4-1 Operations Management Forecasting Chapter 4 - Part 2

Operations Operations ManagementManagement

ForecastingForecastingChapter 4 - Part 2Chapter 4 - Part 2

Trend is increasing or decreasing pattern.

First, plot data to verify trend.

If trend exists, then moving averages and exponential smoothing will always lag.

Forecasting a TrendForecasting a Trend

Plot DataPlot Data

Period

Actual

4 532 1 6

MA = 3 period Moving Average

Moving Averages for a TrendMoving Averages for a Trend

Period MA

1 8 2 11 3 13 4 15 10.67 4.33 5 19 13.00 6.00

MAError

6 15.67 ?

Trend Graph – Moving AverageTrend Graph – Moving Average

Period

Actual

MA Forecast

4 532 1 6

ES = Exponential Smoothing with =0.5 (F2=11)

Exponential Smoothing for a Exponential Smoothing for a TrendTrend

Period ES

1 8 2 11 3 13 11 4 15 12 3.0 5 19 13.5 5.5

ESError

Trend Graph – Exponential Trend Graph – Exponential Smoothing and Moving AverageSmoothing and Moving Average

Period

Actual

MA Forecast

4 532 1 6

20ES Forecast

Moving Averages and (simple) Exponential Smoothing are always poor.

For a linear trend can use: Exponential Smoothing with Trend Adjustment

(pp. 115-117). Linear Trend Projection (linear regression).

For non-linear trend can use: Non-linear regression techniques.

Forecasting a TrendForecasting a Trend

Used for forecasting linear trend line.

PLOT TO VERIFY LINEAR RELATIONSHIP

Assumes linear relationship between response variable, Y, and time, X.

Y = a + bX

a = y-axis intercept; b = slope

Estimated by least squares method.

Minimizes sum of squared errors.

Linear Trend ProjectionLinear Trend Projection

Plot of X,Y DataPlot of X,Y Data

Time (x)

Actual observation

Least SquaresLeast Squares

Deviation

Time (x)

bxaY ˆ

Actual observation

Point on regression line

Least squares line minimizes sum of squared deviations. This reduces large errors. Similar to MSE.

Deviations around least squares line are assumed to be random.

Least Squares EquationsLeast Squares Equations

Equation: y = a + bx

Slope (p. 119):

yxnyxb

Y-Intercept: xbya

Linear Trend Projection ExampleLinear Trend Projection Example

Period

(x) 1 8 2 11 3 13 4 15

Sales(y)

Given the sales for last 5 periods, forecast future sales using trend projection.

Linear Trend Projection ExampleLinear Trend Projection Example

2.13352242

Period

(x) 1 8 2 11 3 13 4 15

Sales(y) xy

8 22 39

xy=224

9 16 25

x2=55x=3 y=13.2

TP = Trend Projection: Y = 5.4 + 2.6x

Linear Trend Projection ExampleLinear Trend Projection ExamplePeriod

(x) MA ES

1 2 3 4 5

811131519

MAErr.

10.6713.0015.67

4.336.00

Sales(y)

3.05.5

ESErr.

TPErr.TP

21.018.415.8 -0.8

Small errors!

Trend GraphTrend Graph

MA Forecast

ES Forecast

Period

Actual

4 532 1 6

TP Forecast

Models with SeasonalityModels with Seasonality

Use if data exhibits seasonal patterns.

Daily, weekly, monthly, yearly.

Compute seasonal component.

Remove seasonality and forecast.

Factor in seasonal component.

See pages 120-124.

Identify Independent and Dependent variables. Dependent variable (y): Entity to be forecast (demand). Independent variable (x): Used to predict (or explain)

dependent variable. Determine relationship.

Plot data. Consider time lags.

Calculate parameters. Forecast. Monitor.

Associative Forecasting Methods Associative Forecasting Methods

Linear relationship between dependent & explanatory variables. Example: Sales in month i (Yi ) depends on advertising

in month i (Xi ) (eg. number of ads)

Sales may also depend on advertising in previous months!

Independent variable (number of ads).

Y Xi i= +a b

Dependent variable (sales).

Linear RegressionLinear Regression

Deviation

Values of Independent Variable (x)

bxaY ˆ

Actual observation

Point on regression line

Linear Regression EquationsLinear Regression Equations(same as before)(same as before)

Equation: ii bxaY

Slope:

yxnyxb

Y-Intercept: xbya

Slope (b): Y changes by b units for each 1 unit increase in X. If b = +2, then sales (Y) is forecast to increase by 2

for each 1 unit increase in advertising (X).

Y-intercept (a): Average value of Y when X = 0. If a = 4, then average sales (Y) is expected to be 4

when advertising (X) is 0.

