Month: February 2023

The next step is to consider powers of the independent variable x. A term of the form axm is called a monomial of degree m. Summing monomials, we get a polynomial function: Here n is the degree of the polynomial. A few polynomial functions are shown in Fig. 2.8. A quick glance at the three plots suggests a few observations: We define concepts like local or global minimum…

A linear affine function has the following general form: Figure 2.7 shows a few linear functions. Strictly speaking, only the first function is linear. A function f is linear if the following condition holds: Fig. 2.7 Graphs of linear (affine) functions. for arbitrary numbers αi and xi, i = 1, 2. However, this holds only when the coefficient q in (2.4) is zero. To see…

Functions are rules that map input values to output values in a well determined way. They come in many guises, depending on what is mapped on what. Generally, a function is specified as where D is the domain of the function, i.e., the set of possible input values on which the function is defined, and I is the image or range of the function, i.e.,…

Many practical problems involve permutations and combinations of objects. A first question is: Given a collection of n objects, in how many ways can we permute them? For instance, let us consider the set {a, b, c}. Since the set is quite small, we can enumerate all of the possible permutations systematically. First we consider permutations beginning…

Consider an expression like We will meet similar expressions quite often in the book, and a nice shorthand notation for this expression is which should be read as the sum of “x subscript i,” for i ranging from 1 to 4. Sometimes, the sum limits can be symbolic, as in We may even consider an infinite sum like In…

Inequalities like a ≤ x ≤ b, where a and b are arbitrary real numbers such that a < b, define intervals on the real line. The inequality above defines an interval that includes its extreme points. In such a case, we use the notation [a, b] to denote the interval, and we speak of a closed interval. On the contrary, inequalities a < x < b define the open interval (a, b). For…

If we order cars from a car manufacturer, we cannot order 10.56986 cars; we may order either 10 or 11 cars, but any value in between makes no sense. It should be intuitively clear what we mean by an integer number; integer numbers are used to measure variables that have a intrinsically discrete nature. A real number is…

As we have already pointed out, the reader is assumed to be equipped with a basic mathematical background about sets as well as integer and real numbers. In this section we briefly recall a few basic concepts for convenience.

In plotting the function, we have ignored the purchase cost component cd, which is constant and would just push the graph up a bit. This is not relevant to us, since what we are interested in is finding an order size Q* minimizing total cost. Indeed, since the function goes to infinity for very small and very…

Having figured out a relationship between the order size and the average total cost per year, it would be useful to plot the function in order to see the effect of Q and to figure out a good decision. There are plenty of powerful software packages that, given a range of the independent variable Q, compute the corresponding…