Krajewski Ism Ch13 solution

Chapter

13

Forecasting

DISCUSSION QUESTIONS 1. a. There is no trend in the data. Exponential smoothing or simple moving average would be appropriate for estimating the average. b. The primary external factors that can be forecasted three days in advance and can appreciably affect air quality are wind velocity and temperature inversions. c. Weather conditions cannot be forecast two summers in advance. Medium-term causal factors affecting air quality are population, regulations and policies affecting wood burning, mass transit, use of sand and salt on roads, relocation of the airport, and scheduling of major tourism events such as parades, car races, and stock shows. d. In the area of technological forecasting, qualitative methods of forecasting are best. One such approach is the Delphi method, whereby the consensus of a panel of experts is sought. Here we would survey experts in the fields of electric-powered vehicles, coal-fired combustion for electric utilities, and development of alternatives to sand and salt on roads. We hope to determine whether to expect any technological breakthroughs sufficient to affect air quality within the next 10 years. 2. What’s Happening? Our objective in writing this discussion question is to ensure students recognize the difference between sales and demand. Demand forecasting techniques require demand data. Michael is making the common mistake of using sales data as the basis for demand forecasts. Sales are generally equal to the lesser of demand or inventory. Say that inventory matches average demand at a particular location and is 100 newspapers. However, for the current edition, demand is less than average, say 90. Michael enters sales (which happens to be equal to demand in this period) into the forecasting system, resulting in an inventory reduction at that location for the next edition. Now suppose that demand for the next edition is 110. But because inventory has been reduced to 90, only 90 newspapers will be sold. Michael would then enter sales (which happens to be equal to inventory, not demand) into the forecasting system. This approach ratchets downward and tends to starve the distribution system. Because the publication is not reliably available, some customers eventually stop looking for What’s Happening? and demand truly declines. It is important that data used for demand forecasting are demand data, not sales data.

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PROBLEMS 1. Printer rentals a. The forecast for week 11 is 29 rentals. Forecast for Following Week ( Ft +1 )

Week Forecast Calculated

At

5

23 + 24 + 32 + 26 + 31 5

= 27.2 or 27

6

24 + 32 + 26 + 31 + 28 5

= 28.2 or 28

7

32 + 26 + 31 + 28 + 32 5

= 29.8 or 30

8

26 + 31 + 28 + 32 + 35 5

= 30.4 or 30

9

31 + 28 + 32 + 35 + 26 5

= 30.4 or 30

10

28 + 32 + 35 + 26 + 24 5

= 29.0 or 29

b. The Mean Absolute Deviation is 4 rentals. Week 6 7 8 9 10

Actual 28 32 35 26 24

Forecast 27 28 30 30 30 TOTAL MAD

Absolute Error 1 4 5 4 6 20 20/5 = 4

Forecasting z CHAPTER 13 z 347

2. Dalworth Company a. Three-month simple moving average Month

Actual Sales (Thousands)

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Total Average

20 24 27 31 37 47 53 62 54 36 32 29

Three-Month Simple Moving Average Forecast

Absolute Error

Absolute % Error

Squared Error

(20+24+27)/3 = 23.67 (24+27+31)/3 = 27.33 (27+31+37)/3 = 31.67 (31+37+47)/3 = 38.33 (37+47+53)/3 = 45.67 (47+53+62)/3 = 54.00 (53+62+54)/3 = 56.33 (62+54+36)/3 = 50.67 (54+36+32)/3 = 40.67

7.33 9.67 15.33 14.67 16.33 0.00 20.33 18.67 11.67 114.00 12.67

23.65 26.14 32.62 27.68 26.34 0.00 56.47 58.34 40.24 291.48 32.39

53.73 93.51 235.01 215.21 266.67 0.00 413.31 348.57 136.19 1,762.20 195.80

Such results also can be obtained from the Time Series Forecasting Solver: Actual Data

Three-Period Moving Average

Forecast 1/1/02 2/1/02 3/1/02 4/1/02 5/1/02 6/1/02 7/1/02 8/1/02 9/1/02 10/1/02 11/1/02 12/1/02

20 24 27 31 37 47 53 62 54 36 32 29

Error

23.67 27.33 31.67 38.33 45.67 54.00 56.33 50.67 40.67

7.33 9.67 15.33 14.67 16.33 0.00 -20.33 -18.67 -11.67

Method 1—Moving Average: Three -period moving average Forecast for 1/1/03 CFE MAD MSE MAPE

32.33 12.67 12.67 195.80 32.39%

CFE

7.33 17.00 32.33 47.00 63.33 63.33 43.00 24.33 12.67

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b. Four-month simple moving average Month

Actual Sales (Thousands)

Apr. May June July Aug. Sept. Oct. Nov. Dec. Total Average

31 37 47 53 62 54 36 32 29

Four-Month Simple Moving Average Forecast (20+24+27+31)/4 = 25.5 (24+27+31+37)/4 = 29.75 (27+31+37+47)/4 = 35.5 (31+37+47+53)/4 = 42.00 (37+47+53+62)/4 = 49.75 (47+53+62+54)/4 = 54.00 (53+62+54+36)/4 = 51.25 (62+54+36+32)/4 = 46.00

Absolute Error

Absolute % Error

Squared Error

11.50 17.25 17.50 20.00 4.25 18.00 19.25 17.00 124.75 15.59

31.08 36.70 33.02 32.26 7.87 50.00 60.16 58.62 309.71 38.71

132.25 297.56 306.25 400.00 18.06 324.00 370.56 289.00 2,137.68 267.21

Similarly, using Time Series Forecasting Solver, we get: Actual Data

Four-Period Moving Average

Forecast 1/1/02 2/1/02 3/1/02 4/1/02 5/1/02 6/1/02 7/1/02 8/1/02 9/1/02 10/1/02 11/1/02 12/1/02

20 24 27 31 37 47 53 62 54 36 32 29

Error

25.50 29.75 35.50 42.00 49.75 54.00 51.25 46.00

11.50 17.25 17.50 20.00 4.25 -18.00 -19.25 -17.00

Method 1—Moving Average: Four -period moving average Forecast for 1/1/03 CFE MAD MSE MAPE

37.75 16.25 15.59 267.21 38.71%

CFE

11.50 28.75 46.25 66.25 70.50 52.50 33.25 16.25

Forecasting z CHAPTER 13 z 349

c.−e. Comparison of performance Question c. d. e.

