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Chapter 4: Forecasting

Learning Objectives

Understand the three time horizons and which models apply for each use

Explain when to use each of the four qualitative models

Apply the naive, moving average, exponential smoothing, and trend methods

Compute three measures of forecast accuracy

Develop seasonal indices

Conduct a regression and correlation analysis

Note: It would be very

helpful for you to bring

a highlighter for this

lecture to help clarify

equations!

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Forecasting and Demand Planning

Forecasting: the art and science of predicting future events

Many firms integrate forecasting with supply chain and capacity management systems to make better operational _________

_________ forecasts are needed throughout the supply chain, and are used by all functional areas of the organization, including accounting, finance, marketing, operations, and distribution

Demand planning software: systems that integrate marketing, inventory, sales, operations planning, and financial data to synchronize the supply chain

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Forecasting Time Horizons

Planning horizon: the length of ______ on which a forecast is based

Long-range forecasts, usually 3+ years. Examples: new product planning, facility location, research and development

Medium-range forecasts, usually 3 months to 3 years. Examples: sales and production planning, budgeting

Short-range forecasts, usually up to 1 year, generally less than 3 months. Examples: purchasing, job scheduling, workforce levels, job assignments, production levels

Distinguishing differences:

Medium/long range forecasts deal more with comprehensive issues and support management decisions regarding planning and products, plants and processes

Short-term forecasting usually employ difference methodologies than longer-term forecasting and tend to be more _________ than longer-term forecasts

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Types of Forecasts

Economic forecasts: planning indications that are valuable in helping organizations prepare medium- to long-range forecasts

Address business cycle – _________ rate, money supply, housing starts, etc.

Technological forecasts: long-term forecasts concerned with the rates of technological progress

Impacts development of new products

Demand forecasts: projections of a company’s sales for each time period in the planning horizon

Forecasts of demand drive _________ in many areas, examples:

Supply-Chain Management – good supplier relations, advantages in product innovation, cost and speed to market

Human Resources – hiring, training, laying off workers

Capacity – capacity shortages can result in undependable delivery, loss of customers, loss of market share

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Features of Forecasts

Assumes causal system: past  future

Forecasts are rarely _________ because of randomness

Forecasts more accurate for groups vs. individuals

Forecast accuracy _________ as time horizon increases

I see that you will get an A this semester

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Eight* Steps in Forecasting

Determine the _________ of the forecast

Select the items to be forecast

Determine the time horizon of the forecast

Select the forecasting model(s)

Gather the data

Make the forecast

Validate and implement results

The text only gives the first seven steps, but we also need to:

_________ the forecasting performance

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Basic Forecasting Methods

Time Series

Naïve

Moving averages

Weighted averages

Exponential smoothing

Trend projection

Regression Methods

Linear regression

Multiple regression

Delphi method

Market survey

Qualitative

Methods

Quantitative

Methods

Forecasting Methods

Sales force composite

Jury of executive opinion

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Qualitative (Judgmental) Forecasting

Qualitative forecasts: forecasts that incorporate such factors as the decision maker’s intuition, emotions, personal _________ and value system

Jury of executive opinion: a forecasting technique that uses the opinion of high-level managers to form a group estimate of demand

Delphi method: a forecasting technique using a group process that allows experts to make forecasts

Sales force composite: a forecasting technique based on salespersons’ estimates of expected sales

Consumer Market Survey: a forecasting method that solicits input from customers or potential customers regarding future purchasing _________

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Qualitative (Judgmental) Forecasting

When no _________ data is available, only judgmental forecasting is possible

New products, new technology

The major _________ for using judgmental methods are:

Greater accuracy

Ability to incorporate unusual or one-time events

The difficultly of obtaining the data necessary for quantitative techniques

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Basic Forecasting _________

Time Series

Naïve

Moving averages

Weighted averages

Exponential smoothing

Trend projection

Regression Methods

Linear regression

Multiple regression

Delphi method

Market survey

Qualitative

Methods

Quantitative

Methods

Forecasting Methods

Sales force composite

Jury of executive opinion

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Decomposition of a Time Series

Time series: a forecasting technique that uses a series of _________ data points to make a forecast

Analyze by breaking down into components and projecting them forward – there are four components:

Trend: the underlying _________ of growth or decline

Cyclical patterns: regular patterns in a data series that take place over long periods of time

Random variation (or noise): unexplained deviation from a predictable pattern

Seasonal patterns: repeatable periods of ups and downs over short periods of time

Trend

Cycle

Random Variation

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Seasonal Patterns

There are _________ common seasonality patterns

Period Length	“Season” Length	Number of “Seasons” in Pattern
Week	Day	7
Month	Week	4 – 4.5
Month	Day	28 – 31
Year	Quarter	4
Year	Month	12
Year	Week	52

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Demand with Growth _________ and Seasonality Over Four Years

Demand for product or service

| | | |

1 2 3 4

Time (years)

Average demand over 4 years

Trend component

Actual demand line

Random variation

Seasonal peaks

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Naïve Forecasts

Naïve forecast: The forecast for any period equals the previous period’s actual value

_________

Simple to use

Virtually no cost

Quick and easy to prepare

Data analysis is nonexistent

Easily understandable

Cannot provide high accuracy

Can be good starting point

Uh, give me a minute…

We sold 250 wheels last week...

So next week we should sell

_______ wheels?

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Moving Average Forecasts

Moving Average (MA) – A technique that averages a number of recent actual values, updated as _________ values become available

MA methods work best for short planning horizons when there is no major trend, seasonal, or business cycle pattern

As the value of “n” increases, the forecast reacts _________ to recent changes in the time series data

Where:

n = number of periods (data points) in the moving average

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Moving Average _________

Donna’s Garden Supply wants a 3 month, MA(3) and a 6 month, MA(6) moving average forecast for storage sheds, including a forecast for next January

Storage shed sales are shown below

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Donna’s Garden Supply: MA(3)

A = Actual Sales (Demand)

F = Forecast Sales

First forecast is April

because we need 3

periods of sales to

calculate the first forecast!

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Donna’s Garden Supply: MA(6)

A = Actual Sales (Demand)

F = Forecast Sales

First forecast is July

because we need 6

periods of sales to

calculate the first forecast!

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Weighted Moving Average Forecasts

Weighted Moving Average (WMA): more recent values in a series are given more weight in computing the forecast

The moving average formula implies an equal weight being placed on each value that is being averaged

Choosing weights

_________ and trial-and-error are the simplest ways

Generally, the most recent past is the best _________ of what to expect in the future, and therefore, should get higher weighting

Typically, the weights are given

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Weighted Moving Average _______

Donna’s Garden Supply wants to forecast storage shed sales by weighting the past 3 months with more weight given to recent data to make them more significant

Weights Applied Period

3 Last month

2 Two months ago

1 Three months ago

6 Sum of weights

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Donna’s Garden Supply: WMA

Weights Applied Period

3 Last month

2 Two months ago

1 Three months ago

6 Sum of weights

A = Actual Sales (Demand)

F = Forecast Sales

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Potential Problems With Moving Averages

Increasing “n” smooths the forecast but makes it less _________ to changes

Does not forecast trends well, they lag the actual values

Requires _________ records of past data

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Moving Average and _________ Moving Average

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Exponential Smoothing

Exponential Smoothing: a weighted moving average forecasting technique in which data points are weighted by an exponential function

The forecast “smooths out” the _________ fluctuations in the time series

Note: If starting forecast is not given, assume F1=A1

Where:

Ft = New forecast for period t

Ft-1 = Previous period’s forecast

At-1 = Previous period’s actual demand

α = (“alpha”) the smoothing constant (0 ≤  ≤1)

(A – F) = the error term (Actual – Forecast)

Note: Skip “Exponential Smoothing with Trend Adjustment”, pages 120-124

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Exponential Smoothing _________

In January, a car dealer predicted February demand for 142 Ford Mustangs. Actual February demand was 153 cars. Using a smoothing constant chosen by management of  = 0.20, the dealer want to forecast March demand using the exponential smoothing model.

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Effect of the Smoothing Constant

0.0 ≤ α ≤ 1.0

If α = 0: Ft = Ft-1 + α(At-1 – Ft-1)

Ft = Ft-1 + 0

The forecast does not reflect recent data

If α = 1: Ft = Ft-1 + α(At-1 – Ft-1)

Ft = Ft-1 + 1(At-1 – Ft-1)

Ft = At-1

Choosing the smoothing constant, α:

The closer its value to zero, the _________ the forecast will be to adjust to forecast errors (the greater the smoothing)

The closer its value to one, the greater the responsiveness and the less the smoothing

The objective is to obtain the most _________ forecast no matter the technique

We generally do this by selecting the model that gives us the lowest forecast error

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_________ of Different 

225 –

200 –

175 –

150 –

| | | | | | | | |

1 2 3 4 5 6 7 8 9

Quarter

Demand

a = 0.1

Actual demand

a = 0.5

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Mean Square Error (MSE) – the square of how much the forecast missed the target:

Measuring Forecast Error

Mean Absolute Deviation Error (MAD) - how much the forecast missed the target:

Mean Absolute Percentage Error (MAPE) - the average percent error:

Forecast error is the difference between the _________ value of the time series and the forecast, or At – Ft

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Forecast Errors and Accuracy

A major difference between MSE and MAD is that MSE is influenced much more by _________ forecast errors than by _________ errors (because the errors are squared)

MAPE is different in that the measurement scale factor is eliminated by dividing the absolute error by the time-series data value. This makes the measure easier to interpret

The selection of the best measure of forecast accuracy is not a simple matter

Forecasting experts often disagree on which measure should be used

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Forecast Error Solved Problem

Develop:

Three-period moving average forecast, MA(3)

Four-period moving-average forecast, MA(4)

Exponential smoothing forecast with α = 0.5

Compute the MAD, MAPE, and MSE for each of the three forecasts

Which method provides a better forecast?

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Forecast Error Solved Problem: MA(3)

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Forecast Error Solved Problem: MA(3)

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Forecast Error Solved Problem: MA(3)

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Forecast Error Solved Problem: MA(3)

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Forecast Error Solved Problem: MA(3)

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Forecast Error Solved Problem

Develop:

Three-period moving average forecast, MA(3)

Four-period moving-average forecast, MA(4)

Exponential smoothing forecast with α = 0.5

Compute the MAD, MAPE, and MSE for each of the three forecasts

Which method provides a better forecast?

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Forecast Error Solved Problem: MA(4)

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Forecast Error Solved Problem

Develop:

Three-period moving average forecast, MA(3)

Four-period moving-average forecast, MA(4)

Exponential smoothing forecast with α = 0.5

Compute the MAD, MAPE, and MSE for each of the three forecasts

Which method provides a better forecast?

