# A Study of an EOQ Model of Growing Items with Parabolic Dense Fuzzy Lock Demand Rate
## 3 Arithmetic Operation over Parabolic Dense Fuzzy Numbers
\begin{array}{l}\left\{\frac{z-a\big{(}1-\frac{ \rho}{1+n}\big{)}+b\big{(}1+\frac{\sigma^{\prime}}{1+n}\big{)}}{\frac{\sigma} {1+n}+\frac{b^{\prime}}{1+n}}\right\}\right\}^{2},\ for\ a\big{(}1-\frac{\rho}{1+n} \big{)}-b\Big{(}1+\frac{\sigma^{\prime}}{1+n}\big{)}\leq z\leq a-b\\ \left\{\frac{a\big{(}1+\frac{\sigma}{1+n}\big{)}-b\big{(}1-\frac{ \sigma^{\prime}}{1+n}\big{)}-z}{\frac{\sigma}{1+n}+\frac{b^{\prime}}{1+n}} \right\}^{2},\ for\ a-b\leq z\leq a\big{(}1+\frac{\sigma}{1+n}\big{)}-b\Big{(}1- \frac{\sigma^{\prime}}{1+n}\big{)}\\ 0,\ for\ z<a\big{(}1-\frac{\rho}{1+n}\big{)}-b\Big{(}1+\frac{\sigma^{\prime}}{ 1+n}\Big{)}\ or\ z>\!\!a\big{(}1+\frac{\sigma}{1+n}\big{)}-b\Big{(}1-\frac{ \sigma^{\prime}}{1+n}\big{)}\end{array} \right. \tag{9}\]  
Graphically, the subtraction of two PDFNs is given in Figure 4.  
(iii): **Scalar multiplication of Parabolic Dense Fuzzy Numbers**  
Let \(\widetilde{A}=a\big{(}1-\frac{\rho}{1+n}\big{)}\), \(a,a\big{(}1+\frac{\sigma}{1+n}\big{)}\) be a PDFN with the membership function given in Equation (6) and \(z=kx\) be the transformation; then, \(k\widetilde{A}\) is also a PDFN given by  
\[k\widetilde{A}=\left\{ \begin{array}{l}\langle ka\big{(}1-\frac{\rho}{1+n} \big{)},\ ka,ka\big{(}1+\frac{\sigma}{1+n}\big{)}\rangle\ if\ k>0\\ \langle ka\big{(}1+\frac{\rho}{1+n}\big{)},ka,\ ka\big{(}1-\frac{\rho}{1+n} \big{)}\rangle\ if\ k<0\end{array} \right. \tag{10}\]  
Graphically, the scalar multiplication of two PDFNs is given in Figure 5.  
Figure 4: Subtraction of two parabolic dense fuzzy numbers. Figure 5: Scaler multiplication of two parabolic dense fuzzy numbers. 4. **Multiplication of Two Parabolic Dense Fuzzy Numbers**  
Let us consider two parabolic dense fuzzy numbers \(\widetilde{A}\) and \(\widetilde{B}\) whose membership functions are given in Equations (6) and (7). Then, the multiplication of these two fuzzy numbers, i.e., the fuzzy variable \(\widetilde{C}=\widetilde{A}\cdot\widetilde{B}\), is also a PDFN and the corresponding membership function is given by  
\[\mu_{\widetilde{A}\cdot\widetilde{B}}\left(z\right)=\left\{ \begin{array}{l} \left\{\frac{-Q_{1}+\sqrt{Q_{1}^{2}-4P_{1}\left(R_{1}-z\right)}}{2P_{1}} \right\}^{2},\;for\;a\big{(}1-\frac{\rho}{1+n}\big{)}b\big{(}1-\frac{\rho^{ \prime}}{1+n}\big{)}\leq z\leq ab\\ \left\{\frac{-Q_{2}+\sqrt{Q_{2}^{2}-4P_{2}\left(R_{2}-z\right)}}{2P_{2}} \right\}^{2},\;for\;ab\leq z\leq a\big{(}1+\frac{\sigma}{1+n}\big{)}b\Big{(}1+ \frac{\sigma^{\prime}}{1+n}\Big{)}\\ 0,\;for\;z<a\big{(}1-\frac{\rho}{1+n}\big{)}b\big{(}1-\frac{\rho^{\prime}}{1+n} \big{)}\;or\;z>\!\!a\big{(}1+\frac{\sigma}{1+n}\big{)}b\Big{(}1+\frac{\sigma^ {\prime}}{1+n}\big{)}\end{array} \right. \tag{11}\]  
where \(P_{1}=\frac{ab\rho^{1}}{\left(1+n\right)^{2}},\;Q_{1}=\frac{ab}{1+n}\Big{[} \big{(}1-\frac{\rho}{1+n}\big{)}\rho^{\prime}+\big{(}1-\frac{\rho^{\prime}}{ 1+n}\big{)}\rho\Big{]}\), \(R_{1}=ab\big{(}1-\frac{\rho}{1+n}\big{)}\big{(}1-\frac{\rho^{\prime}}{1+n} \big{)}\), \(P_{2}=\frac{ab\rho^{1}}{\left(1+n\right)^{2}},\;Q_{2}=-\frac{ab}{1+n}\Big{[} \big{(}1+\frac{\sigma}{1+n}\big{)}\sigma^{\prime}+\big{(}1+\frac{\sigma^{ \prime}}{1+n}\big{)}\sigma\Big{]}\), and \(R_{2}=ab\big{(}1+\frac{\sigma}{1+n}\big{)}\left(1+\frac{\sigma^{\prime}}{1+n} \right)\). Graphically, the multiplication of two PDFNs is given in Figure 6.  
(v) **Inverse of a Parabolic Dense Fuzzy Number**  
Let \(\widetilde{A}\) be a parabolic dense fuzzy number whose membership function is given in Equation (6). Then, the inverse of \(\widetilde{A}\), i.e., \(\widetilde{A}^{-1}=\langle a^{-1}\big{(}1+\frac{\sigma}{1+n}\big{)}^{-1},\;a^ {-1},\;a^{-1}\big{(}1-\frac{\rho}{1+n}\big{)}^{-1}\rangle\), is also a parabolic dense fuzzy number whose membership function is  
\[\mu_{\widetilde{A}^{-1}}(z)=\left\{