'type' or 'types'

Question

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)

Answer 1

I think you have almost answered your own question for yourself in your own lucid explanation of what you are trying to say. Let’s look at your draft text.

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."

In your numbered list you have correctly used the singular type. You are listing three and only three types, so that each type counts as just one and must be singular.

But in that case, each of the types must also be singular. But each type happens to be a set of more than one members. Is you put them in the plural, it makes the singular type look weird; but if you put type in the plural, that is weird also (because then there must be more than three types.

Most readers will see what you mean. But you need to avoid this dissonance. (No, to be honest, I want you to avoid it!).

The first sentence is easy. approach should be singular. It is quite normal to use approach in the singular to refer to the category of approaches.

In this thesis three types of approach are developed.

The first type you refer to is the heuristic algorithm. The addition of the definite article allows you to use the noun as a general abstract category of algorithms. Moreover, you do not have to use the word type (or types) again. You have established it as the (singular) topic of discussion.

The first is the heuristic algorithm;

But with the second we are in trouble. Exact methods can be use as above, but does not sound right. It isn’t a recognised category. Surely all methods should be as exact as possible. Is it/them really an alternative/s to the other two? Assuming that, having considered this question you are satisfied that it/they are, you can avoid the plural thus:

The second is the use of exact methods.

Then we come to rule-based methods.

As before you can justify the plural methods by the insertion of the use of.

But in what way is an algorithm not a form of rule, or in some sense rule-based?

The one thing I do not know is what you are trying to optimise. Without this, it is hard to tell what kinds of rule you are seeking to apply. So I cannot tell why not to expect any method, whatever you call it, to be exact, or how any method,whether algorithmic or exact not in some way to be subject to rules of some sort. But that is only a question for you to consider. My suggestion for sorting out the number dilemma still stands.