A Systematic Review on Generalized Fuzzy Numbers and Its Applications: Past, Present and Future

Abstract

Nowadays, the program’s mechanisms are becoming more dynamic. As a result, maintaining performance for a longer duration of time to increase the system’s long-term growth is tough. This is mostly because of the malfunction occurrence during the study, as the machine does not always have all details. In order to tackle this problem, the available data must be used to construct the problems. However, one of the most successful data theories is fuzzy set theory. The concept of fuzzy logic has recently grown in favour, and it plays an important role in engineering and management. Fuzzy arithmetic was very significant in research fields including decision-making problems, confidence analysis, optimization etc. as compare to others. Fuzzy numbers came into existence to perform operations on fuzzy observations. The distinction between generalised fuzzy and classic fuzzy arithmetic operations is that the former can handle both non-normal and normalised fuzzy, while the latter can only handle normalised fuzzy.

The goal of this research is to give a broad overview of current techniques in this field. The methodology reported in this article focuses on improving the arithmetic process in a fuzzy environment. The current arithmetic operations take into account the same degree of precision with specific fuzzy numbers, it is found that the lack of knowledge is responsible for incorrect performance. To avoid and maintain the uniformity of the fuzzy numbers, an improved operator like adding, scaling, subtracting, multiplication has been derived for generalized trapezoidal (triangular), sigmoidal and parabolic fuzzy numbers.

Appendices

Appendix A

Proof—Assume P and Q are two generalised fuzzified values with varying levels of confidence, such that $\varepsilon_{P} { < }\varepsilon_{Q}$. Take $\varepsilon = \min \left( {\varepsilon_{P} ,\varepsilon_{Q} } \right)$ i.e. ε = ε_P then α− cut of P and Q are.

$$P_{\alpha } = \left[ {p_{1} + \alpha \left( {\frac{{p_{2 } - p_{1 } }}{{\varepsilon_{P} }}} \right), p_{4} - \alpha \left( {\frac{{p_{4 } - p_{3 } }}{{\varepsilon_{P} }}} \right)} \right] \, ; \, \forall_{\alpha } \in \left[ {0, \varepsilon_{P} } \right] s.t. 0 \le \varepsilon_{P} \le 1$$

$$Q_{\alpha } = \left[ {q_{1} + \alpha \left( {\frac{{q_{2}^{*} - q_{1} }}{\varepsilon }} \right), q_{4} - \alpha \left( {\frac{{q_{4 } - q_{3}^{*} }}{\varepsilon }} \right)} \right] \, ; \, \forall_{\alpha } \in \left[ {0, \varepsilon } \right] s.t. 0 \le \varepsilon \le 1$$

Let, R=P+Q={x | x ∈ R_α} for all α ∈ [0, ε]. Here R_α=[R^L(α), R^U(α)] be its α− cuts such that, ${\text{R}}^{{\text{L}}} \, \, \left( { \, \alpha \, } \right) = {\text{P}}^{{\text{L}}} \left( { \, \alpha \, } \right) + {\text{Q}}^{{\text{L}}} \left( { \, \alpha \, } \right){\text{, and R}}^{{\text{U}}} \left( { \, \alpha \, } \right) = {\text{P}}^{{\text{U}}} \left( { \, \alpha \, } \right) + {\text{Q}}^{{\text{U}}} \left( { \, \alpha \, } \right)$ i.e.

$$R_{\alpha } = \left[ { P^{L} \left( \alpha \right) + Q^{L} \left( \alpha \right), P^{U} \left( \alpha \right) + Q^{U} \left( \alpha \right)} \right]$$

(57)

$$= \left[ {p_{1} + \alpha \left( {\frac{{p_{2 } - p_{1 } }}{{\varepsilon_{P} }}} \right) + q_{1} + \alpha \left( {\frac{{q_{2}^{*} - q_{1} }}{\varepsilon }} \right), p_{4} - \alpha \left( {\frac{{p_{4 } - p_{3 } }}{{\varepsilon_{P} }}} \right) + q_{4} - \alpha \left( {\frac{{q_{4 } - q_{3}^{*} }}{\varepsilon }} \right)} \right]$$

$$= \left[ { p_{1} + q_{1} + \alpha \left( {\frac{{p_{2 } - p_{1 } }}{{\varepsilon_{P} }} + \frac{{q_{2}^{*} - q_{1} }}{\varepsilon }} \right), p_{4} + q_{4} - \alpha \left( {\frac{{p_{4 } - p_{3 } }}{{\varepsilon_{P} }} + \frac{{q_{4 } - q_{3}^{*} }}{\varepsilon }} \right)} \right]$$

Now,

$p_{1} + q_{1} + \alpha \left( {\frac{{p_{2 } - p_{1 } }}{{\varepsilon_{P} }} + \frac{{q_{2}^{*} - q_{1} }}{\varepsilon }} \right) - x = 0$ and

$$p_{4} + q_{4} - \alpha \left( {\frac{{p_{4 } - p_{3 } }}{{\varepsilon_{P} }} + \frac{{q_{4} - q_{3}^{*} }}{\varepsilon }} \right) - x = 0$$

As a result, R’s left and right membership functions are.

$${\text{f}}\frac{{\text{L}}}{{\text{R}}}\left( {\text{x}} \right) = \frac{{{\text{x}} - {\text{ p}}_{{1{ }}} - {\text{ q}}_{{1{ }}} { }}}{{\frac{{{\text{p}}_{{2{ }}} { } - {\text{ p}}_{{1{ }}} }}{{{\upvarepsilon }_{{\text{P}}} }}{ } - { }\frac{{{\text{q}}_{2}^{*} { } - {\text{q}}_{1} }}{{\upvarepsilon }}}}$$

(58)

$${\text{f}}\frac{{\text{L}}}{{\text{R}}}\left( {\text{x}} \right) = \frac{{{\text{p}}_{{4}} + { }_{{4}} - {\text{x}}}}{{\frac{{{\text{p}}_{{4 }} - {\text{p}}_{{3 }} }}{{{\upvarepsilon }_{{\text{P}}} }} + { }\frac{{{\text{q}}_{{4 }} - {\text{q}}_{{3}}^{*} }}{{\upvarepsilon }}}}$$

(59)

Since $\varepsilon = \varepsilon_{P}$ and $q_{2}^{*} = q_{1} + \varepsilon \left( {\frac{{q_{2} - q_{1} }}{{\varepsilon_{Q} }}} \right)$ and $q_{3}^{*} = q_{4} - \varepsilon \left( {\frac{{q_{4} - q_{3} }}{{\varepsilon_{Q} }}} \right)$, Thus, above ${\text{f}}_{{\text{R}}}^{{\text{L}}}$ and ${\text{f}}_{{\text{R}}}^{{\text{R}}}$ becomes,

$$f\frac{L}{R}\left( x \right) = \frac{{x - p_{1 } - q_{1 } }}{{\frac{{p_{2 } - p_{1 } }}{{\varepsilon_{P} }} - \frac{{q_{2}^{*} - q_{1} }}{\varepsilon }}}$$

$$= \varepsilon \left( {\frac{{x - p_{1} - q_{1 } }}{{p_{2 } - p_{1} - q_{1} + q_{2}^{*} }}} \right)$$

$$= \varepsilon \left( { \frac{{x - ( p_{1 } + q_{1} )}}{{p_{2 } + q_{1} + \varepsilon \left( { \frac{{q_{2} - q_{1} }}{{\varepsilon_{Q} }} } \right) - ( p_{1} + q_{1} )}} } \right)$$

(60)

For, $p_{1} + q_{1} \le x\, \le \,p_{2} + q_{1} + \varepsilon \left( { \frac{{q_{2} - q_{1} }}{{\varepsilon_{Q} }} } \right)$Similarly,

$$f_{R}^{R} \left( x \right) = \varepsilon \left( { \frac{{ ( p_{4} + q_{4} ) - x}}{{q_{4 } + p_{3} - \varepsilon \left( { \frac{{q_{2} - q_{3} }}{{\varepsilon_{Q} }} } \right) - ( p_{4} + q_{4} )}} } \right)$$

(61)

For, $q_{4} + p_{3} - \varepsilon \left( { \frac{{q_{2} - q_{3} }}{{\varepsilon_{Q} }} } \right) \le x\, \le \,( p_{4} + q_{4} )$

Thus, adding two generalized fuzzy values is another generalized fuzzy, whose membership function is explained as,

$$\mu_{R} \left( x \right) = \left\{ \begin{gathered} \varepsilon \left( { \frac{{x - ( p_{1 } + q_{1} )}}{{p_{2 } + q_{1} + \varepsilon \left( { \frac{{q_{2} - q_{1} }}{{\varepsilon_{Q} }} } \right) - ( p_{1} + q_{1} )}} } \right) ;\quad p_{1} + q_{1} \le x \le p_{2 } + q_{1} + \varepsilon \left( { \frac{{q_{2} - q_{1} }}{{\varepsilon_{Q} }} } \right) \hfill \\ \varepsilon ;\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad p_{2 } + q_{1} + \varepsilon \left( { \frac{{q_{2} - q_{1} }}{{\varepsilon_{Q} }} } \right) \le x \le q_{4 } + p_{3} - \varepsilon \left( { \frac{{q_{2} - q_{3} }}{{\varepsilon_{Q} }} } \right) \hfill \\ \varepsilon \left( { \frac{{ ( p_{4} + q_{4} ) - x}}{{q_{4 } + p_{3} - \varepsilon \left( { \frac{{q_{2} - q_{3} }}{{\varepsilon_{Q} }} } \right) - ( p_{4} + q_{4} )}} } \right) ;\quad q_{4 } + p_{3} - \varepsilon \left( { \frac{{q_{2} - q_{3} }}{{\varepsilon_{Q} }} } \right) \le x \le ( p_{4} + q_{4} ) \hfill \\ 0; \quad \quad \quad \quad \quad \quad {\text{otherwise}} \hfill \\ \end{gathered} \right.$$

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To put in another way, that’s the addition of two generalised fuzzy numbers, given by:

R = P + Q = (${\mathrm{r}}_{1}, {\mathrm{ r}}_{2}, {\mathrm{ r}}_{3},{\mathrm{ r}}_{4} ;\upvarepsilon )=$ min (${\upvarepsilon }_{\mathrm{P}}$,${\upvarepsilon }_{\mathrm{Q}}$) is a generalized fuzzy number where,

$$r_{1} = p_{1} \, + \,q_{1}$$

$$r_{2} = q_{1} + p_{2} \, + \,\frac{{\varepsilon \left( {q_{2} - q_{1} } \right) }}{{\varepsilon_{Q} }}$$

$$r_{3} = q_{4} \, + \,p_{3} - \frac{{\varepsilon \left( {q_{4} - q_{3} } \right) }}{{\varepsilon_{Q} }}$$

$$r_{4} = p_{4} \, + \,q_{4}$$

Appendix B

Proof—When k > 0, the α − cut for P’s membership function is.

