Inverting the multiple-assisting tool network problem to solve for optimality

Many network problems deal with the routing of a main tool comprised of several parallel assisting tools. These problems can be found with multi-tool-head routing of CNC machines, waterjets, plasma sprayers, cutting machines, and routers. Additional applications involve logistics, distribution and material handling systems such that the assisting tool remains in proximity and constrained by the main tool. Currently no studies exist that optimally routes the main tool with several assisting tools, nor do any studies evaluate the impact of adding additional capabilities to the main tool. Herein we define the network routing problem for the main tool with its constrained assisting tools. We mathematically define the problem as a mixed integer optimization problem as well as develop an ~optimization procedure using a genetic algorithm routine. Further, we discuss the fundamentals surrounding the constrained relationship between the main tool and assisting tools. We introduce a geometric ‘practical best case’ analysis of the tool paths and invert the problem to extract out ‘best case’ assisting tool range, speed, and number such that the system is not oversaturated. Computational studies reveal that the theorems introduced here improve the overall delivery performance of the system without saturating the system with unused resources.