BUS 644 Week 3 Discussion 1 & 2

6

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Models and Forecasting

Learning Objec�ves A�er comple�ng this chapter, you should be able to:

Define a model and describe how models can be used to analyze opera�ng problems. Discuss the nature of forecas�ng. Explain how forecas�ng can be applied to problems. Describe methods of forecas�ng, including judgment and experience, �me-series analysis, and regression and correla�on. Construct forecas�ng models. Es�mate forecas�ng errors.

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

Uwe Lein/ASSOCIATED PRESS/AP Images

6.1 Introduction to Models and Decision Making

In order for an organiza�on to design, build, and operate a produc�on facility that is capable of mee�ng customer demand for services (such as health care) or goods (such as ceiling fans), it is necessary for management to obtain an es�mate or forecast of demand for its products. A forecast is a predic�on of the future. It o�en examines historical data to determine rela�onships among key variables in a problem and uses those rela�onships to make statements about the future value of one or more of the variables. Once an organiza�on 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 opera�ng variables. A model is an abstrac�on from the real problem of the key variables and rela�onships in order to simplify the problem. The purpose of modeling is to provide the user with a be�er understanding of the problem and with a means of manipula�ng the results for what-if analyses. Forecas�ng uses models to help organiza�ons 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 forecas�ng and its role in opera�ons.

Stages in Decision Making

Organiza�onal performance is a result of the decisions that management makes over a period of �me: decisions about what markets to enter, what products to produce, what types of equipment and facili�es to acquire, and where to locate facili�es. The quality of these decisions is a func�on 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 objec�ves

The hospital selects cost per meal and pa�ent sa�sfac�on as the criteria. The objec�ves are to reduce meal cost by 15% and improve pa�ent sa�sfac�on to 90%, based upon the hospital's weekly surveys.

Formulate a model or models The model includes mathema�cal rela�onships 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. Pa�ent preferences are inves�gated so that meals meet nutri�onal requirements and taste good.

Iden�fy and evaluate alterna�ves

Alterna�ves include subcontrac�ng food prepara�on, considering new food suppliers, establishing be�er training programs for the staff, and changing management.

Select the best alterna�ve One of the alterna�ves or some combina�on of alterna�ves is selected.

Implement the alterna�ve, and reevaluate

The selected alterna�ve is implemented, and the problem is reevaluated through monitoring costs and the pa�ent survey data to see if the objec�ves have been achieved.

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 rela�onships. The model also allows managers to try different op�ons 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 characteris�cs. Model airplanes can be tested in wind tunnels to determine aerodynamic proper�es, and a model of a hydroelectric dam can help architects and engineers find ways of integra�ng the structure with the landscape. These models have physical characteris�cs similar to those of the real thing. Experiments can be performed on this type of model to see how it may perform under opera�ng condi�ons. With technology, such as computer simula�on systems, virtual models can be rendered and tested quickly and less expensively. The aerodynamic proper�es of an airplane can be tested in a virtual wind tunnel that exists only inside the memory of a computer. Models also include the drawings of a building that display the physical rela�onships between the various parts of the structure. All of these models are simplifica�ons of the real thing used to help designers make be�er decisions.

Computer-based technology has been used for many years to design cars, buildings, furniture, 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 skeleton. The so�ware can show each separately and put them all together in one 3-D picture. The so�ware can take input from various medical tests and generate 3-D models of a pa�ent to diagnose medical condi�ons faster and be�er.

In addi�on to these physical and virtual models, managers use mathema�cal abstrac�on to model important rela�onships. The break-even point calcula�on that is taught in accoun�ng and finance is an example of applying a mathema�cal model. The use of drawings and diagrams is also modeling. The newspaper graph that illustrates stock market price

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changes in the last six months is a way to help the reader see trends in the market. Models do not have to be sophis�cated to be useful. Most models can be grouped into four categories, and computers play a cri�cal role in the development and use of each type.

Mathema�cal models include algebraic models such as break-even analysis, sta�s�cal models used in forecas�ng and quality control, mathema�cal programming models, and calculus-based models. Graphs and charts are pictorial representa�ons of mathema�cal rela�onships. They include a visual representa�on of break-even analysis, a pie chart that illustrates market share, a graph of stock prices over �me, or a bar graph that indicates the demand for energy for the last five years. Diagrams and drawings are pictorial representa�ons of conceptual rela�onships. 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 aircra� carrier. Scale models and prototypes are physical representa�ons of an item. They include a scale model of an airplane and the first part produced (prototype), which is normally used for tes�ng purposes. These models are o�en built and analyzed inside a computer system. Three-dimensional technology called stereolithography allows computers to create solid models of parts. This is done by successively "prin�ng" very thin layers of a material, which cures quickly to form a sold part.

Mathema�cal models, graphs and charts, and diagrams are most commonly used by business 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 mul�plying the number of people expected to a�end by the amount each person is expected to consume. The host is likely to then mul�ply the an�cipated cost per pizza by the number ordered to determine the cost. This is a simple mathema�cal model that can be used to plan a small party or major social event.

In mathema�cal models, symbols and algebra are used to show rela�onships. Mathema�cal models can be simple or complex. For example, suppose a family is planning a trip to Walt Disney World in Orlando, Florida. To es�mate gasoline costs for the trip, family members check a road atlas (one type of model), or go online to get direc�ons 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 es�mates that the car will achieve 23 miles per gallon (mpg) on the highway. The average cost of a gallon of gasoline is es�mated at $3.80. Using the following model, they make an es�mate of gasoline cost.

*Throughout this text, to enlarge the size of the math equa�ons, please right click on the equa�on and choose "se�ngs" then "scale all math" to increase the viewing percentage.

