# Introduction to Models and Decision Making

Learning Objectives

After completing this chapter, you should be able to:

• Define a model and describe how models can be used to analyze operating problems.

• Discuss the nature of forecasting.

• Explain how forecasting can be applied to problems.

• Describe methods of forecasting, including judgment and experience, time-series analysis, and regression and correlation.

• Construct forecasting models.

• Estimate forecasting errors.

6 .Thinkstock

Models and Forecasting

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CHAPTER 6Section 6.1 Introduction to Models and Decision Making

6.1 Introduction to Models and Decision Making

In order for an organization to design, build, and operate a production facility that is capable of meeting customer demand for services (such as health care) or goods (such as ceiling fans), it is necessary for management to obtain an estimate or forecast of demand for its products. A forecast is a prediction of the future. It often examines historical data to determine relationships among key variables in a problem and uses those relationships to make statements about the future value of one or more of the variables. Once an organiza- tion has a forecast of demand, it can make decisions regarding the volume of product that needs to be produced, the number of workers to hire, and other key operating variables. A model is an abstraction from the real problem of the key variables and relationships in order to simplify the problem. The purpose of modeling is to provide the user with a bet- ter understanding of the problem and with a means of manipulating the results for what- if analyses. Forecasting uses models to help organizations predict important parameters. Demand is one of those parameters, but cost, revenue, profits, and other variables can also be forecasted. The purpose of this chapter is to discuss models and describe how they can be applied to business problems, and to explain forecasting and its role in operations.

Stages in Decision Making Organizational performance is a result of the decisions that management makes over a period of time: decisions about what markets to enter, what products to produce, what types of equipment and facilities to acquire, and where to locate facilities. The quality of these decisions is a function of how well managers perform (see Table 6.1).

Table 6.1: Stages in decision making

Stage Example

Define the problem and the factors that influence it

A hospital is having difficulty maintaining high-quality, low-cost food service. The quality and cost of incoming food and the training of staff are influencing factors.

Select criteria to guide the decision; establish objectives

The hospital selects cost per meal and patient satisfaction as the criteria. The objectives are to reduce meal cost by 15% and improve patient satisfaction to 90%, based upon the hospital’s weekly surveys.

Formulate a model or models

The model includes mathematical relationships that indicate how materials (food) and labor are converted into meals. This model includes an analysis of wasted food and the standard amount of labor required to prepare a meal.

Collect relevant data Data on food costs, the amount of food consumed, the number of meals served, and the amount of labor are collected. Patient preferences are investigated so that meals meet nutritional requirements and taste good.

Identify and evaluate alternatives

Alternatives include subcontracting food preparation, considering new food suppliers, establishing better training programs for the staff, and changing management.

Select the best alternative One of the alternatives or some combination of alternatives is selected.

Implement the alternative, and reevaluate

The selected alternative is implemented, and the problem is reevaluated through monitoring costs and the patient survey data to see if the objectives have been achieved.

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CHAPTER 6Section 6.1 Introduction to Models and Decision Making

A model is a way of thinking about a problem. Decision makers use models to increase their understanding of the problem because it helps to simplify the problem by focusing on the key variables and relationships. The model also allows managers to try different options quickly and inexpensively. In these ways, decision making can be improved.

Types of Models Models are commonly seen for airplanes, cars, dams, or other structures. These models can be used to test design characteristics. Model airplanes can be tested in wind tun- nels to determine aerodynamic properties, and a model of a hydroelectric dam can help architects and engineers find ways of integrating the structure with the landscape. These models have physical characteristics similar to those of the real thing. Experiments can be performed on this type of model to see how it may per- form under operating condi- tions. With technology, such as computer simulation systems, virtual models can be rendered and tested quickly and less expensively. The aerodynamic properties of an airplane can be tested in a virtual wind tun- nel that exists only inside the memory of a computer. Models also include the drawings of a building that display the physi- cal relationships between the various parts of the structure. All of these models are simpli- fications of the real thing used to help designers make better decisions.

Computer-based technology has been used for many years to design cars, buildings, fur- niture, and other products. It is moving quickly into the field of medicine. Medical schools teach students about anatomy using 3-D computer generated models. Students can see the nervous system, the blood vessels, the lymph nodes, and glands along with the skel- eton. The software can show each separately and put them all together in one 3-D picture. The software can take input from various medical tests and generate 3-D models of a patient to diagnose medical conditions faster and better.

In addition to these physical and virtual models, managers use mathematical abstraction to model important relationships. The break-even point calculation that is taught in account- ing and finance is an example of applying a mathematical model. The use of drawings and diagrams is also modeling. The newspaper graph that illustrates stock market price changes in the last six months is a way to help the reader see trends in the market. Models do not have to be sophisticated to be useful. Most models can be grouped into four categories, and computers play a critical role in the development and use of each type.

.Associated Press/AP Images

Model airplanes and buildings have physical characteristics similar to full-scale versions and can be used to test design characteristics.

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CHAPTER 6Section 6.1 Introduction to Models and Decision Making

• Mathematical models include algebraic models such as break-even analysis, statis- tical models used in forecasting and quality control, mathematical programming models, and calculus-based models.

• Graphs and charts are pictorial representations of mathematical relationships. They include a visual representation of break-even analysis, a pie chart that illus- trates market share, a graph of stock prices over time, or a bar graph that indi- cates the demand for energy for the last five years.

• Diagrams and drawings are pictorial representations of conceptual relationships. They include a precedence diagram that represents the sequence required to assemble a building, a drawing of a gear that is part of a transmission in a car, a diagram that represents the logic of a computer program, and a drawing of an aircraft carrier.

• Scale models and prototypes are physical representations of an item. They include a scale model of an airplane and the first part produced (prototype), which is normally used for testing purposes. These models are often built and analyzed inside a computer system. Three-dimensional technology called stereolithog- raphy allows computers to create solid models of parts. This is done by succes- sively “printing” very thin layers of a material, which cures quickly to form a sold part.

Mathematical models, graphs and charts, and diagrams are most commonly used by busi- ness and management professionals, so the discussion in this chapter focuses on these types of models.

Application of Models Many people use models frequently without realizing it. At a pizza party, the host will probably determine how much pizza to order by multiplying the number of people expected to attend by the amount each person is expected to consume. The host is likely to then multiply the anticipated cost per pizza by the number ordered to determine the cost. This is a simple mathematical model that can be used to plan a small party or major social event.

In mathematical models, symbols and algebra are used to show relationships. Mathemati- cal models can be simple or complex. For example, suppose a family is planning a trip to Walt Disney World in Orlando, Florida. To estimate gasoline costs for the trip, fam- ily members check a road atlas (one type of model), or go online to get directions and a map (another type of model). They determine that Orlando is approximately a 2,200-mile round trip from their home. From knowledge of the family car (a database), the family estimates that the car will achieve 23 miles per gallon (mpg) on the highway. The average cost of a gallon of gasoline is estimated at \$3.80. Using the following model, they make an estimate of gasoline cost.

Cost 5 (trip miles)(cost per gallon)/miles per gallon

5 12,200 miles 2 1\$3.80 per gallon 2

23 mpg

5 \$363.48

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CHAPTER 6Section 6.2 Forecasting

A mathematical model can be used to answer what-if questions. In the previous example, costs could be estimated with a \$.30 increase in the price of a gallon of gas, as shown in the following:

Cost 5 12,200 miles 2 1\$4.10 per gallon 2

23 mpg

5 \$392.17

The model could also be used to estimate the cost of the trip if the car averaged only 20 miles per gallon, as shown in the following:

Cost 5 12,200 miles 2 1\$3.80 per gallon 2

20 mpg

5 \$418.00

Models cannot include all factors that affect the outcome because many factors cannot be defined precisely. Also, adding too many variables can complicate the model without significantly increasing the accuracy of the prediction. For example, on the trip to Florida, the number of miles driven is affected by the number of rest stops made, the number of unexpected detours taken, and the number of lane changes made. The number of miles per gallon is influenced by the car’s speed, the rate of acceleration, and the amount of time spent idling in traffic. These variables are not in the model. The model builder should ask if adding the variables would significantly improve the model’s accuracy and usefulness.

