# On the Use of Importance Measures in the Reliability of Inventory Systems, Considering the Cost
## 2 Reliability Analysis in Inventory Systems  
Many logistics units constitute the logistics system organically according to a certain link mode. The premise of analyzing the reliability of the whole logistics system is to determine the calculation method of the reliability of a single logistics unit. The reliability of a logistics unit refers to the probability that the service provided by logistics units remains within the specified error limit under certain conditions and time. In the service capacity curve of the logistics unit in Figure 1, the \(M\) curve represents the standard service curve that the logistics unit is expected to achieve, the \(N\) curve represents the error limit of the logistics service specified for the logistics unit, and the \(P\) curve represents the real logistics service curve provided by the logistics unit. In Figure 1, \(L\) is the level of the service capacity of the logistics unit. In the intervals \([t1-t2]\),\([t3-t4]\), \([t5-t6]\), the logistics service provided by the logistics unit exceeds the specified error limit. Therefore, it is considered that the work of the logistics unit in this situation is unreliable. In the intervals \([t2-t3]\), \([t4-t5]\), the logistics service provided by the logistics unit does not exceed the specified error limit. Therefore, it is considered that the operation of the logistics unit is reliable here. Assuming that the reliability of the logistics unit is \(M_{U}\), then  
\[M_{U}=1-\frac{\sum\limits_{k=1}^{n}\left(t_{k2}-t_{k1}\right)}{T}\]  
where \(t_{k1}\), \(t_{k2}\) are the starting and ending time, respectively, of the kth observation that the logistics service provided by the logistics unit exceeds the allowable deviation range, \((k=1,2,3,\ldots,n)\). \(T\) is the total observation time, and \(n\) is the number of times that the logistics service provided by the logistics unit exceeds the allowable deviation range. In a typical tandem logistics system, such as the one shown in Figure 2, the reliability of the five logistics units of transportation, storage, circulation processing, loading and unloading handling, and distribution are \(M_{Y}\), \(M_{C}\), \(M_{L}\), \(M_{Z}\) and \(M_{P}\), respectively. Figure 1: Service capacity curve of a logistics unit. Figure 2: Structural diagram of a typical tandem logistics system. This logistics system is composed of five logistics units in series. According to the mathematical reliability model of series system, the reliability of typical series logistics system is  
\[M_{T}=M_{Y}M_{C}M_{L}M_{Z}M_{P}\]  
The logistics system cannot be fully paralleled; therefore the reliability calculation can only be carried out for one parallel subsystem. The reliability of circulation processing, \(L1\), and circulation processing, \(L2\), are \(M_{L1}\) and \(M_{L2}\), respectively, as shown in Figure 3.  
Taking the parallel subsystem of circulation processing as an example, according to the mathematical reliability model of the parallel system, the reliability of the parallel subsystem is  
\[M_{LT}=1-(1-M_{L1})(1-M_{L2})=M_{L1}+M_{L2}-M_{L1}M_{L2}\]  
In a typical series parallel logistics system, as shown in Figure 4, the reliability of transportation \(Y1\), transportation \(Y2\), storage \(C1\), storage \(C2\), circulation processing \(L1\), circulation processing \(L2\), loading and unloading handling \(Z1\), loading and unloading handling \(Z2\), distribution \(P1\) and distribution \(P2\) are \(M_{Y1}\), \(M_{Y2}\), \(M_{C1}\), \(M_{C2}\), \(M_{L1}\), \(M_{L2}\), \(M_{Z1}\), \(M_{Z2}\), \(M_{P1}\), and \(M_{P2}\), respectively. The first group series system consists of five logistics units: transportation \(Y1\), storage \(C1\), circulation processing \(L1\), loading and unloading handling \(Z1\), and distribution \(P1\).