What is the approximate forecast for May using a two month weighted moving average

This appendix contains these topics:

• Section A.1, "Forecast Calculation Methods,"

• Section A.2, "Forecast Performance Evaluation Criteria,"

• Section A.3, "Method 1 - Specified Percent Over Last Year,"

• Section A.4, "Method 2 - Calculated Percent Over Last Year,"

• Section A.5, "Method 3 - Last year to This Year,"

• Section A.6, "Method 4 - Moving Average,"Section A.7, "Method 5 - Linear Approximation,"

• Section A.7, "Method 5 - Linear Approximation,"

• Section A.8, "Method 6 - Least Square Regression,"

• Section A.9, "Method 7 - Second Degree Approximation,"

• Section A.10, "Method 8 - Flexible Method,"

• Section A.11, "Method 9 - Weighted Moving Average,"

• Section A.12, "Method 10 - Linear Smoothing,"

• Section A.13, "Method 11 - Exponential Smoothing,"

• Section A.14, "Method 12 - Exponential Smoothing with Trend and Seasonality,"

• Section A.15, "Evaluating the Forecasts,"

• Section A.16, "Mean Absolute Deviation (MAD)."

A.1 Forecast Calculation Methods

Twelve methods of calculating forecasts are available. Most of these methods provide for limited user control. For example, the weight placed on recent historical data or the date range of historical data used in the calculations might be specified. The following examples show the calculation procedure for each of the available forecasting methods, given an identical set of historical data.

The following examples use the same 2004 and 2005 sales data to produce a 2006 sales forecast. In addition to the forecast calculation, each example includes a simulated 2005 forecast for a three month holdout period (processing option 19 = '3') which is then used for percent of accuracy and mean absolute deviation calculations (actual sales compared to simulated forecast).

A.2 Forecast Performance Evaluation Criteria

Depending on your selection of processing options and on the trends and patterns existing in the sales data, some forecasting methods will perform better than others for a given historical data set. A forecasting method that is appropriate for one product may not be appropriate for another product. It is also unlikely that a forecasting method that provides good results at one stage of a product's life cycle will remain appropriate throughout the entire life cycle.

You can choose between two methods to evaluate the current performance of the forecasting methods. These are Mean Absolute Deviation (MAD) and Percent of Accuracy (POA). Both of these performance evaluation methods require historical sales data for a user specified period of time. This period of time is called a holdout period or periods best fit (PBF). The data in this period is used as the basis for recommending which of the forecasting methods to use in making the next forecast projection. This recommendation is specific to each product, and may change from one forecast generation to the next. The two forecast performance evaluation methods are demonstrated in the pages following the examples of the twelve forecasting methods.

A.3 Method 1 - Specified Percent Over Last Year

This method multiplies sales data from the previous year by a user specified factor; for example, 1.10 for a 10% increase, or 0.97 for a 3% decrease.

Required sales history: One year for calculating the forecast plus the user specified number of time periods for evaluating forecast performance (processing option 19).

Month	2004 Sales	2005 Sales	2006 Forecast	Simulated 2005 Forecast
January	125	128	147
February	132	117	135
March	115	115	132
April	137	125	144
May	122	122	140
June	130	137	158
July	141	129	148
August	128	140	161
September	118	131	151
October	123	114	131	141.45
November	139	119	137	159.85
December	133	137	158	152.95

A.3.1 Forecast Calculation

User specified factor (processing option 1a) = 1.15 in this example.

A.3.2 Simulated Forecast Calculation

October, 2004 sales = 123 * 1.15 = 141.45

November, 2004 sales = 139 * 1.15 = 159.85

December, 2004 sales = 133 * 1.15 = 152.95

A.3.3 Percent of Accuracy Calculation

POA = (141.45 + 159.85 + 152.95) / (114 + 119 + 137) * 100 = 454.25 / 370 = 122.770

A.3.4 Mean Absolute Deviation Calculation

MAD = (|141.45 - 114| + |159.85 - 119| + |152.95 - 137|) / 3 = (27.45 + 40.85 + 15.95) / 3 = 84.25/3 = 28.08

A.4 Method 2 - Calculated Percent Over Last Year

This method multiplies sales data from the previous year by a factor calculated by the system.

Required sales history: One year for calculating the forecast plus the user specified number of time periods for evaluating forecast performance (processing option 19).

Month	2004 Sales	2005 Sales	2006 Forecast	Simulated 2005 Forecast
January	125	128	120
February	132	117	110
March	115	115	108
April	137	125	117
May	122	122	114
June	130	137	128
July	141	129	121
August	128	140	131
September	118	131	123
October	123	114	107	127.13178
November	139	119	111	143.66925
December	133	137	128	137.4677

A.4.1 Forecast Calculation

Range of sales history to use in calculating growth factor (processing option 2a) = 3 in this example.