6.2 Forecasting

Forecasting is an attempt to predict the future. Forecasts are usually the result of examining past experiences to gain insights into the future. These insights often take the form of mathematical models that are used to project future sales, product costs, advertising costs, and more. The application of forecasting is not limited to predicting factors needed to operate a business. Forecasting can also be used to estimate the cost of living, housing prices, the federal debt, and the average family income in the year 2025. For organizations, forecasts are an essential part of planning. It would be illogical to plan for tomorrow without some idea of what could happen.

The critical word in the last sentence is “could.” Any competent forecaster knows that the future holds many possibilities and that a forecast is only one of those possibilities. The difference between what actually happens and what is predicted is forecasting error, which is discussed later in this chapter. In spite of this potential error, management should recognize the need to proceed with planning using the best possible forecast and should develop contingency plans to deal with the possible error. Management should not assume that the future is predetermined, but should realize that its actions can help to shape future events. With the proper plans and execution of those plans, an organization can have some control over its future.

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CHAPTER 6Section 6.2 Forecasting

Stages of Forecast Development The forecasting process consists of the following steps: determining the objectives of the forecast, developing and testing a model, applying the model, considering real-world constraints on the model’s application, and revising and evaluating the forecast (human judgment). Figure 6.1 illustrates these steps.

Figure 6.1: Steps in forecasting

Determining the objectives. What kind of information does the manager need? The fol- lowing questions should be considered:

1. What is the purpose of the forecast? 2. What variables are to be forecast? 3. Who will use the forecast? 4. What is the time frame of the forecast—long or short term? 5. How accurate should the forecast be? 6. When is the forecast needed?

Determine objectives

Develop and test model

Apply the model

Consider constraints

Revise and evaluate

the forecast

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CHAPTER 6Section 6.2 Forecasting

Highlight: Forecasting for Quarry-Front Ice Cream Stand

In a small Midwest town, the Quarry-Front Ice Cream Stand operates in a small spot of land that is adja- cent to an old stone quarry now used for swimming, and baseball fields used for T-ball, Pee Wee, Little League, and PONY league baseball. The owner is preparing a plan to operate the stand for the coming summer months, which she is basing upon information gathered about prior years of operation.

1. Objective: The owner needs to forecast demand, so she can order enough milk product, sprin- kles, and other items as well as schedule enough staff to meet demand. As expected for an ice cream stand in the Midwest, the demand is highly seasonal, so the time period for the forecast is from early in May when baseball begins until Labor Day. This stand closes for the rest of the year.

2. Developing and Testing the Model: The owner has sales receipts by day for the last five sum- mers. The owner decides to use a simple average to project demand for the coming year. She averages the daily receipts for the 5-year period. As she tests her forecast with the actual sales data over the past five years, she finds that her projections are not very good. (continued)

Developing and testing a model. A model should be developed and then tested to ensure that it is as accurate as possible. Several techniques including moving average, weighted moving average, exponential smoothing, and regression analysis for developing fore- casting models are discussed later in this chapter. In addition to these quantitative approaches, it is often useful to consider qualitative factors, which are also discussed later in this chapter.

Applying the model. After the model is tested, historical data about the problem are col- lected. These data are applied to the model, and the forecast is obtained. Great care should be taken so that the proper data are used and the model is applied correctly.

Real-world constraints. Applying any model requires consideration of real-world con- straints. A model may predict that sales will double in the next three years. Management, therefore, adds the needed personnel and facilities to produce the service or good, but does not consider the impact this increase will have on the distribution system. A software company expands its product offerings by hiring additional programmers and analysts, but it does not provide the capability to install the software on customers’ systems. If a manufacturer is planning to expand production to address an increase in demand: Should it consider raw-material availability? Will competitors react by cutting prices so that demand is less than expected? Where can the firm find the skilled labor to do the work? Forecast should not be taken as fact. A forecast is one scenario that managers must ground in reality. A forecast is not a complete answer, but rather one more piece of information.

Revising and evaluating the forecast. The technical forecast should be tempered with human judgment. What relationships may have changed? In the case of the electric util- ity industry, a fundamental change in the rate of growth greatly affected the accuracy of estimates for future consumption. Forecasts should not be treated as complete or static. Revisions should be made as changes take place within the firm or the environment. The need for revision may be occasioned by changes in price, product characteristics, advertising expenditures, or actions by competitors. Evaluation is the ongoing exercise of comparing the forecast with the actual results. This control process is necessary to attain accurate forecasts.

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CHAPTER 6Section 6.2 Forecasting

Real World Scenarios: 1973 Oil Embargo

In 1973, an oil embargo hit the United States, and energy prices climbed substantially in only a few weeks. The costs of all forms of energy increased, including gasoline, natural gas, and electricity. The embargo caused a nationwide effort to conserve energy. The demand for fiberglass insulation soared; fiberglass companies did not have sufficient capacity because their planning models were based upon much slower growth rates. Higher energy prices made spending money to conserve energy an attractive investment. Conversely, the growth in demand for electricity dropped from about 3% annually, to near zero. In a relatively short time it rebounded to about 1% per year. The embargo changed the pattern of growth in the industry. Electrical utilities had planned for a signifi- cantly higher growth rate and did not react quickly enough to the change. Many utilities continued to build new power plants. The result was a surplus of electrical generation capacity and the cancel- lation of orders for nuclear power plants.

In the 1990s, the growth rate for electricity rebounded in part because of the growing demand for computer technology, including the proliferation of computer servers. Once again, the forecasting models, this time using the slower growth rates of the late 1970s and 1980s, underestimated the need for electricity. This resulted in a brownout in some parts of the United States in the late 1990s and early 2000s.

Highlight: Forecasting for Quarry-Front Ice Cream Stand (continued)

As she examines the data, she sees that there are major differences among the days of the week. For example, demand on Sunday is much lower. She recalculates the averages by day of the week, so she has a projection for Monday based upon the average of all Mondays, for Tuesdays based upon all Tuesdays, etc. Demand on Mondays, shows big differences; some Mondays are very busy, but others are not. She is unsure how to utilize this data, but she moves forward with a plan based upon the daily forecast.

3. Applying the Model: As the ice cream stand opens, the owner decides to ask her staff to keep a simple tally for the first month of operations. She provides each of them with a sheet that is has a single column with the rows designated by 30-minute increments starting at 11:00 a.m. when the Quarry-Front Ice Cream Stand opens, and ending when it closes at night 10:00 p.m. The staff is to place a tally mark for each customer served. As she studies the results, she notices strong demand in the early afternoon, which she deduces is most likely driven by kids from the quarry who want lunch or a snack. She also notices a strong demand in the evenings, which is associated with teams and baseball players’ parents purchasing a postgame ice cream treat. There is also a very big demand in early June when the small town has its homecoming parade and festival. The owner gets the operating schedule from the quarry and for the Base- ball Association to use that data to adjust her inventory and staffing to better meet the pat- terns of demand.

4. Real World Constraints: The quarry and the baseball leagues are part of real world constraints, but there are other factors as well. Weather greatly reduces demand because the quarry may be closed and the baseball games rained out. Games scheduled before school is dismissed also cut demand because parents want their kids home early on weeknights.

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CHAPTER 6Section 6.2 Forecasting

Application of Models Before becoming immersed in the details involved with preparing a forecast, it is impor- tant to know that forecasting requires more than developing the model and performing an analysis. The results from the model should be tempered with human judgment. The future is never perfectly represented by the past, and relationships change over time.

