!
7.	MCSAMPLE—Download a small subset of MonteCarloSimulationS which is a free, complimentary software. This demonstration software shows how to use random numbers in a spreadsheet environment to simulate the reliability of simple and complicated systems. If you find these freebie models are helpful for solving your problems, see details below for purchasing the full set of software. The MCSAMPLE.XLS file contains: · Why use Monte Carlo simulations? · Series model using reliability numbers · Series model using age-to-failure information · Parallel model using reliability numbers · Complex model using reliability numbers · Simple business failure model using Weibull failure data · Air compressor simulation · More information about MonteCarloSimulationS
8.	Paycheck Simulation- This simulation is described in the April ‘04 problem concerning Process Reliability Line Segments. When you have a production process, it will demonstrate variability in output which is described by the Weibull beta value. The Weibull characteristic value eta will describe the magnitude of the output. A process reliability value describes the point where common-cause variability gives way to special-cause variability. Daily production output drives the monthly paycheck for the company. The paycheck simulation is intended to show individuals the connection with the process output measures and how similar variability would appear in their monthly paycheck. [If you’ve just declared you don’t want variability in your monthly paycheck maybe the light bulb is about to glow over your head as you make the connection between expectations for the company to control output variability!---yes, I know you want the big positive variations but not the downside variations. Got it?] Take your net paycheck value for the eta value. Use the beta from your production process. Include the reliability of your production process along with the process reliability to demonstrate the high variability that would exist for your personal paycheck if it varied as much as your process. Do you like the variability in your monthly paycheck? Most operations have the mistaken opinion they are #1 in their field. If you think you’re #1, why should you change? The phenomenon that we’re #1 is call denial. For example of denial, read the book by Dr. Elisabeth Kübler Ross, M.D., on Death and Dying.  Her book was based on observations of patients dying in Cook County Charity Hospital in Chicago.  She identified 5 stages that most people go through enroute to the next adventure at end of human life: 1. Denial, ßWhy should we change? Don’t you know we’re #1! This program will blow over. Sit tight.             2. Anger, ßThe white shirts hate us, we’re doing the best we can! They’re punishing us. 3. Bargaining, ßWould 10% more output make you happy, and then get off our back? 4. Depression, and ßThey don’t appreciate me, and I work so hard—the bastards! 5. Acceptance. ßI’ve got to get a new job, I can’t live in this environment! The same stages are required for burying an inferior (remember, we’re #1!) manufacturing process so you can get a break thorough.  While Dr. Ross, a Swiss physician, defined these stages for clarity, she was a pragmatist and not nearly as rigid as some of her critics want to describe her based on the late 1960’s which I attended one of her seminars. Manufacturing teams go through each of the stages of denial…acceptance as propelled by Corporate for making major improvements. Not all individuals and not all process survive the transition. The purpose of the paycheck simulation is to show you how your process is performing based on your personal paycheck—ask this question: How would your mate view the need for making process improvements if your paycheck varied as much as your manufacturing process?
9.	Here are some MonteCarloSimulationS™ to illustrate how the use of random numbers can solve problems in Excel^® to produce useful results. Montecar.XLS is a collection of early Monte Carlo simulations in Excel to show how random numbers and Weibull technology go hand-in-hand to solve engineering problems. The models are intended to simulate your thinking LCC100.XLS is a life cycle cost program allowing changes in equipment grade, maintenance replacement strategies, and installation strategies for a pumping system which influence the life of components. The results from the Monte Carlos simulation go into a net present value worksheet. FailureGroups.XLS develops Crow-AMSAA reliability growth plots by way of Monte Carlo techniques using three different regression technologies (IEC 61164, MLE as described in MIL-HDBK-189, and conventional least squares regression. For small datasets, MLE produces trend lines which are too steep, the simple regression produces lines which are often too flat because of early failures, while the IEC method is less biased and the trend line always goes through the last data point.
	Page under construction—more later.

Other simulation software is available for:

1. life cycle cost models, and
2. reliability block diagrams.
   
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Last revised 10/07/2008