Operations and Supply Chain Management MGMT 3306 Lecture 03 Instructor: Dr. Yan Qin

Outline – Demand Forecasting What is Forecasting Qualitative forecasting methods Quantitative forecasting methods Time-series models Associative models Linear regression Trend and Seasonal factors Trend-adjusted exponential smoothing method Seasonal indices

Forecasting Forecasting is the art and science of predicting future events. Three major types of forecasts in planning future operations: Economic forecasts: address the business cycle by predicting inflation rates, money supplies, and etc.; Technological forecasts: predict rates of technological progress; Demand forecasts: projections of demand for a company’s products or services. (What we’ll discuss in this lecture)

Seven Steps in Forecasting Determine the use of the forecast Select the items to be forecasted Determine the time horizon of the forecast Select the forecasting model(s) Gather the data Make the forecast Validate and implement results

Demand Forecasts There are two general approaches to forecasting: Qualitative, which incorporate factors such as decision maker’s intuition, emotions, personal experience, and so on. Quantitative, which uses a variety of models that rely on historical data to forecast demand.

Common types of Qualitative Methods Jury of execution opinion: The opinions of a group of high-level experts or mangers are pooled together to arrive at a group estimate of demand. This approach is used widely because it is relatively quick, inexpensive and uses the solid understanding of participants who are presumed to have relevant knowledge. Consumer market survey: Collect information from current or potential customers about future purchasing plan.

Common types of Qualitative Methods Delphi method: Surveys and questionnaires are distributed to a group of selected respondents. Then a group of 5-10 experts make the decision based on the responses. The state of Alaska used Delphi method to develop its long- range economic forecast. Sales force composite: Combine each sales person’s estimate of future demand. Lexus dealers meet every quarter to talk about the sales, such as what is selling, in what colors, so the manufacturer has an idea what to build.

Common Quantitative Methods Time series models predict on the assumption that the future is a function of the past. Naïve approach Moving averages Exponential smoothing Associative Models incorporate variables that might influence the quantity being forecast. For example, an associative model for lawn mowers sales might use factors such as new housing starts, advertising budget and so on. Linear regressions

Naïve Approach Assume that demand in the next period will be equal to the most recent period. For example, if the sales in Feb was 56 units, the forecast for March would be 56 under the naïve approach. This approach takes a temporary change as a permanent change. Bull-whip effect will be maximized under this forecasting method.

Moving averages A moving-average forecast uses a number of historical actual data values to generate a forecast. It is useful if we can assume that market demands will stay fairly steady over time. An n-period moving-average model: 𝐹 𝑡 = 𝐴 𝑡−1 + 𝐴 𝑡−2 +⋯+ 𝐴 𝑡−𝑛 𝑛 Usually the n is 3, 6, 9 periods. n : # of periods to be averaged 𝐹 𝑡 : Forecast for period t 𝐴 𝑡−1 : Actual demand for period t-1

Example: 3-/6-month moving average Actual Sales 3-Month 6-Month Jan 10 Feb 12 March 13 April 16 (10+12+13)/3=11.7 May 19 (12+13+16)/3=13.7 June 23 (13+16+19)/3=16 July 26 (16+19+23)/3=19.3 (10+12+13+16+19+23)/6=15.5 Aug 30 (19+23+26)/3=22.7 (12+13+16+19+23+26)/6=18.2 Sep 28 (23+26+30)/3=26.3 (13+16+19+23+26+30)/6=21.2

Weighted moving averages Weights can be used to place more emphasis on recent values or older values. A n-period weighted moving average: 𝐹 𝑡 = 𝑤 𝑡−1 𝐴 𝑡−1 + 𝑤 𝑡−2 𝐴 𝑡−2 +⋯+ 𝑤 𝑡−𝑛 𝐴 𝑡−𝑛 𝑤 𝑡−1 + 𝑤 𝑡−2 +⋯+ 𝑤 𝑡−𝑛 𝑤 𝑖 : Weight attached to the actual demand in period i

Example: Weighted moving average Suppose we use the 3-period weighted moving average method to make demand forecasts. An importance weight of 0.6 is assigned to the most recent actual demand, 0.3 to the second most recent actual demand, and 0.1 to the oldest observation. Then what are the 3-period forecasts for demands in April and May? Month Actual Sales Forecasts Jan 10 Feb 12 March 13 April 16 ? May 19

