Enhancing The Reliability Of Measurements And Evaluations Based On Service Quality Models

Fuzzy Numbers as Intersections of Two Soft Inequalities

In our approach, the values on a Likert-scale are represented by fuzzy numbers; that is, instead of expressing an opinion by selecting a particular x crisp value on the scale, we allow the evaluator to select an “approximately x“ value that is given by a fuzzy number. We will use sigmoid functions to compose the membership functions of fuzzy numbers.

Definition 1. The sigmoid function ${\sigma }_a^{\left(\lambda \right)}\left(x\right)$ with parameter $a$ and $\lambda$ is given by

$${\sigma }_a^{\left(\lambda \right)}\left(x\right)=\frac{\mathrm{1}}{\mathrm{1}+e^{-\lambda \left(x-a\right)}},$$

where $x,a,\lambda \mathbb{}\in \mathbb{R}$ and $\lambda$ is nonzero.

The main properties of the sigmoid function ${\sigma }_a^{\left(\lambda \right)}\left(x\right)$ are as follows.

• Range. The range of ${\sigma }_a^{\left(\lambda \right)}\left(x\right)$ is the interval $\left(\mathrm{0,1}\right)$

• Continuity. ${\sigma }_a^{\left(\lambda \right)}\left(x\right)$ is continuous in $\mathbb{R}$

• Monotony.

  • If $\lambda >0$, then ${\sigma }_a^{\left(\lambda \right)}\left(x\right)$ is strictly monotonously increasing

  • If $\lambda <0$, then ${\sigma }_a^{\left(\lambda \right)}\left(x\right)$ is strictly monotonously decreasing

• Limits.

$\underset{x\to +\mathrm{\infty }}{\lim }{\sigma }_a^{\left(\lambda \right)}\left(x\right)=\left\{\begin{array}{c}\mathrm{1},\mathrm{}\text{if}\lambda >\mathrm{0}\mathrm{}\\ \mathrm{0},\text{if}\lambda <\mathrm{0}\end{array}\right.$, $\underset{x\to -\mathrm{\infty }}{\lim }{\sigma }_a^{\left(\lambda \right)}\left(x\right)=\left\{\begin{array}{c}\mathrm{1},\mathrm{}\text{if}\lambda <\mathrm{0}\mathrm{}\\ \mathrm{0},\text{if}\lambda >\mathrm{0}\end{array}\right.$

• Role of parameters.

  • Parameter $a$ is the locus at which ${\sigma }_a^{\left(\lambda \right)}\left(x\right)$ has the value 0.5

  • The slope of ${\sigma }_a^{\left(\lambda \right)}\left(x\right)$ at $a$ is $\lambda /4$; that is, the parameter $\lambda$ determines the gradient of function curve at $a$

Figure 1 shows examples of sigmoid function graphs. Let us assume, that we have two sigmoid functions, ${\sigma }_{a_l}^{\left({\lambda }_l\right)}\left(x\right)$ and ${\sigma }_{a_r}^{\left({\lambda }_r\right)}\left(x\right)$, where ${\lambda }_l>\mathrm{0}$ and ${\lambda }_r<\mathrm{0}$, that is, ${\sigma }_{a_l}^{\left({\lambda }_l\right)}\left(x\right)$ is strictly monotonously increasing, while ${\sigma }_{a_r}^{\left({\lambda }_r\right)}\left(x\right)$ $)$ is strictly monotonously decreasing. Then, we will use the Dombi conjunction operator (Dombi intersection) to implement intersection of two fuzzy sets (Dombi, 2008).

Definition 2. The Dombi intersection of the fuzzy sets ${\mathit{A}}_{\mathrm{1}}$ and ${\mathit{A}}_{\mathrm{2}}$ that are given by the membership functions ${\mu }_{A_{\mathrm{1}}}\left(x\right)$ and ${\mu }_{A_{\mathrm{2}}}\left(x\right)$, respectively, is the fuzzy set with membership function ${\mu }_{A_{\mathrm{1}}\cap A_{\mathrm{2}}}\left(x\right)$:

$${\mu }_{A_{\mathrm{1}}\cap A_{\mathrm{2}}}\left(x\right)={\mu }_{A_{\mathrm{1}}}\left(x\right){\mathrm{*}}_{\left(D\right)}{\mu }_{A_{\mathrm{2}}}\left(x\right)=\mathrm{}\frac{\mathrm{1}}{\mathrm{1}+{\left({\left(\frac{\mathrm{1}-{\mu }_{A_{\mathrm{1}}}\left(x\right)}{{\mu }_{A_{\mathrm{1}}}\left(x\right)}\right)}^{\alpha }+{\left(\frac{\mathrm{1}-{\mu }_{A_{\mathrm{2}}}\left(x\right)}{{\mu }_{A_{\mathrm{2}}}\left(x\right)}\right)}^{\alpha }\right)}^{\mathrm{1}/\alpha }},$$

where ${\mu }_{A_{\mathrm{1}}}\left(x\right)$, ${\mu }_{A_{\mathrm{2}}}\left(x\right)$ $\in \left(\mathrm{0,1}\right)$, $\alpha \in \mathbb{R}$, $\alpha >\mathrm{0}$ and ${\mathrm{*}}_{\left(D\right)}$ denotes the Dombi intersection operator.

If we apply the Dombi intersection to ${\sigma }_{a_l}^{\left({\lambda }_l\right)}\left(x\right)$ and ${\sigma }_{a_r}^{\left({\lambda }_r\right)}\left(x\right)$ with $\alpha =\mathrm{1}$, we get

$$\mathrm{}{\sigma }_{a_l}^{\left({\lambda }_l\right)}\left(x\right){\mathrm{*}}_{\left(D\right)}{\sigma }_{a_r}^{\left({\lambda }_r\right)}\left(x\right)=\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}-{\sigma }_{a_l}^{\left({\lambda }_l\right)}\left(x\right)}{{\sigma }_{a_l}^{\left({\lambda }_l\right)}\left(x\right)}+\frac{\mathrm{1}-{\sigma }_{a_r}^{\left({\lambda }_r\right)}\left(x\right)}{{\sigma }_{a_r}^{\left({\lambda }_r\right)}\left(x\right)}}.$$

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Utilizing Definition 1 we get:

$$\mathrm{}{\sigma }_{a_l}^{\left({\lambda }_l\right)}\left(x\right){\mathrm{*}}_{\left(D\right)}{\sigma }_{a_r}^{\left({\lambda }_r\right)}\left(x\right)=\mathrm{}\frac{\mathrm{1}}{\mathrm{1}+e^{-{\lambda }_l\left(x-a_l\right)}+e^{-{\lambda }_r\left(x-a_r\right)}}.$$

Figure 01 . shows the intersection of two fuzzy sets given by an increasing and a decreasing sigmoid membership function. The following theorem allows us to separately aggregate the left hand sides and right hand sides of fuzzy numbers.

Theorem 1. If ${\mathit{A}}_{\mathrm{1}},{\mathit{A}}_{\mathrm{2}},\dots ,{\mathit{A}}_n\mathrm{}$are fuzzy sets with the membership functions ${\sigma }_{a_{\mathrm{1}}}^{\left({\lambda }_{\mathrm{1}}\right)},{\sigma }_{a_{\mathrm{2}}}^{\left({\lambda }_{\mathrm{2}}\right)},\dots ,{\sigma }_{a_n}^{\left({\lambda }_n\right)}$, respectively, $sgn\left({\lambda }_{\mathrm{1}}\right)=sgn\left({\lambda }_{\mathrm{2}}\right)=\dots =sgn\left({\lambda }_n\right)$, and the fuzzy set $\mathit{A}$ is given by the linear combination $\mathit{A}=\underset{i=\mathrm{1}}{\overset{n}{\sum }}{w}_i{\mathit{A}}_i,$ where ${\sum }_{i=\mathrm{1}}^n{w}_i=\mathrm{1}$, then $\mathit{A}$ is also sigmoid-shaped with the membership function ${\sigma }_a^{\left(\lambda \right)}$, where $a=\underset{i=\mathrm{1}}{\overset{n}{\sum }}{w}_i{a}_i,$ $\frac{\mathrm{1}}{\lambda }=\underset{i=\mathrm{1}}{\overset{n}{\sum }}{w}_i\frac{\mathrm{1}}{{\lambda }_i}.$

Proof. See Dombi (2009).