${\text{P}}_{{\upalpha }} = [{\text{p}}_{1} +$α($\frac{{{\text{p}}_{{2{ }}} { } - {\text{ p}}_{{1{ }}} }}{{{\upvarepsilon }_{{\text{P}}} }}$), ${\text{p}}_{4} -$α($\frac{{{\text{p}}_{{4{ }}} { } - {\text{ p}}_{{3{ }}} }}{{{\upvarepsilon }_{{\text{P}}} }}$)] ; $\forall_{{\upalpha }}$ ∈ [0, ${\upvarepsilon }_{{\text{P}}}$] s.t. 0 ≤ ${\upvarepsilon }_{{\text{P}}}$  ≤ 1

This is,therefore,

$$x \in \left[ {p_{1} + \alpha \left( {\frac{{p_{2} - p_{1} }}{{\varepsilon_{p} }}} \right),p4 - \alpha \left( {\frac{{p_{4} - p_{3} }}{{\varepsilon_{p} }}} \right)} \right]$$

$$y = kx \in \left[ { kp_{1} + \alpha \left( { \frac{{kp_{2 } - kp_{1 } }}{{\varepsilon_{P} }} } \right), kp_{4} - \alpha \left( { \frac{{kp_{4} - kp_{3 } }}{{\varepsilon_{P} }} } \right) } \right]$$

(63)

As a conclusion, the scalar product’s membership function is:

$${\upmu }_{{{\text{kP}}}} \left( {\text{x}} \right) = \left\{ \begin{gathered} {\upvarepsilon }_{{\text{P }}} \left( {{ }\frac{{{\text{x}} - {\text{kp}}_{{1 }} }}{{{\text{kp}}_{{2 }} - {\text{kp}}_{{1 }} }}{ }} \right)\,{;}\quad {\text{if kp}}_{{1 }} \le {\text{x }} \le {\text{ kp}}_{{2 }} \hfill \\ {\upvarepsilon }_{{\text{P }}} ;\quad \quad \quad \quad \quad \quad \quad {\text{if kp}}_{{2 }} \le {\text{x}} \le {\text{kp}}_{{3 }} \hfill \\ {\upvarepsilon }_{{\text{P }}} \left( {{ }\frac{{{\text{kp}}_{{4 }} - {\text{x}}}}{{{\text{kp}}_{{4 }} - {\text{kp}}_{{3 }} }}{ }} \right){;}\quad {\text{if kp}}_{{3 }} \le {\text{x}} \le {\text{kp}}_{{4 }} \hfill \\ {0;}\quad \quad \quad \quad \quad \quad \quad \,{\text{otherwise}} \hfill \\ \end{gathered} \right.$$

Appendix C

Proof—Because there are two distinct fuzzy numbers with confidence levels, ${\upvarepsilon }_{\mathrm{P}}$ and ${\upvarepsilon }_{\mathrm{q}}$ defined as ${\upvarepsilon }_{\mathrm{P}}$ ≤${\upvarepsilon }_{\mathrm{q}}$. So, first of all we’ll turn the fuzzy Q into ${\text{Q*}} = {\text{Q}} = (q_{1} , q_{2}^{*} , q_{3}^{*} , q_{4} ; \varepsilon$, where.

$q_{2}^{*} = q_{1} + \varepsilon \left( {\frac{{q_{2 } - q_{1 } }}{{\varepsilon_{Q} }}} \right){\text{and }}q_{3}^{*} = q_{4} - \varepsilon \left( {\frac{{q_{4 } - q_{3 } }}{{\varepsilon_{Q} }}} \right)$, Now the α − cuts that suit P and Q* are equivalent to.

$$P_{\alpha } = \left[ {p_{1} + \alpha \left( {\frac{{p_{2} - p_{1 } }}{{\varepsilon_{P} }}} \right), p_{4} - \alpha \left( {\frac{{p_{4} - p_{3 } }}{{\varepsilon_{P} }}} \right)} \right]\quad ;\quad \forall_{\alpha } \in \left[ {0, \, \varepsilon_{P} } \right] \, s.t. \, 0 \le \varepsilon_{P} \le 1$$

$$Q_{\alpha }^{*} = \left[ {q_{1} + \alpha \left( {\frac{{q_{2}^{*} - q_{1 } }}{\varepsilon }} \right), q_{4} - \alpha \left( {\frac{{q_{4} - q_{3}^{*} }}{\varepsilon }} \right)} \right]\quad ;\quad \forall_{\alpha } \in \left[ {0, \, \varepsilon_{P} } \right] \, s.t. \, 0 \le \varepsilon \le 1$$

Suppose $R = P \times Q = \left\{ {x| x \in R_{\alpha } } \right\}$ for all ${\upalpha } \in \left[ {0, \, \varepsilon } \right]$. Here $R_{\alpha } = \left[ {R_{\alpha }^{L} , R_{\alpha }^{U} } \right]$ be its α − cuts such that $R_{\alpha }^{L} = P_{\alpha }^{L} Q_{\alpha }^{*L}$ and $R_{\alpha }^{U} = P_{\alpha }^{U} Q_{\alpha }^{*U}$ i.e.

$$R_{\alpha } = \left[ {P_{\alpha }^{L} Q_{\alpha }^{*L} , P_{\alpha }^{U} Q_{\alpha }^{*U} } \right]$$

$$= \left[ {\left\{ {p_{1} + \alpha \left( {\frac{{p_{2} - p_{1 } }}{{\varepsilon_{P} }}} \right)} \right\}, \left\{ {q_{1} + \alpha \left( {\frac{{q_{2}^{*} - q_{1 } }}{\varepsilon }} \right)} \right\}, \left\{ {p_{4} - \alpha \left( {\frac{{p_{4 } - p_{3 } }}{{\varepsilon_{P} }}} \right)} \right\} \left\{ {q_{4} - \alpha \left( {\frac{{q_{4} - q_{3}^{*} }}{\varepsilon }} \right)} \right\}} \right]$$

$$= \left[ {p_{1 } q_{1} + \alpha \left( {\frac{{p_{1 } \left( {q_{2}^{*} - q_{1} } \right) + q_{1 } \left( {p_{2 } - p_{1} } \right)}}{\varepsilon }} \right) + \frac{{\alpha^{2} }}{{\varepsilon^{2} }} \left( {p_{2} - p_{1} } \right) \left( {q_{2}^{*} - q_{1} } \right)} \right]$$

$$\left[ {p_{4 } q_{4} + \alpha \left( {\frac{{p_{4 } \left( { q_{4} - q_{3}^{*} } \right) + q_{1 } \left( { p_{4} - p_{3 } } \right)}}{\varepsilon }} \right) + \frac{{\alpha^{2} }}{{\varepsilon^{2} }} \left( {p_{4} - p_{3} } \right)\left( {q_{4} - q_{3}^{*} } \right)} \right]$$

Therefore,

$$p_{1 } q_{1} + \alpha \left( {\frac{{p_{1 } \left( { q_{2}^{*} - q_{1 } } \right) + q_{1 } \left( { p_{2 } - p_{1 } } \right)}}{\varepsilon }} \right) + \frac{{\alpha^{2} }}{{\varepsilon^{2} }} \left( {p_{2 } - p_{1} } \right)\left( {q_{2}^{*} - q_{1} } \right) - x = {\text{0, and}}$$

$p_{4 } q_{4} + \alpha \left( {\frac{{p_{4 } \left( { q_{4 } - q_{3}^{*} } \right) + q_{1 } \left( { p_{4 } - p_{3 } } \right)}}{\varepsilon }} \right) + \frac{{\alpha^{2} }}{{\varepsilon^{2} }} \left( {p_{4} - p_{3} } \right)\left( {q_{4} - q_{3}^{*} } \right) - x = 0$

Which are the quadratic functions in the α and therefore its roots give the left as well as right membership of R,

$$f_{R}^{L} \left( x \right) = \frac{{ - N_{1} + \sqrt {N_{1}^{2} + 4M_{1} \left( { x - J_{1} } \right)} }}{{2M_{1} }}; \, r_{1} \le x \le r_{2}$$

(64)

$$f_{R}^{U} \left( x \right) = \frac{{ - N_{2} + \sqrt {N_{2}^{2} + 4M_{2} \left( { x - J_{2} } \right)} }}{{2M_{2} }}; \, r_{3} \le x \le r_{4}$$

(65)