Example: Solution We first calculate the forecast for April. Let t = 4. Then, period 1 is Jan. Period 2 is Feb. And period 3 is Mar in this case. 𝐹 4 = 𝑤 3 𝐴 3 + 𝑤 2 𝐴 2 + 𝑤 1 𝐴 1 𝑤 3 + 𝑤 2 + 𝑤 1 = 0.6∗13+0.3∗12+0.1∗10 0.6+0.3+0.1 =12.4 We now calculate the forecast for May. Let t = 5. 𝐹 5 = 𝑤 4 𝐴 4 + 𝑤 3 𝐴 3 + 𝑤 2 𝐴 2 𝑤 4 + 𝑤 3 + 𝑤 2 = 0.6∗16+0.3∗13+0.1∗12 0.6+0.3+0.1 =14.7

Limitations of moving averages Increasing the size of n makes the method less sensitive to real changes in the data; Moving averages cannot predict changes to either higher or lower levels. They lag the actual values. Moving averages require extensive records of past data.

Exponential smoothing Exponential smoothing is a sophisticated weighted- moving-average method. Call 𝛼 a smoothing constant, valued between 0 an 1. Usually from 0.05 to 0.50 for business application. 𝐹 𝑡 = 𝐹 𝑡−1 +𝛼∙( 𝐴 𝑡−1 − 𝐹 𝑡−1 ) That is, the forecast for period t is the sum of the forecast for period t -1 and the amount of adjustment determined by the forecast error in period t-1 and the smoothing constant.

Example: Exponential Smoothing Let 𝛼=0.2. Then what are the forecasts for the demands in Feb, Mar, and April using Exponential Smoothing? Month Actual Sales Forecast Jan 10 8 Feb 12 ? March 13 April 16 May 19

Example: Solution Let Jan be period 1, Feb be period 2, Mar be period 3, and so on. Then the forecast for Feb can be computed as follows: 𝐹 2 = 𝐹 1 +𝛼∙ 𝐴 1 − 𝐹 1 =8+0.2∙ 10−8 =8.4 The forecast for March: 𝐹 3 = 𝐹 2 +𝛼∙ 𝐴 2 − 𝐹 2 =8.4+0.2∙ 12−8.4 =9.12 The forecast for April: 𝐹 4 = 𝐹 3 +𝛼∙ 𝐴 3 − 𝐹 3 =9.12+0.2∙ 13−9.12 = 9.896

Impact of Smoothing Constant 𝛼 225 – 200 – 175 – 150 – | | | | | | | | | 1 2 3 4 5 6 7 8 9 Quarter Demand Chose high values of α when underlying average is likely to change Actual demand a = .5 a = .1

Choosing Smoothing Constant 𝛼 The objective is to obtain the most accurate forecast no matter the technique We generally do this by selecting the model that gives us the lowest forecast error Forecast error = Actual demand - Forecast value = At - Ft

Common Measures of Error Mean Absolute Deviation (MAD): Average of the absolute values of the individual forecast errors. 𝑀𝐴𝐷= 𝐹𝑜𝑟𝑒𝑐𝑎𝑠𝑡 𝐸𝑟𝑟𝑜𝑟 𝑛 Mean Squared Error (MSE): The average of the squared forecast errors. 𝑀𝑆𝐸= (𝐹𝑜𝑟𝑒𝑐𝑎𝑠𝑡 𝐸𝑟𝑟𝑜𝑟) 2 𝑛

Example: Measures of Error Rounded Absolute Rounded Absolute Actual Forecast Forecast error Forecast Forecast error Tonnage with for with for Quarter Unloaded a = .10 a = .10 a = .50 a = .50 1 180 175 5.00 175 5.00 2 168 175.5 7.50 177.50 9.50 3 159 174.75 15.75 172.75 13.75 4 175 173.18 1.82 165.88 9.12 5 190 173.36 16.64 170.44 19.56 6 205 175.02 29.98 180.22 24.78 7 180 178.02 1.98 192.61 12.61 8 182 178.22 3.78 186.30 4.30 82.45 98.62 Sum Sum

