Friday, October 28, 2016

Friday, October 28, 2016
Modeling uncertainty in risk assessment: An integrated approach
with fuzzy set theory and Monte Carlo simulation


This journal article uses a fuzzy set theory and Monte Carlo Simulations to model and evaluate uncertainty and risk to a benzene extraction unit (BEU) of a chemical plant in India. They first described the situation that risk plays to many industries, and then went into a literature review of studies using Bayesian Network analysis, and other methods used to reduce uncertainty in analysis.

  1. After reviewing, other methods of analysis to reduce uncertainty and risk, the scientists then moved into their methodology. First they outlined the three major components for risk modeling which were 1) estimation/probability of undesired outcome/situation; 2) estimation of losses due to undesired outcomes/situations; and 3) modeling the risk while including variability and uncertainty in the probability of failure and its resultant consequences. From here the scientists then moved into the method they would use which was a simulation analysis using a Monte Carlo analysis (MCA) simulation technique. Specifically, MCA is used commonly in risk assessment circumstances due to its ability to quantify uncertainty or variability in a probabilistic frameworks.

  1. The particular MCA used by the scientists in this study was a hybrid MCA called 2-dimensional fuzzy MCA or 2D FCMA. In this MCA, 2 loops are used with the inner loop models consisting of the random variables for each fuzzy membership value, leaving the outer loop to model the parameters. The equation used for this is g(R)=f1(P)*f2(C), with P=probability of failure; C=consequences/loss due to failure; and f1 and f2 and g being the functional forms.

  1. Moving to the next step after the scientists used their equation, was the use of a vertex method while substituting a DSW algorithm. These algorithms reduce the computational effort used in estimating the upper and lower intervals, while using a form of standard interval analysis with α-cut concept.

  1. Through a number of mathematical equations the scientists would produce their “1) estimation of fuzzy cumulative distribution function (CDF) of failure probability, 2) estimation of fuzzy consequence intervals, 3) estimation of fuzzy risk, and 4) estimation of support, uncertainty, possibility and necessity measures”(Arunraj, Mandal, & Maiti 2013). All of which would be used to produce the lower and upper bounds of risk.

  1.  Applied to the BEU and its 8 section failures, the scientists used the standard deviation and mean of lognormal distribution of likely failure as the fuzzy numbers. Which were then put into DSWs and came out as 5 different combinations (Table 4). For the 5 pairs of means and standard deviations, 5000 Monte Carlo simulations were used to create the CDFs. Which were then split into 100 numbers of percentiles, and applied into the 8 sections of the BEU for evaluation (Table 5). All of which were set to a benchmark of a compliance guideline of industry operations, or some regulatory authority (i.e. the plant management), and printed in the Table 7 results.

Table 6 Most Likely Value of Risk
Table 7 Final Results For Measures to Compliance Benchmark

In conclusion, the scientists acknowledge that evaluating a point risk is difficult and has serious limitations for decision makers. Yet, the use of interval risk values that use variability and estimation reduce the uncertainty for a decision maker. With the use of the 2D FMCA, it uses two forms of uncertainty assessment models, which are the combination of fuzzy set theory and probability theory. The 2D FMCA method reduced more uncertainty than any of the other methods described in the literature review of past studies, making it a stronger piece of support to aiding a decision maker’s capabilities of making the right decision, particularly in regards to the BEU. Which for the BEU the uncertainty index showed the highest degree of uncertainty for the process condensate system, followed by the solvent regeneration section, benzene stripper column section, and lastly the storage and slop drums when put against the high risk sections (See Table 7 results).


Due to limited knowledge on the topic of MCA and the resultant other theories used in this piece, I would say the track record for MCA is credible in being able to reduce uncertainty. This is assuming that the person doing all the mathematical equations behind it knows exactly what they are doing. I found it interesting that like intelligence the chemical sectors try to keep their failures from happening for like intelligence their failures are known not their successes. For the researchers acknowledged that finding backing data for their study was difficult to obtain. I personally think the article was well rounded in that it evaluated all methods before going into the methodology section that the researchers selected. It allowed one to see and compare, and MCA by my understanding and by the researchers results proved the better method to reduce uncertainty, particularly if it is for a decision maker.


Arunraj, N. S., Mandal, S., & Maiti, J. (2013). Modeling uncertainty in risk assessment: An integrated approach with fuzzy set theory and Monte Carlo simulation. Accident Analysis & Prevention, 55, 242-255. <>.


  1. Roland, you are correct in thinking that the person running the Monte Carlo simulations should have a good understanding of the math behind it, although it is not as complicated as it might seem. Monte Carlo methods are incredibly useful for understanding and quantifying uncertainty in a way that makes sense to decision makers. However, it does not take a PhD in statistics to understand how Monte Carlo methods work and to do them yourself. They are excellent methods for risk professionals to have in their toolbox.

  2. Thanks Hank, I did take statistics so I have a basic understanding, but much of statistics was not necessarily my strongest point. Yet, thanks again Hank, and I hope the simulation went well yesterday.