Is there any easy and cheap way to obtain an invitation to any Schengen zone country?

How do I get Schengen invitation?

First of all you need to understand in which category you are applying for a Schengen visa. Normally if you are going to apply as a tourist, you don't need any invitation letter from host. Simply you can provide hotel booking and all the relevant documents required by Schengen embassy/consulate. Further for tourist visas, you can only apply to Schengen embassy/consulate which will be your main destination.

You would definitely need an invitation if you are planning to visit your friends or family member. Remember each Schengen member state has its own requirements for formal invitation from host. For example German Consulate requirement for invitation if visiting friends or family members.

I don't have anyone living there, but all I want to do is travel.

Since you have already mentioned that you don't know anyone there, it means you don't really need an invitation letter for Schengen visas. You can simply apply as a tourist which always required hotel/hostel booking along with all the relevant documents. And you can only apply to the embassy/consulate where you will be spending maximum time. OR in case you are spending equal nights in Schengen countries, you can only apply where you will be entering first.

As I understand, you want to travel to a country located in the Schengen area, for travel.

In this case, you need to present evidence that you have an accomodation prepared for your stay (amongst other requirements for visa application which are irrelevant here). This can indeed be an invitation letter from someone residing in the Schengen area, as you mention. But it can also be a hotel booking confirmation, for example. See this for instance (with caution as it is not an official site).

In short, you don't have to be invited by someone you know in the Schengen area. You can book a hotel (be careful to book accommodations for your entire stay, for consistency in your visa application) and the hotel booking confirmation will be the proof of accommodation you'll provide your visa application with.

I think the best approach is to just apply for a tourist visa from any of the schengen areas countries as all an invite really does is circumvent proof of being able to take care of oneself while travelling there and accommodation. Both of which can be taken care of by one booking a hotel stay which you don't have to purchase outright and 2 proving that you can take care of yourself whilst in the area- each country has a minimum amount required per stay eg. Netherlands is £30 per day so for a 5 day stay you would only need to prove you can provide that for your stay. I hope this helps as a possible alternative.