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Introduction to Mathematical Modeling for Business Analytics
OBJECTIVES
1. Understand the modeling process.
2. Know and use the steps in modeling.
3. Experience a wide variety of examples.
1.1 Introduction
Consider the importance of modeling for decision-making in business (B), industry (I), and government (G), BIG. BIG decision-making is essential for success at all levels. We do not encourage shooting from the hip or simply flipping a coin to make a decision. We recommend good analysis that enables the decision-maker to examine and question results to find the best alternative to choose or decision to make. This book presents, explains, and illustrates a modeling process and provides examples of decision-making analysis throughout.
Let us describe a mathematical model as a mathematical description of a system by using the language of mathematics. Why mathematical modeling? Mathematical modeling, business analytics, and operations research are all similar descriptions that represent the use of quantitative analysis to solve real problems. This process of developing such a mathematical model is termed as mathematical modeling. Mathematical models are used in the natural sciences (such as physics, biology, earth science, and meteorology), engineering disciplines (e.g., computer science, systems engineering, operations research, and industrial engineering), and in the social sciences (such as business, economics, psychology, sociology, political science, and social networks). The professionals in these areas use mathematical models all the time. A mathematical model may be used to help explain a system, to study the effects of different components, and to make predictions about behavior (Giordano et al., 2014, pp. 58–60). So let us make a more formal definition of a mathematical model: a mathematical model is the application of mathematics to a real-world problem.
Mathematical models can take many forms, including but not limited to dynamical systems: statistical models, differential equations, optimization models, or game theoretic models. These and other types of models can overlap, of which one output becomes the input for another similar or different model forms. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with the results of repeatable experiments (Giordano et al., 2014, pp. 58–60). Any lack of agreement between theoretical mathematical models and experimental measurements leads to model refinements and better models. We do not plan to cover all the mathematical modeling processes here. We only provide an overview to the decision-makers. Our goal is to offer competent, confident problem solvers for the twenty-first century. We suggest the books listed in the reference section to become familiar with many more modeling forms.
1.2 Background
1.2.1 Overview and Process of Mathematical Modeling
Bender (2000, pp. 1–8) first introduced a process for modeling. He highlighted the following: formulate the model, outline the model, ask if it is useful, and test the model. Others have expanded this simple outlined process. Giordano et al. (2014, p. 64) presented a six-step process: identify the problem to be solved, make assumptions, solve the model, verify the model, implement the model, and maintain the model. Myer (2004, pp. 13–15) suggested some guidelines for modeling, including formulation, mathematical manipulation, and evaluation. Meerschaert (1999) developed a five-step process: ask the question, select the modeling approach, formulate the model, solve the model, and answer the question. Albright (2010) subscribed mostly to concepts and process described in previous editions of Giordano et al. (2014). Fox (2012, pp. 21–22) suggested an eight-step approach: understand the problem or question, make simplifying assumptions, define all variables, construct the model, solve and interpret the model, verify the model, consider the model’s strengths and weaknesses, and implement the model.
Most of these pioneers in modeling have suggested similar starts in understanding the problem or question to be answered and in making key assumptions to help enable the model to be built. We add the need for sensitivity analysis and model testing in this process to help ensure that we have a model that is performing correctly to answer the appropriate questions.
For example, student teams in the Mathematical Contest in Modeling were building models to determine the all-time best college sports coach. One team picked a coach who coached less than a year, went undefeated for the remaining part of the year, and won their bowl game. Thus, his season was a perfect season. Their algorithm picked this person as the all-time best coach. Sensitivity analysis and model testing could have shown the fallacy to their model.
Someplace between the defining of the variables and the assumptions, we begin to consider the model’s form and technique that might be used to solve the model. The list of techniques is boundless in mathematics, and we will not list them here. Suffice it to say that it might be good to initially decide among the forms: deterministic or stochastic for the model, linear or nonlinear for the relationship of the variables, and continuous or discrete.
For example, consider the following scenarios:
Two observation posts that are 5.43 miles apart pick up a brief radio signal. The sensing devices were oriented at 110° and 119°, respectively, when a signal was detected. The devices are accurate to within 2° (that is of their respective angle of orientation). According to intelligence, the reading of the signal came from a region of active terrorist exchange, and it is inferred that there is a boat waiting for someone to pick up the terrorists. It is dusk, the weather is calm, and there are no currents. A small helicopter leaves a pad from Post 1 and is able to fly accurately along the angle direction. This helicopter has only one detection device, a searchlight. At 200 ft, it can just illuminate a circular region with a radius of 25 ft. The helicopter can fly 225 miles in support of this mission due to its fuel capacity. Where do we search for the boat? How many search helicopters should you use to have a good chance of finding the target (Fox and Jaye, 2011, pp. 82–93)?
The writers of TV and movies decide that they are not receiving fair compensation for their work as their shows continue to be played on cable and DVDs. The writers decide to strike. Management refuses to budge. Can we analyze this to prevent this from reoccurring? Can we build a model to examine this?
Consider locating emergency response teams within a county or region. Can we model the location of ambulances to ensure that the maximum number of potential patients is covered by the emergency response teams? Can we find the minimum number of ambulances required?
You are a new manager of a bank. You set new goals for your tenure as manager. You analyze the current status of service to measure against your goals. Are you meeting demand? If not what can be done to improve service? You want to prevent catastrophic failure at your bank.
You have many alternatives to choose from for your venture. You have certain decision criteria that you consider to sue to help in making this future. Can we build a mathematical model to assist us in this decision?
These are all events that we can model using mathematics. This chapter will help a decision-maker understand what a mathematical modeler might do for them as a confident problem solver using the techniques of mathematical modeling. As a decision-maker, understanding the possibilities and asking the key questions will enable better deci...
