**A. Appendix: computation of the MGF of** *T*~*T*~

The MGF of *T*~*T*~can be represented as follows:

*Reliability-Based Marginal Cost Pricing Problem DOI: http://dx.doi.org/10.5772/intechopen.92844*

$$\begin{split} \mathcal{M}\_{\mathcal{TT}}(\boldsymbol{s}) &= \sum\_{a \in A} \boldsymbol{E} \left[ \exp \left( \boldsymbol{s} \boldsymbol{V}\_{a} \boldsymbol{\varvarproj} \boldsymbol{T}\_{a} \right) \right] \\ &= \sum\_{a \in A} \boldsymbol{E} \left\{ \exp \left[ \boldsymbol{s} \boldsymbol{V}\_{a} (\boldsymbol{T}\_{a} + \boldsymbol{e}\_{a}) \right] \right\} \\ &= \sum\_{a \in A} \boldsymbol{E}\_{T\_{a}} \left\{ \exp \left( \boldsymbol{s} \boldsymbol{V}\_{a} \boldsymbol{T}\_{a} \right) \exp \left( \boldsymbol{s} \boldsymbol{V}\_{a} \boldsymbol{e}\_{a} \right) \right\} \\ &= \sum\_{a \in A} \boldsymbol{E}\_{T\_{a}} \left\{ \exp \left( \boldsymbol{s} \boldsymbol{V}\_{a} \boldsymbol{T}\_{a} \right) \boldsymbol{E}\_{\boldsymbol{e}\_{a} \mid \boldsymbol{r}\_{a}} \left[ \exp \left( \boldsymbol{s} \boldsymbol{V}\_{a} \boldsymbol{e}\_{a} \right) \right] \right\} \\ &= \sum\_{a \in A} \boldsymbol{E}\_{T\_{a}} \left\{ \exp \left( \boldsymbol{s} \boldsymbol{V}\_{a} \boldsymbol{T}\_{a} \right) \boldsymbol{M}\_{\boldsymbol{e}\_{a} \mid \boldsymbol{r}\_{a}} \left( \boldsymbol{s} \boldsymbol{V}\_{a} \right) \right\} \\ &= \sum\_{a \in A} \boldsymbol{E}\_{T\_{a}} \left\{ \exp \left( \boldsymbol{s} \boldsymbol{V}\_{a} \boldsymbol{T}\_{a} \right) \exp \left[ \boldsymbol{s} \boldsymbol{V}\_{a} \boldsymbol{T}\_{a} \left( \boldsymbol{\chi} + \boldsymbol{\varpi}^{2} \boldsymbol{s} \boldsymbol{V}\_{a} / 2 \right) \right] \right\} \end{split} \tag{68}$$

The first-order moment is, from the first derivative evaluated at *s* ¼ 0,

$$E[\tilde{T}\tilde{T}] = \sum\_{a \in A} (1 + \chi) E[V\_a T\_a] \tag{69}$$

Similarly, the second-order moment of *T*~*T*~ can be derived from the second derivative evaluated at*s* ¼ 0,

$$E\left[\left(\ddot{T}\ddot{T}\right)^{2}\right] = \sum\_{a \in A} \left\{ (\mathbf{1} + \boldsymbol{\chi})^{2} E\left[ (V\_{a}T\_{a})^{2} \right] + \varpi^{2} E\left[ V\_{a}^{2}T\_{a} \right] \right\} \tag{70}$$

Then we can obtain the variance of *T*~*T*~ as follows:

$$\begin{split} \text{Var}\left[\tilde{T}\tilde{T}\right] &= E\left[\left(\tilde{T}\tilde{T}\right)^{2}\right] - E\left[\tilde{T}\tilde{T}\right]^{2} \\ &= \sum\_{a \in A} \left\{ (\mathbb{1} + \chi)^{2} \left\{ E\left[ \left(V\_{a}T\_{a}\right)^{2} \right] - E\left[V\_{a}T\_{a}\right]^{2} \right\} + \varpi^{2} E\left[V\_{a}^{2}T\_{a}\right] \right\} \\ &= \sum\_{a \in A} \left\{ (\mathbb{1} + \chi)^{2} \text{Var}[TT] + \varpi^{2} E\left[V\_{a}^{2}T\_{a}\right] \right\} \end{split} \tag{71}$$