Measure

3-Month SMA 12.67 32.39 195.80

MAD MAPE MSE

4-Month SMA 15.59 38.71 267.21

Recommendation 3-month SMA 3-month SMA 3-month SMA

3. Karl’s Copiers Week Forecast Calculated 7/3 7/10 7/17 7/24 7/31

Ft +1 = α D + (1 − α ) Ft

Ft +1

0.20(24) + 0.80(24) 0.20(32) + 0.80(24) 0.20(36) + 0.80(25.6) 0.20(23) + 0.80(27.68) 0.20(25) + 0.80(26.744)

= 24 = 25.6 or 26 = 27.68 or 28 = 26.744 or 27 = 26.3952 or 26

The forecast for the week of August 7 is 26 calls. Similarly, using Time Series Forecasting Solver, we get: Actual Data 7/3/02 7/10/02 7/17/02 7/24/02 7/31/02

24 32 36 23 25

Exponential Smoothing Forecast Error CFE 24.00 0.00 0.00 24.00 8.00 8.00 25.60 10.40 18.40 27.68 -4.68 13.72 26.74 -1.74 11.98

Method 3—Exponential Smoothing: α Initial Forecast

0.20 24.00

Forecast for 8/7/02

26.40

CFE MAD MSE MAPE

11.98 4.96 49.28 20.30%

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4. Dalworth Company (continued) a. Three-month weighted moving average (weights of 3/6, 2/6, and 1/6) Month Actual Sales Three-Month Weighted (000s) Moving Average Forecast Jan. 20 Feb. 24 Mar. 27 Apr. 31 [(3 × 27)+(2 × 24)+(l × 20)]/6 = 24.83 May 37 [(3 × 31)+(2 × 27)+(l × 24)]/6 = 28.50 June 47 [(3 × 37)+(2 × 31)+(l × 27)]/6 = 33.33 July 53 [(3 × 47)+2 × 37)+(l × 31)]/6 = 41.00 Aug. 62 [(3 × 53)+(2 × 47)+(l × 37)]/6 = 48.33 Sept. 54 [(3 × 62)+(2 × 53)+(l × 47)]/6 = 56.50 Oct. 36 [(3 × 54)+(2 × 62)+(l × 53)]/6 = 56.50 Nov. 32 [(3 × 36)+(2 × 54)+(l × 62)]/6 = 46.33 Dec. 29 [(3 × 32)+(2 × 36)+(l × 54)]/6 = 37.00 Total Average

Absolute Absolute % Error Error

6.17 8.50 13.67 12.00 13.67 2.50 20.50 14.33 8.00 99.34 11.04

19.90 22.97 29.09 22.64 22.05 4.63 56.94 44.78 27.59 250.59 27.84

The results from Time Series Forecasting Solver give the same results: Three-Period Weighted Moving Average

Forecast

24.8332 28.4999 33.3332 40.9998 48.333 56.4998 56.4997 46.3336 37.0006

Error

CFE

6.17 8.50 13.67 12.00 13.67 -2.50 -20.50 -14.33 -8.00

6.17 14.67 28.33 40.33 54.00 51.50 31.00 16.67 8.67

Method 2—Weighted Moving Average: Three -Period Weighted Moving Average Forecast for 1/1/03 CFE MAD MSE MAPE

31.17 8.67 11.04 147.09 27.84%

Squared Error

38.07 72.25 186.87 144.00 186.87 6.25 420.25 205.35 64.00 1,323.91 147.09

Forecasting z CHAPTER 13 z 351

b. Exponential smoothing (α = 0.6) Month (t) Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Total Average

Dt (millions) 20 24 27 31 37 47 53 62 54 36 32 29

Ft

Ft+1 = Ft + α(Dt − Ft)

22.00 20.80 22.72 25.29 28.72 33.69 41.67 48.47 56.59 55.04 43.62 36.64

Absolute Absolute Error % Error

20.80 22.72 25.29 28.72 33.69 41.67 48.47 56.59 55.04 43.62 36.64 32.06

5.71 8.28 13.31 11.33 13.53 2.59 19.04 11.61 7.65 93.05 10.34

18.41 22.38 28.32 21.38 21.82 4.80 52.88 36.28 26.38 232.65 25.85

Squared Error

32.60 68.56 177.16 128.37 183.06 6.71 362.52 134.79 58.52 1,152.29 128.03

c.−e. Comparison of performance Question c. d. e.

