# Life-Cycle Oriented Risk Assessment Using a Monte Carlo Simulation
## 4 Results
### Generic System Breakdown  
To understand how potential warranty issues can occur--i.e., differences between the customer's expectations and the performance provided--a generic breakdown of the relevant system dynamics is created. Figure 4 shows the suggested system breakdown structure for the subsequent risk assessment. The structure is generic for a multi-system product, i.e., a system consisting of multiple subsystems and parts. A generic product has specifications that originate from the desired outcome and performance of the system. Ideally those are formulated by or together with the system's user. Based on this general goal, the product's specification is obtained by formulating use-cases and requirements.  
The products need manufacturing and procurement processes to be created during the BOL. These processes will necessarily affect the products' performance. As for the product, its sub-assemblies have specifications, show performances, need manufacturing and procurement processes, and might include (sub-)sub-assemblies or parts, the details of which are covered in the overall specification. Importantly, the overall performance is affected by the performances of the sub-assemblies.  
During its MOL, the product will show a specific performance. The difference between this performance and the specification is denoted as \(\Delta\)y. The goal of the risk management introduced later is to minimize the negative entries of \(\Delta\)y. In an ideal world, \(\Delta\)y can be quantified precisely. However, in practice, two issues must be addressed:  
1. Uncertainties in the product specification: These can be caused by a mismatch between the true customer expectations and the assumed expectations. Either the customer has not been sufficiently involved in the specification synthesis or the customer's desired performance is not correctly achieved. An alternative cause can be differing expectations between different users: what might be good for one customer might be unsuitable for another customer.
2. Uncertainties in the product's performance: The overall performance of a product depends on various factors. A firm must distinguish between a general uncertainty over all manufactured products (general performance offset) and the product's specific offset.  
Given a customer specification \(s_{j}\) and a product with general performance \(p_{i}\) and specific performance offset \(\delta p_{i}\), the resulting specification offset is given as follows:  
\[\Delta\)y\({}_{j,i}=s_{j}-(p_{i}+\delta p_{i}) \tag{1}\]  
As stated above, \(s_{j}\) and \((p_{i}+\delta p_{i})\) are subject to uncertainties. These uncertainties are here known as the probability density functions (PDF) \(s_{j}\sim f_{s}\) and \(p_{i}+\delta p_{i}\sim f_{p}\). Thus, \(\Delta\)y\({}_{j,i}\) can be estimated using its PDF:  
\[f_{\Delta}=f_{s}*f_{p} \tag{2}\]  
\(f_{\Delta}\) represents the probability of having a difference \(\Delta\)y between specified and realized performance. Given the costs of such a difference as \(c_{\Delta}\) for a specific situation, the expected risks in terms of costs can be evaluated, i.e., by providing accurate estimations of \(c_{\Delta}\) and \(f_{\Delta}\) the expected costs and standard deviation of the costs are  
\[\begin{array}{rcl}E_{c}&=&\int_{-\infty}^{\infty}c_{\Delta}(\text{u})\cdot f _{\Delta}(\text{u})\;\text{du}&\text{and}\\ \sigma_{c}&=&\sqrt{\int_{-\infty}^{\infty}(c_{\Delta}(\text{u})-E_{c})^{2}\; \text{du}}.\end{array} \tag{3}\]