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Table 1 Application of two kinds of evaluation factor types in membership function

From: Risk assessment methodology for iron stability under water quality factors based on fuzzy comprehensive evaluation

Membership functionIncremental factorDescending factor
\(\mu_{{\nu_{1} }} \left( {u_{i} } \right)\; = \;\left\{ \begin{aligned} 1 \hfill \\ 0.5 \times \left[ {1 + \frac{{u_{i} - k_{1} }}{{k_{1}^{'} - k_{1} }}} \right] \hfill \\ 0.5 \times \left[ {1 - \frac{{k_{1} - u_{i} }}{{k_{1} - k_{2}^{'} }}} \right] \hfill \\ 0 \hfill \\ \end{aligned} \right.\)\(\mu_{{\nu_{1} }} \left( {u_{i} } \right){ = }\left\{ {\begin{array}{*{20}c} {u_{i} < k^{\prime}_{1} } \\ {k^{\prime}_{1} \le u_{i} < k_{1} } \\ {k_{1} \le u_{i} < k^{\prime}_{2} } \\ {u_{i} \ge k_{2}^{'} } \\ \end{array} } \right.\)\(\mu_{{\nu_{1} }} \left( {u_{i} } \right){ = }\left\{ {\begin{array}{*{20}c} {u_{i} \ge k_{1}^{'} } \\ {k_{1}^{'} > u_{i} \ge k_{1} } \\ {k_{1} > u_{i} \ge k_{2}^{'} } \\ {u_{i} < k_{2}^{'} } \\ \end{array} } \right.\)
\(\mu_{{\nu_{2} }} \left( {u_{i} } \right){ = }\left\{ \begin{aligned} 0 \hfill \\ 0.5 \times \left[ {1 - \frac{{u_{i} - k_{1} }}{{k^{\prime}_{1} - k_{1} }}} \right] \hfill \\ 0.5 \times \left[ {1 + \frac{{k_{1} - u_{i} }}{{k_{1} - k^{\prime}_{2} }}} \right] \hfill \\ 0.5 \times \left[ {1 + \frac{{u_{i} - k_{2} }}{{k^{\prime}_{2} - k_{2} }}} \right] \hfill \\ 0.5 \times \left[ {1 - \frac{{k_{2} - u_{i} }}{{k_{2} - k^{\prime}_{3} }}} \right] \hfill \\ 0 \hfill \\ \end{aligned} \right.\)\(\mu_{{\nu_{2} }} \left( {u_{i} } \right){ = }\left\{ {\begin{array}{*{20}c} {u_{i} < k^{\prime}_{1} } \\ {k^{\prime}_{1} \le u_{i} < k_{1} } \\ {k_{1} \le u_{i} < k^{\prime}_{2} } \\ {k^{\prime}_{2} \le u_{i} < k_{2} } \\ {k_{2} \le u_{i} < k^{\prime}_{3} } \\ {u_{i} \ge k^{\prime}_{3} } \\ \end{array} } \right.\)\(\mu_{{\nu_{2} }} \left( {u_{i} } \right)\; = \;\left\{ {\begin{array}{*{20}c} {u_{i} \ge k^{\prime}_{1} } \\ {k^{\prime}_{1} > u_{i} \ge k_{1} } \\ {k_{1} > u_{i} \ge k^{\prime}_{2} } \\ {k^{\prime}_{2} > u_{i} \ge k_{2} } \\ {k_{2} > u_{i} \ge k^{\prime}_{3} } \\ {u_{i} < k^{\prime}_{3} } \\ \end{array} } \right.\)
\(\mu_{{\nu_{3} }} \left( {u_{i} } \right){ = }\left\{ \begin{aligned} 0 \hfill \\ 0.5 \times \left[ {1 - \frac{{u_{i} - k_{2} }}{{k^{\prime}_{2} - k_{2} }}} \right] \hfill \\ 0.5 \times \left[ {1 + \frac{{u_{i} - k_{3} }}{{k^{\prime}_{3} - k_{3} }}} \right] \hfill \\ 0.5 \times \left[ {1 - \frac{{k_{3} - u_{i} }}{{k_{3} - k^{\prime}_{4} }}} \right] \hfill \\ 0 \hfill \\ \end{aligned} \right.