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hello here is question with the slides, most likely you gone find the answer in the slides if you did not find in the slides you gone find it in the txt bookA consultant has been brought in to a manufacturing plant to help apply Six Sigma principles. Her first task is to work on the production of rubber balls. The upper and lower spec limits are 21 and 19 cm, respectively. A. The consultant takes ten samples of size five and computes the the mean of the sample means to be 19.89 cm and the standard deviation of the process to be 0.7. Is this process capable of meeting the specifications? Why or why not? What would you do bring the process under control? B. If the mean is set to 20cm, what should be the standard deviation of the process to achieve 3 sigma quality, 4 sigma quality and 6 sigma quality? A. Thinking of a university as a production facility that prepares students for the job market, draw a supply chain for this process. Identify the upstream and downstream processes that need to be managed. What information flows back and forth in the supply chain? How would you measure the performance of this supply chain? B. The following data are pulled from a recent Walsh Manufacturing annual report. Assets Raw material inventory $120,000 Work-in-process inventory $50,000 Finished goods inventory $300,000 Property, plant & equipment $500,000 Other assets $200,000 Total assets $1,170,000 Condensed Income Statement Revenue $2,000,000 Cost of goods sold $600,000 Other expenses $1,000,000 Net income $400,000 Calculate: (a) Percent invested in inventory, (b) Inventory turnover, and (c) Weeks of supply.
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Operations Management: Processes and
Supply Chains
Twelfth Edition
Chapter 8
Forecasting
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
What is a Forecast?
• Forecast
– A prediction of future events used for planning
purposes.
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Demand Patterns (1 of 5)
• Time series
– The repeated observations of demand for a service or
product in their order of occurrence
• There are five basic time series patterns
1.
2.
3.
4.
5.
Horizontal
Trend
Seasonal
Cyclical
Random
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Demand Patterns (2 of 5)
Figure 8.1 Patterns of Demand
(a) Horizontal: Data cluster about a horizontal line
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Demand Patterns (3 of 5)
Figure 8.1 [continued]
(b) Trend: Data consistently increase or decrease
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Demand Patterns (4 of 5)
Figure 8.1 [continued]
(c) Seasonal: Data consistently show peaks and valleys
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Demand Patterns (5 of 5)
Figure 8.1 [continued]
(d) Cyclical: Data reveal gradual increases and decreases over
extended periods
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Key Decisions on Making Forecasts
• Deciding What to Forecast
– Level of aggregation
– Units of measurement
• Choosing the Type of Forecasting Technique
– Judgment methods
– Causal methods
– Time-series analysis
– Trend projection using regression
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Forecast Error
• For any forecasting method, it is important to measure the
accuracy of its forecasts.
• Forecast error is simply the difference found by subtracting
the forecast from actual demand for a given period, or
Et = Dt – Ft
where
Et = forecast error for period t
Dt = actual demand in period t
Ft = forecast for period t
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Measures of Forecast Error (1 of 3)
Cumulative sum of
forecast errors (Bias)
? =
CFE = ? Et
Average forecast error
E=
CFE
n
(
? Et – E
2
)
2
n -1
Mean Absolute Deviation
MAD =
Mean Squared Error
? Et
MSE =
n
Standard deviation
? Et
n
Mean Absolute Percent Error
?E
(
MAPE =
t
Dt ) (100 )
n
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Measures of Forecast Error (2 of 3)
Figure 8.2(B) Detailed Calculations of Forecast Errors
Blank
Actual
Forecast
Absolute
Error Error
Error
Absolute
Pct
Pct Error
Error
Error
Errorcap
^2 2
Past period 1
39
41
-2
2
4
5.128%
Past period 2
37
43
-6
6
36
16.216%
Past period 3
55
45
10
10
100
18.182%
Past period 4
40
50
-10
10
100
25%
Past period 5
59
51
8
8
64
13.559%
Past period 6
63
56
7
7
49
11.111%
Past period 7
41
61
-20
20
400
48.78%
Past period 8
57
60
-3
3
9
5.236%
Past period 9
56
62
-6
6
36
10.714%
Past period 10
54
63
-9
9
81
16.667%
Totals
501
Blank
-31
81
879
170.621%
Average
50.1
Blank
-3.1
8.1
87.9
17.062%
Blank
0
(Bias)
(MAD)
(MSE)
(MAPE)
Next period forecast
Blank
Blank
Blank
Blank
std err
9.883
Blank
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Measures of Forecast Error (3 of 3)
Figure 8.2(C) Error Measures
Measure
Value
Error Measures
Blank
CFC (Cumulative Forecast Error)
-31
MAD (Mean Absolute Deviation)
8.1
MSE (Mean Squared Error)
87.9
Standard Deviation of Errors
9.883
MAPE (Mean Absolute Percent Error)
17.062%
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 1 (1 of 4)
The following table shows the actual sales of upholstered chairs for a
furniture manufacturer and the forecasts made for each of the last eight
months.
