A company's supply chain encompasses all of the facilities, functions, and activities involved in producing a product or service from suppliers (and their suppliers) to customers (and their customers). Supply chain functions include purchasing, inventory, production, scheduling, facility location, transportation, and distribution. All these functions are affected in the short run by product demand and in the long run by new products and processes, technology advances, and changing markets.

Forecasts of product demand determine how much inventory is needed, how much product to make, and how much material to purchase from suppliers to meet forecasted customer needs. This in turn determines the kind of transportation that will be needed and where plants, warehouses, and distribution centers will be located so that products and services can be delivered on time. Without accurate forecasts, large stocks of costly inventory must be kept at each stage of the supply chain to compensate for the uncertainties of customer demand. If there are insufficient inventories, customer service suffers because of late deliveries and stockouts. This is especially hurtful in today's competitive global business environment, where customer service and on-time delivery are critical factors. Figure 12.1 illustrates the effects of bad forecasting on the supply chain.

Accurate forecasting determines how much inventory a company must keep at various points along its supply chain.

While accurate forecasts are necessary, completely accurate forecasts are never possible. Hopefully, the forecast will reduce uncertainty about the future as much as possible, but it will never eliminate uncertainty. As such, all of the supply chain processes need to be flexible to respond to some degree of uncertainty.

In Chapter 10 on supply chain management, we talked about the "bullwhip effect" and its negative impact on the supply chain. The bullwhip effect is the distortion of information about product demand (including forecasts) as it is transmitted up the supply chain back toward suppliers. As demand moves further away from the ultimate end-use consumer, the variation in demand becomes greater and demand forecasts become less reliable. This increased variation can result in excessive, costly safety stock inventories at each stage in the supply chain and poorer customer service.

The bullwhip effect is caused when slight demand variability is magnified as information moves back upstream in the supply chain (see Figure 10.4). It is created when supply chain members make ordering decisions with an eye to their own self-interest and/or they do not have accurate demand forecasts from adjacent supply chain members. If each supply chain member is uncertain and not confident about what the actual demand is for the succeeding member it supplies, and it's making its own demand forecast, then it will stockpile extra inventory to compensate for the uncertainty; that is, the member creates a security blanket of inventory. One way to cope with the bullwhip effect is to develop demand forecasts that will reduce uncertainty and for supply chain members to share these forecasts with each other. Ideally, a single forecast of demand for the final customer in the supply chain would drive the development of subsequent forecasts for each supply chain member back up through the supply chain.

Figure 12.1. The Effect of Inaccurate Forecasting on the Supply Chain

One trend in supply chain design is continuous replenishment, wherein continuous updating of data is shared between suppliers and customers. In this system, customers are continuously being replenished, daily or even more often, by their suppliers based on actual sales. Continuous replenishment, typically managed by the supplier, reduces inventory for the company and speeds customer delivery. Variations of continuous replenishment include quick response, just-in-time (JIT), VMI (vendor-managed inventory), and stockless inventory. Such systems rely heavily on accurate short-term forecasts, usually on a weekly basis, of end-use sales to the ultimate customer. The supplier at one end of a company's supply chain must forecast the company's customer demand at the other end of the supply chain in order to maintain continuous replenishment. The forecast also has to be able to respond to sudden, quick changes in demand. Longer forecasts based on historical sales data for 6 to 12 months into the future are also generally required to help make weekly forecasts and suggest trend changes.

In continuous replenishment, the supplier and customer share continuously updated data.

Forecasting is also crucial in a quality management environment. More and more, customers perceive good-quality service to mean having a product when they demand it. This holds true for manufacturing and service companies. When customers walk into a McDonald's to order a meal, they do not expect to wait long to place orders. They expect McDonald's to have the item they want, and they expect to receive their orders within a short period of time. A good forecast of customer traffic flow and product demand enables McDonald's to schedule enough servers, to stock enough food, and to schedule food production to provide high-quality service. An inaccurate forecast causes service to break down, resulting in poor quality. For manufacturing operations, especially for suppliers, customers expect parts to be provided when demanded. Accurately forecasting customer demand is a crucial part of providing the high-quality service.

Forecasting customer demand is a key to providing good-quality service.

There can be no strategic planning without forecasting. The ultimate objective of strategic planning is to determine what the company should be in the future—what markets to compete in, with what products, to be successful and grow. To answer these questions, the company needs to know what new products its customers will want, how much of these products customers will want, and the level of quality and other features that will be expected in these products. Forecasting answers these questions and is a key to a company's long-term competitiveness and success. The determination of future new products and their design subsequently determines process design, the kinds of new equipment and technologies that will be needed, and the design of the supply chain, including the facilities, transportation, and distribution systems that will be required. These elements are ultimately based on the company's forecast of the long-run future.

Successful strategic planning requires accurate forecasts of future products and markets.

The type of forecasting method depends on time frame, demand behavior, and causes of behavior.

• Short- to mid-range forecast: typically encompasses the immediate future–daily up to two years.

Forecasts are either short- to mid-range, or long-range. Short-range (to mid-range) forecasts are typically for daily, weekly, or monthly sales demand for up to approximately two years into the future, depending on the company and the type of industry. They are primarily used to determine production and delivery schedules and to establish inventory levels. At Hewlett-Packard monthly forecasts for printers are constructed from 12 to 18 months into the future, while at Levi Strauss weekly forecasts for jeans are prepared for five years into the future.

• Long-range forecast: usually encompasses a period of time longer than two years.

A long-range forecast is usually for a period longer than two years into the future. A long-range forecast is normally used for strategic planning—to establish long-term goals, plan new products for changing markets, enter new markets, develop new facilities, develop technology, design the supply chain, and implement strategic programs. At Unisys, long-range strategic forecasts project three years into the future; Hewlett-Packard's long-term forecasts are developed for years 2 through 6; and at Fiat, the Italian automaker, strategic plans for new and continuing products go 10 years into the future.

These classifications are generalizations. The line between short- and long-range forecasts is not always distinct. For some companies a short-range forecast can be several years, and for other firms a long-range forecast can be in terms of months. The length of a forecast depends a lot on how rapidly the product market changes and how susceptible the market is to technological changes.

Demand sometimes behaves in a random, irregular way. At other times it exhibits predictable behavior, with trends or repetitive patterns, which the forecast may reflect. The three types of demand behavior are trends, cycles, and seasonal patterns.

• Trend: a gradual, long-term up or down movement of demand.

• Random variations: movements in demand that do not follow a pattern.

• Cycle: an up-and-down repetitivemovement in demand.