Thus, the forecast should take into account judgment and experience.

Many techniques exist for devel- oping a forecast. It is impossible to cover all the techniques effec- tively in a short time. Entire books are devoted to forecasting, and some university students major in forecasting as others major in marketing, accounting, or supply chain management. In the following sections, qualita- tive, time-series, and regression analysis methods of forecasting are discussed. Regression analy- sis can be used to project time- series and cross-sectional data. There are several variations of these methods:

• Qualitative methods • Buildup method • Survey method • Test markets • Panel of experts (Delphi Technique)

• Time-series methods • Simple moving average • Weighted moving average • Exponential smoothing • Regression and correlation analysis (simple and multiple regression)

Qualitative Methods Mathematical models are known as quantitative methods, while more subjective approaches are referred to as qualitative. Although mathematical models are useful because they help management make predictions, qualitative approaches can also be helpful. Qualitative forecasts that are based upon subjective interpretation of historical data and observations are frequently used. A homeowner who decides to refinance his or her home has made an implicit prediction that home mortgage rates cannot be lower, and are likely to remain constant or to increase in the future. Similarly, a manager who decides

.Tyler E. Nixon/Getty Images

Forecasting involves more than developing a model and conducting analysis. Because the future may not accurately represent the past, the results from a model should take into account the forecaster’s judgment and experience.

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CHAPTER 6Section 6.2 Forecasting

to purchase extra materials because of uncertainty in supply has made an implicit predic- tion that a strike or other action may disrupt the flow of materials. There are many differ- ent qualitative methods for making forecasts. The buildup method, surveys, test markets, and the panel of experts are discussed briefly, next.

Buildup Method

The buildup method requires starting at the bottom of an organization and making an overall estimate by adding together estimates from each element. For example, a broker- age firm could use this approach to forecast revenues from stock market transactions. If the buildup method is used for predicting revenue, the first step is to ask each representa- tive to estimate his or her revenue. These estimates are passed on to the next-higher level in the organization for review and evaluation. Estimates that appear too high or too low are discussed with the representative so that management can understand the logic that supports the prediction. If the representative cannot convince the supervisor, a new pre- diction based upon this discussion is made. The prediction is then passed on to the next level in the organization.

As these subjective judgments are passed up the organization, they are reviewed and refined until they become, in total, the revenue forecast for the entire organization. It is top management’s responsibility to make the final judgment about the forecast’s validity. Once top management has decided on the forecast, it becomes an input used in making capacity, production planning, and other decisions.

Survey Method

In some cases, organizations use surveys to gather information from external sources. A survey is a systematic effort to elicit information from specific groups and is usually conducted via a written questionnaire, a phone interview, or the Internet. The target of the survey could be consumers, purchasing agents, economists, or others. A survey may attempt to determine how many consumers would buy a new flavor of toothpaste, or consider a maintenance service that comes to their home to complete minor repairs on their car. Currently, surveys of purchasing agents are conducted to assess the health of the economy. Surveys are often used to prepare forecasts when historical data are not avail- able, or when historical data are judged not to be indicative of the future. Surveys can also be used to verify the results of another forecasting technique.

Test Markets

Test marketing is a special kind of survey. In a test market, the forecaster arranges for the placement of a new or redesigned product in a city believed to be representative of the organization’s overall market. For example, an organization that wants to test the “at-home” and “at-work” market for an oil change service could offer the service in one or two cities to determine how customers may respond. The analyst examines the sales behavior in the test market and uses it to predict sales in other markets. Test marketing can be expensive, but the results tend to be more accurate than those complied from a survey because the consumers in a test market actually use the product.

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CHAPTER 6Section 6.2 Forecasting

Highlight: Assessing Demand for Car Repair Services

If Jordan decided to launch this at-home service, he would do this in a limited way. For example, he could limit the geography to provide only routine maintenance to part of his current service area. He could also limit the services offered to oil changes, air filters, and lubrication. This would allow him to keep his initial investment low and also gather data about demand, which could be used to project demand for his full-service operation. A smaller investment reduces his risk.

Panel of Experts

A panel of experts is comprised of people who are knowledgeable about the subject being considered. This group attempts to make a forecast by building consensus. In an organiza- tion, this process may involve executives who are trying to predict the level of information technology applied to banking operations, or store managers who are trying to estimate labor costs in retail operations. The panel can be used for a wide variety of forecasts, and with this method, forecasts can often be made very quickly.

The Delphi Technique uses a panel of experts and surveys in a particular manner. The members of the panel provide a sequence of forecasts through responses to questionnaires. This sequence of questionnaires is directed at the same item or set of items. After each fore- cast, results are compiled, and the individuals are given summary statistics such as the median response and the 50th percentile of the item or items being forecasted. This pro- vides a reference point for the participants, who can decide whether or not to change their estimate based upon this information. Because responses are gathered by questionnaire

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CHAPTER 6Section 6.2 Forecasting

rather than by group interaction, the participants do not meet face-to-face. As a result, a few participants, who may be overly conservative or overly optimis- tic, cannot dominate the discus- sion and bias the results. The Delphi process assumes that as each forecast is conducted and the results disseminated among the panel members, the range of responses diminishes and the median represents the “true” consensus of the group.

Time-Series Methods The historical data used in fore- casting can be cross-sectional data, time-series data, or a com- bination of the two. Cross-sectional data samples across space, such as height of adults in the United States, Europe, and Asia. The simplest way to illustrate the differences in these data is with an example. One Pacific Coast Bank wants to project usage of its automated teller service. It has collected data from ATM systems in Stockton, San Jose, Santa Cruz, and Berkeley for the last two years. The study has both time-series and cross-sectional elements, as shown in Table 6.2. The time-series data are the two years of data that are available for the banks. The cross-sectional element is represented by the data from more than one bank.

Table 6.2: Time-series and cross-sectional data

Jan. Feb. Mar. . . . Dec. Jan. Feb. Mar. . . . Dec.

Stockton

San Jose

Santa Cruz

Berkeley

Forecasting sales, costs, and other relevant estimates usually involves time-series data, and the techniques discussed here are useful in predicting such data. See Figure 6.2 for the time line and notation used in forecasting. Each point on the time line has associated with it an actual value, which is represented by x and a subscript. Each point on the line also has a forecasted value, represented by f and a subscript. Every period has a forecasted value when it is in the future; as time passes, it will have an actual value.

Organizations often employ subject experts who attempt to make forecasts by building consensus.

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CHAPTER 6Section 6.2 Forecasting

Figure 6.2: Forecasting time line

Simple Moving Average

One approach to forecasting is to use only the most recent time period to project the next time period. This system, however, can introduce a significant error into a forecast because any odd occurrence in the previous period will be completely reflected in the prediction. Suppose that in one month a temporary price cut caused sales to be significantly greater than normal. If these actions are not repeated in the next month, then using the previous month’s sales as the forecast will provide a biased prediction.

The purpose of the simple moving average is to smooth out the peaks and valleys in the data. In the data set shown in Figure 6.3, the data fluctuate significantly. Basing a projec- tion on the prior quarter’s result could provide a significant error. A moving average will smooth these peaks and valleys and provide a more reasoned prediction. In the moving average model, the forecast for the next period is equal to the average of recent periods.

ft11 5 a

n21 i50 1xt2 i 2

n

where

ft11 5 the forecast for time period t 1 i, that is, the next time period when i 5 1

xt2i 5 the observed value for period t 2 i, where t is the last period for which data are available and i 5 0, . . ., n21

n 5 the number of time periods in the average

XtXt – 1Xt – 2Xt – 3 ft + 1 ft + 2 ft + 3

Xt = the actual value of the item to be forecast for the most recent time period t. Prior observations are noted by subtracting 1 from time period t.

ft + 1 = the forecasted value for the next period. Following periods are designated by adding 1 to time period t + 1.

