**5. Discussion**

According to the present method and the methods proposed in [25,43,44], to reach the process consensus, Yang et al. [25] use the "AND" operator, while methods in [43,44] did not discuss the aggregation problem. In addition, Example 2 shows that all the methods have the same option *o*8, which is the best object. Consequently, algorithms in methods [25,43,44] select just one option, which is the optimum, and do not select the worst option, while the proposed algorithm selects two options—the optimum and as well as the worst option. However, the methods in [25,43,44], rank the objects based on a linear ordering system (see Example 3), while the present method ranks the objects based on preorder relation and a preference relationship, which allows one to have some incomparable objects (nonlinear ordering system). For example, in the Example 2, the objects *o*12 and *o*5 have the same overall score values, which means that these objects cannot be compared with all of the others.

This is the same for the objects *o*13, *o*1 and *o*14, *o*4 (see Figure 2). The comparison results between the new proposed method and methods in [25,43,44] are also given in Table 11.



$$
\begin{pmatrix} \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \\ \bullet \end{pmatrix}$$

**Figure 2.** Nonlinear ordering system.