\)\(\mu_{{\nu_{3} }} \left( {u_{i} } \right){ = }\left\{ {\begin{array}{*{20}c} {u_{i} < k^{\prime}_{2} } \\ {k^{\prime}_{2} \le u_{i} < k_{2} } \\ {k_{2} \le u_{i} < k^{\prime}_{3} } \\ {k^{\prime}_{3} \le u_{i} < k_{3} } \\ {k_{3} \le u_{i} < k^{\prime}_{4} } \\ {u_{i} \ge k^{\prime}_{4} } \\ \end{array} } \right.\)\(\mu_{{\nu_{3} }} \left( {u_{i} } \right){ = }\left\{ {\begin{array}{*{20}c} {u_{i} \ge k^{\prime}_{2} } \\ {k^{\prime}_{2} > u_{i} \ge k_{2} } \\ {k_{2} > u_{i} \ge k^{\prime}_{3} } \\ {k^{\prime}_{3} > u_{i} \ge k_{3} } \\ {k_{3} > u_{i} \ge k^{\prime}_{4} } \\ {u_{i} < k^{\prime}_{4} } \\ \end{array} } \right.\)
\(\mu_{{\nu_{4} }} \left( {u_{i} } \right){ = }\left\{ \begin{aligned} 0 \hfill \\ 0.5 \times \left[ {1 - \frac{{u_{i} - k_{3} }}{{k^{\prime}_{3} - k_{3} }}} \right] \hfill \\ 0.5 \times \left[ {1 + \frac{{k_{3} - u_{i} }}{{k_{3} - k^{\prime}_{4} }}} \right] \hfill \\ 0.5 \times \left[ {1 + \frac{{u_{i} - k_{4} }}{{k^{\prime}_{4} - k_{4} }}} \right] \hfill \\ 0.5 \times \left[ {1 - \frac{{k_{4} - u_{i} }}{{k_{4} - k^{\prime}_{5} }}} \right] \hfill \\ \end{aligned} \right.\)\(\mu_{{\nu_{4} }} \left( {u_{i} } \right){ = }\left\{ {\begin{array}{*{20}c} {u_{i} < k^{\prime}_{3} } \\ {k^{\prime}_{3} \le u_{i} < k_{3} } \\ {k_{3} \le u_{i} < k^{\prime}_{4} } \\ {k^{\prime}_{4} \le u_{i} < k_{4} } \\ {k_{4} \le u_{i} < k^{\prime}_{5} } \\ {u_{i} \ge k^{\prime}_{5} } \\ \end{array} } \right.\)\(\mu_{{\nu_{4} }} \left( {u_{i} } \right){ = }\left\{ {\begin{array}{*{20}c} {u_{i} \ge k^{\prime}_{3} } \\ {k^{\prime}_{3} > u_{i} \ge k_{3} } \\ {k_{3} > u_{i} \ge k^{\prime}_{4} } \\ {k^{\prime}_{4} > u_{i} \ge k_{4} } \\ {k_{4} > u_{i} \ge k^{\prime}_{5} } \\ {u_{i} < k^{\prime}_{5} } \\ \end{array} } \right.\)
\(\mu_{{\nu_{5} }} \left( {u_{i} } \right){ = }\left\{ \begin{aligned} 0 \hfill \\ 0.5 \times \left[ {1 - \frac{{u_{i} - k_{4} }}{{k_{4}^{'} - k_{4} }}} \right] \hfill \\ 0.5 \times \left[ {1 + \frac{{k_{4} - u_{i} }}{{k_{4} - k_{5}^{'} }}} \right] \hfill \\ 0 \hfill \\ \end{aligned} \right.\)\(\mu_{{\nu_{5} }} \left( {u_{i} } \right){ = }\left\{ {\begin{array}{*{20}c} {u_{i} < k^{\prime}_{4} } \\ {k^{\prime}_{4} \le u_{i} < k_{4} } \\ {k_{4} \le u_{i} < k^{\prime}_{5} } \\ {u_{i} \ge k^{\prime}_{5} } \\ \end{array} } \right.\)\(\mu_{{\nu_{5} }} \left( {u_{i} } \right){ = }\left\{ {\begin{array}{*{20}c} {u_{i} \ge k^{\prime}_{4} } \\ {k^{\prime}_{4} > u_{i} \ge k_{4} } \\ {k_{4} > u_{i} \ge k^{\prime}_{5} } \\ {u_{i} < k^{\prime}_{5} } \\ \end{array} } \right.\)
  1. k1, k2, k3, k4, k5 represented the critical value of the adjacent classification interval; k1, k2, k3, k4 and k5 were the midpoint of the rank of the corresponding level v1, v2, v3, v4, and v5, respectively