## 2 Cost Model of Predictive Maintenance
### Risks
#### 2.3.1 Human Risks  
**Value of statistical life (VSL):** a challenging task is to evaluate risks without considering the financial aspect. Risks to life and health are usually distinct from financial risks. They cannot be evaluated in money. To deal with this issue, economists have introduced first the measure of human capital to assess human life on the basis of its contribution to the well-being of the society [27]. From the 1960s, a new approach appeared in the literature to replace the human capital. This approach is called willingness to pay. The value of life is thus measured by the amount a person is willing to pay to reduce their exposure to risk. There are two main methods for measuring the willingness to pay: the method of revealed preferences which measures the individuals' preferences for risk by observing only their behavior on the market and the method of contingent valuation which measures the individuals' preferences for risk in hypothetical market situations. The empirical estimation of the willingness to pay brought out the concept of VSL [27; 28; 29]. The VSL is the most widespread terminology to represent the rate of trade-off between fatality and money: it reflects the worker's consent to pay to accept risk and for more safety. The VSL terminology highlights the valuation probabilistic aspect because at the moment of decision, the lives to be saved are only known in a probabilistic way. The VSL has attractive properties: according to [30], it gives a cardinal measure for the life valuation rather than an ordinal measure. Moreover, it is applied not only to estimate the value of the willingness-to-pay but also the disposal to accept risk changes. For example, in France, the VSL has been estimated to three million euros according to a governmental report on elements of review on the value of human life [31]. It can be around 4.5 million euros for OECD countries [32,33]. The data on VSL have been actualized to 5 million euros (the value adopted in this paper).  
Let us note \(\{p_{j,i}^{d}\}_{1\leq j\leq n,1\leq i\leq N_{in}}\): the matrix of death probabilities, where \(p_{j,i}^{d}\) is the probability of death of the person \(j\) in the case of occurrence of failure between inspection \(i\) and \(i\)+1. The human risks \(R_{h}\) during the planning horizon \(D_{T}\) by taking into account the expected number of failures can be evaluated by:  
\[R_{h}=VSL\cdot\sum_{j=1}^{j=n}\sum_{i=1}^{N_{in}}p_{j,i}^{d}. \tag{6}\]  
It is obvious that the term \(p_{j,i}^{d}\) contains the probability of death of \(j\)th person due to system failure multiplied by the probability of system failure between inspection \(i\) and \(i\)+1. This latest probability is of course equal to \((1-N_{i})\int_{t_{i}}^{t_{i+1}}f_{i}(t\backslash T>t_{i})\cdot dt\), where, as a reminder, \(f_{i}\) is the failure probability density function of the system and \(N_{i}\) is the decision variable indicator of PdM between inspection \(i\) and \(i\)+1.  
This method can be applied, similarly, to evaluate the human injury risks: different grades of injury are to be considered with their equivalent compensation costs.