Cost = (trip miles)(cost per gallon)/miles per gallon = = $363.48

A mathema�cal model can be used to answer what-if ques�ons. In the previous example, costs could be es�mated with a $.30 increase in the price of a gallon of gas, as shown in the following:

Cost = = $392.17

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

Cost =

= $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 predic�on. 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 accelera�on, and the amount of �me 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.

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Technology Forecas�ng; TEDTalks: Chris Anderson— Technology's Long Tail

6.2 Forecasting

Forecas�ng is an a�empt to predict the future. Forecasts are usually the result of examining past experiences to gain insights into the future. These insights o�en take the form of mathema�cal models that are used to project future sales, product costs, adver�sing costs, and more. The applica�on of forecas�ng is not limited to predic�ng factors needed to operate a business. Forecas�ng can also be used to es�mate the cost of living, housing prices, the federal debt, and the average family income in the year 2025. For organiza�ons, forecasts are an essen�al part of planning. It would be illogical to plan for tomorrow without some idea of what could happen.

The cri�cal word in the last sentence is "could." Any competent forecaster knows that the future holds many possibili�es and that a forecast is only one of those possibili�es. The difference between what actually happens and what is predicted is forecas�ng error, which is discussed later in this chapter. In spite of this poten�al error, management should recognize the need to proceed with planning using the best possible forecast and should develop con�ngency plans to deal with the possible error. Management should not assume that the future is predetermined, but should realize that its ac�ons can help to shape future events. With the proper plans and execu�on of those plans, an organiza�on can have some control over its future.

Stages of Forecast Development

The forecas�ng process consists of the following steps: determining the objec�ves of the forecast, developing and tes�ng a model, applying the model, considering real-world constraints on the model's applica�on, and revising and evalua�ng the forecast (human judgment). Figure 6.1 illustrates these steps.

Figure 6.1: Steps in forecas�ng

Determining the objec�ves. What kind of informa�on does the manager need? The following ques�ons 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 �me frame of the forecast—long or short term? 5. How accurate should the forecast be? 6. When is the forecast needed?

Developing and tes�ng 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, exponen�al smoothing, and regression analysis for developing forecas�ng models are discussed later in this chapter. In addi�on to these quan�ta�ve approaches, it is o�en useful to consider qualita�ve factors, which are also discussed later in this chapter.

Applying the model. A�er the model is tested, historical data about the problem are collected. 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 considera�on of real-world constraints. A model may predict that sales will double in the next three years. Management, therefore, adds the needed personnel and facili�es to produce the service or good, but does not consider the impact this increase will have on the distribu�on system. A so�ware company expands its product offerings by hiring addi�onal programmers and analysts, but it does not provide the capability to install the so�ware on customers' systems. If a manufacturer is planning to expand produc�on to address an increase in demand: Should it consider raw-material availability? Will compe�tors react by cu�ng prices so that

Technology Forecasting From Title:

TEDTalks: Chris Anderson—Technology’s Long Tai... (https://fod.infobase.com/PortalPlaylists.aspx? wID=100753&xtid=48088)

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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 informa�on.

Revising and evalua�ng the forecast. The technical forecast should be tempered with human judgment. What rela�onships may have changed? In the case of the electric u�lity industry, a fundamental change in the rate of growth greatly affected the accuracy of es�mates for future consump�on. Forecasts should not be treated as complete or sta�c. 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 characteris�cs, adver�sing expenditures, or ac�ons by compe�tors. Evalua�on is the ongoing exercise of comparing the forecast with the actual results. This control process is necessary to a�ain accurate forecasts.

Highlight: Forecas�ng 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 adjacent to an old stone quarry now used for swimming, and baseball fields used for T-ball, Pee Wee, Li�le 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 informa�on gathered about prior years of opera�on.

1. Objec�ve: The owner needs to forecast demand, so she can order enough milk product, sprinkles, 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 �me period for the forecast is from early in May when baseball begins un�l Labor Day. This stand closes for the rest of the year.

2. Developing and Tes�ng the Model: The owner has sales receipts by day for the last five summers. 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 projec�ons are not very good. 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 projec�on 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 u�lize 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 opera�ons. She provides each of them with a sheet that is has a single column with the rows designated by 30-minute increments star�ng 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 no�ces strong demand in the early a�ernoon, which she deduces is most likely driven by kids from the quarry who want lunch or a snack. She also no�ces 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 fes�val. The owner gets the opera�ng schedule from the quarry and for the Baseball Associa�on to use that data to adjust her inventory and staffing to be�er meet the pa�erns 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.

Real World Scenarios: 1973 Oil Embargo

In 1973, an oil embargo hit the United States, and energy prices climbed substan�ally in only a few weeks. The costs of all forms of energy increased, including gasoline, natural gas, and electricity. The embargo caused a na�onwide effort to conserve energy. The demand for fiberglass insula�on 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 a�rac�ve investment. Conversely, the growth in demand for electricity dropped from about 3% annually, to near zero. In a rela�vely short �me it rebounded to about 1% per year. The embargo changed the pa�ern of growth in the industry. Electrical u�li�es had planned for a significantly higher growth rate and did not react quickly enough to the change. Many u�li�es con�nued to build new power plants. The result was a surplus of electrical genera�on capacity and the cancella�on 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 prolifera�on of computer servers. Once again, the forecas�ng models, this �me using the slower growth rates of the late 1970s and 1980s, underes�mated the need for electricity. This resulted in a brownout in some parts of the United States in the late 1990s and early 2000s.

Application of Models

Before becoming immersed in the details involved with preparing a forecast, it is important to know that forecas�ng 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 rela�onships change over �me. Thus, the forecast should take into account judgment and experience.