A trend is a gradual, long-term up or down movement of demand. For example, the demand for houses has followed an upward trend during the past few decades, without any sustained downward movement in the market. Trends are often the starting points for developing forecasts. Figure 12.2 (a) illustrates a demand trend in which there is a general upward movement, or increase. Notice that Figure 12.2(a) also includes several random movements up and down. Random variations are movements that are not predictable and follow no pattern (and thus are virtually unpredictable). They are routine variations that have no "assignable" cause.

A cycle is an up-and-down movement in demand that repeats itself over a lengthy time span (i.e., more than a year). For example, new housing starts and, thus, construction-related products tend to follow cycles in the economy. Automobile sales also tend to follow cycles. The demand for winter sports equipment increases every four years before and after the Winter Olympics. Fig-ure 12.2(b) shows the behavior of a demand cycle.

Figure 12.2. Forms of Forecast Movement

A seasonal pattern is an oscillating movement in demand that occurs periodically (in the short run) and is repetitive. Seasonality is often weather-related. For example, every winter the demand for snowblowers and skis increases, and retail sales in general increase during the holiday season. However, a seasonal pattern can occur on a daily or weekly basis. For example, some restaurants are busier at lunch than at dinner, and shopping mall stores and theaters tend to have higher demand on weekends. At FedEx seasonalities include the month of the year, day of the week, and day of the month, as well as various holidays. Figure 12.2c illustrates a seasonal pattern in which the same demand behavior is repeated each year at the same time.

• Seasonal pattern: an up-and-down repetitive movement in demand occurring periodically.

Demand behavior frequently displays several of these characteristics simultaneously. Although housing starts display cyclical behavior, there has been an upward trend in new house construction over the years. Demand for skis is seasonal; however, there has been an upward trend in the demand for winter sports equipment during the past two decades. Figure 12.2d displays the combination of two demand patterns, a trend with a seasonal pattern.

Instances when demand behavior exhibits no pattern are referred to as irregular movements, or variations. For example, a local flood might cause a momentary increase in carpet demand, or a competitor's promotional campaign might cause a company's product demand to drop for a time. Although this behavior has a cause and, thus, is not totally random, it still does not follow a pattern that can be reflected in a forecast.

The factors discussed previously in this section determine to a certain extent the type of forecasting method that can or should be used. In this chapter we are going to discuss three basic types of forecasting: time series methods, regression methods, and qualitative methods.

Types of methods: time series, causal, and qualitative.

Time series methods are statistical techniques that use historical demand data to predict future demand. Regression (or causal) forecasting methods attempt to develop a mathematical relationship (in the form of a regression model) between demand and factors that cause it to behave the way it does. Most of the remainder of this chapter will be about time series and regression forecasting methods. In this section we will focus our discussion on qualitative forecasting.

• Regression forecasting methods: relate demand to other factors that cause demand behavior.

Qualitative (or judgmental) methods use management judgment, expertise, and opinion to make forecasts. Often called "the jury of executive opinion," they are the most common type of forecasting method for the long-term strategic planning process. There are normally individuals or groups within an organization whose judgments and opinions regarding the future are as valid or more valid than those of outside experts or other structured approaches. Top managers are the key group involved in the development of forecasts for strategic plans. They are generally most familiar with their firms' own capabilities and resources and the markets for their products.

Management, marketing and purchasing, and engineering are sources for internal qualitative forecasts.

The sales force of a company represents a direct point of contact with the consumer. This contact provides an awareness of consumer expectations in the future that others may not possess. Engineering personnel have an innate understanding of the technological aspects of the type of products that might be feasible and likely in the future.

Consumer, or market, research is an organized approach using surveys and other research techniques to determine what products and services customers want and will purchase, and to identify new markets and sources of customers. Consumer and market research is normally conducted by the marketing department within an organization, by industry organizations and groups, and by private marketing or consulting firms. Although market research can provide accurate and useful forecasts of product demand, it must be skillfully and correctly conducted, and it can be expensive.

• Delphi method: involves soliciting forecasts about technological advances from experts.

The Delphi method is a procedure for acquiring informed judgments and opinions from knowledgeable individuals using a series of questionnaires to develop a consensus forecast about what will occur in the future. It was developed at the Rand Corporation shortly after World War II to forecast the impact of a hypothetical nuclear attack on the United States. Although the Delphi method has been used for a variety of applications, forecasting has been one of its primary uses. It has been especially useful for forecasting technological change and advances.

Technological forecasting has become increasingly crucial to compete in the modern international business environment. New enhanced computer technology, new production methods, and advanced machinery and equipment are constantly being made available to companies. These advances enable them to introduce more new products into the marketplace faster than ever before. The companies that succeed manage to get a "technological" jump on their competitors by accurately predicting what technology will be available in the future and how it can be exploited. What new products and services will be technologically feasible, when they can be introduced, and what their demand will be are questions about the future for which answers cannot be predicted from historical data. Instead, the informed opinion and judgment of experts are necessary to make these types of single, long-term forecasts.

Forecasting is a continuous process.

Forecasting is not simply identifying and using a method to compute a numerical estimate of what demand will be in the future. It is a continuing process that requires constant monitoring and adjustment illustrated by the steps in Figure 12.3.

Figure 12.3. Steps of the Forecasting Process

In the next few sections, we present several different forecasting methods applicable for different patterns of demand behavior. Thus, one of the first steps in the forecasting process is to plot the available historical demand data and, by visually looking at them, to attempt to determine the forecasting method that best seems to fit the patterns the data exhibit. Historical demand is usually past sales or orders data. There are several measures for comparing historical demand with the forecast to see how accurate the forecast is. Following our discussion of the forecasting methods, we present several measures of forecast accuracy. If the forecast does not seem to be accurate, another method can be tried until an accurate forecast method is identified. After the forecast is made over the desired planning horizon, it may be possible to use judgment, experience, knowl-edge of the market, or even intuition to adjust the forecast to enhance its accuracy. Finally, as demand actually occurs over the planning period, it must be monitored and compared with the forecast in order to assess the performance of the forecast method. If the forecast is accurate, then it is appropriate to continue using the forecast method. If it is not accurate, a new model or adjust-ing the existing one should be considered.

In a naive forecast demand in the current period is used as the next period's forecast.

A time series forecast can be as simple as using demand in the current period to predict demand in the next period. This is sometimes called a naive or intuitive forecast. For example, if demand is 100 units this week, the forecast for next week's demand is 100 units; if demand turns out to be 90 units instead, then the following week's demand is 90 units, and so on. This type of forecasting method does not take into account historical demand behavior; it relies only on demand in the current period. It reacts directly to the normal, random movements in demand.