Interpretation of CoefficientsInterpretation of Coefficients

Least SquaresLeast Squares Plot data to verify linearity!

If curve is present, use non-linear regression.

Forecast only in (or near) range of observed values!

May need future values of independent variable to make forecast. Example: Summer hotel demand may depend on

summer gasoline price.

Monthly Sales vs. Number of AdsMonthly Sales vs. Number of Ads

Number of TV ads per month

Least Squares LineLeast Squares Line

bxaY ˆ0

What is sales forecast for small number of ads?

Forecasting Outside Range of Forecasting Outside Range of Observed Values is UnreliableObserved Values is Unreliable

bxaY ˆ0

Forecast is for negative sales!

Answers: ‘How strong is the linear relationship between the variables?’

Coefficient of correlation - r Measures degree of association; ranges from -1 to +1

Coefficient of determination - r2

Amount of variation explained by regression equation

CorrelationCorrelation

Sample Coefficient of CorrelationSample Coefficient of Correlation

yynxxn

yxyxnr

r = +1 r = -1

r = .89 r = 0

Coefficient of CorrelationCoefficient of Correlation

A good forecast has: No pattern or direction in forecast error.

Error = Actual - Forecast

A small forecast error. Mean square error (MSE). Mean absolute deviation (MAD). Mean absolute percentage error (MAPE).

Guidelines for Selecting Guidelines for Selecting Forecasting ModelForecasting Model

Desired Pattern

Trend Not Fully Accounted for

Pattern of Forecast ErrorPattern of Forecast Error

Suppose you have forecast sales with a linear regression model & exponential smoothing. Which model do you use?

Linear Regression Exponential Actual Model Smoothing

Year Sales Forecast Forecast (.9)1 1 0.6 1.002 1 1.3 1.003 2 2.0 1.004 2 2.7 1.905 4 3.4 1.99

Selecting Forecasting Model Selecting Forecasting Model ExampleExample

MSE = Σ Error2 / n = 1.10 / 5 = 0.220MAD = Σ |Error| / n = 2.0 / 5 = 0.400MAPE = Σ[|Error|/Actual]/n = 1.2/5 = 0.24 = 24%

Linear Regression ModelLinear Regression Model

Year Actual F’cast

1 1 0.6 0.4 0.16 0.4 2 1 1.3 -0.3 0.09 0.3 3 2 2.0 0.0 0.00 0.0 4 2 2.7 -0.7 0.49 0.7 5 4 3.4 0.6 0.36 0.6Total 0.0 2.0

Error Error2 |Error|

MSE = Σ Error2 / n = 5.05 / 5 = 1.01

MAD = Σ |Error| / n = 3.11 / 5 = 0.622

MAPE = Σ[|Error|/Actual]/n = 1.0525/5 = 0.2105 = 21%

Exponential Smoothing ModelExponential Smoothing Model

Year Y i F’cast

1 1 1.00 0.0 0.00 0.0 2 1 1.00 0.0 0.00 0.0 3 2 1.00 1.0 1.00 1.0 4 2 1.90 0.1 0.01 0.1 5 4 2.01 4.04 2.01Total 0.3 5.05 3.11

Error Error2 |Error|

Which is Better???Which is Better???

Linear Regression Model:MSE = Σ Error2 / n = 1.10 / 5 = 0.220MAD = Σ |Error| / n = 2.0 / 5 = 0.400MAPE = Σ[|Error|/Actual]/n = 1.2/5 = 0.24 = 24%

Exponential Smoothing Model:MSE = Σ Error2 / n = 5.05 / 5 = 1.01MAD = Σ |Error| / n = 3.11 / 5 = 0.622MAPE = Σ[|Error|/Actual]/n = 1.0525/5 = 0.2105 = 21%

Measures how well the forecast is predicting actual values.

To use: Calculate tracking signal each time period.

Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD).

Plot tracking signal on graph. Signal should be within upper and lower control

limits based on MAD.

Tracking SignalTracking Signal

Plot of a Tracking SignalPlot of a Tracking Signal

Lower control limit

Upper control limit

Signal exceeded limit

Tracking signal

Acceptable rangeMAD

Tracking Signal EquationTracking Signal Equation

RSFETS

Based on Normal Distribution of forecast errors: 1 MAD = approximately 0.8 standard deviations. Limits at ±3 MAD (±2.4 std. dev.) mean that 98% of

values should be within limits. Limits at ±4 MAD (±3.2 std. dev.) mean that 99.9% of

values should be within limits.

Use smaller limits to better control important items. (For example: ±2 MAD)

Patterns, even if within limits, indicate better forecasts can be made.