Calculate CFE, MSE, s, MAD, and MAPE for this product.
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 1 (2 of 4)
Using the formulas for the measures, we get:
Cumulative forecast error (mean bias)
CFE = -15
Average forecast error (mean bias):
E=
CFE 15
=
= -1.875
n
8
Mean squared error:
2
5,275
? Et
MSE =
=
= 659.4
n
8
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 1 (3 of 4)
Standard deviation:
? ?Et – ( -1.875)?
= 27.4
n -1
2
? =
Mean absolute deviation:
? Et
195
MAD =
=
= 24.4
n
8
Mean absolute percent error:
MAPE =
(? E
t
Dt ) (100 )
n
=
81.3%
= 10.2%
8
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 1 (4 of 4)
• A CFE of -15 indicates that the forecast has a slight bias to
overestimate demand.
• The MSE, s, and MAD statistics provide measures of forecast error
variability.
• A MAD of 24.4 means that the average forecast error was 24.4 units
in absolute value.
• The value of s, 27.4, indicates that the sample distribution of forecast
errors has a standard deviation of 27.4 units.
• A MAPE of 10.2 percent implies that, on average, the forecast error
was about 10 percent of actual demand.
These measures become more reliable as the number of periods of
data increases.
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Judgment Methods
• Other methods (casual, time-series, and trend projection
using regression) require an adequate history file, which
might not be available.
• Judgmental forecasts use contextual knowledge gained
through experience.
– Salesforce estimates
– Executive opinion
– Market research
– Delphi method
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Causal Methods: Linear Regression
• Dependent variable – The variable that one wants to forecast
• Independent variable – The variable that is assumed to affect
the dependent variable and thereby “cause” the results
observed in the past
• Simple linear regression model is a straight line
Y = a + bX
where
Y = dependent variable
X = independent variable
a = Y-intercept of the line
b = slope of the line
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Linear Regression (1 of 2)
Figure 8.3 Linear Regression Line Relative to Actual Demand
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Linear Regression (2 of 2)
• The sample correlation coefficient, r
– Measures the direction and strength of the relationship between
the independent variable and the dependent variable.
– The value of r can range from -1.00 ? r ? 1.00
• The sample coefficient of determination, r 2
– Measures the amount of variation in the dependent variable
about its mean that is explained by the regression line
– The values of r 2 range from0.00 ? r ² ? 1.00
• The standard error of the estimate, syx
– Measures how closely the data on the dependent variable
cluster around the regression line
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 2 (1 of 4)
The supply chain manager seeks a better way to forecast the demand
for door hinges and believes that the demand is related to advertising
expenditures. The following are sales and advertising data for the past
5 months:
Month
Sales (thousands of units)
Advertising (thousands of $)
1
264
2.5
2
116
1.3
3
165
1.4
4
101
1.0
5
209
2.0
The company will spend $1,750 next month on advertising for the
product. Use linear regression to develop an equation and a forecast
for this product.
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 2 (2 of 4)
We used POM for Windows to determine the best values
of a, b, the correlation coefficient, the coefficient of
determination, and the standard error of the estimate
a = -8.135
b = 109.229
r = 0.980
r 2 = 0.960
syx = 15.603
The regression equation is Y = -8.135 + 109.229X
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 2 (3 of 4)
The r of 0.98 suggests an unusually strong positive relationship
between sales and advertising expenditures. The coefficient of
determination, r 2 , implies that 96 percent of the variation in sales
is explained by advertising expenditures.
Figure 8.4 Linear
Regression Line
for the Sales and
Advertising Data
Using POM for
Windows
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 2 (4 of 4)
• Forecast for month 6:
Y = -8.135 + 109.229X
Y = -8.135 + 109.229 (1.75 )
Y = 183.016 or 183,016 units
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Time Series Methods
• Naïve forecast
– The forecast for the next period equals the demand
for the current period (Forecast = Dt)
• Horizontal Patterns: Estimating the average
– Simple moving average
– Weighted moving average
– Exponential smoothing
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Simple Moving Averages
• Specifically, the forecast for period t + 1 can be
calculated at the end of period t (after the actual demand
for period t is known) as
Ft +1
Sum of last n demands Dt + Dt -1 + Dt -2 + ? ? ? + Dt -n+1
=
=
n
n
where
Dt = actual demand in period t
n = total number of periods in the average
Ft+1 = forecast for period t + 1
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 3 (1 of 2)
a. Compute a three-week moving average forecast for the
arrival of medical clinic patients in week 4. The numbers
of arrivals for the past three weeks were as follows:
Week
Patient Arrivals
1
400
2
380
3
411
b. If the actual number of patient arrivals in week 4 is 415,
what is the forecast error for week 4?
c. What is the forecast for week 5?