Past

Present

Future

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CHAPTER 6Section 6.2 Forecasting

The longer the time—that is, the greater the n—the more smoothing that will take place. The selection of n is a management decision based upon the amount of smoothing desired. A small value of n will put more emphasis on recent predictions and will more completely reflect fluctuations in actual sales. In fact, if n 5 1, then the most recent time period’s actual results will become the next period’s forecast.

Figure 6.3: Graph of imports

\$10,000

9,000

8,000

7,000

6,000

5,000

4,000

3,000

2,000

1,000

12:1 12:2 12:3 12:4 13:1 13:2 13:3 13:4

Year: Quarter

Im p

o rt

s (\$

0 0

0, 0

0 0)

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CHAPTER 6Section 6.2 Forecasting

Example: Following are the data shown in Figure 6.3:

Year: Quarter Imports (\$000,000)

2012:1 4,100

2012:2 2,000

2012:3 5,700

2012:4 2,500

2013:1 7,300

2013:2 9,200

2013:3 6,300

To calculate a seven-quarter moving average for imports, sum the most recent seven quarters, and divide by seven. Please observe that the notation “year: quarter” is used in the subscript here. The fourth quarter of 2013 is noted as “13:4.”

f13:4 5 14,100 1 2,000 1 5,700 1 2,500 1 7,300 1 9,200 1 6,300 2

7

5 5,300

A three-quarter moving average is calculated as follows:

f13:4 5 17,300 1 9,200 1 6,300 2

3

5 7,600

Which estimate is likely to better represent the future? Which prediction should be used? It depends upon whether the forecaster feels the last three quarters better predict what is to come than the prior seven months. If so, use the 3-month moving average. If the last three months reflect some unusual conditions that are unlikely to recur, use the 7-month moving average to smooth the high values in the last three quarters. Forecasting models do not provide complete answers to questions. Managerial judgment plays a critical role.

This technique is called a moving average because to forecast the next quarter, the most recent quar- ter’s actual imports are added and the oldest quarter’s actual imports are subtracted from the total. In a way, the average moves. Refer again to the import example. Assume that actual imports for the fourth quarter of 2013 are \$7,500 million. A three-quarter moving average for the first quarter of 2014 would drop the \$7,300 million, which is the actual value for the first quarter of 2013, and add the most recent quarter. The following illustrates the calculation for the first quarter of 2014:

f14:1 5 19,200 1 6,300 1 7,500 2

3

5 7,667

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CHAPTER 6Section 6.2 Forecasting

Example: Five-period weighted moving average for the fourth quarter of 2013

Year: Quarter Weight Imports (\$000,000)

2012:1 — 4,100

2012:2 — 2,000

2012:3 0.10 5,700

2012:4 0.15 2,500

2013:1 0.20 7,300

2013:2 0.25 9,200

2013:3 0.30 6,300

F13:4 5 0.1(5,700) 1 0.15(2,500) 1 0.2(7,300) 1 0.25(9,200) 1 0.30(6,300) 5 6,595

If the weights for each period are set at 0.20, then the weighted moving average and the simple mov- ing average using five quarters will be equal. Try this for yourself.

Weighted Moving Average

In a simple moving average, each time period has the same weight. With a weighted moving average, it is possible to assign different weights to each period. The equation for determining the weighted moving average is:

ft11 5 a n21

i50 1wt2 i 2 1xt2 i 2

where

wt2i 5 the weight for period t–i, where t is the last period for which data are available and i 5 0,. . . , n21. The weights for all n periods must sum to 1.0.

The weights for each period need to be selected in some logical way. Usually the most recent periods are weighted more heavily because these periods are thought to be more representative of the future. If there is an up or down trend in the data, a weighted mov- ing average can adjust more quickly than a simple moving average. Still, this form of the weighted moving average is not as accurate as regression analysis (discussed later in this chapter) is in adapting to trends.

Exponential Smoothing

Exponential smoothing is another form of a weighted moving average. It is a procedure for continually revising an estimate to include more recent data. The method is based upon averaging (smoothing) past values. To start a forecast using exponential smoothing,

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CHAPTER 6Section 6.2 Forecasting

the forecast for the first period, ft11 would be based upon the actual value for the most recent period, xt. (See equation 6.1.) The forecast for the second period, ft12 is equal to the actual value of the previous period, xt11 times the smoothing constant, A, plus (1 2 A) times the prior period’s forecast, ft11. (See equation 6.2.) Remember, the prior forecast, ft11, is simply the actual value from period t. The forecast in equation 6.2 is A times the prior period’s actual value plus (1 2 A) times the prior period’s forecast, ft11.

ft11 5 xt (6.1)

ft12 5 A(xt11) 1 (1 2 A)ft11 (6.2)

ft13 5 A(xt12) 1 (1 2 A)ft12 (6.3)

. . .

. . .

. . .

. . .

ft1n 5 A(xt1n21) 1 (1 2 A)ft1n21

where

n 5 some number of periods in the future

0  A  1

Consider one more equation in detail. Equation 6.3 uses the prior period’s actual value times the weighting factor, A, plus (1 2 A) times the prior period’s forecast. Exponential smoothing carries all the historical actual data in the prior period’s forecast.

How should the smoothing constant A be selected? First, A must be greater than or equal to zero and less than or equal to one. Within this range, a manager has discretion. What will happen if a manager selects a smoothing constant at an extreme? If A 5 1, then according to equation 6.2, the forecast will be based solely on the actual value from the prior period. In this case, no smoothing takes place. The forecast for the next period is always the last period’s actual value. If the smoothing constant is set to 0, then the prior period’s actual value is ignored. Once the forecasting pattern gets started, the forecast is so smooth that it will never change. No actual amounts can enter the equation because A 5 0. Neither of these alternatives is acceptable.

There are no specific rules about choosing the value of A. If the forecaster wants to put more weight on the most recent time period, then A should be set closer to 1. If the man- ager desires a smoother forecast that will not react drastically to short-term change, A should be set closer to 0. Values between 0.1 and 0.3 are most commonly used. Typi- cally, values in this range are selected so the forecast does not overcompensate for sudden

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CHAPTER 6Section 6.2 Forecasting

Example: Exponential Smoothing

Use exponential smoothing to forecast imports from the previous example. To illustrate the impact of the smoothing constant, use A 5 0.1 and A 5 0.6. To begin, there can be no forecast for the first quarter of available data because no history is available. The forecast for the second quarter is the prior quarter’s actual value because no forecast is available for the first quarter. In Figure 6.4, 4,100 is the forecast for both A 5 0.1 and A 5 0.6. After that, the forecasts are significantly different because of the large difference in smoothing constants and the large fluctuations in demand. The third quar- ter’s forecast follows the equations described previously because an actual value and a forecasted value are available for the prior quarter. Despite that the actual demand is available through the third quarter of 2013, the forecasted values were calculated to illustrate the difference between the two forecasts and the potential for inaccurate forecasts when historical demand varies significantly.

