EBK PRINCIPLES OF OPERATIONS MANAGEMENT
EBK PRINCIPLES OF OPERATIONS MANAGEMENT
11th Edition
ISBN: 9780135175859
Author: Munson
Publisher: VST
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Chapter 4, Problem 17P
Summary Introduction

To determine: Find the forecast of sales using exponential smoothing with smoothing constant 0.6 and 0.9 and infer the effect of exponential smoothing on forecast. Using MAD, determine the accurate forecast of exponential smoothing with given smoothing constant 0.3, 0.6 and 0.9.

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 a uniform time interval. Moving average method and trend projections are two of the time series methods which use weights to prioritize past data.

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Answer to Problem 17P

On Comparing MAD from exponential smoothing with smoothing constant 0.3, 0.6 and 0.9 (refer to equations (1), (2) and (3)), it can be inferred that the MAD of exponential smoothing with smoothing constant is most accurate because of least value of MAD.

Explanation of Solution

Forecast of sales using exponential smoothing with smoothing constant 0.6:

Given information:

Year Sales
1 450
2 495
3 518
4 563
5 584

Initial forecast for year 1=410Smoothingconstant=0.6

Formula to calculate the forecasted demand:

Ft=Ft-1+α(At-1-Ft-1)

Where,

Ft=newforecastFt-1=Previousperiod'sforecastα=smoothingconstantAt-1=PreviousperiodactualDemand

Smoothing constant=0.6
Year Sales Forecast Absolute error
1 450 410 40
2 495 434 61
3 518 470.6 47.4
4 563 499.04 63.96
5 584 537.416 46.584
6 565.3664
Total 258.944
MAD 51.7888

Excel worksheet:

EBK PRINCIPLES OF OPERATIONS MANAGEMENT, Chapter 4, Problem 17P , additional homework tip  1

Calculation of the forecast for year 2:

F2=F1+α(A1-F1)=410+0.6(450410)=434

To calculate the forecast for year 2, substitute the value of forecast of year 1, smoothing constant and the difference of actual and forecasted demand in the above formula. The result of the forecast for year 2 is 434.

Calculation of the forecast for year 3:

F3=F2+α(A2-F2)=434+0.6(495434)=470.6

To calculate the forecast for year 3, substitute the value of forecast of year 2, smoothing constant and the difference of actual and forecasted demand in the above formula. The result of the forecast for year 3 is 470.6.

Calculation of the forecast for year 4:

F4=F3+α(A3-F3)=470.60+0.6(518470.60)=499.04

To calculate the forecast for year 4, substitute the value of forecast of year 3, smoothing constant and the difference of actual and forecasted demand in the above formula. The result of the forecast for year 4 is 499.04.

Calculation of the forecast for year 5:

F5=F4+α(A4-F4)=499.04+0.6(563499.04)=537.416

To calculate the forecast for year 5, substitute the value of forecast of year 4, smoothing constant and the difference of actual and forecasted demand in the above formula. The result of the forecast for year 5 is 537.416.

Calculation of the forecast for year 6:

F6=F5+α(A5F5)=537.416+0.6(584537.416)=565.36

To calculate the forecast for year 5, substitute the value of forecast of year 5, smoothing constant and the difference of actual and forecasted demand in the above formula. The result of the forecast for year 6 is 565.36.

Calculation of MAD using exponential smoothing with smoothing constant α=0.6:

Formula to calculate the Mean Absolute Deviation:

MAD=|ActualForecast|n

Calculation of the absolute error for year 1:

Absoluteerror=|ActualForecast|=|450410|=|40|=40

The absolute error for year 1 is the modulus of the difference between 450 and 410, which corresponds to 40. Therefore, the absolute error for year 1 is 40.

Calculation of the absolute error for year 2:

Absoluteerror=|Actual-Forecast|=|495434|=|61|=61

The absolute error for year 2 is the modulus of the difference between 495 and 434, which is correspond to 61. Therefore, the absolute error for year 2 is 61.

Calculation of the absolute error for year 3:

Absoluteerror=|Actual-Forecast|=|518470.6|=|47.4|=47.4

The absolute error for year 3 is the modulus of the difference between 518and 470.6, which is correspond to 47.4. Therefore, the absolute error for year 3 is 47.4.

Calculation of the absolute error for year 4:

Absoluteerror=|Actual-Forecast|=|563499.04|=|63.96|=63.96

The absolute error for year 4 is the modulus of the difference between 563and499.04, which is correspond to 63.96. Therefore, the absolute error for year 4 is 63.96.

