Chapter 03 - Forecasting
CHAPTER 03
FORECASTING
Forecasting is placed early in the text mainly because it is a point of departure. Some instructors like to
emphasize the operations part of operations management and de-emphasize the design part. Other
instructors prefer to blend the two. However, forecasting is an important input for both, and for that
reason, it is presented as early as possible.
Teaching Notes
This is a long chapter, so you may want to be selective about the topics covered to shorten the time
devoted to it. I tend to devote more time to the time series methods than I do to regression analysis, for
several reasons. One is that students often are exposed to regression in their stat course(s). Another is that
time series models are used more than associative models are. Other optional materials that can be
mentioned briefly, but not explored in detail, include trend-adjusted exponential smoothing (mentioned so
that students will realize that exponential smoothing does not work well if there is trend present) and
computation of seasonal relatives (you may want to explain how relatives are used without getting into
how they are derived).
I try to emphasize an intuitive approach to forecasting, with frequent reference to the importance of
plotting the data to assist the decision-maker in determining which forecasting technique may be more
appropriate to use.
In operations management, we forecast a wide range of future events, which could significantly affect the
long-term success of the firm. Most often, the basic need for forecasting arises in estimating customer
demand for a firm’s products and services. However, we may need aggregate estimates of demand as well
as estimates for individual products. In most cases, a firm will need a long-term estimate of overall
demand as well as a shorter-run estimate of demand for each individual product or service. Short-term
demand estimates for individual products are necessary to determine daily or weekly management of the
firm’s activities such as scheduling personnel and ordering materials. On the other hand, long-term
estimates of overall or aggregate demand can be used in determining company strategy, planning longterm capacity and establishing facility location needs of the firm.
Finally, it is important to point out the difference between forecasting and planning. Planning is often in
response to a forecast. A passive response would be to reduce output because of a predicted decrease in
demand, while an active response would be to advertise in an effort to offset the predicted decrease in
demand.
Reading: Gazing at the Crystal Ball
1.
Demand forecasting (DF) is part science and part art (intuition) for estimating what future demand
for a product or service will be. The science part uses information technology to generate demand
forecasts using existing data from a variety of sources, e.g., distribution channels, factory outlets,
value-added resellers, historical sales data, and macroeconomic data. The art/intuition part involves
subject matter experts (SMEs) making educated guesses about future demand.
2.
A company executive might make bold predictions about future demand to Wall Street analysts to
maintain the company’s stock price.
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Chapter 03 - Forecasting
3.
An executive’s comments to Wall Street analysts may result in the company changing its demand
forecast to reflect the comments made by the executive. The result often is excessive inventory
build-up starting at the distribution channels to the upstream suppliers.
Answers to Discussion and Review Questions
1.
It depends on the situation at hand. In certain situations, one approach will be superior to the
other.
Quantitative techniques lend themselves to computerization, they are less subject to personal
biases, and they may force managers to quantify information. On the other hand, the results are
only as good as the data, and in many cases, insufficient data exist to use a quantitative technique.
In addition, use of the computer sometimes creates an impression of “preciseness,” which is
misleading.
2.
Poor forecasting leads to poor planning. This could result in offering products and services that
customers do not want. Poor forecasting and planning would negatively affect budgeting and
planning for capacity, sales, production and inventory, labor, purchasing, energy requirements,
capital requirements, and materials requirements.
3.
a. Consumer surveys may be invalid if they are not carefully constructed, administered, and
interpreted. Moreover, respondents may be ill informed or otherwise formulate answers that
do not correctly reflect their future actions.
b. Salespeople often tend to be overly optimistic or pessimistic. They may attempt to use
estimates to influence quotas.
c. Committees of managers or executives can be expensive, diffuse responsibility for a forecast,
and reflect the opinions of a few dominant members.
4.
Forecasts generally are wrong due to the use of an incorrect model to forecast, random variation,
or unforeseen events.
5.
Control limits reveal the bounds of random errors; they enable managers to judge if a forecasting
technique is performing as well as it might (and hence, when a technique should be reevaluated).
6.
The relative costs of reevaluating a forecast when nothing is wrong versus not reevaluating it
when something is wrong. (Can also explain in terms of relative costs of Type I and Type II
errors.)
7.
MAD focuses on average error while MSE focuses on squared errors. (MSE gives considerably
more weight to large forecast errors.)
8.
Exponential smoothing: requires less data storage, gives more weight to recent data, and is easier
to change to make it more responsive to changes in demand.
9.
The fewer the periods in a moving average, the greater the responsiveness.
10.
The choice of alpha in exponential smoothing depends on how responsive a forecast the manager
desires. This, in turn, relates to the cost of not responding to a real change relative to the cost of
responding to what are merely random variations in the data.
