Comparison among the Different Methods

This section presents a comparison among the results of the statistical approach, probabilistic approach, and possibilistic approach. There are different approaches to quantify the uncertainty, namely, statistics, probability (Dempster, 1968), possibility (Zadeh, 1978; Dubois and Prade, 1988), imprecise probability (Walley, 1991; Walley 2000), evidence (Shafer, 1976; Klir, 1990), and random interval (Joslyn and Booker, 2004). As there are many methods to quantify the uncertainty, which method should be selected? In this study, uncertainty is

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quantified using three mostly used approaches and investigated a reliable approach to quantify the uncertainty in the data. And it is concluded that, possibility distribution is better method and reliable compared to other methods. The quantified data using three different methods are listed in Table 3-g.

Table 3-g. Probabilistic and possibilistic uncertainties of jute yarn.

Properties Parameters Statistical Method

Distribution Weibull Possibility TS [MPa]

Expected value ( ) [37.76, 45.60] 41.60 41.75 Standard deviation (sd) 4.01 4.35

at 95% confidence level/ Alpha cut

[2.94, 6.32] [25.07, 69.00] [39.1, 44.4]

E [GPa]

Expected Value ( ) [0.64, 0.72] 41.60 41.75 Standard deviation (sd) 0.04 4.35

at 95% confidence level/ Alpha cut

[0.03, 0.06] [25.07, 69.00] [39.1, 44.4]

s [%]

Expected Value ( ) [5.53, 6.87] 0.67 0.67 Standard deviation (sd) 0.69 0.43

at 95% confidence level/ Alpha cut

[0.51, 1.09] [0.408, 1.123] [0.642, 0.7]

In case of statistical approach, the expected values for the TS, E,and s are in the form of range [37.76, 45.60], [0.64, 0.72], and [5.53, 6.87], respectively as exposed in Table 3-g. For example, the expected value of the TS, E, and s of jute yarn are [37.76, 45.6], [0.64, 0.72] and [5.53, 6.87], respectively as shown in Table 3-g. As previously explained when there is uncertainty, statistical analyses estimate both highly pessimistic and optimistic values of TS, E, and s of jute yarns. That means some data are highly emphasized however, all data are not equally emphasized. When the uncertainty is represented by the statistical method the confidence interval is considered as range. This range may or may not be included in the experimental data points. Though, statistical approach is easy and familiar, one cannot rely on the statistical approach for such types of uncertainty quantification. Moreover, statistical approach deals with finite data, however, for infinite series (general assumption) the probability distribution is

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an alternative option. Last but not least, the true behavior of a variable is described by its infinite statistics, finite statistics describe only the behavior of the finite data set (Figliola and Beasley, 2000).

Apart from the statistical approaches to quantify the uncertainty associated with material properties, an alternative method (Weibull distribution) is considered.

Statistical method estimates the expected value as a range. However, the Weibull and possibility distributions estimate a specific value for a certain group of uncertainty in the data of the TS, E,and s. In addition, probability distribution estimated the expected value of the TS is 41.60 MPa, whereas the possibility estimated the most possible value (expected value) of TS is 41.75 MPa for of jute yarn. Both of the quantification approaches give the same result for TS and E except s. Therefore, it can be informed that possibility distribution can quantify the data as probabilistic approach. Thus, one can rely on the possibility distribution to quantify the uncertainty. Therefore, it can be inferred that probabilistic and possibilistic approaches are better compared to the statistical method.

Generally, the Weibull distribution is used to observe the life-cycle of the product. The Weibull distribution is one of the special forms of the binomial distribution, and follows the binomial theorem. Nevertheless, most of the researchers use this distribution to quantify the variability for natural materials.

