An integrated labor management system for taco bell

Tenth Edition

Bernard W. Taylor III Virginia Polytechnic Institute and State University

Prentice Hall Upper Saddle River, New Jersey 07458

Introduction to Management Science

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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.

Library of Congress Cataloging-in-Publication Data

Taylor, Bernard W. Introduction to management science / Bernard W. Taylor III.—10th ed.

p. cm. Includes bibliographical references and index. ISBN-13: 978-0-13-606436-7 (alk. paper) ISBN-10: 0-13-606436-1 (alk. paper)

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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.

1

Chapter 1

Management Science

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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.

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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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

Time Out 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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6 Chapter 1 Management Science

Some management science techniques do not generate an answer or a recommended decision. Instead, they provide descriptive results: results that describe the system being modeled. For example, suppose the business firm in our example desires to know the aver- age number of units sold each month during a year. The monthly data (i.e., sales) for the past year are as follows:

Management Science Application Management Science at Taco Bell

Taco Bell, an international fast-food chain with annual salesof approximately $4.6 billion, operates more than 6,500 locations worldwide. In the fast-food business the operating objective is, in general, to provide quality food, good service, and a clean environment. Although Taco Bell sees these three attributes as equally important, good service, as measured by its speed, has the greatest impact on revenues.

The 3-hour lunch period 11:00 A.M. to 2:00 P.M. accounts for 52% of Taco Bell’s daily sales. Most fast-food restaurants have lines of waiting customers during this period, and so speed of service determines sales capacity. If service time decreases, sales capacity increases, and vice versa. However, as speed of service increases, labor costs also increase. Because very few food items can be pre- pared in advance and inventoried, products must be prepared when they are ordered, making food preparation very labor inten- sive. Thus, speed of service depends on labor availability.

Taco Bell research studies showed that when customers are in line up to 5 minutes only, their perception of that waiting time is only a few minutes. However, after waiting time exceeds 5 minutes, customer perception of that waiting time increases exponentially. The longer the perceived waiting time, the more likely the customer is to leave the restaurant without ordering. The company determined that a 3-minute average waiting time would result in only 2.5% of customers leaving. The company believed this was an acceptable level of attrition, and it estab- lished this waiting time as its service goal.

To achieve this goal Taco Bell developed a labor-management system based on an integrated set of management science mod- els to forecast customer traffic for every 15-minute interval

during the day and to schedule employees accordingly to meet customer demand. This labor-management system includes a forecasting model to predict customer transactions; a simulation model to determine labor requirements based on these trans- actions; and an integer programming model to schedule employees and minimize payroll. From 1993 through 1997 the labor-management system using these models saved Taco Bell over $53 million.

Source: J. Heuter and W. Swart, “An Integrated Labor- Management System for Taco Bell,” Interfaces 28, no. 1 (January– February 1998): 75–91.

Month Sales Month Sales

January 30 July 35

February 40 August 50

March 25 September 60

April 60 October 40

May 30 November 35

June 25 December 50

Total 480 units

A management science solution can be either a recommended

decision or information that helps a manager make a decision.

Monthly sales average 40 units . This result is not a decision; it is informa- tion that describes what is happening in the system. The results of the management science

(480 , 12)

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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.

Model Building: Break-Even Analysis 7

techniques in this text are examples of the two types shown in this section: (1) solutions/ decisions and (2) descriptive results.

Implementation The final step in the management science process for problem solving described in Figure 1.1 is implementation. Implementation is the actual use of the model once it has been developed or the solution to the problem the model was developed to solve. This is a critical but often overlooked step in the process. It is not always a given that once a model is developed or a solution found, it is automatically used. Frequently the person responsible for putting the model or solution to use is not the same person who developed the model, and thus the user may not fully understand how the model works or exactly what it is sup- posed to do. Individuals are also sometimes hesitant to change the normal way they do things or to try new things. In this situation the model and solution may get pushed to the side or ignored altogether if they are not carefully explained and their benefit fully demon- strated. If the management science model and solution are not implemented, then the effort and resources used in their development have been wasted.