• Moving average: method uses average demand for a fixed sequence of periods.

The simple moving average method uses several demand values during the recent past to develop a forecast. This tends to dampen, or smooth out, the random increases and decreases of a forecast that uses only one period. The simple moving average is useful for forecasting demand that is stable and does not display any pronounced demand behavior, such as a trend or seasonal pattern.

Moving average is good for stable demand with no pronounced behavioral patterns.

Moving averages are computed for specific periods, such as three months or five months, depending on how much the forecaster desires to "smooth" the demand data. The longer the moving average period, the smoother it will be. (Alternatively, a shorter moving average is more susceptible to simple random variation.) The formula for computing the simple moving average is

where

Computing a Simple Moving Average

The Heartland Produce Company sells and delivers food produce to restaurants and catering services within a 100-mile radius of its warehouse. The food supply business is competitive, and the ability to deliver orders promptly is a factor in getting new customers and keeping old ones. The manager of the company wants to be certain enough drivers and vehicles are available to deliver orders promptly and they have adequate inventory in stock. Therefore, the manager wants to be able to forecast the number of orders that will occur during the next month (i.e., to forecast the demand for deliveries).

From records of delivery orders, management has accumulated the following data for the past 10 months, from which it wants to compute three- and five-month moving averages.

Month	Orders
January	120
February	90
March	100
April	75
May	110
June	50
July	75
August	130
September	110
October	90

Solution

Let us assume that it is the end of October. The forecast resulting from either the three- or five-month moving average is typically for the next month in the sequence, which in this case is November. The moving average is computed from the demand for orders for the prior three months in the sequence according to the following formula:

The five-month moving average is computed from the prior five months of demand data as follows:

The three- and five-month moving average forecasts for all the months of demand data are shown in the following table. Actually, the manager would use only the forecast for November based on the most recent monthly demand. However, the earlier forecasts for prior months allow us to compare the forecast with actual demand to see how accurate the forecasting method is—that is, how well it does.

Three- and Five-Month Averages

Month	Orders per Month	Three-Month Moving Average	Five-Month Moving Average
January	120	—	—
February	90	—	—
March	100	—	—
April	75	103.3	—
May	110	88.3	—
June	50	95.0	99.0
July	75	78.3	85.0
August	130	78.3	82.0
September	110	85.0	88.0
October	90	105.0	95.0
November	—	110.0	91.0

Both moving average forecasts in the preceding table tend to smooth out the variability occurring in the actual data. This smoothing effect can be observed in the following figure in which the three-month and five-month averages have been superimposed on a graph of the original data:

The five-month moving average in the previous figure smooths out fluctuations to a greater extent than the three-month moving average. However, the three-month average more closely reflects the most recent data available to the produce company manager. In general, forecasts using the longer-period moving average are slower to react to recent changes in demand than would those made using shorter-period moving averages. The extra periods of data dampen the speed with which the forecast responds. Establishing the appropriate number of periods to use in a moving average forecast often requires some amount of trial-and-error experimentation.

Longer-period moving averages react more slowly to recent demand changes than shorter-period moving averages; shorter-period moving averages are more susceptible to simple random variation.

The disadvantage of the moving average method is that it does not react to variations that occur for a reason, such as cycles and seasonal effects. Factors that cause changes are generally ignored. It is basically a "mechanical" method, which reflects historical data in a consistent way. However, the moving average method does have the advantage of being easy to use, quick, and relatively inexpensive. In general, this method can provide a good forecast for the short run, but it should not be pushed too far into the future.

The moving average method can be adjusted to more closely reflect fluctuations in the data. In the weighted moving average method, weights are assigned to the most recent data according to the following formula:

where

Determining the precise weights to use for each period of data usually requires some trial-and-error experimentation, as does determining the number of periods to include in the moving aver-age. If the most recent periods are weighted too heavily, the forecast might overreact to a random fluctuation in demand. If they are weighted too lightly, the forecast might underreact to actual changes in demand behavior.

Weighted moving average: weights are assigned to the most recent data.

Computing a Weighted Moving Average

The Heartland Produce Company in Example 12.1 wants to compute a three-month weighted moving average with a weight of 50% for the October data, a weight of 33% for the September data, and a weight of 17% for the August data. These weights reflect the company's desire to have the most recent data influence the forecast most strongly.

Solution

The weighted moving average is computed as

Notice that the forecast includes a fractional part, 0.4. In general, the fractional parts need to be included in the computation to achieve mathematical accuracy, but when the final forecast is achieved, it must be rounded up or down.

This forecast is slightly lower than our previously computed three-month average forecast of 110 orders, reflecting the lower number of orders in October (the most recent month in the sequence).

Exponential smoothing is also an averaging method that weights the most recent data more strongly. As such, the forecast will react more to recent changes in demand. This is useful if the recent changes in the data are significant and unpredictable instead of just random fluctuations (for which a simple moving average forecast would suffice).

• Exponential smoothing: an averaging method that reacts more strongly to recent changes in demand.

Exponential smoothing is one of the more popular and frequently used forecasting techniques, for a variety of reasons. Exponential smoothing requires minimal data. Only the forecast for the current period, the actual demand for the current period, and a weighting factor called a smoothing constant are necessary. The mathematics of the technique are easy to understand by management. Virtually all forecasting computer software packages include modules for exponential smoothing. Most importantly, exponential smoothing has a good track record of success. It has been employed over the years by many companies that have found it to be an accurate method of forecasting.

The exponential smoothing forecast is computed using the formula

where

F_{t + 1} = the forecast for the next period

D_t = actual demand in the present period

F_t = the previously determined forecast for the present period

α = a weighting factor referred to as the smoothing constant

• Smoothing constant: the weighting factor given to the most recent data in exponential smoothing forecasts.

The smoothing constant, α, is between 0.0 and 1.0. It reflects the weight given to the most recent demand data. For example, if α = 0.20,

F_t+1 = 0.20D_t + 0.80F_t

which means that our forecast for the next period is based on 20% of recent demand (D_t) and 80% of past demand (in the form of forecast F_t, since F_t is derived from previous demands and fore-casts). If we go to one extreme and let α = 0.0, then

At many service-oriented businesses like fast food restaurants, quality service often equates with fast service. Taco Bell needs an accurate forecast of customer demand at different times during the day in order to provide fast, good quality service during peak demand periods around lunch and dinner.

and the forecast for the next period is the same as the forecast for this period. In other words, the forecast does not reflect the most recent demand at all.

