# Quantitative methods homework

2 Chapter 1 Management Science

Management science is a scientific approach to solving

management problems.

Management science is the application of a scientific approach to solving managementproblems in order to help managers make better decisions. As implied by this defini- tion, management science encompasses a number of mathematically oriented techniques that have either been developed within the field of management science or been adapted from other disciplines, such as the natural sciences, mathematics, statistics, and engineer- ing. This text provides an introduction to the techniques that make up management science and demonstrates their applications to management problems.

Management science is a recognized and established discipline in business. The applica- tions of management science techniques are widespread, and they have been frequently credited with increasing the efficiency and productivity of business firms. In various sur- veys of businesses, many indicate that they use management science techniques, and most rate the results to be very good. Management science (also referred to as operations research, quantitative methods, quantitative analysis, and decision sciences) is part of the fundamental curriculum of most programs in business.

As you proceed through the various management science models and techniques con- tained in this text, you should remember several things. First, most of the examples pre- sented in this text are for business organizations because businesses represent the main users of management science. However, management science techniques can be applied to solve problems in different types of organizations, including services, government, mili- tary, business and industry, and health care.

Second, in this text all of the modeling techniques and solution methods are mathemat- ically based. In some instances the manual, mathematical solution approach is shown because it helps one understand how the modeling techniques are applied to different problems. However, a computer solution is possible for each of the modeling techniques in this text, and in many cases the computer solution is emphasized. The more detailed math- ematical solution procedures for many of the modeling techniques are included as supple- mental modules on the companion Web site for this text.

Finally, as the various management science techniques are presented, keep in mind that management science is more than just a collection of techniques. Management science also involves the philosophy of approaching a problem in a logical manner (i.e., a scientific approach). The logical, consistent, and systematic approach to problem solving can be as useful (and valuable) as the knowledge of the mechanics of the mathematical techniques themselves. This understanding is especially important for those readers who do not always see the immediate benefit of studying mathematically oriented disciplines such as manage- ment science.

The Management Science Approach to Problem Solving

As indicated in the previous section, management science encompasses a logical, systematic approach to problem solving, which closely parallels what is known as the scientific method for attacking problems. This approach, as shown in Figure 1.1, follows a generally recognized and ordered series of steps: (1) observation, (2) definition of the problem, (3) model construction, (4) model solution, and (5) implementation of solution results. We will analyze each of these steps individually.

Observation The first step in the management science process is the identification of a problem that exists in the system (organization). The system must be continuously and closely observed

Management science can be used in a variety of organizations to

solve many different types of problems.

Management science encompasses a logical approach to problem

solving.

The steps of the scientific method are (1) observation, (2) problem

definition, (3) model construction, (4) model solution, and

(5) implementation.

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Introduction to Management Science, Tenth Edition, by Bernard W. Taylor III. Published by Prentice Hall. Copyright © 2010 by Pearson Education, Inc.

so that problems can be identified as soon as they occur or are anticipated. Problems are not always the result of a crisis that must be reacted to but, instead, frequently involve an anticipatory or planning situation. The person who normally identifies a problem is the manager because managers work in places where problems might occur. However, prob- lems can often be identified by a management scientist, a person skilled in the techniques of management science and trained to identify problems, who has been hired specifically to solve problems using management science techniques.

Definition of the Problem Once it has been determined that a problem exists, the problem must be clearly and con- cisely defined. Improperly defining a problem can easily result in no solution or an inap- propriate solution. Therefore, the limits of the problem and the degree to which it pervades other units of the organization must be included in the problem definition. Because the existence of a problem implies that the objectives of the firm are not being met in some way, the goals (or objectives) of the organization must also be clearly defined. A stated objective helps to focus attention on what the problem actually is.

Model Construction A management science model is an abstract representation of an existing problem situa- tion. It can be in the form of a graph or chart, but most frequently a management science model consists of a set of mathematical relationships. These mathematical relationships are made up of numbers and symbols.

As an example, consider a business firm that sells a product. The product costs $5 to produce and sells for $20. A model that computes the total profit that will accrue from the items sold is

In this equation x represents the number of units of the product that are sold, and Z rep- resents the total profit that results from the sale of the product. The symbols x and Z are variables. The term variable is used because no set numeric value has been specified for these items. The number of units sold, x, and the profit, Z, can be any amount (within limits); they can vary. These two variables can be further distinguished. Z is a dependent variable because

Z = $20x – 5x

The Management Science Approach to Problem Solving 3

Management science techniques

Observation

Problem definition

Model construction

Solution

Feedback

Information

Implementation

Figure 1.1

The management science process

A management scientist is a person skilled in the application of

management science techniques.

A variable is a symbol used to represent an item that can take on

any value.

A model is an abstract mathematical representation of a

problem situation.

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Introduction to Management Science, Tenth Edition, by Bernard W. Taylor III. Published by Prentice Hall. Copyright © 2010 by Pearson Education, Inc.

4 Chapter 1 Management Science

its value is dependent on the number of units sold; x is an independent variable because the number of units sold is not dependent on anything else (in this equation).