Many techniques exist for developing a forecast. It is impossible to cover all the techniques effec�vely in a short �me. En�re books are devoted to forecas�ng, and some university students major in forecas�ng as others major in marke�ng, accoun�ng, or supply chain management. In the following sec�ons, qualita�ve, �me-series, and regression analysis methods of forecas�ng are discussed. Regression analysis can be used to project �me-series and cross-sec�onal data. There are several varia�ons of these methods:

Qualita�ve methods Buildup method Survey method Test markets Panel of experts (Delphi Technique)

Time-series methods Simple moving average Weighted moving averageProcessing math: 0%

Forecas�ng involves more than developing a model and conduc�ng 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.

©Tyler E Nixon/Flickr/Ge�y Images

Exponen�al smoothing Regression and correla�on analysis (simple and mul�ple regression)

Qualitative Methods

Mathema�cal models are known as quan�ta�ve methods, while more subjec�ve approaches are referred to as qualita�ve. Although mathema�cal models are useful because they help management make predic�ons, qualita�ve approaches can also be helpful. Qualita�ve forecasts that are based upon subjec�ve interpreta�on of historical data and observa�ons are frequently used. A homeowner who decides to refinance his or her home has made an implicit predic�on that home mortgage rates cannot be lower, and are likely to remain constant or to increase in the future. Similarly, a manager who decides to purchase extra materials because of uncertainty in supply has made an implicit predic�on that a strike or other ac�on may disrupt the flow of materials. There are many different qualita�ve 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 star�ng at the bo�om of an organiza�on and making an overall es�mate by adding together es�mates from each element. For example, a brokerage firm could use this approach to forecast revenues from stock market transac�ons. If the buildup method is used for predic�ng revenue, the first step is to ask each representa�ve to es�mate his or her revenue. These es�mates are passed on to the next-higher level in the organiza�on for review and evalua�on. Es�mates that appear too high or too low are discussed with the representa�ve so that management can understand the logic that supports the predic�on. If the representa�ve cannot convince the supervisor, a new predic�on based upon this discussion is made. The predic�on is then passed on to the next level in the organiza�on.

As these subjec�ve judgments are passed up the organiza�on, they are reviewed and refined un�l they become, in total, the revenue forecast for the en�re organiza�on. 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, produc�on planning, and other decisions.

Survey Method

In some cases, organiza�ons use surveys to gather informa�on from external sources. A survey is a systema�c effort to elicit informa�on from specific groups and is usually conducted via a wri�en ques�onnaire, a phone interview, or the Internet. The target of the survey could be consumers, purchasing agents, economists, or others. A survey may a�empt 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 o�en used to prepare forecasts when historical data are not available, or when historical data are judged not to be indica�ve of the future. Surveys can also be used to verify the results of another forecas�ng technique.

Test Markets

Test marke�ng 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 representa�ve of the organiza�on's overall market. For example, an organiza�on that wants to test the "at-home" and "at-work" market for an oil change service could offer the service in one or two ci�es 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 marke�ng 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.

Highlight: Assessing Demand for Car Repair Services

Jordan's car repair service center is planning to launch its "At-Home – Car Services" business beginning in the summer of the coming year. The business model is based upon providing car repairs and rou�ne service at a customer's home or work place instead of at a repair shop. Before launching the new business, Jordan would like to know something about demand such as the kind of at-home services customers want, the level of demand for these desired services, if there is a seasonal or other pa�ern to the demand, and whether customers would be willing to pay a small premium for this convenient service. Using mathema�cal modeling to project demand will not likely provide a good forecast because Jordan has no history of demand for this new business and there are no other businesses like it; therefore there is no demand data. Jordan has decided to design a short survey to collect data about demand from three different groups of poten�al customers. First, he will seek input from his ac�ve customers to see if they would like to use the new service. While this group is easy to access because they use the service center regularly, the group provides only li�le, if any, new revenue because they are already supplying Jordan with their business. He may a�ract, at best, a small increase in business from this group, or he may prevent them from choosing a compe�tor in the future. Second, and more financially lucra�ve, Jordan would like to iden�fy people who are not currently using his services. This is new business that is likely to support the at-home service, and if the new customers like the at-home service, they may bring their vehicle to the service center for work that cannot be easily performed at-home. This creates synergy between the two parts of his business. Third, if the business is ini�ally successful, Jordan would like to expand the at-home service to include neighboring towns. If he can build an at-home service in these towns, he may be able to open an addi�onal service center there.

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 rou�ne maintenance to part of his current service area. He could also limit the services offered to oil changes, air filters, and lubrica�on. This would allow him to keep his ini�al investment low and also gather data about demand, which could be used to project demand for his full-service opera�on. A smaller investment reduces his risk.

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Organiza�ons o�en employ subject experts who a�empt to make forecasts by building consensus.

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Panel of Experts

A panel of experts is comprised of people who are knowledgeable about the subject being considered. This group a�empts to make a forecast by building consensus. In an organiza�on, this process may involve execu�ves who are trying to predict the level of informa�on technology applied to banking opera�ons, or store managers who are trying to es�mate labor costs in retail opera�ons. The panel can be used for a wide variety of forecasts, and with this method, forecasts can o�en be made very quickly.

The Delphi Technique uses a panel of experts and surveys in a par�cular manner. The members of the panel provide a sequence of forecasts through responses to ques�onnaires. This sequence of ques�onnaires is directed at the same item or set of items. A�er each forecast, results are compiled, and the individuals are given summary sta�s�cs such as the median response and the 50th percen�le of the item or items being forecasted. This provides a reference point for the par�cipants, who can decide whether or not to change their es�mate based upon this informa�on. Because responses are gathered by ques�onnaire rather than by group interac�on, the par�cipants do not meet face-to-face. As a result, a few par�cipants, who may be overly conserva�ve or overly op�mis�c, cannot dominate the discussion 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 forecas�ng can be cross-sec�onal data, �me-series data, or a combina�on of the two. Cross-sec�onal 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 �me-series and cross-sec�onal elements, as shown in Table 6.2. The �me-series data are the two years of data that are available for the banks. The cross- sec�onal element is represented by the data from more than one bank.