The parameters $a$ and $\lambda$ of the sigmoid function ${\sigma }_a^{\left(\lambda \right)}$ can be unambiguously given by determining two points of the function curve. The sigmoid function neither takes the value 0, nor the value 1, these are its limits. In practical applications, it may be useful if the function is given by two points which have vertical coordinates close to 0 and 1. Let $\epsilon$ be a small positive value, for example $\epsilon =0.001$, and $y_0=\epsilon$, $y_1=1-\epsilon .$

If we wish ${\sigma }_a^{\left(\lambda \right)}$ to take the values of $y_0$ and $y_1$ at $x_0$ and $x_1$, respectively, the parameters $a$ and $\lambda$ need to be set as follows:

$a=\frac{x_{\mathrm{0}}ln\left(\frac{\mathrm{1}-y_{\mathrm{1}}}{y_{\mathrm{1}}}\right)-x_{\mathrm{1}}ln\left(\frac{\mathrm{1}-y_{\mathrm{0}}}{y_{\mathrm{0}}}\right)}{ln\left(\frac{\mathrm{1}-y_{\mathrm{1}}}{y_{\mathrm{1}}}\right)-ln\left(\frac{\mathrm{1}-y_{\mathrm{0}}}{y_{\mathrm{0}}}\right)}$, $\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\lambda =-\frac{ln\left(\frac{\mathrm{1}-y_{\mathrm{0}}}{y_{\mathrm{0}}}\right)}{x_{\mathrm{0}}-a},$

where $x_{\mathrm{0}}\ne a$.

This approach enables us to represent the “approximately $m$” value by the parameter triple $l,m,r$. Namely, “approximately $m$” can be given as the Dombi intersection of the increasing sigmoid fuzzy membership function ${\sigma }_{a_l}^{\left({\lambda }_l\right)}\left(x\right)$ and the decreasing sigmoid fuzzy membership function ${\sigma }_{a_r}^{\left({\lambda }_r\right)}\left(x\right)$, where

$a_l=\frac{lln\left(\frac{\epsilon }{\mathrm{1}-\epsilon }\right)-mln\left(\frac{\mathrm{1}-\epsilon }{\epsilon }\right)}{ln\left(\frac{\epsilon }{\mathrm{1}-\epsilon }\right)-ln\left(\frac{\mathrm{1}-\epsilon }{\epsilon }\right)}$ ${\lambda }_l=-\frac{ln\left(\frac{\mathrm{1}-\epsilon }{\epsilon }\right)}{l-a_l}$

$a_r=\frac{rln\left(\frac{\epsilon }{\mathrm{1}-\epsilon }\right)-mln\left(\frac{\mathrm{1}-\epsilon }{\epsilon }\right)}{ln\left(\frac{\epsilon }{\mathrm{1}-\epsilon }\right)-ln\left(\frac{\mathrm{1}-\epsilon }{\epsilon }\right)}$ ${\lambda }_r=-\frac{ln\left(\frac{\mathrm{1}-\epsilon }{\epsilon }\right)}{r-a_r}.$

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Figure 02 . shows an example of the fuzzy number “approximately $m"$ that is composed of a left hand side and a right hand side sigmoid functions. Based on the favourable properties of fuzzy Likert-scales that can help to overcome the difficulties associated with student satisfaction measurement, a pilot fuzzy number based evaluation has been launched at the Faculty during the fall semester 2016. In order to gain experiences with fuzzy evaluation of teaching performance, the questionnaires used to evaluate the midterm tests/exams by students have been selected to be evaluated by a fuzzy Likert scale. In case of 5 courses 7 midterm tests have been selected after which students have been asked to use a fuzzy Likert scale to evaluate the midterm test. The eight dimensions used to evaluate the midterm tests are as follows: D1 availability of instructional materials, D2 midterm test, exam circumstances, D3 review the course of tests, exams, D4 clarity of exam questions, D5 consonance of exam questions with requirements, D6 clarity of result calculation, D7 standard of consultation opportunities, D8 standard of midterm test/ exam viewing opportunities. The scale was applied with a division of 0.25 units in order to allow students a more detailed reflection of their judgement. After each midterm test, 10-15 students evaluated the lecturers’ performance on this fuzzy scale, so that altogether 85 fuzzy questionnaires have been filled out, while most of the students did not experience any difficulties when making their evaluation. The experiences were in accordance with the results of Gil et al. (2015). Parallel to the fuzzy number based evaluation, the traditional Likert scale based questionnaires have been launched as well, which allowed us to compare the results.