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Chapter 03 - Forecasting
Of course, the accuracy of your five-day weather forecast will depend on a number of variables
such as time of year, where you live, etc. However, there is one trend that will establish itself and
that is as time passes from the first day to the fifth day, the accuracy of the forecast will decline.
The amount of random variation about the forecast (actual vs. forecast) would increase over time
somewhat like the following:
Weekly cost
11.
Minutes of daytime calls
If the normal random variation inherent in the forecast technique being used is deemed too great,
then try another technique. Note: Students answers will vary depending on what actual data they
obtain.
12.
For example, if each average is based on 12 months, as each new data point is added to the
moving average, its counterpart is removed from the other end of the series.
13.
Sales indicate how much customers bought, while demand indicates how much they wanted. The
distinction is important when demand exceeds supply, because supply places an upper bound on
the data.
14.
A reactive approach takes the forecast as a “given” while a proactive approach takes an
unacceptable forecast and attempts to alter demand. An example of the reactive approach is a
highway department preparing snow removal equipment for a predicted storm. Another is
evacuation of residents due to hurricane warnings. An example of the proactive approach is
colleges adopting a more aggressive stance towards applicants due to a forecast of a declining
college-age population base. Generally, firms that use advertising, promotions, discounts, and so
on tend to be proactive in dealing with forecasts.
15.
There is always going to be a certain amount of random variation about the forecast. The amount
of this random variation about the forecast (actual vs. forecast) will increase as the forecasting
horizon is extended. In other words, forecasting accuracy tends to decline over time.
Consequently, one of two approaches might be employed to handle the problem. One would be to
pick out some reasonable future point in time; then, based on past forecasting data, estimate the
amount of random variation that occurs over this period. The next step would be to build or
develop enough flexibility into your production system to be able to adjust to the extremes of the
random variation.
The other approach would be to estimate the flexibility of your production system and then see
how far into the future your production system would be able to handle the random variation that
is inherent in your forecasting approach. This would give you an indication of the amount of
change your system could handle over a period of time. Then adjustments could be made to
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Chapter 03 - Forecasting
increase your capabilities to respond to change or you could try another forecasting approach to
see if its inherent variation is less.
16.
Forecasting in the context of supply chain involves connection and communication between the
supply chain databases. For example, assume that Company X is a durable goods manufacturer.
Based on the market and historical sales information, Company X determines short and
intermediate term multi-period forecasts for its products and provides this forecast information to
its suppliers’ databases. Let us also assume that Company Y supplies Company X with parts and
components. Company Y uses the forecasting information from Company X, as well as from
other companies it supplies, and develops its own forecasts and provides all the forecasting
information it possesses to its suppliers. This type of cooperation and communication among the
supply chain databases provides all of the companies on the supply chain with additional
information to generate better forecasts. Potential difficulties in doing supply chain forecasting
include creating the ability for sharing of data between different information systems and
establishing trust between supply chain partners so that they are willing to share data.
17.
It depends on the situation. Sometimes one approach is better, sometimes the other is better, and
sometimes both are used. Considerations include the importance of the forecasts, how quickly the
forecasts are needed, the cost of obtaining the forecasts, the availability of resources, and the
availability of data. The qualitative approach is generally more popular with smaller companies
because they generally cannot afford to install a sophisticated quantitative technique. Larger
companies tend to utilize more sophisticated quantitative techniques due to the availability of
resources and the need to generate a large number of forecasts.
18.
In forecasting initial sales for the new version of its software, the software producer should
consider:
a. The historical demand information for the old version.
b. The features of the new version of the software in comparison to the features of the old
version.
c. The price of the new version of the software in comparison to the price of the old version.
d. Market/consumer information and response about the new version of the software based on
the results of a market survey and the beta testing of the new version of the software.
e. The features of competitors’ similar software packages.
f.
19.
The price of competitors’ software packages.
a. Demand for Mother’s Day greeting cards: Naïve using last year’s demand. Alternatively,
because greeting cards have seasonal demand, we could use a seasonal model where the
season begins a few weeks before Mother’s Day and ends just after Mother’s Day.
b. Popularity of a new TV series: Delphi or associative based on features of existing series.
c. Demand for vacations on the moon: Delphi.
d. The impact of a price increase: Associative.
e. Demand for toothpaste: Naïve or averaging.
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Chapter 03 - Forecasting
Taking Stock
1.
If the forecast system is too responsive and it becomes too sensitive to the changes in actual
demand, it will have a tendency to overreact, resulting in too much adjustment to the forecasted
