# Information Fusion for Multi-Source Material Data: Progress and Challenges
## 3 Progress of Information Fusion for Multi-Source Material Data
### Algorithms for Material Data Fusion
#### 3.4.3 Decision-Level Fusion  
Decision-level fusion aims to construct appropriate expert systems and knowledge bases, and to develop fusion rules suitable for decision-making. The fusion result is to provide decision basis and decision support for command and control decisions, to analyze the site situation, support and assist decision-making, the evaluation of decisions, evaluation of system performance, and so on.  
The advantage is that different types of data can be fused, the calculation amount is small, the fault tolerance ability and the anti-interference ability are strong, but the disadvantages are also obvious, e.g., the loss of data information will bring the precision down. Decision-level fusion often uses the DS (Dempster-Shafer) evidence theory, the fuzzy theory, statistical decision-making, production rules and neural networks to make the final comprehensive decision.  
1. DS Evidence Theory [51]. The DS Evidence Theory is a typical and widely applicable decision making-method. The DS Evidence Theory [52; 53] is favored for its ability of dealing with uncertainty, integration of measurement information, and reasonable theoretical derivation. In the DS model, the frame of discernment (FoD) is denoted by \(\Theta\), which is a set of \(N\) hypotheses that are mutually exclusive and exhaustive, i.e., \(\Theta=\left\{H_{1},H_{2},\ldots,H_{N}\right\}=\left\{H_{i}|i=1,2,\ldots,N\right\}\). The basic probability assignment (BPA) is a mass function \(m\): \(2^{\Theta}\rightarrow\left[0,1\right],\) which satisfies Equation (4).  
\[\sum_{A\subseteq\Theta}m(A)=1 \tag{4}\]  
The belief function (Bel) is defined by Equation (5).  
\[Bel(A)=\sum_{B\subseteq A}m(B) \tag{5}\]and the plausibility function (PI) is defined by Equation (6).  
\[Pl(A)=\sum_{B\cap A\
eq\varnothing}m(B) \tag{6}\]  
Finally, the DS combination rule can be represented by Equation (7).  
\[m(A)=[m_{1}\oplus m2](A)=\frac{1}{1-k}\sum_{A_{i}\cap B_{j}}m_{1}(A_{i})\cdot m _{2}\Big{(}B_{j}\Big{)} \tag{7}\]  
where \(\oplus\) represents the orthogonal sum operator, and \(k\) is the global conflict factor demonstrating the conflict degree between \(m_{1}\)_and_\(m_{2}\).  
2. Production rules [54; 55]: Production rules are a common means of knowledge representation, which represents the causality in the form of \"IF-THEN\". This form of rules reflects the behavioral characteristics of humans solving a class of problems, which can be solved by applying these rules cyclically. Material science can give us many scientific rules, which can be expressed in the form of \"IF-THEN\". Production rules can increase the automation degree of material data fusion.  
3. Fuzzy theory [56; 57]: Fuzzy theory refers to the theory that uses the basic concept of fuzzy sets or continuous degree of membership functions. Fuzzy sets provide a form for dealing with less rigorous information. Let \(Z\) be the set of elements, and \(z\) denotes a class of elements of \(Z\), i.e., \(Z\)= \(\{z\}\). The fuzzy subset \(A\) in \(Z\) is represented by the membership function \(\mu_{A}(z)\), which is between [0,1]. In the literature [58], the AHP (Analytic Hierarchy Process) approach [59] was introduced to determine the priorities of different monitoring models, after which a fuzzy decision fusion strategy was developed for process monitoring applications. The overall priority index \(PRT_{k}\) for each model has been obtained based on monitoring a specific fault in the process by a monitoring statistic, e.g., squared prediction error (SPE). Then, based on the determined posterior probabilities, a fuzzy discriminant matrix corresponding to SPE can be formulated for fusing the probabilities generated through different monitoring models. A fuzzy decision system could be considered to handle the material data-fusion process, which can satisfy the purpose of online process monitoring.