This work presents an algorithmic framework that uses linear programming to construct \emph{addition-based Private Information Retrieval (AB-PIR)} schemes, where retrieval is performed by downloading only linear combinations of message symbols with coefficients set to 0 or 1. The AB-PIR schemes generalize several existing capacity-achieving PIR schemes and are of practical interest because they use only addition operations -- avoiding multiplication and other complex operations -- and are compatible with any finite field, including binary. Our framework broadens the search space to include all feasible solutions and can be used to construct optimal AB-PIR schemes for the entire range of problem parameters, including the number of servers, the total number of messages, and the number of messages that need to be retrieved. The framework enables us to identify schemes that outperform the previously proposed PIR schemes in certain cases and, in other cases, achieve performance on par with the best-known AB-PIR solutions. Additionally, the schemes generated by our framework can be integrated into existing solutions for several related PIR scenarios, improving their overall performance.