From bean to cup how starbucks transformed its supply chain

Operations Forecasting

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18 FORECASTING Learning Objectives LO18–1 Understand how forecasting is essential to supply chain planning. LO18–2 Evaluate demand using quantitative forecasting models. LO18–3 Apply qualitative techniques to forecast demand. LO18–4 Apply collaborative techniques to forecast demand.

FROM BEAN TO CUP: STARBUCKS GLOBAL SUPPLY CHAIN CHALLENGE Starbucks Corporation is the largest coffeehouse company in the world with over 17,000 stores in more than 50 countries. The company serves some 50 million customers each week. Forecasting demand for a Starbucks is an amazing challenge. The product line goes well beyond drip-brewed coffee sold on demand in the stores. It includes espresso-based hot drinks, other hot and cold drinks, coffee beans, salads, hot and cold sandwiches and panini, pastries, snacks, and items such as mugs and tumblers. Through its entertainment division and the Hear Music brand, the company also markets books, music, and videos. Many of the company’s products are seasonal or specific to the locality of the store. Starbucks-branded ice cream and coffee are also offered at grocery stores around the world.

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STARBUCKS CAFE IN BUR JUMAN CENTER SHOPPING MALL, DUBAI, UNITED ARAB EMIRATES. The creation of a single, global logistics system was important for Starbucks because of its far-flung supply chain. The company generally brings coffee beans from Latin America, Africa, and Asia to the United States and Europe in ocean containers. From the port of entry, the “green” (unroasted) beans are trucked to six storage sites, either at a roasting plant or nearby. After the beans are roasted and packaged, the finished product is trucked to regional distribution centers, which range from 200,000 to 300,000 square feet in

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443 size. Starbucks runs five regional distribution centers (DCs) in the United States. It has two DCs in Europe and two more in Asia. Coffee, however, is only one of the many products held at these warehouses. They also handle other items required by Starbucks retail outlets, everything from furniture to cappuccino mix.

In the Analytics Exercise at the end of the chapter, we consider the challenging demand forecasting problem that Starbucks must solve to be able to successfully run this complex supply chain.

Source: Based on James A. Cooke, “From bean to cup: How Starbucks transformed its supply chain,” CSCUPs Supply Chain Quarterly, Fourth Quarter, 2010.

LO18–1 Understand how forecasting is essential to supply chain planning.

FORECASTING IN OPERATIONS AND SUPPLY CHAIN MANAGEMENT Forecasts are vital to every business organization and for every significant management decision. Forecasting is the basis of corporate planning and control. In the functional areas of finance and accounting, forecasts provide the basis for budgetary planning and cost control. Marketing relies on sales forecasting to plan new products, compensate sales personnel, and make other key decisions. Production and operations personnel use forecasts to make periodic decisions involving supplier selection, process selection, capacity planning, and facility layout, as well as for continual decisions about purchasing, production planning, scheduling, and inventory.

In considering what forecasting approach to use, it is important to consider the purpose of the forecast. Some forecasts are for very high-level demand analysis. What do we expect the demand to be for a group of products over the next year, for example? Some forecasts are used to help set the strategy of how, in an aggregate sense, we will meet demand. We will call these strategic forecasts. Relative to the material in the book, strategic forecasts are most appropriate when making decisions related to overall strategy (Chapter 2), capacity (Chapter 5), manufacturing process design (Chapter 7), service process design (Chapter 9), location and distribution design (Chapter 15), sourcing (Chapter 16), and in sales and operations planning (Chapter 19). These all involve medium and long-term decisions that relate to how demand will be met strategically.

Strategic forecasts Medium and long-term forecasts that are used for decisions related to strategy and aggregate demand.

Forecasts are also needed to determine how a firm operates processes on a day-to-day basis. For example, when should the inventory for an item be replenished, or how much production should we schedule for an item next week? These are tactical forecasts where the goal is to estimate demand in the relatively short term, a few weeks or months. These forecasts are important to ensure that in the short term we are able to meet customer lead time expectations and other criteria related to the availability of our products and services.

Tactical forecasts Short-term forecasts used for making day-to-day decisions related to meeting demand.

In Chapter 7, the concept of decoupling points is discussed. These are points within the supply chain where inventory is positioned to allow processes or entities in the supply chain to operate independently. For example, if a product is stocked at a retailer, the customer pulls the item from the shelf and the manufacturer never sees a customer order. Inventory acts as a buffer to separate the customer from the manufacturing process. Selection of decoupling points is a strategic decision that determines customer lead times and can greatly impact inventory investment. The closer this point is to the customer, the quicker the customer can be served. Typically, a trade-off is involved where quicker response to customer demand comes at the expense of greater inventory investment because finished goods inventory is more expensive than raw material inventory.

