Question: What problem/s does a Rule Engine Algorithm solve?  

Which are the recommended ones?

Answer: 
A rules engine can encapsulate business logic, often in a declarative manner. This avoids the need to custom-code business logic in software.

A rules engine algorithm is how the rule matching and triggering is implemented, so it's an implementation detail rather that actually solving a problem itself. Algorithms might have different non-functional strengths or weaknesses. A common example of a rules engine algorithm is the [Rete Algorithm](http://en.wikipedia.org/wiki/Rete_algorithm), used in the Java rules engines [JESS](http://www.jessrules.com/jess/docs/52/rete.html) and [Drools](http://legacy.drools.codehaus.org/Rete).
A rule engine calculates a decision, based on one or several input parameters.