# Entropy-Based Algorithm for Supply-Chain Complexity Assessment
## 2 Basic Definitions
For any node \(u\) from \(s\), we can define corresponding probabilities \(p_{s}(u,f)\) of the event that a driver \(f\) is the source of adverse events in node \(u\)  
\[p_{s}(u,f)=N_{s}(u,f)/R_{s}(u). \tag{13}\]  
This paper treats the \(p_{s}(u,f)\) values defined by Equation(3) as probabilities of events participating in calculation of the entropy function in Equation (1). The main idea of the suggested entropic approach is that the information entropy in this study estimates the average amount of information contained in a stream of critical events of the risk protocol. Thus the entropy characterizes our uncertainty, or the absence of knowledge, about the risks. The idea here is that the less the entropy is, the more information and knowledge about risks is available for the decision makers. The entropy value can be computed iteratively for each cut of the SC. Assume that the nodes of a cut \(C_{s-1}\), are defined at step (iteration) \(s-1\). Denote by \(T_{s}\) all supplier-nodes in the supply layer \(s\) of the given tree. Let \(L_{s}(T_{s})\) denote the total losses defined by the risk protocol and summed up for all nodes of cut \(C_{s}\) in tiers \(T_{s}\): \(L_{s}(T_{s})=\sum_{u\in T_{s}}c_{s}(u)\). Further, let \(LT\) denote the total losses for all nodes of the entire chain. Thus, \(L_{s}(T_{s})\) are contributions of the suppliers of cut \(C_{s}\) into the total losses \(LT\). Then \(L_{s}(T_{s})/LT\) define the relative losses in the \(s\)-truncated supply chain. The relative contribution of lower tiers, that is, of those with larger \(s\) values, are, respectively, \((LT-L_{s}(T_{s}))/LT\). One can observe that the larger is the share \((LT-L_{s}(T_{s}))/LT\) in comparison with \(L_{s}(T_{s})/LT\), the less is the available information about the losses in the cut \(C_{s}\). For example, if the ratio \(L_{s}(T_{s})/LT=0.2\), this case give us less information about the losses in cut \(C_{s}\) in comparison with the opposite case of \(L_{s}(T_{s})/LT=0.8\). This argument motivates us to take the ratios \((LT-L_{s}(T_{s}))/LT\) as the coefficients (weights) of the entropy (or, of our unawareness) about the economic losses incurred by adverse events affecting the environmental quality. In other words, the latter coefficients weigh the lack of our knowledge about the losses; as far as the number \(s\) grows, these coefficients become less and less significant. Then the total entropy of all nodes \(u\) included into the cut \(s\) will be defined as  
\[H(s)=\sum_{u}H(u) \tag{14}\]  
where the weighted entropy in each node \(u\) is computed as  
\[H(u)=-U_{s}\sum_{i}p_{s}(u,f)\log p_{s}(u,f), \tag{15}\]  
\[U_{s}=(LT-L_{s}(T_{s}))/LT, \tag{16}\]  
Let \(x_{u}=1\) if node \(u\) from \(T_{s}\) is included into \(s\), and \(x_{u}=0\), otherwise. Then the total entropy of all nodes included into the cut \(s\) will be defined as  
\[H(s)=\sum_{u}H(u)\;x_{u}, \tag{17}\]  
where \(H(u)\) is defined in (15). Computations of \(p_{s}(u,f)\) as well as the summation over risk factors \(f\) are taken in the risk event protocols for all the events related to nodes \(u\) from \(T_{s}\). As far as entropy values are found for each node, the vulnerability to risks over the supply chain is measured as a total entropy of the \(s\)-truncated supply chain subject to the restricted losses. Define the weighted entropy for each cut \(s\) as  
\[H(C_{s})=c(s)H^{*}(C_{s}) \tag{18}\]  
where  
\[H^{*}(C_{s})=-\sum_{j=1}^{N}p_{j}(C_{s})\log p_{j}(C_{s})\text{ is the entropy of cut }C_{s}\]  
We assume that the weight \(c(s)\) satisfies the following conditions:  
1. \(c(s)\) is decreasing;
2. \(c(0)=L\);
3. \(\lim_{s\to\infty}c(s)=0\). Define the \"variation of relative entropy\" depending upon the cut number is  
\[REV(s)=\frac{H(s-1)-\frac{c(s-1)}{c(s)}H(s)}{H(1)-\frac{c(s-1)}{c(s)}H(s)}. \tag{19}\]  
The following claim is valid:  
**Theorem**.