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# Ch04pp

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### Ch04pp

1. 1. Practice Problems: Chapter 4, Forecasting Problem 1: Auto sales at Carmen’s Chevrolet are shown below. Develop a 3-week moving average. Week Auto Sales 1 8 2 10 3 9 4 11 5 10 6 13 7 - Problem 2: Carmen’s decides to forecast auto sales by weighting the three weeks as follows: Weights Applied Period 3 Last week 2 Twoweeks ago 1 Three weeks ago 1
2. 2. 6 Total Problem 3: A firm uses simple exponential smoothing with α = 0.1 to forecast demand. The forecast for the week of January 1 was 500 units whereas the actual demand turned out to be 450 units. Calculate the demand forecast for the week of January 8. Problem 4: Exponential smoothing is used to forecast automobile battery sales. Two value of α are examined, α = 0.8 and α = 0.5. Evaluate the accuracy of each smoothing constant. Which is preferable? (Assume the forecast for January was 22 batteries.) Actual sales are given below: Month Actual Forecast Battery Sales January 20 22 February 21 March 15 April 14 May 13 June 16 2
3. 3. Problem 5: Use the sales data given below to determine: (a) the least squares trend line, and (b) the predicted value for 2003 sales. Year Sales (Units) 1996 100 1997 110 1998 122 1999 130 2000 139 2001 152 2002 164 To minimize computations, transform the value of x (time) to simpler numbers. In this case, designate year 1996 as year 1, 1997 as year 2, etc. 3
4. 4. Problem 6: Given the forecast demand and actual demand for 10-foot fishing boats, compute the tracking signal and MAD. Year Forecast Actual Demand Demand 1 78 71 2 75 80 3 83 101 4 84 84 5 88 60 6 85 73 Problem: 7 Over the past year Meredith and Smunt Manufacturing had annual sales of 10,000 portable water pumps. The average quarterly sales for the past 5 years have averaged: spring 4,000, summer 3,000, fall 2,000 and winter 1,000. Compute the quarterly index. Problem: 8 Using the data in Problem, Meredith and Smunt Manufacturing expects sales of pumps to grow by 10% next year. Compute next year’s sales and the sales for each quarter. 4
5. 5. ANSWERS: Problem 1: Moving average = ∑ demand in previous n periods Three-Week Average n Week Auto Sales Moving 1 8 2 10 3 9 4 11 (8 + 9 + 10) / 3 = 9 5 10 (10 + 9 + 11) / 3 = 10 6 13 (9 + 11 + 10) / 3 = 10 7 - (11 + 10 + 13) / 3 = 11 1/3 5
6. 6. Problem 2: Weighted moving average = ∑ (weight for period n)(demand in period n) ∑ weights Week Auto Sales Three-Week Moving Average 1 8 2 10 3 9 4 11 [(3*9) + (2*10) + (1*8)] / 6 = 9 1/6 5 10 [(3*11) + (2*9) + (1*10)] / 6 = 10 1/6 6 13 [(3*10) + (2*11) + (1*9)] / 6 = 10 1/6 7 - [(3*13) + (2*10) + (1*11)] / 6 = 11 2/3 Problem 3: Ft = Ft −1 + α ( A t −1 − Ft −1 ) = 500 + 0.1( 450 − 500) = 495 units 6
7. 7. Problem 4: Month Actual Rounded Battery Sales Forecast with a =0.8 Absolute Deviation with a =0.8 Rounded Forecast with a =0.5 Absolute Deviation with a =0.5 January 20 22 2 22 2 February 21 20 1 21 0 March 15 21 6 21 6 April 14 16 2 18 4 May 13 14 1 16 3 June 16 13 3 14.5 1.5 S = 15 2.56 2.95 3.5 SE S = 16 3.9 On the basis of this analysis, a smoothing constant of a = 0.8 is preferred to that of a = 0.5 because it has a smaller MAD. 7
8. 8. Problem 5: Year Time Sales X2 Period (Units) (X) (Y) XY 1996 1 100 1 100 1997 2 110 4 220 1998 3 122 9 366 1999 4 130 16 520 2000 5 139 25 695 2001 6 152 36 912 2002 7 164 49 1148 S X = S Y S S XY 28 =917 X2=140 = 3961 x= ∑ x = 28 = 4 y= ∑ y = 917 = 131 b= n 7 n 7 ∑ xy − nxy = 3961 − (7)(4)(131) = 293 = 10.46 140 − (7)( 4 ) 28 ∑ x − nx 2 2 2 a = y − bx = 131 − (10.46 × 4) = 89.16 Therefore, the least squares trend equation is:  y = a + bx = 89.16 + 10.46 x To project demand in 2003, we denote the year 2003 as x = 8, and: Sales in 2003 = 89.16 + 10.46 * 8 = 172.84 8
9. 9. Problem 6: Year Forecast Actual Error RSFE Demand Demand 1 78 71 -7 -7 2 75 80 5 -2 3 83 101 18 16 4 84 84 0 16 5 88 60 -28 -12 6 85 73 -12 -24 MAD = ∑ Forecast errors n = 70 = 11.7 6 Year Forecast Actual |Forecast Cumulative MAD Tracking Demand Demand Error| Error Signal 1 78 71 7 7 7.0 -1.0 2 75 80 5 12 6.0 -0.3 3 83 101 18 30 10.0 +1.6 4 84 84 0 30 7.5 +2.1 5 88 60 28 58 11.6 -1.0 6 85 73 12 70 11.7 -2.1 Tracking Signal = RFSE −24 = = 2.1 MADs MAD 11.7 9
10. 10. Problem 7: Sales of 10,000 units annually divided equally over the 4 seasons is 10,000 / 4 = 2,500 and the seasonal index for each quarter is: spring 4,000 / 2,500 = 1.6; summer 3,000 / 2,500 = 1.2; fall 2,000 / 2,500 =.8; winter 1,000 / 2,500 =.4. Problem 8: . Next years sales should be 11,000 pumps (10,000 * 110 = 11,000). Sales for each quarter should be 1/4 of the annual sales * the quarterly index. Spring = (11,000 / 4) *1.6 = 4,400; Summer = (11,000 / 4) *1.2 = 3,300; Fall = (11,000 / 4) *.8 = 2,200; Winter = (11,000 / 4) *.4.= 1,100. 10