math guru

jackson

Chapter8.pdf

Home >Business & Finance homework help >Accounting homework help >math guru

Forecasting

(Chapter 8)

Production & Operations Management

INFO 335-71

Week 1

Learning Objectives

⚫ Identify Principles of Forecasting

⚫ Explain the steps in the forecasting process

⚫ Identify types of forecasting methods and their

characteristics

⚫ Describe time series and causal models

⚫ Generate forecasts for data with different patterns: level, trend, seasonality, and cyclical

⚫ Describe causal modeling using linear regression

⚫ Compute forecast accuracy

⚫ Explain how forecasting models should be selected

Principles of Forecasting

Many types of forecasting models differ in complexity and amount of data & way they generate forecasts.

Common features include:

1. Forecasts are rarely perfect

2. Forecasts are more accurate for grouped data than for individual items

3. Forecast are more accurate for shorter than longer time periods

Steps in the Forecasting Process

1. Determine the purpose of the forecast

2. Establish a time horizon

3. Select a forecasting technique

4. Obtain, clean, and analyze appropriate data

5. Make the forecast

6. Monitor the forecast

Types of Forecasting Models

⚫ Qualitative methods – judgmental methods

• Forecasts generated subjectively by the forecaster • Educated guesses

⚫ Quantitative methods – based on mathematical

modeling:

• Forecasts generated through mathematical modeling

Executive Decisions : Market Research : Delphi Method

Time Series : Causal/Associative Method

Time Series Models

⚫ Forecaster looks for data patterns as

• Data = historic pattern + random variation

⚫ Historic pattern to be forecasted:

• Level (long-term average) – data fluctuates around a constant mean

• Trend – data exhibits an increasing or decreasing pattern • Seasonality – any pattern that regularly repeats itself and is of

a constant length

• Cycle – patterns created by economic fluctuations

⚫ Random Variation cannot be predicted

P A

T T

E R

Time Series Models

⚫ Naive:

⚫ Simple Mean:

⚫ Simple Moving Average:

tA=+1tF

n/AF t1t =+

Time Series Models cont'd

⚫ Weighted Moving Average:

• Method in which “n” of the most recent observations are averaged and past observations may be

weighted differently

• All weights must add to 100% or 1.00 e.g. Ct .5, Ct-1 .3, Ct-2 .2 (weights add to 1.0)

• Allows emphasizing one period over others; above indicates more weight on recent data (Ct=.5)

• Differs from the simple moving average that weighs all periods equally - more responsive to trends

=+ tt1t ACF

Weights

⚫ Future period is

Wednesday – create a

forecast (t+1)

⚫ Tuesday’ Demand = 50

widgets (t)

⚫ Monday’ Demand = 40

widgets (t-1)

Simple 2-period Moving Average

= (50+40)/2 = 45

=50/2 + 40/2

=(1/2)*50 + (1/2)*40

=.5*50 + .5*40

2-period Weighted Moving Average

60% weight to the most recent period of

demand, and 40% to the next most recent

Weights in the ratio 6:4 with greater

weight to the more recent period

=0.6*50 + 0.4*40 = 30 + 16 = 46 widgets

Time Series Models cont'd

⚫ Exponential Smoothing:

• Most frequently used time series method because of ease of use and minimal amount of data needed

• Need just three pieces of data to start: • Last period’s forecast (Ft) • Last periods actual value (At) • Select value of smoothing coefficient, ,between 0 and 1.0

• If no last period forecast is available, average the last few periods or use naive method

• Higher values (e.g. .7 or .8) place a lot of weight on current periods actual demand and influenced by

random variation

( ) tt1t Fα1αAF −+=+



Exponentially Weighted Moving Average (EWMA)

Time Series Problem

Determine forecast for periods 7 &

⚫ 2-period moving average - 340

⚫ 4-period moving average

⚫ 2-period weighted moving average

with t-1 weighted 0.6 and t-2

weighted 0.4 - 344

⚫ Exponential smoothing with

alpha=0.2 and the period 6 forecast

being 375

Period Actual

1 300

2 315

3 290

4 345

5 320

6 360

7 375

8 ( ) tt1t Fα1αAF −+=+

375

372

372.6

Time Series Problem Solution

Period Actual 2-Period 4-Period 2-Per.Wgted.