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Forecast Error Solved Problem: Exponential Smoothing ( = 0.5)

Note: If starting forecast is

not given, assume F1=A1

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Forecast Error Solved Problem: Exponential Smoothing ( = 0.5)

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Forecast Error Solved Problem: _________

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Trend Projections

Trend projection: a time-series forecasting method that fits a trend line to a series of historical data points and then _________ the line into the future for forecasts

Linear trends can be found using the least squares technique

y = a + bx

where y = (“y hat”) computed value of the variable to be predicted (dependent variable)

a = y-axis intercept

b = slope of the regression line

x = the independent variable (in this case is time)

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Least Squares _________

The least squares method _________ the sum of the squared errors (deviations):

Time period

Values of Dependent Variable

Deviation1

(error)

Deviation5

Deviation7

Deviation2

Deviation6

Deviation4

Deviation3

Actual observation (y-value)

Trend line, y = a + bx

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Trend Projections

Statisticians have developed equations that we can use to find the values of a and b for any regression line

The slope (b) and the y intercept (a) can be calculated with the following equations:

We can express the line with the _________ :

y = a + bx

where b = slope of the regression line

x = known values of the independent variable

y = known values of the dependent variable

x = (“x-bar”) average of the x-values

y = (“y-bar”) average of the y-values

n = number of data points or observations

a = y-axis intercept

b =

(Sxy) – (n)(x)(y)

(Sx2) – (n)(x)2

a = y – (b)x

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Least Squares _________

The demand for electric power at N.Y. Edison over the past 7 years is shown in the following table, in megawatts. The firm wants to forecast next year’s demand by fitting a straight line trend to these data.

Year Electrical Power Demand (megawatts)

1 74

2 79

3 80

4 90

5 105

6 142

7 122

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Least Squares Example

Time Electrical Power

Year Period (x) Demand (y)

1 1 74 2 2 79

3 3 80

4 4 90

5 5 105 6 6 142 7 7 122

n=7 ∑x = 28 ∑y = 692

x = 28/7 y = 692/7

x = 4 y = 98.86

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Least Squares Example

Time Electrical Power

Year Period (x) Demand (y) x2

1 1 74 1

2 2 79 4

3 3 80 9

4 4 90 16

5 5 105 25

6 6 142 36

7 7 122 49

n=7 ∑x = 28 ∑y = 692 ∑x2 = 140

x = 4 y = 98.86

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Least Squares Example

Time Electrical Power

Year Period (x) Demand (y) x2 xy

1 1 74 1 74

2 2 79 4 158

3 3 80 9 240

4 4 90 16 360

5 5 105 25 525

6 6 142 36 852

7 7 122 49 854

n=7 ∑x = 28 ∑y = 692 ∑x2 = 140 ∑xy = 3,063

x = 4 y = 98.86

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Least Squares Example

| | | | | | | | |

1 2 3 4 5 6 7 8 9

160 –

150 –

140 –

130 –

120 –

110 –

100 –

90 –

80 –

70 –

60 –

50 –

Year

Power demand

Trend line,

y = 56.70 + 10.54x

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Least Squares Requirements

Always plot the data to insure a linear relationship

Do not predict time periods far _________ the database

Deviations around the least squares line are assumed to be _________

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Regression Analysis

Regression analysis: a straight-line mathematical model to describe the functional relationships between independent and dependent variables

The dependent variable will still be y but the independent variable (x) need no longer be _________

_________ provides a very simple tool to find the best-fitting regression model for a time series by selecting the “Add Trendline” option

y = a + bx

where y = (“y hat”) computed value of the variable to be predicted (dependent variable)

a = y-axis intercept

b = slope of the regression line

x = the independent variable

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_________ Regression Forecasting Example

Steps:

Calculate: n, x, y, xy, x2

Substitute into equations to solve for a and b

Substitute a and b into regression equation

Use regression equation to forecast

Nodel Construction Company renovates old homes in West Bloomfield, Michigan. Over time, the company has found that its dollar volume of renovation work is dependent on the West Bloomfield area payroll. Management wants to establish a mathematical relation to help predict sales.

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Linear Regression Forecasting Solution

Steps:

Calculate: n, x, y, xy, x2

Substitute into equations to solve for a and b

Substitute a and b into regression equation

Use regression equation to forecast

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Linear Regression Example Using Excel

Steps to use Excel:

Open new spreadsheet and input x and y data

Insert: Chart

Select chart type: Scatter

Data indicates fairly linear relationship

“Select” the plotted data, right click and select “Add Trendline”

Select: “Linear”

Select Options: Display equation and R-squared value

Calculate forecast and evaluate correlation

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Plot Data and Create Scatter Diagram

Open new spreadsheet and input x and y data

Insert: Chart

Select chart type: Scatter

Data indicates fairly linear relationship

If it is not linear, you should NOT use this method

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Add Trendline and Display Output

“Select” the plotted data, right click and select “Add Trendline”

Select: “Linear”

Select Options: Display equation and R-squared value

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Calculate Forecast and _________ Correlation

Result: y = 0.25x + 1.75, R2 = 0.8125

If payroll is predicted to be $6B, estimated sales:

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1 3 4 2 1 7 2 3 2.5 2 2 3.5

_________ Coefficient

Correlation Coefficient (r):

A measure of the strength of the linear relationship between independent and dependent variables

Varies between -1.00 and + 1.00

-1.0

-0.5

0.0

+0.5

+1.0

Perfect Negative

Correlation

Perfect Positive

Correlation

Increasing degree of

Negative Correlation

Increasing degree of

Positive Correlation

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Correlation Coefficient (r)

(a) Perfect negative correlation

(d) Positive correlation

(e) Perfect positive correlation

(b) Negative correlation

High

Moderate

Low

Correlation coefficient values

High

Moderate

Low

| | | | | | | | |

–1.0 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0

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Correlation Coefficient

Correlation Coefficient (r):

r = n(Σxy) – (Σx)(Σy)

n(Σx2) – (Σx)2 n(Σy2) – (Σy)2

Nasty equation but

you don’t need to

know how to use it,

just be sure you can

_________ the correlation

coefficient!

Coefficient of Determination (r2):

The percentage of the variation in the dependent variable that results from the independent variable

Varies between 0 and 1

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Calculate Forecast and Interpret Correlation

If payroll is predicted to be $6B, estimated sales:

y = 0.25x + 1.75  y = 0.25 (6) + 1.75 = $3.25 M

R2 = 0.8125 ….what does this mean???

Coefficient of Determination (r2):

The percentage of the variation in the dependent variable that results from the independent variable.

Correlation Coefficient (r):

A measure of the strength of the linear relationship between independent and dependent variables

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1 3 4 2 1 7 2 3 2.5 2 2 3.5

Seasonal Variations In Data

The _________ seasonal model can adjust trend data for seasonal variations in demand

_________ in process

Find average historical demand for each season

Compute the average demand over all seasons

Compute a seasonal index for each season

Estimate next year’s total demand

Divide this estimate of total demand by the number of seasons, then multiply it by the seasonal index for that season  this provides the seasonal forecast

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Seasonal Index Example

A Des Moines distributor of Sony laptop computers wants to develop monthly indices for sales. Data from the past three years are shown below:

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Seasonal Index Example

A Des Moines distributor of Sony laptop computers wants to develop monthly indices for sales. Data from the past three years are shown below:

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Seasonal Index Example

A Des Moines distributor of Sony laptop computers wants to develop monthly indices for sales. Data from the past three years are shown below:

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Seasonal Index Example

A Des Moines distributor of Sony laptop computers wants to develop monthly indices for sales. Data from the past three years are shown below:

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Seasonal Index Example

If we expected annual demand for next year for computers to be 1150 units, we would use these seasonal indices to forecast the monthly demand

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Seasonal Index Example

140 –

130 –

120 –

110 –

100 –

90 –

80 –

70 –

| | | | | | | | | | | |

J F M A M J J A S O N D

Time

Demand

Avg Next Year Fcst

Next Year Forecast

Year 3 Demand

Year 2 Demand

Year 1 Demand

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Forecasting in Practice

Managers use a variety of judgmental and quantitative forecasting techniques

Statistical methods _________ cannot account for such factors as sales promotions, competitive strategies, unusual economic disturbances, new products, large one-time orders, natural disasters, or labor complications

The first step in developing a practical forecast is to understand the purpose, time horizon, and level of aggregation

Different forecasting methods require different levels of technical _________ and understanding of mathematical principles and assumptions

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periods

Demand

MA(n)

average

Moving

MonthActual Sales

January10

February12

March13

April 16

May19

June 23

July26

August30

September28

October 18

November 16

December14

January

Sheet1

Month	Actual Sales	MA(3) Forecast	MA(6) Forecast	WMA Forecast
January	10
February	12
March	13
April	16	11.67		12.17
May	19	13.67		14.33
June	23	16.00		17.00
July	26	19.33	15.50	20.50
August	30	22.67	18.17	23.83
September	28	26.33	21.17	27.50
October	18	28.00	23.67	28.33
November	16	25.33	24.00	23.33
December	14	20.67	23.50	18.67
January		16.00	22.00	15.33

Sheet1

Actual Sales

MA(3) Forecast

MA(6) Forecast

WMA Forecast

Month

Sales (# Sheds)

Sheet2

Sheet3

MonthActual SalesMA(3) Forecast

January10

February12

March13

April 1611.67

May1913.67

June 2316.00

July2619.33

August3022.67

September2826.33

October 1828.00

November 1625.33

December1420.67

January 16.00

periods

demand

MA(n)

average

Moving

Sheet1

Month	Actual Sales	MA(3) Forecast	MA(6) Forecast	WMA Forecast
January	10
February	12
March	13
April	16	11.67		12.17
May	19	13.67		14.33
June	23	16.00		17.00
July	26	19.33	15.50	20.50
August	30	22.67	18.17	23.83
September	28	26.33	21.17	27.50
October	18	28.00	23.67	28.33
November	16	25.33	24.00	23.33
December	14	20.67	23.50	18.67
January		16.00	22.00	15.33
Month	Actual Sales	MA(6) Forecast
January	10
February	12
March	13
April	16
May	19
June	23
July	26	15.50
August	30	18.17
September	28	21.17
October	18	23.67
November	16	24.00
December	14	23.50
January		22.00
Month	Actual Sales	WMA Forecast
January	10
February	12
March	13
April	16	12.17
May	19	14.33
June	23	17.00
July	26	20.50
August	30	23.83
September	28	27.50
October	18	28.33
November	16	23.33
December	14	18.67
January		15.33

Sheet1

Actual Sales

MA(3) Forecast

MA(6) Forecast

WMA Forecast

Month

Sales (# Sheds)