On the other hand, if α = 1.0, then

The closer α is to 1.0, the greater the reaction to the most recent demand.

and we have considered only the most recent data (demand in the present period) and nothing else. Thus, the higher α is, the more sensitive the forecast will be to changes in recent demand, and the smoothing will be less. The closer α is to zero, the greater will be the dampening, or smoothing, effect. As α approaches zero, the forecast will react and adjust more slowly to differences between the actual demand and the forecasted demand. The most commonly used values of α are in the range of 0.01 to 0.50. However, the determination of α is usually judgmental and subjective and is often based on trial-and-error experimentation. An inaccurate estimate of α can limit the usefulness of this forecasting technique. (As α approaches 1.0, the forecast is the same as the naive result.)

Computing an ExponentiallySmoothed Forecast

HiTek Computer Services repairs and services personal computers at its store, and it makes local service calls. It primarily uses part-time State University students as technicians. The company has had steady growth since it started. It purchases generic computer parts in volume at a discount from a variety of sources whenever they see a good deal. Thus, they need a good forecast of demand for repairs so that they will know how many computer component parts to purchase and stock, and how many technicians to hire.

The company has accumulated the demand data shown in the accompanying table for repair and service calls for the past 12 months, from which it wants to consider exponential smoothing forecasts using smoothing constants (α) equal to 0.30 and 0.50.

Demand for Repair and Service Calls

Period	Month	Demand
1	January	37
2	February	40
3	March	41
4	April	37
5	May	45
6	June	50
7	July	43
8	August	47
9	September	56
10	October	52
11	November	55
12	December	54

Solution

To develop the series of forecasts for the data in this table, we will start with period 1 (January) and compute the forecast for period 2 (February) using α = 0.30. The formula for exponential smoothing also requires a forecast for period 1, which we do not have, so we will use the demand for period 1 as both demand and forecast for period 1. (Other ways to determine a starting forecast include averaging the first three or four periods or making a subjective estimate.) Thus, the forecast for February is

The forecast for period 3 is computed similarly:

The remainder of the monthly forecasts are shown in the following table. The final forecast is for period 13, January, and is the forecast of interest to HiTek:

Exponential Smoothing Forecasts, α = 0.30 and α = 0.50

Period	Month	Demand	Forecast, Ft+1
			α = 0.30	α = 0.50
1	January	37	—	—
2	February	40	37.00	37.00
3	March	41	37.90	38.50
4	April	37	38.83	39.75
5	May	45	38.28	38.37
6	June	50	40.29	41.68
7	July	43	43.20	45.84
8	August	47	43.14	44.42
10	October	52	47.81	50.85
11	November	55	49.06	51.42
12	December	54	50.84	53.21
13	January	—	51.79	53.61

This table also includes the forecast values using α = 0.50. Both exponential smoothing forecasts are shown in the following figure together with the actual data.

In Example 12.3, the forecast using the higher smoothing constant, α = 0.50, seems to react more strongly to changes in demand than does the forecast with α = 0.30, although both smooth out the random fluctuations in the forecast. Notice that both forecasts lag behind the actual demand. For example, a pronounced downward change in demand in July is not reflected in the forecast until August. If these changes mark a change in trend (i.e., a long-term upward or downward movement) rather than just a random fluctuation, then the forecast will always lag behind this trend. We can see a general upward trend in service calls throughout the year. Both forecasts tend to be consistently lower than the actual demand; that is, the forecasts lag the trend.

Based on simple observation of the two forecasts in Example 12.3, α = 0.50 seems to be the more accurate of the two in the sense that it seems to follow the actual data more closely. (Later in this chapter we discuss several quantitative methods for determining forecast accuracy.) When demand is relatively stable without any trend, a small value for α is more appropriate to simply smooth out the forecast. When actual demand displays an increasing (or decreasing) trend, as is the case in the figure, a larger value of α is better. It will react more quickly to more recent upward or downward movements in the actual data. In some approaches to exponential smoothing, the accuracy of the forecast is monitored in terms of the difference between the actual values and the forecasted values. If these differences become larger, then α is changed (higher or lower) in an attempt to adapt the forecast to the actual data. However, the exponential smoothing forecast can also be adjusted for the effects of a trend.

In Example 12.3, the final forecast computed was for one month, January. A forecast for two or three months could have been computed by grouping the demand data into the required number of periods and then using these values in the exponential smoothing computations. For example, if a three-month forecast were needed, demand for January, February, and March could be summed and used to compute the average forecast for the next three-month period, and so on, until a final three-month forecast results. Alternatively, if a trend is present, the final period forecast can be used for an extended forecast by adjusting it by a trend factor.

The adjusted exponential smoothing forecast consists of the exponential smoothing forecast with a trend adjustment factor added to it:

• Adjusted exponential smoothing forecast: an exponential smoothing forecast with an adjustment for a trend added to it.

where

T = an exponentially smoothed trend factor

The trend factor is computed much the same as the exponentially smoothed forecast. It is, in effect, a forecast model for trend:

where

The closer β is to 1.0 the stronger the trend is reflected.

β is a value between 0.0 and 1.0. It reflects the weight given to the most recent trend data. β is usually determined subjectively based on the judgment of the forecaster. A high β reflects trend changes more than a low β. It is not uncommon for β to equal α in this method.

Notice that this formula for the trend factor reflects a weighted measure of the increase (or decrease) between the next period forecast, F_{t + 1}, and the current forecast, F_t.

Computing an Adjusted Exponentially Smoothed Forecast

HiTek Computer Services now wants to develop an adjusted exponentially smoothed forecas using the same 12 months of demand shown in the table for Example 12.3. It will use the exponentially smoothed forecast with α = 0.5 computed in Example 12.3 with a smoothing constant for trend, β, of 0.30.

Solution

The formula for the adjusted exponential smoothing forecast requires an initial value for T_t to start the computational process. This initial trend factor is often an estimate determined subjectively or based on past data by the forecaster. In this case, since we have a long sequence of demand data (i.e., 12 months) we will start with the trend T_t equal to zero. By the time the forecast value of interest F₁₃ is computed, we should have a relatively good value for the trend factor.