Table 6.2: Time-series and cross-sec�onal data

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

Stockton

San Jose

Santa Cruz

Berkeley

Forecas�ng sales, costs, and other relevant es�mates usually involves �me-series data, and the techniques discussed here are useful in predic�ng such data. See Figure 6.2 for the �me line and nota�on used in forecas�ng. Each point on the �me 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 �me passes, it will have an actual value.

Figure 6.2: Forecas�ng �me line

Simple Moving Average

One approach to forecas�ng is to use only the most recent �me period to project the next �me 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 predic�on. Suppose that in one month a temporary price cut caused sales to be significantly greater than normal. If these ac�ons are not repeated in the next month, then using the previous month's sales as the forecast will provide a biased predic�on.

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�on 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 predic�on. In theProcessing math: 0%

ft+1 = the forecast for �me period t + i, that is, the next �me period when i = 1

xt–i = the observed value for period t – i, where t is the last period for which data are available and i = 0, . . ., n–1

n = the number of �me periods in the average

moving average model, the forecast for the next period is equal to the average of recent periods.

where

The longer the �me—that is, the greater the n—the more smoothing that will take place. The selec�on of n is a management decision based upon the amount of smoothing desired. A small value of n will put more emphasis on recent predic�ons and will more completely reflect fluctua�ons in actual sales. In fact, if n = 1, then the most recent �me period's actual results will become the next period's forecast.

Figure 6.3: Graph of imports

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 nota�on "year: quarter" is used in the subscript here. The fourth quarter of 2013 is noted as "13:4."

f13:4 =

= 5,300

A three-quarter moving average is calculated as follows:

f13:4 =

= 7,600

Which es�mate is likely to be�er represent the future? Which predic�on should be used? It depends upon whether the forecaster feels the last three quarters be�er 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 condi�ons that are unlikely to recur, use the 7-month moving average to smooth the high values in the last three quarters. Forecas�ng models do not provide complete answers to ques�ons. Managerial judgment plays a cri�cal role.

This technique is called a moving average because to forecast the next quarter, the most recent quarter's actual imports are added ande 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 calcula�on for the first quarter of 2014:

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f14:1 =

= 7,667

Weighted Moving Average

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

where

wt–i = the weight for period t–i, where t is the last period for which data are available and i = 0,. . . , n–1. The weights for all n periods must sum to 1.0.

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

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

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 representa�ve of the future. If there is an up or down trend in the data, a weighted moving average can adjust more quickly than a simple moving average. S�ll, this form of the weighted moving average is not as accurate as regression analysis (discussed later in this chapter) is in adap�ng to trends.

Exponen�al Smoothing

Exponen�al smoothing is another form of a weighted moving average. It is a procedure for con�nually revising an es�mate to include more recent data. The method is based upon averaging (smoothing) past values. To start a forecast using exponen�al smoothing, the forecast for the first period, Exponen�al smoothing is another form of a weighted moving average. It is a procedure for con�nually revising an es�mate to include more recent data. The method is based upon averaging (smoothing) past values. To start a forecast using exponen�al smoothing, the forecast for the first period, ft+1  would be based upon the actual value for the most recent period, xt. (See equa�on 6.1.) The forecast for the second

period, ft+2 is equal to the actual value of the previous period, xt+1 �mes the smoothing constant, A, plus (1 – A) �mes the prior period's forecast, ft+1. (See equa�on 6.2.)

Remember, the prior forecast, ft+1, is simply the actual value from period t. The forecast in equa�on 6.2 is A �mes the prior period's actual value plus (1 – A) �mes the prior

period's forecast, ft+1.

ft+1 = xt (6.1)

ft+2 = A(xt+1) + (1 – A)ft+1 (6.2)

ft+3 = A(xt+2) + (1 – A)ft+2 (6.3)

.       .         .

.       .         .

.       .         .

.       .         . ft+n = A(xt+n – 1) + (1 – A)ft+n – 1

where

n = some number of periods in the future

Consider one more equa�on in detail. Equa�on 6.3 uses the prior period's actual value �mes the weigh�ng factor, A, plus (1 – A) �mes the prior period's forecast. Exponen�al 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 discre�on. What will happen if a manager selects a smoothing constant at an extreme? If A = 1, then according to equa�on 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 forecas�ng pa�ern gets started, the forecast is so smooth that it will never change. No actual amounts can enter the equa�on because A = 0. Neither of these alterna�ves is acceptable.

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There are no specific rules about choosing the value of A. If the forecaster wants to put more weight on the most recent �me period, then A should be set closer to 1. If the manager desires a smoother forecast that will not react dras�cally to short-term change, A should be set closer to 0. Values between 0.1 and 0.3 are most commonly used. Typically, values in this range are selected so the forecast does not overcompensate for sudden changes in the data. For example, if weather was extremely warm in the last �me period and demand was high, the forecast for the next �me 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 �me period.

Example: Exponen�al Smoothing

Use exponen�al smoothing to forecast imports from the previous example. To illustrate the impact of the smoothing constant, use A = 0.1 and A = 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 = 0.1 and A = 0.6. A�er that, the forecasts are significantly different because of the large difference in smoothing constants and the large fluctua�ons in demand. The third quarter's forecast follows the equa�ons 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 poten�al for inaccurate forecasts when historical demand varies significantly.

For A = 0.1

f12:3 = A(x12:2) + (1 – A)f12:2

= 0.1(2,000) + 0.9(4,100)

= 3,890

For A = 0.6

f12:3 = 0.6(2,000) + 0.4(4,100)

= 2,840

Year: Quarter Imports ($000,000) Forecast A = 0.1 Forecast A = 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 = 0.1 does not react abruptly to sudden changes. The forecast with A = 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 plo�ed.

