**6. Conclusions**

For the constrained OWA aggregation problem with one constraint on the sum of all variables, this paper introduces some constraints to reduce the multiple solution problem. For the three-dimensional constrained OWA aggregation problem with the same lower bounds, by using the change of variables, the optimal solution is the same as that of the constrained OWA aggregation problem without lower bounded variables. For the three-dimensional constrained OWA aggregation problem with lower bounded variables, this paper presents four types (I, II, III and IV) of solutions depending on the number of zero elements. When the number of zero elements of solution is two (type I), there are three closed-form expressions of *X* and six closed-form expressions of *Y*. When the number of zero elements of the solution is one (types II and III), there are three closed-form expressions of *X* and six closed-form expressions of *Y* for type II, and six closed-form expressions of *X* and six closed-form expressions of *Y* for type III. When the number of zero elements of the solution is zero (type IV), there is only one closed-form expression of *X* and one closed-form expression of *Y*. According to the computerized experiment we perform for the three-dimensional constrained OWA aggregation problem with lower bounded variables, the optimal solution type is I for *w*1 = max *i*=1,2,3 *wi* and *w*1 > max *i*=2,3 *wi*, and types I, II, *wi*.

III and IV for *w*2 = max *i*=1,2,3 *wi*, *w*3 = max *i*=1,2,3 *wi*, *w*2 > max *i*=1,3 *wi* and *w*3 > max *i*=2,3

Worthy of future research is that the analysis is extended to the lower and upper bounded variables for the constrained OWA aggregation problem, especially for the three-dimensional constrained OWA aggregation problem with upper bounded variables. Thus, the analysis of the constrained OWA aggregation problem with bounded variables is a subject of considerable ongoing research.

**Author Contributions:** Y.-F.C. conceived the study and performed the experiments. H.-C.T. analyzed the method and wrote the paper.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare that they have no competing interests.