Example: MAD 𝑀𝐴𝐷= 𝐹𝑜𝑟𝑒𝑐𝑎𝑠𝑡 𝐸𝑟𝑟𝑜𝑟 𝑛 For α = 0.1, 𝑀𝐴𝐷= 82.45 8 =10.31 𝑀𝐴𝐷= 𝐹𝑜𝑟𝑒𝑐𝑎𝑠𝑡 𝐸𝑟𝑟𝑜𝑟 𝑛 For α = 0.1, 𝑀𝐴𝐷= 82.45 8 =10.31 For α = 0.5, 𝑀𝐴𝐷= 98.45 8 =12.33 Lower overall forecast error based on MAD when α=0.1

Example: MSE 𝑀𝑆𝐸= (𝐹𝑜𝑟𝑒𝑐𝑎𝑠𝑡 𝐸𝑟𝑟𝑜𝑟) 2 𝑛 𝑀𝑆𝐸= (𝐹𝑜𝑟𝑒𝑐𝑎𝑠𝑡 𝐸𝑟𝑟𝑜𝑟) 2 𝑛 For α = 0.1, 𝑀𝑆𝐸= 1526.54 8 =190.82 For α = 0.5, 𝑀𝐴𝐷= 1561.91 8 =195.24 Lower overall forecast error based on MSE when α=0.1

Associative Model – Linear Regression Regression is used to predict one variable given the other variable(s). The relationship is usually developed from observed data. Linear regression is a special class of regression where the relationship between variables is linear. 𝑌=𝑎+𝑏𝑋 a: Intercept; b: Slope Y: dependent variable we are tying to make forecast for X: independent variable whose value has a linear impact on the value of Y.

Least squares method It is a method to estimate the a and b values based on historic data. The objective is to minimize the sum of the squares of the differences between forecasts and actual observations. Let 𝑦 𝑖 denote the y value of the 𝑖 𝑡ℎ observation and 𝑌 𝑖 be the corresponding forecast using a certain set of a and b values. Suppose there are n data points. 𝑀𝑖𝑛 ( 𝑦 1 − 𝑌 1 ) 2 +( 𝑦 2 − 𝑌 2 ) 2 +⋯+( 𝑦 𝑛 − 𝑌 𝑛 ) 2 Forecast error of observation 1 Forecast error of observation 2 Forecast error of observation n

Figure: Least squares method We want to find a straight line that minimizes the sum of the squared forecast errors.

Least squares method (Cont.) Using the least squares method, 𝑏= 𝑖=1 𝑛 (𝑥 𝑖 𝑦 𝑖 ) −𝑛 𝑥 𝑦 𝑖=1 𝑛 (𝑥 𝑖 2 )−𝑛 𝑥 2 𝑎= 𝑦 −𝑏 𝑥 where 𝑥 𝑖 is the x value of observation i; 𝑦 𝑖 is the y value of observation i; 𝑥 is the mean of all x values; 𝑦 is the mean of all y values; n is the number of observations in the data set

Least squares method (Cont.) Standard error of estimates (How well the estimated line fits the data) where 𝑌 𝑖 is the forecast for the y value of observation i.

Example: Least squares Consider a situation where the sales of a product is time- dependent. What is the forecast for the demand in month 6 using the least squares Linear Regression method? Month ( 𝒙 𝒊 ) Actual Sales ( 𝒚 𝒊 ) 1 10 2 12 3 13 4 16 5 19

Example: Solution We first calculate the squares and products required in the calculation of a and b. Month ( 𝑥 𝑖 ) Actual Sales ( 𝑦 𝑖 ) 𝒙 𝒊 𝒚 𝒊 𝒙 𝒊 𝟐 1 10 2 12 24 4 3 13 39 9 16 64 5 19 95 25 𝑥 =3 𝑦 =14 𝑏= 𝑖=1 𝑛 (𝑥 𝑖 𝑦 𝑖 ) −𝑛 𝑥 𝑦 𝑖=1 𝑛 (𝑥 𝑖 2 )−𝑛 𝑥 2 = 232−5∗3∗14 55−5∗9 =2.2 𝑎= 𝑦 −𝑏 𝑥 =14−2.2∗3=7.4

