# A Study of an EOQ Model of Growing Items with Parabolic Dense Fuzzy Lock Demand Rate
## 2 Preliminaries  
**Definition 1**. : _Dense fuzzy number (De and Beg [7]). Let \(\widetilde{B}=b_{n},b_{n-1},\ldots,b_{1},b,b^{\prime}_{1},\ldots,b^{\prime}_{n-1},b^{\prime}_{n}\) be a \(2n+1\) tuple fuzzy number whose components are \(b_{n}=bg_{n},b_{n-1}=bg_{n-1},\ldots,b_{1}=bg_{1},b^{\prime}_{1}=bf_{1},b^{ \prime}_{n-1}=bf_{n-1},b^{\prime}_{n}=bf_{n},\) where \(f_{i}\) and \(g_{i}\) are Cauchy sequences. If the components \(f_{i}\) and \(g_{i}\) converge to \(1\) as \(n\rightarrow\infty\), then the fuzzy number \(\widetilde{B}\) will also converge to \(b\). Then, the fuzzy number \(\widetilde{B}=b_{n},b_{n-1},\ldots,b_{1},b,b^{\prime}_{1},\ldots,b^{\prime}_{ n-1},b^{\prime}_{n}\) will be called a dense fuzzy number._  
**Example 1**. : _If \(\widetilde{B}=\langle b\big{(}1-\frac{\rho}{1+n}\big{)},b,b\big{(}1+\frac{ \sigma}{1+n}\big{)}\rangle\) is a triangular dense fuzzy number. Then, the membership function of \(\widetilde{B}\) is given by_  
\[\mu(y,n)= \begin{cases}&\frac{y-b\big{(}1-\frac{\rho}{1+n}\big{)}}{\frac{d\rho }{1+n}},\;for\;b\big{(}1-\frac{\rho}{1+n}\big{)}\leq y\leq b\\ &\frac{b\big{(}1+\frac{\rho}{1+n}\big{)}-y}{\frac{d\sigma}{1+n}},\;for\;b\leq y \leq b\big{(}1+\frac{\sigma}{1+n}\big{)}\\ 0\;,\;for\;y<b\big{(}1-\frac{\rho}{1+n}\big{)}\;or\;b\big{(}1+\frac{\sigma}{1 +n}\big{)}<y\end{cases}  \tag{1}\]  
_Here, \(\rho\) and \(\sigma\) are fuzzy deviation parameters. The values of \(\rho\) and \(\sigma\) lie within \((0,\;1)\). When the values of \(\rho\) and \(\sigma\) become zero, then the triangular fuzzy number is converted to a crisp number. Two-dimensional and three-dimensional graphs of the membership function of a triangular dense fuzzy number are given in Figure 1a,b, respectively. In Figure 1b, the arrow sign indicates the convergence of triangular dense fuzzy numbers._  
**Definition 2**. : _Parabolic fuzzy number (PFN). [PERSON] and [PERSON] [11] first defined arithmetic operations on parabolic fuzzy numbers. Let \(\widetilde{B}=b(1-\rho),b,b(1+\sigma)\) be a parabolic fuzzy number whose membership function is given by_  
\[\mu_{B}(y)= \begin{cases}&\Big{\{}\frac{y-b(1-\rho)}{b\rho}\Big{\}}^{2},\;for \;b(1-\rho)\leq y\leq b\\ &\Big{\{}\frac{b(1+\sigma)-y}{b\rho}\Big{\}}^{2},\;for\;b\leq y\leq b(1+\sigma )\\ 0\;,\;for\;y<b\big{(}1-\rho\big{)}\;or\;b(1+\sigma)<y\end{cases}  \tag{2}\]  
_Here, \(\rho\) and \(\sigma\) are fuzzy deviation parameters. The values of \(\rho\) and \(\sigma\) lie within \((0,\;1)\)._  
Figure 1: (a). Graph of membership function of triangular dense fuzzy number against learning frequency; (b) graph of membership function of triangular dense fuzzy number against learning frequency. **Definition 3**. : _Parabolic dense fuzzy number (PDFN). The general form of a triangular parabolic dense fuzzy number is defined as \(\widetilde{A}=\langle a\big{(}1-\frac{\rho}{1+n}\big{)},\ a,a\big{(}1+\frac{ \sigma}{1+n}\big{)}\rangle\) associated with the membership function_  
\[\mu\left(\widetilde{A}\right)=\left\{ \begin{array}{l}\left\{\frac{x-a\big{(}1 -\frac{\rho}{1+n}\big{)}}{\frac{\sigma}{1+n}}\right\}^{2},\ for\ a\big{(}1-\frac{ \rho}{1+n}\big{)}\leq x\leq a\\ \left\{\frac{a\big{(}1+\frac{\sigma}{1+n}\big{)}-x}{\frac{\sigma}{1+n}}\right\} ^{2},\ for\ a\leq x\leq a\big{(}1+\frac{\sigma}{1+n}\big{)}\\ 0,\ for\ x<a\big{(}1-\frac{\rho}{1+n}\big{)}\ or\ a\big{(}1+\frac{\sigma}{1+n} \big{)}<x\end{array} \right. \tag{3}\]  
_Here, the values of fuzzy deviation parameters \(\rho\) and \(\sigma\) lie within \((0,\ 1)\). The plot of a triangular parabolic dense fuzzy number is given in Figure 2. This figure shows the variation in membership function with respect to learning experience._  
**Definition 4**. : _Parabolic dense fuzzy lock set (PDFLS). Let \(\widetilde{B}=\langle b(1-\rho u_{n}),\ b,b(1+\sigma v_{n})\rangle\) for \(0<\rho,\sigma<1\) be a fuzzy set, where \(u_{n}\) and \(v_{n}\) are Cauchy sequences.