Algorithm 2. Incremental algorithm when adding a conditional value in C-SODS

Input: (1) The set-valued ordered decision information system at time t + 1;

(2) The dominating sets and dominated sets for each x at time t: ${\left[x\right]}_B^{+,\left(t\right)}$ , ${\left[x\right]}_B^{-,\left(t\right)}$ , $\forall x\in U$ ;

(3) The upward unions and downward unions: $C{l}_n^{\ge }$ , $C{l}_n^{\le }\left(0\le n\le t\right)$ ;

(4) The approximations at time t: ${\underset{\_}{R}}_B^{\left(t\right)}\left(C{l}_n^{\ge }\right)$ , ${\overline{R}}_B^{\left(t\right)}\left(C{l}_n^{\ge }\right)$ , ${\underset{\_}{R}}_B^{\left(t\right)}\left(C{l}_n^{\le }\right)$ , ${\overline{R}}_B^{\left(t\right)}\left(C{l}_n^{\le }\right)$ ;

(5) A object sets which add a conditional value: U⁺; (6) A object sets in which the addition of a conditional value has not occurred: $\overline{U}$ .

Output: ${\underset{\_}{R}}_B^{\left(t+1\right)}\left(C{l}_n^{\ge }\right)$ , ${\overline{R}}_B^{\left(t+1\right)}\left(C{l}_n^{\ge }\right)$ , ${\underset{\_}{R}}_B^{\left(t+1\right)}\left(C{l}_n^{\le }\right)$ , ${\overline{R}}_B^{\left(t+1\right)}\left(C{l}_n^{\le }\right)$

1: if $v_a^{+}\in f^{\left(t+1\right)}\left(y,a\right)$ then

2: $U^{+}\leftarrow U^{+}\cup \left\{y\right\}$

3: else

4: $\overline{U}\leftarrow \overline{U}\cup \left\{x\right\}$

5: for $\forall y\in U^{+}$

6: ${\left[y\right]}_B^{+,\left(t+1\right)}\leftarrow {\left[y\right]}_B^{+,\left(t\right)}-\overline{U}$

7: if $\exists y\notin {\underset{\_}{R}}_B^{\left(t\right)}\left(C{l}_n^{\ge }\right)$ and $\overline{U}\cancel{\subseteq }C{l}_n^{\ge }$ and ${\left[y\right]}_B^{+,\left(t+1\right)}\subseteq C{l}_n^{\ge }$ then

8: ${\underset{\_}{R}}_B^{\left(t+1\right)}\left(C{l}_n^{\ge }\right)\leftarrow {\underset{\_}{R}}_B^{\left(t\right)}\left(C{l}_n^{\ge }\right)\cup \left\{y\right\}$

9: if $\exists y\in {\overline{R}}_B^{\left(t\right)}\left(C{l}_n^{\le }\right)$ and $\overline{U}\subseteq C{l}_n^{\le }$ and ${\left[y\right]}_B^{+,\left(t+1\right)}\cap C{l}_n^{\le }=\varnothing$ then

10: ${\overline{R}}_B^{\left(t+1\right)}\left(C{l}_n^{\le }\right)\leftarrow {\overline{R}}_B^{\left(t\right)}\left(C{l}_n^{\le }\right)-\left\{y\right\}$

11: for $\forall x\in \overline{U}$

12: ${\left[x\right]}_B^{-,\left(t+1\right)}\leftarrow {\left[x\right]}_B^{-,\left(t\right)}-U^{+}$

13: if $\exists x\notin {\underset{\_}{R}}_B^{\left(t\right)}\left(C{l}_n^{\le }\right)$ and $U^{+}\cancel{\subseteq }C{l}_n^{\le }$ and ${\left[x\right]}_B^{-,\left(t+1\right)}\subseteq C{l}_n^{\le }$ then

14: ${\underset{\_}{R}}_B^{\left(t+1\right)}\left(C{l}_n^{\le }\right)\leftarrow {\underset{\_}{R}}_B^{\left(t\right)}\left(C{l}_n^{\le }\right)\cup \left\{x\right\}$

15: if $\exists x\in {\overline{R}}_B^{\left(t\right)}\left(C{l}_n^{\ge }\right)$ and $U^{+}\subseteq C{l}_n^{\ge }$ and ${\left[x\right]}_B^{-,\left(t+1\right)}\cap C{l}_n^{\le }=\varnothing$ then

16: ${\overline{R}}_B^{\left(t+1\right)}\left(C{l}_n^{\ge }\right)\leftarrow {\overline{R}}_B^{\left(t\right)}\left(C{l}_n^{\ge }\right)-\left\{x\right\}$

17: Return ${\underset{\_}{R}}_B^{\left(t+1\right)}\left(C{l}_n^{\ge }\right)$ , ${\overline{R}}_B^{\left(t+1\right)}\left(C{l}_n^{\ge }\right)$ , ${\underset{\_}{R}}_B^{\left(t+1\right)}\left(C{l}_n^{\le }\right)$ , ${\overline{R}}_B^{\left(t+1\right)}\left(C{l}_n^{\le }\right)$