Measure MAD MAPE MSE

3-Month WMA 11.04 27.84 147.10

Exponential Smoothing 10.34 25.85 128.03

Recommendation Exponential smoothing Exponential smoothing Exponential smoothing

5. Convenience Store At = 0.2 Dt + 0.8 ( At −1 + Tt −1 ) Tt = 0.1( Averagethisperiod − Averagelastperiod ) + 0.9(Trendlastperiod ) Ft +1 = At + Tt

May AMay = 0.2 ( 760 ) + 0.8 ( 700 + 50 ) = 752

TMay = 0.1( 752 − 750 ) + 0.9 ( 50 ) = 50.2 Forecast for June = 752 + 50.2 = 802.2 or 802 June AJune = 0.2 ( 800 ) + 0.8 ( 752 + 50.2 ) = 801.76 or 802

TJune = 0.1( 801.76 − 752 ) + 0.9 ( 50.2 ) = 50.16 or 50 Forecast for July = 801.76 + 50.16 = 851.92 or 852

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July AJuly = 0.2 ( 820 ) + 0.8 ( 801.76 + 50.16 ) = 845.54 or 846

TJuly = 0.1( 845.54 − 801.76 ) + 0.9 ( 50.16 ) = 49.52 Forecast for August = 845.54 + 49.52 = 895.06 or 895 6. Community Federal AMay = 0.3 ( 680 ) + 0.7 ( 600 + 60 ) = 666 TMay = 0.2 ( 666 − 600 ) + 0.8 ( 60 ) = 61.2 FJune = 666 + 61.2 = 727.2 June AJune = 0.3 ( 710 ) + 0.7 ( 666 + 61.2 ) = 722.04

TJune = 0.2 ( 722.04 − 666 ) + 0.8 ( 61.2 ) = 60.17 Forecast for July = 722.04 + 60.17 = 782.21

July AJuly = 0.3 ( 790 ) + 0.7 ( 722.04 + 60.17 ) = 784.55

TJuly = 0.2 ( 784.55 − 722.04 ) + 0.8 ( 60.17 ) = 60.64 Forecast for August = 784.55 + 60.64 = 845.18 or 845 7. Heartville General Hospital i. Exponential smoothing, α = 0.6 Year Demand 1 2 3 4 5

45 50 52 56 58

Exponential Smoothing 41 41 + .6(45 – 41) = 43.4 43.4 + .6(50 – 43.4) = 47.4 47.4 + .6(52 – 47.4) = 50.2 50.2 + .6(56 – 50.2) = 53.7 Total Average

Absolute Deviation

Absolute % Deviation

Square Error

6.60 4.60 5.80 4.30 21.30 5.33

13.20 8.85 10.36 7.41 39.82 9.96

43.56 21.16 33.64 18.49 116.85 29.2125

Forecasting z CHAPTER 13 z 353

ii. Exponential smoothing, α = 0.9 Year Demand 1 2 3 4 5

Exponential Smoothing

45 50 52 56 58

41 41 + .9(45 – 41) = 44.6 44.6 + .9(50 – 44.6) = 49.5 49.5 + .9(52 – 49.5) = 51.8 51.8 + .9(56 – 51.8) = 55.6 Total Average

Absolute Deviation

Absolute % Deviation

5.40 2.50 4.20 2.40 14.50 3.63

10.80 4.81 7.50 4.14 27.25 6.81

Squared Error 29.16 6.25 17.64 5.76 58.81 14.7025

iii. Trend-adjusted exponential smoothing ( α = 0.6, β = 01 . ) Year

Demand

At

Tt

1 2 3 4 5

45 50 52 56 58

43.40 48.26 51.50 55.23 57.97

2.24 2.50 2.58 2.69 2.70

Ft 41.00 45.64 50.76 54.08 57.92 Total Average

Absolute Deviation 4.00 4.36 1.24 1.92 0.08 11.60 2.32

Calculations by year: Year 1 A1 : 0.6 ( 45 ) + 0.4 ( 39 + 2 ) = 27.0 + 16.4 = 43.40 T1 : 0.1 + ( 43.4 − 39.00 ) + 0.9 ( 2 ) = 0.44 + 1.80 = 2.24 F2 + T1 = 45.64 Year 2 A2 : 0.6 ( 50 ) + 0.4 ( 43.4 + 2.24 ) = 30.0 + 18.26 = 48.26

T2 : 0.1( 48.26 − 43.40 ) + 0.9 ( 2.24 ) = 0.49 + 2.02 = 2.50 F3 : A2 + T2 = 50.76

Year 3 A3 : 0.6 ( 52 ) + 0.4 ( 48.26 + 2.50 ) = 31.2 + 20.30 = 51.50

T3 : 0.1( 51.50 − 48.26 ) + 0.9 ( 2.50 ) = 0.32 + 2.25 = 2.58 F4 : A3 + T3 = 54.08

Year 4 A4 : 0.6 ( 56 ) + 0.4 ( 51.50 + 2.58 ) = 33.6 + 21.63 = 55.23

T4 : 0.1( 55.23 − 51.50 ) + 0.9 ( 2.58 ) = 0.37 + 2.32 = 2.69 F5 : A4 + T4 = 57.92

Absolute % Deviation 8.89 8.72 2.38 3.43 0.14 23.56 4.71

Squared Error 16.00 19.01 1.54 3.69 0.01 40.24 8.05

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A5 : 0.6 ( 58 ) + 0.4 ( 55.23 + 2.69 ) = 34.8 + 23.17 = 57.97

T5 : 0.1( 57.97 − 55.23) + 0.9 ( 2.69 ) = 0.27 + 2.42 = 2.70 iv. Three-year moving average Year

Demand

1 2 3 4 5

45 50 52 56 58

3-Year Moving Average

(45 + 50 + 52)/3 = 49 (50 + 52 + 56)/3 = 52.7 Total Average

Absolute Deviation

Absolute % Deviation

Square Error

7.00 5.30 12.30 6.15

12.50 9.14 21.64 10.82

49.00 28.09 77.09 38.55

Absolute Deviation

Absolute % Deviation

Squared Error

6 4 10 5

10.71 6.90 17.61 8.81

36 16 52 26

v. Three-year weighted moving average Year Demand 1 2 3 4 5

45 50 52 56 58

3-Year Weighted Moving Average

(45(1/6) + 50(2/6) + 52(3/6)) = 50 (50(1/6) + 52(2/6) + 56(3/6)) = 54 Total Average

vi. Regression model Y = 42.6 + 32 . X Year

Demand

1 2 3 4 5

45 50 52 56 58

Trend Projection 42.6 + 3.2 × 1 = 45.8 42.6 + 3.2 × 2 = 49.0 42.6 + 3.2 × 3 = 52.2 42.6 + 3.2 × 4 = 55.4 42.6 + 3.2 × 5 = 58.6