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Strategy Forecasting is needed at these decoupling points to set appropriate inventory levels for these buffers. The actual setting of these

levels is the topic of Chapter 20, “Inventory Management,” but an essential input into those decisions is a forecast of expected demand and the expected error associated with that demand. If, for example, we are able to forecast demand very accurately, then inventory levels can be set precisely to expected customer demand. On the other hand, if predicting short-term demand is difficult, then extra inventory to cover this uncertainty will be needed.

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FORECASTING IS CRITICAL IN DETERMINING HOW MUCH INVENTORY TO KEEP TO MEET CUSTOMER NEEDS.

The same is true relative to service settings where inventory is not used to buffer demand. Here capacity availability relative to expected demand is the issue. If we can predict demand in a service setting very accurately, then tactically all we need to do is ensure that we have the appropriate capacity in the short term. When demand is not predictable, then excess capacity may be needed if servicing customers quickly is important.

Bear in mind that a perfect forecast is virtually impossible. Too many factors in the business environment cannot be predicted with certainty. Therefore, rather than search for the perfect forecast, it is far more important to establish the practice of continual review of forecasts and to learn to live with inaccurate forecasts. This is not to say that we should not try to improve the forecasting model or methodology or even to try to influence demand in a way that reduces demand uncertainty. When forecasting, a good strategy is to use two or three methods and look at them for the commonsense view. Will expected changes in the general economy affect the forecast? Are there changes in our customers’ behaviors that will impact demand that are not being captured by our current approaches? In this chapter, we look at both qualitative techniques that use managerial judgment and also quantitative techniques that rely on mathematical models. It is our view that combining these techniques is essential to a good forecasting process that is appropriate to the decisions being made.

LO18–2 Evaluate demand using quantitative forecasting models.

QUANTITATIVE FORECASTING MODELS Forecasting can be classified into four basic types: qualitative, time series analysis, causal relationships, and simulation. Qualitative techniques are covered later in the chapter. Time series analysis, the primary focus of this chapter, is based on the idea that data relating to past demand can be used to predict future demand. Past data may include several components, such as trend, seasonal, or cyclical influences, and are described in the following section. Causal forecasting, which we discuss using the linear regression technique, assumes that demand is related to some underlying factor or factors in the environment. Simulation models allow the

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forecaster to run through a range of assumptions about the condition of the forecast. In this chapter we focus on qualitative and time series techniques since these are most often used in supply chain planning and control.

Time series analysis A forecast in which past demand data is used to predict future demand.

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Components of Demand In most cases, demand for products or services can be broken down into six components: average demand for the period, a trend, seasonal element, cyclical elements, random variation, and autocorrelation. Exhibit 18.1 illustrates a demand over a four-year period, showing the average, trend, and seasonal components and randomness around the smoothed demand curve.

Cyclical factors are more difficult to determine because the time span may be unknown or the cause of the cycle may not be considered. Cyclical influence on demand may come from such occurrences as political elections, war, economic conditions, or sociological pressures.

Random variations are caused by chance events. Statistically, when all the known causes for demand (average, trend, seasonal, cyclical, and autocorrelative) are subtracted from total demand, what remains is the unexplained portion of demand. If we cannot identify the cause of this remainder, it is assumed to be purely random chance.

Autocorrelation denotes the persistence of occurrence. More specifically, the value expected at any point is highly correlated with its own past values. In waiting line theory, the length of a waiting line is highly autocorrelated. That is, if a line is relatively long at one time, then shortly after that time, we would expect the line still to be long.

When demand is random, it may vary widely from one week to another. Where high autocorrelation exists, the rate of change in demand is not expected to change very much from one week to the next.

Trend lines are the usual starting point in developing a forecast. These trend lines are then adjusted for seasonal effects, cyclical elements, and any other expected events that may influence the final forecast. Exhibit 18.2 shows four of the most common types of trends. A linear trend is obviously a straight continuous relationship. An S-curve is typical of a product growth and maturity cycle. The most important point in the S-curve is where the trend changes from slow growth to fast growth or from fast to slow. An asymptotic trend starts with the highest demand growth at the beginning but then tapers off. Such a curve could happen when a firm enters an existing market with the objective of saturating and capturing a large share of the market. An exponential curve is common in products with explosive growth. The exponential trend suggests that sales will grow at an ever-increasing rate—an assumption that may not be safe to make.