The numbers $20 and $5 in the equation are referred to as parameters. Parameters are constant values that are generally coefficients of the variables (symbols) in an equation. Parameters usually remain constant during the process of solving a specific problem. The parameter values are derived from data (i.e., pieces of information) from the problem envi- ronment. Sometimes the data are readily available and quite accurate. For example, pre- sumably the selling price of $20 and product cost of $5 could be obtained from the firm’s accounting department and would be very accurate. However, sometimes data are not as readily available to the manager or firm, and the parameters must be either estimated or based on a combination of the available data and estimates. In such cases, the model is only as accurate as the data used in constructing the model.

The equation as a whole is known as a functional relationship (also called function and relationship). The term is derived from the fact that profit, Z, is a function of the number of units sold, x, and the equation relates profit to units sold.

Because only one functional relationship exists in this example, it is also the model. In this case the relationship is a model of the determination of profit for the firm. However, this model does not really replicate a problem. Therefore, we will expand our example to create a problem situation.

Let us assume that the product is made from steel and that the business firm has 100 pounds of steel available. If it takes 4 pounds of steel to make each unit of the product, we can develop an additional mathematical relationship to represent steel usage:

This equation indicates that for every unit produced, 4 of the available 100 pounds of steel will be used. Now our model consists of two relationships:

We say that the profit equation in this new model is an objective function, and the resource equation is a constraint. In other words, the objective of the firm is to achieve as much profit, Z, as possible, but the firm is constrained from achieving an infinite profit by the limited amount of steel available. To signify this distinction between the two relation- ships in this model, we will add the following notations:

subject to

This model now represents the manager’s problem of determining the number of units to produce. You will recall that we defined the number of units to be produced as x. Thus, when we determine the value of x, it represents a potential (or recommended) decision for the manager. Therefore, x is also known as a decision variable. The next step in the manage- ment science process is to solve the model to determine the value of the decision variable.

Model Solution Once models have been constructed in management science, they are solved using the management science techniques presented in this text. A management science solution technique usually applies to a specific type of model. Thus, the model type and solution method are both part of the management science technique. We are able to say that a model

4x = 100

maximize Z = $20x – 5x

4x = 100 Z = $20x – 5x

4x = 100 lb. of steel

Parameters are known, constant values that are often coefficients of

variables in equations.

Data are pieces of information from the problem environment.

A model is a functional relationship that includes

variables, parameters, and equations.

A management science technique usually applies to a specific

model type.

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Introduction to Management Science, Tenth Edition, by Bernard W. Taylor III. Published by Prentice Hall. Copyright © 2010 by Pearson Education, Inc.

The Management Science Approach to Problem Solving 5

is solved because the model represents a problem. When we refer to model solution, we also mean problem solution.

For the example model developed in the previous section,

subject to

the solution technique is simple algebra. Solving the constraint equation for x, we have

Substituting the value of 25 for x into the profit function results in the total profit:

Thus, if the manager decides to produce 25 units of the product and all 25 units sell, the business firm will receive $375 in profit. Note, however, that the value of the decision vari- able does not constitute an actual decision; rather, it is information that serves as a recom- mendation or guideline, helping the manager make a decision.

= $375 = 20(25) – 5(25)

Z = $20x – 5x

x = 25 units x = 100>4

4x = 100

4x = 100

maximize Z = $20x – 5x

Ti m e O u t for Pioneers in Management Science Throughout this text TIME OUT boxes introduce you to the individuals who developed the various techniques that are described in the chapters. This will provide a historical per- spective on the development of the field of management science. In this first instance we will briefly outline the develop- ment of management science.

Although a number of the mathematical techniques that make up management science date to the turn of the twentieth century or before, the field of management science itself can trace its beginnings to military operations research (OR) groups formed during World War II in Great Britain circa 1939. These OR groups typically consisted of a team of about a dozen individuals from different fields of science, mathematics, and the military, brought together to find solutions to military- related problems. One of the most famous of these groups— called “Blackett’s circus” after its leader, Nobel laureate P. M. S. Blackett of the University of Manchester and a former naval officer—included three physiologists, two mathematical physi- cists, one astrophysicist, one general physicist, two mathemati- cians, an Army officer, and a surveyor. Blackett’s group and the other OR teams made significant contributions in improving Britain’s early-warning radar system (which was instrumental in their victory in the Battle of Britain), aircraft gunnery, anti- submarine warfare, civilian defense, convoy size determination, and bombing raids over Germany.

The successes achieved by the British OR groups were observed by two Americans working for the U.S. military, Dr. James B. Conant and Dr. Vannevar Bush, who recommended that OR teams be established in the U.S. branches of the military. Subsequently, both the Air Force and Navy created OR groups.

After World War II the contributions of the OR groups were considered so valuable that the Army, Air Force, and Navy set up various agencies to continue research of military problems. Two of the more famous agencies were the Navy’s Operations Evaluation Group at MIT and Project RAND, established by the Air Force to study aerial warfare. Many of the individuals who developed operations research and management science techniques did so while working at one of these agencies after World War II or as a result of their work there.

As the war ended and the mathematical models and tech- niques that were kept secret during the war began to be released, there was a natural inclination to test their applicabil- ity to business problems. At the same time, various consulting firms were established to apply these techniques to industrial and business problems, and courses in the use of quantitative techniques for business management began to surface in American universities. In the early 1950s the use of these quan- titative techniques to solve management problems became known as management science, and it was popularized by a book of that name by Stafford Beer of Great Britain.

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