Sum the final three months of 2005: 114 + 119 + 137 = 370

Sum the same three months for the previous year: 123 + 139 + 133 = 395

The calculated factor = 370/395 = 0.9367

Calculate the forecasts:

January, 2005 sales = 128 * 0.9367 = 119.8036 or about 120

February, 2005 sales = 117 * 0.9367 = 109.5939 or about 110

March, 2005 sales = 115 * 0.9367 = 107.7205 or about 108

A.4.2 Simulated Forecast Calculation

Sum the three months of 2005 prior to holdout period (July, Aug, Sept):

129 + 140 + 131 = 400

Sum the same three months for the previous year:

141 + 128 + 118 = 387

The calculated factor = 400/387 = 1.033591731

Calculate simulated forecast:

October, 2004 sales = 123 * 1.033591731 = 127.13178

November, 2004 sales = 139 * 1.033591731 = 143.66925

December, 2004 sales = 133 * 1.033591731 = 137.4677

A.4.3 Percent of Accuracy Calculation

POA = (127.13178 + 143.66925 + 137.4677) / (114 + 119 + 137) * 100 = 408.26873 / 370 * 100 = 110.3429

A.4.4 Mean Absolute Deviation Calculation

MAD = (|127.13178 - 114| + |143.66925 - 119| + |137.4677- 137|) / 3 = (13.13178+ 24.66925 + 0.4677)/3 = 12.75624

A.5 Method 3 - Last year to This Year

This method copies sales data from the previous year to the next year.

Required sales history: One year for calculating the forecast plus the number of time periods specified for evaluating forecast performance (processing option 19).

Month	2004 Sales	2005 Sales	2006 Forecast	Simulated 2005 Forecast
January	125	128	128
February	132	117	117
March	115	115	115
April	137	125	125
May	122	122	122
June	130	137	137
July	141	129	129
August	128	140	140
September	118	131	131
October	123	114	114	123
November	139	119	119	139
December	133	137	137	133

A.5.1 Forecast Calculation

January 2005 sales = January 2006 forecast = 128

February 2005 sales = February 2006 forecast = 117

March 2005 sales = March 2006 forecast = 115

A.5.2 Simulated Forecast Calculation

October 2004 sales = 123

November 2004 sales = 139

December 2004 sales = 133

A.5.3 Percent of Accuracy Calculation

POA = (123 + 139 + 133) / (114 + 119 + 137) * 100 = 395/370 * 100 = 106.7567

A.5.4 Mean Absolute Deviation Calculation

MAD = (|123-114| + |139 - 119| + |133 - 137|) / 3 = (9 + 20 + 4)/3 = 11

A.6 Method 4 - Moving Average

This method averages a user specified number of months (processing option 4a) to project the next months demand.

Required sales history: Twice the number of periods to be included in the average (processing option 4a), plus number of time periods for evaluating forecast performance (processing option 19).

Month	2004 Sales	2005 Sales	2006 Forecast	Simulated 2005 Forecast
January	125	128	123
February	132	117	126
March	115	115	129
April	137	125	126
May	122	122	127
June	130	137	127
July	141	129	127
August	128	140	127
September	118	131	127
October	123	114	127	133.3333
November	139	119	127	128.3333
December	133	137	127	121.3333

A.6.1 Forecast Calculation

Number of periods to be included in the average (processing option 4a) = 3 in this example

For each month of the forecast, average the previous three month's data.

January forecast: 114 + 119 + 137 = 370, 370 / 3 = 123.333 or 123

February forecast: 119 + 137 + 123 = 379, 379 / 3 = 126.333 or 126

March forecast: 137 + 123 + 126 = 379, 386 / 3 = 128.667 or 129

A.6.2 Simulated Forecast Calculation

October 2005 sales = (129 + 140 + 131)/3 = 133.3333

November 2005 sales = (140 + 131 + 114)/3 = 128.3333

December 2005 sales = (131 + 114 + 119)/3 = 121.3333

A.6.3 Percent of Accuracy Calculation

POA = (133.3333 + 128.3333 + 121.3333) / (114 + 119 + 137) * 100 = 103.513

A.6.4 Mean Absolute Deviation Calculation

MAD = (|133.3333 - 114| + |128.3333 - 119| + |121.3333 - 137|) / 3 = 14.7777

A.7 Method 5 - Linear Approximation

Linear Approximation calculates a trend based upon two sales history data points.

Those two points define a straight trend line that is projected into the future. Use this method with caution, as long range forecasts are leveraged by small changes in just two data points.

Required sales history: The number of periods to include in regression (processing option 5a), plus 1 plus the number of time periods for evaluating forecast performance (processing option 19).

Month	2004 Sales	2005 Sales	2006 Sales	Simulated 2005 Forecast
January	125	128	149
February	132	117	160
March	115	115	172
April	137	125	183
May	122	122	195
June	130	137	206
July	141	129	218
August	128	140	229
September	118	131	241
October	123	114	252	132
November	139	119	264	101
December	133	137	275	113

A.7.1 Forecast Calculation

Number of periods to include in regression (processing option 5a) = 3 in this example

For each month of the forecast, add the increase or decrease during the specified periods prior to holdout period the previous period.