Where, $N_{1} = \frac{{p_{1 } \left( { q_{2}^{*} - q_{1 } } \right) + q_{1 } \left( { p_{2 } - p_{1 } } \right)}}{\varepsilon }$, $N_{2} = \frac{{p_{4 } \left( { q_{4 } - q_{3}^{*} } \right) + q_{1 } \left( { p_{4 } - p_{3 } } \right)}}{\varepsilon }$, $M_{1} = \frac{{\left( { p_{2 } - p_{1 } } \right) \left( { q_{2}^{*} - q_{1 } } \right) }}{{\varepsilon^{2} }}$, $M_{2} = \frac{{\left( { p_{4 } - p_{3 } } \right) \left( {q_{4 } - q_{3}^{*} } \right) }}{{\varepsilon^{2} }}$, $J_{1} = p_{1 } q_{1} , J_{2} = p_{4 } q_{4}$

By Substituting the value of ${\mathrm{q}}_{2}^{*}$ and ${\mathrm{q}}_{3}^{*}$, we get.

$$N_{1} = \frac{{\left( {p_{2} - p_{1 } } \right) \left( { q_{2} - q_{1 } } \right)}}{{\varepsilon \varepsilon_{Q} }} \, ; \, N_{2} = \frac{{\left( {p_{4} - p_{3 } } \right) \left( { q_{4} - q_{3 } } \right)}}{{\varepsilon \varepsilon_{Q} }}$$

(66)

$$M_{1} = \frac{{ p_{1 } \left( { q_{2} - q_{1 } } \right)}}{{\varepsilon_{Q} }} + \frac{{q_{1 } \left( { p_{2} - p_{1 } } \right) }}{\varepsilon } \, ; \, M_{2} = \frac{{ p_{4 } \left( { q_{4} - q_{3 } } \right)}}{{\varepsilon_{Q} }} + \frac{{q_{4 } \left( { p_{4} - p_{3 } } \right) }}{\varepsilon }$$

(67)

As an outcome, the product of two generalised fuzzy numbers is a fuzzy number with the membership,

$$\mu_{R} \left( x \right) = \left\{ {\begin{array}{*{20}c} {\frac{{ - N_{1} + \sqrt {N_{1}^{2} + 4M_{1} \left( { x - J_{1} } \right)} }}{{2M_{1} }}\quad ;\quad r_{1 } \le x \le r_{2 } } \\ {\varepsilon \quad \quad \quad \quad \quad \quad \quad ; \quad r_{2 } \le x \le r_{3 } } \\ {\frac{{ - N_{2} + \sqrt {N_{2}^{2} + 4M_{2} \left( { x - J_{2} } \right)} }}{{2M_{2} }}\quad ; \quad r_{3 } \le x \le r_{4 } } \\ \end{array} } \right.$$

(68)

where,

$$r_{1} = p_{1} q_{1}$$

$$r_{2} = \frac{{\varepsilon \left( {p_{2} q_{2} - p_{2} q_{1} } \right) }}{{\varepsilon_{Q} }} + p_{2} q_{1}$$

$$r_{3} = \frac{{\varepsilon \left( {p_{3} q_{3} - p_{3} q_{4} } \right) }}{{\varepsilon_{Q} }} + p_{3} q_{4}$$

$$r_{4} = p_{4} q_{4}$$

Appendix D

Proof—Consider the membership of two parabolic fuzzy numbers, X and Y be.

$$\mu_{X} \left( x \right) = \left\{ {\begin{array}{*{20}c} {\varepsilon_{1} L_{1} \left( x \right){\text{; if }}p_{1 } \le x < p_{2,} } \\ {\varepsilon_{1} {\text{; if }}x = p_{2,} } \\ {\varepsilon_{1} R_{1} \left( x \right){\text{; if }}p_{2} \le x < p_{3} } \\ { \quad \quad 0{;}\quad {\text{if otherwise}}} \\ \end{array} } \right.$$

And,

$$\mu_{Y} \left( y \right) = \left\{ {\begin{array}{*{20}c} {\varepsilon_{1} L_{1} \left( y \right){;}\quad {\text{if }}q_{1 } \le y < q_{2,} } \\ {\varepsilon_{1} {;}\quad {\text{if }}y = q_{2,} } \\ {\varepsilon_{1} R_{1} \left( y \right){;}\quad {\text{if }}q_{2} \le y < q_{3} } \\ { \quad 0{;}\quad \quad {\text{if otherwise}}} \\ \end{array} } \right.$$

To add the fuzzy numbers ${\text{X}}$ and ${\text{Y}}$, the next fuzzy number of them, ${\text{Z}} = {\text{X}} + {\text{Y}}$

Implies $Z = \left[ {p_{1} + q_{1} ,p_{2} + q_{2} ,p_{3} + q_{3} } \right]$

As $z = x + y$ at $y = \varphi_{1} \left( {\text{x}} \right)$ and $\varphi_{2} \left( {\text{x}} \right)$ we get,

$z = x + \varphi_{1} \left( x \right)$ and ${\text{z}} = {\text{x}} + \varphi_{2} \left( {\text{x}} \right)$ respectively.that specifies that $x = \varphi_{1} \left( z \right)$ and $x = \varphi_{2} \left( z \right)$ where

$$x = \varphi_{1} \left( z \right) = \frac{{z - \frac{{p_{2} q_{1} }}{{p_{2} - p_{1} }} + \frac{{p_{1} q_{2} }}{{p_{2} - p_{1} }}}}{{1 + \frac{{ \left( {q_{2} - q_{1} } \right)}}{{p_{2} - p_{1} }}}}$$

(69)

Hence, $\eta_{1} \left( z \right) = \left( {\frac{2}{{ (p_{2} - p_{1} )^{2} }}} \right)\left( {\frac{{z - p_{1} - q_{1} }}{{1 + \frac{{\left( {q_{2} - q_{1} } \right)}}{{p_{2} - p_{1} }}}}} \right)$, $m_{1} \left( z \right) = 1 + \frac{{\left( {q_{2} - q_{1} } \right)}}{{p_{2} - p_{1} }}$so, the left sided distribution function

$$\mathop \smallint \limits_{{p_{1} + q_{1} }}^{X} \eta_{1} \left( z \right)m_{1} \left( z \right)dz = \mathop \smallint \limits_{{p_{1} + q_{1} }}^{X} \left( {\frac{2}{{ (p_{2} - p_{1} )^{2} }}} \right) \left( {\frac{{z - p_{1} - q_{1} }}{{1 + \frac{{ \left( {q_{2} - q_{1} } \right)}}{{p_{2} - p_{1} }}}}} \right) \times \left( {\frac{1}{{1 + \frac{{ \left( {q_{2} - q_{1} } \right)1}}{{p_{2} - p_{1} }}}}} \right)dz$$

$$= \left( {\frac{2}{{ (p_{2} - p_{1} )^{2} }}} \right) \left( {\frac{1}{{1 + \frac{{ \left( {q_{2} - q_{1} } \right)}}{{p_{2} - p_{1} }}}})^{2} \times \mathop \smallint \limits_{{p_{1} + q_{1} }}^{X} (z - p_{1} - q_{1} } \right)dz = (\frac{1}{{p_{2} - p_{1} }})^{2} (\frac{{x - \left( {p_{1} + q_{1} } \right)}}{{1 + \frac{{\left( {q_{2} - q_{1} } \right)}}{{p_{2} - p_{1} }}}})^{2}$$

$$= (\frac{{x - \left( {p_{1} + q_{1} } \right)}}{{p_{2} + q_{2} - p_{1} - q_{1} }})^{2} \, ; \, p_{1} + q_{1} \le x < p_{2} + q_{2}$$

Similarly, if ${\text{y}} = \phi_{2} { }\left( {\text{x}} \right)$ then ${\text{z}} = {\text{x}} + {\text{y}}$ becomes ${\text{x}} = \varphi_{2} { }\left( {\text{z}} \right)$ where

$$\varphi_{2} \left( x \right) = \frac{{z + \frac{{ - p_{2} q_{3} }}{{p_{3} - p_{2} }} + \frac{{p_{3} q_{2} }}{{p_{3} - p_{2} }}}}{{1 + \frac{{q_{3} - q_{2} }}{{p_{3} - p_{2} }}}}$$

(70)

Here, in this case

$$\eta_{2} \left( x \right) = \left( {\frac{ - 2}{{ (p_{3} - p_{2} )^{2} }}} \right) \left( {\frac{{p_{3} + q_{3} - z}}{{1 + \frac{{ \left( {q_{3} - q_{2} } \right)}}{{p_{3} - p_{2} }}}}} \right)$$

(71)

so, the right sided distribution function,

$$\mathop \smallint \limits_{{p_{3} + q_{3} }}^{X} \eta_{2} \left( z \right)m_{2} \left( z \right)dz = \mathop \smallint \limits_{{p_{3} + q_{3} }}^{X} \left( {\frac{ - 2}{{ (p_{3} - p_{2} )^{2} }}} \right) \left( {\frac{{p_{3} + q_{3} - z}}{{1 + \frac{{ \left( {q_{3} - q_{2} } \right)}}{{p_{3} - p_{2} }}}}} \right) \times \left( {\frac{1}{{1 + \frac{{ \left( {q_{3} - q_{2} } \right)}}{{p_{3} - p_{2} }}}}} \right)dz$$

(72)

$$= \left( {\frac{ - 2}{{ (p_{3} - p_{2} )^{2} }}} \right) \left( {\frac{1}{{1 + \frac{{ \left( {q_{3} - q_{2} } \right)}}{{p_{3} - p_{2} }}}})^{2} \times \mathop \smallint \limits_{{p_{3} + q_{3} }}^{X} (p_{3} + q_{3} - z} \right)dz$$

$$= \left( {\frac{ - 2}{{ (p_{3} - p_{2} )^{2} }}} \right) (\frac{1}{{1 + \frac{{ \left( {q_{3} - q_{2} } \right)}}{{p_{3} - p_{2} }}}})^{2} \times \frac{{ (p_{3} + q_{3} - x)^{2} }}{ - 2}$$

$$= (\frac{1}{{p_{3} - p_{2} }})^{2} (\frac{{ \left( {p_{3} + q_{3} } \right) - x}}{{1 + \frac{{ \left( {q_{3} - q_{2} } \right)}}{{p_{3} - p_{2} }}}})^{2}$$

$$= (\frac{{ \left( {p_{3} + q_{3} } \right) - x}}{{p_{3} - p_{2} + q_{3} - q_{2} }})^{2} ; p_{2} + q_{2} \le x < p_{3} + q_{3}$$

