## 2 The Proposed Method for Integration of Uncertain Expert Knowledge to the System Model  
System models can be formalized mathematically in general, where the outputs (\(\mathbf{y}\)) of the system are functions of the inputs (\(\mathbf{u}\)) and the parameters (\(\mathbf{\Theta}\)):  
\[\mathbf{y}=f(\mathbf{u},\mathbf{\Theta}) \tag{1}\]  
If the model structure of the complex system (\(f()\)) is known, it has to be identified to describe the real operation of the system. This means that the model parameters should be determined. They are usually fitted to measurement data if available, or the knowledge of workers who directly interact with the technology and operate it (called experts in this paper) can be integrated into it.  
Experts sometimes cannot provide exact assumptions about the missing information regarding parameters, unmeasured outputs or inputs, but only intervals. By integrating the available uncertain expert knowledge into the system model, the unknown parameters or inputs of the system can be estimated as will be introduced, thus refining the model, and outputs can also be calculated. In the next part of the paper, the possibly uncertain variables and parameters of the model needed to be estimated are gathered in a vector \(\mathbf{x}\):  
\[\mathbf{x}=[\mathbf{u}^{T},\mathbf{\Theta}^{T},\mathbf{y}^{T}]^{T} \tag{2}\]  
and elements of \(\mathbf{x}\) is denoted by \(x_{i}\) in the paper.  
From the members of \(\mathbf{x}\), in our case study, the inputs are considered known, while the parameters and outputs are only available in the form of uncertain expert knowledge. Multiple expert judgments are available regarding all unknown variables in the form of intervals, and these data are even conflicting sometimes. Thereby, the available expert knowledge is represented by probability distributions, and the opinions of multiple experts are aggregated.  
The gained probability distributions make it possible to estimate the parameters and outputs of the system if the model structure is known. The system outputs can be evaluated by a Monte Carlo (MC) simulation using the expert-based probability distributions of the parameters and the system model, and the results can be compared to those suggested by the experts. Thus, the consistency of the expert knowledge about the parameters and outputs can be examined. Based on the probability distributions of expert-based outputs, the parameters can be estimated by the importance sampling (IS) technique. Executing these two steps iteratively, the imprecise information of the parameters and outputs can be brought to a consensus, and the calculations converge to a result in the form of probability distributions, as seen in Figure 1.  
Figure 2 outlines the methodological background of this paper, whose elements are introduced in detail in this section. The system model and expert assessments are available as initial data. The uncertain expert knowledge about the parameters and outputs has to be represented mathematically and transformed into probability distributions. These pre-processing steps are described in Section 2.1. The proposed methodology that uses MC simulation and IS to handle expert knowledge and estimate the most likely probability distributions of the parameters and outputs based on it is introduced in Section 2.2. To evaluate expert judgments, the distributions created from the intervals provided by them are weighted compared to the estimated distributions; thus, goodness values (\(g_{i,e}\)) are generated, which can be used to eliminate incorrect expert judgments, as is described in Section 2.3.