For A 5 0.1

f12:3 5 A(x12:2) 1 (12A)f12:2

5 0.1(2,000) 1 0.9(4,100)

5 3,890

For A 5 0.6

f12:3 5 0.6(2,000) 1 0.4(4,100)

5 2,840

Year: Quarter Imports (\$000,000) Forecast A 5 0.1 Forecast A 5 0.6

2012:1 4,100

2012:2 2,000 4,100 4,100

2012:3 5,700 3,890 2,840

2012:4 2,500 4,071 4,556

2013:1 7,300 3,914 3,322

2013:2 9,200 4,253 5,709

2013:3 6,300 4,748 7,804

2013:4 4,903 6,902

The forecasts are significantly different. The forecast with A 5 0.1 does not react abruptly to sud- den changes. The forecast with A 5 0.6 does respond but the response is delayed. This can be seen graphically in Figure 6.4 where the actual value and the two forecasts are plotted.

changes in the data. For example, if weather was extremely warm in the last time period and demand was high, the forecast for the next time period would not be pushed to an extreme level if a value of A is selected that is within this range. The forecast would be smoothed because more weight is placed on the historical data, meaning the data that are prior to the sales value for the most recent time period.

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CHAPTER 6Section 6.3 Advanced Statistical Methods

Figure 6.4: Exponential smoothing examples

Correlation analysis measures the degree of relationship between two variables, and regression analysis is a method to predict the value of one variable based upon the value of other variables. The coefficient of correlation is a measure of the strength of linear relationship between variables. If there is no relationship, then the coefficient of correlation is 0. Perfect positive correlation is 1.0, and perfect negative correlation is –1.0 (see Figure 6.5). Between the limits of perfect positive and perfect negative correlation, there are many levels of strength. Examples are shown in Figures 6.5 and 6.6.

\$10,000

9,000

8,000

7,000

6,000

5,000

4,000

3,000

2,000

1,000

12:1 12:2 12:3 12:4 13:1 13:2 13:3 13:4

Year: Quarter

Actual Forecast: A = .1 Forecast: A = .6

Im p

o rt

s (\$

0 0

0, 0

0 0)

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CHAPTER 6Section 6.3 Advanced Statistical Methods

Figure 6.5: Scatter diagrams showing zero, perfect positive, and perfect negative correlations

High

High

Low

Low

N u

m b

er o

f q

u al

it y

d ef

ec ts

Average weight of sales force

Zero Correlation

High

High

Low

Low

Perfect Positive Correlation

High

High

Low

Low

Perfect Negative Correlation

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CHAPTER 6Section 6.3 Advanced Statistical Methods

Regression analysis can be used to forecast both time-series and cross-sectional data. Regression analysis is often used to estimate the slope of a trend line for time-series data. Regression analysis can be either simple or multiple. Simple regression analysis involves the prediction of only one variable (the dependent variable) and uses only one variable for prediction (the independent variable). Multiple regression analysis has only one dependent variable, but can have more than one independent variable.

Figure 6.6: Scatter diagrams showing examples of correlation P

ro d

u ct

co st

S h

ip p

in g

d am

ag e

C u

st o

m er

sa ti

sf ac

ti o

n

S al

es o

f eg

g b

ea te

rs

High

High

Low

Low

Strong Negative Correlation

High

High

Low

Low

Weak Negative Correlation

Material costs

Distance shipped

Defective parts produced and distributed

Time

High

High

Low

Low

Strong Positive Correlation

High

High

Low

Low

Weak Positive Correlation

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CHAPTER 6Section 6.3 Advanced Statistical Methods

Example: Regression

When working with time-series data, it is usually easier to convert the time variable from the month/ day/year format to simpler numbers. There are many possible ways of coding. Here, the number 1 is used to represent the first time period for which data are available. Following periods will be con- secutively numbered. In this example, the assumption is that demand (Y) depends on time (X), the independent variable. The import data from an earlier example are used for analysis. (continued)

Regression and Correlation Analysis The equation for simple regression follows. Y is the dependent variable, and X is the inde- pendent variable. The variable b is the slope of the line, which is estimated by equation 6.4, and variable a is the Y-intercept, which is estimated by equation 6.5.

Y 5 a 1 b(X)

where

Y 5 the dependent variable. It depends on the variables X, a, and b.

X 5 the independent variable

n 5 the number of data points in the sample

r 5 the coefficient of correlation

b 5 naXY

2 aXaY

naX2 2 aaXb 2 (6.4)

a 5 aY

n 2 b aX

n (6.5)

r 5 naXY2aXaY

Ä cnaX 2 2 aaXb

2

d cnaY2 2 aaYb 2

d (6.6)

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CHAPTER 6Section 6.3 Advanced Statistical Methods

Example: Regression (continued)

Year: Quarter

Coded Value for Year: Quarter (X)

Imports (\$000,000) (Y)

XY X2 Y2

2012:1 1 4,100 4,100 1 16,810,000

2012:2 2 2,000 4,000 4 4,000,000

2012:3 3 5,700 17,100 9 32,490,000

2012:4 4 2,500 10,000 16 6,250,000

2013:1 5 7,300 36,500 25 53,290,000

2013:2 6 9,200 55,200 36 84,640,000

2013:3 7 6,300 44,100 49 39,690,000

Sum 28 37,100 171,000 140 237,170,000

b 5 naXY 2 aXaY

naX2 2 aaXb 2

5 7 1171,000 2 2 28 137,100 2

7 1140 2 2 282

5 158,200

196

5 807.1

a 5 aY

n 2 b

aX

n

5 37,100

7 2

807.1 128 2 7

5 2,071.6

r 5 naXY

2 aXaY

Å cnaX 2 2 aaXb

2

d cnaY2 2 aaYb 2

d

(continued)

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CHAPTER 6Section 6.3 Advanced Statistical Methods

Example: Regression (continued)

7 1171,000 2 2 28 137,100 2 !37 1140 2 2 282 4 37 1237,170,000 2 2 37,1002 4

5 158,200

!3196 4 3283,780,000 4

5 0.671

Interpreting the results of the model requires an understanding of the original units of the data as well as the slope/intercept method of representing a straight line. The last quarter of 2013 is coded as “8” because the quarters are consecutively numbered. The imports are given in millions of dollars. As a result, the imports are projected to increase \$807.1 million per quarter. The intercept is \$2,072 million, and it represents the point on the regression line for the quarter prior to the first quarter of 2012. Project the imports for the last quarter of 2013 where the estimated value is represented by Ye. The predictive model follows.

Ye5 2,071.6 1 807.1X

5 2,071.6 1 807.1(8)

5 8,528

Thus, the projection for imports is \$8,528 million.

Goodness of Fit

How well does the equation determined by regression analysis fit the data? The principles on which simple regression analysis and multiple regression analysis are constructed are similar. The regression model estimates the Y-intercept (a) and the slope of the line (b) that best fits the data. The criterion that is used to determine the “best fit” line minimizes the squared distance from each point to the line. This is often called the least squares method. These distances are labeled di in Figure 6.7, with i equal to 1, . . ., n. The method used to derive the parameters of the best fit line is based upon differential calculus and is not covered in this text. The equations that determine the parameters of the slope (b) and the Y- intercept (a) are 6.4 and 6.5, respectively.

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CHAPTER 6Section 6.3 Advanced Statistical Methods

Figure 6.7: Regression line

The coefficient of correlation calculated in the prior example (r 5 0.671) indicates a high degree of relationship between the dependent and independent variables. The higher the coefficient of correlation (closer to 1), the more confident the forecaster can be that varia- tion in the dependent variable (imports) is explained by the independent variable (time). This can be observed by looking at the scatter diagram in Figure 6.8. A measurement of this variation about the regression line is the standard error of the estimate, sy/x. It is the difference between each observed value, Yo, and the estimated value, Ye. The equation for the standard error of the estimate follows. An alternative formula that is easier to use with a calculator is provided in Figure 6.8.

sy/x 5 ä a 1Yo 2 Ye 2 2

n 2 2

Simple regression models can be constructed for cross-sectional data. The mechanics are similar.