(73)

so, the membership functions of the fuzzy variable ${\text{Z}} = {\text{X}} + {\text{Y}}$ is,

$$\mu_{Z} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\varepsilon (\frac{{x - \left( {p_{1} + q_{1} } \right)}}{{p_{2} - p_{1} + q_{2} - q_{1} }})^{2} } \hfill & {;\quad p_{1} + q_{1} \le x < p_{2} + q_{2} } \hfill \\ \varepsilon \hfill & {;\quad x = p_{2} + q_{2} } \hfill \\ {\varepsilon (\frac{{ \left( {p_{3} + q_{3} } \right) - x}}{{p_{3} - p_{2} + q_{3} - q_{2} }})^{2} } \hfill & {;\quad p_{2} + q_{2} \le x < p_{3} + q_{3} } \hfill \\ 0 \hfill & {;\quad otherwise} \hfill \\ \end{array} } \right.$$

(74)

where $\upvarepsilon =\mathrm{ min }({\upvarepsilon }_{1}, {\upvarepsilon }_{2})$

Appendix E

Proof—Using the change z = kx, we receive x = z/k as well as x = z/k and thus $\varphi { }\left( {\text{z}} \right) = \frac{{\text{z}}}{{\text{k}}}$

so, $\left| {\frac{dx}{{dz}}} \right| = \frac{1}{k} = m \left( z \right)$.

Therefore,

$$\mathop \smallint \limits_{{kp_{1} }}^{X} \eta_{1} \left( z \right)m \left( z \right)dz = \mathop \smallint \limits_{{kp_{1} }}^{X} (\frac{{2 \left( {z - kp_{1} } \right)}}{{k (p_{2} - p_{1} )^{2} }}) \left( \frac{1}{k} \right)dz = (\frac{{x - kp_{1} }}{{kp_{2} - kp_{1} }})^{2}$$

(75)

$$\mathop \smallint \limits_{{kp_{3} }}^{X} \eta_{2} \left( z \right)m \left( z \right)dz = \mathop \smallint \limits_{{kp_{3} }}^{X} (\frac{{ - 2 \left( {kp_{3} - z} \right)}}{{k (p_{3} - p_{2} )^{2} }}) \left( \frac{1}{k} \right)dz = (\frac{{kp_{3} - x}}{{kp_{3} - kp_{2} }})^{2}$$

Therefore, $\mathrm{at}$ $\mathrm{k}>0$ is

$$\mu_{kX} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\varepsilon_{1} (\frac{{x - kp_{1} }}{{kp_{2} - kp_{1} }})^{2} } \hfill & {;\quad kp_{1} \le x < kp_{2} } \hfill \\ {\varepsilon_{1} } \hfill & {;\quad x = kp_{2} } \hfill \\ {\varepsilon_{1} (\frac{{kp_{3} - x}}{{kp_{3} - kp_{2} }})^{2} } \hfill & {;\quad kp_{2} \le x < kp_{3} } \hfill \\ 0 \hfill & {;\quad otherwise} \hfill \\ \end{array} } \right.$$

(76)

Also, at ${\text{k}} < 0$ is

$$\mu_{kX} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\varepsilon_{1} (\frac{{x - kp_{3} }}{{kp_{2} - kp_{3} }})^{2} } \hfill & {;\quad kp_{3} \le x < kp_{2} } \hfill \\ {\varepsilon_{1} } \hfill & {;\quad x = kp_{2} } \hfill \\ {\varepsilon_{1} (\frac{{kp_{1} - x}}{{kp_{1} - kp_{2} }})^{2} } \hfill & {;\quad kp_{2} \le x < kp_{1} } \hfill \\ 0 \hfill & {;\quad otherwise} \hfill \\ \end{array} } \right.$$

(77)

Appendix F

Proof—Because X and Y have a parabolic membership,

$$\mu_{x} \left( x \right) = \left\{ {\begin{array}{*{20}c} {\varepsilon_{1} L_{1} \left( x \right)\quad ;\quad if \, p_{1 } \le x < p_{2,} } \\ {\varepsilon_{1} \quad ;\quad if \, x = p_{2.} } \\ {\varepsilon_{1} R_{1} \left( x \right)\quad ;\quad if \, p_{2} \le x < p_{3} } \\ { 0\quad \quad \quad ;\quad if \, otherwise} \\ \end{array} } \right.$$

And,

$$\mu_{Y} \left( y \right) = \left\{ {\begin{array}{*{20}c} {\varepsilon_{1} L_{1} \left( y \right) \quad ;\quad if\quad q_{1 } \le y < q_{2,} } \\ {\varepsilon_{1} ; \quad if\quad y = q_{2,} } \\ {\varepsilon_{1} R_{1} \left( y \right)\quad ; \quad if\quad q_{2} \le y < q_{3} } \\ { 0\quad \quad \quad ;\quad if\quad otherwise} \\ \end{array} } \right.$$

Therefore, to find the membership functions${\text{Z}} = {\text{XY}}$

at ${\text{y}} = \emptyset_{1} { }\left( {\text{x}} \right),$ ${\text{z}} = {\text{xy}}$ gives

$$x = \frac{{ \left( {p_{1} q_{2} - p_{2} q_{1} } \right) \pm \sqrt { (p_{1} q_{2} - p_{2} q_{1} )^{2} + 4 \left( {q_{2} - q_{1} } \right) \left( {p_{2} - p_{1} } \right)z} }}{{2 \left( {q_{2} - q_{1} } \right)}} = \varphi_{1} \left( z \right)$$

(78)

Take,

$$P_{1} = \left( {p_{2} - p_{1} } \right) \left( {q_{2} - q_{1} } \right)$$

$$Q_{1} = p_{1} \left( {q_{2} - q_{1} } \right) + q_{1} \left( {p_{2} - p_{1} } \right)$$

$$R_{1} = p_{1} q_{1}$$

Hence,

$$\eta_{1} \left( z \right) = \frac{2}{{ (p_{2} - p_{1} )^{2} }} \times \left[ {\frac{{ - p_{1} q_{2} - p_{2} q_{1} + 2p_{1} q_{1} + \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} }}{{2 \left( {q_{2} - q_{1} } \right)}}\left] { = \frac{1}{{p_{2} - p_{1} }}} \right[\frac{{ - Q_{1} + \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} }}{{P_{1} }}} \right]$$

$$m \left( z \right) = \left| {\frac{dx}{{dz}}} \right| = \frac{{p_{2} - p_{1} }}{{\sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} }}$$

(79)

therefore,

$$\mathop \smallint \limits_{{p_{1} b_{1} }}^{X} \eta_{1} \left( z \right)m_{1} \left( z \right)dz = \mathop \smallint \limits_{{p_{1} b_{1} }}^{X} \frac{1}{{p_{2} - p_{1} }}\left[ {\frac{{ - Q_{1} + \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} }}{{P_{1} }}} \right] \times \frac{{p_{2} - p_{1} }}{{\sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} }}dz = \mathop \smallint \limits_{{p_{1} b_{1} }}^{X} \frac{1}{{P_{1} }}\left[ {\frac{{ - Q_{1} + \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} }}{{\sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - x} \right)} }}} \right]dz$$

$$= \mathop \smallint \limits_{{p_{1} b_{1} }}^{X} \frac{1}{{P_{1} }}\left[ {\frac{{ - Q_{1} }}{{\sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} }} + 1\left] {dz = \frac{1}{{P_{1} }}} \right[\frac{{ ( - Q_{1} )^{2} - Q_{1} \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - x} \right)} + 2P_{1} x - 2P_{1} R_{1} }}{{2P_{1} }}} \right]$$

$$= [\frac{{ - Q_{1} + \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - x} \right)} }}{{2P_{1} }}]^{2} \, ; \, p_{1} q_{1} \le x < p_{2} q_{2}$$

(80)

Similarly, by taking.

$$P_{2} = \left( {p_{3} - p_{2} } \right) \left( {q_{3} - q_{2} } \right)$$

$$Q_{2} = - p_{3} \left( {q_{3} - q_{2} } \right) - q_{3} \left( {p_{3} - p_{2} } \right)$$

$$R_{2} = p_{3} q_{3}$$

the membership function for the corresponding dispersal functions as.

$$\mathop \smallint \limits_{{p_{3} q_{3} }}^{X} \eta_{1} \left( z \right)m_{1} \left( z \right)dz = [\frac{{ - Q_{2} + \sqrt {Q_{2}^{2} - 4P_{2} \left( {R_{2} - x} \right)} }}{{2P_{2} }}]^{2} ; p_{2} q_{2} \le x < p_{3} q_{3}$$

(81)

As a result, the fuzzy variable Z’s membership is given by:

$$\mu_{XY} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\varepsilon (\frac{{ - Q_{1} + \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - x} \right)} }}{{2P_{1} }})^{2} } \hfill & {p_{1} q_{1} \le x < p_{2} q_{2} } \hfill \\ \varepsilon \hfill & {x = p_{2} q_{2} } \hfill \\ {\varepsilon (\frac{{ - Q_{2} + \sqrt {Q_{2}^{2} - 4P_{2} \left( {R_{2} - x} \right)} }}{{2P_{2} }})^{2} } \hfill & {p_{2} q_{2} \le x < p_{3} q_{3} } \hfill \\ 0 \hfill & {otherwise} \hfill \\ \end{array} } \right.$$

(82)

where, Z = XY.

