Mathematics is the abstract science of quantity, and quantity is measured in numbers. So, since mathematics is really about numbers the first thing we have to ask ourselves is: what is a number? What, for instance, does the number 5,334 really mean?
Our dependence on computers as tools for analysis and calculation hardly needs comment. Soon no area of life will be free from their influence. It might seem to follow from this that we would all seek to understand and come to terms with them.
The manipulation of symbols is helped by the use of abbreviations. The commonly accepted abbreviation for a × a is a2, for a × a × a is a3 and more generally, for a × a × a… × a (n times) is an, and n is called the exponent or index.
Any collection of objects is a set. The objects are called the elements or members of the set. It is normal to use upper case letters (i.e. capitals) to denote the sets and lower case letters for the elements of the sets. If an element p is a member of a set A, we write:
$$
p \in A
$$
and if p is not a member we write:
$$
p \notin A
$$
A set is completely determined or defined when all its members are specified. This is called the principle of extension. There are two ways in which a particular set can be defined. The first is by specifying its members:
$$
B = \left\{ {2,4,6} \right\}
$$
Note that the members are enclosed in brackets and separated by commas. The second method is to state the property characterising the elements. We could therefore express this set B equivalently by writing:
$$
B = \left\{ {x:x\,is\,a\,positive\,even\,number\,less\,than\,8} \right\}
$$
What is a matrix? Almost every text dealing with this subject approaches it in a different way. In Chapter 4 we saw the matrix format as simply a way of representing a relationship. Let us follow up this approach by way of an example.
Frequently in life we meet problems which, because of their very size and complexity, are difficult for us to solve. One way to cope with these sorts of problems is to break them down into smaller problems each of which can be solved by the application of logic. Breaking down the problem and using a step-by-step approach results in a series of logical instructions one following from the other. We call this an algorithm.
We have already used equations on a number of occasions, and throughout this book we shall continue to use them. Equations frequently summarise important facts. Thus the fact that the simple interest (I) at an interest rate (r) on a principal (P) is the rate multiplied by the principal can be summarized as I = rP. Equations can also be descriptive and useful for management decisions. In Chapter 3 we saw how the I = rP equation can be helpful in capital investment appraisal decisions. Equations can provide models either at macroeconomic level — for instance, the input output model for the economy (Chapter 5) — or at the microeconomic level. One example of the latter is that for a product produced by a firm, total costs are a + bx, where a are the fixed costs, b the variable costs per unit and x the number of units produced (provided that certain basic assumptions and conditions are met). If we call total costs y, these costs are then given by the equation y = a + bx.
In Chapter 4 we said that if R represents the set of all real numbers, then R2 will be the set of all ordered pairs of all real numbers. The visual representation of this was all the points in a two-dimensional Cartesian plane—i.e., the plane itself (this page is part of one such plane).
For explaining how total cost (y) varies with throughput (x) we have generally used two basic functions. They are the straight line:
$$
y = a + bx
$$
(9.1)
and an equation of the second degree:
$$
y = a + bx + {c^2}
$$
(9.2)
If all variable costs are simply and directly variable with output we will use equation (9.1). If not equation (9.2) may give a better and more accurate representation. It all depends on the particular business situation.
Probability theory has long been part of most mathematical texts. The way it has been approached has varied very much with the prevailing attitude. At one time a very traditional approach was used. Recently with the increased use of modern maths, the approach has been through the use of sets. In line with our general views we use set theory, supported by traditional mathematics.
In all life we see examples of how one thing depends on another. In a factory direct material costs go up with throughput, at home, electricity consumption with the number of lights an hour turned on. Many examples have been already met with in this book.
Quantitative methods are becoming an increasingly valuable part of industrial and commercial organisation. The trend for a more numerate management and cheaper computing resources will ensure that they will become more widespread in the future. Any quantitative technique requires some mathematical model of all or part of a commercial organisation, its competitors, customers or supplies, or some aspect of the environment within which it operates. These models may be very simple or highly complex, and the construction and study of them is becoming increasingly important. One that has already achieved widespread popularity is that of linear programming. It is a technique that is prescriptive rather than descriptive and its prescriptions can yield substantial dividends. Reliable computer software exists to exploit it and this has become so efficient that models with five thousand decisions can be used routinely. Thus it is an exciting and powerful tool of business analysis.
In this chapter we examine the solution of linear programs in more detail, consider concepts of duality and look at a special form of the algorithm for solving transportation problems.
