Optimization Concepts and Applications in Engineering

Introduction

There are many problems where the variables are not divisible as fractions. Some examples are the number of operators that can be assigned to jobs, the number of airplanes that can be purchased, the number of plants operating, and so on. These problems form an important class called integer programming problems. Further, several decision problems involve binary variables that take the values 0 or 1. If some variables must be integers and others allowed to take fractional values, the problem is of the mixed integer type. Discrete programming problems are those problems where the variables are to be chosen from a discrete set of available sizes, such as available shaft sizes and beam sections, engine capacities, pump capacities.

In integer programming problems, treating the variable as continuous and rounding off the optimum solution to the nearest integer is easily justified if the involved quantities are large. Consider the example of the number of barrels of crude processed where the solution results in 234566.4 barrels. This can be rounded to 234566 barrels or even 234570 without significantly altering the characteristics of the solution. When the variable values are small, such as the number of aircraft in a fleet, rounding is no longer intuitive and may not even yield a feasible solution. In problems with binary variables, rounding makes no sense as the choice between 0 or 1 is a choice between two entirely different decisions. The following two variable example provides some characteristics of integer requirements.