## 3 Cost Model of Predictive Maintenance Considering Prospect Theory
### Prospect Theory
#### 3.1.3. Probability Transformation Function  
\(\pi(.)\) in Equation (12) is not a measure of probability but a continuous, increasing and generally nonlinear function verifying the following: \(\pi(0)=0\) and \(\pi(1)=1\). This function is sub-additive: \(\pi(p)+\pi(1-p)<1\). Finally, this function tends to overestimate the small probabilities: \(\pi(p)>p\) for small probabilities (Figure 3). As mentioned previously, we  
Figure 2: Valuation function.  
distinct positive consequences \(x_{i,i\in[0..,n]}\) from negative consequences \(x_{ij,i\in[-m,..,0]}\) with their respectively transformed probabilities \(\pi_{i}^{+}\) and \(\pi_{j}^{-}\):  
\[\begin{array}{l}\pi_{i}^{+}=w^{+}(p_{n}),\tau_{-m}^{-}=w^{-}(p_{-m})\\ \pi_{i}^{+}=w^{+}(\sum_{j=i}^{n}p_{j})-w^{+}(\sum_{j=i+1}^{n}p_{j})\text{ for }i\in[0,..,n-1]\\ \pi_{i}^{-}=w^{-}(\sum_{j=-m}^{i-1}p_{j})-w^{-}(\sum_{j=-m}^{i-1}p_{j})\text{ for }i\in[1-m,..,0]\end{array} \tag{16}\]  
where \(w^{+}\) and \(w^{-}\) are strictly increasing functions from the unit interval to itself verifying the following: \(w^{+}(0)=w^{-}(0)=0\) and \(w^{+}(1)=w^{-}(1)=1\). According to [PERSON] and [PERSON], \(w^{+}\) and \(w^{-}\) can be expressed as follows:  
\[\begin{array}{l}w^{+}(p)=\frac{p^{\gamma_{1}}}{(p^{\gamma_{1}}+(1-p)^{\gamma _{1}})^{\frac{1}{\gamma_{1}}}}\\ w^{-}(p)=\frac{p^{\gamma_{2}}}{(p^{\gamma_{2}}+(1-p)^{\gamma_{2}})^{\frac{1}{ \gamma_{2}}}}\end{array} \tag{17}\]  
The parameters \(\gamma_{1}\) and \(\gamma_{2}\) of this proposed probability transformation function have been estimated empirically using the same method as to estimate the parameters of the valuation function. The mean values of the estimated parameters of the weighting function are: \(\tilde{\gamma_{1}}=0.61\) and \(\tilde{\gamma_{2}}=0.69\)[11,44].  
Several empirical studies have been conducted, with other values than the ones above. While it is clear that such parameters depend on the individuals, they confirm stable features of human behavior. For example, [PERSON] and [PERSON] (1992) conducted the same kind of experience on Chinese students, which allowed, thanks to the differences in standards of living, to expand the scale of the monetary consequences [45]. The main features were confirmed in this new context.  
For the value function, first, with a concave shape in gains and a median estimate with regard to the loss domain confirming the convex shape, please refer to these two works [46,47]. For the probability weighting function, the pattern reflects an overweighting of less likely events and an underweighting of rather likely events in the gains as well as in the losses. This finding is broadly consistent with an inverse S-shaped transformation function. [PERSON] and [PERSON] (1994) find a value of 0.56 in the gains [44], [PERSON] and [PERSON] (1996) of 0.71 in the gains [48]. Values of 0.60 in the gains and 0.70 in the losses were found in [46], while values between 0.67 and 0.71 were found in in [49] according to the chosen approximation model. In this context, the empirical studies confirm that small probabilities are amplified and that the parameters are closed to the ones proposed initially.  
Figure 3: Probability transformation function.