demand. On the other hand, if the forecasts are too stable, they do not react fast enough to
changes in demand, resulting in insufficient response to changes in actual demand.
2.
Forecasting needs to be a collaborative effort involving marketing, operations, and technical
people. In addition, the forecasting group should include employees from different levels of
management.
3.
The technology has had tremendous impact on forecasting mainly because of the advancement of
the computer technology. Computer technology plays a very important role in preparing forecasts
based on quantitative data. Computer technology allows companies to generate forecasts quickly
due to the computer system’s enhanced ability to update information on prices, demand, and other
variables. In addition, the ability to integrate databases along the supply chain has proved to be an
invaluable asset to companies because this feature increases the communication between
suppliers and their customers, resulting in better management of inventories and purchase orders.
Critical Thinking Exercises
1.
The conditions that would have to exist for driving a car that are analogous to the assumptions
made when using exponential smoothing are that the immediate future will be like the recent past.
This would suggest:
a. No sharp curves or turns on the road
b. Constant traffic conditions
c. No traffic lights or stop signs
d. Constant road conditions
2.
Instantaneous re-supply and/or completely flexible capacity.
3.
Potential investors would expect information on the current and future size of the market, the
expected initial market share and growth rate for 5-10 years, profit/loss projections for the
forecast horizon, and the likelihood of achieving the projected results.
4.
How to handle a poor forecast (i.e., one that is substantially above or below actual demand)
would depend on what the items is, and on a number of factors. For example, a low forecast
would lead to a stockout. How critical that is would relate to how important that a stockout is to
the customers or to internal operations, how quickly the item could be restocked, whether
substitutes are available, whether it is a seasonal item, and so on. For a high forecast, the key
issues might be storage space and whether the items can be carried over to the following period(s)
or they are perishable. For perishable items, a price reduction might be an option. Another
possibility might be to return items to the vendor. If the poor forecast affects capacity required,
handling that would depend on the ability or inability to adjust capacity in the short run, or
finding other uses for an over-supply, or finding other ways to make up for an undersupply.
5.
Although understandable, Omar’s approach is not ethical. He should turn in the forecast based on
the information he has and tell his superiors that he thinks he can get those numbers up. The only
pro to Omar’s optimistic forecast might be preventing Oscar from being laid off over the nearterm if he can convince customers not to cut back on orders. The cons are that if sales do not
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Chapter 03 - Forecasting
materialize, Oscar will be laid off and inventories are going to be high at both his company and at
his customers.
6.
Student answers will vary. Some possibilities follow:
a. If an executive lied and was overly optimistic about demand forecasts, this would violate the
Utilitarian Principle and the Virtue Principle.
b. If an executive forced a subordinate to adjust a forecast based on arbitrary reasons, this would
violate the Rights Principle.
c. If a company used a committee of executives to forecast and one executive’s views
dominated the process, this would be a violation of the Fairness Principle.
d. If a city manager lied about forecasts for demand for a new water park considered in a
referendum, this act would violate the Common Good Principle.
e. If a salesperson intentionally understated future demand to be able to earn a bonus more
easily, this would violate the Virtue Principle.
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Chapter 03 - Forecasting
Solutions
1.
a. Plotting each data set reveals that blueberry muffin orders are stable, varying around an
average. Therefore, the naïve forecast is the last value, 33. The demand for cinnamon buns
has a trend. The last change was from 31 to 33 (33 – 31 = 2). Using the last value and adding
the last trend change, the forecast is 33 + 2 = 35. Demand for cupcakes has an apparent
seasonal variation with peaks every five days. Day 1 = 45, Day 6 = 48, and Day 11 = 47.
Since the peaks occur every five days, the next peak would be at Day 16. We could predict
that demand will be the same as it was the last season—here this value would equal 47.
b. The use of sales data instead of demand implies that sales adequately reflect demand. We are
assuming that no stockouts occurred because demand equals sales if there are no shortages.
2.
Given:
Month
Feb.
Mar.
Apr.
May
Jun.
Jul.
Aug.
Sales (000 units)
19
18
15
20
18
22
20
a.
Sales
20
0
b.
F
M
A
M
Month
J
J
A
S
1)
Using the naïve approach, the forecast for the next month (September) will equal 20.
2)
A five-month moving average is shown below:
MA5 
15  20  18  22  20
 19.00 (round to two decimals)
5
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Chapter 03 - Forecasting
3) A weighted using average using 0.60 for August, 0.30 for July, and 0.10 for June is shown
below:
0.10(18) + 0.30(22) + 0.60(20) = 20.40 (round to two decimals)
4) Exponential smoothing, with alpha = 0.20 and an initial forecast for March of 19 are shown
below (round to two decimals):
Month
Forecast
April
18.80
F(old) + .20[Actual – F(old)]
=
= 19
+ .20[ 18 – 19
]
May
18.04
= 18.80
+ .20[ 15
– 18.80 ]
June
18.43
= 18.04
+ .20[ 20
– 18.04 ]
July
18.34
= 18.43
+ .20[ 18
– 18.43 ]
August
19.07
= 18.34
+ .20[ 22
– 18.34 ]
September 19.26
= 19.07
+ .20[ 20
– 19.07 ]
5) A linear trend forecast is shown below (round b & a to two decimals):
t
1
Y
19
t*Y
19
t2
1
2
18
36
4
3
15
45
9
4
20
80
16
5
18
90
25
6
22
132
36
7
20
140
49
28
132
542
140
b
n  tY   t  Y 7(542)  28(132)