Model Building: Break-Even Analysis

In the previous section we gave a brief, general description of how management science models are formulated and solved, using a simple algebraic example. In this section we will continue to explore the process of building and solving management science models, using break-even analysis, also called profit analysis. Break-even analysis is a good topic to expand our discussion of model building and solution because it is straightforward, rela- tively familiar to most people, and not overly complex. In addition, it provides a convenient means to demonstrate the different ways management science models can be solved— mathematically (by hand), graphically, and with a computer.

The purpose of break-even analysis is to determine the number of units of a product (i.e., the volume) to sell or produce that will equate total revenue with total cost. The point where total revenue equals total cost is called the break-even point, and at this point profit is zero. The break-even point gives a manager a point of reference in determining how many units will be needed to ensure a profit.

Components of Break-Even Analysis The three components of break-even analysis are volume, cost, and profit. Volume is the level of sales or production by a company. It can be expressed as the number of units (i.e., quantity) produced and sold, as the dollar volume of sales, or as a percentage of total capac- ity available.

Two type of costs are typically incurred in the production of a product: fixed costs and variable costs. Fixed costs are generally independent of the volume of units produced and sold. That is, fixed costs remain constant, regardless of how many units of product are pro- duced within a given range. Fixed costs can include such items as rent on plant and equip- ment, taxes, staff and management salaries, insurance, advertising, depreciation, heat and light, and plant maintenance. Taken together, these items result in total fixed costs.

Variable costs are determined on a per-unit basis. Thus, total variable costs depend on the number of units produced. Variable costs include such items as raw materials and resources, direct labor, packaging, material handling, and freight.

Fixed costs are independent of volume and remain constant.

Variable costs depend on the number of items produced.

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8 Chapter 1 Management Science

Total variable costs are a function of the volume and the variable cost per unit. This relationship can be expressed mathematically as

where and (number of units) sold. The total cost of an operation is computed by summing total fixed cost and total vari-

able cost, as follows:

or

where . As an example, consider Western Clothing Company, which produces denim jeans. The

company incurs the following monthly costs to produce denim jeans:

If we arbitrarily let the monthly sales volume, v, equal 400 pairs of denim jeans, the total cost is

The third component in our break-even model is profit. Profit is the difference between total revenue and total cost. Total revenue is the volume multiplied by the price per unit,

where . For our clothing company example, if denim jeans sell for $23 per pair and we sell

400 pairs per month, then the total monthly revenue is

Now that we have developed relationships for total revenue and total cost, profit (Z) can be computed as follows:

Computing the Break-Even Point For our clothing company example, we have determined total revenue and total cost to be $9,200 and $13,200, respectively. With these values, there is no profit but, instead, a loss of $4,000:

We can verify this result by using our total profit formula,

and the values , , , and :

= - $4,000 = $9,200 - 10,000 - 3,200 = $(400)(23) - 10,000 - (400)(8)

Z = vp - cf - vcv

cv = $8cf = $10,000p = $23v = 400

Z = vp - cf - vcv

total profit = total revenue - total cost = $9,200 - 13,200 = - $4,000

= vp - cf - vcv Z = vp - (cf + vcv)

total profit = total revenue - total cost

total revenue = vp = (400)(23) = $9,200

p = price per unit

total revenue = vp

TC = cf + vcv = $10,000 + (400)(8) = $13,200

variable cost = cv = $8 per pair fixed cost = cf = $10,000

cf = fixed cost

TC = cf + vcv

total cost = total fixed cost + total variable cost

v = volumecv = variable cost per unit

total variable cost = vcv

Total cost (TC) equals the fixed cost plus the variable cost

per unit multiplied by volume (v).

(cv) (cf )

Profit is the difference between total revenue (volume multiplied

by price) and total cost.