Total Average

Absolute Deviation 0.80 1.00 0.20 0.60 0.60 3.20 0.64

Absolute % Deviation 1.78 2.00 0.38 1.07 1.03 6.26 1.25

Squared Error 0.64 1.00 0.04 0.36 0.36 2.40 0.48

a.-c. Comparison of the forecasting methodologies Forecast Methodology Exponential smoothing α = .6 Exponential smoothing α = .9 Trend-adjusted exp. smoothing Three-year moving average Three-year weighted moving average Regression model

MAD

MAPE

MSE

5.33 3.63 2.32 6.15 5.00 0.64

9.96 6.81 4.71 10.82 8.81 1.25

29.21 14.70 8.05 38.55 26.00 0.48

Regression model methodology works best in this case under all performance criteria.

Forecasting z CHAPTER 13 z 355

8. Calculator sales A1 = 0.2 ( 46 ) + 0.8 ( 45 + 2 ) = 46.8

T1 = 0.2 ( 46.8 − 45 ) + 0.8 ( 2 ) = 1.96

F2 = 46.8 + 1.96 = 48.76 or 48.8 A2 = 0.2 ( 49 ) + 0.8 ( 46.8 + 1.96 ) = 48.81

T2 = 0.2 ( 48.81 − 46.8 ) + 0.8 (1.96 ) = 1.97 F3 = 48.81 + 1.97 = 50.78 or 51

A3 = 0.2 ( 43) + 0.8 ( 48.81 + 1.97 ) = 49.224

T3 = 0.2 ( 49.224 − 48.81) + 0.8 (1.97 ) = 1.659

F4 = 49.22 + 1.66 = 50.88 or 51 A4 = 0.2 ( 50 ) + 0.8 ( 49.22 + 1.66 ) = 50.7

T4 = 0.2 ( 50.7 − 49.22 ) + 0.8 (1.66 ) = 1.624

F5 = 50.7 + 1.62 = 52.32 or 52 A5 = 0.2 ( 53) + 0.8 ( 50.7 + 1.62 ) = 52.456

T5 = 0.2 ( 52.456 − 50.7 ) + 0.8 (1.62 ) = 1.648

F6 = 52.456 + 1.65 = 54.11 or 54

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9. Forrest’s boxes of chocolates a. One possible estimated forecast for Year 4: Quarter 1 2 3 4

Forecast 3,700 2,700 1,900 6,500 14,800

b. Multiplicative seasonal method Quarter 1 2 3 4 Total Average

Year 1 3,000 1,700 900 4,400 10,000 2,500

Seasonal Factor 1.20 0.68 0.36 1.76

Year 2 3,300 2,100 1,500 5,100 12,000 3,000

Seasonal Factor 1.1 0.7 0.5 1.7

Year 3 3502 2448 1768 5882 13,600 3,400

Seasonal Factor 1.03 0.72 0.52 1.73

Average Seasonal Factor 1.11 0.70 0.46 1.73

Forecast for year 4, 14,800. Average = 3,700. Quarter 1 2 3 4

Average 3,700 3,700 3,700 3,700

Factor 1.11 0.70 0.46 1.73

Forecast 4,107 2,590 1,702 6,401 14,800

This technique forecasts that the third-quarter sales will decrease compared to sales for the third quarter of the third year. Betcha thought it would increase. Mamma always said: “Life is full of surprises!” Just to make sure, we find confirmation of our calculations using the Seasonal Forecasting Solver: Quarter 1 2 3 4

Seasonal Index 1.1100 0.7000 0.4600 1.7300

Forecast 4107 2590 1702 6401

Forecasting z CHAPTER 13 z 357

10. Snyder’s Garden Center Quarter 1 2 3 4 Total Average

Seasonal Factor 0.179 1.573 1.303 0.944

Year 1 40 350 290 210 890 222.50

Year 2 60 440 320 280 1,100 275.00

Seasonal Factor 0.218 1.600 1.164 1.018

Average Seasonal Factor 0.199 1.587 1.234 0.981

Average quarterly sales in year 3 are expected to be 287.50 (1,150/4). Using the average seasonal factors, the forecasts for year 3 are: Quarter 1 2 3 4

0.199(287.50) 1.587(287.50) 1.234(287.50) 0.981(287.50)

Forecast 57 456 355 282

With the Seasonal Forecasting Solver, we get the same results Quarter 1 2 3 4

Seasonal Index 0.1990 1.5865 1.2335 0.9810

Forecast 57.213 456.119 354.631 282.038

11. Utility company Quarter 1 2 3 4 Total Average

Year 1 103.5 126.1 144.5 166.1 540.2 135.05

Year 2 94.7 116.0 137.1 152.5 500.3 125.075

Year 3 118.6 141.2 159.0 178.2 597.0 149.25

Year 4 109.3 131.6 149.5 169.0 559.4 139.85

Quarter

Year 1

Year 2

Year 3

Year 4

1 2 3 4 Total

0.7664 0.9337 1.0700 1.2299 4.0

0.7571 0.9274 1.0961 1.2193 4.0

0.7946 0.9410 1.0653 1.1940 4.0

0.7816 0.9410 1.0690 1.2084 4.0

Average Seasonal Index 0.7749 0.9371 1.0751 1.2129 4.0

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Forecast for Year 5 Quarter 1 2 3 4

Average Demand per Quarter 150 150 150 150 600

Adjusted Demand 116.235 140.565 161.265 181.935 600

= = = =

116 141 161 182

Turning to the Seasonal Forecasting Solver, we get the same results: Quarter 1 2 3 4