Figure 6.4: Exponen�al smoothing examples

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6.3 Advanced Statistical Methods

Correla�on analysis measures the degree of rela�onship 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 correla�on is a measure of the strength of linear rela�onship between variables. If there is no rela�onship, then the coefficient of correla�on is 0. Perfect posi�ve correla�on is 1.0, and perfect nega�ve correla�on is –1.0 (see Figure 6.5). Between the limits of perfect posi�ve and perfect nega�ve correla�on, there are many levels of strength. Examples are shown in Figures 6.5 and 6.6.

Figure 6.5: Sca�er diagrams showing zero, perfect posi�ve, and perfect nega�ve correla�ons

Figure 6.6: Sca�er diagrams showing examples of correla�on

Regression analysis can be used to forecast both �me-series and cross-sec�onal data. Regression analysis is o�en used to es�mate the slope of a trend line for �me-series data. Regression analysis can be either simple or mul�ple. Simple regression analysis involves the predic�on of only one variable (the dependent variable) and uses only one variable for predic�on (the independent variable). Mul�ple regression analysis has only one dependent variable, but can have more than one independent variable.

Regression and Correlation Analysis

The equa�on for simple regression follows. Y is the dependent variable, and X is the independent variable. The variable b is the slope of the line, which is es�mated by equa�on 6.4, and variable a is the Y-intercept, which is es�mated by equa�on 6.5.

where

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

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n = the number of data points in the sample

r = the coefficient of correla�on

b = (6.4) a = (6.5) r = (6.6)

Example: Regression

When working with �me-series data, it is usually easier to convert the �me 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 �me period for which data are available. Following periods will be consecu�vely numbered. In this example, the assump�on is that demand (Y) depends on �me (X), the independent variable. The import data from an earlier example are used for analysis.

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

Interpre�ng the results of the model requires an understanding of the original units of the data as well as the slope/intercept method of represen�ng a straight line. The last quarter of 2013 is coded as "8" because the quarters are consecu�vely 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 es�mated value is represented by Ye. The predic�ve model follows.

Thus, the projec�on for imports is $8,528 million.

Goodness of Fit

How well does the equa�on determined by regression analysis fit the data? The principles on which simple regression analysis and mul�ple regression analysis are constructed are similar. The regression model es�mates 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 o�en 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 differen�al calculus and is not covered in this text. The equa�ons that determine the parameters of the slope (b) and the Y- intercept (a) are 6.4 and 6.5, respec�vely.

Figure 6.7: Regression line

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The coefficient of correla�on calculated in the prior example (r = 0.671) indicates a high degree of rela�onship between the dependent and independent variables. The higher the coefficient of correla�on (closer to 1), the more confident the forecaster can be that varia�on in the dependent variable (imports) is explained by the independent variable (�me). This can be observed by looking at the sca�er diagram in Figure 6.8. A measurement of this varia�on about the regression line is the standard error of the es�mate, sy/x. It is the

difference between each observed value, Yo, and the es�mated value, Ye. The equa�on for the standard error of the es�mate follows. An alterna�ve formula that is easier to use

with a calculator is provided in Figure 6.8.

Simple regression models can be constructed for cross-sec�onal data. The mechanics are similar.

Figure 6.8: Sca�er diagram and regression line for import problem

Computer Applica�on of Simple Regression Analysis

Many different computer so�ware packages are available for doing both simple and mul�ple regression analyses. Table 6.3 is the computer-based output for regression analysis. The coefficients calculated by so�ware are the same (with allowances for rounding) as the coefficients calculated by hand. The standard error of the es�mate is also the same as the value calculated by hand. The computer output also provides addi�onal informa�on. The standard error of the coefficients, 1,784.800 and 399.093, are standard devia�ons 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-sta�s�cs for the hypothesis tests. The two-sided significant probabili�es 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 coefficient 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 es�mate = 2111.804Processing math: 0%

Mul�ple Regression Model

Mul�ple regression has only one dependent variable, but can have many independent variables.

Y = a + b1 x1 + b2 x2 + . . . + bk xk

where

Y = 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.

(Equa�ons for the parameters are not given here. There are many available computer packages, such as EXCEL, SPSSX, SAS, or MINITAB to do the necessary calcula�ons.)

xi = an independent variable, with i = 1, . . ., k. Each independent variable will have n observa�ons or data points.

Minimizing squared distances from each observed point to the best fit regression line is s�ll useful. However, because mul�ple regression requires more than two dimensions, two- dimensional graphs cannot be used. Computerized sta�s�cal models are used to make the calcula�ons.

Problem

The prior examples use only one independent variable (�me) to predict imports. Most rela�onships 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.

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 mul�ple regression output is shown in Figure 6.9. The equa�on for predic�ng imports is

ye = –141,000 – 1,387.6x1 + 206.35x2 + 1,205.4x3

Figure 6.9: Mul�ple regression output

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 = –141,000 – 1,387.6(8) + 206.35(78) + 1,205.4(118) = 6,232

The predic�on is $6,232 million worth of imports in the fourth quarter of 2013.

Measuring Forecasting Error

Regardless of which forecas�ng model is used, it is important to have some way to determine the model's propensity for error. If an organiza�on has been using a par�cular model to forecast sales for some �me, has the model been performing well? How large is the error? One approach is to simply subtract the forecast for one �me period from the actual

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When using a forecas�ng model, it is cri�cal to have a way to determine the model's propensity for error.

iStockphoto/Thinkstock

value for the same �me period. This can be repeated so that the forecaster has differences for many periods. Differences are posi�ve when the forecast is less than the actual value, and nega�ve when the forecast is greater than the actual value. In raw form, these differences tell the forecaster li�le. If these are added, the nega�ve errors and the posi�ve error will cancel and therefore underes�mate the error. A common method used by forecasters 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 observa�ons. The following calcula�ons help illustrate the method.