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Model Building: Break-Even Analysis 9

Obviously, the clothing company does not want to operate with a monthly loss of $4,000 because doing so might eventually result in bankruptcy. If we assume that price is static because of market conditions and that fixed costs and the variable cost per unit are not subject to change, then the only part of our model that can be varied is volume. Using the modeling terms we developed earlier in this chapter, price, fixed costs, and variable costs are parameters, whereas the volume, v, is a decision variable. In break-even analysis we want to compute the value of v that will result in zero profit.

At the break-even point, where total revenue equals total cost, the profit, Z, equals zero. Thus, if we let profit, Z, equal zero in our total profit equation and solve for v, we can deter- mine the break-even volume:

In other words, if the company produces and sells 666.7 pairs of jeans, the profit (and loss) will be zero and the company will break even. This gives the company a point of refer- ence from which to determine how many pairs of jeans it needs to produce and sell in order to gain a profit (subject to any capacity limitations). For example, a sales volume of 800 pairs of denim jeans will result in the following monthly profit:

In general, the break-even volume can be determined using the following formula:

For our example,

Graphical Solution It is possible to represent many of the management science models in this text graphically and use these graphical models to solve problems. Graphical models also have the advan- tage of providing a “picture” of the model that can sometimes help us understand the mod- eling process better than the mathematics alone can. We can easily graph the break-even model for our Western Clothing Company example because the functions for total cost and total revenue are linear. That means we can graph each relationship as a straight line on a set of coordinates, as shown in Figure 1.2.

In Figure 1.2, the fixed cost, , has a constant value of $10,000, regardless of the volume. The total cost line, TC, represents the sum of variable cost and fixed cost. The total cost line

cf

= 666.7 pairs of jeans

= 10,000

23 - 8

v = cf

p - cv

v = cf

p - cv

v(p - cv) = cf 0 = v(p - cv) - cf Z = vp - cf - vcv

= $(800)(23) - 10,000 - (800)(8) = $2,000 Z = vp - cf - vcv

v = 666.7 pairs of jeans 15v = 10,000

0 = 23v - 10,000 - 8v 0 = v(23) - 10,000 - v(8) Z = vp - cf - vcv

The break-even point is the volume (v) that equates total revenue with total cost where

profit is zero.

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10 Chapter 1 Management Science

10

20

30

40

50

200 0

400 600 800 1,000 1,200 1,400 1,600 Volume, v

Total cost

R ev

en ue

, c os

t, an

d pr

of it

($ 1,

00 0s

)

Variable cost

Fixed cost

Total revenue

Loss

Profit

Break-even point

Figure 1.2

Break-even model

increases because variable cost increases as the volume increases. The total revenue line also increases as volume increases, but at a faster rate than total cost. The point where these two lines intersect indicates that total revenue equals total cost. The volume, v, that corresponds to this point is the break-even volume. The break-even volume in Figure 1.2 is 666.7 pairs of denim jeans.

Sensitivity Analysis We have now developed a general relationship for determining the break-even volume, which was the objective of our modeling process. This relationship enables us to see how the level of profit (and loss) is directly affected by changes in volume. However, when we developed this model, we assumed that our parameters, fixed and variable costs and price, were constant. In reality such parameters are frequently uncertain and can rarely be assumed to be constant, and changes in any of the parameters can affect the model solu- tion. The study of changes on a management science model is called sensitivity analysis— that is, seeing how sensitive the model is to changes.

Sensitivity analysis can be performed on all management science models in one form or another. In fact, sometimes companies develop models for the primary purpose of experi- mentation to see how the model will react to different changes the company is contemplat- ing or that management might expect to occur in the future. As a demonstration of how sensitivity analysis works, we will look at the effects of some changes on our break-even model.

