**Algorithm 1: Algorithm based on The quadripartitioned weighted similarty measure**

Step 1. Give the QBN-multi-attribute decision making matrix [*bij*]*m*×*<sup>n</sup>* to the decision maker.

Step 2. Compute the positive ideal QBN solution ¯*b* ∗ *j* and negative ideal QBN solution *b* ∗ *j* for the decision matrix [*bij*]*m*×*n*.

Step 3. Determine T *w* 2 ( ¯*b* ∗ *j* , *bi* ) for *j* = 1, 2, ..., *m*, the weighted quadripartitioned similarity measure between positive ideal solution ¯*b* ∗ *j* and *<sup>b</sup><sup>i</sup>* = [*bij*]1×*<sup>n</sup>* for *<sup>i</sup>* = 1, 2, ..., *<sup>m</sup>* and *<sup>j</sup>* = 1, 2, ..., *<sup>n</sup>* and T *w* 2 (*b* ∗ *j* , *bi* ) for *j* = 1, 2, ..., *m*, the weighted quadripartitioned similarity measure between negative ideal solution *b* ∗ *j* and *<sup>b</sup><sup>i</sup>* = [*bij*]1×*<sup>n</sup>* for *i* = 1, 2, ..., *m* and *j* = 1, 2, ..., *n* as:

$$\begin{split} \bullet \ \mathsf{T}\_{2}^{w}(\bar{b}\_{j}^{\*},h\_{i}) &= \mathbbm{1} - \left[ \frac{1}{n} \sum\_{k=1}^{n} w\_{k} \left( \frac{1}{\mathsf{4}} \left( \bar{\delta}\_{1}^{\flat\_{j},h\_{i}}(\mathsf{x}\_{k}) + \bar{\delta}\_{2}^{\flat\_{j},h\_{i}}(\mathsf{x}\_{k}) + \lambda\_{1}^{\flat\_{j},h\_{i}}(\mathsf{x}\_{k}) + \lambda\_{2}^{\flat\_{j},h\_{i}}(\mathsf{x}\_{k}) \right) \right)^{p} \right]^{\frac{1}{p}} \\ \bullet \ \ \mathsf{T}^{w}(h^{\*},h\_{i}) &= \mathbbm{1} - \left[ \mathbbm{1} \sum\_{m=1}^{n} \left( \mathbbm{1}\_{\{\mathsf{A}^{\flat}\_{j},h\_{i}^{\*}(\mathsf{x}\_{m}) + \mathsf{A}^{\sharp\_{j},h\_{i}^{\*}(\mathsf{x}\_{m})}(\mathsf{x}\_{k}) + \mathsf{A}^{\sharp\_{j},h\_{i}^{\*}(\mathsf{x}\_{m})}(\mathsf{x}\_{k}) + \mathsf{A}^{\sharp\_{j},h\_{i}^{\*}(\mathsf{x}\_{m})}(\mathsf{x}\_{k}) \right) \right]^{\frac{1}{p}} \end{split}$$

$$\begin{split} \bullet \ \mathcal{T}\_{2}^{w}(\underline{b}\_{j}^{\*},h\_{l}) &= 1 - \left[ \frac{1}{n} \sum\_{k=1}^{n} w\_{k} \left( \frac{1}{4} \left( \boldsymbol{\delta}\_{1}^{\underline{k}\_{1}^{\*},h\_{l}}(\boldsymbol{\chi}\_{k}) + \boldsymbol{\delta}\_{2}^{\underline{k}\_{1}^{\*},h\_{l}}(\boldsymbol{\chi}\_{k}) + \boldsymbol{\lambda}\_{1}^{\underline{k}\_{1}^{\*},h\_{l}}(\boldsymbol{\chi}\_{k}) + \boldsymbol{\lambda}\_{2}^{\underline{k}\_{1}^{\*},h\_{l}}(\boldsymbol{\chi}\_{k}) \right) \right)^{p} \right] \\ \bullet \ \mathsf{C} \ \mathsf{s} \ \ \mathsf{d} \ \mathsf{Fin} \ \mathsf{s} \ \mathsf{d} \ \mathsf{the} \ \mathsf{c} \ \mathsf{i} \ \mathsf{end} \ \mathsf{end} \ \mathsf{c} \ \mathsf{d} \ \mathsf{the} \ \mathsf{c} \ \mathsf{s} \ \mathsf{d} \ \mathsf{end} \ \mathsf{end} \ \qquad \qquad \mathcal{T}\_{2}^{w}(\boldsymbol{b}\_{r}^{\*},h\_{l}) + \mathcal{T}\_{2}^{w}(\underline{b}\_{r}^{\*},h\_{l}) \ \mathsf{d} \ \mathsf{and} \ \mathsf{d} \ \mathsf{int} \ \mathsf{s} \ \mathsf{d} \ \mathsf{int} \ \mathsf{d} \ \mathsf{int} \ \mathsf{end} \ \qquad \qquad \mathsf{s} \ \mathsf{d} \ \mathsf{end} \ \mathsf{d} \ \mathsf{end}$$

Step 4. Figure out the non-increasing order of the average ideal solution, <sup>T</sup> 2 *j* ,*bi*)+T 2 *j* 2 for *j* = 1, 2, ..., *m* and select the best alternatives.