Sheet2

Sheet3

MonthActual SalesMA(6) Forecast

January10

February12

March13

April 16

May19

June 23

July2615.50

August3018.17

September2821.17

October 1823.67

November 1624.00

December1423.50

January 22.00

Sheet1

Month	Actual Sales	MA(3) Forecast	MA(6) Forecast	WMA Forecast
January	10
February	12
March	13
April	16	11.67		12.17
May	19	13.67		14.33
June	23	16.00		17.00
July	26	19.33	15.50	20.50
August	30	22.67	18.17	23.83
September	28	26.33	21.17	27.50
October	18	28.00	23.67	28.33
November	16	25.33	24.00	23.33
December	14	20.67	23.50	18.67
January		16.00	22.00	15.33
Month	Actual Sales	MA(6) Forecast
January	10
February	12
March	13
April	16
May	19
June	23
July	26	15.50
August	30	18.17
September	28	21.17
October	18	23.67
November	16	24.00
December	14	23.50
January		22.00
Month	Actual Sales	WMA Forecast
January	10
February	12
March	13
April	16	12.17
May	19	14.33
June	23	17.00
July	26	20.50
August	30	23.83
September	28	27.50
October	18	28.33
November	16	23.33
December	14	18.67
January		15.33

Sheet1

Actual Sales

MA(3) Forecast

MA(6) Forecast

WMA Forecast

Month

Sales (# Sheds)

Sheet2

Sheet3

Weights

)

period

Demand

)(

period

for

(Weight

Average

Moving

Weighted

Weights

)

period

Demand

)(

period

for

(Weight

Average

Moving

Weighted

MonthActual Sales

January10

February12

March13

April 16

May19

June 23

July26

August30

September28

October 18

November 16

December14

January

Sheet1

Month	Actual Sales	MA(3) Forecast	MA(6) Forecast	WMA Forecast
January	10
February	12
March	13
April	16	11.67		12.17
May	19	13.67		14.33
June	23	16.00		17.00
July	26	19.33	15.50	20.50
August	30	22.67	18.17	23.83
September	28	26.33	21.17	27.50
October	18	28.00	23.67	28.33
November	16	25.33	24.00	23.33
December	14	20.67	23.50	18.67
January		16.00	22.00	15.33
Month	Actual Sales	MA(6) Forecast
January	10
February	12
March	13
April	16
May	19
June	23
July	26	15.50
August	30	18.17
September	28	21.17
October	18	23.67
November	16	24.00
December	14	23.50
January		22.00
Month	Actual Sales	WMA Forecast
January	10
February	12
March	13
April	16	12.17
May	19	14.33
June	23	17.00
July	26	20.50
August	30	23.83
September	28	27.50
October	18	28.33
November	16	23.33
December	14	18.67
January		15.33

Sheet1

Actual Sales

MA(3) Forecast

MA(6) Forecast

WMA Forecast

Month

Sales (# Sheds)

Sheet2

Sheet3

MonthActual SalesWMA Forecast

January10

February12

March13

April 1612.17

May1914.33

June 2317.00

July2620.50

August3023.83

September2827.50

October 1828.33

November 1623.33

December1418.67

January 15.33

Weights

)

period

Demand

)(

period

for

(Weight

average

moving

Weighted

Sheet1

Month	Actual Sales	MA(3) Forecast	MA(6) Forecast	WMA Forecast
January	10
February	12
March	13
April	16	11.67		12.17
May	19	13.67		14.33
June	23	16.00		17.00
July	26	19.33	15.50	20.50
August	30	22.67	18.17	23.83
September	28	26.33	21.17	27.50
October	18	28.00	23.67	28.33
November	16	25.33	24.00	23.33
December	14	20.67	23.50	18.67
January		16.00	22.00	15.33
Month	Actual Sales	MA(6) Forecast
January	10
February	12
March	13
April	16
May	19
June	23
July	26	15.50
August	30	18.17
September	28	21.17
October	18	23.67
November	16	24.00
December	14	23.50
January		22.00
Month	Actual Sales	WMA Forecast
January	10
February	12
March	13
April	16	12.17
May	19	14.33
June	23	17.00
July	26	20.50
August	30	23.83
September	28	27.50
October	18	28.33
November	16	23.33
December	14	18.67
January		15.33

Sheet1

Actual Sales

MA(3) Forecast

MA(6) Forecast

WMA Forecast

Month

Sales (# Sheds)

Sheet2

Sheet3

Chart1

January	January	January	January
February	February	February	February
March	March	March	March
April	April	April	April
May	May	May	May
June	June	June	June
July	July	July	July
August	August	August	August
September	September	September	September
October	October	October	October
November	November	November	November
December	December	December	December
January	January	January	January

Actual Sales

MA(3) Forecast

MA(6) Forecast

WMA Forecast

Month

Sales (# Sheds)

11.67

12.17

13.67

14.33

19.33

15.5

20.5

22.67

18.17

23.83

26.33

21.17

27.5

23.67

28.33

25.33

23.33

20.67

23.5

18.67

15.33

Sheet1

Month	Actual Sales	MA(3) Forecast	MA(6) Forecast	WMA Forecast
January	10
February	12
March	13
April	16	11.67		12.17
May	19	13.67		14.33
June	23	16.00		17.00
July	26	19.33	15.50	20.50
August	30	22.67	18.17	23.83
September	28	26.33	21.17	27.50
October	18	28.00	23.67	28.33
November	16	25.33	24.00	23.33
December	14	20.67	23.50	18.67
January		16.00	22.00	15.33
Month	Actual Sales	MA(6) Forecast
January	10
February	12
March	13
April	16
May	19
June	23
July	26	15.50
August	30	18.17
September	28	21.17
October	18	23.67
November	16	24.00
December	14	23.50
January		22.00
Month	Actual Sales	WMA Forecast
January	10
February	12
March	13
April	16	12.17
May	19	14.33
June	23	17.00
July	26	20.50
August	30	23.83
September	28	27.50
October	18	28.33
November	16	23.33
December	14	18.67
January		15.33

Sheet1

Actual Sales

MA(3) Forecast

MA(6) Forecast

WMA Forecast

Month

Sales (# Sheds)

Sheet2

Sheet3

)

α(A

alue

Forecast v

demand

Actual

error

Forecast

Actual

MAD

errors)

(Forecast

MSE

Actual

Forecast

Actual

100

MAPE

PeriodSales (A)

186

293

388

489

592

694

791

893

996

1097

1193

1295

OM 11-2

Month	Sales	MA(3)	MA(4)	Error	Error ^2	Abs Error	Abs Error/Sale	Error	Error ^2	Abs Error	Abs Error/Sale
March	170
April	229
May	192
Jun	271	197.00		74.00	5476.00	74.00	27.31
Jul	238	230.67	215.50	7.33	53.78	7.33	3.08	22.50	506.25	22.50	9.45
Aug	255	233.67	232.50	21.33	455.11	21.33	8.37	22.50	506.25	22.50	8.82
Sep	290	254.67	239.00	35.33	1248.44	35.33	12.18	51.00	2601.00	51.00	17.59
Oct	279	261.00	263.50	18.00	324.00	18.00	6.45	15.50	240.25	15.50	5.56
Nov	301	274.67	265.50	26.33	693.44	26.33	8.75	35.50	1260.25	35.50	11.79
		T = 6	T = 5		8250.78	182.33	66.14		5114.00	147.00	53.21
					1375.13	30.39	11.02		1022.80	29.40	10.64
					MSE	MAD	MAPE		MSE	MAD	MAPE

Example

Period	Sales (A)	MA(3)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89	89.00	0.00	0.00	0.00	0.00
5	92	90.00	2.00	4.00	2.00	2.17
6	94	89.67	4.33	18.78	4.33	4.61
7	91	91.67	-0.67	0.44	0.67	0.74
8	93	92.33	0.67	0.44	0.67	0.72
9	96	92.67	3.33	11.11	3.33	3.47
10	97	93.33	3.67	13.44	3.67	3.78
11	93	95.33	-2.33	5.44	2.33	2.51
12	95	95.33	-0.33	0.11	0.33	0.35
	n =	9		53.78	17.33	18.34
				5.98	1.93	2.04
				MSE	MAD	MAPE
Period	Sales (A)	MA(4)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89
5	92	89.00	3.00	9.00	3.00	3.26
6	94	90.50	3.50	12.25	3.50	3.72
7	91	90.75	0.25	0.06	0.25	0.27
8	93	91.50	1.50	2.25	1.50	1.61
9	96	92.50	3.50	12.25	3.50	3.65
10	97	93.50	3.50	12.25	3.50	3.61
11	93	94.25	-1.25	1.56	1.25	1.34
12	95	94.75	0.25	0.06	0.25	0.26
	T =	8.00		49.69	16.75	17.73
				6.21	2.09	2.22	Diff from book
				MSE	MAD	MAPE
Period	Sales (A)	ES (0.5)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86	86.00	0.00	0.00	0.00	0.00
2	93	86.00	7.00	49.00	7.00	7.53
3	88	89.50	-1.50	2.25	1.50	1.70
4	89	88.75	0.25	0.06	0.25	0.28
5	92	88.88	3.13	9.77	3.13	3.40
6	94	90.44	3.56	12.69	3.56	3.79
7	91	92.22	-1.22	1.49	1.22	1.34
8	93	91.61	1.39	1.93	1.39	1.49
9	96	92.30	3.70	13.66	3.70	3.85
10	97	94.15	2.85	8.11	2.85	2.94
11	93	95.58	-2.58	6.64	2.58	2.77
12	95	94.29	0.71	0.51	0.71	0.75
	T =	12.00		106.10	27.89	29.85
				8.84	2.32	2.49
				MSE	MAD	MAPE

Sheet2

Sheet3

PeriodSales (A)MA(3)

186

293

388

48989.00

59290.00

69489.67

79191.67

89392.33

99692.67

109793.33

119395.33

129595.33

n = 9

OM 11-2

Month	Sales	MA(3)	MA(4)	Error	Error ^2	Abs Error	Abs Error/Sale	Error	Error ^2	Abs Error	Abs Error/Sale
March	170
April	229
May	192
Jun	271	197.00		74.00	5476.00	74.00	27.31
Jul	238	230.67	215.50	7.33	53.78	7.33	3.08	22.50	506.25	22.50	9.45
Aug	255	233.67	232.50	21.33	455.11	21.33	8.37	22.50	506.25	22.50	8.82
Sep	290	254.67	239.00	35.33	1248.44	35.33	12.18	51.00	2601.00	51.00	17.59
Oct	279	261.00	263.50	18.00	324.00	18.00	6.45	15.50	240.25	15.50	5.56
Nov	301	274.67	265.50	26.33	693.44	26.33	8.75	35.50	1260.25	35.50	11.79
		T = 6	T = 5		8250.78	182.33	66.14		5114.00	147.00	53.21
					1375.13	30.39	11.02		1022.80	29.40	10.64
					MSE	MAD	MAPE		MSE	MAD	MAPE