The first thing we will analyze is price. As an example, we will increase the price for denim jeans from $23 to $30. As expected, this increases the total revenue, and it therefore reduces the break-even point from 666.7 pairs of jeans to 454.5 pairs of jeans:

The effect of the price change on break-even volume is illustrated in Figure 1.3. Although a decision to increase price looks inviting from a strictly analytical point of

view, it must be remembered that the lower break-even volume and higher profit are possible but not guaranteed. A higher price can make it more difficult to sell the product. Thus, a change in price often must be accompanied by corresponding increases in costs, such as those for advertising, packaging, and possibly production (to enhance quality). However, even such direct changes as these may have little effect on product demand

= 10,000

30 - 8 = 454.5 pairs of denim jeans

v = cf

p - cv

In general, an increase in price lowers the break-even point, all

other things held constant.

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because price is often sensitive to numerous factors, such as the type of market, monopo- listic elements, and product differentiation.

When we increased price, we mentioned the possibility of raising the quality of the product to offset a potential loss of sales due to the price increase. For example, suppose the stitching on the denim jeans is changed to make the jeans more attractive and stronger. This change results in an increase in variable costs of $4 per pair of jeans, thus raising the variable cost per unit, , to $12 per pair. This change (in conjunction with our previous price change to $30) results in a new break-even volume:

This new break-even volume and the change in the total cost line that occurs as a result of the variable cost change are shown in Figure 1.4.

= 10,000

30 - 12 = 555.5 pairs of denim jeans

v = cf

p - cv

cv

Model Building: Break-Even Analysis 11

10

20

30

40

50

200 0

400 600 800 1,000 1,200 1,400 1,600 Volume, v

Total cost

R ev

en ue

, c os

t, an

d pr

of it

($ 1,

00 0s

)

New total revenue

Fixed cost

Old total revenue

Old B-E point

New B-E point

Figure 1.3

Break-even model with an increase in price

In general, an increase in variable costs will decrease the break-even

point, all other things held constant.

10

20

30

40

50

200 0

400 600 800 1,000 1,200 1,400 1,600 Volume, v

Old total cost

R ev

en ue

, c os

t, an

d pr

of it

($ 1,

00 0s

)

Total revenue

Fixed cost

New total cost

New B-E point

Old B-E point

Figure 1.4

Break-even model with an increase in variable cost

Next let’s consider an increase in advertising expenditures to offset the potential loss in sales resulting from a price increase. An increase in advertising expenditures is an addition to fixed costs. For example, if the clothing company increases its monthly advertising bud- get by $3,000, then the total fixed cost, , becomes $13,000. Using this fixed cost, as well ascf

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12 Chapter 1 Management Science

10

20

30

40

50

200 0

400 600 800 1,000 1,200 1,400 1,600 Volume, v

Old total cost

R ev

en ue

, c os

t, an

d pr

of it

($ 1,

00 0s

)

Total revenue

Old fixed cost

New total cost

New fixed cost

New B-E point

Old B-E point

Figure 1.5

Break-even model with a change in fixed cost

the increased variable cost per unit of $12 and the increased price of $30, we compute the break-even volume as follows:

This new break-even volume, representing changes in price, fixed costs, and variable costs, is illustrated in Figure 1.5. Notice that the break-even volume is now higher than the origi- nal volume of 666.7 pairs of jeans, as a result of the increased costs necessary to offset the potential loss in sales. This indicates the necessity to analyze the effect of a change in one of the break-even components on the whole break-even model. In other words, generally it is not sufficient to consider a change in one model component without considering the over- all effect.

= 722.2 pairs of denim jeans

= 13,000

30 - 12

v = cf

p - cv

In general, an increase in fixed costs will increase the break-even

point, all other things held constant.

Computer Solution

Throughout the text we will demonstrate how to solve management science models on the computer by using Excel spreadsheets and QM for Windows, a general-purpose quantita- tive methods software package by Howard Weiss. QM for Windows has program modules to solve almost every type of management science problem you will encounter in this book. There are a number of similar quantitative methods software packages available on the market, with characteristics and capabilities similar to those of QM for Windows. In most cases you simply input problem data (i.e., model parameters) into a model template, click on a solve button, and the solution appears in a Windows format. QM for Windows is included on the companion Web site for this text.