January forecast: (137 - 114)/2 + 137 = 148.5 or 149

February forecast: (137 - 114)/2 * 2 + 137 = 160

March forecast: (137 - 114)/2 * 3 + 137 = 171.5 or 172

A.7.2 Simulated Forecast Calculation

October 2004 sales = (131 - 129) / 2 + 131 = 132

November 2004 sales = (114 - 140) / 2 + 114 = 101

December 2004 sales = (119 - 131) /2 + 119 = 113

A.7.3 Percent of Accuracy Calculation

POA = (132 + 101 + 113) / (114 + 119 + 137) * 100 = 93.5135

A.7.4 Mean Absolute Deviation Calculation

MAD = (|132 - 114| + |101 - 119| + |113 - 137|) / 3 = 20

A.8 Method 6 - Least Square Regression

Linear Regression or Least Squares Regression (LSR) is the most popular method for identifying a linear trend in historical sales data. The method calculates the values for "a" and "b" to be used in the formula: Y = a + bX. The equation describes a straight line where Y represents sales, and X represents time. Linear regression is slow to recognize turning points and step function shifts in demand. Linear regression fits a straight line to the data, even when the data is seasonal or would better be described by a curve. When the sales history data follows a curve or has a strong seasonal pattern, forecast bias and systematic errors occur.

Forecast specifications: n = identifies the periods of sales history that will be used in calculating the values for a and b. For example, specify n = 3 to use the history from October through December, 2005 as the basis for the calculations. When data is available a larger n (such as n = 24) would ordinarily be used. LSR will define a line for as few as two data points. For this example, a small value for n (n = 3) was chosen to reduce the manual calculations required to verify the results.

Required sales history: The number of periods to include in regression (processing option 6a) plus the number of time periods for evaluating forecast performance (processing option 19).

Month	2004 Sales	2005 Sales	2006 Forecast	Simulated 2005 Forecast
January	125	128	146
February	132	117	158
March	115	115	169
April	137	125	181
May	122	122	192
June	130	137	204
July	141	129	215
August	128	140	227
September	118	131	238
October	123	114	250	135.333
November	139	119	261	102.333
December	133	137	273	109.333

A.8.1 Forecast Calculation

Number of periods to include in regression (processing option 6a) = 3 in this example

For each month of the forecast, add the increase or decrease during the specified periods prior to holdout period the previous period.

January forecast:

Average of the previous three months = (114 + 119 + 137)/3 = 123.3333

Summary of the previous three months with weight considered

= (114 * 1) + (119 * 2) + (137 * 3) = 763

Difference between the values

= 763 - 123.3333 * (1 + 2 + 3) = 23

Ratio = (1^2 + 2^2 + 3^2) - {(1 + 2 + 3)/3}^2 * 3 = 14 - 12 = 2

Value1 = Difference/Ratio = 23/2 = 11.5

Value2 = Average - value1 * ratio = 123.3333 - 11.5 * 2 = 100.3333

Forecast = (1 + n) * value1 + value2 = 4 * 11.5 + 100.3333 = 146.333 or 146

February forecast:

Forecast = 5 * 11.5 + 100.3333 = 157.8333 or 158

March forecast:

Forecast = 6 * 11.5 + 100.3333 = 169.3333 or 169

A.8.2 Simulated Forecast Calculation

October 2004 sales:

Average of the previous three months

= (129 + 140 + 131)/3 = 133.3333

Summary of the previous three months with weight considered

= (129 * 1) + (140 * 2) + (131 * 3) = 802

Difference between the values

= 802 - 133.3333 * (1 + 2 + 3) = 2

Ratio = (1^2 + 2^2 + 3^2) - {(1 + 2 + 3)/3}^2 * 3 = 14 - 12 = 2

Value1 = Difference/Ratio = 2/2 = 1

Value2 = Average - value1 * ratio = 133.3333 - 1 * 2 = 131.3333

Forecast = (1 + n) * value1 + value2 = 4 * 1 + 131.3333 = 135.3333

November 2004 sales

Average of the previous three months

= (140 + 131 + 114)/3 = 128.3333

Summary of the previous three months with weight considered

= (140 * 1) + (131 * 2) + (114 * 3) = 744

Difference between the values = 744 - 128.3333 * (1 + 2 + 3) = -25.9999

Value1 = Difference/Ratio = -25.9999/2 = -12.9999

Value2 = Average - value1 * ratio = 128.3333 - (-12.9999) * 2 = 154.3333

Forecast = 4 * -12.9999 + 154.3333 = 102.3333

December 2004 sales

Average of the previous three months

= (131 + 114 + 119)/3 = 121.3333

Summary of the previous three months with weight considered

= (131 * 1) + (114 * 2) + (119 * 3) = 716

Difference between the values

= 716 - 121.3333 * (1 + 2 + 3) = -11.9999

Value1 = Difference/Ratio = -11.9999/2 = -5.9999

Value2 = Average - value1 * ratio = 121.3333 - (-5.9999) * 2 = 133.3333

Forecast = 4 * (-5.9999) + 133.3333 = 109.3333

A.8.3 Percent of Accuracy Calculation

POA = (135.33 + 102.33 + 109.33) / (114 + 119 + 137) * 100 = 93.78

A.8.4 Mean Absolute Deviation Calculation

MAD = (|135.33 - 114| + |102.33 - 119| + |109.33 - 137|) / 3 = 21.88

A.9 Method 7 - Second Degree Approximation

Linear Regression determines values for a and b in the forecast formula Y = a + bX with the objective of fitting a straight line to the sales history data. Second Degree Approximation is similar. However, this method determines values for a, b, and c in the forecast formula Y = a + bX + cX2 with the objective of fitting a curve to the sales history data. This method may be useful when a product is in the transition between stages of a life cycle. For example, when a new product moves from introduction to growth stages, the sales trend may accelerate. Because of the second order term, the forecast can quickly approach infinity or drop to zero (depending on whether coefficient c is positive or negative). Therefore, this method is useful only in the short term.

Forecast specifications: The formulae finds a, b, and c to fit a curve to exactly three points. You specify n in the processing option 7a, the number of time periods of data to accumulate into each of the three points. In this example n = 3. Therefore, actual sales data for April through June are combined into the first point, Q1. July through September are added together to create Q2, and October through December sum to Q3. The curve will be fitted to the three values Q1, Q2, and Q3.

Required sales history: 3 * n periods for calculating the forecast plus the number of time periods required for evaluating the forecast performance (PBF).

Month	2004 Sales	2005 Sales	2006 Forecast	Simulated 2005 Forecast
January	125	128	98
February	132	117	98
March	115	115	98
April	137	125	57
May	122	122	57
June	130	137	57
July	141	129	1
August	128	140	1
September	118	131	1
October	123	114		136
November	139	119		136
December	133	137		136

A.9.1 Forecast Calculation

Number of periods to include (processing option 7a) = 3 in this example

Use the previous (3 * n) months in three-month blocks:

Q1(Apr - Jun) = 125 + 122 + 137 = 384

Q2(Jul - Sep) = 129 + 140 + 131 = 400

Q3(Oct - Dec) = 114 + 119 + 137 = 370

The next step involves calculating the three coefficients a, b, and c to be used in the forecasting formula Y = a + bX + cX^2

(1) Q1 = a + bX + cX^2 (where X = 1) = a + b + c

(2) Q2 = a + bX + cX^2 (where X = 2) = a + 2b + 4c

(3) Q3 = a + bX + cX^2 (where X = 3) = a + 3b + 9c

Solve the three equations simultaneously to find b, a, and c:

Subtract equation (1) from equation (2) and solve for b

(2) - (1) = Q2 - Q1 = b + 3c

b = (Q2 - Q1) - 3c

Substitute this equation for b into equation (3)

(3) Q3 = a + 3[(Q2 - Q1) - 3c] + c

a = Q3 - 3(Q2 - Q1)

Finally, substitute these equations for a and b into equation (1)

[Q3 - 3(Q2 - Q1)] + [(q2 - Q1) - 3c] + c = Q1

c = [(Q3 - Q2) + (Q1 - Q2)]/2

The Second Degree Approximation method calculates a, b, and c as follows:

a = Q3 - 3(Q2 - Q1) = 370 - 3(400 - 384) = 322

c = [(Q3 - Q2) + (Q1 - Q2)]/2 = [(370 - 400) + (384 - 400)]/2 = -23

b = (Q2 - Q1) - 3c = (400 - 384) - (3 * -23) = 85

Y = a + bX + cX^2 = 322 + 85*X + (-23)X^2

January thru March forecast (X=4):

(322 + 340 - 368)/3 = 294/3 = 98 per period

April thru June forecast (X=5):

(322 + 425 - 575)/3 = 57.333 or 57 per period

July thru September forecast (X=6):

(322 + 510 - 828)/3 = 1.33 or 1 per period

October thru December (X=7)

(322 + 595 - 1127/3 = -70

A.9.2 Simulated Forecast Calculation

October, November and December, 2004 sales:

Q1(Jan - Mar) = 360

Q2(Apr - Jun) = 384

Q3(Jul - Sep) = 400

a = 400 - 3(384 - 360) = 328

c = [(400 - 384) + (360 - 384)]/2 = -4

b = (384 - 360) - 3 * (-4) = 36

[328 + 36 * 4 + (-4) * 16]/3 = 136

A.9.3 Percent of Accuracy Calculation

POA = (136 + 136 + 136) / (114 + 119 + 137) * 100 = 110.27

A.9.4 Mean Absolute Deviation Calculation

MAD = (|136 - 114| + |136 - 119| + |136 - 137|) / 3 = 13.33

A.10 Method 8 - Flexible Method

The Flexible Method (Percent Over n Months Prior) is similar to Method 1, Percent Over Last Year. Both methods multiply sales data from a previous time period by a user specified factor, then project that result into the future. In the Percent Over Last Year method, the projection is based on data from the same time period in the previous year. The Flexible Method adds the capability to specify a time period other than the same period last year to use as the basis for the calculations.

