This course is a basic introduction to important models and solution methods in Operations Research (OR). Operations Research is concerned with the modeling and analysis of complex decision problems that arise in the management of an organization. The basic theme in OR is optimization; that is, the goal is to improve the efficiency of operations through detailed modeling and analysis.
What is Operations Research?
Operations Research originated in Great Britain during World War II to bring mathematical or quantitative approaches to bear on military operations. Thus, Operations Research is concerned with research on operations; and the word "research" means that mathematical methods are used to conduct rigorous scientific analyses of complex problems.
Since its birth in the 1940's, OR has been widely recognized as an important approach to decision-making in the management of all aspects of an organization. (Consequently, OR is also referred to as Management Science.) Hence, the word "operations" should be interpreted broadly as any operation that requires the efficient allocation of limited resources. Specific areas of application include, for example, manufacturing, transportation and distribution, finance, marketing, health care, and telecommunications.
The mathematical methods most widely used in OR include mathematical programming, probability and statistics, and computer science. Some areas of application of OR, such as inventory and production control, queueing theory, scheduling theory, and simulation have, in fact, grown into subdisciplines in their own right.
For an overview of the history and the impact of Operations Research, read Chapter 1 of the text. In particular, Table 1.1 provides a good summary of applications of OR in industry.
What is a Mathematical Model?
The majority of practical decision problems are described in very vague terms. Therefore, a most-important step in a scientific or quantitative analysis of a problem is to formulate a model that adequately captures the essence of a problem. The result of such a formulation, or an abstraction, is called a mathematical optimization model.
Generally speaking, a mathematical optimization model has the following typical components:
Two concrete examples are: in production planning, one could be interested in setting a production schedule (the decision variables) that minimizes the total cost (the objective function) of operation over a given planning horizon, for a given production capacity (the constraint); and in investment management, one could be interested in assembling a collection of securities (the decision variables) that maximizes the expected yield (the objective function) of a portfolio, under a limited budget (the constraint).
The OR Modeling Approach
In addition to the formulation of a model, the OR approach involves several additional steps. The next step, of course, is to apply (or, sometimes, to develop) a solution method to derive a set of values for the decision variables that "optimizes" the objective function subject to the given constraints. After having formulated and solved a problem, the remaining steps then consist of model validation and implementation.
Thus, the basic steps in the OR modeling approach are:
It is important to realize that before an actual implementation, one should cycle through the first four steps a sufficient number of times so as to ensure that a reasonably valid model has been achieved.
For a detailed discussion of these steps, read Chapter 2 of the text.
Course Coverage
OR models are generally classified as either deterministic or stochastic. A deterministic model is one in which all problem parameters are assumed to be known with certainty; whereas a stochastic model is one in which some of the parameters are assumed to be random.
The emphasis in this (short) course is on deterministic models. In particular, we will focus on two topics, namely linear programming and dynamic programming. Linear programming refers to the formulation and solution of mathematical optimization models that have a linear objective function and a set of linear constraints; and dynamic programming refers to a versatile mathematical method that can be used to solve optimization problems that involve a sequence of interrelated decisions, usually over time. These two topics are chosen mainly for their wide range of applicability.
Our coverage corresponds to Chapters 1-5, 6 (Sections 6.6 and 6.7), 8, and 11 in the text.
Despite our limited scope, it is hoped that the course will help you acquire a solid understanding of how complex problems can be modeled and solved with these methods.