Appendix G

Proof—let a fuzzy inconstant be $X = \left[ {p_{1} , p_{2} , p_{3} , \varepsilon_{1} } \right]$ whose membership is given in.

$$\mu_{x} \left( x \right) = \left\{ {\begin{array}{*{20}c} {\varepsilon_{1} L_{1} \left( x \right) ;\quad if\quad p_{1 } \le x < p_{2,} } \\ {\varepsilon_{1} ; \quad if\quad x = p_{2,} } \\ {\varepsilon_{1} R_{1} \left( x \right) ; \quad if\quad p_{2} \le x < p_{3} } \\ { \quad 0\quad ; \quad if\quad otherwise} \\ \end{array} } \right.$$

Then let ${\text{z}} = \frac{1}{{\text{X}}}$ so that $\left| {\frac{{{\text{dx}}}}{{{\text{dz}}}}} \right| = \frac{1}{{{\text{z}}^{2} }}$. So, for ${\text{X}}^{ - 1}$

$$\mathop \smallint \limits_{X}^{{p_{1}^{ - 1} }} \eta_{1} \left( z \right)m \left( z \right)dz = \mathop \smallint \limits_{X}^{{p_{1}^{ - 1} }} (\frac{2}{{ (p_{2} - p_{1} )^{2} }} \left( {\frac{1}{z} - p_{1} } \right)) \left( {\frac{1}{{z^{2} }}} \right)dz = (\frac{{1 - p_{{1^{X} }} }}{{x \left( {p_{2} - p_{1} } \right)}})^{2}$$

And,

$$\mathop \smallint \limits_{{p_{3}^{ - 1} }}^{X} \eta_{2} \left( z \right)m \left( z \right)dz = \mathop \smallint \limits_{{p_{3}^{ - 1} }}^{X} (\frac{2}{{ (p_{3} - p_{2} )^{2} }} \left( {\frac{1}{z} - p_{3} } \right)) \left( {\frac{1}{{z^{2} }}} \right)dx = (\frac{{xp_{3} - 1}}{{x \left( {p_{3} - p_{2} } \right)}})^{2}$$

So, constructed on the distribution, fuzzy membership of ${\text{X}}^{ - 1}$ is

$$\mu_{{X^{ - 1} }} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\varepsilon_{1} (\frac{{xp_{3} - 1}}{{x \left( {p_{3} - p_{2} } \right)}})^{2} } \hfill & {if\quad p_{3}^{ - 1} \le x < p_{2}^{ - 1} } \hfill \\ {\varepsilon_{1} } \hfill & {if\quad x = p_{2}^{ - 1} } \hfill \\ {\varepsilon_{1} (\frac{{1 - p_{{1^{X} }} }}{{x \left( {p_{2} - p_{1} } \right)}})^{2} } \hfill & {if\quad p_{2}^{ - 1} \le x < p_{1}^{ - 1} } \hfill \\ 0 \hfill & {otherwise} \hfill \\ \end{array} } \right.$$

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Appendix H

Proof—Deliberate two fuzzy Sigmoid numbers X and Y having membership function as,

$$\mu_{X} \left( x \right) = \left\{ {\begin{array}{*{20}c} {\varepsilon_{1} L_{1} \left( x \right) ; \quad if\quad p_{1 } \le x < p_{2,} } \\ {\varepsilon_{1} ;\quad if\quad x = p_{2,} } \\ {\varepsilon_{1} R_{1} \left( x \right) ; \quad if\quad p_{2} \le x < p_{3} } \\ { \quad 0 ;\quad if\quad otherwise} \\ \end{array} } \right.$$

And,

$$\mu_{Y} \left( y \right) = \left\{ {\begin{array}{*{20}c} {\varepsilon_{1} L_{1} \left( y \right) ;\quad if\quad q_{1 } \le y < q_{2,} } \\ {\varepsilon_{1} ;\quad if\quad y = q_{2,} } \\ {\varepsilon_{1} R_{1} \left( y \right) ; \quad if\quad q_{2} \le y < q_{3} } \\ { 0 ;\quad if\quad otherwise} \\ \end{array} } \right.$$

We have Z = X + Y = $\left( {{\text{p}}_{1} + {\text{ q}}_{{1{ }}} ,{\text{ p}}_{2} + {\text{ q}}_{{2{ }}} ,{\text{ p}}_{3} + {\text{ q}}_{3} } \right)$ as the result of combining these fuzzy sigmoid numbers, and if we let z = x + y, we obtain z = x + ${\mathrm{\varnothing }}_{1}$ (x) and z = x + ${\mathrm{\varnothing }}_{2}$ (x) which means x = ${\upxi }_{1}$ (z) and x = ${\upxi }_{2}$ (z) where,

$${\text{x}} = {\upxi }_{1} { }\left( {\text{z}} \right) = \frac{{({\text{ p}}_{2} - {\text{ p}}_{1} ){\text{ z }} - { }\left( {{\text{q}}_{{1{ }}} {\text{p}}_{2} { } - {\text{ p}}_{1} {\text{ q}}_{{2{ }}} } \right){ }}}{{{\text{p}}_{2} + {\text{ q}}_{{2{ }}} - {\text{ p}}_{1} - {\text{ q}}_{1} }}$$

hence,

$${\text{m}}_{1} \left( {\text{z}} \right) = \frac{{\text{d}}}{{{\text{dz}}}}{ }\left( {{\upxi }_{1} { }\left( {\text{z}} \right)} \right) = \frac{{({\text{ p}}_{2} - {\text{ p}}_{1} )}}{{{ }({\text{ p}}_{2} - {\text{ p}}_{1} + {\text{ q}}_{{2{ }}} - {\text{ q}}_{{1{ }}} )}}$$

As in addition,

$$n_{1} \left( z \right) = \frac{d}{dz} \left( {L_{1} } \right) {\text{ at }} x = \xi_{1} \left( z \right)$$

$$= \frac{10}{{ \left( {p_{2} - p_{1} } \right) \left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)}} \left[ {\varphi \left( { x - \frac{{p_{1 } + p_{2} }}{2} } \right) \left( { \frac{10}{{p_{2} - p_{1} }} } \right)} \right]$$

$$\times \left[ {1 - \varphi \left( { x - \frac{{p_{1 } + p_{2} }}{2} } \right) \left( { \frac{10}{{p_{2} - p_{1} }} } \right)} \right]_{{x = \xi_{1} \left( z \right)}}$$

By solving above equation by part

$$\varphi \left( {x - \frac{{p_{1} + p_{2} }}{2}} \right)\left( {\frac{10}{{p_{2} - p_{1} }}} \right) = \varphi \left( { \frac{{ (p_{2} - p_{1} ) z - \left( {q_{1 } p_{2} - p_{1} q_{2 } } \right) }}{{p_{2} + q_{2} - p_{1} - q_{1 } }} - \frac{{p_{1} + p_{2} }}{2}} \right)\left( { \frac{10}{{p_{2} - p_{1} }}} \right)$$

$$= \varphi \left( {\frac{{2( p_{2} - p_{1} ) z - 2 \left( {q_{1 } p_{2} - p_{1} q_{2 } } \right) - \left( {p_{1} + p_{2} } \right) \left( {p_{2} + q_{2} - p_{1 } + q_{1} } \right) }}{{2 \left( {p_{2} + q_{2 } - p_{1} - q_{1 } } \right)}} - \frac{{p_{1 } + p_{2} }}{2}} \right)\left( { \frac{10}{{p_{2} - p_{1} }}} \right)$$

$$= \varphi \left( {\frac{{2(p_{2} - p_{1} )z + (p_{1} + p_{2} )(p_{1} - p_{2} ) + q_{2} (\left( {p_{1} - p_{2} } \right) + q_{1} ((p_{1} - p_{2} ))}}{{2(p_{2} + q_{2} - p_{1} - q_{1} )}} - \frac{{p_{1} + p_{2} }}{2}} \right)\left( {\frac{10}{{p_{2} - p_{1} }}} \right)$$

$$= \varphi \left( {\frac{{2z - (p_{1} + p_{2} + q_{1} + q_{2} )}}{{2(p_{2} + q_{2} - p_{1} - q_{1} )}}} \right)$$

$$= \varphi \left( {z - \frac{{\left( {p_{1} + p_{2} + q_{1} + q_{2} } \right)}}{2}} \right)\left( {\frac{10}{{\left( {p_{2} + q_{2} - p_{1} - q_{1} } \right)}}} \right)$$

Thus,

$$n_{1} \left( z \right) = \frac{10}{{ \left( {p_{2} - p_{1} } \right)\left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)}} \times \varphi \left( {z - \frac{{ \left( {p_{1} + p_{2} + q_{1} + q_{2 } } \right)}}{2}} \right)\left( {\frac{10}{{ \left( {p_{2} + q_{2 } - p_{1} - q_{1 } } \right)}}} \right)$$

$$\times \left[ {1 - \varphi \left( {z - \frac{{ \left( {p_{1} + p_{2} + q_{1} + q_{2 } } \right) }}{2}} \right) \left( {\frac{10}{{ \left( {p_{2} + q_{2 } - p_{1} - q_{1 } } \right)}} } \right)} \right]_{{x = \xi_{1} \left( z \right)}}$$

