# Industrial Dynamics

By Jay W. Forrester | Go to book overview

CHAPTER 7
System of Equations

The model structure described in Chapter 6 leads to a simple system of equations that suffices for representing information-feedback systems. The equations tell how to generate the system conditions for a new point in time, given the conditions known from the previous point in time. The equations of the model are evaluated repeatedly to generate a sequence of steps equally spaced in time. Level equations and rate equations generate the levels and rates of the basic model structure. In addition, auxiliary, supplementary, and initial-value equations are used. The interval of time between solutions must be relatively short, determined by the dynamic characteristics of the real system that is being modeled. Simple first-order integration of level equations is satisfactory.

IN the preceding chapter an arrangement for the basic structure of a model was illustrated in terms of levels (reservoirs) interconnected by rates of flow. To describe this general structure, we must have a suitable system of equations. The equations should be compatible with the five objectives of a model listed at the beginning of Chapter 6.

The system of equations should be adequate to describe the situations, concepts, interactions, and decision processes that constitute the. world of management and economics. The model should be able to fit our concepts of reality rather than requiring violent and unacceptable simplifications to bring those concepts into the bounds of a model structure.

The equations to be described here form a basic system going with, and elaborating slightly on, the model structure already described. Classes of equations are discussed in this chapter but not the special forms that particular equations of these classes may assume.1

Basically, the equation system consists of two types of equations corresponding to the levels and the rates discussed in the preceding chapters. In a following section will be introduced other incidental types of equations which add convenience and clarity to complex systems but which are not fundamental to the model structure. Before doing that, the fundamental time sequence of computation can be described in terms of levels and rates.

7.1 Computing Sequence

A system of equations is written in the context of certain conventions that state how the equations are to be evaluated. We are dealing here with a system of equations that controls the changing interactions of a set of variables as time advances. This system evolution implies that the equations will be computed periodically to yield the successive new states of the system.

At each point in time, there may be a particular sequence of computation implied by the equation system. The sequence to be used here is shown in Figure 7-1. The continuous advance of time is broken into small intervals of equal length DT (Standing for "delta time," or time

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1
In particular, this discussion omits various intermittent operations like periodic sampling and "box- car" delays that are useful in segregating and saving historical information, as in ringing forward annual sales curves from past years and separating capital equipment by age and productivity.

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