Tracking Signal LimitsTracking Signal Limits

Tracking Signal - Month 1Tracking Signal - Month 1

MoMo F’cstF’cst ActAct ErrorError RSFERSFE MADMAD TSTS

11 100100 9090

CumCum|Error||Error|

11 100100 9090 -10-10 -10-10

RSFE = Errors = -10RSFE = Errors = -10

Error = Actual - Forecast = 90 - 100 = -10

11 100100 9090 -10-10 -10-10 1010

Cum |Error| = |Errors| = 10

11 100100 9090 -10-10 -10-10 1010 10.010.0

MAD = |Errors|/n = 10/1 = 10

11 100100 9090 -10-10 -10-10 1010 10.010.0 -1-1

TS = RSFE/MAD = -10/10 = -1

11 100100 9090

22 99 99 9494

-10-10 -10-10 1010 10.010.0 -1-1

11 100100 9090

22 9999 9494

-10-10 -10-10 1010 10.010.0 -1-1

Error = Actual - Forecast = 94 - 99 = -5

11 100100 9090

22 9999 9494

-10-10 -10-10 1010 10.010.0 -1-1

-5-5 -15-15

RSFE = Errors = (-10) + (-5) = -15

11 100100 9090

22 9999 9494

-10-10 -10-10 1010 10.010.0 -1-1

-5-5 -15-15 1515

Cum Error = |Errors| = 10 + 5 = 15

11 100100 9090

22 9999 9494

-10-10 -10-10 1010 10.010.0 -1-1

-5-5 -15-15 1515 7.57.5

MAD = |Errors|/n = 15/2 = 7.5

11 100100 9090

22 9999 9494

-10-10 -10-10 1010 10.010.0 -1-1

-5-5 -15-15 1515 7.57.5 -2-2

TS = RSFE/MAD = -15/7.5 = -2

11 100100 9090

22 9999 9494

33 9898 113113

-10-10 -10-10 1010 10.010.0 -1-1

-5-5 -15-15 1515 7.57.5 -2-2

1515 00 3300

Tracking Signal - Months 4-6Tracking Signal - Months 4-6

11 100100 9090

22 9999 9494

33 9898 113113

44 105105 9595

55 104104 119119

66 110110 140140

-10-10 -10-10 1010 10.010.0 -1-1

-5-5 -15-15 1515 7.57.5 -2-2

1515 00 3300

-10-10 -10-10 4400

1010 -1-1

1515 55 5555

.45.45

3030 3535 8855

14.214.2 2.472.47

Demand and ForecastDemand and Forecast

708090

100110120130140

0 1 2 3 4 5 6 7

Forecast

Actual demand

Tracking SignalTracking Signal

0 1 2 3 4 5 6 7

98% of points should be between these limits.

Suppose you have forecast sales with a linear regression model & exponential smoothing. Which model do you use?

Linear Regression Exponential Actual Model Smoothing

Year Sales Forecast Forecast (.9)1 1 0.6 1.002 1 1.3 1.003 2 2.0 1.004 2 2.7 1.905 4 3.4 1.99

Selecting Forecasting Model Selecting Forecasting Model Example - RevisitedExample - Revisited

Linear Regression Model Linear Regression Model Tracking SignalTracking Signal

Year Y i F’cast

1 1 0.6 0.4 0.4 1.0 2 1 1.3 -0.3 0.35 0.29 3 2 2.0 0.0 0.233 0.43 4 2 2.7 -0.7 0.35 -1.71 5 4 3.4 0.6 0.40 0.0

Error MAD TS

Exponential Smoothing Model Exponential Smoothing Model Tracking SignalTracking Signal

Year Y i F’cast

1 1 1.00 0.0 0.0 0.0 2 1 1.00 0.0 0.0 0.0 3 2 1.00 1.0 0.33 3.0 4 2 1.90 0.1 0.275 4.0 5 4 2.01 0.622 5.0

Error MAD TS

Tracking SignalsTracking Signals

50 1 2 3 4

Exponential Smoothing

Linear Regression

Forecasting in the Service SectorForecasting in the Service Sector

Examples: For staffing hospitals, fast-food restaurants, banking, etc.

Presents unusual challenges: Large variability (during day, week, etc.). Special need for short term forecasting. Needs differ greatly as function of industry and

product. Issues of holidays and calendar.

Forecasting SummaryForecasting Summary

Determine purpose of forecast first.

Plot data.

Use several appropriate methods.

Continually monitor, evaluate and adjust methods to improve forecasts.

4-1 Operations Management Forecasting Chapter 4 - Part 2

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