Calculation of the absolute error for year 5:

Absoluteerror=|Actual-Forecast|=|584537.416|=|46.584|=46.584

The absolute error for year 5 is the modulus of the difference between 584and537.416, which is correspond to 46.584. Therefore, the absolute error for year 5 is 46.584.

Calculation of the Mean Absolute Deviation using exponential smoothing with smoothing constant 0.6:

MAD=|Actual-Forecast|n=40+61+47.4+63.96+46.5845=258.9445=51.7888 (1)

Upon the substitution of summation value of the absolute error for 5 years, that is, 258.944 is divided by the number of years. That is, 5 yields MAD of 51.7888.

The forecast of sales using exponential smoothing with 0.6 as smoothing constant is 565.36 and MAD is 51.7888.

Forecast of sales using exponential smoothing with smoothing constant 0.9:

Given information:

Year Sales
1 450
2 495
3 518
4 563
5 584

Initial forecast for year 1=410Smoothingconstant=0.9

Formula to calculate the forecasted demand:

Ft=Ft-1+α(At-1Ft-1)

Where,

Ft=newforecastFt-1=Previousperiod'sforecastα=smoothingconstantAt-1=PreviousperiodactualDemand

Smoothing constant=0.9
Year Sales Forecast Absolute error
1 450 410 40
2 495 446 49
3 518 490.1 27.9
4 563 515.21 47.79
5 584 558.221 25.779
6 581.4221
Total 190.469
MAD 38.0938

Excel worksheet:

EBK PRINCIPLES OF OPERATIONS MANAGEMENT, Chapter 4, Problem 17P , additional homework tip  2

Calculation of the forecast for year 2:

F2=F1+α(A1-F1)=410+0.9(450410)=446

To calculate the forecast for year 2, substitute the value of forecast of year 1, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 2 is 446.

Calculation of the forecast for year 3:

F3=F2+α(A2-F2)=446+0.9(495446)=490.1

To calculate the forecast for year 3, substitute the value of forecast of year 2, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 3 is 490.1.

Calculation of the forecast for year 4:

F4=F3+α(A3-F3)=490.1+0.9(518490.1)=515.21

To calculate the forecast for year 4, substitute the value of forecast of year 3, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 4 is 515.21.

Calculation of the forecast for year 5:

F5=F4+α(A4-F4)=515.21+0.9(563515.21)=558.221

To calculate the forecast for year 5, substitute the value of forecast of year 4, smoothing constant and difference of actual and forecasted demand in the above formula. The result of forecast for year 5 is 558.221.

Calculation of the forecast for year 6:

F6=F5+α(A5-F5)=558.221+0.9(584558.221)=581.42

To calculate the forecast for year 6, substitute the value of forecast of year 5, smoothing constant and difference of actual and forecasted demand in the above formula. The result of forecast for year 6 is 581.42.

Calculation of MAD using exponential smoothing with smoothing constant α=0.9:

Formula to calculate the Mean Absolute Deviation:

MAD=|ActualForecast|n

Calculation of the absolute error for year 1:

Absoluteerror=|Actual-Forecast|=|450410|=|40|=40

The absolute error for year 1 is the modulus of the difference between 450 and 410, which corresponds to 40. Therefore, the absolute error for year 1 is 40.

Calculation of the absolute error for year 2:

Absoluteerror=|Actual-Forecast|=|495446|=|49|=49

The absolute error for year 2 is the modulus of the difference between 495 and 446, which corresponds to 49. Therefore, the absolute error for year 2 is 49.

Calculation of the absolute error for year 3:

Absoluteerror=|Actual-Forecast|=|518490.1|=|27.9|=27.9

The absolute error for year 3 is the modulus of the difference between 518and490.1, which corresponds to 27.9. Therefore, the absolute error for year 3 is 27.9.

Calculation of the absolute error for year 4:

Absoluteerror=|Actual-Forecast|=|563515.21|=|47.79|=47.79

The absolute error for year 4 is the modulus of the difference between 563and515.21, which corresponds to 4.254. Therefore, the absolute error for year 4 is 47.79.

Calculation of the absolute error for year 5:

Absoluteerror=|Actual-Forecast|=|584558.221|=|25.779|=25.779

The absolute error for year 5 is the modulus of the difference between 584and558.221, which corresponds to 25.779. Therefore, the absolute error for year 5 is 25.779.

Calculation of the Mean Absolute Deviation using exponential smoothing:

MAD=|Actual-Forecast|n=40+49+27.9+47.79+25.7795=190.4695=38.093 (2)

Upon the substitution of summation value of absolute error for 5 years, that is, 190.469 is divided by the number of years. That is, 5 yields MAD of 38.093.