Problem

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

January 419,000 448,000 – 29,000 841,000,000

February 480,000 481,000 – 1,000 1,000,000

March 601,000 563,000 + 38,000 1,444,000,000

April 505,000 525,000 – 20,000 400,000,000

May 462,000 490,000 – 28,000 784,000,000

June 567,000 519,000 + 48,000 2,304,000,000

TOTAL 5,774,000,000

Some�mes the square root of the mean squared error is used to measure the error. This is analogous to the standard error of the es�mate, which is discussed in the sec�on on regression analysis.

It is also possible to use mean absolute devia�on (MAD), which is similar to MSE, to es�mate forecas�ng error. MAD is calculated by adding together the differences between the actual and forecasted value once the nega�ve and posi�ve signs are removed. If MAD is not calculated, a large nega�ve error would offset a large posi�ve error, so the total error would be greatly underes�mated. Try summing the "Error" column in this example with the signs included. The total error is $8,000 because the nega�ve and posi�ve 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 es�mate future forecas�ng errors.

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Chapter Summary

A model is an important way to think about problems. It is an abstrac�on from a real problem of the key variables and rela�onships in order to simplify the problem 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�sts, engineers, managers, and others; and mathema�cal models used in many disciplines. Models are used to assist managers and others in answering what-if ques�ons by changing a parameter in the model. There are qualita�ve and quan�ta�ve methods for developing a forecast. Forecas�ng is a type of mathema�cal model that can be used to predict the future. It is an important part of the planning process in an organiza�on The forecas�ng process consists of determining the objec�ves of the forecast, developing and tes�ng a model, applying the model, considering real-world constraints in the applica�on of the model, and revising and evalua�ng the forecast. Forecas�ng techniques typically use historical data to develop the model that is used to make the projec�on. If the rela�onships in the data change over �me, the model may no longer predict the future accurately. Forecasters require a way to measure the amount of forecas�ng error.

Case Studies

Blast-Away Housecleaning Service

Blast-Away Housecleaning Service uses powerful water jets to clear loose paint from residen�al buildings and to clean aluminum siding. The company is trying to arrive at a fast and accurate way of es�ma�ng cleaning jobs. The following simple formula is its first a�empt. It includes a fixed charge for coming to the job plus �me requirements, which are a func�on of the exterior of the house measured in square feet (sf).

Es�mated cost = $15 + ($.06/sf)(sf)

A�er 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 planning to add another crew this year, but if he cannot determine the reason for the losses and find a solu�on, his investors would be reluctant to provide him with addi�onal 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 recommended 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 �ckets, 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 �me and the �me necessary to complete each house. As you analyze the job �ckets, you no�ce that a substan�al 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 sa�sfac�on.

1. In addi�on 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 recommenda�ons?

Lucy's Lamps-R-Us

Lucy Mertz has opened a specialty lamp shop in a suburban shopping mall. Mertz's shop has an excellent loca�on next to the entrance to the largest and most popular department store in the five-county area. A�er 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 judgment. Mertz wants you to use a quan�ta�ve 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 disappoin�ng. In January, sales declined because of returns from the Christmas buying season and an increase in bargain hun�ng. Also, the large department store that anchored Mertz's end of the shopping mall closed at the end of January because of opera�ng losses by the parent company.

Time Period Actual Sales January 7,500 February 6,000

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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?

Discussion Ques�ons

Click on each ques�on to reveal the answer.

1. What is model building, and why is model building important for managers? (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644

A model is an abstrac�on from the real problem of the cri�cal variables and rela�onships in order to simplify the problem. Managers should use models to gain a be�er understanding of the problem. Models also permit the manager to analyze a given situa�on by asking "what if" ques�ons.

A model is an important way of thinking about a problem. Decision makers use models to increase their understanding of a problem. A model helps managers simplify the problem by focusing on the key variables and rela�onships in the problem.

2. Discuss the different types of models. (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644

Many different types of models are possible. Models can be used to test design characteris�cs such as tes�ng the aerodynamic proper�es of an airplane in a wind tunnel. Mathema�cal and graphical models are also possible.

a. Mathema�cal models include algebraic models such as break-even analysis, sta�s�cal models used in forecas�ng and quality control, and calculus based models such as the economic order quan�ty.

b. Graphs and Charts are pictorial representa�ons of mathema�cal rela�onships. They include a graphical representa�on of break-even analysis, a pie chart that illustrates market share in an industry, and a graph of stock prices for the past four months.

c. Diagrams and drawings are pictorial representa�ons of conceptual rela�onships. They include such things as precedence diagrams that represent the sequence required to assemble a building, a drawing of a gear that might be part of a transmission in a car, and a diagram that represents the logic of a computer program.

d. Scale models and prototypes are physical representa�ons of an item. They include a scale model of an airplane and the first part produced (prototype) which is used for tes�ng purposes.

3. Describe how models can be used to answer what-if ques�ons. (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644

A model can be used to answer "what if" ques�ons by changing one of the assump�ons in the model. For example, in the model of gasoline costs on the trip to Florida, an increase in the cost per gallon of gasoline caused the cost of the trip to increase and a decrease in the miles per gallon, which the car would a�ain, caused the cost to increase. In a physical model, such as the airplane example, the designers could change the angle or shape of the wing to determine the impact of the change on the performance of the plane.

