Principles Of Operations Management
Principles Of Operations Management
11th Edition
ISBN: 9780135173930
Author: RENDER, Barry, HEIZER, Jay, Munson, Chuck
Publisher: Pearson,
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Chapter 4, Problem 20P

Resolve Problem 4.19 with α = .1 and β =.8. Using MSE, determine which smoothing constants provide a better forecast.

Expert Solution & Answer
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Summary Introduction

To determine: Compute MSE using the given smoothing constants and find the better forecasting smoothing constants using trend-adjusted exponential smoothing method.

Introduction: A sequence of data points in successive order is known as time series. Time series forecasting is the prediction based on past events which are at uniform time interval. Moving average method and trend projections are two of the time series methods which use weights to prioritize past data.

Answer to Problem 20P

On comparing MSE from two smoothing constants (refer to equations (1) and (2)), it can be inferred that smoothing constant with α=0.1 and β=0.8 provides better forecast because it minimizes the error.

Explanation of Solution

Given information:

Time period Month Income ($ in thousands)
1 February 70
2 March 68.5
3 April 64.8
4 May 71.7
5 June 71.3
6 July 72.8

Formula to calculate the MSE &forecasted demand

MSE=(Actual-Forecast)2n

Forecastincludingtrend(FITt)=(Exponentiallysmoothedforecastaverage(Ft)+ExponetiallysmoothedTrend(Tt))FITt=Ft+Tt

Ft=α(At-1)+1α(Ft-1+Tt-1)Tt=β(F-Ft-1)+(1β)Tt-1

Where,

Ft=ExponentiallySmoothedforecastaverageofthedataseriesinperiodtTt=ExponentiallySmoothedforecastaverageofthedataseriesinperiodtAt=Actualdemandinperiodt

α=smoothingconstantβ=smoothingconstantfortrend

Calculation of FIT and MSE usingα=0.1 andβ=0.2:

Time period Month Income ($ in thousands) Ft ($ in thousands) Tt FIT Error Sq. Error
1 February 70 65 0 65 5 25
2 March 68.5 65.500 0.100 65.600 2.9 8.41
3 April 64.8 65.890 0.158 66.048 -1.25 1.56
4 May 71.7 65.923 0.133 66.056 5.64 31.85
5 June 71.3 66.621 0.246 66.867 4.43 19.66
6 July 72.8 67.310 0.335 67.644 5.16 26.58
Total 113.05
MSE 18.84

Table 1

Excel worksheet:

Principles Of Operations Management, Chapter 4, Problem 20P , additional homework tip  1

Calculation of FIT for February:

FIT=F1+T1=65+0=$65

To calculate FIT for February, compute F1 and T1. The forecast F1 is $ 65 and trend T1 is 0. The sum of both values gives FIT, which is equal to $ 65.

Calculation of FIT for March:

F2=α(A1)+1-α(F1+T1)=0.1×70+1-0.1(65+0)=$65.5

T2=β(F2-F1)+(1-β)T1=0.2(65.5-65)+(1-0.2)0=0.1

FIT=F2+T2=65.5+0.10=$65.60

To calculate FIT for February, compute F2 and T2. The forecast F2 is $ 65.5 and trend T2 is 0.1. The sum of both values gives FIT, which is equal to $ 65.60.

Calculation of FIT for April:

F3=α(A2)+1-α(F2+T2)=0.1×68.5+1-0.1(65.5+0.1)=$65.890

T3=β(F3F2)+(1β)T2=0.2(65.7165.5)+(10.2)0.1=0.158

FIT=F3+T3=65.890+0.158=$66.048

To calculate FIT for March, compute F3 and T3. The forecast F3 is $ 65.890 and trend T3 is 0.158. The sum of both values gives FIT, which is equal to $ 66.048.

Calculation of FIT for May:

F4=α(A3)+1α(F3+T3)=0.1×64.8+10.1(65.710+0.122)=$65.923

T4=β(F4F3)+(1β)T3=0.2(65.50965.710)+(10.2)0.122=0.133

FIT=F4+T4=65.923+0.133=$66.056

To calculate FIT for May, compute F4 and T4. The forecast F4 is $ 65.923 and trend T4 is 0.133. The sum of both values gives FIT, which is equal to $ 66.056.

Calculation of FIT for June:

F5=α(A4)+1α(F4+T4)=0.1×71.7+1-0.1(65.509+0.057)=$66.621

T5=β(F5F4)+(1β)T4=0.2(66.07765.509)+(10.2)0.057=0.246

FIT=F5+T5=66.621+0.246=$66.867

To calculate FIT for June, compute F5 and T5. The forecast F5 is $ 66.621 and trend T5 is 0.246. The sum of both values gives FIT, which is equal to $ 66.867.