The adjusted forecast for February, AF₂, is the same as the exponentially smoothed forecast, since the trend computing factor will be zero (i.e., F₁ and F₂ are the same and T₂ = 0). Thus, we compute the adjusted forecast for March, AF₃, as follows, starting with the determination of the trend factor, T₃:

and

This adjusted forecast value for period 3 is shown in the accompanying table, with all other adjusted forecast values for the 12-month period plus the forecast for period 13, computed as follows:

and

Adjusted Exponential Smoothing Forecast Values

Period	Month	Demand	Forecast Ft + 1	Trend Tt+1	Adjusted Forecast AFt+1
1	January	37	37.00	—	—
2	February	40	37.00	0.00	37.00
3	March	41	38.50	0.45	38.95
4	April	37	39.75	0.69	40.44
5	May	45	38.37	0.07	38.44
6	June	50	41.68	1.04	42.73
7	July	43	45.84	1.97	47.82
8	August	47	44.42	0.95	45.37
9	September	56	45.71	1.05	46.76
10	October	52	50.85	2.28	53.13
11	November	55	51.42	1.76	53.19
12	December	54	53.21	1.77	54.98
13	January	—	53.61	1.36	54.96

The adjusted exponentially smoothed forecast values shown in the above table are compared with the exponentially smoothed forecast values and the actual data in the figure on the following page. Notice that the adjusted forecast is consistently higher than the exponentially smoothed forecast and is thus more reflective of the generally increasing trend of the actual data. However, in general, the pattern, or degree of smoothing, is very similar for both forecasts.

Linear regression is a method of forecasting in which a mathematical relationship is developed between demand and some other factor that causes demand behavior. However, when demand displays an obvious trend over time, a least squares regression line, or linear trend line, that relates demand to time, can be used to forecast demand.

• Linear trend line: a linear regression model relating demand to time.

A linear trend line relates a dependent variable, which for our purposes is demand, to one in-dependent variable, time, in the form of a linear equation:

y = a + bx

where

a = intercept (at period 0)

b = slope of the line

x = the time period

y = forecast for demand for period x

These parameters of the linear trend line can be calculated using the least squares formulas for linear regression:

where

Computing a Linear Trend Line

The data for HiTek Computer Services (shown in the table for Example 12.3) appears to follow an increasing linear trend. The company wants to compute a linear trend line to see if it is more accurate than the exponential smoothing and adjusted exponential smoothing forecasts developed in Examples 12.3 and 12.4.

Solution

The values required for the least squares calculations are as follows:

Least Squares Calculations

x (period)	y (demand)	xy	x2
1	37	37	1
2	40	80	4
3	41	123	9
4	37	148	16
5	45	225	25
6	50	300	36
7	43	301	49
8	47	376	64
9	56	504	81
10	52	520	100
11	55	605	121
12	54	648	144
78	557	3867	650

Using these values, we can compute the parameters for the linear trend line as follows:

Therefore, the linear trend line equation is

y =35.2 +1.72x

To calculate a forecast for period 13, let x = 13 in the linear trend line:

The graph on the following page shows the linear trend line compared with the actual data. The trend line appears to reflect closely the actual data—that is, to be a good fit—and would thus be a good forecast model for this problem. However, a disadvantage of the linear trend line is that it will not adjust to a change in the trend, as the exponential smoothing forecast methods will; that is, it is assumed that all future forecasts will follow a straight line. This limits the use of this method to a shorter time frame in which you can be relatively certain that the trend will not change.

A seasonal pattern is a repetitive increase and decrease in demand. Many demand items exhibit seasonal behavior. Clothing sales follow annual seasonal patterns, with demand for warm clothes increasing in the fall and winter and declining in the spring and summer as the demand for cooler clothing increases. Demand for many retail items, including toys, sports equipment, clothing, electronic appliances, hams, turkeys, wine, and fruit, increase during the holiday season. Greeting card demand increases in conjunction with special days such as Valentine's Day and Mother's Day. Seasonal patterns can also occur on a monthly, weekly, or even daily basis. Some restaurants have higher demand in the evening than at lunch or on weekends as opposed to weekdays. Traffic—hence sales—at shopping malls picks up on Friday and Saturday.

Snow skiing is an industry that exhibits several different patterns of demand behavior. It is primarily a seasonal (i.e., winter) industry and over a long period of time the snow skiing industry has exhibited a generally increasing growth trend. Random factors can cause variations, or abrupt peaks and valleys, in demand. For example, demand for skiing products always show a pronounced increase after the Winter Olympics.

• Seasonal factor: adjust for seasonality by multiplying the normal forecast by a seasonal factor.

There are several methods for reflecting seasonal patterns in a time series forecast. We will describe one of the simpler methods using a seasonal factor. A seasonal factor is a numerical value that is multiplied by the normal forecast to get a seasonally adjusted forecast.

One method for developing a demand for seasonal factors is to divide the demand for each seasonal period by total annual demand, according to the following formula:

The resulting seasonal factors between 0 and 1.0 are, in effect, the portion of total annual demand assigned to each season. These seasonal factors are multiplied by the annual forecasted demand to yield adjusted forecasts for each season.

Computing a Forecast with Seasonal Adjustments

Wishbone Farms grows turkeys to sell to a meat-processing company throughout the year. However, its peak season is obviously during the fourth quarter of the year, from October to December. Wishbone Farms has experienced the demand for turkeys for the past three years shown in the following table:

Demand for Turkeys at Wishbone Farms

Year	Demand (1000s) per Quarter
	1	2	3	4	Total
2008	12.6	8.6	6.3	17.5	45.0
2009	14.1	10.3	7.5	18.2	50.1
2010	15.3	10.6	8.1	19.6	53.6
Total	42.0	29.5	21.9	55.3	148.7

Solution

Because we have three years of demand data, we can compute the seasonal factors by dividing total quarterly demand for the three years by total demand across all three years:

Next, we want to multiply the forecasted demand for the next year, 2008, by each of the seasonal factors to get the forecasted demand for each quarter. To accomplish this, we need a demand forecast for 2011. In this case, since the demand data in the table seem to exhibit a generally increasing trend, we compute a linear trend line for the three years of data in the table to get a rough forecast estimate:

Thus, the forecast for 2011 is 58.17, or 58,170 turkeys.

Using this annual forecast of demand, we find that the seasonally adjusted forecasts, SF_i, for 2011 are

Comparing these quarterly forecasts with the actual demand values in the table, we see that they would seem to be relatively good forecast estimates, reflecting both the seasonal variations in the data and the general upward trend.

Turkeys are an example of a product with a long-term trend for increasing demand with a seasonal pattern. Turkey sales show a distinct seasonal pattern by increasing markedly during the Thanksgiving holiday season. For example, turkey sales are lowest from January to May, they begin to rise in June and July and peak in August when distributors begin to build up their inventory of frozen turkeys for increased sales in November. Sales remain high for September, October, and November and then begin to decline in December and January.

