Problems 1 The Saki motorcycle dealer in the MinneapolisSt Paul area wants to make an accurate forecast of demand for the Saki Super TXII motorcycle during the next month Because the manufacturer is in Japan, it is difficult to send motorcycles back or reorder if the proper number is not ordered a month ahead From sales records, the dealer has accumulated the following data for the past year: Month Motorcycle Sales January 9 February 7 March 10 April 8 May 7 June 12 July 10 August 11 September 12 October 10 November 14 December 16 a Compute a 3-month moving average forecast of demand for April through January (of the next year) b Compute a 5-month moving average forecast for June through January c Compare the two forecasts computed in (a) and (b), using MAD, MSE and MAPE Which one should the dealer use for January of the next year? 2 The manager of the Carpet City outlet needs to make an accurate forecast of the demand for Soft Shag carpet (its biggest seller) If the manager does not order enough carpet from the carpet mill, customers will buy their carpet from one of Carpet City's many competitors The manager has collected the following demand data for the past 8 months:
Month Demand for Soft Shag Carpet (1,000 yd) 1 8 2 12 3 7 4 9 5 15 6 11 7 10 8 12 a Compute a 3-month moving average forecast for months 4 through 9 b Compute a weighted 3-month moving average forecast for months 4 through 9 Assign weights of 55, 33, and 12 to the months in sequence, starting with the most recent month c Compare the two forecasts by using MAD, MSE and MAPE Which forecast appears to be more accurate? 3 The Fastgro Fertilizer Company distributes fertilizer to various lawn and garden shops The company must base its quarterly production schedule on a forecast of how many tons of fertilizer will be demanded from it The company has gathered the following data for the past 3 years from its sales records: Year Quarter Demand for Fertilizer (tons) 1 1 105 2 150 3 93 4 121 2 5 140
6 170 7 105 8 150 3 9 150 10 170 11 110 12 130 a Compute a three-quarter moving average forecast for quarters 4 through 13 and compute the forecast error for each quarter b Compute a five-quarter moving average forecast for quarters 6 through 13 and compute the forecast error for each quarter c Compute a weighted three-quarter moving average forecast, using weights of 50, 33, and 17 for the most recent, next recent, and most distant data, respectively, and compute the forecast error for each quarter d Compare the forecasts developed in (a), (b), and (c), using MAD, MSE and MAPE Which forecast appears to be most accurate? Do any of them exhibit any bias? 4 The chairperson of the department of management at State University wants to forecast the number of students who will enroll in production and operations management (POM) next semester, in order to determine how many sections to schedule The chair has accumulated the following enrollment data for the past eight semesters: Semester Students Enrolled in POM 1 400 2 450 3 350 4 420
5 500 6 575 7 490 8 650 a Compute a three-semester moving average forecast for semesters 4 through 9 b Compute the exponentially smoothed forecast (α = 20) for the enrollment data c Compare the two forecasts by using MAD, MSE and MAPE and indicate the more accurate of the three 5 The manager of the Petroco Service Station wants to forecast the demand for unleaded gasoline next month so that the proper number of gallons can be ordered from the distributor The owner has accumulated the following data on demand for unleaded gasoline from sales during the past 10 months: [ Month Gasoline Demanded (gal) October 800 November 725 December 630 January 500 February 645 March 690 April 730 May 810 June 1,200 July 980 a Compute an exponentially smoothed forecast, using and value of 30 b Compute an adjusted exponentially smoothed forecast (with α = 30 and β = 20)
see appendix for information on this forecast c Compare the two forecasts by using MAD, MSE, and MAPE and indicate which seems to be more accurate 6 The Victory Plus Mutual Fund of growth stocks has had the following average monthly price for the past 10 months: Month Fund Price 1 627 2 639 3 680 4 664 5 672 6 658 7 682 8 693 9 672 10 701 Compute the exponentially smoothed forecast with α = 40, the adjusted exponential smoothing forecast with α = 40 and β = 30, and the linear trend line forecast Compare the accuracy of the three forecasts, using MAD, MSE and MAPE, and indicate which forecast appears to be most accurate 7 The Bayside Fountain Hotel is adjacent to County Coliseum, a 24,000-seat arena that is home to the city's professional basketball and ice hockey teams and that hosts a variety of concerts, trade shows, and conventions throughout the year The hotel has experienced the following occupancy rates for the past 9 years, since the coliseum opened: Year Occupancy Rate (%) 1 83 2 78 3 75
4 81 5 86 6 85 7 89 8 90 9 86 Compute an exponential smoothing forecast with α = 20, an adjusted exponential smoothing forecast with α = 20 and β = 20, and a linear trend line forecast Compare the three forecasts, using MAD, MSE and MAPE, and indicate which seems to be most accurate 8 Eurotronics manufactures components for use in small electronic products such as computers, CD players, and radios at plants in Belgium, Germany, and France The parts are transported by truck to Hamburg, where they are shipped overseas to customers in Mexico, South America, the United States, and the Pacific Rim The company has to reserve space on ships months and sometimes years in advance This requires an accurate forecasting model Following are the number of cubic feet of container space the company has used in each of the past 18 months: Month Space (1,000s ft 3 ) 1 106 2 127 3 98 4 113 5 136 6 144 7 122
