Evaluation of the performance of a distribution system by Q-Analysis

The performance of an existing goods distribution system is evaluated by means of a modern system-theoretic tool called Q-analysis. It is assumed that the system has been developed without an initial master plan, so that potential difficulties or breakdowns may occur. The case of a water distribution network which has been expanded stagewise in a southwestern U.S. city is used to illustrate the approach. The system consists of interconnected sources (reservoirs or ''vertices'') supplying a set of sinks (users or ''simplices'') through a capacitated network. Q-analysis, a technique which stems from algebraic topology, may be visualized as dealing with multidimensional graph theory, and yields indices that are used to measure performance. Specifically, the following indices are used: the q-vector, indicating the connectivity level between sources; the eccentricity, measuring whether or not elements are well integrated in the system (represented as a complex); the pattern, introducing dynamics into the analysis; the obstruction vector, identifying potential bottlenecks; and the complexity, measuring the length of various distribution paths and, indirectly, the stability. The analysis, first done from the viewpoint of source adequacy, is repeated by considering the conjugate complex, in which sources and sinks are interchanged: this introduces the user's viewpoint into the performance analysis. A numerical example shows how the above concepts provide a simple way to compare designs, identify problem areas, and improve operation characteristics of a distribution system.