Example: Solution The linear regression function is: Y = 7.4 + 2.2∗X Then the forecast for the demand in month 6 = 7.4 + 2.2∗6 = 20.6 Note that 𝑥 6 =6

Decomposition of a time series A time series can have 4 components: Trend: the gradual upward or downward movement of the data over time. (Can be estimated using Least squares Linear Regression method) Seasonality: a data pattern that repeats itself after a period of days, weeks, months, or quarters. Cycle: patterns in the data that occur every several years. Random variations: “blips” in the data caused by chance and unusual situations.

Figure: 4 major components

Exponential smoothing with Trend adjustment Adjust exponential smoothing forecasts for positive or negative lag in trends. (requires two smoothing constants: 𝛼, 𝛽) Steps to take: Step 1: Compute 𝐹 𝑡 , the exponential smoothing forecast for period t. 𝐹 𝑡 =𝛼 𝐴 𝑡−1 +(1−𝛼)( 𝐹 𝑡−1 + 𝑇 𝑡−1 ) Step 2: Compute the trend estimate for period t, 𝑇 𝑡 . 𝑇 𝑡 =𝛽 𝐹 𝑡 − 𝐹 𝑡−1 +(1−𝛽) 𝑇 𝑡−1 Step 3: Calculate the forecast for period t including trend 𝐹𝐼 𝑇 𝑡 = 𝐹 𝑡 + 𝑇 𝑡 .

Example: Trend adjustment Let the smoothing constants 𝛼=0.2 and 𝛽=0.4. The company assumes that the initial forecast for Jan was 10 and the trend over that period was 2 units. What are the trend-adjusted forecasts for Feb in this case? Month Actual Sales Forecast t Trend t Jan 10 9 2 Feb 12 ?

Example: Solution For Feb, =12.72 𝐹 2 =𝛼 𝐴 1 + 1−𝛼 𝐹 1 + 𝑇 1 𝐹 2 =𝛼 𝐴 1 + 1−𝛼 𝐹 1 + 𝑇 1 =0.2∗10+ 1−0.2 ∗ 9+2 =10.8 𝑇 2 =𝛽 𝐹 2 − 𝐹 1 + 1−𝛽 𝑇 1 =0.4∗ 10.8−9 + 1−0.4 ∗2 =1.92. 𝐹𝐼 𝑇 𝑡 = 𝐹 𝑡 + 𝑇 𝑡 =10.8+1.92 =12.72

Seasonal index A seasonal index is the amount of correction needed in a time series to adjust for the season of the year. Example 1: Assume that in past years, a firm sold an average of 1,000 units of a product each year. On average, 200 units were sold in the spring, 350 units in the summer, 300 units in the fall, and 150 units in the winter. Then the seasonal index is the ratio of the amount sold during each season divided by the average for all seasons.

Example: Seasonal index Question 1: For the previous example, what are the four seasonal indices? Seasonal average = 1,000 / 4 = 250 Seasonal index = average sales in a season / seasonal average Seasonal index for spring = 200/250 = 0.8 Seasonal index for summer = 350/250=1.4 Seasonal index for fall = 300/250 = 1.2 Seasonal index for winter = 150/250 = 0.6

Example: Seasonal index (Cont.) Question 2: Now suppose the expected demand for next year is 1,100 units. What are the forecasts for each season next year? Forecast for a season = seasonal average for next year * seasonal index for that season Seasonal average for next year = 1100 / 4 = 275 Forecast for spring =275*0.8 = 220 Forecast for summer =275*1.4 = 385 Forecast for fall = 275*1.2 = 330 Forecast for winter = 275*0.6 = 165

Example 2: Seasonal index What if we have more than one year of data? Answer: Divide the average for each season by the general seasonal average for all years. (Assuming constant seasonal indices for each year or additive seasonal effect) Year 1 Actual Sales Year 2 Spring 300 520 Summer 200 420 Fall 220 400 Winter 530 700 The general average is 411.25 (Average of the 8 numbers) Seasonal index for Spring = ((300+520)/2)/411.25 = 0.99