Example

Period	Sales (A)	MA(3)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89	89.00	0.00	0.00	0.00	0.00
5	92	90.00	2.00	4.00	2.00	2.17
6	94	89.67	4.33	18.78	4.33	4.61
7	91	91.67	-0.67	0.44	0.67	0.74
8	93	92.33	0.67	0.44	0.67	0.72
9	96	92.67	3.33	11.11	3.33	3.47
10	97	93.33	3.67	13.44	3.67	3.78
11	93	95.33	-2.33	5.44	2.33	2.51
12	95	95.33	-0.33	0.11	0.33	0.35
	n =	9		53.78	17.33	18.34
				5.98	1.93	2.04
				MSE	MAD	MAPE
Period	Sales (A)	MA(4)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89
5	92	89.00	3.00	9.00	3.00	3.26
6	94	90.50	3.50	12.25	3.50	3.72
7	91	90.75	0.25	0.06	0.25	0.27
8	93	91.50	1.50	2.25	1.50	1.61
9	96	92.50	3.50	12.25	3.50	3.65
10	97	93.50	3.50	12.25	3.50	3.61
11	93	94.25	-1.25	1.56	1.25	1.34
12	95	94.75	0.25	0.06	0.25	0.26
	T =	8.00		49.69	16.75	17.73
				6.21	2.09	2.22	Diff from book
				MSE	MAD	MAPE
Period	Sales (A)	ES (0.5)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86	86.00	0.00	0.00	0.00	0.00
2	93	86.00	7.00	49.00	7.00	7.53
3	88	89.50	-1.50	2.25	1.50	1.70
4	89	88.75	0.25	0.06	0.25	0.28
5	92	88.88	3.13	9.77	3.13	3.40
6	94	90.44	3.56	12.69	3.56	3.79
7	91	92.22	-1.22	1.49	1.22	1.34
8	93	91.61	1.39	1.93	1.39	1.49
9	96	92.30	3.70	13.66	3.70	3.85
10	97	94.15	2.85	8.11	2.85	2.94
11	93	95.58	-2.58	6.64	2.58	2.77
12	95	94.29	0.71	0.51	0.71	0.75
	T =	12.00		106.10	27.89	29.85
				8.84	2.32	2.49
				MSE	MAD	MAPE

Sheet2

Sheet3

PeriodSales (A)MA(3)Error

186

293

388

48989.000.00

59290.002.00

69489.674.33

79191.67-0.67

89392.330.67

99692.673.33

109793.333.67

119395.33-2.33

129595.33-0.33

n = 9

OM 11-2

Month	Sales	MA(3)	MA(4)	Error	Error ^2	Abs Error	Abs Error/Sale	Error	Error ^2	Abs Error	Abs Error/Sale
March	170
April	229
May	192
Jun	271	197.00		74.00	5476.00	74.00	27.31
Jul	238	230.67	215.50	7.33	53.78	7.33	3.08	22.50	506.25	22.50	9.45
Aug	255	233.67	232.50	21.33	455.11	21.33	8.37	22.50	506.25	22.50	8.82
Sep	290	254.67	239.00	35.33	1248.44	35.33	12.18	51.00	2601.00	51.00	17.59
Oct	279	261.00	263.50	18.00	324.00	18.00	6.45	15.50	240.25	15.50	5.56
Nov	301	274.67	265.50	26.33	693.44	26.33	8.75	35.50	1260.25	35.50	11.79
		T = 6	T = 5		8250.78	182.33	66.14		5114.00	147.00	53.21
					1375.13	30.39	11.02		1022.80	29.40	10.64
					MSE	MAD	MAPE		MSE	MAD	MAPE

Example

Period	Sales (A)	MA(3)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89	89.00	0.00	0.00	0.00	0.00
5	92	90.00	2.00	4.00	2.00	2.17
6	94	89.67	4.33	18.78	4.33	4.61
7	91	91.67	-0.67	0.44	0.67	0.74
8	93	92.33	0.67	0.44	0.67	0.72
9	96	92.67	3.33	11.11	3.33	3.47
10	97	93.33	3.67	13.44	3.67	3.78
11	93	95.33	-2.33	5.44	2.33	2.51
12	95	95.33	-0.33	0.11	0.33	0.35
	n =	9		53.78	17.33	18.34
				5.98	1.93	2.04
				MSE	MAD	MAPE
Period	Sales (A)	MA(4)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89
5	92	89.00	3.00	9.00	3.00	3.26
6	94	90.50	3.50	12.25	3.50	3.72
7	91	90.75	0.25	0.06	0.25	0.27
8	93	91.50	1.50	2.25	1.50	1.61
9	96	92.50	3.50	12.25	3.50	3.65
10	97	93.50	3.50	12.25	3.50	3.61
11	93	94.25	-1.25	1.56	1.25	1.34
12	95	94.75	0.25	0.06	0.25	0.26
	T =	8.00		49.69	16.75	17.73
				6.21	2.09	2.22	Diff from book
				MSE	MAD	MAPE
Period	Sales (A)	ES (0.5)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86	86.00	0.00	0.00	0.00	0.00
2	93	86.00	7.00	49.00	7.00	7.53
3	88	89.50	-1.50	2.25	1.50	1.70
4	89	88.75	0.25	0.06	0.25	0.28
5	92	88.88	3.13	9.77	3.13	3.40
6	94	90.44	3.56	12.69	3.56	3.79
7	91	92.22	-1.22	1.49	1.22	1.34
8	93	91.61	1.39	1.93	1.39	1.49
9	96	92.30	3.70	13.66	3.70	3.85
10	97	94.15	2.85	8.11	2.85	2.94
11	93	95.58	-2.58	6.64	2.58	2.77
12	95	94.29	0.71	0.51	0.71	0.75
	T =	12.00		106.10	27.89	29.85
				8.84	2.32	2.49
				MSE	MAD	MAPE

Sheet2

Sheet3

PeriodSales (A)MA(3)ErrorError ^2

186

293

388

48989.000.000.00

59290.002.004.00

69489.674.3318.78

79191.67-0.670.44

89392.330.670.44

99692.673.3311.11

109793.333.6713.44

119395.33-2.335.44

129595.33-0.330.11

n = 953.78

5.98

MSE

OM 11-2

Month	Sales	MA(3)	MA(4)	Error	Error ^2	Abs Error	Abs Error/Sale	Error	Error ^2	Abs Error	Abs Error/Sale
March	170
April	229
May	192
Jun	271	197.00		74.00	5476.00	74.00	27.31
Jul	238	230.67	215.50	7.33	53.78	7.33	3.08	22.50	506.25	22.50	9.45
Aug	255	233.67	232.50	21.33	455.11	21.33	8.37	22.50	506.25	22.50	8.82
Sep	290	254.67	239.00	35.33	1248.44	35.33	12.18	51.00	2601.00	51.00	17.59
Oct	279	261.00	263.50	18.00	324.00	18.00	6.45	15.50	240.25	15.50	5.56
Nov	301	274.67	265.50	26.33	693.44	26.33	8.75	35.50	1260.25	35.50	11.79
		T = 6	T = 5		8250.78	182.33	66.14		5114.00	147.00	53.21
					1375.13	30.39	11.02		1022.80	29.40	10.64
					MSE	MAD	MAPE		MSE	MAD	MAPE

Example

Period	Sales (A)	MA(3)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89	89.00	0.00	0.00	0.00	0.00
5	92	90.00	2.00	4.00	2.00	2.17
6	94	89.67	4.33	18.78	4.33	4.61
7	91	91.67	-0.67	0.44	0.67	0.74
8	93	92.33	0.67	0.44	0.67	0.72
9	96	92.67	3.33	11.11	3.33	3.47
10	97	93.33	3.67	13.44	3.67	3.78
11	93	95.33	-2.33	5.44	2.33	2.51
12	95	95.33	-0.33	0.11	0.33	0.35
	n =	9		53.78	17.33	18.34
				5.98	1.93	2.04
				MSE	MAD	MAPE
Period	Sales (A)	MA(4)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89
5	92	89.00	3.00	9.00	3.00	3.26
6	94	90.50	3.50	12.25	3.50	3.72
7	91	90.75	0.25	0.06	0.25	0.27
8	93	91.50	1.50	2.25	1.50	1.61
9	96	92.50	3.50	12.25	3.50	3.65
10	97	93.50	3.50	12.25	3.50	3.61
11	93	94.25	-1.25	1.56	1.25	1.34
12	95	94.75	0.25	0.06	0.25	0.26
	T =	8.00		49.69	16.75	17.73
				6.21	2.09	2.22	Diff from book
				MSE	MAD	MAPE
Period	Sales (A)	ES (0.5)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86	86.00	0.00	0.00	0.00	0.00
2	93	86.00	7.00	49.00	7.00	7.53
3	88	89.50	-1.50	2.25	1.50	1.70
4	89	88.75	0.25	0.06	0.25	0.28
5	92	88.88	3.13	9.77	3.13	3.40
6	94	90.44	3.56	12.69	3.56	3.79
7	91	92.22	-1.22	1.49	1.22	1.34
8	93	91.61	1.39	1.93	1.39	1.49
9	96	92.30	3.70	13.66	3.70	3.85
10	97	94.15	2.85	8.11	2.85	2.94
11	93	95.58	-2.58	6.64	2.58	2.77
12	95	94.29	0.71	0.51	0.71	0.75
	T =	12.00		106.10	27.89	29.85
				8.84	2.32	2.49
				MSE	MAD	MAPE

Sheet2

Sheet3

PeriodSales (A)MA(3)ErrorError ^2Abs Error

186

293

388

48989.000.000.000.00

59290.002.004.002.00

69489.674.3318.784.33

79191.67-0.670.440.67

89392.330.670.440.67

99692.673.3311.113.33

109793.333.6713.443.67

119395.33-2.335.442.33

129595.33-0.330.110.33

n = 953.7817.33

5.981.93

MSEMAD

OM 11-2

Month	Sales	MA(3)	MA(4)	Error	Error ^2	Abs Error	Abs Error/Sale	Error	Error ^2	Abs Error	Abs Error/Sale
March	170
April	229
May	192
Jun	271	197.00		74.00	5476.00	74.00	27.31
Jul	238	230.67	215.50	7.33	53.78	7.33	3.08	22.50	506.25	22.50	9.45
Aug	255	233.67	232.50	21.33	455.11	21.33	8.37	22.50	506.25	22.50	8.82
Sep	290	254.67	239.00	35.33	1248.44	35.33	12.18	51.00	2601.00	51.00	17.59
Oct	279	261.00	263.50	18.00	324.00	18.00	6.45	15.50	240.25	15.50	5.56
Nov	301	274.67	265.50	26.33	693.44	26.33	8.75	35.50	1260.25	35.50	11.79
		T = 6	T = 5		8250.78	182.33	66.14		5114.00	147.00	53.21
					1375.13	30.39	11.02		1022.80	29.40	10.64
					MSE	MAD	MAPE		MSE	MAD	MAPE