\$10,000

9,000

8,000

7,000

6,000

5,000

4,000

3,000

2,000

1,000

12:1 12:2

a = \$2,071.6

b = 807.1 \$/Quarter Each

Quarter

12:3 12:4

d3

d5

d6

d7

d4

d2

d1

13:1 13:2 13:3 13:4

Year: Quarter

Im p

o rt

s (\$

0 0

0, 0

0 0)

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CHAPTER 6Section 6.3 Advanced Statistical Methods

Figure 6.8: Scatter diagram and regression line for import problem

Computer Application of Simple Regression Analysis

Many different computer software packages are available for doing both simple and multiple regression analyses. Table 6.3 is the computer-based output for regres- sion analysis. The coefficients calculated by software are the same (with allowances for rounding) as the coefficients calculated by hand. The standard error of the esti- mate is also the same as the value calculated by hand. The computer output also provides additional information. The standard error of the coefficients, 1,784.800 and 399.093, are standard deviations for the coefficients. They can be used to test the null hypotheses that the actual values of the coefficients are equal to zero. The t-values are the calculated t-statistics for the hypothesis tests. The two-sided

\$10,000

9,000

8,000

7,000

6,000

5,000

4,000

3,000

2,000

1,000

12:1 12:2

Ye/x = 7 = 2,071.6 + 807.1(7) = 7,721

Sy/x =

Sy/x = 2,111.8

∑Y2 – a∑Y – b∑XY n – 2

Ye/x = 2,071.6 + 807.1X

Yo = 6,300

12:3 12:4 13:1 13:2 13:3 13:4

Year: Quarter

Im p

o rt

s (\$

0 0

0, 0

0 0)

d7

Sy/x = 237,170,000 – 2,071.6(37,100) – 807.1(171,000)

7 – 2

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CHAPTER 6Section 6.3 Advanced Statistical Methods

Problem

The prior examples use only one independent variable (time) to predict imports. Most relationships are not that simple, because other factors will also affect the dependent variable. To expand the previous example, disposable income and the consumer price index are added to the model. (continued)

significant probabilities are the levels that alpha or Type 1 error would have to be set at in order to fail to reject the null hypothesis. In this example, the trend coeffi- cient would be significant if alpha error is set at 0.1 or higher. On the other hand, the coefficient for the intercept would be significant if alpha error is set at 0.3 or higher.

Table 6.3: Regression coefficients for imports

Variable Coefficient Std. Error t-value Two-sided Sig. Prob.

Constant 2,071.42900 1,784.8000 1.16059 0.298204

YRS/OUT 807.14290 399.09340 2.02244 0.099061

Standard error of estimate 5 2111.804

Multiple Regression Model

Multiple regression has only one dependent variable, but can have many independent variables.

Y 5 a 1 b1 x1 1 b2 x2 1 . . . 1 bk xk

where

Y 5 the dependent variable. It depends on the variables X1 through Xk and the model parameters a, b1, b2. . ., bk , where k is the number of independent variables. (Equations for the parameters are not given here. There are many available computer packages, such as EXCEL, SPSSX, SAS, or MINITAB to do the necessary calculations.)

xi 5 an independent variable, with i 5 1, . . ., k. Each independent variable will have n observations or data points.

Minimizing squared distances from each observed point to the best fit regression line is still useful. However, because multiple regression requires more than two dimensions, two-dimensional graphs cannot be used. Computerized statistical models are used to make the calculations.

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CHAPTER 6Section 6.3 Advanced Statistical Methods

Problem (continued)

Imports (\$000,000)

Year: Quarter

Code Value For Year: Quarter (Xi)

Disposable Income (Billions of \$) (X2)

Consumer Price Index (X3)

4,100 2012:1 1 65 110

2,000 2012:2 2 60 111

5,700 2012:3 3 73 113

2,500 2012:4 4 61 113

7,300 2013.1 5 70 117

9,200 2013.2 6 77 118

6,300 2013:3 7 78 117

The multiple regression output is shown in Figure 6.9. The equation for predicting imports is

ye 5 2141,000 2 1,387.6×1 1 206.35×2 1 1,205.4×3

Figure 6.9: Multiple regression output

THE REGRESSION EQUATION IS

y = – 141,000 – 1,388 x1 = 206 x2 + 1,205 x3

PREDICTOR

B1 B2 B3

S = 534.6

S = standard error of the estimate R – SQ = (coefficient of correlation)2

R – SQ = 97.9%

–141,000 –1,387.6 206.35 1,205.4

COEF

(continued)

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CHAPTER 6Section 6.3 Advanced Statistical Methods

Measuring Forecasting Error Regardless of which forecasting model is used, it is important to have some way to deter- mine the model’s propensity for error. If an organization has been using a particular model to forecast sales for some time, has the model been performing well? How large is the error? One approach is to simply subtract the forecast for one time period from the actual value for the same time period. This can be repeated so that the forecaster has differences for many periods. Differences are positive when the forecast is less than the

actual value, and negative when the forecast is greater than the actual value. In raw form, these differences tell the forecaster little. If these are added, the negative errors and the posi- tive error will cancel and there- fore underestimate the error. A common method used by fore- casters to avoid this problem is to calculate the mean squared error. The mean squared error (MSE) is the average of all the squared errors. The differences are squared and added together, and then that total is divided by the number of observations. The following calculations help illustrate the method.

.Thinkstock

When using a forecasting model, it is critical to have a way to determine the model’s propensity for error.

Problem (continued)

To predict imports for the fourth quarter of 2013, assume that disposable income is \$78 billion and the consumer price index is 118 for the fourth quarter.

ye 5 2141,000 2 1,387.6(8) 1 206.35(78) 1 1,205.4(118) 5 6,232

The prediction is \$6,232 million worth of imports in the fourth quarter of 2013.

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CHAPTER 6Chapter Summary

Problem

Month Actual Sales (\$) Forecasted Sales (\$) Error (\$) Squared Error

January 419,000 448,000 229,000 841,000,000

February 480,000 481,000 21,000 1,000,000

March 601,000 563,000 138,000 1,444,000,000

April 505,000 525,000 220,000 400,000,000

May 462,000 490,000 228,000 784,000,000

June 567,000 519,000 148,000 2,304,000,000

TOTAL 5,774,000,000

MSE 5 a n t21 1xt 2 ft 2 2

n

5 5,774,000,000

6

5 962,333,333

Sometimes the square root of the mean squared error is used to measure the error. This is analogous to the standard error of the estimate, which is discussed in the section on regression analysis.

It is also possible to use mean absolute deviation (MAD), which is similar to MSE, to estimate fore- casting error. MAD is calculated by adding together the differences between the actual and forecasted value once the negative and positive signs are removed. If MAD is not calculated, a large negative error would offset a large positive error, so the total error would be greatly underestimated. Try summing the “Error” column in this example with the signs included. The total error is \$8,000 because the nega- tive and positive errors cancel each other. Once the signs are removed, the total error is \$164,000, which is divided by the number of data points n, as was done for MSE. The MAD is \$164,000/6, which equals \$27,333. The MAD is easier to interpret than MSE because MAD is the average error for the prior six forecasts. As a result, MAD can be used to estimate future forecasting errors.

Chapter Summary

• A model is an important way to think about problems. It is an abstraction from a real problem of the key variables and relationships in order to simplify the prob- lem and improve understanding.

• There are many different types of models, including prototypes used in product design; scale models used in architecture; diagrams and drawings used by scien- tists, engineers, managers, and others; and mathematical models used in many disciplines.

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CHAPTER 6Case Studies

• Models are used to assist managers and others in answering what-if questions by changing a parameter in the model.

• There are qualitative and quantitative methods for developing a forecast. • Forecasting is a type of mathematical model that can be used to predict the

future. It is an important part of the planning process in an organization • The forecasting process consists of determining the objectives of the forecast,

developing and testing a model, applying the model, considering real-world con- straints in the application of the model, and revising and evaluating the forecast.