Therefore,

$$L_{1} \left( x \right) = \mathop \smallint \limits_{{p_{1} + q_{1} }}^{X} m_{1} \left( z \right)\eta_{1} \left( z \right)dz$$

$$= \mathop \smallint \limits_{{p_{1} + q_{1} }}^{X} \frac{{10 \left( {p_{2} - p_{1} } \right)}}{{ \left( {p_{2} - p_{1} + q_{2} - q_{1} } \right)\left( {p_{2} - p_{1} } \right)\left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)}}\left[ {\varphi \left( \cdot \right)\left( {1 - \varphi \left( \cdot \right)} \right)} \right]dz$$

where $\varphi \left( \cdot \right) = \varphi \left( {z - \frac{{ \left( {p_{1} + p_{2} + q_{1} + q_{2 } } \right) }}{2}} \right) \left( {\frac{10}{{ \left( {p_{2} + q_{2 } - p_{1} - q_{1 } } \right)}}} \right)$

$$= \frac{1}{{ \left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)}}\left[ {\varphi \left( {\left\{ {z - \frac{{p_{1} + p_{2} + q_{1} + q_{2} }}{2}} \right\}\frac{10}{{ \left( {p_{2} + q_{2} - p_{1} - q_{1} } \right)}}} \right)} \right]_{{p_{1} + q_{1} }}^{X}$$

$$= \frac{1}{{ \left( {\varphi \left( 5 \right) - \psi \left( { - 5} \right)} \right)}}\left[ {\varphi \left( {\left\{ {x - \frac{{p_{1} + p_{2} + q_{1} + q_{2} }}{2}} \right\}\frac{10}{{ \left( {p_{2} + q_{2} - p_{1} - q_{1} } \right)}}} \right)} \right.$$

$$- \left. {\varphi \left( {\left\{ {p_{1} + q_{1} - \frac{{p_{1} + p_{2} + q_{1} + q_{2} }}{2}} \right\}\frac{10}{{ \left( {p_{2} + q_{2} - p_{1} - q_{1} } \right)}}} \right)} \right]$$

$$= \frac{1}{{ \left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)}}\left[ {\varphi \left( {\left\{ {x - \frac{{p_{1} + p_{2} + q_{1} + q_{2} }}{2}} \right\}\frac{10}{{ \left( {p_{2} + q_{2} - p_{1} - q_{1} } \right)}}} \right) - \varphi \left( { - 5} \right)} \right]$$

So, the distribution function of left side fuzzy number Z = X + Y is;

$$L_{1} \left( z \right) = \frac{{\varphi \left( {\left\{ {z - \frac{{p_{1} + p_{2} + q_{1} + q_{2} }}{2}} \right\}\frac{10}{{\left( {p_{2} + q_{2} - p_{1} - q_{1} } \right)}}} \right) - \varphi \left( { - 5} \right)}}{{\varphi \left( 5 \right) - \varphi \left( { - 5} \right)}}$$

Likewise, if $y = \varphi_{2} \left( x \right)$, then ${\text{x}} = {\text{x}} + {\text{y}}$ becomes ${\text{x}} = {\upxi }_{2} { }\left( {\text{z}} \right)$, where

$$\xi_{2} \left( z \right) = \frac{{ \left( {p_{3} - p_{2} } \right)z - \left( {q_{2} p_{3} - p_{2} q_{3} } \right)}}{{ \left( {p_{3} - p_{2} } \right) + \left( {q_{3} - q_{2} } \right)}}$$

Thus, $m_{2} \left( z \right) = \frac{{d \left( {\xi_{2} \left( z \right)} \right)}}{dx} = \frac{{p_{3} - p_{2} }}{{ \left( {p_{3} - p_{2} } \right) + \left( {q_{3} - q_{2} } \right)}}$

and,

$$\eta_{2} \left( x \right) = \frac{d}{dx} \left( {R_{1} \left( x \right)} \right) at x = \xi_{2} \left( z \right)$$

$$= \frac{ - 10}{{ \left( {p_{3} - p_{2} } \right) \left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)}}\varphi \times \left( {\left\{ {x - \frac{{p_{2} + p_{3} }}{2}} \right\}\frac{10}{{p_{3} - p_{2} }}} \right) \times \left( {1 - \varphi \left( {\left\{ {x - \frac{{p_{2} + p_{3} }}{2}} \right\}\frac{10}{{p_{3} - p_{2} }}} \right)} \right)$$

At $x = \xi_{2} \left( z \right) = \frac{{\left( {p_{3} - p_{2} } \right)z - \left( {q_{2} p_{3} - p_{2} q_{3} } \right)}}{{\left( {p_{3} - p_{2} } \right) + \left( {q_{3} - q_{2} } \right)}}, {\text{we have}}$

By solving above equation by part

$$\varphi \left( {\left\{ {x - \frac{{p_{2} + p_{3} }}{2}} \right\}\frac{10}{{p_{3} - p_{2} }}} \right) = \varphi \left( {\left\{ {\frac{{ \left( {p_{3} - p_{2} } \right)x - \left( {q_{2} p_{3} - p_{2} q_{3} } \right)}}{{ \left( {p_{3} - p_{2} } \right) + \left( {q_{3} - q_{2} } \right)}} - \frac{{p_{2} + p_{3} }}{2}} \right\} \times \frac{10}{{p_{3} - p_{2} }}} \right)$$

$$= \varphi \left( {\frac{{2 \left( {p_{3} - p_{2} } \right)x - 2 \left( {q_{2} p_{3} - p_{2} q_{3} } \right) - \left( {p_{2} + p_{3} } \right) \left( {p_{3} + q_{3} - p_{2} - q_{2} } \right)}}{{2 \left( {p_{3} + q_{3} - p_{2} - q_{2} } \right)}} \times \frac{10}{{p_{3} - p_{2} }}} \right)$$

$$= \varphi \left( {\frac{{2 \left( {p_{3} - p_{2} } \right)x + \left( {p_{2} + p_{3} } \right) \left( {p_{2} - p_{3} } \right) + q_{2} \left( {p_{2} - p_{3} } \right) + q_{3} \left( {p_{2} - p_{3} } \right)}}{{2 \left( {p_{3} + q_{3} - p_{2} - q_{2} } \right)}} \times \frac{10}{{p_{3} - p_{2} }}} \right)$$

$$= \varphi \left( {\left\{ {\frac{{2x - \left( {p_{2} + p_{3} + q_{2} + q_{3} } \right)}}{{2 \left( {p_{3} + q_{3} - p_{2} - q_{2} } \right)}}} \right\} \times 10} \right)$$

$$= \varphi \left( {\left\{ {z - \frac{{p_{3} + p_{2} + q_{2} + q_{3} }}{2}} \right\}\frac{10}{{p_{3} + q_{3} - p_{2} - q_{2} }}} \right)$$

$$\begin{gathered} \eta_{2} { }\left( z \right) = \frac{10}{{{ }\left( {p_{3} - p_{2} } \right){ }\left( {\varphi { }\left( 5 \right) - \varphi { }\left( { - 5} \right)} \right)}}\varphi { }\left( {\left\{ {z - \frac{{p_{3} + p_{2} + q_{2} + q_{3} }}{2}} \right\}\frac{10}{{p_{3} + q_{3} - p_{2} - q_{2} }}} \right) \hfill \\ \times \left[ {1 - \varphi { }\left( {\left\{ {z - \frac{{p_{3} + p_{2} + q_{2} + q_{3} }}{2}} \right\}\frac{10}{{p_{3} + q_{3} - p_{2} - q_{2} }}} \right)} \right] \hfill \\ \end{gathered}$$

Therefore,

$$R_{1} { }\left( x \right) = \mathop \smallint \limits_{x}^{{p_{3} + q_{3} }} \eta_{2} { }\left( z \right)m_{2} { }\left( z \right)dz$$

$$= \mathop \smallint \limits_{x}^{{p_{3} + q_{3} }} \frac{{10 \left( {p_{3} - p_{2} } \right)}}{{ \left( {p_{3} - p_{2} + q_{3} - q_{2} } \right)\left( {p_{3} - p_{2} } \right)\left( {\psi \left( 5 \right) - \psi \left( { - 5} \right)} \right)}}\left[ {\varphi \left( \cdot \right)\left( {1 - \varphi \left( \cdot \right)} \right)} \right]dz$$

where

$$\varphi \left( \cdot \right) = \varphi \left( {\left\{ {z - \frac{{p_{3} + p_{2} + q_{2} + q_{3} }}{2}} \right\}\frac{10}{{p_{3} + q_{3} - p_{2} - q_{2} }}} \right)$$

$$= \frac{1}{{\left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)}}\left[ {\varphi \left( {{ }\left( {z - \frac{{p_{3} + p_{2} + q_{2} + q_{3} }}{2}} \right)\frac{10}{{p_{3} + q_{3} - p_{2} - q_{2} }}} \right)} \right]_{x}^{{p_{3} + q_{3} }}$$

$$= \frac{1}{{\left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)}}\left[ {\varphi \left( {{ }\left( {p_{3} + q_{3} - \frac{{p_{3} + p_{2} + q_{2} + q_{3} }}{2}} \right)\frac{10}{{p_{3} + q_{3} - p_{2} - q_{2} }}} \right)} \right.$$

$$\left. { - \varphi \left( {\left( {x - \frac{{p_{3} + p_{2} + q_{2} + q_{3} }}{2}} \right)\frac{10}{{p_{3} + q_{3} - p_{2} - q_{2} }}} \right)} \right]$$

$$= \frac{{\varphi { }\left( 5 \right) - \varphi { }\left( {\left\{ {x - \frac{{p_{3} + p_{2} + q_{2} + q_{3} }}{2}} \right\}\frac{10}{{p_{3} + q_{3} - p_{2} - q_{2} }}} \right)}}{{\varphi { }\left( 5 \right) - \varphi { }\left( { - 5} \right)}}$$

As a result, the right-side fuzzy number (Z = X + Y)’s distribution function is;

Similarly, the non-complementary function ${R}_{1} (z)$, gives.

$$R_{1} { }\left( z \right) = \frac{{\varphi \left( 5 \right) - \varphi \left( {\left\{ {z - \frac{{p_{3} + p_{2} + q_{2} + q_{3} }}{2}} \right\}\frac{10}{{p_{3} + q_{3} - p_{2} - q_{2} }}} \right)}}{{\varphi \left( 5 \right) - \varphi { }\left( { - 5} \right)}}$$

