A Mathematical Logic Approach for the Transformation of the Linear Conditional Piecewise Functions of Dispersion-and-Store and Cell Transmission Traffic Flow Models into Linear Mixed-Integer Form

The modeling of traffic control systems for solving such problems as surface street signalization, dynamic traffic assignment, etc., typically results in the appearance of a conditional function. For example, the consistent representation of the outflow discharge at an approach of a signalized intersection implies a function that is conditional on the signal indication and the prevailing traffic conditions. Representing such functions by some sort of constraint(s), ideally linear, so as to be considered in the context of a mathematical programming problem, is a nontrivial task, most often resolved by adopting restrictive assumptions regarding real-life process behavior. To address this general problem, we develop two methodologies that are largely based on analogies from mathematical logic that provide a practical device for the transformation of a specific form of a linear conditional piecewise function into a mixed integer model (MIM), i.e., a set of mixed-integer linear inequality constraints. We show the applicability of these methodologies to transforming into a MIM virtually every possible conditional piecewise function that can be found when one is modeling transportation systems based on the widely adopted dispersion-and-store and cell transmission traffic flow models, as well as to analyzing existing MIMs for identifying and eliminating redundancies.