Overview of the Operations Research Modeling Approach:formulating a mathematical model

formulating a mathematical model

After the decision-maker’s problem is defined, the next phase is to reformulate this problem in a form that is convenient for analysis. The conventional OR approach for doing this is to construct a mathematical model that represents the essence of the problem. Before discussing how to formulate such a model, we first explore the nature of models in general and of mathematical models in particular.

Models, or idealized representations, are an integral part of everyday life. Common examples include model airplanes, portraits, globes, and so on. Similarly, models play an important role in science and business, as illustrated by models of the atom, models of genetic structure, mathematical equations describing physical laws of motion or chemical reactions, graphs, organizational charts, and industrial accounting systems. Such models are invaluable for abstracting the essence of the subject of inquiry, showing interrelation- ships, and facilitating analysis.

Mathematical models are also idealized representations, but they are expressed in terms of mathematical symbols and expressions. Such laws of physics as F = ma and E = mc2 are familiar examples. Similarly, the mathematical model of a business problem is the system of equations and related mathematical expressions that describe the essence of the problem. Thus, if there are n related quantifiable decisions to be made, they are represented as decision variables (say, x1, x2, . . . , xn) whose respective values are to be determined. The appropriate measure of performance (e.g., profit) is then expressed as a mathematical func- tion of these decision variables (for example, P = 3x1 + 2x2 + . . . + 5xn). This function is called the objective function. Any restrictions on the values that can be assigned to these decision variables are also expressed mathematically, typically by means of inequalities or equations (for example, x1 + 3x1x2 + 2x2 :s 10). Such mathematical expressions for the restrictions often are called constraints. The constants (namely, the coefficients and right- hand sides) in the constraints and the objective function are called the parameters of the model. The mathematical model might then say that the problem is to choose the values of the decision variables so as to maximize the objective function, subject to the specified con- straints. Such a model, and minor variations of it, typifies the models used in OR.

Determining the appropriate values to assign to the parameters of the model (one value per parameter) is both a critical and a challenging part of the model-building process. In contrast to textbook problems where the numbers are given to you, determining para- meter values for real problems requires gathering relevant data. As discussed in the pre- ceding section, gathering accurate data frequently is difficult. Therefore, the value assigned to a parameter often is, of necessity, only a rough estimate. Because of the uncer- tainty about the true value of the parameter, it is important to analyze how the solution derived from the model would change (if at all) if the value assigned to the parameter were changed to other plausible values. This process is referred to as sensitivity analysis, as discussed further in the next section (and much of Chap. 7).

Although we refer to “the” mathematical model of a business problem, real problems normally don’t have just a single “right” model. Section 2.4 will describe how the process of testing a model typically leads to a succession of models that provide better and better representations of the problem. It is even possible that two or more completely different types of models may be developed to help analyze the same problem.

You will see numerous examples of mathematical models throughout the remainder of this book. One particularly important type that is studied in the next several chapters is the linear programming model, where the mathematical functions appearing in both the objective function and the constraints are all linear functions. In Chap. 3, specific linear programming models are constructed to fit such diverse problems as determining (1) the

mix of products that maximizes profit, (2) the design of radiation therapy that effectively attacks a tumor while minimizing the damage to nearby healthy tissue, (3) the allocation of acreage to crops that maximizes total net return, and (4) the combination of pollution abatement methods that achieves air quality standards at minimum cost.

Mathematical models have many advantages over a verbal description of the problem. One advantage is that a mathematical model describes a problem much more concisely. This tends to make the overall structure of the problem more comprehensible, and it helps to reveal important cause-and-effect relationships. In this way, it indicates more clearly what additional data are relevant to the analysis. It also facilitates dealing with the problem in its entirety and considering all its interrelationships simultaneously. Finally, a mathematical model forms a bridge to the use of high-powered mathematical techniques and computers to analyze the problem. Indeed, packaged software for both personal computers and main- frame computers has become widely available for solving many mathematical models.