As a result, the fuzzy variable Z = X + Y membership is given by

$$\mu_{z} { }\left( {\text{Z}} \right) = \left\{ \begin{gathered} \varepsilon { }\frac{{\varphi { }\left[ {{ }\left( {{ }z{ } - { }\frac{{p_{{1{ }}} + { }p_{2} + { }q_{{1{ }}} + { }q_{2} }}{2}{ }} \right){ }\left( {{ }\frac{10}{{p_{2} { } + { }q_{2} { } - { }p_{1} - { }q_{{1{ }}} }}{ }} \right){ }} \right]{ } - { }\varphi { }\left( { - 5} \right){ }}}{{{ }\varphi { }\left( 5 \right){ } - { }\varphi { }\left( { - 5} \right)}}\quad \quad ;\quad if\quad p_{1} + { }q_{{1{ }}} \le z < p_{2} + { }q_{{2{ }}} \hfill \\ \varepsilon { }\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ;\quad if\quad z{ } = { }p_{{\begin{array}{*{20}c} {2 } \\ \\ \end{array} }} + { }q_{2} \hfill \\ \varepsilon { }\frac{{\varphi { }\left( 5 \right){ } - { }\varphi { }\left[ {{ }\left( {{ }z{ } - { }\frac{{p_{{2{ }}} + { }p_{3} { } + { }q_{{2{ }}} + { }q_{3} }}{2}{ }} \right){ }\left( {{ }\frac{10}{{p_{3} { } + { }q_{3} { } - { }p_{2} - { }q_{2} { }}}{ }} \right){ }} \right]{ }}}{{{ }\varphi { }\left( 5 \right){ } - { }\varphi { }\left( { - 5} \right)}}\quad \, ;\quad if\quad p_{2} + { }q_{{2{ }}} { } \le z < p_{3} + { }q_{3} \hfill \\ 0\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ;\quad \quad otherwise \hfill \\ \end{gathered} \right.$$

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Appendix I

Proof—By the use of transformation given by z = kx, we get x = $\frac{\mathrm{z}}{\mathrm{k}}$, which can be written as $\upxi (\mathrm{z})$

so, $\left| {\frac{{{\text{dx}}}}{{{\text{dz}}}}} \right| = \frac{1}{{\text{k}}} = {\text{m }}\left( {\text{z}} \right)$. Then, at ${\text{x}} = {\xi }\left( {\text{z}} \right)$

$$\varphi \left( z \right) = \varphi \left( {\left\{ {\frac{z}{k} - \frac{{p_{1} + p_{2} }}{2}} \right\}\frac{10}{{ \left( {p_{2} - p_{1} } \right)}}} \right)$$

In addition, we have.

$${\upeta }_{1} { }\left( {\text{z}} \right) = \frac{{\text{d}}}{{{\text{dx}}}}{ }\left( {{\text{L}}_{1} } \right){\text{ at x}} = {\upxi }_{1} { }\left( {\text{z}} \right) = \frac{{\text{z}}}{{\text{k}}}$$

$$= \frac{10}{{ \left( {p_{2} - p_{1} } \right) \left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)}} \times \left( {\varphi \left( {\left\{ {\frac{z}{k} - \frac{{p_{1} + p_{2} }}{2}} \right\}\frac{10}{{p_{2} - p_{1} }}} \right)} \right) \times \left( {1 - \varphi \left( {\left\{ {\frac{z}{k} - \frac{{p_{1} + p_{2} }}{2}} \right\}\frac{10}{{p_{2} - p_{1} }}} \right)} \right)$$

Hence,

$$L_{1} \left( x \right) = \mathop \smallint \limits_{{kp_{1} }}^{x} \eta_{1} \left( z \right)m \left( z \right)dz$$

$$= \mathop \smallint \limits_{{kp_{1} }}^{x} \left[ \begin{gathered} \frac{10}{{k \left( {p_{2} - p_{1} } \right) \left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)}} \left( {\varphi \left( {\left\{ {\frac{z}{k} - \frac{{p_{1} + p_{2} }}{2}} \right\}\frac{10}{{p_{2} - p_{1} }}} \right)} \right) \hfill \\ \times \left( {1 - \varphi \left( {\left\{ {\frac{z}{k} - \frac{{p_{1} + p_{2} }}{2}} \right\}\frac{10}{{p_{2} - p_{1} }}} \right)} \right) \hfill \\ \end{gathered} \right]dz$$

$$= \frac{{\varphi \left( {\left\{ {x - \frac{{kp_{1} + kp_{2} }}{2}} \right\}\frac{10}{{ \left( {kp_{2} - kp_{1} } \right)}}} \right) - \varphi \left( { - 5} \right)}}{{\varphi \left( 5 \right) - \varphi \left( { - 5} \right)}}$$

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$\eta_{2} \left( z \right) = \frac{d}{dx} \left( {R_{1} \left( x \right)} \right)$ at $x = \xi \left( z \right) = \frac{z}{k}$

$$= \frac{10}{{ \left( {p_{3} - p_{2} } \right) \left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)}}\varphi \left( {\left\{ {\frac{z}{k} - \frac{{p_{2} + p_{3} }}{2}} \right\}\frac{10}{{p_{3} - p_{2} }}} \right) \times \left( {1 - \varphi \left( {\left\{ {\frac{z}{k} - \frac{{p_{2} + p_{3} }}{2}} \right\}\frac{10}{{p_{3} - p_{2} }}} \right)} \right)$$

And therefore,

$$R_{1} \left( x \right) = \mathop \smallint \limits_{X}^{{kp_{3} }} \eta_{2} \left( z \right)m \left( z \right)dx$$

$$= \mathop \smallint \limits_{X}^{{kp_{3} }} \left[ {\frac{10}{{ \left( {p_{3} - p_{2} } \right) \left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)}} \times \left( {\varphi \left( {\left\{ {\frac{z}{k} - \frac{{p_{2} + p_{3} }}{2}} \right\}\frac{10}{{p_{3} - p_{2} }}} \right)} \right) \times \left( {1 - \varphi \left( {\left\{ {\frac{z}{k} - \frac{{p_{2} + p_{3} }}{2}} \right\}\frac{10}{{p_{3} - p_{2} }}} \right)} \right)} \right]dz$$

$$= \frac{{\varphi \left( 5 \right) - \varphi \left( {\left\{ {x - \frac{{kp_{2} + kp_{3} }}{2}} \right\}\frac{10}{{ \left( {kp_{3} - kp_{2} } \right)}}} \right)}}{{\varphi \left( 5 \right) - \psi \left( { - 5} \right)}}$$

The membership for the fuzzy number ${\text{kX}}$ at ${\text{k}} > 0$ is:

$$\mu_{kX} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\varepsilon_{1} \frac{{\varphi \left( {\left\{ {x - \frac{{kp_{1} + kp_{2} }}{2}} \right\}\frac{10}{{ \left( {kp_{2} - kp_{1} } \right)}}} \right) - \varphi \left( { - 5} \right)}}{{\varphi \left( 5 \right) - \varphi \left( { - 5} \right)}};} \hfill & {kp_{1} \le x < kp_{2} } \hfill \\ {\varepsilon_{1} ;} \hfill & {x = kp_{2} } \hfill \\ {\varepsilon_{1} \frac{{\varphi \left( 5 \right) - \varphi \left( {\left\{ {x - \frac{{kp_{2} + kp_{3} }}{2}} \right\}\frac{10}{{ \left( {kp_{3} - kp_{2} } \right)}}} \right)}}{{\varphi \left( 5 \right) - \varphi \left( { - 5} \right)}};} \hfill & {kp_{2} \le x < kp_{3} .} \hfill \\ \end{array} } \right.$$

Likewise, the membership for the fuzzy number kX at k < 0, is:

$$\mu_{kX} \left( x \right) = \left\{ \begin{gathered} \varepsilon_{1} \frac{{\varphi \left( {\left\{ {x - \frac{{kp_{2} + kp_{3} }}{2}} \right\}\frac{10}{{ \left( {kp_{2} - kp_{3} } \right)}}} \right) - \varphi \left( { - 5} \right)}}{{\varphi \left( 5 \right) - \varphi \left( { - 5} \right)}};\quad kp_{3} \le x < kp_{2} \hfill \\ \varepsilon_{1} ; \quad \quad \quad \quad \quad \quad \quad \quad x = kp_{2} \hfill \\ \varepsilon_{1} \frac{{\varphi \left( 5 \right) - \varphi \left( {\left\{ {x - \frac{{kp_{1} + kp_{2} }}{2}} \right\}\frac{10}{{ \left( {kp_{1} - kp_{2} } \right)}}} \right)}}{{\varphi \left( 5 \right) - \varphi \left( { - 5} \right)}}; \quad kp_{2} \le x < kp_{1} \hfill \\ \end{gathered} \right.$$

Appendix J

Proof—The fuzzy number $Z = XY$, associate $L_{1} \left( x \right)$ with $L_{1} \left( y \right)$ and $R_{1} \left( x \right)$ with $R_{2} \left( y \right)$ and get $y = \emptyset_{1} \left( x \right)$ and $y = \emptyset_{2} \left( x \right)$,

Where,$\emptyset_{1} \left( x \right) = \frac{{q_{1} + q_{2} }}{2} + \frac{{q_{2} - q_{1} }}{{a_{2} - a_{1} }}\left( {x - \frac{{a_{1} + a_{2} }}{2}} \right)$;$\emptyset_{2} \left( x \right)$$= \frac{{q_{2} + q_{3} }}{2} + \frac{{q_{3} - q_{2} }}{{a_{3} - a_{2} }}\left( {x - \frac{{a_{2} + a_{3} }}{2}} \right)$.

