Knowledge representation of mathematical optimization problems and constructs for modeling

Abstract

We introduce two new knowledge bases that we have developed to categorize mixed-integer linear programming (MILP) problems and standardize the element definitions of MILP models. MILP is a commonly used mathematical programming technique for modeling and solving real-life scheduling, routing, planning, resource allocation, and timetabling optimization problems providing optimized business solutions for industry sectors such as manufacturing, agriculture, defense, healthcare, medicine, energy, finance, and transportation among others. Despite the numerous real-life combinatorial optimization problems (COPs) found and solved, and many yet to be discovered and formulated, the number of types of constraints (the building blocks of a MILP) is relatively small. In the search for a uniform categorization of MILP problems and a machine-readable knowledge representation structure for MILP models, we have developed an optimization modeling tree (OMT) and a MILP model ontology. The two knowledge structures can serve as a standardized, uniform representation in understanding MILP problems and developing MILP models for combinatorial business optimization problems. While there are several algebraic modeling languages (AMLs) for developing and solving MILP models, the semantic correctness of such models cannot be guaranteed using the syntactic grammar of such AMLs. The MILP model ontology will act as the main resource for semantic validation of MILP models through ontology instantiations and axiom assertions.