Calculation of FIT for July:

F6=α(A5)+1α(F5+T5)=0.1×71.3+10.1(66.077+0.159)=$67.310

T6=β(F6F5)+(1β)T5=0.2(66.45566.077)+(10.2)0.159=0.335

FIT=F6+T6=67.310+0.335=$67.644

To calculate FIT for July, compute F6 and T6. The forecast F6 is $ 67.310 and trend T6 is 0.335. The sum of both values gives FIT, which is equal to $ 67.644.

Calculation of MSE:

MSE is obtained by dividing the summation value of the square of the difference between actual and forecasted sales with the number of years n; n=6.

Table 1provides the values for square of the difference between actual and forecasted sales.

MSE=(Actual-Forecast)2n=25+8.41+1.56+31.85+19.66+26.586=113.056=18.84 (1)

MSE using α=0.1 and β=0.2 is 18.84.

Calculation of FIT and MSE using α=0.1 and β=0.8:

Time period Month Income ($ in thousands) Ft ($ in thousands) Tt FIT Error Sq. Error
1 February 70 65 0 65 5 25
2 March 68.5 65.500 0.400 65.900 2.6 6.76
3 April 64.8 66.160 0.608 66.768 -1.968 3.87
4 May 71.7 66.571 0.451 67.022 4.678 21.89
5 June 71.3 67.490 0.825 68.314 2.986 8.91
6 July 72.8 68.613 1.064 69.677 3.123 9.76
Total 76.19
MSE 12.70

Table 2

Excel worksheet:

Principles Of Operations Management, Chapter 4, Problem 20P , additional homework tip  2

Calculation of FIT for February:

FIT=F1+T1=65+0=$65

To calculate FIT for February, compute F1 and T1. The forecast F1 is $ 65 and trend T1 is 0. The sum of both values gives FIT, which is equal to $ 65.

Calculation of FIT for March:

F2=α(A1)+1α(F1+T1)=0.1×70+10.1(65+0)=$65.5

T2=β(F2F1)+(1β)T1=0.8(65.565)+(10.8)0=0.4

FIT=F2+T2=65.5+0.4=$65.90

To calculate FIT for February, compute F2 and T2. The forecast F2 is $ 65.5 and trend T2 is 0.4. The sum of both values gives FIT, which is equal to $ 65.90.

Calculation of FIT for April:

F3=α(A2)+1α(F2+T2)=0.1×68.5+10.1(65.5+0.4)=$66.160

T3=β(F3-F2)+(1-β)T2=0.8(65.440-65.5)+(1-0.8)0.4=0.608

FIT=F3+T3=66.160+0.608=$66.786

To calculate FIT for March, compute F3 and T3. The forecast F3 is $ 66.160 and trend T3 is 0.608. The sum of both values gives FIT, which is equal to $ 66.786.

Calculation of FIT for May:

F4=α(A3)+1α(F3+T3)=0.1×64.8+10.1(65.440+0.608)=$66.571

T4=β(F4F3)+(1β)T3=0.8(65.34765.440)+(10.8)0.608=0.451

FIT=F4+T4=65.347+0.451=$67.022

To calculate FIT for May, compute F4 and T4. The forecast F4 is $66.571 and trend T4 is 0.451. The sum of both values gives FIT, which is equal to $ 67.022.

Calculation of FIT for June:

F5=α(A4)+1α(F4+T4)=0.1×71.7+10.1(65.347+0.451)=$67.490

T5=β(F5F4)+(1β)T4=0.8(67.49066.571)+(10.8)0.451=0.825

FIT=F5+T5=67.490+0.825=$68.314

To calculate FIT for June, compute F5 and T5. The forecast F5 is $ 67.490 and trend T5 is 0.825. The sum of both values gives FIT, which is equal to $ 68.314.

Calculation of FIT for July:

F6=α(A5)+1α(F5+T5)=0.1×71.3+10.1(67.490+0.825)=$68.613

T6=β(F6F5)+(1β)T5=0.8(68.61367.490)+(10.8)0.825=1.064

FIT=F6+T6=68.613+1.064=$69.677

To calculate FIT for July, compute F6 and T6. The forecast F6 is $ 68.613 and trend T6 is 1.064. The sum of both values gives FIT, which is equal to $ 69.677.

Calculation of MSE:

MSE is obtained by dividing the summation value of the square of the difference between actual and forecasted sales with the number of years n=6.

Table 2provides the values for square of the difference between actual and forecasted sales.

MSE=(ActualForecast)2n=25+6.76+3.87+21.89+8.91+9.766=76.196=12.70 (2)

MSE using α=0.1 and β=0.8 is 12.70.

On comparing MSE from two smoothing constants (refer to equations(1)&(2)), it can be inferred that smoothing constant with α=0.1 and β=0.8 provides better forecast because it minimizes the error.

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Principles Of Operations Management

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