• Forecasting techniques typically use historical data to develop the model that is used to make the projection. If the relationships in the data change over time, the model may no longer predict the future accurately.

• Forecasters require a way to measure the amount of forecasting error.

Case Studies

Blast-Away Housecleaning Service Blast-Away Housecleaning Service uses powerful water jets to clear loose paint from resi- dential buildings and to clean aluminum siding. The company is trying to arrive at a fast and accurate way of estimating cleaning jobs. The following simple formula is its first attempt. It includes a fixed charge for coming to the job plus time requirements, which are a function of the exterior of the house measured in square feet (sf ).

Estimated cost 5 \$15 1 (\$.06/sf )(sf )

After one year of experience, Blast-Away has lost \$50,000 on sales of \$250,000. At first, the owner, Hadley Powers, could not understand the reasons for his losses. His employees worked hard, and Blast-Away could barely meet with demand. In fact, Powers was plan- ning to add another crew this year, but if he cannot determine the reason for the losses and find a solution, his investors would be reluctant to provide him with additional capital. What caused the loss?

He learned from his accountant that the model he used had not included a recovery of his investment in the equipment used on the jobs. Powers had invested \$60,000 in equipment at the beginning of the first year and expected it to last three years. His accountant recom- mended that Powers increase the price charged per job to generate an extra \$20,000 per year to cover equipment costs. If Powers were able to do this, his losses would be \$30,000 if all other factors remained the same. He had to look further for answers to the problem.

Powers has hired you to carefully examine last year’s job tickets, which contain the quoted price; distance from headquarters; size of the house; type of exterior, such as painted wood, aluminum, or brick; and style of the house, such as ranch, two-story, or split-level. You also have the operator’s logbook that lists travel time and the time necessary to com- plete each house. As you analyze the job tickets, you notice that a substantial number of the jobs that Blast-Away gets are for small, split-level or two-story homes located in the suburbs and surrounding rural area. Many of the homes have wooden siding, which is the most difficult type of siding to clean to the customer’s satisfaction.

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CHAPTER 6Case Studies

1. In addition to the equipment recovery problem, what is causing Blast-Away to lose money?

2. What would you recommend Powers do to correct the problem? 3. What data would you want to collect to verify your recommendations?

Lucy’s Lamps-R-Us Lucy Mertz has opened a specialty lamp shop in a suburban shopping mall. Mertz’s shop has an excellent location next to the entrance to the largest and most popular department store in the five-county area. After a slow beginning, business picked up nicely, and the lamp shop had made a nice profit. To plan for the next year, Mertz decided to use sales for the last eight months to forecast next year’s sales. She has asked you to use the following data to project sales. The forecast listed here, which is for last year, was based upon judg- ment. Mertz wants you to use a quantitative approach.

Time Period Forecasted Sales Actual Sales

May \$ 5,000 \$ 8,300

June 5,200 10,200

July 5,600 9,900

August 6,200 10,200

September 6,900 9,800

October 7,800 11,400

November 8,500 12,800

December 9,000 14,500

1. How much error existed in the old forecast? 2. Project the sales for January, February, and March of next year.

It is now the end of March, and the actual sales for the first three months are available. The results are disappointing. In January, sales declined because of returns from the Christmas buying season and an increase in bargain hunting. Also, the large department store that anchored Mertz’s end of the shopping mall closed at the end of January because of operat- ing losses by the parent company.

Time Period Actual Sales

January 7,500

February 6,000

March 6,100

3. Why did the model give Mertz a poor forecast? 4. What would you recommend to Mertz regarding the forecast for the next three

months?

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CHAPTER 6Problems

Discussion Questions

1. What is model building, and why is model building important for managers? 2. Discuss the different types of models. 3. Describe how models can be used to answer what-if questions. 4. How are models used in business and operations? 5. What is forecasting, and why is it important to an organization? 6. Describe the forecasting process. 7. Discuss the qualitative approaches to forecasting. 8. How does the Delphi Technique work? What are its advantages? 9. How is regression analysis different from the moving average, the weighted

moving average, and exponential smoothing? 10. What is forecasting error, and why should it be measured?

Problems

1. Blast-Away Housecleaning Service uses powerful water jets to clear loose paint from residential buildings and to clean aluminum siding. The company is trying to arrive at a fast and accurate way of estimating cleaning jobs. The following simple formula is its first attempt. It includes a fixed charge for coming to the job plus time requirements, which are a function of the exterior of the house mea- sured in square feet (sf ).

Estimated cost 5 \$15 1 (\$0.06/sf )(number of sf )

a. How much should Blast-Away charge to clean a house that is a rectangular 40-by-28 feet? The distance from the roof line to the bottom of the siding is 9 feet.

b. Suppose Blast-Away’s labor costs increase and the cost per square foot increases to \$0.064. How much should it charge for the house in Part a?

c. What other factors may Blast-Away include in the pricing model to improve the precision of the model?

2. As a service to its customers, Turbo Natural Gas Company will estimate the amount of natural gas required (NGR) in hundreds of cubic feet (CCF) to heat your home. This is done by a mathematical model that considers the square footage on the first floor (sf1), the square footage on the second floor (sf2), and the temperature setting on the thermostat. The temperature setting entered into the model should be the difference between the temperature setting in the home and 65 degrees (td). Make sure to keep the minus sign if the setting is less than 65 degrees. The model builder assumed that the homes have 8-foot ceilings, an average number of good-quality windows, 3.5 inches of insulation in each wall, 6 inches of insulation in the attic, and a typical Midwestern winter.

NGR 5 (0.50 CCF/sf )(sf1) 1 (0.25 CCF/sf )(sf2)

1 (0.015 CCF/degree/sf )(td)(sf1)

1 (0.0075 CCF/degree/sf )(td)( sf2)

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CHAPTER 6Problems

a. How much natural gas will an 1,800-square-foot ranch home (one floor only) use if the thermostat is set at 70 degrees?

b. How much natural gas will a two-story home with a total of 2,400 square feet use if the thermostat is set at 63 degrees? There is 1,000 square feet on the sec- ond floor.

c. What happens to the natural gas cost in Parts a and b if the model is revised and the usage for the first floor increases to 0.60 CCF/sf from 0.50 CCF/sf.

3. It appears that the imports of beef have been increasing about 10% annually on the average. Project the 2002 imports using linear regression.

Year Imports of Beef (Thousands of Tons)

2003 82

2004 101

2005 114

2006 126

2007 137

2008 151

2009 164

2010 182

2011 189

4. Mighty-Maid Home Cleaning Service has been in operation for eight months, and demand for its products has grown rapidly. The owner, who is also the man- ager, of Mighty-Maid is trying to keep pace with demand, which means hiring and training more workers. She believes that demand will continue at the same pace. She needs an estimate of demand so she can recruit and train the work- force. The following represents the history of Mighty-Maid:

Time Hours of Service Rendered

December 300

January 750

February 650

March 920

April 1,300

May 1,400

June 1,200

July 1,500

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CHAPTER 6Problems

Estimate the trend in the data using regression analysis.

5. Use the regression model calculated in the Mighty-Maid problem to estimate the hours of service for December through July. Now that both the actual and the forecasted values are available, answer the following questions: a. What are the MSE and MAD for the forecast? b. Is the forecasting model a “good” model?

6. The figures below indicate the number of mergers that took place in the savings and loan industry over a 12-year period.

Year Mergers Year Mergers

2000 46 2006 83

2001 46 2007 123

2002 62 2008 97

2003 45 2009 186

2004 64 2010 225

2005 61 2011 240

a. Calculate a 5-year moving average to forecast the number of mergers for 2012. b. Use the moving average technique to determine the forecast for 2005 to 2011.