The mean absolute deviation, or MAD, is one of the most popular and simplest to use measures of forecast error. MAD is an average of the difference between the forecast and actual demand, as computed by the following formula:

• mean absolute deviation (MAD): the average, absolute difference between the forecast and demand.

where

t = the period number

D_t = demand in period t

F_t = the forecast for period t

n = the total number of periods

|| = absolute value

Measuring Forecasting Accuracy with MAD

In Examples 12.3, 12.4, and 12.5, forecasts were developed using exponential smoothing, (α = 0.30 and α = 0.50), adjusted exponential smoothing (α = 0.50, β = 0.30), and a linear trend line, respectively, for the demand data for HiTek Computer Services. The company wants to compare the accuracy of these different forecasts using MAD.

Solution

We will compute MAD for all four forecasts; however, we will present the computational detail for the exponential smoothing forecast with α = 0.30 only. The following table shows the values necessary to compute MAD for the exponential smoothing forecast:

Computational Values for MAD

Period	Demand, Dt	Forecast Ft (αi = 0.30)	Error (et) (Dt – Fi)	Dt – Ft
1	37	37.00	—	—
2	40	37.00	3.00	3.00
3	41	37.90	3.10	3.10
4	37	38.83	−1.83	1.83
5	45	38.28	6.72	6.72
6	50	40.29	9.69	9.69
7	43	43.20	−0.20	0.20
8	47	43.14	3.86	3.86
9	56	44.30	11.70	11.70
10	52	47.81	4.19	4.19
11	55	49.06	5.94	5.94
12	54	50.84	3.15	3.15
	557		49.32	53.38

The computation of MAD will be based on 11 periods, periods 2 through 12, excluding the initial demand and forecast values for period 1 since they both equal 37.

Using the data in the table, MAD is computed as

The smaller the value of MAD, the more accurate the forecast, although viewed alone, MAD is difficult to assess. In this example, the data values were relatively small, and the MAD value of 4.85 should be judged accordingly. Overall, it would seem to be a "low" value; that is, the forecast appears to be relatively accurate. However, if the magnitude of the data values were in the thousands or millions, then a MAD value of a similar magnitude might not be bad either. The point is, you cannot compare a MAD value of 4.85 with a MAD value of 485 and say the former is good and the latter is bad; they depend to a certain extent on the relative magnitude of the data.

The lower the value of MAD, relative to the magnitude of the data, the more accurate the forecast.

One benefit of MAD is to compare the accuracy of several different forecasting techniques, as we are doing in this example. The MAD values for the remaining forecasts are as follows:

Since the linear trend line has the lowest MAD value of 2.29, it would seem to be the most accurate, although it does not appear to be significantly better than the adjusted exponential smoothing forecast. Furthermore, we can deduce from these MAD values that increasing α from 0.30 to 0.50 enhanced the accuracy of the exponentially smoothed forecast. The adjusted forecast is even more accurate.

The mean absolute percent deviation (MAPD) measures the absolute error as a percentage of demand rather than per period. As a result, it eliminates the problem of interpreting the measure of accuracy relative to the magnitude of the demand and forecast values, as MAD does. The mean absolute percent deviation is computed according to the following formula:

• mean absolute percent deviation (MAPD): the absolute error as a percentage of demand.

Using the data from the table in Example 12.7 for the exponential smoothing forecast (α = 0.30) for HiTek Computer Services,

A lower percent deviation implies a more accurate forecast. The MAPD values for our other three forecasts are

Cumulative error is computed simply by summing the forecast errors, as shown in the following formula.

• Cumulative error: the sum of the forecast errors.

A large positive value indicates that the forecast is probably consistently lower than the actual demand, or is biased low. A large negative value implies that the forecast is consistently higher than actual demand, or is biased high. Also, when the errors for each period are scrutinized, a preponderance of positive values shows the forecast is consistently less than the actual value and vice versa.

The cumulative error for the exponential smoothing forecast (α = 0.30) for HiTek Computer Services can be read directly from the table in Example 12.7; it is simply the sum of the values in the "Error" column:

Large +E indicates forecast is biased low; large −E, forecast is biased high.

This large positive error for cumulative error, plus the fact that the individual errors for all but two of the periods in the table are positive, indicates that this forecast is consistently below the actual demand. A quick glance back at the plot of the exponential smoothing (α = 0.30) forecast in Example 12.3 visually verifies this result.

The cumulative error for the other forecasts are

Exponential smoothing(α = 0.50): E = 33.21

Adjusted exponential smoothing(α = 0.50, β = 0.30): E = 21.14

We did not show the cumulative error for the linear trend line. E will always be near zero for the linear trend line.

A measure closely related to cumulative error is the average error, or bias. It is computed by averaging the cumulative error over the number of time periods:

• Average error: the per-period average of cumulative error.

For example, the average error for the exponential smoothing forecast (α = 0.30) is computed as follows. (Notice a value of 11 was used for n, since we used actual demand for the first-period forecast, resulting in no error, that is, D₁ = F₁ = 37.)

The average error is interpreted similarly to the cumulative error. A positive value indicates low bias, and a negative value indicates high bias. A value close to zero implies a lack of bias.

Table 12.1 summarizes the measures of forecast accuracy we have discussed in this section for the four example forecasts we developed in Examples 12.3, 12.4, and 12.5 for HiTek Computer Services. The results are consistent for all four forecasts, indicating that for the HiTek Computer Services example data, a larger value of is preferable for the exponential smoothing forecast. The adjusted forecast is more accurate than the exponential smoothing forecasts, and the linear trend is more accurate than all the others. Although these results are for specific examples, they indicate how the different forecast measures for accuracy can be used to adjust a forecasting method or select the best method.

There are several ways to monitor forecast error over time to make sure that the forecast is performing correctly—that is, the forecast is in control. Forecasts can go "out of control" and start providing inaccurate forecasts for several reasons, including a change in trend, the unanticipated appearance of a cycle, or an irregular variation such as unseasonable weather, a promotional campaign, new competition, or a political event that distracts consumers.

A tracking signal indicates if the forecast is consistently biased high or low. It is computed by dividing the cumulative error by MAD, according to the formula

• Tracking signal: monitors the forecast to see if it is biased high or low.

The tracking signal is recomputed each period, with updated, "running" values of cumulative error and MAD. The movement of the tracking signal is compared to control limits; as long as the tracking signal is within these limits, the forecast is in control.

Typically, forecast errors are normally distributed, which results in the following relationship between MAD and the standard deviation of the distribution of error, σ:

This enables us to establish statistical control limits for the tracking signal that corresponds to the more familiar normal distribution. For example, statistical control limits of ±3 standard deviations, corresponding to 99.7% of the errors, would translate to ±3.75 MADs; that is, 3σ ÷ 0.8 = 3.75 MADs. Control limits of ±2 to ±5 MADs are used most frequently.

