# The Development of a Cold-Chain-Packaging Risk Management Model Based on Fuzzy Bayesian Network
## 3 Fuzzy Bayesian Network-Based Risk Management Model for Cold Chain Packaging  
BN is a probabilistic graphical model for uncertainty modelling that contains nodes and directional arcs pointing between nodes. The directional arcs represent the casual relationships of different nodes. When two nodes are not directly linked to each other, they are assumed to be conditionally independent. The degree of dependence between nodes is represented by conditional probabilities, which are usually determined by historical data and expert opinions. The conditional probabilities of the BN nodes are calculated based on the formular below:  
\[\mathrm{P}(\mathrm{A}|\mathrm{B})=\frac{\mathrm{P}(\mathrm{B}|\mathrm{A}) \mathrm{P}(\mathrm{A})}{\mathrm{P}(\mathrm{B})} \tag{1}\]  
P(A) is the prior probability of event A, and P(A\(\,\mathrm{B}\)) is the posterior probability. P(B) is the probability of event B, and P(B\(\,\mathrm{A}\)) indicates the probability of A when B already happens. Conditional probability tables (CPT) are needed for all the non-root nodes. Then, the probability value of each node is determined based on the data derived from expert opinions and practitioners in the field. A triangular fuzzy number will be used to account for any linguistic ambiguity to expand the BN model to a fuzzy BN model.  
Fuzzy logic models use mathematical methods to represent vagueness and imprecise data and information. They are very flexible and highly dependent on the specific scenarios. Fuzzy logic defines which linguistic terms are suitable to describe a variable and converts them into numerical values. A fuzzy membership function defines the associated reliability of the values. A fuzzy membership function map inputs the numbers into a range from 0 to 1 based on the possibility of being a member of the set. A combination of fuzzy logic reasoning and Bayesian Network modelling method is used to deal with high uncertainty and complex casual relationships due to the complicated interdependence of risk factors and insufficient data of the cold chain packaging systems.  
The flow diagram of the proposed BN-based risk management model is shown in Figure 1. The first step is to define the problem and identify all the risk factors and casual relationships. The data used for the study are collected from past data and expert opinions. Data from company handbooks, documents, practical guidelines, etc. are used for the input of the model. The variables and causal relationships of variables can be verified by experts. Then, the main variables can be defined and inputted into the fuzzy BN-based risk management model. The prior and conditional probabilities of variables will be defined by experts. The results will be fuzzified. Next, a sensitivity analysis will be conducted to test the sensitivity of the parameters by changing parameters' values and see how it affects the outcomes. Defuzzification is performed by transforming fuzzy number sets back into crisp numbers, which can be then inputted into Bayesian Network modelling. In this study, GeNie is used for Bayesian Network modelling and inferencing.