8 167 9 181 10 192 11 163 12 147 13 182 14 196 15 214 16 228 17 206 18 187 Develop a forecasting model that you believe would provide the company with relatively accurate forecasts for the next year and indicate the forecasted shipping space required for the next 3 months 9 The Whistle Stop Cafe in Weems, Georgia, is well known for its popular homemade ice cream, made in a small plant in back of the cafe People drive all the way from Atlanta and Macon to buy the ice cream The two women who own the cafe want to develop a forecasting model so they can plan their ice cream production operation and determine the number of employees they need to sell ice cream in the cafe They have accumulated the following sales records for their ice cream for the past 12 quarters: Year Quarter Ice Cream Sales (gal) 2003 1 350 2 510 3 750 4 420 2004 5 370
6 480 7 860 8 500 2005 9 450 10 550 11 820 12 570 Develop an adjusted exponential smoothing model with α = 50 and β = 50 to forecast demand and assess its accuracy using MAD, MSE and MAPE? 10 For the demand data in Problem 9, develop a seasonally adjusted forecast for 2004 (Use a linear trend line model to develop a forecast estimate for 2006) Which forecast model do you perceive to be more accurate: the exponential smoothing model from Problem 9 or the seasonally adjusted forecast? 11 Monaghan's Pizza delivery service has randomly selected 8 weekdays during the past month and recorded orders for pizza at four different time periods per day: Day Time Period 1 2 3 4 5 6 7 8 10:00 AM3:00 PM 62 49 53 35 43 48 56 43 3:00 PM7:00 PM 73 55 81 77 60 66 85 70
7:00 PM11:00 PM 42 38 45 50 29 37 35 44 11:00 PM12:00 AM 35 40 36 39 26 25 36 31 Develop a seasonally adjusted forecasting model for daily pizza demand and forecast demand for each of the time periods for a single upcoming day 12 The Cat Creek Mining Company mines and ships coal It has experienced the following demand for coal during the past 8 years: Year Coal Sales (tons) 1 4,260 2 4,510 3 4,050 4 3,720 5 3,900 6 3,470 7 2,890 8 3,100 Develop an adjusted exponential smoothing model (α = 30, β = 20) and a linear trend line model and compare the forecast accuracy of the two by using MAD, MSE and MAPE Indicate which forecast seems to be more accurate 13 The Northwoods Outdoor Company is a catalog sales operation that specializes in outdoor recreational clothing Demand for its items is very seasonal, peaking during the Christmas season and during the spring It has accumulated the following data for orders per season (quarter) during the past 5 years: Orders (1,000s) Quarter 2001 2002 2003 2004 2005 JanuaryMarch 186 181 224 232 245 AprilJune 235 247 288 276 310
JulySeptember 204 195 210 244 237 OctoberDecember 419 463 455 471 528 a Develop a seasonally adjusted forecast model for these order data Forecast demand for each quarter for 2006 (using a linear trend line forecast estimate for orders in 2006) 14 Metro Food Vending operates vending machines in office buildings, the airport, bus stations, colleges, and other businesses and agencies around town, and it operates vending trucks for building and construction sites The company believes its sandwich sales follow a seasonal pattern It has accumulated the following data for sandwich sales per season during the past 4 years: Sandwich Sales (1,000s) Season 2002 2003 2004 2005 Fall 427 443 457 406 Winter 369 427 348 415 Spring 513 556 493 473 Summer 629 648 712 745 Develop a seasonally adjusted forecast model for these sandwich sales data Forecast demand for each season for 2006 by using a linear trend line estimate for sales in 2006 Do the data appear to have a seasonal pattern? 15 The emergency room at the new Community Hospital selected every other week during the past 5 months to observe the number of patients during two parts of each week :the weekend (Friday through Sunday) and weekdays (Monday through Thursday) They typically experience greater patient traffic on weekends than during the week: Number of Patients Week Weekend Weekdays
1 116 83 2 126 92 3 125 97 4 132 91 5 128 103 6 139 88 7 145 96 8 137 106 9 151 95 10 148 102 Develop a seasonally adjusted forecasting model for the number of patients during each part of the week for week 11 16 Aztec Industries has developed a forecasting model that was used to forecast during a 10- month period The forecasts and actual demand were as follows: Month Actual Demand Forecast Demand 1 160 170 2 150 165 3 175 157 4 200 166 5 190 183 6 220 186 7 205 203 8 210 204 9 200 207