Example

Period	Sales (A)	MA(3)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89	89.00	0.00	0.00	0.00	0.00
5	92	90.00	2.00	4.00	2.00	2.17
6	94	89.67	4.33	18.78	4.33	4.61
7	91	91.67	-0.67	0.44	0.67	0.74
8	93	92.33	0.67	0.44	0.67	0.72
9	96	92.67	3.33	11.11	3.33	3.47
10	97	93.33	3.67	13.44	3.67	3.78
11	93	95.33	-2.33	5.44	2.33	2.51
12	95	95.33	-0.33	0.11	0.33	0.35
	n =	9		53.78	17.33	18.34
				5.98	1.93	2.04
				MSE	MAD	MAPE
Period	Sales (A)	MA(4)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89
5	92	89.00	3.00	9.00	3.00	3.26
6	94	90.50	3.50	12.25	3.50	3.72
7	91	90.75	0.25	0.06	0.25	0.27
8	93	91.50	1.50	2.25	1.50	1.61
9	96	92.50	3.50	12.25	3.50	3.65
10	97	93.50	3.50	12.25	3.50	3.61
11	93	94.25	-1.25	1.56	1.25	1.34
12	95	94.75	0.25	0.06	0.25	0.26
	T =	8.00		49.69	16.75	17.73
				6.21	2.09	2.22	Diff from book
				MSE	MAD	MAPE
Period	Sales (A)	ES (0.5)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86	86.00	0.00	0.00	0.00	0.00
2	93	86.00	7.00	49.00	7.00	7.53
3	88	89.50	-1.50	2.25	1.50	1.70
4	89	88.75	0.25	0.06	0.25	0.28
5	92	88.88	3.13	9.77	3.13	3.40
6	94	90.44	3.56	12.69	3.56	3.79
7	91	92.22	-1.22	1.49	1.22	1.34
8	93	91.61	1.39	1.93	1.39	1.49
9	96	92.30	3.70	13.66	3.70	3.85
10	97	94.15	2.85	8.11	2.85	2.94
11	93	95.58	-2.58	6.64	2.58	2.77
12	95	94.29	0.71	0.51	0.71	0.75
	T =	12.00		106.10	27.89	29.85
				8.84	2.32	2.49
				MSE	MAD	MAPE

Sheet2

Sheet3

PeriodSales (A)MA(3)ErrorError ^2Abs Error

(Abs

Error/A)*100

186

293

388

48989.000.000.000.000.00

59290.002.004.002.002.17

69489.674.3318.784.334.61

79191.67-0.670.440.670.74

89392.330.670.440.670.72

99692.673.3311.113.333.47

109793.333.6713.443.673.78

119395.33-2.335.442.332.51

129595.33-0.330.110.330.35

n = 953.7817.3318.34

5.981.932.04

MSEMADMAPE

OM 11-2

Month	Sales	MA(3)	MA(4)	Error	Error ^2	Abs Error	Abs Error/Sale	Error	Error ^2	Abs Error	Abs Error/Sale
March	170
April	229
May	192
Jun	271	197.00		74.00	5476.00	74.00	27.31
Jul	238	230.67	215.50	7.33	53.78	7.33	3.08	22.50	506.25	22.50	9.45
Aug	255	233.67	232.50	21.33	455.11	21.33	8.37	22.50	506.25	22.50	8.82
Sep	290	254.67	239.00	35.33	1248.44	35.33	12.18	51.00	2601.00	51.00	17.59
Oct	279	261.00	263.50	18.00	324.00	18.00	6.45	15.50	240.25	15.50	5.56
Nov	301	274.67	265.50	26.33	693.44	26.33	8.75	35.50	1260.25	35.50	11.79
		T = 6	T = 5		8250.78	182.33	66.14		5114.00	147.00	53.21
					1375.13	30.39	11.02		1022.80	29.40	10.64
					MSE	MAD	MAPE		MSE	MAD	MAPE

Example

Period	Sales (A)	MA(3)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89	89.00	0.00	0.00	0.00	0.00
5	92	90.00	2.00	4.00	2.00	2.17
6	94	89.67	4.33	18.78	4.33	4.61
7	91	91.67	-0.67	0.44	0.67	0.74
8	93	92.33	0.67	0.44	0.67	0.72
9	96	92.67	3.33	11.11	3.33	3.47
10	97	93.33	3.67	13.44	3.67	3.78
11	93	95.33	-2.33	5.44	2.33	2.51
12	95	95.33	-0.33	0.11	0.33	0.35
	n =	9		53.78	17.33	18.34
				5.98	1.93	2.04
				MSE	MAD	MAPE
Period	Sales (A)	MA(4)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89
5	92	89.00	3.00	9.00	3.00	3.26
6	94	90.50	3.50	12.25	3.50	3.72
7	91	90.75	0.25	0.06	0.25	0.27
8	93	91.50	1.50	2.25	1.50	1.61
9	96	92.50	3.50	12.25	3.50	3.65
10	97	93.50	3.50	12.25	3.50	3.61
11	93	94.25	-1.25	1.56	1.25	1.34
12	95	94.75	0.25	0.06	0.25	0.26
	T =	8.00		49.69	16.75	17.73
				6.21	2.09	2.22	Diff from book
				MSE	MAD	MAPE
Period	Sales (A)	ES (0.5)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86	86.00	0.00	0.00	0.00	0.00
2	93	86.00	7.00	49.00	7.00	7.53
3	88	89.50	-1.50	2.25	1.50	1.70
4	89	88.75	0.25	0.06	0.25	0.28
5	92	88.88	3.13	9.77	3.13	3.40
6	94	90.44	3.56	12.69	3.56	3.79
7	91	92.22	-1.22	1.49	1.22	1.34
8	93	91.61	1.39	1.93	1.39	1.49
9	96	92.30	3.70	13.66	3.70	3.85
10	97	94.15	2.85	8.11	2.85	2.94
11	93	95.58	-2.58	6.64	2.58	2.77
12	95	94.29	0.71	0.51	0.71	0.75
	T =	12.00		106.10	27.89	29.85
				8.84	2.32	2.49
				MSE	MAD	MAPE

Sheet2

Sheet3

OM 11-2

Month	Sales	MA(3)	MA(4)	Error	Error ^2	Abs Error	Abs Error/Sale	Error	Error ^2	Abs Error	Abs Error/Sale
March	170
April	229
May	192
Jun	271	197.00		74.00	5476.00	74.00	27.31
Jul	238	230.67	215.50	7.33	53.78	7.33	3.08	22.50	506.25	22.50	9.45
Aug	255	233.67	232.50	21.33	455.11	21.33	8.37	22.50	506.25	22.50	8.82
Sep	290	254.67	239.00	35.33	1248.44	35.33	12.18	51.00	2601.00	51.00	17.59
Oct	279	261.00	263.50	18.00	324.00	18.00	6.45	15.50	240.25	15.50	5.56
Nov	301	274.67	265.50	26.33	693.44	26.33	8.75	35.50	1260.25	35.50	11.79
		T = 6	T = 5		8250.78	182.33	66.14		5114.00	147.00	53.21
					1375.13	30.39	11.02		1022.80	29.40	10.64
					MSE	MAD	MAPE		MSE	MAD	MAPE

Example

Period	Sales (A)	MA(3)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89	89.00	0.00	0.00	0.00	0.00
5	92	90.00	2.00	4.00	2.00	2.17
6	94	89.67	4.33	18.78	4.33	4.61
7	91	91.67	-0.67	0.44	0.67	0.74
8	93	92.33	0.67	0.44	0.67	0.72
9	96	92.67	3.33	11.11	3.33	3.47
10	97	93.33	3.67	13.44	3.67	3.78
11	93	95.33	-2.33	5.44	2.33	2.51
12	95	95.33	-0.33	0.11	0.33	0.35
	n =	9		53.78	17.33	18.34
				5.98	1.93	2.04
				MSE	MAD	MAPE
Period	Sales (A)	MA(4)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89
5	92	89.00	3.00	9.00	3.00	3.26
6	94	90.50	3.50	12.25	3.50	3.72
7	91	90.75	0.25	0.06	0.25	0.27
8	93	91.50	1.50	2.25	1.50	1.61
9	96	92.50	3.50	12.25	3.50	3.65
10	97	93.50	3.50	12.25	3.50	3.61
11	93	94.25	-1.25	1.56	1.25	1.34
12	95	94.75	0.25	0.06	0.25	0.26
	T =	8.00		49.69	16.75	17.73
				6.21	2.09	2.22	Diff from book
				MSE	MAD	MAPE
Period	Sales (A)	ES (0.5)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86	86.00	0.00	0.00	0.00	0.00
2	93	86.00	7.00	49.00	7.00	7.53
3	88	89.50	-1.50	2.25	1.50	1.70
4	89	88.75	0.25	0.06	0.25	0.28
5	92	88.88	3.13	9.77	3.13	3.40
6	94	90.44	3.56	12.69	3.56	3.79
7	91	92.22	-1.22	1.49	1.22	1.34
8	93	91.61	1.39	1.93	1.39	1.49
9	96	92.30	3.70	13.66	3.70	3.85
10	97	94.15	2.85	8.11	2.85	2.94
11	93	95.58	-2.58	6.64	2.58	2.77
12	95	94.29	0.71	0.51	0.71	0.75
	T =	12.00		106.10	27.89	29.85
				8.84	2.32	2.49
				MSE	MAD	MAPE

Sheet2

Sheet3

PeriodSales (A)MA(4)ErrorError ^2Abs Error(Abs Error/A)*100

186

293

388

489

59289.003.009.003.003.26

69490.503.5012.253.503.72

79190.750.250.060.250.27

89391.501.502.251.501.61

99692.503.5012.253.503.65

109793.503.5012.253.503.61

119394.25-1.251.561.251.34

129594.750.250.060.250.26

n = 849.6916.7517.73

6.212.092.22

MSEMADMAPE

OM 11-2

Month	Sales	MA(3)	MA(4)	Error	Error ^2	Abs Error	Abs Error/Sale	Error	Error ^2	Abs Error	Abs Error/Sale
March	170
April	229
May	192
Jun	271	197.00		74.00	5476.00	74.00	27.31
Jul	238	230.67	215.50	7.33	53.78	7.33	3.08	22.50	506.25	22.50	9.45
Aug	255	233.67	232.50	21.33	455.11	21.33	8.37	22.50	506.25	22.50	8.82
Sep	290	254.67	239.00	35.33	1248.44	35.33	12.18	51.00	2601.00	51.00	17.59
Oct	279	261.00	263.50	18.00	324.00	18.00	6.45	15.50	240.25	15.50	5.56
Nov	301	274.67	265.50	26.33	693.44	26.33	8.75	35.50	1260.25	35.50	11.79
		T = 6	T = 5		8250.78	182.33	66.14		5114.00	147.00	53.21
					1375.13	30.39	11.02		1022.80	29.40	10.64
					MSE	MAD	MAPE		MSE	MAD	MAPE

