1 Introduction

To work with a mass of data, a concise formalism and notation is needed. In mathematics, the so-called indexed notation is the appropriate notation. This notation is an integral and fundamental part of every mathematical modeling activity – and they are also used in mathematical modeling languages as in LPL [4] – to group various objects and entities. It is surprising how little has been published in the community of modeling language design on this concept. Even in mathematical textbooks, the indexed notation used in formulae is often taken for granted and not much thought is given to it. This fact contrasts with my experience that students often have difficulties with the indexed notation in mathematics.

To represent single data, called scalars, names and expressions are used, such as:

x,  y  or   x - y

To specify further what we mean by a symbol, we write x,y , for example, saying that x and y represent any real numbers. We can attach a new symbol (z) to an expression as in:

z = x - y

meaning that z is a new number which is defined as the difference of x and y. We say “z substitutes the expression x-y”. In this way, we can build complex expressions. We learnt this algebraic notation already in school. These symbols have no specific “meaning” besides the fact that they represent numbers. To use them in a modeling context, we then attach a meaning. For example, in an economical context we might interpret “x” as “total revenue” and “y” as “total costs”. Then “z” might be interpreted as “total profit”. There is nothing magic about these symbols, instead of writing z = x - y we can also write:

total profit = total revenue  - total costs

However, to economize our writings we prefer short “names”, like x, y, and z. But there is nothing wrong with longer names to make expressions more readable in a specific context.

It seems to be a little bit less familiar that in the same way as using symbols for scalars, symbols can represent mass of data. Suppose that we want to express the profit of several profit centers in a company. Then we could write:

profit at center 1 = revenue at center 1 - costs at center 1
profit at center 2 = revenue at center 2 - costs at center 2

profit at center 3 = revenue at center 3 - costs at center 3
                     ...   etc.   ...

or (using the “names” x1, x2, y1, and so on)

 z1 = x1 - y1

 z2 = x2 - y2
 z3 = x3 - y3

...   etc.   ...

The “etc.” means that we have to continue writing as many lines as we have profit centers, there may be dozens – a rather boring task!

There is, however, a much more economical way in mathematics to represent all these expressions in one single statement. It is the indexed notation. To formulate them, we first introduce a set: the set I of all profit centers:

I = {center 1 ,  center 2  ,  ...  }

here again the “” means that we continue writing all centers in a list. Often a short name form is used as follows:

I =  {1...n }    where   n ∈ ℤ+

In the previous set definition we just mean that there are n centers, n being a positive number. We are not concerned right now of how many centers there are, we just say n – it can be 5 or 1000. Of course, in a concrete context we must specify the number n, but it is part of the data of that specific context.

In a second step, we introduce symbols for all profits, revenues and costs. Now, instead of using each time a new symbol, we just attach an subscript to the symbols: zi, xi, yi to express the fact that they “mean” the profit, the revenue and the cost of the i-th center, together with the notation i I, which means that i just designates an arbitrary (i-th) element in I. Hence, All data can be written in a concise way as follows:

xi,  yi,  zi    where   i ∈ I

Mathematically speaking, three vectors are declared to represent all data. The list of expression then can be written in a single statement as follows:

zi = xi - yi   with   i ∈ I

This notation is in fact nothing else than an economical way of the following n expressions:

z1 = x1 - y1
z  = x  - y
 2    2    2
     ...
zn = xn - yn

where n is some positive integer.

In a similar way, we can concisely build an expression that sums all profits, for instance. For this purpose we use the mathematical operator , the Greek sigma symbol. The total profit p of all profit centers can be formulated as follows:

     ∑
p =     zi
     i∈I

Again, this formmula is nothing else than a shortcut for the long expression:

p = z1 + z2 + ⋅⋅⋅ + zn

After this introductory example, this paper presents now a more precise way, on how the indexed notation is defined and how it can be used in modeling.