 0.50
n  t 2  ( t ) 2
7(140)  (28) 2
a
 Y  b  t 132  0.50(28)

 16.86
n
7
For Sept., t = 8, and Yt = 16.86 + 0.50(8) = 20.86 = 20,860
c. The linear trend approach seems to be the least appropriate because the data appear to vary
around an average of about 19 [18.86] and because the slope is close to zero (0.50).
d. Sales are reflective of demand (i.e., no stockouts or backorders occurred).
3.
a. Exponential smoothing forecast for September with alpha = 0.10:
88 + 0.10(89.6 – 88) = 88.16 (round to two decimals)
b. Exponential smoothing forecast for October with alpha = 0.10:
88.16 + 0.10(92 – 88.16) = 88.54 (round to two decimals)
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Chapter 03 - Forecasting
4.
Given:
Week
Requests
1
20
2
22
3
18
4
21
5
22
a. Naïve approach forecast for Week 6 = Demand in Week 5 = 22
b. Four-period moving average forecast for Week 6:
22  18  21  22
 20 .75 (round to two decimals)
4
c. Exponential smoothing with alpha = 0.30 and a Week 2 Forecast = 20 (round to two
decimals):
F3 = 20 + 0.30(22 – 20) = 20.60
F4 = 20.60 + 0.30(18 – 20.6) = 19.82
F5 = 19.82 + 0.30(21 – 19.82) = 20.17
F6 = 20.17 + 0.30(22 – 20.17) = 20.72
5.
a. Annual sales are increasing by 15,000 bottles per year (the slope of the line)
b. Forecast for Year 6:
t = 6, Yt = 80 + 15(6) = 170, which is 170,000 bottles.
6.
Slope of the line is estimated by Rise/Run = (500-300)/(10-0) = 200/10 = 20.00. The Y Intercept =
500.
Y  500.00  20.00t
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Chapter 03 - Forecasting
7.
a.
t
1
Y
220
t*Y
220
t2
1
2
245
490
4
3
280
840
9
4
275
1,100
16
5
300
1,500
25
6
310
1,860
36
7
350
2,450
49
8
360
2,880
64
9
400
3,600
81
10
380
3,800
100
11
420
4,620
121
12
450
5,400
144
13
460
5,980
169
14
475
6,650
196
15
500
7,500
225
16
510
8,160
256
17
525
8,925
289
18
541
9,738
324
171
7,001
75,713 2,109
b
n  tY   t  Y 18(75,713)  171(7,001)