KEY IDEA

Remember that we are illustrating long-term patterns so they should be considered when making long-term forecasts. When making short-term forecasts, these patterns are often not so strong.

A widely used forecasting method plots data and then searches for the curve pattern (such as linear, S-curve, asymptotic, or exponential) that fits best. The attractiveness of this method is that because the mathematics for the curve are known, solving for values for future time periods is easy.

Sometimes our data do not seem to fit any standard curve. This may be due to several causes essentially beating the data from several directions at the same time. For these cases, a simplistic but often effective forecast can be obtained by simply plotting data.

exhibit 18.1 Historical Product Demand Consisting of a Growth Trend and Seasonal Demand

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For the Excel template, visit www.mhhe.com/jacobs14e.

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exhibit 18.2 Common Types of Trends

Time Series Analysis Time series forecasting models try to predict the future based on past data. For example, sales figures collected for the past six weeks can be used to forecast sales for the seventh week. Quarterly sales figures collected for the past several years can be used to forecast future quarters. Even though both examples contain sales, different forecasting time series models would likely be used.

Exhibit 18.3 shows the time series models discussed in the chapter and some of their characteristics. Terms such as short, medium, and long are relative to the context in which they are used. However, in business forecasting short term usually refers to under three months; medium term, three months to two years; and long term, greater than two years. We would generally use short-

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1. 2. 3.

term forecasts for tactical decisions such as replenishing inventory or scheduling employees in the near term, and medium-term forecasts for planning a strategy for meeting demand over the next six months to a year and a half. In general, the short-term models compensate for random variation and adjust for short-term changes (such as consumers’ responses to a new product). They are especially good for measuring the current variability in demand, which is useful for setting safety stock levels or estimating peak loads in a service setting. Medium-term forecasts are useful for capturing seasonal effects, and long-term models detect general trends and are especially useful in identifying major turning points.

Which forecasting model a firm should choose depends on:

Time horizon to forecast Data availability Accuracy required

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4. 5.

447

exhibit 18.3 A Guide to Selecting an Appropriate Forecasting Method

Size of forecasting budget Availability of qualified personnel

To view a tutorial on Forecasting, visit www.mhhe.com/jacobs14e_tutorial_ch18.

In selecting a forecasting model, there are other issues such as the firm’s degree of flexibility. (The greater the ability to react quickly to changes, the less accurate the forecast needs to be.) Another item is the consequence of a bad forecast. If a large capital investment decision is to be based on a forecast, it should be a good forecast.

Simple Moving Average When demand for a product is neither growing nor declining rapidly, and if it does not have seasonal characteristics, a moving average can be useful in removing the random fluctuations for forecasting. The idea here is to simply calculate the average demand over the most recent periods. Each time a new forecast is made, the oldest period is discarded in the average and the newest period included. Thus, if we want to forecast June with a five-month moving average, we can take the average of January, February, March, April, and May. When June passes, the forecast for July would be the average of February, March, April, May, and June. An example using weekly demand is shown in Exhibit 18.4. Here, 3-week and 9-week moving average forecasts are calculated. Notice how the forecast is shown in the period following the data used. The 3-week moving average for week 4 uses actual demand from weeks 1, 2, and 3.

Moving Average A forecast based on average past demand.

Selecting the period length should be dependent on how the forecast is going to be used. For example, in the case of a medium- term forecast of demand for planning a budget, monthly time periods might be more appropriate, whereas, if the forecast were being used for a short-term decision related to replenishing inventory, a weekly forecast might be more appropriate. Although it is important to select the best period for the moving average, the number of periods to use in the forecast can also have a major impact on the accuracy of the forecast. As the moving average period becomes shorter, and fewer periods are used, and there is more oscillation, there is a closer following of the trend. Conversely, a longer time span gives a smoother response, but lags the trend. The formula for a simple moving average is

+ + + ⋯ +

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[18.1]Ft =

where

Ft= Forecast for the coming period n= Number of periods to be averaged

At21= Actual occurrence in the past period At−2, At−3, and At−n= Actual occurrences two periods ago, three periods ago, and so on, up to n periods ago

At−1 + At−2 + At−3 + ⋯ + At−n n

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exhibit 18.4 Forecast Demand Based on a Three- and a Nine-Week Simple Moving Average

For the Excel template, visit www.mhhe.com/jacobs14e.