Therefore, at $y = \emptyset_{1} \left( x \right),$ $z = xy$ becomes

$$x = \frac{{p_{1} q_{2} - p_{2} q_{1} \pm \sqrt {(p_{1} q_{2} - p_{2} q_{1} )^{2} + 4\left( {p_{2} - p_{1} } \right)\left( {q_{2} - q_{1} } \right)x} }}{{2\left( {q_{2} - q_{1} } \right)}} = \xi_{1} \left( z \right)$$

Therefore,

$$m_{1} \left( z \right) = |\frac{dx}{{dz}}|_{{x = \xi_{1} \left( z \right)}} = \frac{{4\left( {p_{2} - p_{1} } \right)\left( {q_{2} - q_{1} } \right)}}{{4\left( {q_{2} - q_{1} } \right)\sqrt {(p_{1} q_{2} - p_{2} q_{1} )^{2} + 4\left( {p_{2} - p_{1} } \right)\left( {q_{2} - q_{1} } \right)z} }}$$

$$= \frac{{p_{2} - p_{1} }}{{\sqrt {(p_{1} q_{2} - p_{2} q_{1} )^{2} + 4\left( {p_{2} - p_{1} } \right)\left( {q_{2} - q_{1} } \right)z} }} = \frac{{p_{2} - p_{1} }}{{\sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} }}$$

where $P_{1} = \left( {p_{2} - p_{1} } \right)\left( {q_{2} - q_{1} } \right)$; $Q_{1} = p_{1} \left( {q_{2} - q_{1} } \right) +$;$R_{1} = p_{1} q_{1}$.and in addition,

$$\eta_{1} \left( z \right) = \frac{d}{dx}\left( {L_{1} } \right){\text{ at, }}x = \xi_{1} \left( z \right)$$

$$= \frac{10}{{\left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)dx}}\left( {\varphi \left( {\left\{ {x - \frac{{p_{1} + p_{2} }}{2}} \right\}\frac{10}{{\left( {p_{2} - p_{1} } \right)}}} \right)} \right)$$

$$= \frac{10}{{\left( {p_{2} - p_{1} } \right)\left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)}}\varphi \left( {\left\{ {x - \frac{{p_{1} + p_{2} }}{2}} \right\}\frac{10}{{\left( {p_{2} - p_{1} } \right)}}} \right) \times \left( {1 - \varphi \left( {\left\{ {x - \frac{{p_{1} + p_{2} }}{2}} \right\}\frac{10}{{\left( {p_{2} - p_{1} } \right)}}} \right)} \right)$$

At,

$$X = \xi_{1} \left( z \right) = \frac{{p_{1} q_{2} - p_{2} q_{1} \pm \sqrt {(p_{1} q_{2} - p_{2} q_{1} )^{2} + 4\left( {p_{2} - p_{1} } \right)\left( {q_{2} - q_{1} } \right)z} }}{{2\left( {q_{2} - q_{1} } \right)}}$$

$$\varphi \left( {\left\{ {x - \frac{{p_{1} + p_{2} }}{2}} \right\}\frac{10}{{\left( {p_{2} - p_{1} } \right)}}} \right) = \varphi \left( {\left\{ {\frac{{p_{1} q_{2} - p_{2} q_{1} \pm \sqrt {\left( {p_{1} q_{2} - p_{2} q_{1} } \right)^{2} + 4\left( {p_{2} - p_{1} } \right)\left( {q_{2} - q_{1} } \right)z} p_{1} + p_{2} }}{{2\left( {q_{2} - q_{1} } \right)2}}} \right\} \times \frac{10}{{p_{2} - p_{1} }}} \right)$$

$$= \varphi \left( {\left\{ {\frac{{p_{1} q_{2} - p_{2} q_{1} \pm \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} p_{1} + p_{2} }}{{2\left( {q_{2} - q_{1} } \right)2}}} \right\}\frac{10}{{p_{2} - p_{1} }}} \right)$$

$$= \varphi \left( {\left\{ {\frac{{p_{1} q_{1} - p_{2} q_{2} \pm \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} }}{{2\left( {q_{2} - q_{1} } \right)}}} \right\}\frac{10}{{p_{2} - p_{1} }}} \right)$$

Hence

$$\eta_{1} \left( z \right) = \frac{10}{{\left( {p_{2} - p_{1} } \right)\left( {\psi \left( 5 \right) - \psi \left( { - 5} \right)} \right)}}\varphi \left( {\left\{ {\frac{{p_{1} q_{1} - p_{2} q_{2} \pm \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} }}{{2\left( {q_{2} - q_{1} } \right)}}} \right\}\frac{10}{{p_{2} - p_{1} }}} \right)$$

$$\times \left[ {1 - \varphi \left( {\left\{ {\frac{{p_{1} q_{1} - p_{2} q_{2} \pm \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} }}{{2\left( {q_{2} - q_{1} } \right)}}} \right\}\frac{10}{{p_{2} - p_{1} }}} \right)} \right]$$

Therefore,

$$L_{1} \left( x \right) = \mathop \smallint \limits_{{p_{1} q_{1} }}^{X} m_{1} \left( z \right)\eta_{1} \left( z \right)dz = \mathop \smallint \limits_{{p_{1} q_{1} }}^{X} \frac{10}{{\left( {\varphi \left( 5 \right) - \psi \left( { - 5} \right)} \right)\sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} }}\varphi \left( \cdot \right)\left( {1 - \varphi \left( \cdot \right)} \right)dz$$

where $\varphi \left( \cdot \right) = \varphi \left( {\left\{ {\frac{{p_{1} q_{1} - p_{2} q_{2} \pm \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} }}{{2\left( {q_{2} - q_{1} } \right)}}} \right\}\frac{10}{{p_{2} - p_{1} }}} \right)$

$$= \frac{1}{{\left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)}}\left[ {\varphi \left( {\left\{ {\frac{{p_{1} q_{1} - p_{2} q_{2} \pm \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} }}{{2\left( {q_{2} - q_{1} } \right)}}} \right\}\frac{10}{{p_{2} - p_{1} }}} \right)} \right]_{{p_{1} b_{1} }}^{X}$$

$$= \frac{1}{{\left( {\varphi \left( 5 \right) - \varphi \left( { - 5} \right)} \right)}}\left[ {\varphi \left( {\left\{ {\frac{{p_{1} q_{1} - p_{2} q_{2} \pm \sqrt {(Q_{1} )^{2} - 4P_{1} \left( {R_{1} - x} \right)} }}{{2\left( {q_{2} - q_{1} } \right)\left( {p_{2} - p_{1} } \right)}}} \right\}10} \right)} \right. - \varphi \left( {\left\{ {\frac{{p_{1} q_{1} - p_{2} q_{2} \pm \sqrt {Q_{1}^{2} - 4P_{1} \left( {Q_{1} - p_{1} q_{1} } \right)} }}{{2\left( {q_{2} - b_{1} } \right)\left( {p_{2} - p_{1} } \right)}}} \right\}} \right.$$

$$= \frac{{\varphi \left( {\left\{ {\frac{{p_{1} q_{1} - p_{2} q_{2} \pm \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - x} \right)} }}{{2\left( {q_{2} - q_{1} } \right)\left( {p_{2} - p_{1} } \right)}}} \right\}10} \right) - \varphi \left( { - 5} \right)}}{{\varphi \left( 5 \right) - \varphi \left( { - 5} \right)}}$$

$= \frac{{\varphi \left( {\tau_{1} } \right) - \varphi \left( { - 5} \right)}}{{\varphi \left( 5 \right) - \varphi \left( { - 5} \right)}}$, where, $\tau_{1} = 10\left( {\frac{{ - Q_{1} + \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - x} \right)} - P_{1} }}{{2P_{1} }}} \right)$.

By taking, $P_{2} = \left( {p_{3} - p_{2} } \right)\left( {q_{3} - q_{2} } \right)$; $Q_{2} = - p_{3} \left( {q_{3} - q_{2} } \right) - q_{3} \left( {p_{3} - p_{2} } \right)$; $R_{2} = p_{3} q_{3}$,

We're having the accurate membership of $Z = XY$ as:

$R_{1} \left( x \right) = l^{{p_{3} q_{3} }} \eta_{2} \left( z \right)m_{2} \left( z \right)dz = \frac{{\varphi \left( 5 \right) - \varphi \left( {\tau_{2} } \right)}}{{\varphi \left( 5 \right) - \varphi \left( { - 5} \right)}}$, where $\tau_{2} = 10\left( {\frac{{ - Q_{2} - \sqrt {Q_{2}^{2} - 4P_{2} \left( {R_{2} - x} \right)} + P_{2} }}{{2P_{2} }}} \right)$consequently, the membership of the fuzzy number $Z = XY$ is:

$$\mu_{XY} \left( z \right) = \left\{ {\begin{array}{*{20}l} {\varepsilon \frac{{\varphi \left( {\tau_{1} } \right) - \varphi \left( { - 5} \right)}}{{\varphi \left( 5 \right) - \varphi \left( { - 5} \right)}};} \hfill & {p_{1} q_{1} \le z < p_{2} q_{2} } \hfill \\ {\varepsilon ;} \hfill & {z = p_{2} q_{2} } \hfill \\ {\varepsilon \frac{{\varphi \left( 5 \right) - \varphi \left( {\tau_{2} } \right)}}{{ \varphi \left( 5 \right) - \varphi \left( { - 5} \right)}};} \hfill & {p_{2} q_{2} \le z < p_{3} q_{3} } \hfill \\ \end{array} } \right.$$

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where $\tau_{1} = 10\left( {\frac{{ - Q_{1} + \sqrt {Q_{1}^{2} - 4P_{1} \left( {R_{1} - z} \right)} - P_{1} }}{{2P_{1} }}} \right)$ and $\tau_{2} = 10\left( {\frac{{ - Q_{2} - \sqrt {Q_{2}^{2} - 4P_{2} \left( {R_{2} - z} \right)} + P_{2} }}{{2P_{2} }}} \right)$