Download
1 / 35
350 likes | 519 Vues
Chapter 1. Introduction to Modeling. Introduction. Today’s business problems tend to be very complex. Businesses turn to algorithms to solve problems. Management science models become useful when common sense and intuition fail to solve the problems.
E N D
Chapter 1 Introduction to Modeling
Introduction • Today’s business problems tend to be very complex. • Businesses turn to algorithms to solve problems. • Management science models become useful when common sense and intuition fail to solve the problems. • This book describes how quantitative methods can be used to solve business problems. • The methods in this book are powerful because they apply to many problems and environments.
Introduction continued • The purpose of this book is to demonstrate a variety of problems that have been solved successfully with management science methods. Specifically, the book: • emphasizes both the applied and mathematical aspects of management science; • discusses many successful management science applications; • leads through the solution procedures of many interesting and realistic problems; • uses Excel spreadsheets to solve problems, which makes the quantitative analysis understandable and intuitive.
Mathematical model • A mathematical model is a quantitative representation, or idealization, of a real problem. • It is a key to virtually every management science application. • It can be phrased in terms of mathematical expressions (equations and inequalities) or a series of interrelated cells in a spreadsheet.
Mathematical model continued • The purpose of a mathematical model is to represent the essence of a problem in a concise form, providing several advantages: • Enables managers to understand the problem better; • Helps to define the scope of the problem, the possible solutions, and the data requirements • Allows analysts to employ a variety of the mathematical solution procedures that have been developed over the last 50 years; • The modeling process itself, if done correctly, often helps to “sell” the solution to the people who must work with the system that is eventually implemented.
Chapter outline • Example of a relatively simple mathematical model. • Discussion of the distinction between modeling and a collection of models. • Discussion of a seven-step modeling process that is used in most successful management science applications. • Discussion of why the study of management science is valuable, not only to large corporations, but also to students who are about to enter the business world.
Model types • Descriptive models: models that simply describe a situation. • Optimization models: models that suggest a desirable course of action. • Example • Waiting line: Convenience store with a single cash register. • The manager suspects that excessive waiting times in lines to the register hurt the business. • The manager builds a mathematical model to help understand the problem, and suggest improvements to the current situation.
Descriptive model • Waiting line example is a typical queuing problem. • Two important inputsto the problem: • The arrival rate of potential customers to the store • The rate at which customers can be served by a single cashier • As the arrival rate increases and/or service rate decreases, the waiting line will tend to increase, and customers will wait longer or not enter the line. • Length of waiting line, time in line per customer, and fraction of customers who do not enter are referred to as outputs.
Descriptive model continued • More complex models do not have equations that relate inputs to outputs, but there may be mathematical procedures for calculating outputs from inputs implemented in Excel. • Convenience store problem: • Inputs: the arrival rate A, the service rate S, and the number in the store, N. • First two inputs can be estimated; N can be assumed.
Descriptive model continued • Input estimates are entered in the spreadsheet model. • Formulas entered into the spreadsheet reflect an adequate approximation of the convenience store’s situation. • This model allows the manager to enter any sensible values for the inputs in cells B4 through B6 and observe the resulting outputs in cells B9 through B11. • The power of the model is that it allows the manager to ask many what-if questions.
Descriptive model continued • In reality, the manager would attempt to validate the spreadsheet model before trusting its answers to what-if questions. • At the very least, examine the reasonableness of the assumptions. • Is the customer arrival rate constant? • Check the outputs when the current inputs are used from the data currently available, and modify the model if outputs are not corresponding to current data.
Optimization model • Descriptive models fail to reflect any economic information, such as the cost of speeding up service, making customers wait in line, and losing customers. • Given the spreadsheet model developed previously, it is relatively easy to incorporate economic information and then make rational choices.
Optimization model continued • To incorporate economic information into the previous example, the manager has three choices: • Leave the system as it is; • Hire a second person to help the first cashier, with an effect of decreasing average service time from 2.5 to 1.8 minutes; • Lease a new model of cash register to speed up the service process with an effect of decreasing average service time from 2.5 to 1.25 minutes. • Manager’s choice depends on the costs of hiring or leasing, both of which are probably known.
Optimization model continued • Optimization model allows to incorporate all the costs and compare the outputs for all decisions simultaneously • The option to leasethe new cash registeris the clear winnerfrom a cost standpoint • Entering new valuesand checking outputsis easy with this model
Modeling versus models • Management science is a collection of mathematical tools. • Linear programming models (the transportation model, the diet model, the shortest route model, etc) • Inventory models • Queuing models • Management science practitioners argue that majority of real-life management science models cannot be neatly categorized as one of the handful of models from the textbook.
Modeling versus models continued • Emphasis on specific models has been changing in the past two decades, and the goal of this book is to continue this change. • This book stresses modeling, not models. • Learning specific models is essentially a memorization process. • Modeling is a process where you abstract the essence of a real problem into a model. • Successful modelers treat each problem on its own merits and model it appropriately, using all the logical, analytical, and spreadsheet skills they have.
The seven-step modeling process • Step 1: Problem definition • The analyst first defines the organization’s problem. • Defining the problem includes specifying the organization’s objectives and the parts of the organization that must be studied before the problem can be solved. • Step 2: Data collection • After defining the problem, the analyst collects data to estimate the value of parameters that affect the organization’s problem.
The seven-step modeling process continued • Step 3: Model development • In the third step, the analyst develops a model of the problem. • Models where an equation is used to regulate inputs are called analytical models. • Most situations are too complex to be solved with an equation or a system of equations – they are intractable. When no tractable analytical model exists, analysts rely instead on a simulation model, which approximates the behavior of the actual system.