Seasonal Index 0.7749 0.9371 1.0751 1.2129

Forecast 116.235 140.565 161.265 181.935

12. Garcia’s Garage a. The results, using the Regression Analysis Solver, are: R-squared

0.668

R

0.817

Constant Standard Error of Estimate Trial X1 Value

42.464 4.572 9

X1 Coefficient Predicted Y Value

The regression equation is Y = 42.464 + 2.452X b. Forecasts Y (Sep) = 42.464 + 2.452 (9) = 64.532 or 65 Y (Oct) = 42.464 + 2.452 (10) = 66.984 or 67 Y (Nov) = 42.464 + 2.452 (11) = 71.888 or 72

2.452 64.532

Forecasting z CHAPTER 13 z 359

13. Hydrocarbon Processing Factory Using the Regression Analysis Solver, we get: R-squared

0.450

R

0.671

Constant

0.888

Standard Error of Estimate

0.331

Trial X1 Value

3.5

X1 Coefficient Predicted Y Value

0.622 3.065

a. Relationship to forecast Y from X Y = 0.888 + 0.622 X b. Strength of relationship between Y and X is moderate as indicated by R2 = 0.450 R = 0.671 Standard Error of Estimate = 0.331 14. Ohio Swiss Milk The results from the Regression Analysis Solver are: R-squared R

0.888 -0.942

Constant Standard Error of Estimate Trial X1 Value

1121.212 12.342 325

X1 Coefficient Predicted Y Value

-0.282 1029.562

a. Y = 1,121.212. − 0.282 X b. R2 = 0.888 R = −0.942 indicates a fairly strong negative relationship. Increases in costs explain 89% of the decreases in gallons sold c. Y = 1,121.212 − 0.282 (325) = 1,029.562

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ADVANCED PROBLEMS

15. Large Public Library Using the Time Series Forecasting Solver, we get the following results by varying the number of periods (n) in the Simple Moving Average method: n 1 2 3 4 5 10

Forecast for January, Year 4 2,451 2,299 2,221 2,127 2,037 2,189

MAD

CFE

282 267 257 242 242 216

604 –68 –291 –524 –846 –444

In general, as n increases, MAD decreases and CFE (bias) increases. The two-month moving average appears to be the best because of its relatively low MAD and CFE. 16. Large Public Library (continued, using 1,847 as initial average) Using the Time Series Forecasting Solver, we get the following results by varying α in the exponential smoothing model: α 0.10 0.20 0.30 0.50 0.65 0.70 0.80 1.00

Forecast for January, Year 4 2,193 2,178 2,186 2,259 2,325 2,346 2,385 2,451

MAD 257 249 249 251 248 249 254 274

CFE 3,458 1,655 1,131 823 736 713 673 604

As α increases, CFE (bias) generally decreases and MAD generally increases. When alpha = 0.65 there seems to be a good combination of low bias and low MAD.

Forecasting z CHAPTER 13 z 361

17. Large Public Library (continued, using 1,847 as initial average and 0 as initial trend) Using the Time Series Forecasting Solver, we get the following results by varying α and β in the Trend-Adjusted Exponential Smoothing model: α

β

0.10 0.20 0.20 0.30 0.35 0.40 0.45

0.1 0.1 0.2 0.1 0.1 0.1 0.1

Forecast for January, Year 4 2,257 2,144 2,077 2,139 2,153 2,173 2,198

MAD

CFE

286 271 271 274 276 278 279

983 876 876 1,102 1,058 1,092 1,113

Equating both α and β to 0.2 gives fairly low values for both MAD and CFE (bias). The two month moving average seems to be the best forecast because of the relatively low CFE and MAD results. 18. Cannister Inc. a. Multiplicative Seasonal Method Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Total Average

1 742 697 776 898 1,030 1,107 1,165 1,216 1,208 1,131 971 783 11,724 977

2 741 700 774 932 1,099 1,223 1,290 1,349 1,341 1,296 1,066 901 12,712 1,059.333

3 896 793 885 1,055 1,204 1,326 1,303 1,436 1,473 1,453 1,170 1,023 14,017 1,168.083

4 951 861 938 1,109 1,274 1,422 1,486 1,555 1,604 1,600 1,403 1,209 15,412 1,284.333

5 1,030 1,032 1,126 1,285 1,468 1,637 1,611 1,608 1,528 1,420 1,119 1,013 15,877 1,323.083

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Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Managing Value Chains

1

2

3

4

5

0.759 0.713 0.794 0.919 1.054 1.133 1.192 1.245 1.236 1.158 0.994 0.801

0.699 0.661 0.731 0.880 1.037 1.154 1.218 1.273 1.266 1.223 1.006 0.851

0.767 0.679 0.758 0.903 1.031 1.135 1.116 1.229 1.261 1.244 1.002 0.876

0.740 0.670 0.730 0.863 0.992 1.107 1.157 1.211 1.249 1.246 1.092 0.941

0.778 0.780 0.851 0.971 1.110 1.237 1.218 1.215 1.155 1.073 0.846 0.766

Average Seasonal Index 0.749 0.701 0.773 0.907 1.045 1.153 1.180 1.235 1.233 1.189 0.988 0.847

b. Simple Linear Regression Model to forecast annual sales Y = 10, 646.6 + 1,100.6 X c. Forecast for Year 6 (or X = 6 ) is: Y = 17, 250.2 d. Monthly Seasonal Forecast for Year 6 (17,250/12)*Average Seasonal Index Jan Feb Mar Apr May Jun

