Supply chain management strategy planning and operation 7th edition chopra solutions manual

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Chapter 7: Demand Forecasting in a Supply Chain Exercise Solutions: Problem 7-1: We utilize a static model with level, trend, and seasonality components to evaluate the forecasts for year 6. Initially, we deseasonalize the demand and utilize regression in estimating the trend and level components. We then estimate the seasonal factors for each period and evaluate forecasts. Spreadsheet Exercise 7-1 provides the solution to this problem.

The model utilized for forecasting is: Ft +l = [ L + (t + l )T ]St +l The deseasonalized regression model is (see Cells T52 and T53 for values of deseasonalized level at period 0 and trend at period 0): _

D t = 5997.261 + 70.245 t

The seasonal indices for each of the 12 months are (see Cells T59-70): Month S.I JAN 0.427 FEB 0.475 MAR 0.463 APR 0.398 MAY 0.621 JUN 0.834 JUL 0.853 AUG 1.151 SEP 1.733 OCT 1.778 NOV 2.124 DEC 1.095

For example, the forecast for January of Year 6 is obtained by the following calculation (Cell H81): F61 = [5997.261 + (61) * 70.245] * 0.4266 = 4386

The forecast for year 6 is obtained in Cells H81:92 and is as follows:

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The various measures of forecast errors are as follows (Cells L80 – Q80).

The overall quality of the forecast is reasonably good.

Problem 7-2: Worksheet Exercise 7-2 compares the four-week moving average approach with the exponential smoothing model (alpha = 0.1). In a four-week moving average model, the weight assigned to the most recent data is 0.25 whereas in the case of the exponential smoothing model the weight assigned is 0.1. The following graphs depict the results from the two models.

Unit Demand

Moving Average

Actual Demand

Forecasted Demand

130 125 120 115 110 105 100 95 90 85 80 1

2

3

4

5

6

7

8 Periods

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9

10 11 12 13 14 15 16

EXPONENTIAL SMOOTHING

Unit Demand

Actual Demand

Forecasted Demand

130 125 120 115 110 105 100 95 90 85 80 1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16

Periods

For this specific problem, it is evident that the moving average model is more responsive than the exponential smoothing approach due the difference in weights allocation (0.25 and 0.1). The two methods have very similar errors (the errors for Moving Average are shown in Cells G14:N26; the errors for Exponential Smoothing are shown in Cells G40:N53). Using MAD as a measure for forecast accuracy, it can be concluded that the moving average model (MAD = 9) is slightly more accurate than the exponential smoothing model (MAD = 10) in evaluating forecasts.

Problem 7-3: The simple exponential smoothing model only considers the level component and does not include a trend component in the analysis. However, Holt’s model allows for the incorporation of the trend component into the analysis. Worksheet Exercise 7-3 provides the results of the two approaches.

The forecast and errors using simple exponential smoothing (alpha = 0.1) are as shown below:

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By investigating the relationship between sales and period (shown in the above graph), it is evident that the data exhibits both random fluctuation and trend. We run a linear regression between Sales and Period to estimate the initial level (intercept value) and initial trend (coefficient of X Variable 1) for Holt’s model. The initial level and trend are used as L0 and T0 for Holt’s model. The results using Holt’s model are as shown below:

Given the obvious trend in the data, it is not surprising in the analysis that Holt’s model (alpha = 0.1, beta = 0.1, MAD = 8) is a better approach than the simple exponential smoothing model (alpha = 0.1, MAD = 21).

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Problem 7-4: Worksheet Exercise 7-4 evaluates demand forecasts for the ABC Corporation using moving average (see sheet MovingAverage), simple exponential smoothing (see sheet ExponentialSmoothing), Holt’s model (see sheet Holt’sModel), and Winter’s model (see sheet Winter’sModel). Note that solver is utilized for simple exponential smoothing, Holt’s and Winter’s models in determining the optimal values for the smoothing constants by minimizing the MAD subject to the constraint that the smoothing constant values are < 1. The forecasts for Year 6 using the various methods are as in the Table below:

Month

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

Moving Average

Simple Exponential Smoothing

9,417 9,417 9,417 9,417 9,417 9,417 9,417 9,417 9,417 9,417 9,417 9,417

8,000 8,000 8,000 8,000 8,000 8,000 8,000 8,000 8,000 8,000 8,000 8,000

Holt’s Model 8,170 8,240 8,310 8,380 8,451 8,521 8,591 8,661 8,732 8,802 8,872 8,942

Winter’s Model 4,420 4,949 4,856 4,207 6,607 8,929 9,185 12,476 18,901 19,514 23,452 12,164

The MAD for the various methods is as follows: Method Moving Average Simple Exponential Smoothing Holt’s Model Winter’s Model

MAD 3,590 3,004 2,961 1,028

It is evident that Winter’s model is preferable in this case with the lowest MAD value, that is, lowest forecast error. It is also important to note that Winter’s model allows for the incorporation of level, trend, and seasonality, which are evident in the demand data for this case.

