Two criteria in a problem that involves multiple-criteria decision-making

Question

I would like your valuable opinion on one issue:

I created a decision matrix considering two criteria and 10 alternatives, as below. My problem involves determining the optimal number of clusters from a dataset that has information about rural properties. So, basically, I use two criteria to know the ideal number of clusters. The alternatives are the number of clusters it can be. This decision matrix, I obtained using agglomerative hierarchical clustering algorithm.

nclusters(Alternatives) Criterion1 Criterion2
              3           427.7037 1216.07483
              4           2067.8457  315.11427
              5           1797.6554  295.33840
              6           1797.6554  295.33840
              7           1798.7378  229.82400
              8           1468.3216  229.82400
              9           942.4955  229.82400
             10           937.1313  229.31030
             11           937.1313  229.31030
             12           629.0550   29.85698

After obtaining this matrix, I used some multiple-criteria decision-making(MCDM), which are the weighted sum method (WSM), TOPSIS and VIKOR, to generate rankings of the alternatives. Then, I compare the rankings of both MCDM methods, and manage to generate a final ranking. So by chance, if rank = 1 is in the first row, it means that the ideal number of clusters is 3.

Now, I would like to know your opinion if it makes sense to use MCDM using only two criteria, even though there are 10 alternatives? I know that to be MCDM, you need to have more than one criterion. Of course, more criteria would help, but I only got those two.

What do you think?

Answer 1

More criteria would not necessarily help, and in fact might make it harder to make a selection (for instance, if every alternative excels on some criterion and does badly on some other criterion). Ultimately, the relevant criteria (and, by implication, the relevant number of criteria) should be determined by the decision maker(s), based on the context of the problem.

You might start by ruling out dominated alternatives (alternatives that are worse than at least one other alternative in both criteria). After that, you can try various ranking schemes, or you can ask the decision maker to do pairwise comparisons (e.g., "Between 5 and 7 clusters which do you prefer?"), or you can plot the Pareto frontier and ask the decision maker what looks good to them.