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Analytics

A plot of the data in Exhibit 18.4 shows the effects of using different numbers of periods in the moving average. We see that the growth trend levels off at about the 23rd week. The three-week moving average responds better in following this change than the nine-week average, although overall, the nine-week average is smoother.

The main disadvantage in calculating a moving average is that all individual elements must be carried as data because a new forecast period involves adding new data and dropping the earliest data. For a three- or six-period moving average, this is not too severe. But plotting a 60-day moving average for the usage of each of 100,000 items in inventory would involve a significant amount of data.

Weighted Moving Average Whereas the simple moving average assigns equal importance to each component of the moving average database, a weighted moving average allows any weights to be placed on each element, provided, of course, that the sum of all weights equals 1. For example, a department store may find that, in a four-month period, the best forecast is derived by using 40 percent of the actual sales for the most recent month, 30 percent of two months ago, 20 percent of three months ago, and 10 percent of four months ago. If actual sales experience was

Weighted moving average A forecast made with past data where more recent data is given more significance than older data.

MONTH 1 MONTH 2 MONTH 3 MONTH 4 MONTH 5

100 90 105 95 ?

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[18.2]

449 the forecast for month 5 would be

F5 = 0.40(95) + 0.30(105) + 0.20(90) + 0.10(100) = 38 + 31.5 + 18 + 10 = 97.5

The formula for a weighted moving average is

Ft = W1At−1 + W2At−2 + … + Wn At−n

where

W1 = Weight to be given to the actual occurrence for the period t − 1 W2 = Weight to be given to the actual occurrence for the period t − 2 Wn = Weight to be given to the actual occurrence for the period t − n n = Total number of prior periods in the forecast

Although many periods may be ignored (that is, their weights are zero) and the weighting scheme may be in any order (for example, more distant data may have greater weights than more recent data), the sum of all the weights must equal 1.

n

∑ i = 1

Wi = 1

Suppose sales for month 5 actually turned out to be 110. Then the forecast for month 6 would be

F6 = 0.40(110) + 0.30(95) + 0.20(105) + 0.10(90) = 44 + 28.5 + 21 + 9 = 102.5

Experience and trial and error are the simplest ways to choose weights. As a general rule, the most recent past is the most important indicator of what to expect in the future, and, therefore, it should get higher weighting. The past month’s revenue or plant capacity, for example, would be a better estimate for the coming month than the revenue or plant capacity of several months ago.

However, if the data are seasonal, for example, weights should be established accordingly. Bathing suit sales in July of last year should be weighted more heavily than bathing suit sales in December (in the Northern Hemisphere).

The weighted moving average has a definite advantage over the simple moving average in being able to vary the effects of past data. However, it is more inconvenient and costly to use than the exponential smoothing method, which we examine next.

Exponential Smoothing In the previous methods of forecasting (simple and weighted moving averages), the major drawback is the need to continually carry a large amount of historical data. (This is also true for regression analysis techniques, which we soon will cover.) As each new piece of data is added in these methods, the oldest observation is dropped and the new forecast is calculated. In many applications (perhaps in most), the most recent occurrences are more indicative of the future than those in the more distant past. If this premise is valid—that the importance of data diminishes as the past becomes more distant—then exponential smoothing may be the most logical and easiest method to use.

Exponential smoothing A time series forecasting technique using weights that decrease exponentially (1 – α) for each past period.

Exponential smoothing is the most used of all forecasting techniques. It is an integral part of virtually all computerized forecasting programs, and it is widely used in ordering inventory in retail firms, wholesale companies, and service agencies.

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1. 2. 3. 4. 5. 6.

[18.3]

450 Exponential smoothing techniques have become well accepted for six major reasons:

Exponential models are surprisingly accurate. Formulating an exponential model is relatively easy. The user can understand how the model works. Little computation is required to use the model. Computer storage requirements are small because of the limited use of historical data. Tests for accuracy as to how well the model is performing are easy to compute.

In the exponential smoothing method, only three pieces of data are needed to forecast the future: the most recent forecast, the actual demand that occurred for that forecast period, and a smoothing constant alpha (β). This smoothing constant determines the level of smoothing and the speed of reaction to differences between forecasts and actual occurrences. The value for the constant is determined both by the nature of the product and by the manager’s sense of what constitutes a good response rate. For example, if a firm produced a standard item with relatively stable demand, the reaction rate to differences between actual and forecast demand would tend to be small, perhaps just 5 or 10 percentage points. However, if the firm were experiencing growth, it would be desirable to have a higher reaction rate, perhaps 15 to 30 percentage points, to give greater importance to recent growth experience. The more rapid the growth, the higher the reaction rate should be. Sometimes users of the simple moving average switch to exponential smoothing but like to keep the forecasts about the same as the simple moving average. In this case, is approximated by 2 ÷ (n + 1), where n is the number of time periods in the corresponding simple moving average.