Exponential

Smoothing

1 300

2 315

3 290

4 345

5 320

6 360

7 375 340.0 328.8 344.0 372.0

8 367.5 350.0 369.0 372.6

Questions? Which of the following is the least useful sales forecasting model to use

when sales are increasing?

a) Simple mean

b) Exponential smoothing

c) Weighted moving average

d) Naïve

Over the long term, which of the following forecasting models will likely

require carrying the least amount of data?

a) Naïve

b) Simple mean

c) Exponential smoothing

d) Weighted moving average

e) Moving average

Questions? Suppose that Sally’s company uses exponential smoothing to make forecasts. Further

suppose that last period’s demand forecast was for 20,000 units and last period’s

actual demand was 21,000 units. Sally’s company uses a smoothing constant (α)

equal to 40%. What should be the forecast for this period?

a) 20,000

b) 21,000

c) 20,600

d) 20,400

e) 19,600

Suppose that you are using the four-period weighted moving average forecasting

method to forecast sales and you know that sales will be increasing every period for

the foreseeable future. What of the following would be the best set of weights to use

(listed in order from the most recent period to four periods ago, respectively)?

a) 0.25, 0.25, 0.25, 0.25

b) 0.40, 0.30, 0.20, 0.10

c) 1.00, 0.00, 0.00, 0.00

d) 0.10, 0.20, 0.30, 0.40

e) 0.00, 0.00, 0.00, 1.00.

Questions? ⚫ In exponential smoothing, what values can the smoothing constant, , have?

⚫ a) [−1, 1]

⚫ b) [1, ]

⚫ c) [0, ]

⚫ d) [0, 1]

⚫ e) [−, ]

Linear Trend Line

⚫ A time series technique that computes a forecast with trend by drawing a straight line through a set of data using this formula:

Y = a + bx where

Y = forecast for period X

X = the number of time periods from X = 0

A = value of y at X = 0 (Y intercept)

B = slope of the line

Causal Model - Linear Regression

( )( ) ( )( ) 

  −

− =

XXX

YXXY b

⚫ Identify dependent (y) and

independent (x) variables

⚫ Solve for the slope of the line

⚫ Solve for the y intercept

⚫ Develop your equation for

the trend line

Y=a + bX

XbYa −=



 −

− =

2 2 XnX

YXnXY b

Linear Regression Problem: A maker of golf shirts has

been tracking the relationship between sales and

advertising dollars. Use linear regression to find out

what sales might be if the company invested $53,000

in advertising next year.



 −

− =

2 2 XnX

YXnXY bSales $

(Y) Adv.$ (X)

XY X^2 Y^2

1 130 32 4160 1024 16,900

2 151 52 7852 2704 22,801

3 150 50 7500 2500 22,500

4 158 55 8690 3025 24964

5 153.85 53

Tot 589 189 28202 9253 87165

Avg 147.25 47.25

( )( ) ( )

( )

( ) 153.85531.1592.9Y 1.15X92.9bXaY

92.9a

47.251.15147.25XbYa

1.15 47.2549253

147.2547.25428202 b

=+=

+=+=

−=−=

= −

− =

Correlation Coefficient -

How Good is the Fit?

⚫ Correlation coefficient (r) measures the direction and strength of the linear

relationship between two variables. The closer the r value is to 1.0 the better

the regression line fits the data points.

⚫ Coefficient of determination ( ) measures the amount of variation in the

dependent variable about its mean that is explained by the regression line.

Values of ( ) close to 1.0 are desirable.

( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

( ) .964.982r

.992 58987,1654*(189)-4(9253)

58918928,2024 r

YYn*XXn

YXXYn r

2 2

= −

− =

−−

− =



2 r

Questions? What are the two categories of quantitative models?

a) Delphi and non-causal

b) Causal and non-causal

c) Delphi and time series

d) Causal and time series

e) Causal and Delphi

A causal research model is based on the assumption that

a) the independent variable is related to the dependent variable

b) there is a relationship between the time series and the dependent variable

c) the variable being forecast is related to other variables in the environment

d) there is a relationship between the time series and the independent

variable

e) the information is contained in a time series of data

Techniques for Seasonality

⚫ Seasonality – regularly repeating movements in series values that can be tied to recurring events

• Expressed in terms of the amount that actual values deviate from the average value of a series

• Models of seasonality • Additive

• Seasonality is expressed as a quantity that gets added to or subtracted from the time-series average in order to incorporate seasonality

• Multiplicative • Seasonality is expressed as a percentage of the average (or

trend) amount which is then used to multiply the value of a series in order to incorporate seasonality

Models of Seasonality

⚫ A coffee shop owner wants to predict quarterly demand for

hot chocolate for periods 9 and 10, which happen to be the

1st and 2nd quarters of a particular year. The sales data

consist of both trend and seasonality. The trend portion

of demand is projected using the equation Ft = 124 + 7.5 t.

Quarter relatives are

Q1 = 1.20, Q2 = 1.10, Q3 = 0.75, Q4 = 0.95,

Seasonal Relatives Example

⚫ Use this information to predict for periods 9 and 10.

⚫ F9 = 124 +7.5( 9) = 191.5

F10= 124 +7.5(10) = 199.0

Multiplying the trend value by the appropriate quarter relative

yields a forecast that includes both trend and seasonality.

Given that t =9 is a 1st quarter and t = 10 is a 2nd quarter.

The forecast demand for period 9 = 191.5(1.20) = 229.8

The forecast demand for period 10 = 199.0(1.10) = 218.9

Seasonal Relatives Example

(cont’d)

Measuring Forecast Error

⚫ Forecasts are never perfect

⚫ Need to measure over time

⚫ Need to know how much we should rely on our chosen forecasting method

⚫ Measuring forecast error:

⚫ Note that over-forecasts = negative errors and under-forecasts = positive errors

ttt FAE −=

Measuring Forecasting Accuracy

⚫ Mean Absolute Deviation (MAD) • measures the total error in a

forecast without regard to sign

⚫ Cumulative Forecast Error (CFE) • Measures any bias in the forecast

⚫ Mean Square Error (MSE) • Penalizes larger errors

⚫ Tracking Signal • Measures if your model is working;

quality

( )

forecast - actual MSE

 =

MAD

CFE TS =

forecastactual MAD

 − =

( ) −= forecastactualCFE

Selecting the Right Forecasting

Model

1. The amount & type of available data

▪ Some methods require more data than others

2. Degree of accuracy required

▪ Increasing accuracy means more data

3. Length of forecast horizon

▪ Different models for 3 month vs. 10 years

4. Presence of data patterns

▪ Lagging will occur when a forecasting model

meant for a level pattern is applied with a trend

Collaborative Planning

Forecasting & Replenishment

(CPFR)

• Establish collaborative relationships between buyers and sellers

• Create a joint business plan • Create a sales forecast • Identify exceptions for sales forecast • Resolve/collaborate on exception items • Create order forecast • Identify exceptions for order forecast • Resolve/collaborate on exception items • Generate order

CPFR is an iterative process.

Backup Slides (to circle back at

the end of the term based on

time)

Accuracy & Tracking Signal Problem: A company is

comparing the accuracy of two forecasting methods.

Forecasts using both methods are shown below along with

the actual values for January through May. The company

also uses a tracking signal with ±4 limits to decide when a

forecast should be reviewed. Which forecasting method is

best?

Month Actual sales

Method A Method B

F’cast Error Cum.

Error

Tracking Signal

F’cast Error Cum. Error

Tracking Signal

Jan. 30 28 2 2 2 27 3 3 1

Feb. 26 25 1 3 3 25 1 4 2

March 32 32 0 3 3 29 3 7 3

April 29 30 -1 2 2 27 2 9 4

May 31 30 1 3 3 29 2 11 5

MAD 1 2.2

MSE 1.4 5.4