1,076.7 1,007.3 1,110.9 1,304.4 1,501.9 1,658.1

Jul Aug Sep Oct Nov Dec

1,696.4 1,774.9 1,773.1 1,708.9 1,420.2 1,217.5

19. Midwest Computer Company a. Trend-adjusted exponential smoothing α = 0.35, β = 015 . α = 0.70, β = 10 .

MAD 65.17 90.67

CFE –18.67 –1.22*

The trend-adjusted exponential smoothing model provides the lowest CFE (bias) with a reasonable MAD value when α = 0.7 and β = 10 . . The forecasts for weeks 51–53 using this method are Month 51 52 53

Forecast Sales 2,452 2,493 2,535

b. The linear regression model is Y = -152.578 + 4.910 X or Sales = -152.578 + 4.910 (number of leases)

Forecasting z CHAPTER 13 z 363

The forecast for months 51–53 can be based on actual lease information for months 48–50. Month 51 52 53

Forecast Sales 2,283 2,347 2,411

These results are confirmed by the following output from the Regression Analysis Solver, including the forecasts for month 51 (when X = 496):

Results Solver - Regression Analysis R-squared

0.988

R

0.994

Constant

-152.578

Standard Error of Estimate

81.186

Trial X1 Value Trial X2 Value Trial X3 Value Trial X4 Value Trial X5 Value

496 X1 Coefficient X2 Coefficient X3 Coefficient X4 Coefficient X5 Coefficient Predicted Y Value

4.910 0.000 0.000 0.000 0.000 2282.782

c. The linear regression model has a MAD of 632.81 and a CFE (bias) of –603.1. Therefore, the trend-adjusted exponential smoothing model of part (a) will still provide the best forecasts. Bias (Mean Error) -603.1 MAD (Mean Absolute Deviation) 632.81 MSE (Mean Squared Error) 648630.6 Standard Error (denom=n-2=48) 821.98 MAPE (Mean Absolute Percent Error) 1.9 The actual data are (give to students for comparison): Month 51 52 53

Actual Sales 2,450 2,497 2,526

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20. P&Q Supermarkets Numerous methods could be used. Moving Average Number of Periods 2 3 4 5 6 7 12

MAD 7.09 6.87 6.21 4.52 4.80 4.94 5.15

CFE 5.00 13.67 12.75 1.00 7.00 7.43 3.08

Forecast 39.50 41.33 41.50 44.80 44.17 43.43 43.08

Exponential smoothing Use α = 0.20 for the lowest MAD. MAD 5.34 7.52

α = 0.20 α = 100 .

CFE 47.62 5.00

Forecast 42.53 38

Use α = 1.00 (equivalent of naïve model) for the lowest bias. Exponential smoothing with trend Best MAD

MAD 5.34

CFE 47.64

Forecast 42.53

Best Bias

MAD 6.72

Bias –0.03

Forecast 39.81

α = 0.20 β = 0.00

α = 0.63 β = 0.05

A graphic plot of the data reveals a spike every fifth data point. The best of the preceeding methods would appear to be the moving average with five periods, due to its relatively low MAD and CFE (bias). But none of these methods reasonably account for the fifth-period spike. Another way to deal with this cyclical data is to use the multiplicative seasonal method with five periods in a cycle. With this method, the forecast for the 25th month would be 60.

Period 1 2 3 4 5

Cycle 1 33 37 31 39 54 194

Seasonal Index 0.1701 0.1907 0.1598 0.2010 0.2784

Cycle 2 38 42 40 41 54 215

Seasonal Index 0.1767 0.1953 0.1860 0.1907 0.2512

Cycle 3 43 39 37 43 56 218

Seasonal Index 0.1972 0.1789 0.1697 0.1972 0.2569

Cycle 4 41 36 39 41 58 215

Seasonal Index 0.1907 0.1674 0.1814 0.1907 0.2698

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Month 21 22 23 24 25

Actual Cycle 5 42 45 41 38 ??

Average Seasonal Index 0.1837 0.1831 0.1742 0.1949 0.2640

Forecast Cycle 5 42 42 40 44 60 227

The demand forecast (227 units) for all of cycle 5, which was used as the basis for the forecasts for months 21−25, comes from the following regression model: Linear Regression, Y = 38.989 + .241X When X = 25 , Y = 45.014 . These regression results are obtained from the Regression Analysis Solver, which gives the following output:

Results Solver - Regression Analysis R-squared R Constant Standard Error of Estimate Trial X1 Value

0.060 0.246 38.989 6.873 25 X1 Coefficient Predicted Y Value

0.241 45.014

21. Air visibility a. These data contain no detectable trend component over the two-summer history. Either the simple moving average or exponential smoothing should give unbiased results. As long as forecast parameters are chosen for stability (high n in simple moving average, or low α in exponential smoothing), the forecast for August 31 will be about 120.5. Because the data are stable, linear regression will yield a nearly zero slope with a forecast of about 120.5 as well. b. There are no historical data to support expectations for air quality in the third year to be any different than that of the first two years. It will average about 120.5 unless public policy affects transportation, population growth, or utilities burning natural gas rather than coal. That might make a detectable difference in air quality. However, the effects of public policy are not recorded in the database, so qualitative forecasting methods are needed.

