The Fitting of Fuzzy Data and Intervals

There are three intervals which need to be fitting for its convergence with the probability and possibility theories. This was done by conducting a Chi-Sq test to fit these intervals suitable for the fuzzy condition. As a result of the fitting process under probability pattern, i.e., arrival rates and service rates were obtained with one crisp value for each interval. Consequently, these crisp averages were compared to the respective averages under the fuzzy condition which is under possibility pattern. If the averages from both patterns are approximately similar, this means that there is no need to fuzzify those values. Consequently, this is the first indicator that shows the convergence between probability and possibility patterns.

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On the other hand, if these single values are significantly different from the fuzzy data for each interval, this means that the fuzzification process must be done by choosing the linguistic terms and constructing the membership function for each interval. This is another indicator that shows the convergence between the statistical behavior and fuzzy behavior. The steps to fitting the fuzzy subsets interval to make convergence between possibility and probability pattern are as follows:

Step 1: Test each fuzzy subset i.e., each interval low, medium and high. This can be done for each basic element, which are the arrival rates for Class One and Class Two, and average service rates between Class One and Class Two.

Step 2: From Step 1 we have two situations if the results obtained from Chi-Sq conducted are still under fuzzy process. This comparison is based on averages for each interval of the basic elements. If the results of the averages via Chi-Sq conducted are approximately same to the averages of fuzzy process, then go to Step 7. Otherwise, fuzzify these new single crisp values (i.e., the average).

Step 3: Fuzzification of the single crisp values is always in terms of linguistic expressions, such as „approximately‟ or „around‟. These descriptions are suitable and allow capturing of the relevant measurements of uncertainties in real data as mentioned by Chen (2005). Thus, each interval can be represented as linguistic expression.

Step 4: Construct the trapezoidal membership functions (TpMF) for each interval. Therefore, through the statistical measures from the average obtained via fitting of each interval, this single value for each interval (i.e., the average) had to be fuzzified.

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Thus, choosing the main points for each trapezoidal membership function can be done through the process of located number inside the central form as suggested by Bojadziev & Bojadziev, (2007). Note that the two points for each interval are boundaries as the closed interval where the average is located inside this interval. Hence, this process led to the construction of the trapezoidal fuzzy numbers. In this research, there are two types of basic elements of arrival rates of Class one and Class Two and average service rates.

 The Arrival rates for Class One and Class Two were tested for their fitness as mentioned in subsection 4.5.1. As a result of the fitting process for arrival rates for both classes, a poisson distribution was obtained with one crisp value (i.e., the average) for each interval. Consequently, these crisp averages were compared to respective averages to find whether or not they are under fuzzy condition.

 The process of fitting the second element is the service rates. We established two continuous distributions, i.e., exponential service rates and gamma service rates, as mentioned in subsection 4.5.2. Therefore, according to these three fuzzy subsets, which are the service rates for each interval, fitting of the intervals is conducted. The exponential service time distribution has one crisp parameter, µ for each interval via the fitting process. In addition, for the gamma service rates under probability pattern based on Chi-Sq test conducted, we obtained two crisp parameters, α and β, for each interval. These crisp parameters were compared under the fuzzification process. This was done through the averages of these distributions via parameters and Equations (3.15) and (3.20) as described in

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subsections 3.1.4.1 and 3.1.4.2. Consequently, single crisp values were obtained for each interval and it was also compared to respective averages under the fuzzy condition.

Step 5: Construct the triangular membership functions (TrMF) through averages obtained by fitting each interval as suggested by Li (2016) as in subsection 3.2.4.2.

Step 6: Plot three groups of fuzzy numbers, both TpMF and TrMF for each basic element as three intervals of the fuzzy subsets.

Step 7: All the basic elements which includes the arrival rates for Class One and Class Two, and average service rates between Class One and Class Two for each interval are under fuzzy process. Figure 4.4 shows the steps of fitting fuzzy data and intervals are as follows:

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Figure 4.4 Steps to Fitting the Fuzzy Subsets Interval

Furthermore, these scenarios of mapping all basic elements can be presented as the design of intervals for the three fuzzy subsets to achieve Objective 1 of the research, which can be indicated in terms of average. Hence, the next step in this research was phase three of Figure 4.2 which explains the development of the multiple channel queueing model with multiclass arrivals under an uncertain environment.

Step 6

Fuzzify the single crisp Avg. obtained for each interval

All crisp data are under fuzzy process for each interval of the basic element (i.e., arrival rates Class One and Class Two, and service rates)

No

Yes

Calculate the Avg. for each interval (i.e., low, med and high) for each basic element

Step 3

Step 7 Step 1

Step 2

Test each interval of basic elements, which are the arrival rates for Class One and Class Two, and average service rates between

Class One and Class Two.

End Start

Construct TrMF for each interval Construct TpMF for each interval

Plot three groups of fuzzy numbers both trapezoidal and triangular membership functions as intervals low, medium and high for each basic

element (i.e., arrival rates Class One and Class Two, and service rates)

Is Avg. convergence?

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