An Extended PROMETHEE-II-Based Risk Prioritization Method for Equipment Failures in the Geothermal Power Plant

Abstract

The equipment failure is an important factor that may cause accidents in the geothermal power plant. Failure mode and effect analysis (FMEA) has been widely adopted to prevent potential failures in various process enterprises. However, current FMEA models are inefficient to determine risk priority by considering the risk preference and reliability of expert within the context of interaction among risk factors. To overcome this shortcoming, this paper develops a hybrid risk prioritization method by integrating PROMETHEE-II method, Bonferroni mean operator and linguistic Z-number. Firstly, the linguistic Z-number is utilized to depict the reliability and uncertainty of experts. Then, the linguistic Z-number ordered weighted averaging operator is developed to aggregate the risk evaluation information from experts, which can reflect the actual performance of experts. In addition, the extended PROMETHEE-II method with distance-based weighting method and Bonferroni mean operator is proposed to calculate the risk priority, in which the interactions among risk factors and risk preference of expert are considered. Finally, the risk analysis of equipment failures in the geothermal power plant is selected to illustrate the proposed method. The comparison and sensitivity analyses are conducted to further validate its effectiveness.

Appendices

Appendix A

Proof

The mathematical induction on $k$ is adopted to prove Theorem 1 expressed by Eq. (16); the proving process is presented as follows:

• (1) When $k = 1$, $\omega^{(1)} = 1$, it is right obviously.

• (2) Assume that Eq. (16) holds for $k = q$, i.e.,

  $${\text{LZ-OWA}}_{\omega } \left( {a_{1} ,a_{2} , \ldots ,a_{q} } \right) = \sum\limits_{{i^{\prime} = 1}}^{q} {\left( {f^{ * - 1} \left( {\omega^{{i^{\prime}}} f^{ * } \left( {A_{{\phi \sigma (i^{\prime})}} } \right)} \right),B_{{\varphi \sigma (i^{\prime})}} } \right)}$$

  Then, when $k = q + 1$, by the algorithms of linguistic Z-numbers in Definition 2, we have

  $$\begin{aligned} & {\text{LZ-OWA}}_{\omega } \left( {a_{1} ,a_{2} , \ldots ,a_{q + 1} } \right) \\ & \quad = \omega^{(1)} a_{\sigma (1)} \oplus \omega^{(2)} a_{\sigma (2)} \oplus \cdots \oplus \omega^{(q + 1)} a_{\sigma (q + 1)} \\ & \quad = \left( {f^{ * - 1} \left( {\omega^{(1)} f^{ * } \left( {A_{{\phi \left( {\sigma (1)} \right)}} } \right)} \right),B_{{\varphi \left( {\sigma (1)} \right)}} } \right) \oplus \left( {f^{ * - 1} \left( {\omega^{(2)} f^{ * } \left( {A_{{\phi \left( {\sigma (2)} \right)}} } \right)} \right),B_{{\varphi \left( {\sigma (2)} \right)}} } \right) \\ & \quad \quad \oplus \cdots \oplus \left( {f^{ * - 1} \left( {\omega^{(q + 1)} f^{ * } \left( {A_{{\phi \left( {\sigma (q + 1)} \right)}} } \right)} \right),B_{{\varphi \left( {\sigma (q + 1)} \right)}} } \right) \\ & \quad = \sum\limits_{{i^{\prime} = 1}}^{q + 1} {\left( {f^{ * - 1} \left( {\omega^{{(i^{\prime})}} f^{ * } \left( {A_{{\phi \left( {\sigma (i^{\prime})} \right)}} } \right)} \right),B_{{\varphi \left( {\sigma (i^{\prime})} \right)}} } \right)} \\ \end{aligned}$$

  So, when $k = q + 1$, Eq. (16) is also right.

Consequently, according to mathematical induction presented in (1) and (2), Eq. (14) holds for all $k$.$\square$

Appendix B

Proof

• (1) As $a_{{i^{\prime}}} = a_{0} = \left( {A_{\phi (0)} ,B_{\varphi (0)} } \right)\left( {i^{\prime} = 1, \ldots k, \ldots ,q} \right)$, and $\sum\limits_{{i^{\prime} = 1}}^{q} {\omega^{{(i^{\prime})}} } = 1$, by the algorithms of linguistic Z-numbers in Definition 2, we have

  $$\begin{aligned} & {\text{LZ-OWA}}_{\omega } \left( {a_{1} , \ldots, a_{k} , \ldots ,a_{q} } \right) \\ & \quad = {\text{LZ-OWA}}_{\omega } \left( {a_{0} , \ldots, a_{0} , \ldots ,a_{q} } \right) \\ & \quad = \sum\limits_{{i^{\prime} = 1}}^{q} {\left( {f^{ * - 1} \left( {\omega^{{(i^{\prime})}} f^{ * } \left( {A_{\phi (0)} } \right)} \right),B_{\varphi (0)} } \right)} \\ & \quad = \left( {f^{ * - 1} \left( {\sum\nolimits_{{i^{\prime} = 1}}^{q} {\omega^{{(i^{\prime})}} } \times f^{ * } \left( {A_{\phi (0)} } \right)} \right),g^{ * - 1} \left( {\frac{{\left( {\sum {f^{ * } \left( {A_{\phi (0)} } \right)} } \right) \times g^{ * } \left( {B_{\varphi (0)} } \right)}}{{\sum {f^{ * } \left( {A_{\phi (0)} } \right)} }}} \right)} \right) \\ & \quad = \left( {f^{ * - 1} \left( {\sum\nolimits_{{i^{\prime} = 1}}^{q} {\omega^{{(i^{\prime})}} } \times f^{ * } \left( {A_{\phi (0)} } \right)} \right),g^{ * - 1} \left( {g^{ * } \left( {B_{\varphi (0)} } \right)} \right)} \right) \\ \end{aligned}$$