The forecast of sales using exponential smoothing with 0.9 as smoothing constant is 581.4221 and MAD is 38.093.

Forecast of sales using exponential smoothing with smoothing constant 0.3:

Given information:

Year Sales
1 450
2 495
3 518
4 563
5 584

Initial forecast for year 1=410Smoothingconstant=0.3

Formula to calculate the forecasted demand:

Ft=Ft-1+α(At-1-Ft-1)

Where,

Ft=newforecastFt-1=Previousperiod'sforecastα=smoothingconstantAt-1=PreviousperiodactualDemand

Smoothing constant=0.3
Year Sales Forecast Absolute error
1 450 410 40
2 495 422 73
3 518 443.9 74.1
4 563 466.13 96.87
5 584 495.191 88.809
6 521.8337
Total 372.779
MAD 74.5558

Excel worksheet:

EBK PRINCIPLES OF OPERATIONS MANAGEMENT, Chapter 4, Problem 17P , additional homework tip  3

Calculation of the forecast for year 2:

F2=F1+α(A1-F1)=410+0.3(450410)=422

To calculate the forecast for year 2, substitute the value of forecast of year 1, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 2 is 422.

Calculation of the forecast for year 3:

F3=F2+α(A2-F2)=422+0.3(495422)=443.9

To calculate the forecast for year 3, substitute the value of forecast of year 2, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 3 is 443.9.

Calculation of the forecast for year 4:

F4=F3+α(A3-F3)=443.9+0.3(518443.9)=466.13

To calculate the forecast for year 4, substitute the value of forecast of year 3, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 4 is 466.13.

Calculation of the forecast for year 5:

F5=F4+α(A4-F4)=466.13+0.3(563466.13)=495.191

To calculate the forecast for year 5, substitute the value of forecast of year 4, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 5 is 495.191.

Calculation of the forecast for year 6:

F6=F5+α(A5-F5)=495.191+0.9(584495.191)=521.833

To calculate the forecast for year 6, substitute the value of forecast of year 5, smoothing constant and difference of actual and forecasted demand in the above formula. The result of the forecast for year 6 is 521.833.

Calculation of MAD using exponential smoothing with smoothing constant α=0.3:

Formula to calculate the Mean Absolute Deviation:

MAD=|Actual-Forecast|n

Calculation of the absolute error for year 1:

Absoluteerror=|Actual-Forecast|=|450410|=|40|=40

The absolute error for year 1 is the modulus of the difference between 450 and 410, which corresponds to 40. Therefore, the absolute error for year 1 is 40.

Calculation of the absolute error for year 2:

Absoluteerror=|Actual-Forecast|=|495422|=|73|=73

The absolute error for year 2 is the modulus of the difference between 495 and 422, which corresponds to 73. Therefore, the absolute error for year 2 is 73.

Calculation of the absolute error for year 3:

Absoluteerror=|Actual-Forecast|=|518443.9|=|74.1|=74.1

The absolute error for year 3 is the modulus of the difference between 518 and 443.9, which corresponds to 74.1. Therefore, the absolute error for year 3 is 74.1.

Calculation of the absolute error for year 4:

Absoluteerror=|Actual-Forecast|=|563466.13|=|96.87|=96.87

The absolute error for year 4 is the modulus of the difference between 563 and 466.13, which corresponds to 96.87. Therefore, the absolute error for year 4 is 96.87.

Calculation of the absolute error for year 5:

Absoluteerror=|Actual-Forecast|=|584495.191|=|88.809|=88.809

The absolute error for year 5 is the modulus of the difference between 584 and 495.191, which corresponds to 88.809. Therefore, the absolute error for year 5 is 88.809.

Calculation of the Mean Absolute Deviation using exponential smoothing with 0.3 as smoothing constant:

MAD=|Actual-Forecast|n=40+73+74.1+96.87+88.8095=372.7795=74.5558 (3)

Upon the substitution of summation value of absolute error for 5 years, that is, 372.779 are divided by the number of years. That is, 5 yields MAD of 74.5558.

The forecast of sales using exponential smoothing with 0.3 as smoothing constant is 521.833 and MAD is 74.5558.

Hence, on comparing MAD from exponential smoothing with smoothing constant 0.3, 0.6 and 0.9 (refer to equations (1), (2) and (3)), it can be inferred that MAD of exponential smoothing with smoothing constant is most accurate because of least MAD.

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Chapter 4 Solutions

EBK PRINCIPLES OF OPERATIONS MANAGEMENT

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