# Entropy-Based Algorithm for Supply-Chain Complexity Assessment
## 1 Introduction  
This paper presents the entropy-based optimization model for estimating structural complexity of supply chain (SC), and, in particular, complexity of relationship between the principal equipment manufacturer (a root node) and its suppliers (preceding supply nodes). The information entropy is used as a measure of decision maker's knowledge about the risks of shortages and pitfalls in relations between the supply chain components under uncertainty, the main attention being paid to relationships between the principal equipment manufacturer (a root node of the corresponding graph model) and its suppliers. A concept of the conditional (relative) entropy is introduced which is a generalization of the conventional (non-relative) entropy that provides more precise estimation of complexity in supply chains as it takes into account information flows between the components of different layers. A main advantage of the suggested entropy-based approach is that it can essentially simplify the hierarchical tree-like model of the supply chain, at the same time retaining the basic knowledge about main sources of risks in inter-layer relationships.  
Processes of production, storage, transportation, and utilization of products in SCs may lead to many negative effects on the environment, such as emissions of pollutants into the air and soil; discharges of pollutants into surface and groundwater basins; pollution of soil and water with waste products of production, all these phenomena in their entirety deteriorating the relationships between manufacturing components and their suppliers. These and many other risky situations for manufacturing lead to the uncertainty in SC's phases, i.e., engineering, procurement, production, and distribution [1]. The entropic approach developed in this paper aims at defining the best (minimal but sufficient) level of information going through the several supply chain phases. This paper analyzes the structural complexity of the supply chain affected by technological, organizational and environmental adverse events in the SCs, consequences of which lead, as a result, to violations of correct functioning of the SC. The detailed definition and analysis of the supply-chain structural complexity can be found in [2; 3; 4; 5; 6; 7; 8; 9; 10].  
Similar to many other researchers (see, e.g., [11; 12; 13]), in order to accurately measure the risks of pitfalls and cases of failed supplies, this paper tends to determine the probabilities of undesirable events and their negative impacts on material, financial, and information flows. The economic losses are caused by failures to supply, shipments of inappropriate products, incomplete deliveries, delays in deliveries, etc., and the harmful effect of the adverse events may be expressed in monetary form by relevant penalties.  
The main differences of the present work in comparison with close earlier papers (see, e.g., [7; 12; 14; 15]) also exploiting the entropy approach for the SC complexity analysis are the following:  
* This paper develops a new graph-theoretic model as a tool for selecting the most vulnerable disruption risks inside the SC which, in turn, can essentially decrease the size of the initial SC model without sacrificing essential knowledge about the risks;
* This paper introduces the conditional entropy as a tool for integrated analysis of the SC complexity under uncertainty, provides more precise estimation of the supply chain complexity taking into account links between the nodes of different layers;
* This paper suggests a new fast entropy-based algorithm for minimizing the SC size.  
This paper is structured as follows. The related definitions from graph theory are presented in the next section. The definition of information entropy and the detailed problem description are given in Section 3. Section 4 describes the entropy-based algorithm permitting to reduce the SC model size without a loss of essential information. Section 5 describes the numerical example. Section 6 concludes the paper.