However, the Weibull distribution may not be the appropriate approach to handle the uncertainty of the natural material. Because before quantifying the uncertainty using probability distribution, one needs to know that the data are following a distribution. In this study, before calculating the Weibull distribution of jute yarn, it is assumed that data of jute yarn properties are following the Weibull distribution. In case of the jute fiber from jute growth to fiber collection, the growth and collection conditions are unknown because they grow naturally and rotted, washed, dried and finally collected by farmers. Thus, it is difficult to say which distribution they are following. In engineering practice, it is truly impossible to control the operation conditions of the growth in the field

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naturally, that means the distribution is not known. Therefore, it is difficult to consider that jute is following a specific distribution. Thus, Weibull distribution above all probability distributions is not appropriate for such types of uncertainties quantification.

Besides, when the distribution is unknown, it is required to check through different categories of probability distribution for curve fitting, which is time-consuming. As possibility distribution deal with the uncertainty, in that sense possibility distribution is a better choice to calculate the uncertainty compared to other methods.

In addition, a graphical representation is better for human understanding compared to the average value. However, for a small range of the data, it is tough to calculate and draw the probability distribution. For drawing histogram of the data, it is needed to obtain a minimum number of intervals (K) (Figliola and Beasley, 2000; Ashby, 2007) in the range (N) of data shown by equation (3.13).

1 ) 1 ( 87 .

1  ^0.40 

 ^N

K (3.13)

If the number of data and interval are large, the histogram will be better.

Therefore, large number data (N) are better or required for probabilistic approach. On the other hand, possibility distribution can be used for a small number of data as well as big data. This type of evidence is obtained in case of Weibull distribution as there is a limited number of data the error estimation is higher as shown in Table 3-h. From Table 3-h, it is clear that the error estimation is high for TS (10.74), E (17.4), and s (1.57) of the jute yarn.

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Table 3-h. Error estimation of quantified data for mechanical properties of jute yarn.

Parameters Properties

TS [MPa] E [GPa] s[%]

Expected value ( ) Standard Error

41.60 10.74

0.677 17.4

6.15 1.58 Standard deviation (sd)

Standard Error

4.35 1.123

0.043 0.011

0.74 0.19

As the distribution jute yarn is unknown, possibility distribution may be better option to quantify the uncertainty. Moreover, this is one of the better ways to handle the uncertainty in the natural material (jute fiber). In possibility distribution, it is not mandatory to know that they are following any distribution or not. Moreover, a small range of data is able to give us a distribution or conclusion in case of possible distribution. Moreover, since the number of data points was small (15 data points), error estimation behind the probability distribution (i.e., Weibull distribution) becomes high. In fact, when it is unknown which distribution should be used to quantify the uncertainty, or even when there is insufficient data to deduce a probability distribution, the answer is to use a probability distribution-neutral representation (Ullah and Shamsuzzaman, 2013) of uncertainty. This was quite relevant to the case in this study, because of the limited number of data points (15 data points for each property, as shown in Table 3-c). As such, the concept of possibility (Zadeh, 1999) or possibility distribution (Dubois, et al., 2004) could be used. A possibility distribution is popularly referred as a fuzzy number (see (Ullah, 2016) for a definition). A possibility distribution entails a family of probability distributions (e.g., a triangular fuzzy number can entail a set of unimodal probability distributions, e.g., normal distribution, triangular distribution, and uniform distribution, (Ullah and Shamsuzzaman, 2013; Masson and Denœux, 2006)). In addition, a possibility distribution can also be deduced from a limited number of data points

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(Ullah and Shamsuzzaman, 2013; Masson and Denœux, 2006). Such a distribution also provides a reliable representation of the uncertainty, which is compatible with the general concept of uncertainty (Mauris, et al., 2001).

Therefore, it can be suggested that possibility distribution is better to quantify the uncertainty associated with the natural material properties.

Furthermore, in case of possibilistic approach, the upper limits of the alpha cut ranges are greater than the expected value of the respective material properties whereas the lower limits of the logically consistent ranges are smaller than the values of the respective material properties. This means that the lower limit of a logically consistent range is the most conservative estimation of the underlying material property. Therefore, one may consider the lower limit of a logically consistent range to be the design limit of the material property.