The Irreducible Element

1 Those that constitute a translation in space (transport) 

3 Those that constitute a transformation or change (process) 

We note that other areas of activity may have other or additional groups. For example, commerce will have actions related to change of ownership, such as sell and barter, which are not found in engineering. 

Within each of the three groups we can further group the verbs according to the objects they can take. The following grouping suggests itself  

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The cost precedes the revenue, at least to some extent, and so it is in the nature of an investment, and the quantity of interest is the revenue relative to the investment, or the Return on Investment (ROI). The exact definition of this relationship in accounting terms may vary somewhat between various application domains and, in particular, due to the different compositions of the stakeholder groups. This will be discussed in Sec. C4.5, but for the time being the simplest relationship, such as revenue minus cost, divided by cost, is an adequate definition of ROI. The function of generating a return on investment then becomes the universal top element in developing a functional system the starting point of design in the functional domain, and what we shall call the irreducible element. 

A functional element is a real functional element if and only if its included set contains the irreducible element. 

Proof The if part of the theorem follows immediately from the Def C4.8 and Theorem C4.1. The only if part follows from the fact that any functional element that condenses to the irreducible element, as real elements do, must include within it the definition of the purpose of a project, to some level of detail. Consequently, there exists a complete element representing the stakeholder requirements of that project, and by Def C4.8, the real functional element is a member of the included set of that complete element. 

A functional element is either a real or an imaginary element, so that the functional domain consists of two disjoint parts. However, real elements are systems of imaginary elements. This is illustrated in Fig. C4.3, where the functional elements are depicted as triangles in a process of decomposition/ condensation, with the real elements condensing from the stakeholder requirements to the irreducible element. 

To make Fig. C4.3 less abstract, consider the specific case where the stakeholder requirements include a requirement on the maximum value of the penalties incurred per year due to failure to provide the specified service. So, although the stakeholder requirements do not include any requirement on reliability of the plant, the decomposition will, at some level, need to introduce service reliability as a parameter. To what extent this aspect is then further decomposed depends on the nature of the plant and the level of detail in the stakeholder requirements.. But the point is that condensing the element representing the aspect of reliability will never reach the irreducible element. 

Fig. 0X3 As an illustration of decomposition, the figure illustrates the progress from irreducible element to the complete element that represents all aspects of the stakeholder requirements. As an illustration of condensation, the figure illustrates how the irreducible element represents the purpose common to all projects. The lighter coloured slices show the progression of the real elements making up the included set of the complete element, the slices of the darker coloured triangle show the progression of the imaginary elements making up the included set of a particular aspect.

An important example of the distinction between primary and secondary elements is provided by two elements we are quite familiar with by now the Cost and Revenue elements that describe these two dependencies of the irreducible element. The Revenue element is a primary functional element generating a revenue is a measure of what a plant does, of its purpose. The Cost, on the other hand, is a secondary element the purpose of a plant is not to generate a cost the cost is a consequence of having to create and operate a plant in order to provide... 

The above introduction of real and imaginary functional elements leads us to the question of the nature of the irreducible element itself is it a real or an imaginary element On the one hand, it is a characteristic of any project (i.e. of the operation of any plant over its lifetime) and would therefore seem to be an imaginary... 

Because the irreducible element is the same for all projects, we can define it in more detail as long as we do not introduce any variables or functions that would be project-specific. In accordance with the format of functional element introduced in Sec. C3.2, the functional parameter is the return on investment, which we had already decided to identify by U (Sec. B2.4). It is defined in terms of the two dependencies cost, C, and revenue, R, as follows ... 

With this, the symbolic representation of the irreducible element is as follows ... 

In a very simplistic manner, we can visualise this as illustrated in Fig. C4.7. The two sets of parameters, X and Y, have the parameters of the irreducible element in common, and if the level of detail is reduced (i.e. X and Y become smaller), x and y will move towards the irreducible element. It is therefore clear that with the above definition, the distance between two real elements depends no only on the difference in functionality but also on the level of detail of the description. If we, for a moment, interpret Fig. C4.7 in the context of a particular project, then there will be one or more real elements that are complete, and we can visualise them as lying on the boundary of the real part of the functional domain, with the elements between them and the irreducible element being the included sets. 

A further feature of the structure of the functional domain arises from the central role of the irreducible element in the description of functionality and the definition of that element, as depicted in Fig. C4.4. Due to this view of an engineering project as an optimisation of the balance between cost and revenue, the secondary elements fall into two completely separate categories the elements describing aspects of the cost, and the elements describing aspects of the performance, as already mentioned at the end of Sec. C4.3 and illustrated in Fig. C4.6. 

The underlying reason for dwelling on this issue of the Revenue element is, of course, that our application of the system concept to engineering is based on a top-down approach a step-wise development of the system (as a description), starting with the purpose of the plant. The irreducible element is the top element in this development process at the next level down, the Revenue element starts to reflect the specific purpose of a class of plant. The description of that purpose and the revenue generated by it sets the direction for all of the following development of the functionality of the plant i.e. for the process of design in the functional domain. 

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