# An Integrated Risk Index Model Based on Hierarchical Fuzzy Logic for Underground Risk Assessment
## 3 Proposed Integrated Risk Index Model  
The main challenge of the hierarchical fuzzy inference system is to design a full architecture with minimum rules. In this work, we have to take into account 8 parameters; if conventional fuzzy logic is designed for such system, its computational complexity will be high, because computation complexity increases exponentially as new perimeters enter into the system. In this work, we have designed a model named integrated risk index model, which is based on the hierarchical fuzzy model. Due to the hierarchical decomposition, the proposed fuzzy system is called hierarchical fuzzy system. Rendering to their roles, the rules are grouped into the modules in the system. Each module performs the computation of partial solution, and these fractional solutions are used in next succeeding modules to calculate the concluding output of the system. Though calculation in modules is achieved through fuzzy rules, the decomposition of the flat and large rule base into numerous small rule bases (each with only a few variables) shrinks the complete number of rules. The Figure 1 illustrates the hierarchical fuzzy logic architecture for the risk assessment for the UGS system. Four hierarchical levels are present in the proposed architecture for underground risk assessment: input layer (level-3), Blended Layer (layer-2), Collaborative layer (Level-1), and Integrated Risk layer (Layer-0). Total of 7 sub fuzzy logics have been used in this model; the input layer has eight input parameters, namely, water supply risk probability (X\({}_{1}\)), water supply risk severity (X\({}_{2}\)), Sewerage supply risk probability (X\({}_{3}\)), Sewerage supply risk severity (X\({}_{4}\)), metro structure risk probability (X\({}_{5}\)), metro structure risk severity (X\({}_{6}\)), Geo-environmental risk probability (X\({}_{7}\)), and Geo-environmental risk severity (X\({}_{8}\)). The blended layer has four fuzzy inference systems: M1_FIS, M2_FIS, M3_FIS, and G_FIS, which are abbreviated for water supply pipeline risk index (M\({}_{1}\)) fuzzy logic, sewerage supply pipeline risk index (M\({}_{2}\)) fuzzy logic, metro structure risk index (M\({}_{3}\)) fuzzy logic, and Geo-environmental risk index (G) fuzzy logic, respectively. The collaborative layer has two fuzzy logics, namely, collaborative risk index 1 fuzzy logic (CR\({}_{1}\)_FIS) and collaborative risk 2 fuzzy logic (CR\({}_{2}\)_FIS). Similarly, the integrated layer has integrated risk fuzzy logic (FIS_IR); this is the final output of the proposed hierarchical fuzzy inference system architecture. Water supply risk probability and water supply risk severity are inputs to FIS_M1. Inputs to FIS_M2 are sewerage risk probability and sewerage risk severity. The metro structure risk probability and metro structure risk severity are inputs to the FIS_M3; similarly, the FIS_G takes Geo-environmental risk probability and Geo-environmental risk severity as inputs. Further, the output of FIS_M1 and FIS_M2 are inputted to FIS_CR\({}_{1}\); FIS_CR\({}_{2}\)_FIS takes the output of M3_FIS and G_FIS as inputs. The purpose of the proposed model is to take into account all the parameters to calculate a centric risk value using fuzzy logic, hence an integrated risk fuzzy logic (FIS_IR) is used. The output of IR_FIS is the final risk value of hierarchical fuzzy logic for underground risk. The structure diagram for the proposed model is given in Figure 1, and the structure diagram of the conventional logic has the same number of input parameters as given in Figure 2.  
Mamdani fuzzy logic has been used in the proposed hierarchical model. Mamdani fuzzy rules are too simple and more useful as compared to the Takagi-Sugeno-Kang method. Mamdani fuzzy logic is comprised of the fuzzifier, knowledge base, inference engine, and defuzzifier modules. The fuzzifier takes crisp values as input and generates fuzzy values using membership functions.