# A Study of an EOQ Model of Growing Items with Parabolic Dense Fuzzy Lock Demand Rate
## 5 Fuzzy Mathematical Model  
We know that the demand rate is a crucial part of any inventory problem, and it does not remain constant every time. Thus, we considered the demand rate as flexible throughout the proposed model, where it could be assumed as a parabolic dense fuzzy set. Hence, the objective function of the crisp model becomes  
\[\widetilde{Z}=\widetilde{d}\varphi+\psi, \tag{22}\]  
where \(\varphi\) and \(\psi\) are given in Equation (20). The membership functions of the parabolic dense fuzzy demand rate with \(n\) learning frequency are given by  
Figure 8: Diagram of the proposed model. \[\mu\left(\vec{d}\right)=\left\{ \begin{array}{l}\left\{\frac{\chi-d\left(1-\frac{ \rho}{1+n}\right)}{\frac{d\rho}{1+n}}\right\}^{2},\ for\ d\big{(}1-\frac{\rho}{1+n} \big{)}\leq x\leq d\\ \left\{\frac{d\left(1+\frac{\rho}{1+n}\right)-x}{1+n}\right\}^{2},\ for\ d\leq x \leq d\big{(}1+\frac{\sigma}{1+n}\big{)}\\ 0,\ for\ x<d\big{(}1-\frac{\rho}{1+n}\big{)}\ or\ x>\!d\big{(}1+\frac{\sigma}{1+n} \big{)}\end{array} \right. \tag{23}\]  
The left and right \(\alpha\)-cuts of the above membership function are given by  
\[\left[\mu_{L}^{d},\mu_{R}^{d}\right]=\Bigg{[}d+\frac{d\rho\big{(}\sqrt{\alpha} -1\big{)}}{1+n},\ d+\frac{d\sigma\big{(}1-\sqrt{\alpha}\big{)}}{1+n}\Bigg{]}. \tag{24}\]  
Then, the index value of the demand rate is given by  
\[ \begin{array}{rl}I\left(\vec{d}\right)&=\frac{1}{2N}\sum\limits_{n=1}^{N} \int\limits_{\alpha=0}^{1}\left\{d+\frac{d\rho\big{(}\sqrt{\alpha}-1\big{)}} {1+n}+d+\frac{d\sigma\big{(}1-\sqrt{\alpha}\big{)}}{1+n}\right\}d\alpha,\\ &\qquad=\frac{1}{2N}\sum\limits_{n=1}^{N}\Big{\{}2d+\frac{2d(\rho-\sigma)}{3( 1+n)}+\frac{d(\sigma-\rho)}{1+n}\Big{\}},\\ &\qquad=d+\frac{d(\sigma-\rho)}{6N}\Big{\{}\frac{1}{2}+\frac{1}{3}+\ldots+ \frac{1}{N}\Big{\}}.\end{array}  \tag{25}\]  
Again, the equation  
\(Z=d\varphi+\psi\) can be rewritten as  
\[d=\frac{Z-\psi}{\varphi}. \tag{26}\]  
Substituting Equation (26) into Equation (23), we obtain the membership function of the objective function as follows:  
\[\mu\left(\vec{Z}\right)=\left\{\begin{array}{l}\left\{\begin{array}{l} \frac{Z-\psi}{\frac{d\rho}{1+n}}-d\big{(}1-\frac{\rho}{1+n}\big{)}\\ \frac{d\rho}{1+n}\right\}^{2},\ for\ d\varphi\big{(}1-\frac{\rho}{1+n}\big{)}+ \psi\leq x\leq d\varphi+\psi\\ \left\{\frac{d\left(1+\frac{\rho}{1+n}\right)-\frac{2-\psi}{\varphi}}{1+n} \right\}^{2},\ for\ d\varphi+\psi\leq x\leq d\varphi\big{(}1+\frac{\sigma}{1+n} \big{)}+\psi\\ 0,\ for\ x<d\varphi\big{(}1-\frac{\rho}{1+n}\big{)}+\psi\ or\ x>\!d\varphi \big{Next, we considered the demand rate as a parabolic dense fuzzy lock set, where the membership function of the demand rate becomes  
\[\mu\Big{(}\widehat{d}\Big{)}=\left\{ \begin{array}{l}\begin{array}{l}\left\{ \frac{x-d\big{(}1-\rho\Big{(}\frac{1}{l_{1}}-\frac{1}{1+n}\Big{)}\big{)}}{d \rho\big{(}\frac{1}{l_{1}}-\frac{1}{1+n}\Big{)}}\right\}^{2},\ for\ d\big{(}1- \rho\Big{(}\frac{1}{l_{1}}-\frac{1}{1+n}\Big{)}\Big{)}\leq x\leq d\\ \left\{\frac{d\big{(}1+\sigma\Big{(}\frac{1}{l_{2}}-\frac{1}{1+n}\Big{)}\big{)} -x}{d\sigma\big{(}\frac{1}{l_{2}}-\frac{1}{1+n}\Big{)}}\right\}^{2},\ for\ d\leq x\leq d\Big{(}1+\sigma\Big{(}\frac{1}{l_{2}}- \frac{1}{1+n}\Big{)}\Big{)}\\ 0,\ for\ x\!<d\Big{(}1-\rho\Big{(}\frac{1}{l_{1}}-\frac{1}{1+n}\Big{)}\Big{)} \ or\ x\!>\!\!d\Big{(}1+\sigma\Big{(}\frac{1}{l_{2}}-\frac{1}{1+n}\Big{)} \Big{)}\end{array}\right.\end{array} \right. \tag{31}\]  
The left and right \(\alpha\)-cuts of the above membership function are given by  
\[\Big{[}\mu_{L}^{d},\mu_{R}^{d}\Big{]}=\bigg{[}d+d\rho\big{(}\sqrt{\alpha}-1 \big{)}\bigg{(}\frac{1}{l_{1}}-\frac{1}{1+n}\bigg{)},d+d\sigma\big{(}1-\sqrt{ \alpha}\big{)}\bigg{(}\frac{1}{l_{2}}-\frac{1}{1+n}\bigg{)}\bigg{]}. \tag{32}\]  
Thus, the index value of demand rate is given by  
\[\begin{array}{rl}I\Big{(}\widehat{d}\Big{)}&=\frac{1}{2N}\sum\limits_{n=1 }^{N}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
If we substitute \(l_{1}=l_{2}=1\) and \(N\rightarrow\infty\) into Equation (37), then the TDFLS problem becomes a general fuzzy problem.