STATISTICAL FORECASTING and SEASONALITY (M. E. Ippolito; 10-6-13) PART I OVERVIEW The following discussion expands upon exponential smoothing and seasonality as presented in Chapter 11, Forecasting, in the text. There are a variety of forecasting methods and systems. These methods/systems are generally divided by the time frame to be forecasted (short the next few months, intermediate/medium 6-24 months, or strategic/long-range out to five years or more). For short range forecasting, most forecasting software packages utilize some form of exponential smoothing (sometimes along with other short-range methods) to project short-range demand for high volume/low priced products (for example, projecting the demand for corn flakes in 12 oz. boxes and, separately, 8 oz. boxes). As such, these systems utilize mathematical models based upon the exponential smoothing formula discussed on pages 467 through 469. TIME PERIODS/TIME BUCKETS In order to easily handle the history of demand upon which forecasts are based, time is divided into a series of equal time periods or time buckets. Usually, these buckets are set equal to one week or one month. However, if useful, they can be set equal to other periods of time such as bi-weekly, daily, or hourly. Look at following time line of time buckets. t - 3 t - 2 t - 1 t t + 1 t + 2 t + 3 When discussing forecasting, be certain of the definition of the subscript "t" (there is not, unfortunately, a standard definition). As presented in this text, "t" represents the current time period (the period that we are in right now, or have just entered). t-1 represents the last completed time period. As such, "t-2" represents the period before last. "t+1" would represent the "next" time period (after the current time period). For example, let us suppose that it is now August 1 st. "t" would represent August, "t+1" September, t+2 October, "t-1" July, and "t-2" June. A month later, on September 1 st, "t" would now represent September, "t+1" October, t+2 November, "t-1" August, and "t- 2" July. Forecasting systems reforecast at the end of each time period. Thus, the forecast is updated on a regular basis to reflect the most recent actual demand. TREND Trend exists if, through time, average demand (after correction for seasonality) is consistently climbing or dropping. Exponential smoothing has a supplemental algorithm that will provide a trend correction. It is discussed on pages 469-470. We will not utilize trend correction in this course. You should understand, however, that accounting for and projecting trends is a necessary part of forecasting. Forecasting-1
CYCLICAL PATTERNS/SEASONALITY As mentioned on text page 474-5, a repeated pattern of high and low (relative to average) demand periods in a time series indicates a cyclical pattern. A pattern that repeats every twelve months defines seasonality. Seasonality is the most common form of cyclical patterns that show a periodic rise and fall in the overall demand for a product. More generally, these cycles can be of any length. For example: - a fast food restaurant experiences a daily cycle with peaks and valleys in demand occurring on a more or less regular pattern each day. (How do you think such a cycle would look? Would you expect the cycle to be the same on a weekend as on a weekday?) - At the other extreme, a cycle can cover many years. Belden Wire & Cable in Richmond manufactures a wide variety of electrical and electronic wire and cable products. One of these products is a line of cables used by the television industry to hook up cameras and other video equipment. They report experiencing a four-year cycle in demand that peaks in presidential election and Olympic Games years. As noted, the most common type of cyclical pattern is one that repeats every 12 months, which is called seasonality. Part II of this discussion will illustrate seasonality and how it is handled. Forecasting-2
PART II DETERMINING SEASONAL FACTORS Many products exhibit seasonality. Extreme examples are Easter egg dye kits and holiday-specific greeting cards. A more typical example is exterior house paint. The method to be illustrated here (simple proportion) is demonstrated in example 18.3 on pages 475-6. Look at the following data: MONTH DEMAND (gallons) SEASONAL FACTOR Jan 271 0.25 Feb 433 0.40 Mar 1083 Apr 1896 May 2600 Jun 2438 Jul 1354 Aug 1192 Sep 975 Oct 542 Nov 108 Dec 108 Total 13000 The first step in analyzing seasonality is to determine a set of seasonality factors. To do this, divide each month's demand by the average monthly demand. In this case, the average monthly demand is 1083 gallons (13000/12). Dividing 1083 gallons into January's demand of 271 gallons yields a factor of 0.25 (as is shown). Likewise, the factor for February is 0.40 (433/1083) as is shown. Now calculate the seasonal factors for the rest of the months. Finally, add up the factors. What is the sum? NOTE: THESE SEASONAL FACTORS WILL BE USED IN PART III. Forecasting-3