  Since $\sum\limits_{{i^{\prime} = 1}}^{q} {\omega^{{(i^{\prime})}} } = 1$, then we can obtain

  $$\begin{aligned} & \left( {f^{ * - 1} \left( {\sum\nolimits_{{i^{\prime} = 1}}^{q} {\omega^{{(i^{\prime})}} } \times f^{ * } \left( {A_{\phi (0)} } \right)} \right),g^{ * - 1} \left( {g^{ * } \left( {B_{\varphi (0)} } \right)} \right)} \right) \\ & \quad = \left( {f^{ * - 1} \left( {f^{ * } \left( {A_{\phi (0)} } \right)} \right),g^{ * - 1} \left( {g^{ * } \left( {B_{\varphi (0)} } \right)} \right)} \right) \\ & \quad = \left( {A_{\phi (0)} ,B_{\varphi (0)} } \right) = a_{0} \\ \end{aligned}$$

  $\square$

• (2) Assume that $a_{\hbox{min} } = c = \left( {A_{\phi (c)} ,B_{\varphi (c)} } \right)$, $a_{\hbox{max} } = d = \left( {A_{\phi (d)} ,B_{\varphi (d)} } \right)$, then

  $$\begin{aligned} & {\text{LZ-OWA}}_{\omega } \left( {a_{1} , \ldots, a_{k} , \ldots ,a_{q} } \right) \\ & \quad = \sum\limits_{{i^{\prime} = 1}}^{q} {\left( {f^{ * - 1} \left( {\omega^{{(i^{\prime})}} f^{ * } \left( {A_{{\phi (i^{\prime})}} } \right)} \right),B_{{\varphi (i^{\prime})}} } \right)} \\ & \quad \ge \sum\limits_{{i^{\prime} = 1}}^{q} {\left( {f^{ * - 1} \left( {\omega^{{(i^{\prime})}} f^{ * } \left( {A_{\phi (c)} } \right)} \right),B_{\varphi (c)} } \right)} = \left( {A_{\phi (c)} ,B_{\varphi (c)} } \right) \\ \end{aligned}$$

  and

  $$\begin{aligned} & {\text{LZ-OWA}}_{\omega } \left( {a_{1} , \ldots, a_{k} , \ldots ,a_{q} } \right) \\ & \quad = \sum\limits_{{i^{\prime} = 1}}^{q} {\left( {f^{ * - 1} \left( {\omega^{{(i^{\prime})}} f^{ * } \left( {A_{{\phi (i^{\prime})}} } \right)} \right),B_{{\varphi (i^{\prime})}} } \right)} \\ & \quad \le \sum\limits_{{i^{\prime} = 1}}^{q} {\left( {f^{ * - 1} \left( {\omega^{{(i^{\prime})}} f^{ * } \left( {A_{\phi (d)} } \right)} \right),B_{\varphi (d)} } \right)} = \left( {A_{\phi (d)} ,B_{\varphi (d)} } \right) \\ \end{aligned}$$

  Therefore, we can get

  $$a_{\hbox{min} } \le {\text{LZ-OWA}}_{\omega } \left( {a_{1} , \ldots, a_{k} , \ldots ,a_{q} } \right) \le a_{\hbox{max} }$$

  $\square$

• (3) Let ${\text{LZ-OWA}}_{\omega } \left( {a_{1} , \ldots, a_{k} , \ldots ,a_{q} } \right) = \sum\limits_{{i^{\prime} = 1}}^{q} {\left( {f^{ * - 1} \left( {\omega^{{(i^{\prime})}} f^{ * } \left( {A_{{\phi (\sigma (i^{\prime}))}} } \right)} \right),B_{{\varphi (\sigma (i^{\prime}))}} } \right)}$, ${\text{LZ-OWA}}_{\omega } \left( {b_{1} , \ldots b_{k} , \ldots ,b_{q} } \right) = \sum\limits_{{j^{\prime} = 1}}^{q} {\left( {f^{ * - 1} \left( {\omega^{{(j^{\prime})}} f^{ * } \left( {A_{{\phi (\sigma (j^{\prime}))}} } \right)} \right),B_{{\varphi (\sigma (j^{\prime}))}} } \right)} .$ Since $A_{{\phi (i^{\prime})}} \le A_{{\phi (j^{\prime})}}$, $B_{{\varphi (i^{\prime})}} \le B_{{\varphi (j^{\prime})}}$, it follows that $a_{{\sigma (i^{\prime})}} \le b_{{\sigma (j^{\prime})}} \left( {i^{\prime},j^{\prime} = 1,2, \ldots ,q} \right)$, then

  $${\text{LZ-OWA}}_{\omega } \left( {a_{1} , \ldots, a_{k} , \ldots ,a_{q} } \right) \le {\text{LZ-OWA}}_{\omega } \left( {b_{1} , \ldots b_{k} , \ldots ,b_{q} } \right) .$$

  $\square$