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 3 (2 of 2)
a. The moving average forecast at
the end of week 3 is:
F4 =
411 + 380 + 400
= 397.0
3
Week
Patient Arrivals
1
400
2
380
3
411
b. The forecast error for week 4 is
E4 = D4 – F4 = 415 – 397 = 18
c. The forecast for week 5 requires the actual arrivals from
weeks 2 through 4, the three most recent weeks of data
415 + 411 + 380
F5 =
= 402.0
3
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Weighted Moving Averages
In the weighted moving average method, each historical
demand in the average can have its own weight, provided
that the sum of the weights equals 1.0.
The average is obtained by multiplying the weight of each
period by the actual demand for that period, and then
adding the products together
Ft +1 = W1D1 + W2D2 + ? ? ? + Wn Dt – n +1
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Exponential Smoothing (1 of 2)
• A sophisticated weighted moving average that calculates
the average of a time series by implicitly giving recent
demands more weight than earlier demands
• Requires only three items of data
– The last period’s forecast
– The actual demand for this period
– A smoothing parameter, alpha (a), where 0 ? a ? 1.0
• The equation for the forecast is
Ft +1 = a (Demand this period ) + (1 – ? )(Forecast calculated last period)
= aDt + (1 – ? )Ft
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Exponential Smoothing (2 of 2)
• The emphasis given to the most recent demand levels
can be adjusted by changing the smoothing parameter.
• Larger a values emphasize recent levels of demand and
result in forecasts more responsive to changes in the
underlying average.
• Smaller a values are analogous to increasing the value of
n in the moving average method and giving greater
weight to past demand.
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 4 (1 of 3)
a. Reconsider the patient arrival data in Example 14.3. It is
now the end of week 3 so the actual arrivals is known to
be 411 patients. Using a = 0.10, calculate the
exponential smoothing forecast for week 4.
b. What was the forecast error for week 4 if the actual
demand turned out to be 415?
c. What is the forecast for week 5?
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 4 (2 of 3)
a. To obtain the forecast for week 4, using exponential
smoothing with and the initial forecast of 390*, we
calculate the forecast for week 4 as:
F 4 = 0.10 ( 411) + 0.90 ( 390 ) = 392.1
Thus, the forecast for week 4 would be 392 patients.
* POM for Windows and OM Explorer simply use the actual
demand for the first week as the default setting for the
initial forecast for period 1, and do not begin tracking
forecast errors until the second period.
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 4 (3 of 3)
b. The forecast error for week 4 is
E4 = 415 – 392 = 23
c. The new forecast for week 5 would be
F5 = 0.10 ( 415 ) + 0.90 ( 392.1) = 394.4 or 394 patients.
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Trend Patterns: Using Regression
• A trend in a time series is a systematic increase or
decrease in the average of the series over time
• Trend Projection with Regression accounts for the trend
with simple regression analysis.
• The regression equation is Ft = a + bt
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 5 (1 of 4)
• Medanalysis, Inc., provides medical laboratory services.
• Managers are interested in forecasting the number of
blood analysis requests per week.
• There has been a national increase in requests for
standard blood tests.
• The arrivals over the next 16 weeks are given in Table 8.1.
• What is the forecasted demand for the next three periods?
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 5 (2 of 4)
Table 8.1 Arrivals At Medanalysis for Last 16 Weeks
Week
Arrivals
Week
Arrivals
1
28
9
61
2
27
10
39
3
44
11
55
4
37
12
54
5
35
13
52
6
53
14
60
7
38
15
60
8
57
16
75
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 5 (3 of 4)
Figure 8.6(a) Trend Projection with Regression Results
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 5 (4 of 4)
Figure 8.6(b) Detailed Calculations of Forecast Errors
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Seasonal Patterns: Using Seasonal Factors
• Multiplicative seasonal method
– A method whereby seasonal factors are multiplied by
an estimate of average demand to arrive at a
seasonal forecast.
• Additive seasonal method
– A method in which seasonal forecasts are generated
by adding a constant to the estimate of average
demand per season.
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Multiplicative Seasonal Method
Multiplicative seasonal method
1. For each year, calculate the average demand for each
season by dividing annual demand by the number of
seasons per year.