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Problem 7-5: Worksheet Exercise 7-5 compares the two exponential smoothing forecasts with an alpha of 0.1 and alpha of 0.9.

α = 0.1

α = 0.9

While at first glance that the alpha of 0.9 is a better tracker of the forecast, a careful review indicates that the alpha of 0.9 model is essentially using the last period demand to predict next period demand. As a result, during unstable times (periods 4–10) it has large errors. In contrast, the method with alpha of 0.1 is not so reactive to recent changes and provides a more stable forecast where errors are never very large. This difference really shows up in the forecasts from period 13 onwards. The method with alpha of 0.9 behaves as though demand has permanently dropped in periods 11 and 12. In contrast, the method with alpha of 0.1 behaves as though the drop in periods 11 and 12 is part of natural variation. This where forecast becomes as much art as science. An experienced manager will be in a position to decide which of the two methods is more reflective of the current reality.

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Problem 7-6: Worksheet Exercise 7-6 looks at the forecast for A&D Electronics and compares the results of simple exponential smoothing model with the Holt’s model. In looking at the results of these two models, it is evident that the Holt’s model is a better forecasting model. A plot of sales over the 12 periods indicates a clear increasing trend, making Holt’s model the appropriate choice.

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Problem 7-7: Worksheet Exercise 7-7 reexamines the A&D Electronics data with the Holt’s model being run with the original alpha at 0.05 and beta at 0.1 and a revised Holt’s with an alpha and beta both of 0.5.

Observe that increasing the values of α and β results in a more volatile forecast that has larger errors. The smoothing constants α = 0.05 and β = 0.1 are preferred because they result in lower errors and a less volatile forecast.

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Problem 7-8: Worksheet 7-8 compares the five-week moving average approach with the exponential smoothing model (alpha = 0.1). The results are as shown below:

Given a lower MAD and MAPE, the exponential smoothing method seems to do a little better.

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Problem 7-9: Worksheet 7-9 contains the results of the analysis. The first step is to perform a deseasonalized analysis to estimate the initial value of level (for period 0), trend (for period 0), and seasonal factors (for periods 1–4). Our initial estimates (see worksheet Values) are obtained to be: L0 = 921.54; T0 =91.24; S1 = 0.50; S2 = 0.79; S3 = 1.21; S4 = 1.50.

Using these estimates, Winter’s model provides the following forecasts and errors (see worksheet Values):

The values of α, β, and γ that minimize MAD and the associated forecasts are obtained on worksheet Values (optimal MAD). The optimization limits all smoothing constants to be between 0 and 0.9.

Observe that optimizing the smoothing constants lowers the MAD to 27 (compared to 31).

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The values of α, β, and γ that minimize MSE and the associated forecasts are obtained on worksheet Values (optimal MSE). The optimization limits all smoothing constants to be between 0 and 0.9.

Observe that optimizing the smoothing constants lowers the MSE to 1161 (compared to 1466).

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Problem 7-10: Worksheet 7-10 contains the results of the analysis. The first step is to perform a deseasonalized analysis to estimate the initial value of level (for period 0), trend (for period 0), and seasonal factors (for periods 1–4). Our initial estimates (see worksheet Values) are obtained to be: L0 = 1985.68; T0 = 541.43; S1 = 0.94; S2 = 0.45; S3 = 1.17; S4 = 1.46.

Using these estimates, Winter’s model provides the following forecasts and errors (see worksheet Values):

The values of α, β, and γ that minimize MAD and the associated forecasts are obtained on worksheet Values (optimal MAD). The optimization limits all smoothing constants to be between 0 and 0.9.

Observe that optimizing the smoothing constants lowers the MAD to 515 (compared to 592).

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The values of α, β, and γ that minimize MSE and the associated forecasts are obtained on worksheet Values (optimal MSE). The optimization limits all smoothing constants to be between 0 and 0.9.

Observe that optimizing the smoothing constants lowers the MSE to 500,481 (compared to 706,550).

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