The problem of optimal collapsing factor levels in a one-way ANOVA based on information criteria with a fixed number of groups is formulated as a mixed integer linear programming problem. The problem included constraints which fully eliminate all permutation-equivalent solutions. This significantly reduced the computational time and allowed us to explore existing greedy algorithms, in particular, agglomerative merging, from the point of view of the proximity of their solution to the global optimum. The modification of the agglomerative merging method with partial enumeration of solutions, previously proposed by the authors, is improved by sorting the elements of the cluster list and excluding duplicates. A model example is developed to study the efficiency of the methods, in which the number of repeated observations and the error variance are varied. It is found that the proportion of cases when the agglomerative merging algorithm does not find an optimal solution can reach 28% for ten- and 92% for fifty-level factor. This occurs in situations where the optimal number of groups is small and the error variance is large. Different partial enumeration strategies are compared using simulation studies for ten-level factor. It is shown that the tree search does not lead to exponential growth due to the exclusion of identical solutions. Moreover, even for a binary tree it is possible to almost reliably find a global optimum. The agglomerative merging algorithms provide a good approximation for the optimal number of factor level groups, which can be used as an initial value in a mixed integer linear programming problem to avoid exhaustive search.