Developing a Tracking Signal

In Example 12.7, the mean absolute deviation was computed for the exponential smoothing forecast (α = 0.30) for HiTek Computer Services. Using a tracking signal, monitor the forecast accuracy using control limits of ±3 MADs.

Solution

To use the tracking signal, we must recompute MAD each period as the cumulative error is computed.

Using MAD = 3.00, we find that the tracking signal for period 2 is

The tracking signal for period 3 is

The remaining tracking signal values are shown in the following table:

Tracking Signal Values

Period	Demand,Dt	Forecast,Ft	Error,Dt-Fi	E = Σ(Dt-Fi)	MAD	Tracking Signal
1	37	37.00	—	—	—	—
2	40	37.00	3.00	3.00	3.00	1.00
3	41	37.90	3.10	6.10	3.05	2.00
4	37	38.83	−1.83	4.27	2.64	1.62
5	45	38.28	6.72	10.99	3.66	3.00
6	50	40.29	9.69	20.68	4.87	4.25
7	43	43.20	−0.20	20.48	4.09	5.01
8	47	43.14	3.86	24.34	4.06	6.00
9	56	44.30	11.70	36.04	5.01	7.19
10	52	47.81	4.19	40.23	4.92	8.18
11	55	49.06	5.94	46.17	5.02	9.20
12	54	50.84	3.15	49.32	4.85	10.17

The tracking signal values in the table above move outside ±3 MAD control limits (i.e., ±3.00) in period 5 and continue increasing. This suggests that the forecast is not performing accurately or, more precisely, is consistently biased low (i.e., actual demand consistently exceeds the forecast). This is illustrated in the graph on the previous page. Notice that the tracking signal moves beyond the upper limit of 3 following period 5 and continues to rise. For the sake of comparison, the tracking signal for the linear trend line forecast computed in Example 12.5 is also plotted on this graph. Notice that it remains within the limits (touching the upper limit in period 3), indicating a lack of consistent bias.

Another method for monitoring forecast error is statistical control charts. For example, ±3σ control limits would reflect 99.7% of the forecast errors (assuming they are normally distributed). The sample standard deviation, σ, is computed as

This formula without the square root is known as the mean squared error (MSE), and it is sometimes used as a measure of forecast error. It reacts to forecast error much like MAD does. (For our Example 12.8, MSE = 37.57.)

• Mean squared error (MSE): the average of the squared forecast errors.

Forecast Error with Statistical Control Charts

Using the same example for the exponential smoothing forecast (α = 0.30) for HiTek Computer Services, as in Example 12.8, we compute the standard deviation as

Using this value of σ we can compute statistical control limits for forecast errors for our exponential smoothing forecast (α = 0.30) example for HiTek Computer Services. Plus or minus 3σ control limits, reflecting 99.7% of the forecast errors, gives ±3(6.13), or ±18.39. Although it can be observed from the table in Example 12.8 that all the error values are within the control limits, we can still detect that most of the errors are positive, indicating a low bias in the forecast estimates. This is illustrated in the following graph of the control chart with the errors plotted on it.

ALONG THE SUPPLY CHAIN

Forecasting Market Demand at NBC

NBC Universal, a subsidiary of General Electric Company, owns and operates the most profitable television network in the United States with revenues of over $14 billion. Over 60% of these revenues were generated by on-air advertising time on its television networks and stations. The major television networks announce their new programming schedules for the upcoming season (which starts in late September) in mid-May. NBC begins selling their advertising time very soon after the new schedule is announced in May, and 60 to 80% of their airtime inventory is sold in the following two- to three-week period (to approximately 400 advertisers), known as the upfront market. Immediately following the announcement of their new season schedule in May, NBC forecasts ratings and estimates market demand for their shows. The ratings forecasts are estimates of the number of people in several demographic groups who are expected to watch each airing of the shows in the schedule for the entire year, which are, in turn, based on such factors as a show's strength, historical time slot ratings, and the ratings performance of adjacent shows. Total market demand depends primarily on the strength of the economy and the expected performance of the network's schedule. Based on this ratings forecast and market demand, NBC develops pricing strategies and sets the prices for commercials on their shows.

Forecasting the upfront market has always been a challenging process for NBC. In the past the network used historical patterns, expert knowledge, and intuition to forecast demand; then later it used time-series forecasting models based on historical demand. However, these models were unsatisfactory because of the unique nature of NBC's demand population of advertisers. NBC ultimately developed a unique approach to forecasting its upfront market for demand, which includes a combination of the Delphi technique and a "grass roots" forecasting approach. The Delphi technique seeks to develop a consensus (or at least a compromise) forecast from among a group of experts, while a grass roots approach to forecasting asks the individuals closest to the final customer, such as salespeople about the customer's purchasing plans. In their forecasting approach NBC used its (over 100) account executives, who interact closely with the network's (over 400) advertisers, to build a knowledge base in order to estimate individual advertiser's demand, aggregate this demand into a total demand forecast, and then continuously and iteratively update demand estimates based on the account executives expertise. Previous forecasting methods resulted in forecast errors of 5 to 12%, whereas this new forecasting process resulted in a forecast error of only 2.8%, the most accurate forecast NBC had ever achieved for its upfront market.

NBC uses various methods to forecast television ratings that indicate the number of viewers in different demographic groups that will watch their shows, which the network then uses to establish pricing strategies for the shows' commercials it will charge advertisers for.

Why do you think NBC's forecasting approach was more effective than a traditional time series forecasting model based on historical demand?

Source: S. Bollapragada, S. Gupta, B. Hurwitz, P. Miles and R. Tyagi "NBC–Universal Uses a Novel Qualitative Forecasting Technique to Predict Advertising Demand," Interfaces, vol. 38, (2: March–April, 2008), pp. 103–111.

OM Tools has modules for all of the forecasting methods presented in this chapter. As an example, Exhibit 12.6 shows the OM Tools spreadsheet for the exponential smoothing model (α = 0.30) in Example 12.3.

Figure E12.6. Exhibit 12.6

Linear regression is a mathematical technique that relates one variable, called an independent variable, to another, the dependent variable, in the form of an equation for a straight line. A linear equation has the following general form:

y = a + bx

where

y = the dependent variable

a = the intercept

b = the slope of the line

x = the independent variable

Because we want to use linear regression as a forecasting model for demand, the dependent variable, y, represents demand, and x is an independent variable that causes demand to behave in a linear manner.

To develop the linear equation, the slope, b, and the intercept, a, must first be computed using the following least squares formulas:

where

• Linear regression: a mathematical technique that relates a dependent variable to an independent variable in the form of a linear equation.

Linear regression relates demand (dependent variable) to an independent variable.