Spreadsheets are not always easy to use, and you cannot conveniently solve every type of management science model by using a spreadsheet. Most of the time you must not only input the model parameters but also set up the model mathematics, including formulas, as well as your own model template with headings to display your solution output. However, spreadsheets provide a powerful reporting tool in which you can present your model and results in any format you choose. Spreadsheets such as Excel have become almost

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Computer Solution 13

universally available to anyone who owns a computer. In addition, spreadsheets have become very popular as a teaching tool because they tend to guide the student through a modeling procedure, and they can be interesting and fun to use. However, because spread- sheets are somewhat more difficult to set up and apply than is QM for Windows, we will spend more time explaining their use to solve various types of problems in this text.

One of the difficult aspects of using spreadsheets to solve management science problems is setting up a spreadsheet with some of the more complex models and formulas. For the most complex models in the text we will show how to use Excel QM, a supplemental spreadsheet macro that is included on the companion Web site for this text. A macro is a template or an overlay that already has the model format with the necessary formulas set up on the spreadsheet so that the user only has to input the model parameters. We will demonstrate Excel QM in six chapters, including this chapter, Chapter 6 (“Transporta- tion, Transshipment, and Assignment Problems”), Chapter 12 (“Decision Analysis”), Chapter 13 (“Queuing Analysis”), Chapter 15 (“Forecasting”), and Chapter 16 (“Inventory Management”).

Later in this text we will also demonstrate two spreadsheet add-ins, TreePlan and Crystal Ball, which are also included on the companion Web site for this text. TreePlan is a program for setting up and solving decision trees that we use in Chapter 12 (“Decision Analysis”), whereas Crystal Ball is a simulation package that we use in Chapter 14 (“Simulation”). Also, in Chapter 8 (“Project Management”) we will demonstrate Microsoft Project.

In this section we will demonstrate how to use Excel, Excel QM, and QM for Windows, using our break-even model example for Western Clothing Company.

Excel Spreadsheets To solve the break-even model using Excel, you must set up a spreadsheet with headings to identify your model parameters and variables and then input the appropriate mathemati- cal formulas into the cells where you want to display your solution. Exhibit 1.1 shows the spreadsheet for the Western Clothing Company example. Setting up the different headings to describe the parameters and the solution is not difficult, but it does require that you know your way around Excel a little. Appendix B provides a brief tutorial titled “Setting Up and Editing a Spreadsheet” for solving management science problems.

Formula for v, break-even point

Exhibit 1.1

Notice that cell D10 contains the break-even formula, which is displayed on the toolbar near the top of the screen. The fixed cost of $10,000 is typed in cell D4, the variable cost of $8 is in cell D6, and the price of $23 is in cell D8.

As we present more complex models and problems in the chapters to come, the spread- sheets we will develop to solve these problems will become more involved and will enable us to demonstrate different features of Excel and spreadsheet modeling.

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14 Chapter 1 Management Science

Exhibit 1.2

Enter model parameters in cells B10:B13.

Exhibit 1.3

The Excel QM Macro for Spreadsheets Excel QM is included on the companion Web site for this text. You can install Excel QM onto your computer by following a brief series of steps displayed when the program is first accessed.

After Excel is started, Excel QM is normally accessed from the computer’s program files, where it is usually loaded. When Excel QM is activated, “Add-Ins” will appear at the top of the spreadsheet (as indicated in Exhibit 1.2). Clicking on “Excel QM” will pull down a menu of the topics in Excel QM, one of which is break-even analysis. Clicking on “Break- Even Analysis” will result in the window for spreadsheet initialization shown in Exhibit 1.2. Every Excel QM macro listed on the menu will start with a “Spreadsheet Initialization” window similar to this one.