Forecast specifications:

• Multiplication factor. For example, specify 1.15 in the processing option 8b to increase the previous sales history data by 15%.

• Base period. For example, n = 3 will cause the first forecast to be based upon sales data in October, 2005.

Minimum sales history: The user specified number of periods back to the base period, plus the number of time periods required for evaluating the forecast performance (PBF).

Month	2004 Sales	2005 Sales	2006 Sales	Simulated 2005 Forecast
January	125	128	131
February	132	117	137
March	115	115	158
April	137	125	151
May	122	122	157
June	130	137	181
July	141	129	173
August	128	140	181
September	118	131	208
October	123	114	199	148.35
November	139	119	208	161
December	133	137	240	150.65

A.10.1 Forecast Calculation

Number of periods prior (processing option 8a) = 3, and the percent over the previous period (processing option 8b) is 1.15 in this example

For each month of the forecast, multiple the sales history n periods prior by the specified percent

January forecast: (114 * 1.15) = 131.1 or 131

February forecast: (119 * 1.15) = 136.85 or 137

March forecast: (137 * 1.15) = 157.55 or 158

A.10.2 Simulated Forecast Calculation

October 2004 sales = 129 * 1.15 = 148.35

November 2004 sales = 140 * 1.15 = 161

December 2004 sales = 131 * 1.15 = 150.65

A.10.3 Percent of Accuracy Calculation

POA = (148 + 161 + 151) / (114 + 119 + 137) * 100 = 124.32

A.10.4 Mean Absolute Deviation Calculation

MAD = (|148 - 114| + |161 - 119| + |151 - 137|) / 3 = 30

A.11 Method 9 - Weighted Moving Average

The Weighted Moving Average (WMA) method is similar to Method 4, Moving Average (MA). However, with the Weighted Moving Average you can assign unequal weights to the historical data. The method calculates a weighted average of recent sales history to arrive at a projection for the short term. More recent data is usually assigned a greater weight than older data, so this makes WMA more responsive to shifts in the level of sales. However, forecast bias and systematic errors still do occur when the product sales history exhibits strong trend or seasonal patterns. This method works better for short range forecasts of mature products rather than for products in the growth or obsolescence stages of the life cycle.

Forecast specifications:

• n = the number of periods of sales history to use in the forecast calculation. For example, specify n = 3 in the processing option 9a to use the most recent three periods as the basis for the projection into the next time period. A large value for n (such as 12) requires more sales history. It results in a stable forecast, but will be slow to recognize shifts in the level of sales. On the other hand, a small value for n (such as 3) will respond quicker to shifts in the level of sales, but the forecast may fluctuate so widely that production can not respond to the variations.

• The weight assigned to each of the historical data periods. The assigned weights must total to 1.00. For example, when n = 3, assign weights of 0.6, 0.3, and 0.1, with the most recent data receiving the greatest weight.

Minimum required sales history: n plus the number of time periods required for evaluating the forecast performance (PBF).

Month	2004 Sales	2005 Sales	2006 Forecast	Simulated 2005 Forecast
January	125	128	129
February	132	117	131
March	115	115	131
April	137	125	131
May	122	122	131
June	130	137	131
July	141	129	131
August	128	140	131
September	118	131	131
October	123	114	131	133.5
November	139	119	131	121.7
December	133	137	131	118.7

A.11.1 Forecast Calculation

Number of periods prior (processing option 9a) = 3, and the weight for one, two and three periods prior (processing option 9b, 9c and 9d) are 0.6, 03, and 0.1 in this example.

January forecast: 137 * 0.6 + 119 * 0.3 + 114 * 0.1 = 129.3 or 129

February forecast: 129.3 * 0.6 + 137 * 0.3 + 119 * 0.1 = 130.58 or 131

March forecast: 131 * 0.6 + 129 * 0.3 + 137 * 0.1 = 130.748 or 131

A.11.2 Simulated Forecast Calculation

October 2004 sales = 129 * 0.1 + 140 * 0.3 * 131 * 0.6 = 133.5

November 2004 sales = 140 * 0.1 + 131 * 0.3 + 114 * 0.6 = 121.7

December 2004 sales = 131 * 0.1 + 114 * 0.3 + 119 * 0.6 = 118.7

A.11.3 Percent of Accuracy Calculation

POA = (133.5 + 121.7 + 118.7) / (114 + 119 + 137) * 100 = 101.05

A.11.4 Mean Absolute Deviation Calculation

MAD = (|133.5 - 114| + |121.7 - 119| + |118.7 - 137|) / 3 = 13.5

A.12 Method 10 - Linear Smoothing

This method is similar to Method 9, Weighted Moving Average (WMA). However, instead of arbitrarily assigning weights to the historical data, a formula is used to assign weights that decline linearly and sum to 1.00. The method then calculates a weighted average of recent sales history to arrive at a projection for the short term.

As is true of all linear moving average forecasting techniques, forecast bias and systematic errors occur when the product sales history exhibits strong trend or seasonal patterns. This method works better for short range forecasts of mature products rather than for products in the growth or obsolescence stages of the life cycle.

