Modeling and Solving Difficult-to-Represent Optimization Problems

Most optimization problems one encounters in reference books such as this can be, and are generally, formulated as mathematical programming (MP) problems. For some problems, the modeling requires deep insights into mathematical programming techniques or the way the variables of the problems interact with one another. Such exercises, however, involved at the end, yield mathematical programming formulations that have well defined methods for their solution. These solution methods may themselves be complex and require a good understanding of calculus as well as numerical techniques. Often these methods are computationally expensive and in itself requires go-around tools. However, there are some optimization problems that arise in engineering that cannot be formulated as mathematical programming problems. Often these require external procedure-based declarations to evaluate the system performances or define interactions between various subparts of the problem. Such problems are often solved using optimization metaheuristics like genetic algorithms that are structurally quite versatile and typically require information on only system performance to drive the optimization process. In this chapter the simple everyday problem of efficient urban mobility is introduced as an example that poses unique difficulties when modeled as an optimization problem. The question that is sought to be answered is whether a combination of one-way and two-way roads can be obtained such that an urban area provides the least travel time to its users. This chapter presents a formulation as well as a genetic algorithm-based solution methodology for the problem.