Example

Period	Sales (A)	MA(3)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89	89.00	0.00	0.00	0.00	0.00
5	92	90.00	2.00	4.00	2.00	2.17
6	94	89.67	4.33	18.78	4.33	4.61
7	91	91.67	-0.67	0.44	0.67	0.74
8	93	92.33	0.67	0.44	0.67	0.72
9	96	92.67	3.33	11.11	3.33	3.47
10	97	93.33	3.67	13.44	3.67	3.78
11	93	95.33	-2.33	5.44	2.33	2.51
12	95	95.33	-0.33	0.11	0.33	0.35
	T =	9.00		53.78	17.33	18.34
				5.98	1.93	2.04
				MSE	MAD	MAPE
Period	Sales (A)	MA(4)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89
5	92	89.00	3.00	9.00	3.00	3.26
6	94	90.50	3.50	12.25	3.50	3.72
7	91	90.75	0.25	0.06	0.25	0.27
8	93	91.50	1.50	2.25	1.50	1.61
9	96	92.50	3.50	12.25	3.50	3.65
10	97	93.50	3.50	12.25	3.50	3.61
11	93	94.25	-1.25	1.56	1.25	1.34
12	95	94.75	0.25	0.06	0.25	0.26
	n =	8		49.69	16.75	17.73
				6.21	2.09	2.22
				MSE	MAD	MAPE
Period	Sales (A)	ES (0.5)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86	86.00	0.00	0.00	0.00	0.00
2	93	86.00	7.00	49.00	7.00	7.53
3	88	89.50	-1.50	2.25	1.50	1.70
4	89	88.75	0.25	0.06	0.25	0.28
5	92	88.88	3.13	9.77	3.13	3.40
6	94	90.44	3.56	12.69	3.56	3.79
7	91	92.22	-1.22	1.49	1.22	1.34
8	93	91.61	1.39	1.93	1.39	1.49
9	96	92.30	3.70	13.66	3.70	3.85
10	97	94.15	2.85	8.11	2.85	2.94
11	93	95.58	-2.58	6.64	2.58	2.77
12	95	94.29	0.71	0.51	0.71	0.75
	T =	12.00		106.10	27.89	29.85
				8.84	2.32	2.49
				MSE	MAD	MAPE

Sheet2

Sheet3

OM 11-2

Month	Sales	MA(3)	MA(4)	Error	Error ^2	Abs Error	Abs Error/Sale	Error	Error ^2	Abs Error	Abs Error/Sale
March	170
April	229
May	192
Jun	271	197.00		74.00	5476.00	74.00	27.31
Jul	238	230.67	215.50	7.33	53.78	7.33	3.08	22.50	506.25	22.50	9.45
Aug	255	233.67	232.50	21.33	455.11	21.33	8.37	22.50	506.25	22.50	8.82
Sep	290	254.67	239.00	35.33	1248.44	35.33	12.18	51.00	2601.00	51.00	17.59
Oct	279	261.00	263.50	18.00	324.00	18.00	6.45	15.50	240.25	15.50	5.56
Nov	301	274.67	265.50	26.33	693.44	26.33	8.75	35.50	1260.25	35.50	11.79
		T = 6	T = 5		8250.78	182.33	66.14		5114.00	147.00	53.21
					1375.13	30.39	11.02		1022.80	29.40	10.64
					MSE	MAD	MAPE		MSE	MAD	MAPE

Example

Period	Sales (A)	MA(3)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89	89.00	0.00	0.00	0.00	0.00
5	92	90.00	2.00	4.00	2.00	2.17
6	94	89.67	4.33	18.78	4.33	4.61
7	91	91.67	-0.67	0.44	0.67	0.74
8	93	92.33	0.67	0.44	0.67	0.72
9	96	92.67	3.33	11.11	3.33	3.47
10	97	93.33	3.67	13.44	3.67	3.78
11	93	95.33	-2.33	5.44	2.33	2.51
12	95	95.33	-0.33	0.11	0.33	0.35
	n =	9		53.78	17.33	18.34
				5.98	1.93	2.04
				MSE	MAD	MAPE
Period	Sales (A)	MA(4)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89
5	92	89.00	3.00	9.00	3.00	3.26
6	94	90.50	3.50	12.25	3.50	3.72
7	91	90.75	0.25	0.06	0.25	0.27
8	93	91.50	1.50	2.25	1.50	1.61
9	96	92.50	3.50	12.25	3.50	3.65
10	97	93.50	3.50	12.25	3.50	3.61
11	93	94.25	-1.25	1.56	1.25	1.34
12	95	94.75	0.25	0.06	0.25	0.26
	T =	8.00		49.69	16.75	17.73
				6.21	2.09	2.22	Diff from book
				MSE	MAD	MAPE
Period	Sales (A)	ES (0.5)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86	86.00	0.00	0.00	0.00	0.00
2	93	86.00	7.00	49.00	7.00	7.53
3	88	89.50	-1.50	2.25	1.50	1.70
4	89	88.75	0.25	0.06	0.25	0.28
5	92	88.88	3.13	9.77	3.13	3.40
6	94	90.44	3.56	12.69	3.56	3.79
7	91	92.22	-1.22	1.49	1.22	1.34
8	93	91.61	1.39	1.93	1.39	1.49
9	96	92.30	3.70	13.66	3.70	3.85
10	97	94.15	2.85	8.11	2.85	2.94
11	93	95.58	-2.58	6.64	2.58	2.77
12	95	94.29	0.71	0.51	0.71	0.75
	T =	12.00		106.10	27.89	29.85
				8.84	2.32	2.49
				MSE	MAD	MAPE

Sheet2

Sheet3

PeriodSales (A)ES (0.5)

18686.00

29386.00

38889.50

48988.75

59288.88

69490.44

79192.22

89391.61

99692.30

109794.15

119395.58

129594.29

n = 12

)

α(A

OM 11-2

Month	Sales	MA(3)	MA(4)	Error	Error ^2	Abs Error	Abs Error/Sale	Error	Error ^2	Abs Error	Abs Error/Sale
March	170
April	229
May	192
Jun	271	197.00		74.00	5476.00	74.00	27.31
Jul	238	230.67	215.50	7.33	53.78	7.33	3.08	22.50	506.25	22.50	9.45
Aug	255	233.67	232.50	21.33	455.11	21.33	8.37	22.50	506.25	22.50	8.82
Sep	290	254.67	239.00	35.33	1248.44	35.33	12.18	51.00	2601.00	51.00	17.59
Oct	279	261.00	263.50	18.00	324.00	18.00	6.45	15.50	240.25	15.50	5.56
Nov	301	274.67	265.50	26.33	693.44	26.33	8.75	35.50	1260.25	35.50	11.79
		T = 6	T = 5		8250.78	182.33	66.14		5114.00	147.00	53.21
					1375.13	30.39	11.02		1022.80	29.40	10.64
					MSE	MAD	MAPE		MSE	MAD	MAPE

Example

Period	Sales (A)	MA(3)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89	89.00	0.00	0.00	0.00	0.00
5	92	90.00	2.00	4.00	2.00	2.17
6	94	89.67	4.33	18.78	4.33	4.61
7	91	91.67	-0.67	0.44	0.67	0.74
8	93	92.33	0.67	0.44	0.67	0.72
9	96	92.67	3.33	11.11	3.33	3.47
10	97	93.33	3.67	13.44	3.67	3.78
11	93	95.33	-2.33	5.44	2.33	2.51
12	95	95.33	-0.33	0.11	0.33	0.35
	T =	9.00		53.78	17.33	18.34
				5.98	1.93	2.04
				MSE	MAD	MAPE
Period	Sales (A)	MA(4)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89
5	92	89.00	3.00	9.00	3.00	3.26
6	94	90.50	3.50	12.25	3.50	3.72
7	91	90.75	0.25	0.06	0.25	0.27
8	93	91.50	1.50	2.25	1.50	1.61
9	96	92.50	3.50	12.25	3.50	3.65
10	97	93.50	3.50	12.25	3.50	3.61
11	93	94.25	-1.25	1.56	1.25	1.34
12	95	94.75	0.25	0.06	0.25	0.26
	T =	8.00		49.69	16.75	17.73
				6.21	2.09	2.22	Diff from book
				MSE	MAD	MAPE
Period	Sales (A)	ES (0.5)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86	86.00	0.00	0.00	0.00	0.00
2	93	86.00	7.00	49.00	7.00	7.53
3	88	89.50	-1.50	2.25	1.50	1.70
4	89	88.75	0.25	0.06	0.25	0.28
5	92	88.88	3.13	9.77	3.13	3.40
6	94	90.44	3.56	12.69	3.56	3.79
7	91	92.22	-1.22	1.49	1.22	1.34
8	93	91.61	1.39	1.93	1.39	1.49
9	96	92.30	3.70	13.66	3.70	3.85
10	97	94.15	2.85	8.11	2.85	2.94
11	93	95.58	-2.58	6.64	2.58	2.77
12	95	94.29	0.71	0.51	0.71	0.75
	n =	12		106.10	27.89	29.85
				8.84	2.32	2.49
				MSE	MAD	MAPE

Sheet2

Sheet3

PeriodSales (A)ES (0.5)ErrorError ^2Abs Error(Abs Error/A)*100

18686.000.000.000.000.00

29386.007.0049.007.007.53

38889.50-1.502.251.501.70

48988.750.250.060.250.28

59288.883.139.773.133.40

69490.443.5612.693.563.79

79192.22-1.221.491.221.34

89391.611.391.931.391.49

99692.303.7013.663.703.85

109794.152.858.112.852.94

119395.58-2.586.642.582.77

129594.290.710.510.710.75

n = 12106.1027.8929.85

8.842.322.49

MSEMADMAPE

OM 11-2

Month	Sales	MA(3)	MA(4)	Error	Error ^2	Abs Error	Abs Error/Sale	Error	Error ^2	Abs Error	Abs Error/Sale
March	170
April	229
May	192
Jun	271	197.00		74.00	5476.00	74.00	27.31
Jul	238	230.67	215.50	7.33	53.78	7.33	3.08	22.50	506.25	22.50	9.45
Aug	255	233.67	232.50	21.33	455.11	21.33	8.37	22.50	506.25	22.50	8.82
Sep	290	254.67	239.00	35.33	1248.44	35.33	12.18	51.00	2601.00	51.00	17.59
Oct	279	261.00	263.50	18.00	324.00	18.00	6.45	15.50	240.25	15.50	5.56
Nov	301	274.67	265.50	26.33	693.44	26.33	8.75	35.50	1260.25	35.50	11.79
		T = 6	T = 5		8250.78	182.33	66.14		5114.00	147.00	53.21
					1375.13	30.39	11.02		1022.80	29.40	10.64
					MSE	MAD	MAPE		MSE	MAD	MAPE