The seven-step modeling process continued • Step 4: Model verification • The analyst now tries to determine whether the model developed in the previous step is an accurate representation of reality. • A first step in determining how well the model fits reality is to check whether the model is valid for the current situation. • For example, to validate the equation for the waiting time W, the manager might observe actual customer waiting times for several hours.
The seven-step modeling process continued • Step 5: Optimization and decision making • Given a model and a set of possible decisions, the analyst must now choose the decision or strategy that best meets the organization’s objectives. • Many optimization models exist, and they will be discussed throughout the book.
The seven-step modeling process continued • Step 6: Model communication to management • The analyst presents the model and the recommendations from the previous steps to the organization. • Step 7: Model implementation • If the organization has accepted the validity and usefulness of the study, the analyst then helps to implement its recommendations. • The implemented system must be monitored constantly (and updated dynamically as the environment changes) to ensure that the model enables the organization to meet its objectives.
The seven-step modeling process continued • The figure below illustrates the seven-step process. • The discussion that follows explores the seven steps in more detail.
Step 1: Problem definition • Typically, a management science model is initiated when an organization believes it has a problem, and calls in a management scientist (the analyst) to solve it. • In such cases, the problem has probably already been defined by the client, and the client hires the analyst to solve this particular problem. • The task of the analyst is to do some investigating before accepting the client’s claim that the problem has been properly defined. • Failure to do so could mean solving the wrong problem and wasting valuable time, money, and energy. • Volkema (1995) advocates spending as much time thinking about the problem and defining it properly as modeling and solving it.
Step 2: Data collection • This crucial step in the modeling process is often the most tedious. • All organizations keep track of various data on their operations, but the data are often not in the form the analyst requires. • One of the analyst’s first jobs is to gather exactly the right data and put the data into an appropriate and consistent format for use in the model. • This typically requires asking questions of key people (such as the cost accountants) throughout the organization, studying existing organizational databases, and performing time-consuming observational studies of the organization’s processes.
Step 3: Model development • After defining the client’s problem and gathering the necessary data, the analyst must develop a model of the problem. • Several properties are desirable for a good model. • First, it should represent the client’s real problem accurately. • It should take into account all important constraints, such as an upper bound on capacity, or its recommendations might not be possible to implement. • On the other hand, the model should be as simple as possible, without getting bogged down in less important details. • Overly complex models are often of little practical use, because they are too difficult to solve and often incomprehensible to clients. • A good model should neither be too simple nor too complex. This is easier said than done.
Step 4: Model verification • This step is particularly important in real management science applications. • A client is much more likely to accept an analyst’s model if the analyst can provide some type of verification. • Verification can take several forms: • Company’s current value of the inputs is used to check the outputs. • Random inputs are used to check if outputs are reasonable(e.g., use of extreme values). • If the model’s outputs are not as expected, then • the model is a poor approximation of the actual situation, or • the model is fine, but the analyst’s intuition is faulty.
Step 5: Optimization and decision making • To use the model to recommend decisions or strategies, the model has to optimize an objective, such as maximize profit or minimize cost. • The optimization phase is typically the most difficult phase from a mathematical standpoint. • A number of solution algorithms are available to solve real problems. • The most famous is the simplex algorithm for linear optimization problems. • When the problem is too complex, a heuristic is used to solve it. Heuristic is guided by common sense, intuition, and trial-and-error.
Step 6: Model communication to management • The analyst must eventually communicate a model and its recommendations to the client. • A large gap typically exists between management science analysts and the managers of organizations. • Managers know their business, but they often do not understand much about mathematics or mathematical models. • The best strategy for a successful presentation is to involve key people in the organization, including top executives, in the project from the beginning. • The analyst should also try to make the model as intuitive and user-friendly as possible.
Step 7: Model implementation • A real management science application is not complete until it has been implemented. • A successful implementation can occur only when step 6 has been accomplished. • To achieve a successful implementation, it is not enough that the management accepts the model; the people who will run it every day must also be thoroughly trained to use it. • A useful model, once implemented, is likely to be expanded by the organization. • The best analysts often design models that can be expanded.
The seven-step processin real life • In real life, not all steps are employed, or not always in the described order. • Numerous potential applications are never implemented even though the technical aspects of the models are perfectly correct. • The most frequent cause is a failure to communicate. • Company politics can be a model’s downfall, especially if the model recommends a course of action that top management simply does not want to follow – for whatever reasons. • In real life, the analyst often generates several iterations of all or some of the seven steps before the project is considered complete.
The model as a beginning,not an end • This book places heavy emphasis on developing spreadsheet models, which is step 3 of the seven-step modeling process. • However, a completed model is really a starting point. • After you have a working model of the problem, you can – and you should – use it as a tool for gaining insights. • For most models, many what-if questions can be asked. • If the model has been developed correctly, it should be capable of answering such what-if questions fairly easily.
Why study management science? • The modeling approach emphasized throughout this book is an important way to think about problems in general, not just the specific problems we discuss. This approach forces you to think logically. • As you work through the many models in this book, your quantitative skills will be sharpened immensely. • No matter what your spreadsheet abilities are when you enter this course, by the time you are finished, you will be a proficient spreadsheet user. • Management science modeling helps you develop your intuition, and it also indicates where intuition alone sometimes fails.
Conclusion • This chapter introduced the field of management science and the process of mathematical modeling. • It reviewed a simple queuing model to provide a concrete understanding of the book concepts. • It explored a seven-step model-building process from problem definition to final implementation. • It discussed why the study of management science is a valuable experience, even if you do not intend to pursue a professional career in this field.