2. For each year, divide the actual demand for each
season by the average demand per season, resulting in
a seasonal factor for each season.
3. Calculate the average seasonal factor for each season
using the results from Step 2.
4. Calculate each season’s forecast for next year.
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 6 (1 of 5)
The manager of the Stanley Steemer carpet cleaning
company needs a quarterly forecast of the number of
customers expected next year. The carpet cleaning business
is seasonal, with a peak in the third quarter and a trough in
the first quarter.
The manager wants to forecast customer demand for each
quarter of year 5, based on an estimate of total year 5
demand of 2,600 customers.
The table on the following slides shows the quarterly
demand data from the past 4 years along with the calculation
of the seasonal factor for each week.
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 6 (2 of 5)
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 6 (3 of 5)
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 6 (4 of 5)
Average Seasonal Factor
Quarterly Forecasts
Quarter
Average Seasonal
Factor
1
0.2043
2
1.2979
3
2.0001
3
650?times
2.001
1,300.065
650
2.001
= = 1,300.065
4
0.4977
4
650?times
0.4977== 323.505
650
0.4977
323.505
Quarter
Forecast
1
650 ?
times
0.2043==132.795
132.795
650
0.2043
2
650?times
1.2979
843.635
650
1.2979
== 843.635
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Example 6 (5 of 5)
Figure 8.7 Demand Forecasts Using the Seasonal Forecasting
Solver of OM Explorer
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Solved Problem 1 (1 of 2)
Chicken Palace periodically offers carryout five-piece chicken dinners
at special prices. Let Y be the number of dinners sold and X be the
price. Based on the historical observations and calculations in the
following table, determine the regression equation, correlation
coefficient, and coefficient of determination. How many dinners can
Chicken Palace expect to sell at $3.00 each?
Observation
Price (X)
Dinners Sold (Y)
1
$2.70
760
2
$3.50
510
3
$2.00
980
4
$4.20
250
5
$3.10
320
6
$4.05
480
Total
$19.55
3,300
Average
$3.26
550
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Solved Problem 1 (2 of 2)
We use the computer to calculate the best values of a, b, the
correlation coefficient, and the coefficient of determination
a = 1,454.60
b = -277.63
r = -0 .84
r ² = 0.71
The regression line is
Y = a + bX = 1,454.60 – 277.63X
For an estimated sales price of $3.00 per dinner
Y = a + bX = 1,454.60 – 277.63 ( 3.00 )
= 621.71 or 622dinners
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Solved Problem 2 (1 of 3)
The Polish General’s Pizza Parlor is a small restaurant catering to
patrons with a taste for European pizza. One of its specialties is Polish
Prize pizza. The manager must forecast weekly demand for these special
pizzas so that he can order pizza shells weekly. Recently, demand has
been as follows:
Week
Pizzas
Week
Pizzas
June 2
50
June 23
56
June 9
65
June 30
55
June 16
52
July 7
60
a. Forecast the demand for pizza for June 23 to July 14 by using the
simple moving average method with n = 3 then using the weighted
moving average method with weights of 0.50, 0.30, and 0.20, with .50
applying to the most recent demand.
b. Calculate the MAD for each method.
Copyright © 2019, 2016, 2014 Pearson Education, Inc. All Rights Reserved
Solved Problem 2 (2 of 3)
a. The simple moving average method and the weighted
moving average method give the following results:
Current
Week
June 16
June 23
June 30
July 7
Simple Moving Average
Forecast for Next Week
Weighted Moving Average Forecast for Next Week
52 plus 65 plus 50, over 3
52 + 65
+ 50
equals
55.7
or 56.
= 55.7 or 56
3
56 plus 52 plus 65, over 3
56 + 52
+ 65
equals
57.7
or 58.
= 57.7 or 58
3
55
55plus
+ 5656+ plus
52 52, over 3
54.3 or 54
equals 54.3 or =
54.
3
left bracket left paranthesis 0.5 times 52 right paranthesis + left
parantesis
0.3 times
paranthesis
?? 0.5 ? 52
+ 0.365? right
65 +paranthesis
0.2 ? 50 +?? left
= 55.5
or 56 0.2
times 50 right paranthesis right bracket = 55.5 or 56
60
60plus
+ 5555+ plus
56 56, over 3
57.0 or 57
equals 57.0 or =
57.
3
left bracket left paranthesis 0.5 times 60 right paranthesis + left
?? 0.5 ? 60 + 0.3 ? 55 + 0.2 ? 56 ?? = 57.7 or 58
parantesis
0.3 times 55 right paranthesis + left paranthesis 0.2
times 56 right paranthesis right bracket = 57.7 or 58
(
) (
) (

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