4. How are models used in business and opera�ons? (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644

Models are widely used in opera�ons and in other areas within the organiza�on. Financial managers use net present value and internal rate of return models to analyze investment alterna�ves. Informa�on systems personnel use flow diagrams to understand the logic required to develop a computerized inventory control system. Accountants use ra�os such as the current ra�o to measure an organiza�on's ability to pay its short-term bills. Opera�ons managers use models extensively to forecast sales, understand the cost-volume-profit rela�onship, assign tasks to work sta�ons, decide to make or buy a product, and to set the order size.

5. What is forecas�ng, and why is it important to an organiza�on? (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644

Forecas�ng is an a�empt to predict the future. Forecasts are usually the result of examining past experience to gain insight into the future. These insights o�en take the form of mathema�cal models. Forecas�ng is essen�al to the success of an organiza�on because organiza�ons must prepare for the future and a forecast provides some insights as to the condi�ons, which the organiza�ons might encounter.

6. Describe the forecas�ng process. (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644

To begin the forecas�ng process, the modeler should determine the objec�ves of the forecast. This should lead to a discussion of the purpose of the forecast, the variables to be forecast, who will use the forecast, the �me frame of the forecast (long, medium, or short term), the level of accuracy needed, and when the forecast is needed. A technique or method for doing the forecast must be selected and the forecas�ng model should be developed and tested. When the model is applied, the results should be considered with respect to constraint in the "real-world." For example, a forecast of sales that predicts a 20 percent increase in sales must be balanced against a company's ability to produce at that level. Finally, the forecast should be evaluated to determine how well it is predic�ng the future. Have assump�ons upon which the forecas�ng model was built changed to the point that the model is no longer valid? If the assump�ons have changed, then the model should be revised.

7. Discuss the qualita�ve approaches to forecas�ng. (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644

Qualita�ve approaches to forecas�ng are usually based on the judgment and experience of people who are knowledgeable in the area being forecast. Qualita�ve approaches are usually based on historical data because the judgment and experience of the people involved are affected by events in the past. The buildup method for predic�ng sales involves star�ng at the bo�om of an organiza�on. Sales representa�ves from each sales territory predict their sales. These es�mates are passed up to the next level of the organiza�on for review and evalua�on. A�er agreement is reached at that level, the sales are passed up to the next level in the organiza�on. When the sales figures reach the top of the organiza�on, they can become the target for the en�re organiza�on.

A survey is a systema�c effort to elicit informa�on from specific groups and is usually conducted via a wri�en ques�onnaire, Internet survey, or phone interview. The target of a survey could be consumers, purchasing agents, or economists. A survey could a�empt to determine the demand for a new or exis�ng product. The use of a survey is not limited to es�ma�ng demand for product. It could be used to measure the overall health of the economy or the support level of a poli�cal candidate on a par�cular issue.

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In a test market the actual product is distributed in a small geographical area and demand for the product is carefully monitored. The test market area should be representa�ve of the overall market if the results are to be useful. Test markets are used to es�mate sales.

A panel of experts involves people who are knowledgeable about a subject. This group a�empts to make a forecast by building consensus. The Delphi Technique uses a panel of experts in a par�cular manner. The panel of experts can be used to predict a wide variety of items from the cost of a raw material to the chance for nuclear disarmament.

8. How does the Delphi Technique work? What are its advantages? (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644

The Delphi Technique uses a panel of experts and surveys. The members of the panel do not interact directly with one another. They provide a sequence of forecasts through responses to ques�onnaires. A�er each forecast, results are compiled and the individuals on the panel are informed as to the 25th, 50th, 75th percen�le and/or other relevant data points for the item being forecast. Because the response is by ques�onnaire rather than by group interac�on, which can be dominated by a few individuals, undue conserva�sm or op�mism and argumenta�on are avoided. The Delphi Technique assumes that as each forecast is conducted and the results disseminated among the panel members, the range of responses diminishes and the median moves to a posi�on represen�ng the "true" consensus of the group.

9. How is regression analysis different from the moving average, the weighted moving average, and exponen�al smoothing? (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644

There are obvious differences between the way regression analysis parameters are calculated and the ways that moving average, weighted moving average, and exponen�al smoothing are calculated. The difference, which the ques�on is a�emp�ng to address, is that regression analysis actually es�mates the amount of trend in the data. In simple regression analysis, this is the slope coefficient b. The other techniques predict the next �me periods value without es�ma�ng the trend. As a result, these techniques will consistently underes�mate sales when sales are rising at a constant or accelera�ng rate period a�er period and will consistently overes�mate sales when sales are declining in this manner.

Regression analysis (simple or mul�ple regression) is a cause and effect model that considers independent variable(s) to determine the dependent variable. The (simple) moving average, weighted moving average, and exponen�al smoothing models use �me series data to predict the next period. However, �me can be a dependent variable in regression analysis as well. Regression analysis can es�mate a trend.

10. What is forecas�ng error, and why should it be measured? (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644

Forecas�ng error is the difference between the forecasted value for an item and the actual value. There are many different ways to es�mate the error, and Mean Squared Error (MSE) and Mean Absolute Devia�on (MAD) are common ways. Both methods take the difference between the forecast and the actual values for all observa�on. For MSE, these differences are squared and summed; then, the sum is divided by the total number of observa�ons. For MAD, the absolute value of these differences is determined and the errors are summed. The sum is divided by the total number of observa�ons. Without a way to measure forecas�ng error, it would be difficult to determine when a forecast is no longer representa�ve of what is actually driving the factor being forecasted.

Problems

1. Blast-Away Housecleaning Service uses powerful water jets to clear loose paint from residen�al buildings and to clean aluminum siding. The company is trying to arrive at a fast and accurate way of es�ma�ng cleaning jobs. The following simple formula is its first a�empt. It includes a fixed charge for coming to the job plus �me requirements, which are a func�on of the exterior of the house measured in square feet (sf).