PART III UTILIZING THE EXPONENTIAL SMOOTHING FORMULA WITH A SEASONAL PRODUCT Two key terms to keep in mind are seasonal and deseasonal. A seasonal forecast or seasonal demand reflects seasonality, while a deseasonal forecast or deseasonal demand does not. Refer to the data in Part II. The monthly demands as shown are seasonal demands. The average monthly demand of 1083 gallons is deseasonal. The exponential smoothing formula (18.3) on page 468 is applied to DESEASONAL DATA ONLY. Therefore, to apply it to a seasonal product, the demand must first be deseasonalized utilizing the following formula: A = SA/(SFAC) where A is the demand (as used in the formula on page 468), seasonally adjusted. (This can also be referred to as seasonally corrected or deseasonalized demand.) SA is the actual (therefore, seasonal) demand. SFAC is the seasonal factor. For example, from Part II, the deseasonal demand for January is 1083 (271/0.25). This figure is also the value for the average monthly demand, which it should be since we are looking back at the past data upon which the seasonal factors are based. Forecasting is, however, an ongoing process. Let's suppose that the data in Part II are for 2004. Going into January '05, we would expect to see an actual seasonal demand of 271 (assuming no growth in demand from 2004). Let us further suppose that, at the end of January '05, we find that the actual seasonal demand was really 240. In order to revise our forecast, to show the effect of this new data, we would do the following: 1. Deseasonalize the demand. A = SA/SFAC = 240/0.25 = 960 2. Apply the formula on page (277), setting = 0.2 F t = F t-1 + (A t-1 - F t-1 ) = 1083 + 0.2(960-1083) = 1058, or (using the other version of the formula) F t = A t-1 + (1 - ) F t-1 =.2*960 + (1 -.2)*1083 =.2*960 +.8*1083 = 1058 3. Looking forward, our revised seasonal forecast (as of the end of January/beginning of February) for February would be 0.40 x 1058 = 423 (vs. 433 previously) Forecasting-4
The formula for this equation is: SF = SFAC x F where SF is the seasonalized forecast F is the smoothed forecast (deseasonal) as on page 219. (NOTE: the terms smoothed forecast, exponentially smoothed forecast, and deseasonal forecast all mean the same thing in this kind of application.) SFAC is the seasonal factor At this point in time, what would be the seasonal forecasts for March?, April?, July?, December? (You should note that, for the purposes of this course, seasonal factors and the value for " " are assumed to be constant. In actual applications, as discussed in the text, many software packages recalculate seasonal factors and " " at the end of each month before recalculating the forecasts. Whether or not they are automatically recalculated, seasonal factors and the value for " " should be reviewed on a regular basis to ensure that they continue to reasonably represent what is actually happening.) In class exercise: 1. It is now the end of February. Seasonal actual demand came in at 420. Calculate the revised smoothed (deseasonal) forecast. What are the new seasonal forecasts for March?, April?, July?, December? 2. It is now the end of March. Seasonal actual demand came in at 950. Calculate the revised smoothed forecast. What are the new seasonal forecasts for April?, July?, December? Forecasting-5
HOMEWORK EXERCISE The homework exercise is in two parts. The first part appears on this page and is a series of problems using the formulas. The second part is a discussion question that appears on the next page. Solutions to the problems may be turned in hand-written; the discussion question must be typed (although the recommended graph may be hand-drawn). For problems 1 and 2, use the following set of seasonal factors. Use 1200 for the starting deseasonal forecast (meaning, as of January 1, 2013) for problem 1. Seasonal factors for use in this problem set: Jan = 1.5 Feb =.3 Mar =.4 Apr =.6 May = 1.5 Jun = 1.3 Jul = 1.4 Aug = 1.3 Sep =.8 Oct =.5 Nov =.5 Dec = 1.9 1. In January '13, seasonal actual demand came in at 1650 cases. Setting = 0.3, what will the new smoothed forecast be? What will the new seasonal forecasts be for February, April, July, and December? 2. In February '13, seasonal actual demand came in at 410 cases. What will the new smoothed forecast be? What are the new seasonal forecasts for March, April, July, and December? Solutions for 1. and 2. appear in Oncourse under the Resources link. After you try 1. and 2., and review your solutions, then try 3. and 4. YOU WILL TURN IN your solutions to 3 and 4, ALONG WITH your answer to the discussion question on the next page. 3. March Demand comes in at 395. What will be the new smoothed (deseasonal) forecast? What are the new seasonal forecasts for April, July, and December? 4. April demand comes in at 745. What will be the new smoothed (deseasonal) forecast? What are the new seasonal forecasts for May, July, December? Forecasting-6
Discussion Question: You have three years of historical sales data for the product that you forecasted in problems 1 through 4. Based upon this data, do you think that the starting deseasonal forecast (as of January 1, 2013) of 1200 for problem 1 is reasonable? Do you see any year-to-year trends? What about the given seasonality factors? (HINT: calculate the monthly seasonal factors for each of the past three years. Plot these factors on a graph. Use a different color for each year. Do they look consistent? Do you notice any trends?) HISTORICAL DEMAND FOR CASES OF SMART CARDS (USED IN DIGITAL CAMERAS) Month 2010 2011 2012 Jan 1500 1510 1960 Feb 300 350 340 Mar 400 480 450 Apr 600 630 760 May 1500 1700 1760 Jun 1400 1540 1410 Jul 1000 1380 1600 Aug 1400 1540 1410 Sep 900 980 980 Oct 500 530 630 Nov 500 530 630 Dec 2000 2330 2270 Total 12000 13500 14200 Forecasting-7