# On the Use of Importance Measures in the Reliability of Inventory Systems, Considering the Cost
## 4 Numerical Example  
In this section, a shortage and limited inventory capacity is used to demonstrate the proposed model. The inventory models are assumed, and displayed in Table 4.  
Assuming that \([0,T]\) is a time cycle, and when \(t=0\), the instant purchase is \(Q_{1}\), the inventory capacity is \(Q_{0}\), so the capacity of leased warehouses is \(Q_{1}-Q_{0}\). \(Q_{2}\) is the allowable shortage quantity, \(R\) is the demand rate, and it is decreasing constantly during \([0,T]\). Therefore, the change of inventory volume during \([0,T]\) is shown in Figure 6.  
Assuming the total cost of an inventory is \(C\), then  
\[C=\frac{C_{3}(Q_{1}-Q_{0})^{2}}{2Q}+\frac{C_{2}Q_{0}(2Q_{1}-Q_{0})}{2Q}+\frac{ C_{4}(Q-Q_{1})^{2}}{2Q}+\frac{C_{1}R}{Q}\]  
where \(R\) means the demand speed, \(Q\) is the order quantity, and \(Q_{1}\) and \(Q_{0}\) represent the total inventory volume, so the inventory capacity of leased warehouses is \(Q_{1}-Q_{0}\). \(Q_{2}\) is the shortage quantity, so \(Q=Q_{1}+Q_{2}\). \(C_{1}\) is the order cost, \(C_{2}\) is the inventory cost of unit goods per unit time when using their own warehouses, \(C_{3}\) means the inventory cost of unit goods per unit time when using leased warehouses, and \(C_{4}\) represents the shortage cost of unit goods. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case Assumptions** & **Case Assumption Contents** \\ \hline Assumption 1 & A small amount of stockout will not cause much damage to customers and \\  & companies. The loss of unit goods per unit time is \(C_{4}\). \\ \hline Assumption 2 &
\begin{tabular}{c} Companies can store products in their own warehouses or leased warehouses. \\ \(C_{2}\) is the storage fee of unit goods per unit time stored in their own warehouses, \\ \(C_{3}\) is the storage fee in leased warehouses, and \(C_{2}<C_{3}\). \\ \end{tabular} \\ \hline Assumption 3 & The capacity of their own warehouses is \(Q_{0}\). \\ \hline Assumption 4 & When storing, companies firstly store products in their own warehouses until \\  & they are full, and then in leased warehouses. \\ \hline Assumption 5 & When selling, companies will firstly sell the products in leased warehouses \\  & until they are empty, and then from their own warehouses. \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Case assumptions of the inventory models. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Importance** & **Expression** & **Mark** & **Value** & **Order** \\ \hline \(u\) & \(\Gamma_{Ba}(Q^{\prime},\alpha^{*})\) & \(\frac{[R(e^{(Q/\alpha^{*})/T})-Q_{0}]u}{\left\{\frac{Q}{2Q}\right\}}\) & + & \(3.028\times 10^{-3}\) & 4 \\ \(a\) & \(\Gamma_{Ba}(Q^{\prime},\alpha^{*})\) & \(\frac{[\frac{1}{4}R(e^{(Q/\alpha^{*})/T})-\  
each parameter, the variation of Birnbaum importance and the differential importance measure are shown in Figure 7.  
Figure 7: _Cont._  
Figure 7 shows that Birnbaum importance and differential importance measures have similarities in the variation law. The plus-minus and direction of curves are consistent, but the slopes are not the same. In Figure 7e, when the shortage cost is less than 0.5, Birnbaum importance changes almost linearly, otherwise its change is quite slow and tends to be stable. However, the curve of the differential importance measure keeps a steady change rate within the interval. In Figure 7a,e, the two curves even appear to be intersected. At the intersection, the values of Birnbaum importance and the differential importance measure are equivalent. In addition, the differential importance measure is above the Birnbaum importance in Figure 7b, while the exact reverse is the case in Figure 7c,d; the Birnbaum importance of the parameter lies above the differential importance measure. The two importance measures show different changes with the varying parameters, which suggests that there are certain differences between the two importance measures in practice.