Flow system evaluation and optimization

The objective of this research is to investigate the theoretical and computational aspects of the flow system to provide modeling approaches that can be readily applied in industrial usage. A flow system is a network of directed arcs and nodes through which discrete units of material flow. A cost is assigned to the flowing units sequentially as each node is reached. The costs that have been assigned to the units are accumulated as a unit leaves the network. These networks are similar to transportation and GERT networks, but are not treated by linear programming or graphical means. The concept of a deterministic flow system is developed as the condition that a unit is split proportionately among the arcs leaving a particular node in accordance with the standardized weights associated with these arcs. This type of flow system is generalized to include multiple inputs, multiple outputs, by-product or scrap cost allocations, and non-linear unit costs. Matrix equations to describe these generalizations are derived. The concept of a stochastic flow system is also developed under the condition that a unit makes a random choice among the arcs in leaving a node; the selection being made in accordance with probabilities associated with each arc. The generating function method often used with Markov chains is shown to be inadequate and a prior development of the total system cost variance is shown to be generally invalid. A matrix method is developed that generates the cost probability distributions at each flow system output and for the entire system. For multiple units entering the flow system, the total system cost is developed as a convolution of the single input unit cost probability distribution. A method is also presented for expressing any moment of the cost distribution in closed matrix form. With this approach, the unit costs assigned at the nodes can be functionally distributed random variables.