Discuss why the moving average method was used instead of another forecasting method
Discuss why the moving average method was used instead of another forecasting method
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.
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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:
- 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.
- 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.
- 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.
- 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.
- 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
Jordan’s car repair service center is planning to launch its “At-Home – Car Services” business begin- ning in the summer of the coming year. The business model is based upon providing car repairs and routine 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 pattern to the demand, and whether customers would be willing to pay a small premium for this convenient service. Using mathematical 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 potential customers. First, he will seek input from his active 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 little, if any, new revenue because they are already supplying Jordan with their business. He may attract, at best, a small increase in business from this group, or he may prevent them from choosing a competitor in the future. Second, and more financially lucrative, Jordan would like to identify 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 initially 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 additional 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 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.
©Creatas/Thinkstock
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)
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