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22. Flatlands Public Power District The historical data show both trend and seasonal components. We will use the multiplicative seasonal method to forecast demand for the next year, then look for a lowdemand period of two weeks during which the Comstock plant can be serviced. Weeks 7 and 8 look like the best two-week period to schedule maintenance. Demand Seasonal Demand Seasonal Demand Seasonal Demand Seasonal Demand Seasonal Week Year 1 Index Year 2 Index Year 3 Index Year 4 Index Year 5 Index 1 2,050 0.1017 2,000 0.0959 1,950 0.0922 2,100 0.1010 2,275 0.1064 2 1,925 0.0955 2,075 0.0995 1,800 0.0851 2,400 0.1154 2,300 0.1076 3 1,825 0.0906 2,225 0.1067 2,150 0.1017 1,975 0.0950 2,150 0.1006 4 1,525 0.0757 1,800 0.0863 1,725 0.0816 1,675 0.0805 1,525 0.0713 5 1,050 0.0521 1,175 0.0564 1,575 0.0745 1,350 0.0649 1,350 0.0632 6 1,300 0.0645 1,050 0.0504 1,275 0.0603 1,525 0.0733 1,475 0.0690 7 1,200 0.0596 1,250 0.0600 1,325 0.0626 1,500 0.0721 1,475 0.0690 8 1,175 0.0583 1,025 0.0492 1,100 0.0520 1,150 0.0553 1,175 0.0550 9 1,350 0.0670 1,300 0.0624 1,500 0.0709 1,350 0.0649 1,375 0.0643 10 1,525 0.0757 1,425 0.0683 1,550 0.0733 1,225 0.0589 1,400 0.0655 11 1,725 0.0856 1,625 0.0779 1,375 0.0650 1,225 0.0589 1,425 0.0667 12 1,575 0.0782 1,950 0.0935 1,825 0.0863 1,475 0.0709 1,550 0.0725 13 1,925 0.0955 1,950 0.0935 2,000 0.0946 1,850 0.0889 1,900 0.0889 20,850 21,150 20,800 21,375 Total 20,150

Week 1 2 3 4 5 6 7 8 9 10 11 12 13 Total

Demand Year 6 2,147 2,172 2,135 1,707 1,343 1,371 1,396 1,164 1,422 1,475 1,529 1,733 1,992 21,585

Average Seasonal Index 0.0995 0.1006 0.0989 0.0791 0.0622 0.0635 0.0647 0.0539 0.0659 0.0683 0.0708 0.0803 0.0923

Linear Regression Y = 20,145 + 240 ( X ) . When X = 6 , Y = 21,585 .

Forecasting z CHAPTER 13 z 367

23. A manufacturing firm The results from the Regression Analysis Solver are:

Results Solver - Regression Analysis R-squared

0.933

R

0.966

Constant

4.184

Standard Error of Estimate

5.126

Trial X1 Value

80 X1 Coefficient Predicted Y Value

0.943 79.624

a. From the output, the relationship is Rating = 4.184 + 0.943 (Score) b. Score = 80 Rating = 4.184 + 0.943 (80) = 79.624 c. R = R 2 = 0.934 = 0.966 There is a very strong positive relationship. Increases in scores explain 93% of increases in ratings.

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24. Materials handing The results from the Regression Analysis Solver are: R-squared

0.477

R

0.691

Constant

323.791

Standard Error of Estimate

283.358

Trial X1 Value

3

X1 Coefficient Predicted Y Value

a. From the Solver output shown, the relationship is Cost = 323.791 + 131.640 (Age) b. Annual Cost to maintain a three-year old tractor y = 323.791 + 131.640 (3) = $718.71 Annual Cost to maintain a fleet of 20 three-year-old tractors = (20) ($718.711) = $14,374.22

131.640 718.711

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CASE: YANKEE FORK AND HOE COMPANY A. Synopsis Yankee Fork and Hoe is a company that produces garden tools for a mature, pricesensitive market in which customers also want on-time delivery. Recently customers have been complaining about late shipments. The president has hired a consultant to look into the problem. The consultant traces the production planning process and its reliance on accurate forecasts. The consultant must make a recommendation to management. B. Purpose This case provides the basis for a discussion of the need for accurate forecasts in an industry where low-cost production is critical. It also contains sufficient data to enable the student to generate forecasts for each month of the following year. Specifically, the case can be used to: 1. Discuss the effects of poor forecasts on capacities and schedules. 2. Discuss the choice of the proper data to use for forecasts. 3. Quantitatively analyze forecasting data and provide forecasts for the following year. C. Analysis Yankee Fork and Hoe is experiencing two major problems with the current forecasting system. First, the production department is unaware of how marketing arrives at its forecasts. Production views the forecasts as the result of an overinflated estimate of actual customer demand. However, the forecasting technique in use by the marketing department is based on actual shipments rather than on actual demand. Second, marketing, in its desire to reflect production capacity, is compounding the problems experienced by Yankee Fork and Hoe by trying to rectify past problems. Although marketing adjusts for shortages in the actual shipment data, it is still reflecting past problems and not future demand. If Yankee would move to a system that utilizes past demand to forecast future demand, production would be able to schedule bow rake production more effectively. In addition, production must be aware of how the forecasts are made and what information is being provided so that arbitrary adjustments are no longer needed. A forecasting system based on actual demands requires careful analysis of Exhibit TN. 1. It is apparent that the bow rake experiences seasonal demand. It is also obvious that there is an upward trend in the annual demand. A forecasting system that recognizes both of these factors is desirable. To arrive at the average monthly demand for year 5, the average increase in the average monthly demands was determined to be 2,589 units. Therefore the average monthly demand for year five is 45,928 + 2,589 = 48,517. This value is then multiplied by the average seasonal factors (see Exhibit TN.2) to arrive at the forecast shown in Exhibit TN.1. Exhibits TN.3 and TN.4 show graphs of the series.