Developing a Linear Regression Forecast

The State University athletic department wants to develop its budget for the coming year using a forecast for football attendance. Football attendance accounts for the largest portion of its revenues, and the athletic director believes attendance is directly related to the number of wins by the team. The business manager has accumulated total annual average attendance figures for the past eight years.

Wins	Attendance	Wins	Attendance
4	36,300	6	44,000
6	40,100	7	45,600
6	41,200	5	39,000
8	53,000	7	47,500

Given the number of returning starters and the strength of the schedule, the athletic director believes the team will win at least seven games next year. Develop a simple regression equation for this data to forecast attendance for this level of success.

Solution

The computations necessary to compute a and b using the least squares formulas are summarized in the accompanying table. (Note that y is given in 1000s to make manual computation easier.)

Least Squares Computations

x (Wins)	y (Attendance, 1000s)	xy	x2
4	36.3	145.2	16
6	40.1	240.6	36
6	41.2	247.2	36
8	53.0	424.0	64
6	44.0	264.0	36
7	45.6	319.2	49
5	39.0	195.0	25
7	47.5	332.5	49
49	346.9	2167.7	311

Substituting these values for a and b into the linear equation line, we have

y = 18.46 + 4.06x

Thus, for x = 7 (wins), the forecast for attendance is

The data points with the regression line are shown in the following figure. Observing the regression line relative to the data points, it would appear that the data follow a distinct upward linear trend, which would indicate that the forecast should be relatively accurate. In fact, the MAD value for this forecasting model is 1.41, which suggests an accurate forecast.

Correlation in a linear regression equation is a measure of the strength of the relationship between the independent and dependent variables. The formula for the correlation coefficient is

• Correlation: a measure of the strength of the relationship between independent and dependent variables.

The value of r varies between −1.00 and +1.00, with a value of +1.00 indicating a strong linear relationship between the variables. If r = 1.00, then an increase in the independent variable will result in a corresponding linear increase in the dependent variable. If r = −1.00, an increase in the dependent variable will result in a linear decrease in the dependent variable. A value of r near zero implies that there is little or no linear relationship between variables.

We can determine the correlation coefficient for the linear regression equation determined in Example 12.9 by substituting most of the terms calculated for the least squares formula (except for Σy²) into the formula for r:

This value for the correlation coefficient is very close to 1.00, indicating a strong linear relationship between the number of wins and home attendance.

Another measure of the strength of the relationship between the variables in a linear regression equation is the coefficient of determination. It is computed by squaring the value of r. It indicates the percentage of the variation in the dependent variable that is a result of the behavior of the independent variable. For our example, r = 0.947; thus, the coefficient of determination is

This value for the coefficient of determination means that 89.7% of the amount of variation in attendance can be attributed to the number of wins by the team (with the remaining 10.3% due to other unexplained factors, such as weather, a good or poor start, or publicity). A value of 1.00 (or 100%) would indicate that attendance depends totally on wins. However, since 10.3% of the variation is a result of other factors, some amount of forecast error can be expected.

• Coefficient of determination: the percentage of the variation in the dependent variable that results from the independent variable.

The development of the simple linear regression equation and the correlation coefficient for our example was not too difficult because the amount of data was relatively small. However, manual computation of the components of simple linear regression equations can become very time-consuming and cumbersome as the amount of data increases. Excel has the capability of performing linear regression.

Exhibit 12.7 shows a spreadsheet set up to develop the linear regression forecast for Example 12.10 for the State University Athletic Department. Notice that Excel computes the slope directly with the formula " =SLOPE(B5:B12, A5:A12)" entered in cell E7 and shown on the formula bar at the top of the spreadsheet. The formula for the intercept in cell E6 is "=INTERCEPT(B5:B12, A5:A12)." The values for the slope and intercept are subsequently entered into cells E9 and G9 to form the linear regression equation. The correlation coefficient in cell E13 is computed using the formula "=CORREL(B5:B12, A5:A12)." Although it is not shown on the spreadsheet, the coefficient of determination (r²) could be computed using the formula "=RSQ(B5:B12, A5:A12)."

Figure E12.7. Exhibit 12.7

Figure E12.8. Exhibit 12.8

A linear regression forecast can also be developed directly with Excel using the "Data Analysis" option from the Tools menu we accessed previously to develop an exponentially smoothed forecast. Exhibit 12.8 shows the selection of "Regression" from the Data Analysis menu, and Exhibit 12.9 shows the Regression window. We first enter the cells from Exhibit 12.7 that include the y values (for attendance), B5:B12. Next enter the x value cells, A5:A12. The output range is the location on the spreadsheet where you want to put the output results. This range needs to be large (18 cells by 9 cells) and not overlap with anything else on the spreadsheet. Clicking on "OK" will result in the spreadsheet shown in Exhibit 12.10. (Note that the "Summary Output" has been slightly moved around so that all the results could be included on the screen in Exhibit 12.9).

The "Summary Output" in Exhibit 12.10 provides a large amount of statistical information, the explanation and use of which are beyond the scope of this book. The essential items that we are interested in are the intercept and slope (labeled "X Variable 1") in the "Coefficients" column at the bottom of the spreadsheet, and the "Multiple R" (or correlation coefficient) value shown under "Regression Statistics."

Another causal method of forecasting is multiple regression, a more powerful extension of linear regression. Linear regression relates demand to one other independent variable, whereas multiple regression reflects the relationship between a dependent variable and two or more independent variables. A multiple regression model has the following general form:

• Multiple regression: a relationship of demand to two or more independent variables.

Figure E12.9. Exhibit 12.9

Figure E12.10. Exhibit 12.10

where

β0 = the intercept

β₁,..., β_k = parameters representing the contribution of the independent variables

x₁, . . ., x_k = independent variables

For example, the demand for new housing (y) in a region might be a function of several in-dependent variables, including interest rates, population, housing prices, and personal income. Development and computation of the multiple regression equation, including the compilation of data, is more complex than linear regression. The only means for forecasting using multiple regression is with a computer.

Figure E12.11. Exhibit 12.11

A problem often encountered in multiple regression is multicollinearity, or the amount of "overlapping" information about the dependent variable that is provided by several independent variables. This problem usually occurs when the independent variables are highly correlated, as in this example, in which wins and promotional expenditures are both positively correlated (i.e., more wins coincide with higher promotional expenditures and vice versa). (Possibly the athletic department increased promotional expenditures when it thought it would have a better team that would achieve more wins.) The topic of multicollinearity and how to cope with it is beyond the scope of this book and this brief section on multiple regression; however, most statistics textbooks discuss this topic in detail.

Figure E12.12. Exhibit 12.12