Geometrical Modeling for Material Flow Management

Abstract

INTRODUCTION In this paper, we present the theory of connection (ToC) based approach for formulation of mathematical model of material flow processes. The major advantage in using the theory of connection is that, the same set of functions (or procedures) can be used for formulation of mathematical models of many engineering and management processes (Bjorke, 1995; Davidrajuh, 2000). Theory of connection (ToC), previously known as manufacturing system theory, in the form it is presented here, is due to the work of the Scandinavian School of Systems Theory for the past 30 years; for detailed study of ToC, interested readers are referred to Bjorke (1995). The idea behind ToC is to bring geometry and algebra together: first, geometric modeling is used to model physical phenomena, and then a set of algebraic equations are drawn out of the geometric model, so that by using a computer these equations could be solved. The usefulness of ToC is that, different subsystems of different disciplines can be modeled and integrated by performing the same procedure; this is very important for fields like e-commerce as e-commerce involves diverse disciplines like manufacturing, business management, supplier selection, etc. MODELING APPROACH BASED ON ToC First, the concept of system model is introduced; this is done with the help of a simple electrical network known as the inductor-resistor-capacitor (LRC) network. Then, the simple LRC system model is mapped into the geometrical space with the help of properties matrix. System Model A system consists of three fundamental components, such as elements, connections, and sources. The elements carry all the physical or economical properties of the system. Elements are the building blocks of the physical system. For example, in a LRC network, the resistors, capacitors and inductors are the elements; the property of a resistor is its admittance, whereas a machine element's property could be its processing time, ratio between input items and output items, scrap percentage etc. When there is no connection between the elements, the set of isolated elements is called the primitive system. The connections reflect how the elements influence each other and it represents the structure of the system. The set of connected elements is called the connected system. Finally, the sources reflect the influence between the total system and the environment. Sources are the environment's influence on the system; in an electrical circuit, source s could be current or voltage sources; in production planning, demand of products, startup-setup times, costs involved are some of the sources. Geometrical Spaces, Vectors, and Matrices ToC is based on the use of continuous geometrical 3-space or more typically an n-space volume (a Euclidean space [R.sup.n]). A vector (called a contravariant vector) represents a point in the Euclidean space, or primary space, from the origin to the point; A vector in a 3-space is represented by: x = [x.sup.1][[epsilon].sub.1] + [x.sup.2][[epsilon].sub.2] + [x.sup.3][[epsilon].sub.3], where ( [[epsilon].sub.1],[[epsilon].sub.2],[[epsilon].sub.3]) is the basis vector along the three axes. Whereas, a point (called a covariant vector) in the corresponding dual space of the vector space, defines another vector which can be represented by: a = [a.sub.1][[epsilon].sup.1] + [a.sub.2][[epsilon].sup.2] + [a.sub.3][[epsilon].sup.3], Where ([[epsilon].sup.1],[[epsilon].sup.2],[[epsilon].sup.3]) is the basis co-vector along the three axes. A property matrix keeps properties (or characteristics) of primitive elements, whereas a connection matrix represents the connections between the primitive elements. TOOLBOX OF FUNCTIONS The process of mathematical modeling and simulation approach based on ToC is traditionally done by a toolbox of functions implemented in APL (A Programming Language). …