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D. Recommendations The recommendations to management could include the following: 1. Improve the lines of communication between marketing and production regarding the preparation of forecasts. This will eliminate arbitrary adjustments to the forecasts. 2. Use actual demand data rather than shipment data. 3. Use models that somehow handle seasonality, such as the seasonal forecast method, the weighted moving average (with significant weights placed on time periods lagged by one year), or regression a trend variable and also dummy variables for the seasons. 4. Consider a combination forecasting approach or possibly focus forecasting, rather than using a single model. E. Teaching Suggestions: As an Experiential Exercise This case makes for an excellent team-based experiential exercise, spread over two days. Presumably the basic concepts and techniques of forecasting have already been covered. The exercise might take 45 minutes on the first day and 30 minutes on the second day. Day 1 Introduce the exercise after the basic concepts and techniques of forecasting have been covered. Students should have read the case beforehand, and each team should bring at least one laptop to class. To get things started, briefly open up and demonstrate three solvers: 1. Regression Analysis (describe how you should use it with one independent variable for the trend, and dummy variables for some of the major seasons) 2. Seasonal Forecasting 3. Time-Series Forecasting (which represents four basic models and countless options in their use) Have the team members discuss among themselves which forecasting methods might be best, and begin to experiment with some of the models to see how they perform. Have them do their analysis only using data from the first three years, and reserving the fourth year as a holdout sample. They should totally block out that information, as it will provide the “acid test” for their assignment due on the second day. After they get into the project and determine their general approach, give them the assignment for the next day. Day 2 Between the first day and the second session, each team is to develop combination forecasts for the holdout sample (year 4). They must commit to their combination forecasting procedure (such as which methods to include in the combination and their weights) before they evaluate its results for the holdout sample. They are to prepare a short report on their results. On the first page of their report, they should describe the approach taken and indicate why they are confident in their forecasts. On the subsequent page(s) they should show a spreadsheet of actual demand, forecasts (from two or more individual methods and then the combination), period-by-period forecast error terms, and summary error measures

Forecasting z CHAPTER 13 z 371

(CFE, MAD, MAPE, and MSE). They can hand compute the errors, or develop formulas to make the calculations (perhaps borrowing some of the formulas used in the Time Series Forecasting Solver’s “worksheet”). If students use dynamic models, they must “bootstrap” one period at time. If judgment is used as one forecasting technique, the team must control what information the “judgment expert” is given (such as time series model information to date). Actually, a judgment forecasting approach is unlikely to be effective because students have no “contextual knowledge.” It might be convenient to have the teams not only submit hard copy, but also e-mail their results to the instructor before class. If done this way, have the elements in the report combined into one electronic file (such as using the Edit/Paste Special/Picture option to insert spreadsheets and graphs into a Word document. Based on experience to date, a team typically reports CFE values of plus/minus 20,000 for CFE, 6,000 for MAD, 22% for MAPE, and 85,000,000 for MSE. In all cases to date, the combination forecast did better than any individual forecasting method. F. Teaching Suggestions: Out-of-Class Exercise This case should be made an overnight assignment because the student needs to develop forecasts for year 5. A computer program can be used to get the forecasts; however, it is not mandatory. The forecasts contained in Exhibit TN.1 were done manually using the multiplicative seasonal method described in the text. This case is based on an actual company that supplies garden tools to companies such as Sears and Scott’s & Sons. The initial discussion should focus on the competitive priorities for Yankee Fork and Hoe (low costs and on-time delivery) and how operations can support these priorities. The need for accurate forecasts in that sort of competitive environment should be emphasized. The instructor should raise the question, “How would you revise the forecasting system in use at Yankee Fork and Hoe?” This discussion will lead to the issue of which data (shipments or actual demands) to use and how the marketing and production departments can coordinate on the development of the forecasts. Finally, the students can be asked to present their forecasts (perhaps on blank transparencies provided with the assignment). Discuss how each student’s forecast was developed and explore the reasons for the differences between the students’ forecasts. The forecast provided in Exhibit TN. I can be used as a benchmark. G. Board Plan Board 1 Competitive Priorities Low costs On-time delivery

Operations Support Efficient internal schedules Proper inventory levels Good supplier contracts

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Board 2 Current Forecasting System Based on shipments One month’s lead on promotions Marketing passed to production Second-guessing marketing

EXHIBIT TN.1

Month 1 2 3 4 5 6 7 8 9 10 11 12 Averages

Actual Bow Rake Demands and Forecast

Year 1 55,220 57,350 15,445 27,776 21,408 17,118 18,028 19,883 15,796 53,665 83,269 72,991 38,162

EXHIBIT TN.2 Month 1 2 3 4 5 6 7 8 9 10 11 12

Proposed Forecasting System Based on actual demands Several month’s lead on promotions Coordinated between marketing and production Take the forecasts as given

Actual Demands Year 2 Year 3 39,875 32,180 64,128 38,600 47,653 25,020 43,050 51,300 39,359 31,790 10,317 32,100 45,194 59,832 46,530 30,740 22,105 47,800 41,350 73,890 46,024 60,202 41,856 55,200 40,620 44,805

Year 4 62,377 66,501 31,404 36,504 16,888 18,909 35,500 51,250 34,443 68,088 68,175 61,100 45,928

Forecast 70,203 72,911 19,636 35,312 27,217 21,763 22,920 25,278 20,082 68,226 105,862 92,796 48,517

Year 4 1.358 1.448 0.684 0.795 0.368 0.412 0.773 1.116 0.750 1.482 1.484 1.330

Average 1.126 1.348 0.705 0.932 0.652 0.452 0.923 0.867 0.694 1.389 1.536 1.376

Seasonal Factors Year 1 1.447 1.503 0.405 0.728 0.561 0.449 0.472 0.521 0.414 1.406 2.182 1.913

Year 2 0.982 1.579 1.173 1.060 0.969 0.254 1.113 1.145 0.544 1.018 1.133 1.030

Year 3 0.718 0.862 0.558 1.145 0.710 0.694 1.335 0.686 1.067 1.649 1.344 1.232

Forecasting z CHAPTER 13 z 373

EXHIBIT TN.3

Monthly Demands

90000

Year 1 Year 2 Year 3 Year 4

80000 70000 60000 50000 40000 30000 20000 10000 1

2

EXHIBIT TN.4

4

3

6 7 Months

5

8

10

9

11 12

Four-Year Plot

100000 80000 60000 40000 20000

1

6

11

16

21

26

Months

31

36

41

46