Example

Period	Sales (A)	MA(3)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89	89.00	0.00	0.00	0.00	0.00
5	92	90.00	2.00	4.00	2.00	2.17
6	94	89.67	4.33	18.78	4.33	4.61
7	91	91.67	-0.67	0.44	0.67	0.74
8	93	92.33	0.67	0.44	0.67	0.72
9	96	92.67	3.33	11.11	3.33	3.47
10	97	93.33	3.67	13.44	3.67	3.78
11	93	95.33	-2.33	5.44	2.33	2.51
12	95	95.33	-0.33	0.11	0.33	0.35
	T =	9.00		53.78	17.33	18.34
				5.98	1.93	2.04
				MSE	MAD	MAPE
Period	Sales (A)	MA(4)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89
5	92	89.00	3.00	9.00	3.00	3.26
6	94	90.50	3.50	12.25	3.50	3.72
7	91	90.75	0.25	0.06	0.25	0.27
8	93	91.50	1.50	2.25	1.50	1.61
9	96	92.50	3.50	12.25	3.50	3.65
10	97	93.50	3.50	12.25	3.50	3.61
11	93	94.25	-1.25	1.56	1.25	1.34
12	95	94.75	0.25	0.06	0.25	0.26
	T =	8.00		49.69	16.75	17.73
				6.21	2.09	2.22	Diff from book
				MSE	MAD	MAPE
Period	Sales (A)	ES (0.5)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86	86.00	0.00	0.00	0.00	0.00
2	93	86.00	7.00	49.00	7.00	7.53
3	88	89.50	-1.50	2.25	1.50	1.70
4	89	88.75	0.25	0.06	0.25	0.28
5	92	88.88	3.13	9.77	3.13	3.40
6	94	90.44	3.56	12.69	3.56	3.79
7	91	92.22	-1.22	1.49	1.22	1.34
8	93	91.61	1.39	1.93	1.39	1.49
9	96	92.30	3.70	13.66	3.70	3.85
10	97	94.15	2.85	8.11	2.85	2.94
11	93	95.58	-2.58	6.64	2.58	2.77
12	95	94.29	0.71	0.51	0.71	0.75
	n =	12		106.10	27.89	29.85
				8.84	2.32	2.49
				MSE	MAD	MAPE

Sheet2

Sheet3

123456789101112

Sales (A)

MA(3)

MA(4)

ES (0.5)

MSEMADMAPE

MA(3)5.981.932.04

MA(4) 6.212.092.22

ES (0.5)8.842.322.49

Chart2

86			86
93			86
88			89.5
89	89		88.75
92	90	89	88.875
94	89.6666666667	90.5	90.4375
91	91.6666666667	90.75	92.21875
93	92.3333333333	91.5	91.609375
96	92.6666666667	92.5	92.3046875
97	93.3333333333	93.5	94.15234375
93	95.3333333333	94.25	95.576171875
95	95.3333333333	94.75	94.2880859375

Sales (A)

MA(3)

MA(4)

ES (0.5)

OM 11-2

Month	Sales	MA(3)	MA(4)	Error	Error ^2	Abs Error	Abs Error/Sale	Error	Error ^2	Abs Error	Abs Error/Sale
March	170
April	229
May	192
Jun	271	197.00		74.00	5476.00	74.00	27.31
Jul	238	230.67	215.50	7.33	53.78	7.33	3.08	22.50	506.25	22.50	9.45
Aug	255	233.67	232.50	21.33	455.11	21.33	8.37	22.50	506.25	22.50	8.82
Sep	290	254.67	239.00	35.33	1248.44	35.33	12.18	51.00	2601.00	51.00	17.59
Oct	279	261.00	263.50	18.00	324.00	18.00	6.45	15.50	240.25	15.50	5.56
Nov	301	274.67	265.50	26.33	693.44	26.33	8.75	35.50	1260.25	35.50	11.79
		T = 6	T = 5		8250.78	182.33	66.14		5114.00	147.00	53.21
					1375.13	30.39	11.02		1022.80	29.40	10.64
					MSE	MAD	MAPE		MSE	MAD	MAPE

Example

Period	Sales (A)	MA(3)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89	89.00	0.00	0.00	0.00	0.00
5	92	90.00	2.00	4.00	2.00	2.17
6	94	89.67	4.33	18.78	4.33	4.61
7	91	91.67	-0.67	0.44	0.67	0.74
8	93	92.33	0.67	0.44	0.67	0.72
9	96	92.67	3.33	11.11	3.33	3.47
10	97	93.33	3.67	13.44	3.67	3.78
11	93	95.33	-2.33	5.44	2.33	2.51
12	95	95.33	-0.33	0.11	0.33	0.35
	T =	9.00		53.78	17.33	18.34
				5.98	1.93	2.04
				MSE	MAD	MAPE
Period	Sales (A)	MA(4)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89
5	92	89.00	3.00	9.00	3.00	3.26
6	94	90.50	3.50	12.25	3.50	3.72
7	91	90.75	0.25	0.06	0.25	0.27
8	93	91.50	1.50	2.25	1.50	1.61
9	96	92.50	3.50	12.25	3.50	3.65
10	97	93.50	3.50	12.25	3.50	3.61
11	93	94.25	-1.25	1.56	1.25	1.34
12	95	94.75	0.25	0.06	0.25	0.26
	T =	8.00		49.69	16.75	17.73
				6.21	2.09	2.22	Diff from book
				MSE	MAD	MAPE
Period	Sales (A)	ES (0.5)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86	86.00	0.00	0.00	0.00	0.00
2	93	86.00	7.00	49.00	7.00	7.53
3	88	89.50	-1.50	2.25	1.50	1.70
4	89	88.75	0.25	0.06	0.25	0.28
5	92	88.88	3.13	9.77	3.13	3.40
6	94	90.44	3.56	12.69	3.56	3.79
7	91	92.22	-1.22	1.49	1.22	1.34
8	93	91.61	1.39	1.93	1.39	1.49
9	96	92.30	3.70	13.66	3.70	3.85
10	97	94.15	2.85	8.11	2.85	2.94
11	93	95.58	-2.58	6.64	2.58	2.77
12	95	94.29	0.71	0.51	0.71	0.75
	T =	12.00		106.10	27.89	29.85
				8.84	2.32	2.49
				MSE	MAD	MAPE
	MSE	MAD	MAPE	Sales (A)	MA(3)	MA(4)	ES (0.5)
MA(3)	5.98	1.93	2.04	86			86.00
MA(4)	6.21	2.09	2.22	93			86.00
EP (0.5)	8.84	2.32	2.49	88			89.50
				89	89.00		88.75
				92	90.00	89.00	88.88
				94	89.67	90.50	90.44
				91	91.67	90.75	92.22
				93	92.33	91.50	91.61
				96	92.67	92.50	92.30
				97	93.33	93.50	94.15
				93	95.33	94.25	95.58
				95	95.33	94.75	94.29

Example

0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0
0	0	0	0

Sales (A)

MA(3)

MA(4)

ES (0.5)

Sheet2

Sheet3

OM 11-2

Month	Sales	MA(3)	MA(4)	Error	Error ^2	Abs Error	Abs Error/Sale	Error	Error ^2	Abs Error	Abs Error/Sale
March	170
April	229
May	192
Jun	271	197.00		74.00	5476.00	74.00	27.31
Jul	238	230.67	215.50	7.33	53.78	7.33	3.08	22.50	506.25	22.50	9.45
Aug	255	233.67	232.50	21.33	455.11	21.33	8.37	22.50	506.25	22.50	8.82
Sep	290	254.67	239.00	35.33	1248.44	35.33	12.18	51.00	2601.00	51.00	17.59
Oct	279	261.00	263.50	18.00	324.00	18.00	6.45	15.50	240.25	15.50	5.56
Nov	301	274.67	265.50	26.33	693.44	26.33	8.75	35.50	1260.25	35.50	11.79
		T = 6	T = 5		8250.78	182.33	66.14		5114.00	147.00	53.21
					1375.13	30.39	11.02		1022.80	29.40	10.64
					MSE	MAD	MAPE		MSE	MAD	MAPE

Example

Period	Sales (A)	MA(3)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89	89.00	0.00	0.00	0.00	0.00
5	92	90.00	2.00	4.00	2.00	2.17
6	94	89.67	4.33	18.78	4.33	4.61
7	91	91.67	-0.67	0.44	0.67	0.74
8	93	92.33	0.67	0.44	0.67	0.72
9	96	92.67	3.33	11.11	3.33	3.47
10	97	93.33	3.67	13.44	3.67	3.78
11	93	95.33	-2.33	5.44	2.33	2.51
12	95	95.33	-0.33	0.11	0.33	0.35
	T =	9.00		53.78	17.33	18.34
				5.98	1.93	2.04
				MSE	MAD	MAPE
Period	Sales (A)	MA(4)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86
2	93
3	88
4	89
5	92	89.00	3.00	9.00	3.00	3.26
6	94	90.50	3.50	12.25	3.50	3.72
7	91	90.75	0.25	0.06	0.25	0.27
8	93	91.50	1.50	2.25	1.50	1.61
9	96	92.50	3.50	12.25	3.50	3.65
10	97	93.50	3.50	12.25	3.50	3.61
11	93	94.25	-1.25	1.56	1.25	1.34
12	95	94.75	0.25	0.06	0.25	0.26
	T =	8.00		49.69	16.75	17.73
				6.21	2.09	2.22	Diff from book
				MSE	MAD	MAPE
Period	Sales (A)	ES (0.5)	Error	Error ^2	Abs Error	(Abs Error/A)*100
1	86	86.00	0.00	0.00	0.00	0.00
2	93	86.00	7.00	49.00	7.00	7.53
3	88	89.50	-1.50	2.25	1.50	1.70
4	89	88.75	0.25	0.06	0.25	0.28
5	92	88.88	3.13	9.77	3.13	3.40
6	94	90.44	3.56	12.69	3.56	3.79
7	91	92.22	-1.22	1.49	1.22	1.34
8	93	91.61	1.39	1.93	1.39	1.49
9	96	92.30	3.70	13.66	3.70	3.85
10	97	94.15	2.85	8.11	2.85	2.94
11	93	95.58	-2.58	6.64	2.58	2.77
12	95	94.29	0.71	0.51	0.71	0.75
	T =	12.00		106.10	27.89	29.85
				8.84	2.32	2.49
				MSE	MAD	MAPE
	MSE	MAD	MAPE	Sales (A)	MA(3)	MA(4)	ES (0.5)
MA(3)	5.98	1.93	2.04	86			86.00
MA(4)	6.21	2.09	2.22	93			86.00
ES (0.5)	8.84	2.32	2.49	88			89.50
				89	89.00		88.75
				92	90.00	89.00	88.88
				94	89.67	90.50	90.44
				91	91.67	90.75	92.22
				93	92.33	91.50	91.61
				96	92.67	92.50	92.30
				97	93.33	93.50	94.15
				93	95.33	94.25	95.58
				95	95.33	94.75	94.29

Example

Sales (A)

MA(3)

MA(4)

ES (0.5)