Es�mated cost = $15 + ($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 bo�om 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 es�mate the amount of natural gas required (NGR) in hundreds of cubic feet (CCF) to heat your home. This is done by a mathema�cal model that considers the square footage on the first floor (sf1), the square footage on the second floor (sf2), and the temperature se�ng on the

thermostat. The temperature se�ng entered into the model should be the difference between the temperature se�ng in the home and 65 degrees (td). Make sure to keep the minus sign if the se�ng 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 insula�on in each wall, 6 inches of insula�on in the a�c, and a typical Midwestern winter.

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 second 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 opera�on for eight months, and demand for its products has grown rapidly. The owner, who is also the manager, of Mighty-Maid is trying to keep pace with demand, which means hiring and training more workers. She believes that demand will con�nue at the same pace. She needs an es�mate of demand so she can recruit and train the workforce. The following represents the history of Mighty-Maid:

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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

Es�mate the trend in the data using regression analysis.

5. Use the regression model calculated in the Mighty-Maid problem to es�mate the hours of service for December through July. Now that both the actual and the forecasted values are available, answer the following ques�ons:

a. What are the MSE and MAD for the forecast? b. Is the forecas�ng 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 mergers 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 exponen�ally smoothed series for the series in Problem 6, (a) using A = 0.1 and then (b) using A = 0.7, and plot these �me series along with the actual data to see the impact of the smoothing constant.

8. The �me 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:

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 exponen�ally smoothed forecast for this series with A = 0.2. 9. The quarterly data presented here show the number of appliances (in thousands) returned to a par�cular 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 equa�on of the least squares linear trend line that fits this �me series. Let t = 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

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Feb. 2011 61 Oct. 2011 52 Mar. 2011 61 Nov. 2011 51 Apr. 2011 53 Dec. 2011 63

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 exponen�al smoothing technique to forecast sales of AJV's model EM-5V for January 2012 (use A = 0.8). Does the model appear to be appropriate? 12. U�lize the single exponen�al smoothing technique to forecast sales of AJV's model EM-5V for January 2012 (use A = 0.1). How do the results compare with those from problem 11?

Is one be�er 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. Thri�y Bank and Trust is trying to forecast on-the-job performance by its employees. The bank administers an ap�tude test to new employees. A�er the employee training period

and an addi�onal 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 Transac�ons 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 es�mated by the regression analysis model? b. How well does the model fit the data? c. How many transac�ons per hour would you expect from someone who scored 87 on the ap�tude test?

15. The data in the following table were collected during a study of consumer buying pa�erns.

Observa�on 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 correla�on and the standard error of the es�mate. c. How could the coefficient of correla�on and the standard error of the es�mate be used to make a judgment about the model's accuracy?

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

Period Fer�lizer 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

a. Use the least squares method to obtain a linear regression line for the data. b. Calculate the coefficient of correla�on and the standard error of the es�mate. c. Predict lawn mower sales for the first week in August, if two tons of fer�lizer sold six weeks earlier

Click here to see solu�ons to the odd-numbered problems. (h�ps://media.thuze.com/MediaService/MediaService.svc/constella�on/book/AUBUS644.13.2/{pdf}bus644_ch06_odd_problem_solu�ons.pdf)

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Click on each key term to see the defini�on.

buildup method (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

An approach to forecas�ng that starts at the bo�om of an organiza�on and makes an overall es�mate by adding together es�mates from each element.

coefficient of correla�on (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

A measure of the strength of a rela�onship between variables. If there is no rela�onship, the coefficient of correla�on will be zero. A perfect posi�ve correla�on is 1.0 and a perfect nega�ve correla�on is 1.0.

correla�on analysis (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

A measure of the degree of rela�onship between two variables.

Delphi Technique (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

A forecas�ng procedure that uses a panel of experts and surveys to build consensus regarding future events. It is an itera�ve process for consensus building.

dependent variable (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

The variable in regression analysis that is being predicted.

exponen�al smoothing (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

Another form of a weighted moving average. It is a procedure for con�nually revising an es�mate to include more recent data. The method is based on averaging (smoothing) past values.

forecast (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

An es�mate of future events.

forecas�ng (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

The process of a�emp�ng to predict the future.

forecas�ng error (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

The difference between the forecasted value and the actual value.

independent variable (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

A variable in regression analysis which is used to predict the dependent variable.

mean absolute devia�on (MAD) (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

The average of absolute error. The differences between the actual value of a variable and the forecasted value are added a�er the plus and minus signs are removed. This total is divided by the number of observa�ons.

mean squared error (MSE) (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

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 observa�ons.

model (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

An abstrac�on from the real problem of the key variables and rela�onships in order to simplify the problem. The purpose of modeling is to provide the user with a be�er understanding of the problem, and with a way to manipulate the results for "what if" analysis.

mul�ple regression analysis (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

Regression analysis that uses two or more (independent variables) to predict one dependent variable.

panel of experts (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

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An approach to forecas�ng that involves people who are knowledgeable about the subject. This group a�empts to make a forecast by building consensus.

regression analysis (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

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 es�mated regression line.

simple moving average (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

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 �me 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 (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

Regression analysis that uses only one variable (independent variable) to predict a single dependent variable.

survey (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

A systema�c effort to elicit informa�on from specific groups and is conducted via a wri�en ques�onnaire, phone interview or the Internet.

t-sta�s�c (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

Measures the distance from the mean to a point in the t-distribu�on represented by standard devia�ons.

t-value (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

The calculated t-sta�s�c used in hypothesis tes�ng.

test market (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

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

weighted moving average (h�p://content.thuzelearning.com/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/sec�ons/cover/books/AUBUS644.13.2/

A method that is similar to the simple moving average. In the simple moving average, the weight for each historical �me 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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