Forecast specifications:

• n = the number of periods of sales history to use in the forecast calculation. This is specified in the processing option 10a. For example, specify n = 3 in the processing option 10b to use the most recent three periods as the basis for the projection into the next time period. The system will automatically assign the weights to the historical data that decline linearly and sum to 1.00. For example, when n = 3, the system will assign weights of 0.5, 0.3333, and 0.1, with the most recent data receiving the greatest weight.

Minimum required sales history: n plus the number of time periods required for evaluating the forecast performance (PBF).

Month	2004 Sales	2005 Sales	2006 Forecast	Simulated 2005 Forecast
January	125	128
February	132	117	127
March	115	115	129
April	137	125	130
May	122	122	129
June	130	137	129
July	141	129	129
August	128	140	129
September	118	131	129
October	123	114	129	133.6666
November	139	119	129	124
December	133	137	129	119.3333

A.12.1 Forecast Calculation

Number of periods to include in smoothing average (processing option 10a) = 3 in this example

Ratio for one period prior = 3/(n^2 + n)/2 = 3/(3^2 + 3)/2 = 3/6 = 0.5

Ratio for two periods prior = 2/(n^2 + n)/2 = 2/(3^2 + 3)/2 = 2/6 = 0.3333..

Ratio for three periods prior = 1/(n^2 + n)/2 = 1/(3^2 + 3)/2 = 1/6 = 0.1666..

January forecast: 137 * 0.5 + 119 * 1/3 + 114 * 1/6 = 127.16 or 127

February forecast: 127 * 0.5 + 137 * 1/3 * 119 * 1/6 = 129

March forecast: 129 * 0.5 + 127 * 1/3 * 137 * 1/6 = 129.666 or 130

A.12.2 Simulated Forecast Calculation

October 2004 sales = 129 * 1/6 + 140 * 2/6 * 131 * 3/6 = 133.6666

November 2004 sales = 140 * 1/6 + 131 * 2/6 + 114 * 3/6 = 124

December 2004 sales = 131 * 1/6 + 114 * 2/6 + 119 * 3/6 = 119.3333

A.12.3 Percent of Accuracy Calculation

POA = (133.6666 + 124 + 119.3333) / (114 + 119 + 137) * 100 = 101.891

A.12.4 Mean Absolute Deviation Calculation

MAD = (|133.6666 - 114| + |124 - 119| + |119.3333 - 137|) / 3 = 14.1111

A.13 Method 11 - Exponential Smoothing

This method is similar to Method 10, Linear Smoothing. In Linear Smoothing the system assigns weights to the historical data that decline linearly. In exponential smoothing, the system assigns weights that exponentially decay. The exponential smoothing forecasting equation is:

Forecast =a(Previous Actual Sales) + (1 -a) Previous Forecast

The forecast is a weighted average of the actual sales from the previous period and the forecast from the previous period. a is the weight applied to the actual sales for the previous period. (1 -a) is the weight applied to the forecast for the previous period. Valid values for a range from 0 to 1, and usually fall between 0.1 and 0.4. The sum of the weights is 1.00. a+ (1 -a) = 1

You should assign a value for the smoothing constant, a. If you do not assign values for the smoothing constant, the system calculates an assumed value based upon the number of periods of sales history specified in the processing option 11a.

Forecast specifications:

• a = the smoothing constant used in calculating the smoothed average for the general level or magnitude of sales. Valid values for a range from 0 to 1.

• n = the range of sales history data to include in the calculations. Generally one year of sales history data is sufficient to estimate the general level of sales. For this example, a small value for n (n = 3) was chosen in order to reduce the manual calculations required to verify the results. Exponential smoothing can generate a forecast based on as little as one historical data point.

Minimum required sales history: n plus the number of time periods required for evaluating the forecast performance (PBF).

Month	2004 Sales	2005 Sales	2006 Sales	Simulated 2005 Forecasts
January	125	128	127
February	132	117	127
March	115	115	127
April	137	125	127
May	122	122	127
June	130	137	127
July	141	129	127
August	128	140	127
September	118	131	127
October	123	114	127	133.6666
November	139	119	127	124
December	133	137	127	119.3333

A.13.1 Forecast Calculation

Number of periods to include in smoothing average (processing option 11a) = 3, and alpha factor (processing option 11b) = blank in this example

a factor for the oldest sales data = 2/(1+1), or 1 when alpha is specified

a factor for the 2nd oldest sales data = 2/(1+2), or alpha when alpha is specified

a factor for the 3rd oldest sales data = 2/(1+3), or alpha when alpha is specified

a factor for the most recent sales data = 2/(1+n), or alpha when alpha is specified

November Sm. Avg. =a(October Actual) + (1 - a)October Sm. Avg. = 1 * 114 + 0 * 0 = 114

December Sm. Avg. =a(November Actual) + (1 - a)November Sm. Avg. = 2/3 * 119 + 1/3 * 114 = 117.3333

