# A Study of an EOQ Model of Growing Items with Parabolic Dense Fuzzy Lock Demand Rate
## 3 Arithmetic Operation over Parabolic Dense Fuzzy Numbers  
Let \(\widetilde{A}=\langle a\big{(}1-\frac{\rho}{1+n}\big{)},\ a,a\big{(}1+\frac{ \sigma}{1+n}\big{)}\rangle\) and \(\widetilde{B}=\langle b\Big{(}1-\frac{\rho^{\prime}}{1+n}\Big{)},\ b,b\Big{(}1 +\frac{\sigma^{\prime}}{1+n}\big{)}\rangle\) be two parabolic dense fuzzy numbers whose membership functions are given by  
Figure 2: Graph of membership function of PDFN against learning experience. \[\mu_{\widetilde{A}}(x)=\left\{ \begin{array}{l}\left\{\frac{x-a\left(1-\frac{P}{1+ n}\right)}{\frac{P^{\prime}}{1+n}}\right\}^{2},\ for\ a\big{(}1-\frac{P}{1+n}\big{)}\leq x\leq a\\ \left\{\frac{a\big{(}1+\frac{P}{1+n}\big{)}-x}{1+n}\right\}^{2},\ for\ a\leq x \leq a\big{(}1+\frac{\sigma}{1+n}\big{)}\\ 0,\ for\ x<a\big{(}1-\frac{P}{1+n}\big{)}\ or\ a\big{(}1+\frac{\sigma}{1+n} \big{)}<x\end{array} \right. \tag{6}\]  
and  
\[\mu_{\widetilde{B}}(y)=\left\{ \begin{array}{l}\left\{\frac{y-b\left(1-\frac{ P^{\prime}}{1+n}\right)}{\frac{b^{\prime}}{1+n}}\right\}^{2},\ for\ b\big{(}1-\frac{P^{\prime}}{1+n}\big{)}\leq y\leq b\\ \left\{\frac{b\big{(}1+\frac{P^{\prime}}{1+n}\big{)}-y}{\frac{b^{\prime}}{1+n }}\right\}^{2},\ for\ b\leq y\leq b\big{(}1+\frac{\sigma^{\prime}}{1+n}\big{)} \\ 0,\ for\ y<b\big{(}1-\frac{P^{\prime}}{1+n}\big{)}\ or\ b\big{(}1+\frac{\sigma^{ \prime}}{1+n}\big{)}<y\end{array} \right. \tag{7}\]  
Here, \(\sigma\), \(\rho^{\prime}\), and \(\sigma^{\prime}\) are fuzzy deviation parameters. The values of all the fuzzy deviation parameters lie within \((0,\ 1)\). Now, we define some basic arithmetic operations over parabolic dense fuzzy numbers according to the extension of Garg and Ansha [11], as given below. (i) **Addition 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 addition of these two fuzzy numbers, i.e., the fuzzy variable \(\widetilde{\mathcal{C}}=\widetilde{A}+\widetilde{B}\), is also a parabolic dense fuzzy number, and the corresponding membership function is given by  
\[\mu_{\widetilde{C}}(z)=\left\{ \begin{array}{l}\left\{\frac{z-a\big{(}1- \frac{P}{1+n}\big{)}-b\big{(}1-\frac{P^{\prime}}{1+n}\big{)}}{\frac{P^{ \prime}}{1+n}+\frac{P^{\prime}}{1+n}}\right\}^{2},\ for\ a\big{(}1-\frac{P}{1+n} \big{)}+b\Big{(}1-\frac{P^{\prime}}{1+n}\big{)}\leq z\leq a+b\\ \left\{\frac{a\big{(}1+\frac{P}{1+n}\big{)}+b\big{(}1+\frac{P^{\prime}}{1+n} \big{)}-z}{\frac{P^{\prime}}{1+n}+\frac{P^{\prime}}{1+n}}\right\}^{2},\ for\ a+b\leq z\leq a\big{(}1+\frac{\sigma}{1+n}\big{)}+b \Big{(}1+\frac{\sigma^{\prime}}{1+n}\big{)}\\ 0\,,\ for\ z<\ a\big{(}1-\frac{P}{1+n}\big{)}+b\Big{(}1-\frac{P^{\prime}}{1+n} \big{)}\ or\ z>\!\!a\big{(}1+\frac{\sigma}{1+n}\big{)}+b\Big{(}1+\frac{\sigma^{ \prime}}{1+n}\big{)}\end{array} \right. \tag{8}\]  
Graphically, the addition of two PDFNs is given in Figure 3.  
(ii) **Subtraction 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 subtraction of these two fuzzy  
Figure 3: Addition of two parabolic dense fuzzy numbers. numbers, i.e., the fuzzy variable \(\widetilde{C}=\widetilde{A}\ -\widetilde{B}\), is also a parabolic dense fuzzy number, and the corresponding membership function is given by  
\[\mu_{\widetilde{C}}(z)=\left\{