Solving continuous-discrete two-stage logistic problem of optimal partitioning-allocation

The theory of optimal set partitioning is a powerful tool for solving infinite-dimensionaloptimization problems, including infinite-dimensional transportation problems where oneside, producers or consumers, is continuously allocated in a subset of n-dimensional space,for example, over a part ofa plane. A variation of the transportation problem is describedwhere the transportation of products is done in two stages, with intermediate transporta-tion points being integrated intothe model. First, products are transported from a set ofcontinuously distributed producers to a set of waypoints. Then, products are transportedfrom waypoints to a set of ultimate consumers. The defined problem consists of a classicdiscrete transportation problem and an infinite-dimensional transportation problem withoptimal partitioning-allocation of space. Such problem formulation significantly compli-cates the mathematical model but simultaneously allows us to get more optimal solutionsand solve problems that are closer to real-life cases of product transportation.A process of building a mathematical model is described, a method is proposed for solv-ing a two-stage logistic problem of optimal partitioning-allocation, and an algorithm isformulated based on the proposed method. Based on the formulated algorithm, softwareimplementation is developed thatsolves the two-stage problem in the case of two-dimensional space, that is on the plane. The software implementation consists of a compu-tational part that executes all necessary mathematical calculations, a graphical user inter-face part that allows to quickly and conveniently set initial conditions and parameters ofthe problem, and a visualizing part that conducts an interactive demonstration of interme-diate and final solutions of the problem. All parts are integrated into a single system. Ex-amples of solving several problems are shown for demonstration of the workof the pro-posed methodand explanation of key details.