Sheet2

Sheet3

Area Payroll (in

$ billions), x

Nodel's Sales

(in $ millions), y

1.02.0

3.03.0

4.02.5

2.02.0

1.02.0

7.03.5

Area Payroll (in

$ billions), x

Nodel's Sales

(in $ millions), y

x^2xy

11.02.01.02.0

23.03.09.09.0

34.02.516.010.0

42.02.04.04.0

51.02.01.02.0

67.03.549.024.5

n=6

x = 18.0

y= 15.0

x^2 = 80

xy = 51.5

Payroll, xSales, y

1.02.0

3.03.0

4.02.5

2.02.0

1.02.0

7.03.5

Year 1Year 2Year 3

Jan8085105

Feb708585

Mar809382

Apr9095115

May113125131

Jun110115120

Jul100102113

Aug88102110

Sep859095

Oct777885

Nov758283

Dec827880

Demand

Month

TS

Qtr	Actual Demand	Forecast Demand	Error	Cum Error	Abs Fcst Error	Cum Abs Fcst Error	MAD	Tracking Signal
1	90	100
2	95	100
3	115	100
4	100	110
5	125	110
6	140	110
Qtr	Actual Demand	Forecast Demand	Error	Cum Error	Abs Fcst Error	Cum Abs Fcst Error	MAD	Tracking Signal
1	90	100	-10	-10	10	10	10.0	-10/10=-1
2	95	100	-5	-15	5	15	7.5	-15/7.5=-2
3	115	100	15	0	15	30	10.0	0/10=0
4	100	110	-10	-10	10	40	10.0	-10/10=-1
5	125	110	15	5	15	55	11.0	5/11=0.5
6	140	110	30	35	30	85	14.2	35/14.2=2.5

Seasonal Index

Month	Demand			Average
	Year 1	Year 2	Year 3	2007-2009
Jan	80	85	105	90
Feb	70	85	85	80
Mar	80	93	82	85
Apr	90	95	115	100
May	113	125	131	123
Jun	110	115	120	115
Jul	100	102	113	105
Aug	88	102	110	100
Sep	85	90	95	90
Oct	77	78	85	80
Nov	75	82	83	80
Dec	82	78	80	80
Month	Demand			Average	Average	Seasonal
	2007	2008	2009	2007-2009	Monthly	Index
Jan	80	85	105
Feb	70	85	85
Mar	80	93	82
Apr	90	95	115
May	113	125	131
Jun	110	115	120
Jul	100	102	113
Aug	88	102	110
Sep	85	90	95
Oct	77	78	85
Nov	75	82	83
Dec	82	78	80

Sheet3

Average

Year 1Year 2Year 3Yr 1-3

Jan808510590

Feb70858580

Mar80938285

Apr9095115100

May113125131123

Jun110115120115

Jul100102113105

Aug88102110100

Sep85909590

Oct77788580

Nov75828380

Dec82788080

Demand

Month

TS

Qtr	Actual Demand	Forecast Demand	Error	Cum Error	Abs Fcst Error	Cum Abs Fcst Error	MAD	Tracking Signal
1	90	100
2	95	100
3	115	100
4	100	110
5	125	110
6	140	110
Qtr	Actual Demand	Forecast Demand	Error	Cum Error	Abs Fcst Error	Cum Abs Fcst Error	MAD	Tracking Signal
1	90	100	-10	-10	10	10	10.0	-10/10=-1
2	95	100	-5	-15	5	15	7.5	-15/7.5=-2
3	115	100	15	0	15	30	10.0	0/10=0
4	100	110	-10	-10	10	40	10.0	-10/10=-1
5	125	110	15	5	15	55	11.0	5/11=0.5
6	140	110	30	35	30	85	14.2	35/14.2=2.5

Seasonal Index

Month	Demand			Average
	Year 1	Year 2	Year 3	Yr 1-3
Jan	80	85	105	90
Feb	70	85	85	80
Mar	80	93	82	85
Apr	90	95	115	100
May	113	125	131	123
Jun	110	115	120	115
Jul	100	102	113	105
Aug	88	102	110	100
Sep	85	90	95	90
Oct	77	78	85	80
Nov	75	82	83	80
Dec	82	78	80	80
Month	Demand			Average	Average	Seasonal
	2007	2008	2009	2007-2009	Monthly	Index
Jan	80	85	105
Feb	70	85	85
Mar	80	93	82
Apr	90	95	115
May	113	125	131
Jun	110	115	120
Jul	100	102	113
Aug	88	102	110
Sep	85	90	95
Oct	77	78	85
Nov	75	82	83
Dec	82	78	80

Sheet3

Average Average

Year 1Year 2Year 3Yr 1-3Monthly

Jan80851059094

Feb7085858094

Mar8093828594

Apr909511510094

May11312513112394

Jun11011512011594

Jul10010211310594

Aug8810211010094

Sep8590959094

Oct7778858094

Nov7582838094

Dec8278808094

Demand

Month

TS

Qtr	Actual Demand	Forecast Demand	Error	Cum Error	Abs Fcst Error	Cum Abs Fcst Error	MAD	Tracking Signal
1	90	100
2	95	100
3	115	100
4	100	110
5	125	110
6	140	110
Qtr	Actual Demand	Forecast Demand	Error	Cum Error	Abs Fcst Error	Cum Abs Fcst Error	MAD	Tracking Signal
1	90	100	-10	-10	10	10	10.0	-10/10=-1
2	95	100	-5	-15	5	15	7.5	-15/7.5=-2
3	115	100	15	0	15	30	10.0	0/10=0
4	100	110	-10	-10	10	40	10.0	-10/10=-1
5	125	110	15	5	15	55	11.0	5/11=0.5
6	140	110	30	35	30	85	14.2	35/14.2=2.5

Seasonal Index

Month	Demand			Average	Average	Seasonal
	Year 1	Year 2	Year 3	Yr 1-3	Monthly	Index
Jan	80	85	105	90	94	0.957
Feb	70	85	85	80	94	0.851
Mar	80	93	82	85	94	0.904
Apr	90	95	115	100	94	1.064
May	113	125	131	123	94	1.309
Jun	110	115	120	115	94	1.223
Jul	100	102	113	105	94	1.117
Aug	88	102	110	100	94	1.064
Sep	85	90	95	90	94	0.957
Oct	77	78	85	80	94	0.851
Nov	75	82	83	80	94	0.851
Dec	82	78	80	80	94	0.851
	Total Average demand =			1128
	Average monthly demand =			1128/12 = 94
Month	Demand			Average	Average	Seasonal
	2007	2008	2009	2007-2009	Monthly	Index
Jan	80	85	105
Feb	70	85	85
Mar	80	93	82
Apr	90	95	115
May	113	125	131
Jun	110	115	120
Jul	100	102	113
Aug	88	102	110
Sep	85	90	95
Oct	77	78	85
Nov	75	82	83
Dec	82	78	80

Sheet3

Average AverageSeasonal

Year 1Year 2Year 3Yr 1-3MonthlyIndex

Jan808510590940.957

Feb70858580940.851

Mar80938285940.904

Apr9095115100941.064

May113125131123941.309

Jun110115120115941.223

Jul100102113105941.117

Aug88102110100941.064

Sep85909590940.957

Oct77788580940.851

Nov75828380940.851

Dec82788080940.851

Demand

Month

TS

Qtr	Actual Demand	Forecast Demand	Error	Cum Error	Abs Fcst Error	Cum Abs Fcst Error	MAD	Tracking Signal
1	90	100
2	95	100
3	115	100
4	100	110
5	125	110
6	140	110
Qtr	Actual Demand	Forecast Demand	Error	Cum Error	Abs Fcst Error	Cum Abs Fcst Error	MAD	Tracking Signal
1	90	100	-10	-10	10	10	10.0	-10/10=-1
2	95	100	-5	-15	5	15	7.5	-15/7.5=-2
3	115	100	15	0	15	30	10.0	0/10=0
4	100	110	-10	-10	10	40	10.0	-10/10=-1
5	125	110	15	5	15	55	11.0	5/11=0.5
6	140	110	30	35	30	85	14.2	35/14.2=2.5

Seasonal Index

Month	Demand			Average	Average	Seasonal
	Year 1	Year 2	Year 3	Yr 1-3	Monthly	Index
Jan	80	85	105	90	94	0.957
Feb	70	85	85	80	94	0.851
Mar	80	93	82	85	94	0.904
Apr	90	95	115	100	94	1.064
May	113	125	131	123	94	1.309
Jun	110	115	120	115	94	1.223
Jul	100	102	113	105	94	1.117
Aug	88	102	110	100	94	1.064
Sep	85	90	95	90	94	0.957
Oct	77	78	85	80	94	0.851
Nov	75	82	83	80	94	0.851
Dec	82	78	80	80	94	0.851
	Total Average demand =			1128
	Average monthly demand =			1128/12 = 94
Month	Demand			Average	Average	Seasonal
	2007	2008	2009	2007-2009	Monthly	Index
Jan	80	85	105
Feb	70	85	85
Mar	80	93	82
Apr	90	95	115
May	113	125	131
Jun	110	115	120
Jul	100	102	113
Aug	88	102	110
Sep	85	90	95
Oct	77	78	85
Nov	75	82	83
Dec	82	78	80

Sheet3

Average AverageSeasonal

Year 1Year 2Year 3Yr 1-3MonthlyIndex

Jan808510590940.957

Feb70858580940.851

Mar80938285940.904

Apr9095115100941.064

May113125131123941.309

Jun110115120115941.223

Jul100102113105941.117

Aug88102110100941.064

Sep85909590940.957

Oct77788580940.851

Nov75828380940.851

Dec82788080940.851

Demand

Month

TS

Qtr	Actual Demand	Forecast Demand	Error	Cum Error	Abs Fcst Error	Cum Abs Fcst Error	MAD	Tracking Signal
1	90	100
2	95	100
3	115	100
4	100	110
5	125	110
6	140	110
Qtr	Actual Demand	Forecast Demand	Error	Cum Error	Abs Fcst Error	Cum Abs Fcst Error	MAD	Tracking Signal
1	90	100	-10	-10	10	10	10.0	-10/10=-1
2	95	100	-5	-15	5	15	7.5	-15/7.5=-2
3	115	100	15	0	15	30	10.0	0/10=0
4	100	110	-10	-10	10	40	10.0	-10/10=-1
5	125	110	15	5	15	55	11.0	5/11=0.5
6	140	110	30	35	30	85	14.2	35/14.2=2.5

Seasonal Index

Month	Demand			Average	Average	Seasonal
	Year 1	Year 2	Year 3	Yr 1-3	Monthly	Index
Jan	80	85	105	90	94	0.957
Feb	70	85	85	80	94	0.851
Mar	80	93	82	85	94	0.904
Apr	90	95	115	100	94	1.064
May	113	125	131	123	94	1.309
Jun	110	115	120	115	94	1.223
Jul	100	102	113	105	94	1.117
Aug	88	102	110	100	94	1.064
Sep	85	90	95	90	94	0.957
Oct	77	78	85	80	94	0.851
Nov	75	82	83	80	94	0.851
Dec	82	78	80	80	94	0.851
	Total Average demand =			1128
	Average monthly demand =			1128/12 = 94
Month	Demand			Average	Average	Seasonal
	2007	2008	2009	2007-2009	Monthly	Index
Jan	80	85	105
Feb	70	85	85
Mar	80	93	82
Apr	90	95	115
May	113	125	131
Jun	110	115	120
Jul	100	102	113
Aug	88	102	110
Sep	85	90	95
Oct	77	78	85
Nov	75	82	83
Dec	82	78	80