Smoothing constant alpha (α) The parameter in the exponential smoothing equation that controls the speed of reaction to differences between forecasts and actual demand.

The equation for a single exponential smoothing forecast is simply

Ft = Ft−1 + α(At−1 − Ft−1)

where

Ft = The exponentially smoothed forecast for period t Ft−1 = The exponentially smoothed forecast made for the prior period At−1 = The actual demand in the prior period α = The desired response rate, or smoothing constant

This equation states that the new forecast is equal to the old forecast plus a portion of the error (the difference between the previous forecast and what actually occurred).

To demonstrate the method, assume that the long-run demand for the product under study is relatively stable and a smoothing constant (α) of 0.05 is considered appropriate. If the exponential smoothing method were used as a continuing policy, a forecast would have been made for last month. Assume that last month’s forecast (Ft21) was 1,050 units. If 1,000 actually were demanded, rather than 1,050, the forecast for this month would be

Ft = Ft−1 + (At−1 − Ft−1) = 1,050 + 0.05(1,000 − 1,050) = 1,050 + 0.05(−50) = 1,047.5 units

Because the smoothing coefficient is small, the reaction of the new forecast to an error of 50 units is to decrease the next month’s forecast by only 21/2 units.

When exponential smoothing is first used for an item, an initial forecast may be obtained by using a simple estimate, like the

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first period’s demand, or by using an average of preceding periods, such as the average of the first two or three periods. Single exponential smoothing has the shortcoming of lagging changes in demand. Exhibit 18.5 presents actual data plotted as a

smooth curve to show the lagging effects of the exponential forecasts. The forecast lags during an increase or decrease, but overshoots when

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451 a change in direction occurs. Note that the higher the value of alpha, the more closely the forecast follows the actual. To more closely track actual demand, a trend factor may be added. Adjusting the value of alpha also helps. This is termed adaptive forecasting. Both trend effects and adaptive forecasting are briefly explained in following sections.

exhibit 18.5 Exponential Forecasts versus Actual Demand for Units of a Product over Time Showing the Forecast Lag

Exponential Smoothing with Trend Remember that an upward or downward trend in data collected over a sequence of time periods causes the exponential forecast to always lag behind (be above or below) the actual occurrence. Exponentially smoothed forecasts can be corrected somewhat by adding in a trend adjustment. To correct the trend, we need two smoothing constants. Besides the smoothing constant α, the trend equation also uses a smoothing constant delta (δ). Both alpha and delta reduce the impact of the error that occurs between the actual and the forecast. If both alpha and delta are not included, the trend overreacts to errors.

Smoothing constant delta (δ) An additional parameter used in an exponential smoothing equation that includes an adjustment for trend.

To get the trend equation going, the first time it is used the trend value must be entered manually. This initial trend value can be an educated guess or a computation based on observed past data.

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[18.4]

[18.5]

[18.6]

The equations to compute the forecast including trend (FIT) are

Ft = FITt−1 + α(At−1 − FITt−1)

Tt = Tt−1 + δ(Ft + FITt−1)

FITt = Ft + Tt

where

Ft = The exponentially smoothed forecast that does not include trend for period t Tt = The exponentially smoothed trend for period t FITt = The forecast including trend for period t FITt−1 = The forecast including trend made for the prior period

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452

At−1 = The actual demand for the prior period α = Smoothing constant (alpha) δ = Smoothing constant (delta)

To make an exponential forecast that includes trend, step through the equations one at a time.

Step 1:

Using equation 18.4 make a forecast that is not adjusted for trend. This uses the previous forecast and previous actual demand.

Step 2:

Using equation 18.5 update the estimate of trend using the previous trend estimate, the unadjusted forecast just made, and the previous forecast.

Step 3:

Make a new forecast that includes trend by using the results from steps 1 and 2.

EXAMPLE 18.1: Forecast Including Trend Assume a previous forecast including trend of 110 units, a previous trend estimate of 10 units, an alpha of .20, and a delta of .30. If actual demand turned out to be 115 rather than the forecast 110, calculate the forecast for the next period.

