# Entropy-Based Algorithm for Supply-Chain Complexity Assessment
## 2 Basic Definitions  
Wishing to avoid any ambiguity in further discussions, we begin with key definitions of risk and ambiguity in relationships between the manufacturer and its supplier. There is a wide specter of different definitions of risk and uncertainty. In this study, we follow [PERSON]'s [16] view and his numerous followers. The uncertainty is the absence of certainty in our knowledge, or, in other words, a situation wherein it is impossible to precisely describe future outcomes. In the Knightian sense, the risk is measurable uncertainty, which is possible to calculate. Similar to many other risk evaluators, we assume that the notion of risk can be described as the expected value of an undesirable outcome, that is, the product of two characteristics, the probability of an undesirable event (that is, a negative deviation of the delayed supply or failure to reach the planned supply target), and the impact or severity, that is, an expected loss in the case of the disruption affecting the supply of products across organizations in a supply network. In a situation with several possible accidents, we admit that the total risk is the sum of the risks for different accidents (see, e.g., [11; 17; 18]). In the model considered below, an \"event\" is the observable discrete change in the state of the SC or its components. A \"risk driver\" is a factor, a driving force that may be a cause of the undesirable unforeseen event, such as disruptions, breakdowns, defects, mistakes in the design and planning, shortages of material in supply in the SC, etc. In this paper, we study the situations with an \"observable uncertainty\" where there is an objective opportunity to register, for a pre-specified period of time, the adverse events in the relationships between the components in the SC. Such a registration list, called a \"risk protocol\", provides us the information whether or not the events are undesirable and, in the case if the event is undesirable what are its risk drivers and possible loss (see [12; 19]). Such statistics in the risk protocols permits the decision maker to quantitatively evaluate a contribution of each driver and the total (entropy-based) observable information in the SC. There exists a wide diversity of risk types, risk drivers, and options for their mitigation in the SC. Their taxonomy lies beyond the scope of this paper. Many authors noticed that if a researcher tries to analyze potential failures/disruptions of all the suppliers in a SC or their absolute majority, he/she encounters a simply impractical and unrealistic problem demanding an astronomic amount of time and budget. Moreover, the supply chain control of the root node of the supply chain is much more important than the control of any of its successors ([6; 8]). Consider a tree-type graph representing the hierarchical structure of an industrial supply chain Define the \"parent layer\", called also the \"main layer\", as consisting of a single node, as follows: \(L_{0}=\{n_{0}\}\) where \(n_{0}\) is the single node of the parent layer. Called also the original equipment manufacturer (OEM), or the root node. OEM is a firm (or company) that creates an end product, for, instance, assembles and creates an automobile. Define a _layer_\(L_{s}\) (also denoted as layer \(s\)) as the set of nodes which are on the same distance \(s\) from the root node \(n_{0}\) in the underlying graph of the SC. Layer 1 (also called Tier 1) are the companies supplying components directly to the OEM that set up the chain. In a typical supply chain, companies in Tier 2 supply the companies in Tier 1; Tier 3 supplies Tier 2, and so on. Tiered supply chains are common in industries such as aerospace or automotive manufacturing where the final product consists of many complex components and sub-assemblies. Define a \"cut\" (also called a \"cross-section\") \(C_{s}\) as a union of all the layers \(L_{0},L_{1},L_{2},\ldots L_{s}\), from 0 to \(s\).