I have the following sentence and I need your help:
"In this thesis three type(s) of optimization approaches are developed. The first type(s) is/are heuristic algorithms. ... . The second type(s) is/are exact methods."
The question is whether I should use 'type' or 'types' in this context. Note that this question is not answered in these discussions (Agreement in "[Singular Noun] Is/Are [Plural Noun]"?) and (Plural verb used after 'type'). My question is not about the verb (that has to fit to the subject). The question is whether 'type' itself should be singular or plural. I'd appreciate every comment and will be quite thankful if you can help me on that because I have been struggeling with this question.
Here is some background information: I have developed 3 type(s) of optimization approaches. The first type(s) are heuristic algorithms. Here I have developed 3 methods. The second type(s) are exact methods. Here I have developed 2 methods. And the third type(s) are rule-based methods. Here I have developed 4 methods. So overall I have:
Type 1 (Heuristic Algorithms): Heuristic Algorithm 1, Heuristic algorithms 2, Heuristic Algorithm 3
Type 2 (Exact methods): Exact method 1, Exact methods 2
Type 3 (Rule-based methods): Rule-based method 1, Rule-based method 2, Rule-based mehtod 3, Rule-based method 4
Update: Can anybody help me on that. I would like to give you bountry but unfortunately I can only do so tomorrow (but I need the answer as soon as possible)