January Forecast =a(December Actual) + (1 - a)December Sm. Avg. = 2/4 * 137 + 2/4 * 117.3333 = 127.16665 or 127

February Forecast = January Forecast = 127

March Forecast = January Forecast = 127

A.13.2 Simulated Forecast Calculation

July, 2004 Sm. Avg. = 2/2 * 129 = 129

August Sm. Avg. = 2/3 * 140 + 1/3 * 129 = 136.3333

September Sm. Avg. = 2/4 * 131 + 2/4 * 136.3333 = 133.6666

October, 2004 sales = Sep Sm. Avg. = 133.6666

August, 2004 Sm. Avg. = 2/2 * 140 = 140

September Sm. Avg. = 2/3 * 131 + 1/3 * 140 = 134

October Sm. Avg. = 2/4 * 114 + 2/4 * 134 = 124

November, 2004 sales = Sep Sm. Avg. = 124

September 2004 Sm. Avg. = 2/2 * 131 = 131

October Sm. Avg. = 2/3 * 114 + 1/3 * 131 = 119.6666

November Sm. Avg. = 2/4 * 119 + 2/4 * 119.6666 = 119.3333

December 2004 sales = Sep Sm. Avg. = 119.3333

A.13.3 Percent of Accuracy Calculation

POA = (133.6666 + 124 + 119.3333) / (114 + 119 + 137) * 100 = 101.891

A.13.4 Mean Absolute Deviation Calculation

MAD = (|133.6666 - 114| + |124 - 119| + |119.3333 - 137|) / 3 = 14.1111

A.14 Method 12 - Exponential Smoothing with Trend and Seasonality

This method is similar to Method 11, Exponential Smoothing in that a smoothed average is calculated. However, Method 12 also includes a term in the forecasting equation to calculate a smoothed trend. The forecast is composed of a smoothed averaged adjusted for a linear trend. When specified in the processing option, the forecast is also adjusted for seasonality.

Forecast specifications:

• a = the smoothing constant used in calculating the smoothed average for the general level or magnitude of sales. Valid values for alpha range from 0 to 1.

• b = the smoothing constant used in calculating the smoothed average for the trend component of the forecast. Valid values for beta range from 0 to 1.

• Whether a seasonal index is applied to the forecast

Note:

a and b are independent of each other. They do not have to add to 1.0.

Minimum required sales history: two years plus the number of time periods required for evaluating the forecast performance (PBF).

Method 12 uses two exponential smoothing equations and one simple average to calculate a smoothed average, a smoothed trend, and a simple average seasonal factor.

A.14.1 Forecast Calculation

A) An exponentially smoothed average

B) An exponentially smoothed trend

C) A simple average seasonal index

When a is not specified in the processing option, it is calculated.

When b is not specified in the processing option, it is calculated.

Note:

A "t" is considered 6 when it is 6 or greater.

The forecast is then calculated using the results of the three equations:

D)

Where:

• L is the length of seasonality (L=12 months or 52 weeks)

• t is the current time period

• m is the number of time periods into the future of the forecast

• S is the multiplicative seasonal adjustment factor indexed to the appropriate time periods

Month	2003 Sales	2004 Sales	2005 Sales	2006 Forecast	Simulated 2005 Forecast
January		115	115	116
February		137	125	132
March		122	122	123
April		130	137	135
May		141	129	137
June		128	140	136
July		118	131	127
August	118	123	114	121	122.81
September	121	139	119	132	133.14
October	130	133	137	139	135.33
November		1543	1514
December
Total

A.14.2 Forecast Calculation

Alpha, and beta factor (processing option 12a, and 12b) = blank, and seasonality (processing option 13c) is '1' in this example

Description	Calculation
January, 05 Seasonal Index, S1	= (125 + 128)/(1543 + 1514) * 12 = 0.99313
January, 05 Smoothed Average, A1	= Jan, 05 Actual/Jan, 05 Seasonal Index = 128/0.99313 = 128.885
January, 05 Smoothed Trend, T1	= 0 insufficient information to calculate first smoothed trend
February Seasonal Index, S2	=(132 + 117)/(1543 + 1514) * 12 = 0.97742
February Smoothed Average, A2	= 122.7519
February Smoothed Trend, T2	= 2/3 * (122.7519 - 128.885) + 1/3 * 0 = -4.0887333
March Seasonal Index, S3	= (115 + 115)/3057 * 12 = 0.90284
March Smoothed Average, A3	= 2/4 * 115/0.90284 + 2/4 * [122.7519 + (-4.088733)] = 123.01950
March Smoothed Trend, T3	= 2/4 (123.01950 - 122.7519) + 2/4 (-4.0888733) = -1.91063665
(Continued through December '06)
December '06 Seasonal Index, S12	= (133 + 137)/3057 * 12 = 1.05986
December Smoothed Average, A12	= (2/13)137/1.05986 + (11/13)(A11 + T11) = 19.8865 + 107.47247 = 127.35897