However, there are pitfalls to be avoided when you use mathematical models. Such a model is necessarily an abstract idealization of the problem, so approximations and sim- plifying assumptions generally are required if the model is to be tractable (capable of being solved). Therefore, care must be taken to ensure that the model remains a valid rep- resentation of the problem. The proper criterion for judging the validity of a model is whether the model predicts the relative effects of the alternative courses of action with sufficient accuracy to permit a sound decision. Consequently, it is not necessary to include unimportant details or factors that have approximately the same effect for all the alternative courses of action considered. It is not even necessary that the absolute magni- tude of the measure of performance be approximately correct for the various alternatives, provided that their relative values (i.e., the differences between their values) are suffi- ciently precise. Thus, all that is required is that there be a high correlation between the prediction by the model and what would actually happen in the real world. To ascertain whether this requirement is satisfied, it is important to do considerable testing and conse- quent modifying of the model, which will be the subject of Sec. 2.4. Although this testing phase is placed later in the chapter, much of this model validation work actually is con- ducted during the model-building phase of the study to help guide the construction of the mathematical model.

In developing the model, a good approach is to begin with a very simple version and then move in evolutionary fashion toward more elaborate models that more nearly reflect the complexity of the real problem. This process of model enrichment continues only as long as the model remains tractable. The basic trade-off under constant consideration is between the precision and the tractability of the model. (See Selected Reference 9 for a detailed description of this process.)

A crucial step in formulating an OR model is the construction of the objective func- tion. This requires developing a quantitative measure of performance relative to each of the decision maker’s ultimate objectives that were identified while the problem was being defined. If there are multiple objectives, their respective measures commonly are then transformed and combined into a composite measure, called the overall measure of performance. This overall measure might be something tangible (e.g., profit) correspond- ing to a higher goal of the organization, or it might be abstract (e.g., utility). In the latter case, the task of developing this measure tends to be a complex one requiring a careful comparison of the objectives and their relative importance. After the overall measure of performance is developed, the objective function is then obtained by expressing this mea- sure as a mathematical function of the decision variables. Alternatively, there also are methods for explicitly considering multiple objectives simultaneously, and one of these (goal programming) is discussed in the supplement to Chap. 8.

An Application Vignette

Prior to its merger with United Airlines that was com- pleted in 2012, Continental Airlines was a major U.S. air carrier that transported passengers, cargo, and mail. It operated more than 2,000 daily departures to well over 100 domestic destinations and nearly 100 foreign desti- nations. Following the merger under the name of United Airlines, the combined airline has a fleet of over 700 air- craft serving up to 370 destinations.

Airlines like Continental (and now under its reincar- nation as part of United Airlines) face schedule disrup- tions daily because of unexpected events, including inclement weather, aircraft mechanical problems, and crew unavailability. These disruptions can cause flight delays and cancellations. As a result, crews may not be in position to service their remaining scheduled flights. Airlines must reassign crews quickly to cover open flights and to return them to their original schedules in a cost-effective manner while honoring all government regulations, contractual obligations, and quality-of-life requirements.

To address such problems, an OR team at Continental Airlines developed a detailed mathematical model for reassigning crews to flights as soon as such emergencies arise. Because the airline has thousands of crews and daily flights, the model needed to be huge to consider all possible pairings of crews with flights. Therefore, the model has millions of decision variables and many thou- sands of constraints. In its first year of use (mainly in 2001), the model was applied four times to recover from major schedule disruptions (two snowstorms, a flood, and the September 11 terrorist attacks). This led to savings of approximately $40 million. Subsequent appli- cations extended to many daily minor disruptions as well.

Although other airlines subsequently scrambled to apply operations research in a similar way, this initial advantage over other airlines in being able to recover more quickly from schedule disruptions with fewer delays and canceled flights left Continental Airlines in a relatively strong position as the airline industry struggled through a difficult period during the initial years of the 21st century. This initiative led to Continental winning the prestigious First Prize in the 2002 international com- petition for the Franz Edelman Award for Achievement in Operations Research and the Management Sciences.

Example. The Netherlands government agency responsible for water control and public works, the Rijkswaterstaat, commissioned a major OR study to guide the development of a new national water management policy. The new policy saved hundreds of millions of dollars in investment expenditures and reduced agricultural damage by about $15 million per year, while decreasing thermal and algae pollution. Rather than formulating one mathematical model, this OR study developed a comprehensive, integrated system of 50 models! Furthermore, for some of the models, both simple and complex versions were developed. The simple version was used to gain basic insights, including trade-off analyses. The complex version then was used in the final rounds of the analysis or whenever greater accuracy or more detailed outputs were desired. The overall OR study directly involved over 125 person-years of effort (more than one-third in data gathering), created several dozen computer programs, and structured an enormous amount of data. (Selected Reference A8 describes this study in detail. Also see Selected References A3 and A9 for other examples where a large number of mathematical models were effectively integrated in an award-winning OR study.)