1.1

An Introduction to Modeling

Operations research (often referred to as management science) is simply a scientific approach to decision making that seeks to best design and operate a system, usually under conditions requiring the allocation of scarce resources. By a system, we mean an organization of interdependent components that work together to accomplish the goal of the system. For example, Ford Motor Company is a system whose goal consists of maximizing the profit that can be earned by producing quality vehicles. The term operations research was coined during World War II when British military leaders asked scientists and engineers to analyze several military problems such as the deployment of radar and the management of convoy, bombing, antisubmarine, and mining operations. The scientific approach to decision making usually involves the use of one or more mathematical models. A mathematical model is a mathematical representation of an actual situation that may be used to make better decisions or simply to understand the actual situation better. The following example should clarify many of the key terms used to describe mathematical models.

EXAMPLE

1

Maximizing Wozac Yield

Eli Daisy produces Wozac in huge batches by heating a chemical mixture in a pressurized container. Each time a batch is processed, a different amount of Wozac is produced. The amount produced is the process yield (measured in pounds). Daisy is interested in understanding the factors that influence the yield of the Wozac production process. Describe a model-building process for this situation. Solution

Daisy is first interested in determining the factors that influence the yield of the process. This would be referred to as a descriptive model, because it describes the behavior of the actual yield as a function of various factors. Daisy might determine (using regression methods discussed in Chapter 24) that the following factors influence yield: s s s s

container volume in liters (V) container pressure in milliliters (P) container temperature in degrees Celsius (T) chemical composition of the processed mixture

If we let A, B, and C be percentage of mixture made up of chemicals A, B, and C, then Daisy might find, for example, that (1) yield 300 .8V .01P .06T .001T*P .01T2 .001P2 11.7A 9.4B 16.4C 19A*B 11.4A*C 9.6B*C

To determine this relationship, the yield of the process would have to be measured for many different combinations of the previously listed factors. Knowledge of this equation would enable Daisy to describe the yield of the production process once volume, pressure, temperature, and chemical composition were known.

Prescriptive or Optimization Models

Most of the models discussed in this book will be prescriptive or optimization models. A prescriptive model “prescribes” behavior for an organization that will enable it to best meet its goal(s). The components of a prescriptive model include s s s

objective function(s) decision variables constraints

In short, an optimization model seeks to find values of the decision variables that optimize (maximize or minimize) an objective function among the set of all values for the decision variables that satisfy the given constraints.

The Objective Function

Naturally, Daisy would like to maximize the yield of the process. In most models, there will be a function we wish to maximize or minimize. This function is called the model’s objective function. Of course, to maximize the process yield we need to find the values of V P, T, A, B, and C that make (1) as large as possible. , In many situations, an organization may have more than one objective. For example, in assigning students to the two high schools in Bloomington, Indiana, the Monroe County School Board stated that the assignment of students involved the following objectives: s s s

Equalize the number of students at the two high schools. Minimize the average distance...