Calculation of Linear and Seasonal Exponentially Smoothed Forecast

January '06 = (A12+T12)S1 = (127.35897 + 0.28814 * 1) * 0.99313 = 126.77 or 127

February '06 = (A12+T12)S2 = (127.35897 + 0.28814 * 2 ) * 0.9774 = 125.04 or 125

March '06 = (A12+T12)S3 = (127.35897 + 0.28814 * 3) * 0.902845 = 115.77 or 116

December '06 = (A12+T12)S12 = (127.35897 + 0.28814 * 12) * 1.059862 = 138.65 or 139

A.14.3 Simulated Forecast Calculation

Description	Calculation
October, 04 Seasonal Index, S1	= (118 + 123)/(3056) * 12 = 0.94633
October, 04 Smoothed Average, A1	=a * Oct, 04 Actual/Oct, 04 Seasonal Index = 123/0.94633 = 129.9758
October, 04 Smoothed Trend, T1	= 0 insufficient information to calculate first smoothed trend
(Continued through September '05)
September, 05 Seasonal Index, S12	= (118 + 131)/(3056) * 12 = 0.97774869
September, 05 Smoothed Average, A12	= 2/13*131/0.97774869 + 11/13*(A11 + T11) = 129.1410
September, 05 Smoothed Trend, T12	= 2/7 * (129.141025630 - A11) + 5/7 * T11 = 0.6343542

October 2005 sales = (A12 + T12*1)S1 = (129.1410 + 0.6343542 * 1) * 0.94633 = 129.775379872 * 0.94633 = 122.81

November 2005 sales = (A12 + T12*2)S2 = (129.1410 + 0.6343542 * 2) * 1.02094236 = 133.14

December 2005 sales = (A12 + T12*3)S3 = (129.1410 + 0.6343542 * 3) * 1.032722508 = 135.33

A.14.4 Percent of Accuracy Calculation

POA = (122.81 + 133.14 + 135.33) / (114 + 119 + 137) * 100 = 105.75

A.14.5 Mean Absolute Deviation Calculation

MAD = (|122.81 - 114| + |133.14 - 119| + |135.33 - 137|) / 3 = 8.2

A.15 Evaluating the Forecasts

You can select forecasting methods to generate as many as twelve forecasts for each product. Each forecasting method will probably create a slightly different projection. When thousands of products are forecast, it is impractical to make a subjective decision regarding which of the forecasts to use in your plans for each of the products.

The system automatically evaluates performance for each of the forecasting methods that you select, and for each of the products forecast. You can choose between two performance criteria, Mean Absolute Deviation (MAD) and Percent of Accuracy (POA). MAD is a measure of forecast error. POA is a measure of forecast bias. Both of these performance evaluation techniques require actual sales history data for a user specified period of time. This period of recent history is called a "holdout period" or "periods best fit" (PBF).

To measure the performance of a forecasting method, use the forecast formulae to simulate a forecast for the historical holdout period. There will usually be differences between actual sales data and the simulated forecast for the holdout period.

When multiple forecast methods are selected, this same process occurs for each method. Multiple forecasts are calculated for the holdout period, and compared to the known sales history for that same period of time. The forecasting method producing the best match (best fit) between the forecast and the actual sales during the holdout period is recommended for use in your plans. This recommendation is specific to each product, and might change from one forecast generation to the next.

A.16 Mean Absolute Deviation (MAD)

MAD is the mean (or average) of the absolute values (or magnitude) of the deviations (or errors) between actual and forecast data. MAD is a measure of the average magnitude of errors to expect, given a forecasting method and data history. Because absolute values are used in the calculation, positive errors do not cancel out negative errors. When comparing several forecasting methods, the one with the smallest MAD has shown to be the most reliable for that product for that holdout period. When the forecast is unbiased and errors are normally distributed, there is a simple mathematical relationship between MAD and two other common measures of distribution, standard deviation and Mean Squared Error:

A.16.1 Percent of Accuracy (POA)

Percent of Accuracy (POA) is a measure of forecast bias. When forecasts are consistently too high, inventories accumulate and inventory costs rise. When forecasts are consistently two low, inventories are consumed and customer service declines. A forecast that is 10 units too low, then 8 units too high, then 2 units too high, would be an unbiased forecast. The positive error of 10 is canceled by negative errors of 8 and 2.

Error = Actual - Forecast

When a product can be stored in inventory, and when the forecast is unbiased, a small amount of safety stock can be used to buffer the errors. In this situation, it is not so important to eliminate forecast errors as it is to generate unbiased forecasts. However in service industries, the above situation would be viewed as three errors. The service would be understaffed in the first period, then overstaffed for the next two periods. In services, the magnitude of forecast errors is usually more important than is forecast bias.

Note:

The summation over the holdout period allows positive errors to cancel negative errors. When the total of actual sales exceeds the total of forecast sales, the ratio is greater than 100%. Of course, it is impossible to be more than 100% accurate. When a forecast is unbiased, the POA ratio will be 100%. Therefore, it is more desirable to be 95% accurate than to be 110% accurate. The POA criteria select the forecasting method that has a POA ratio closest to 100%.