10 220 203 Measure the accuracy of the forecast by using MAD, MSE,MAPE Which forecast method appear to be most accurate? 17 RAP Computers assembles personal computers from generic parts it purchases at a discount, and it sells the units via phone orders it receives from customers responding to the company's ads in trade journals The business has developed an exponential smoothing forecast model to forecast future computer demand Actual demand for the company's computers for the past 8 months as well as a forecast are shown in the following table: Month Demand Forecast March 120 April 110 1200 May 150 1160 June 130 1296 July 160 1297 August 165 1418 September 140 1511 October 155 1467 November 1500 a Using a measure of forecast accuracy of your choice, ascertain whether the forecast appears to be accurate b Determine whether a 3-month moving average would provide a better forecast 18 19 Develop an exponential smoothing forecast with α = 20 for the demand data in Problem 1 Compare this forecast with the 3-month moving average computed in part (a) of Problem 1, using MAD, and indicate which forecast seems to be more accurate The Jersey Dairy Products Company produces cheese, which it sells to supermarkets and food-processing companies Because of concerns about cholesterol and fat in cheese, the company has seen demand for its products decline during the past decade It is now
considering introducing some alternative low-fat dairy products and wants to determine how much available plant capacity it will have next year The company has developed an exponential smoothing forecast with α = 40 to forecast cheese The actual demand and the forecasts from the model are as follows: Year Demand (1,000 lb) Forecast 1 168 2 141 168 3 153 157 4 127 155 5 119 144 6 123 134 7 115 129 8 108 124 Assess the accuracy of the forecast model by using MAD, MSE and MAPE If the exponential smoothing forecast model does not appear to be accurate, determine whether a linear trend model would provide a more accurate forecast 20 The manager of the Ramona Inn Hotel near Cloverleaf Stadium believes that how well the local Blue Sox professional baseball team is playing has an impact on the occupancy rate at the hotel during the summer months Following are the number of victories for the Blue Sox (in a 162-game schedule) for the past 8 years and the hotel occupancy rates: Year Blue Sox Wins Occupancy Rate (%) 1 75 83 2 70 78 3 85 86 4 91 85 5 87 89 6 90 93
7 87 92 8 67 91 Develop a linear regression model for these data and forecast the occupancy rate for next year if the Blue Sox win 88 games 21 Carpet City wants to develop a means to forecast its carpet sales The store manager believes that the store's sales are directly related to the number of new housing starts in town The manager has gathered data from county records on monthly house construction permits and from store records on monthly sales These data are as follows: Monthly Carpet Sales (1,000 yd) Monthly Construction Permits 5 21 10 35 4 10 3 12 8 16 2 9 12 41 11 15 9 18 14 26 a) Develop a linear regression model for these data and forecast carpet sales if 30 construction permits for new homes are filed b) Determine the strength of the causal relationship between monthly sales and new home construction by using correlation 22 The manager of Gilley's Ice Cream Parlor needs an accurate forecast of the demand for ice cream The store orders ice cream from a distributor a week ahead; if the store orders too little, it loses business, and if it orders too much, the extra must be thrown away The
manager believes that a major determinant of ice cream sales is temperature (ie, the hotter the weather, the more ice cream people buy) Using an almanac, the manager has determined the average daytime temperature for 10 weeks, selected at random, and from store records he has determined the ice cream consumption for the same 10 weeks These data are summarized as follows: Week Average Temperature (degrees) Ice Cream Sold (gal) 1 73 110 2 65 95 3 81 135 4 90 160 5 75 97 6 77 105 7 82 120 8 93 175 9 86 140 10 79 121 Q 23 a Develop a linear regression model for these data and forecast the ice cream consumption if the average weekly daytime temperature is expected to be 85 degrees b Determine the strength of the linear relationship between temperature and ice cream consumption by using correlation For the demand data in Problem 3 ( fertilizer question), develop a seasonally adjusted forecast for year 4 (Use a linear trend line model to develop a forecast estimate for year 4) Appendix Adjusted Exponential Smoothing The adjusted exponential smoothing forecast consists of the exponential smoothing forecast with a trend adjustment factor added to it The formula for the adjusted forecast is
Adjusted exponential smoothing is the exponential smoothing forecast with an adjustment for a trend added to it AF t+1 = F t+1 + T t+1 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: T t+1 = β(f t+1 - F t ) + (1 - β)t t where T t = the last period trend factor β = a smoothing constant for trend Like α, β is a value between zero and one It reflects the weight given to the most recent trend data Also like α, β is often 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 a in this method The closer β is to one, the stronger a trend is reflected Example Alpha 02 beta 02 % input data
Month Demand (gal) October 400 November 425 forecast 400 trend adjusted forecast error ABS error Squared error Absolute % error December 450 40000 0 400 25 25 625 588% January 414 February 460 March 500 April 550 May 600 June 650 July 7680 40500 1 406 44 44 1936 978% 41400 26 4166-26 26 676-063% 41400 208 41608 4392 4392 1928966 955% 42320 3504 426704 73296 73296 5372304 1466% 43856 58752 4444352 1055648 1055648 1114393 1919% 46085 915776 4700058 1299942 1299942 168985 2167% 48868 1289229 5015707 1484293 1484293 2203126 2284% 52094 1676669 5377094 7142291 7142291 51012315 9300% 8572328 5674695 22% alpha 04 MAD mse MAPE The trend value of 1 for December is '=02*(405-400)+08*0