# On the Use of Importance Measures in the Reliability of Inventory Systems, Considering the Cost
## 3 Cost-Based Importance Measures of Inventory Systems Reliability
### Discussions on Importance Measures Based on the Inventory Systems Cost  
In the inventory control model, the optimal lot size, \(Q^{\star}\), is obtained by minimizing the total cost. The function expression of the total cost is as follows. \[\varphi(Q,\alpha)=\frac{(u+a/2)Q+\gamma}{1-e^{-\varphi Q/R}}\]  
where \(Q\) is the selection variable, \(\alpha=(u,a,R,\gamma,\rho)\) is a parameter variable, \(u\) is the unit price of the goods in stock, \(a\) is the unit holding cost, \(R\) is the demand speed, \(\gamma\) is the order cost, and \(\rho\) is the capital cost. Among them, an increase in \(u,a,R\) and \(\gamma\) will cause an increase in the total cost. An increase in \(\rho\) will cause a decrease in total cost. The total cost function takes the first-order derivative of \(Q\) and sets the result to 0, then  
\[\varphi_{Q}(Q^{\star},\alpha)=\Big{(}\frac{1}{2}a+u\Big{)}\Big{(}e^{((Q^{ \star}/R)\rho}-1\Big{)}R-\rho\Big{(}\gamma+Q^{\star}\Big{(}\frac{1}{2}a+u\Big{)} \Big{)}=0.\]  
For the importance of several parameter variables with respect to function \(Q^{\star}(\alpha)\), the implicit importance can be applied to the optimization problem of the model. The final results are shown in Table 1. Assuming that \(u=10\) (yuan/piece), \(a=1\) (yuan/piece), \(R=8000\) (piece), \(\gamma=30\) (yuan/time), and \(\rho=8\%\), the comparative static (CS) can be used to analyze the results as follows. From Table 1, the results can be obtained as follows. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **CS** & **Expression** & **Mark** & **Value** \\ \hline \(u\) & \(Q_{u}\) & \(-\frac{R\big{(}e^{(\zeta Q/R)}-1\big{)}-Q\rho}{\rho\big{(}a+u\big{)}\big{(}e^{( \zeta Q/R)}-1\big{)}}\) & \(-\) & 36 \\ \(a\) & \(Q_{a}\) & \(-\frac{1}{2}R\big{(}e^{(\zeta Q/R)}-1\big{)}-Q\rho}{\rho\big{(}a+u\big{)} \big{(}e^{(\zeta  
1. The increase in \(u\), \(a\) and \(\rho\) leads to a decrease in \(Q^{*}\) in the EOQ model. 2. The increase in \(R\) and \(\gamma\) will cause an increase in \(Q^{*}\) in the EOQ model. 3. It can be seen from the table that \(\rho\) is the most influential parameter in the EOQ results, and its influence degree is far greater than other parameters. On the other hand, the partial derivatives of each parameter have different units of measurement, so they cannot be compared with each other. Similarly, because there are different measurement standards between parameters, assumption 1 of differential importance is not tenable in the application of differential importance. Even if the parameters have the same unit of measurement, but the direction of change is different, and the results of static analysis technology cannot be used as the standard of measurement. According to the relationship between comparative static analysis technology and differential importance, the expression of differential importance measures in implicit models is proposed, and expressed as follows. \[\Gamma(x^{*},\alpha^{*})=\left[\gamma_{j,s}:\gamma_{j,s}=\frac{\left|\Phi^{j }s\right|}{\left|\Phi_{j}\right|}\right],\ \ j=1,\ldots,m,\ \ s=1,\ldots,n.\]  
\(\Gamma(x^{*},\alpha^{*})\) is a matrix, and the elements in the matrix represent the differential importance of the parameter \(\alpha_{s}\) with respect to \(x_{j}\). \(\Phi^{j}=\left[J^{1}_{x}J^{2}_{x}\ldots J^{j-1}_{x}J^{s}_{\alpha}d\alpha_{s} J^{j+1}_{x}\ldots J^{m}_{x}\right]\), \(\Phi_{j}=\left[J^{1}_{x}J^{2}_{x}\ldots J^{j-1}_{x}d\right]\), \(\mu_{J}J^{j+1}_{x}\ldots J^{m}_{x}\). According to the relationship between comparative static analysis technology and the expression of differential importance in the implicit model, the expression of Birnbaum importance is  
\[\Gamma_{B}(x^{*},\alpha^{*})=\left[ \begin{array}{c}\gamma_{j,s}:\gamma_{j,s}= \left[J^{1}_{x}J^{2}_{x}\ldots J^{j-1}_{x}J^{s}_{\alpha}\frac{\alpha}{x^{j}_{ s}}J^{j+1}_{x}\ldots J^{m}_{x}\right]\\ =\frac{\left\|J^{1}_{x}J^{2}_{x}-J^{1}_{x}J^{s}_{\alpha}\frac{\alpha}{x^{j}_{ x}}J^{j+1}_{x}\ldots J^{m}_{x}\right\|\alpha}{x^{j}_{s}}\end{array}