In the window in Exhibit 1.2 you can enter a spreadsheet title and choose under “Options” whether you also want volume analysis and a graph. Clicking on “OK” will result in the spreadsheet shown in Exhibit 1.3. The first step is to input the values for the Western Clothing Company example in cells B10 to B13, as shown in Exhibit 1.3. The spreadsheet shows the break-even volume in cell B17. However, notice that we have also chosen to per- form some volume analysis by entering a hypothetical volume of 800 units in cell B13, which results in the volume analysis in cells B20 to B23.

QM for Windows You begin using QM for Windows by clicking on the “Module” button on the toolbar at the top of the main window that appears when you start the program. This will pull down a window with a list of all the model solution modules available in QM for Windows.

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Management Science Modeling Techniques 15

Exhibit 1.4

Clicking on the “Break-even Analysis” module will access a new screen for typing in the problem title. Clicking again will access a screen with input cells for the model parame- ters—that is, fixed cost, variable cost, and price (or revenue). Next, clicking on the “Solve” button at the top of the screen will provide the solution to the Western Clothing Company example, as shown in Exhibit 1.4.

Exhibit 1.5

You can also get the graphical model and solution for this problem by clicking on “Win- dow” at the top of the solution screen and selecting the menu item for a graph of the prob- lem. The break-even graph for the Western Clothing example is shown in Exhibit 1.5.

Management Science Modeling Techniques

This text focuses primarily on two of the five steps of the management science process described in Figure 1.1—model construction and solution. These are the two steps that use the management science technique. In a textbook, it is difficult to show how an unstructured real-world problem is identified and defined because the problem must be written out.

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16 Chapter 1 Management Science

Management science techniques

Probabilistic techniques

Linear mathematical programming

Linear programming models Graphical analysis Sensitivity analysis Transportation, transshipment, and assignment Integer linear programming Goal programming

Decision analysis

Probability and statistics

Queuing

Network

Text

techniques

Network flow Project

management (CPM/PERT)

Other techniques

Forecasting Simulation

Inventory

Analytical hierarchy process (AHP) Nonlinear programming

Companion Web Site

Branch and bound Markov analysis Game theory

method

Simplex method Transportation

and assignment solution methods

Nonlinear programming

Figure 1.6

Classification of management science techniques

However, once a problem statement has been given, we can show how a model is con- structed and a solution is derived. The techniques presented in this text can be loosely classified into four categories, as shown in Figure 1.6.

Linear Mathematical Programming Techniques Chapters 2 through 6 and 9 present techniques that together make up linear mathematical programming. (The first example used to demonstrate model construction earlier in this chapter is a very rudimentary linear programming model.) The term programming used to identify this technique does not refer to computer programming but rather to a predeter- mined set of mathematical steps used to solve a problem. This particular class of techniques holds a predominant position in this text because it includes some of the more frequently used and popular techniques in management science.

In general, linear programming models help managers determine solutions (i.e., make decisions) for problems that will achieve some objective in which there are restrictions, such as limited resources or a recipe or perhaps production guidelines. For example, you could actually develop a linear programming model to help determine a breakfast menu for yourself that would meet dietary guidelines you may have set, such as number of calo- ries, fat content, and vitamin level, while minimizing the cost of the breakfast. Manufactur- ing companies develop linear programming models to help decide how many units of different products they should produce to maximize their profit (or minimize their cost), given scarce resources such as capital, labor, and facilities.

Six chapters in this text are devoted to this topic because there are several variations of linear programming models that can be applied to specific types of problems. Chapter 4 is devoted entirely to describing example linear programming models for several different types of problem scenarios. Chapter 6, for example, focuses on one particular type of linear programming application for transportation, transshipment, and assignment problems. An example of a transportation problem is a manager trying to determine the lowest-cost routes to use to ship goods from several sources (such as plants or warehouses) to several destinations (such as retail stores), given that each source may have limited goods available and each destination may have limited demand for the goods. Also, Chapter 9 includes the topic of goal programming, which is a form of linear programming that addresses prob- lems with more than one objective or goal.

As mentioned previously in this chapter, some of the more mathematical topics in the text are included as supplementary modules on the companion Web site for the text. Among