Calculate measurement error using MSE and MAD. c. Calculate a 5-year weighted moving average to forecast the number of merg-

ers for 2012. Use weights of 0.10, 0.15, 0.20, 0.25, and 0.30, with the most recent year weighted being the largest.

d. Use regression analysis to forecast the number of mergers in 2012.

7. Find the exponentially smoothed series for the series in Problem 6, (a) using A 5 0.1 and then (b) using A 5 0.7, and plot these time series along with the actual data to see the impact of the smoothing constant.

8. The time series below shows the number of firms in an industry over a 10-year period.

Year Firms Year Firms

2002 441 2007 554

2003 468 2008 562

2004 481 2009 577

2005 511 2010 537

2006 551 2011 589

a. Find the 5-year moving average for this series. b. Find the 3-year weighted moving average for this series. Use the following

scheme to weight the years:

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CHAPTER 6Problems

Weight

Most recent year 0.5

Two years ago 0.3

Three years ago 0.2

c. Determine the amount of measurement error in the forecast. Use the weighted moving average technique (with the weights from Part b to forecast for 2005 to 2011.) Then use that 7-year period to calculate measurement error (both MSE and MAD).

d. Find the exponentially smoothed forecast for this series with A 5 0.2.

9. The quarterly data presented here show the number of appliances (in thousands) returned to a particular manufacturer for warranty service over the last five years.

1st Quarter 2nd Quarter 3rd Quarter 4th Quarter

5 years ago 1.2 0.8 0.6 1.1

4 years ago 1.7 1.2 1.0 1.5

3 years ago 3.1 3.5 3.5 3.2

2 years ago 2.6 2.2 1.9 2.5

1 year ago 2.9 2.5 2.2 3.0

a. Find the equation of the least squares linear trend line that fits this time series. Let t 5 1 be the first quarter five years ago.

b. What would be the trend-line value for the second quarter of the current year—that is, two periods beyond the end of the data provided?

10. The following are AJV Electric’s sales of model EM-5V circuit assemblies over the last 16 months (in thousands of units):

Month Sales (Thousands of Units) Month

Sales (Thousands of Units)

Sept. 2010 55 May 2011 63

Oct. 2010 53 June 2011 53

Nov. 2010 60 July 2011 51

Dec. 2010 49 Aug. 2011 60

Jan. 2011 48 Sept. 2011 58

Feb. 2011 61 Oct. 2011 52

Mar. 2011 61 Nov. 2011 51

Apr. 2011 53 Dec. 2011 63

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CHAPTER 6Problems

Use the moving average technique to forecast sales of AJV’s model EM-5V for January 2012 (use a 3-month base). Does the model appear to be appropriate? Why or why not?

11. Employ the single exponential smoothing technique to forecast sales of AJV’s model EM-5V for January 2012 (use A 5 0.8). Does the model appear to be appropriate?

12. Utilize the single exponential smoothing technique to forecast sales of AJV’s model EM-5V for January 2012 (use A 5 0.1). How do the results compare with those from problem 11? Is one better than another? Why or why not?

13. Using linear regression, forecast the sales of AJV’s model EM-5V for January 2012 through June 2012.

14. Thrifty Bank and Trust is trying to forecast on-the-job performance by its employ- ees. The bank administers an aptitude test to new employees. After the employee training period and an additional six months on the job, the bank measures on- the-job performance. The following data have been gathered from the last eight people hired:

Employee Number Transactions Score per Hour

1 90 36

2 70 29

3 85 40

4 80 32

5 95 42

6 60 23

7 65 29

8 75 33

a. Fit a line to the data using regression analysis. What is the meaning of the parameters that were estimated by the regression analysis model?

b. How well does the model fit the data? c. How many transactions per hour would you expect from someone who

scored 87 on the aptitude test?

15. The data in the following table were collected during a study of consumer buy- ing patterns.

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CHAPTER 6Problems

Observation X Y

1 154 743

2 265 830

3 540 984

4 332 801

5 551 964

6 487 955

7 305 839

8 218 478

9 144 720

10 155 782

11 242 853

12 234 878

13 343 940

a. Fit a linear regression line to the data using the least squares method. b. Calculate the coefficient of correlation and the standard error of the estimate. c. How could the coefficient of correlation and the standard error of the estimate

be used to make a judgment about the model’s accuracy?

16. Perfect Lawns intends to use sales of lawn fertilizer to predict lawn mower sales. The store manager feels that there is probably a six-week lag between fertilizer sales and mower sales. The pertinent data are shown below.

Period Fertilizer Sales (Tons)

Number of Mowers Sold (Six-Week Lag)

1 1.7 11

2 1.4 9

3 1.9 11

4 2.1 13

5 2.3 14

6 1.7 10

7 1.6 9

8 2.0 13

9 1.4 9

10 2.2 16

11 1.5 10

12 1.7 10

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CHAPTER 6Key Terms

buildup method An approach to forecast- ing that starts at the bottom of an organiza- tion and makes an overall estimate by add- ing together estimates from each element.

coefficient of correlation A measure of the strength of a relationship between vari- ables. If there is no relationship, the coef- ficient of correlation will be zero. A perfect positive correlation is 1.0 and a perfect negative correlation is 21.0.

correlation analysis A measure of the degree of relationship between two variables.

Delphi Technique A forecasting proce- dure that uses a panel of experts and sur- veys to build consensus regarding future events. It is an iterative process for consen- sus building.

dependent variable The variable in regression analysis that is being predicted.

exponential smoothing Another form of a weighted moving average. It is a proce- dure for continually revising an estimate to include more recent data. The method is based on averaging (smoothing) past values.

forecast An estimate of future events.

forecasting The process of attempting to predict the future.

forecasting error The difference between the forecasted value and the actual value.

independent variable A variable in regression analysis which is used to pre- dict the dependent variable.

mean absolute deviation (MAD) The average of absolute error. The differences between the actual value of a variable and the forecasted value are added after the plus and minus signs are removed. This total is divided by the number of observations.

mean squared error (MSE) The average of all the squared errors. The differences between the actual value of a variable and the forecasted value are squared, added together, and divided by the number of observations.

model An abstraction from the real prob- lem of the key variables and relationships in order to simplify the problem. The purpose of modeling is to provide the user with a better understanding of the prob- lem, and with a way to manipulate the results for “what if” analysis.

multiple regression analysis Regression analysis that uses two or more (indepen- dent variables) to predict one dependent variable.

panel of experts An approach to forecast- ing that involves people who are knowl- edgeable about the subject. This group attempts to make a forecast by building consensus.

regression analysis A method to predict the value of one variable based upon the value of one or more variables. It is based upon minimizing squared distances from the data points to the estimated regression line.

a. Use the least squares method to obtain a linear regression line for the data. b. Calculate the coefficient of correlation and the standard error of the estimate. c. Predict lawn mower sales for the first week in August, if two tons of fertilizer

sold six weeks earlier

Key Terms

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CHAPTER 6Key Terms

simple moving average A method to smooth out the peaks and valleys in the data by using the most recent actual values to predict the next period. The average moves because as time passes the next period becomes the current period so the actual value for the oldest period is dropped and the most recent actual value is added.

simple regression analysis Regres- sion analysis that uses only one variable (independent variable) to predict a single dependent variable.

survey A systematic effort to elicit infor- mation from specific groups and is con- ducted via a written questionnaire, phone interview or the Internet.

t-statistic Measures the distance from the mean to a point in the t-distribution repre- sented by standard deviations.

t-value The calculated t-statistic used in hypothesis testing.

test market A special kind of survey in which the forecaster arranges for the placement of a new product or an existing product that has been modified and data on actual sales are collected.

weighted moving average A method that is similar to the simple moving average. In the simple moving average, the weight for each historical time period is equal. In the weighted moving average, different weights can be assigned to each historical period. The weights assigned must sum to 1.0.

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