SOLUTION The actual At−1 is given as 115. Therefore,

Ft = FITt−1 + α(At−1 − FITt−1) = 110 + .2(115 − 110) = 111.0

Tt = Tt−1 + δ(Ft − FITt−1) = 10 + .3(111 − 110) = 10.3

FITt = Ft + Tt = 111.0 + 10.3 = 121.3

If, instead of 121.3, the actual turned out to be 120, the sequence would be repeated and the forecast for the next period would be

Ft+1 = 121.3 + .2(120 − 121.3) = 121.04 Tt+1 = 10.3 + .3(121.04 − 121.3) = 10.22 FITt+1 = 121.04 + 10.22 = 131.26

Exponential smoothing requires that the smoothing constants be given a value between 0 and 1. Typically fairly small values are used for alpha and delta in the range of .1 to .3. The values depend on how much random variation there is in demand and how steady the trend factor is. Later in the chapter, error measures are discussed that can be helpful in picking appropriate values for these parameters.

Linear Regression Analysis Regression can be defined as a functional relationship between two or more correlated variables. It is used to predict one variable given the other. The relationship is usually developed from observed data. The data should be plotted first to see if they appear linear or if at least parts of the data are linear. Linear regression refers to the special class of regression where the relationship between variables forms a straight line.

The linear regression line is of the form Y = a + bt, where Y is the value of the dependent variable that we are solving for, a is

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the Y intercept, b is the slope, and t is an index for the time period. Linear regression is useful for long-term forecasting of major occurrences and aggregate planning. For example, linear

regression would be very useful to forecast demands for product

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[18.7]

453 families. Even though demand for individual products within a family may vary widely during a time period, demand for the total product family is surprisingly smooth.

The major restriction in using linear regression forecasting is, as the name implies, that past data and future projections are assumed to fall in about a straight line. Although this does limit its application sometimes, if we use a shorter period of time, linear regression analysis can still be used. For example, there may be short segments of the longer period that are approximately linear.

Linear regression forecasting A forecasting technique that fits a straight line to past demand data.

Linear regression is used both for time series forecasting and for causal relationship forecasting. When the dependent variable (usually the vertical axis on a graph) changes as a result of time (plotted as the horizontal axis), it is time series analysis. If one variable changes because of the change in another variable, this is a causal relationship (such as the number of deaths from lung cancer increasing with the number of people who smoke). We use the following example to demonstrate linear least squares regression analysis:

EXAMPLE 18.2: Least Squares Method A firm’s sales for a product line during the 12 quarters of the past three years were as follows:

QUARTER SALES 1 600 2 1,550 3 1,500 4 1,500 5 2,400 6 3,100 7 2,600 8 2,900 9 3,800 10 4,500 11 4,000 12 4,900

For a step-by-step walkthrough of this example, visit www.mhhe.com/jacobs14e_sbs_ch18.

The firm wants to forecast each quarter of the fourth year—that is, quarters 13, 14, 15, and 16.

SOLUTION The least squares equation for linear regression is

Y = a + bt

where

Y = Dependent variable computed by the equation

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y = The actual dependent variable data point (used below) a = Y intercept b = Slope of the line t = Time period

The least squares method tries to fit the line to the data that minimizes the sum of the squares of the vertical distance between each data point and its corresponding point on the line. If a straight line is drawn through the general area of the points, the difference between the point and the line is y − Y. Exhibit 18.6 shows these differences. The sum of the squares of the differences between the plotted data points and the line points is

(y1 − Y1)2 + (y2 − Y2)2 + … + (y12 − Y12)2

The best line to use is the one that minimizes this total. As before, the straight line equation is

Y = a + bt

In the least squares method, the equations for a and b are

b = ∑ ty − nt̄ ⋅ȳ ∑ t2 − nt̄ 2

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454

exhibit 18.6 Least Squares Regression Line

For the Excel template, visit www.mhhe.com/jacobs14e.

a = ȳ − bt̄

where a = Y intercept b = Slope of the line ȳ = Average of all ys t̄ = Average of all ts t = t value at each data point y = y value at each data point n = Number of data points Y = Value of the dependent variable computed with the regression equation

Exhibit 18.7 shows these computations carried out for the 12 data points in the problem. Note that the final equation for Y shows an intercept of 441.67 and a slope of 359.6. The slope shows that for every unit change in t, Y changes by 359.6. Note that these calculations can be done with the INTERCEPT and SLOPE functions in Microsoft Excel.