 19.00
n  t 2  ( t ) 2
18(2,109)  (171) 2
a
 Y  b  t 7,001  19.00(171)

 208.44
n
18
b. Linear Trend Forecast for Week 20: F = 208.44 + (19.00)(20) = 588.44
Linear Trend Forecast for Week 21: F = 208.44 + (19.00)(21) = 607.44
The forecasted demand for week 20 and 21 is 588.44 and 607.44 respectively.
c. Set the trend equation = 800 and solve for t:
208.44 + 19.00t = 800
19.00t = 800 – 208.44
19.00t = 591.96
t = 591.96 / 19.00
t = 31.13 weeks (during Week 32)
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Chapter 03 - Forecasting
8.
a. There appears to be a long-term upward increasing trend in the data. If we use an averaging
technique, the forecast will underestimate when data values increase.
b.
Trend Analysis for Passengers
Linear Trend Model
Yt = 397.01 + 4.59t
480
 
470
 
460
Passengers
 Actual
Fits
Actual
Fits

450
440

430

420
410

400
0



 





MAPE:
MAD:
MSD:
10
0.6765
2.9086
14.6487
20
Time
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Chapter 03 - Forecasting
t
1
Y
405
t*Y
405
t2
1
2
410
820
4
3
420
1,260
9
4
415
1,660
16
5
412
2,060
25
6
420
2,520
36
7
424
2,968
49
8
433
3,464
64
9
438
3,942
81
10
440
4,400
100
11
446
4,906
121
12
451
5,412
144
13
455
5,915
169
14
464
6,496
196
15
466
6,990
225
16
474
7,584
256
17
476
8,092
289
18
482
8,676
324
171
7,931
77,750
2,109
Round b & a to two decimals:
b
n  tY   t  Y 18(77,750)  171(7,931)

 4.59
n  t 2  ( t ) 2
18(2,109)  (171) 2
a
 Y  b  t 7,931  4.59(171)

 397.01
n
18
Forecasted demand for the next three week (round to two decimals):
Y19 = 397.01 + (4.59)(19) = 484.22
Y20 = 397.01 + (4.59)(20) = 488.81
Y21 = 397.01 + (4.59)(21) = 493.40
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Chapter 03 - Forecasting
9.
t
1
Y
200
t*Y
200
t2
1
2
214
428
4
3
211
633
9
4
228
912
16
5
235
1,175
25
6
232
1,332
36
7
248
1,736
49
8
250
2,000
64
9
253
2,277
81
10
267
2,670
100
11
281
3,091
121
12
275
3,300
144
13
280
3,640
169
14
288
4,032
196
15
310
4,650
225
120
3,772
32,136
1,240
a.
Round b & a to two decimals:
b
n  tY   t  Y 15(32,136)  120(3,772)

 7.00
n  t 2  ( t ) 2
15(1,240)  (120) 2
a
 Y  b  t 3,772  7.00(120)

 195.47
n
15
Forecasts for periods 16 through 19 using Linear Trend are (round to two decimals):
Y16 = 195.47 + (7.00)(16) = 307.47
Y17 = 195.47 + (7.00)(17) = 314.47
Y18 = 195.47 + (7.00)(18) = 321.47
Y19 = 195.47 + (7.00)(19) = 328.47
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Chapter 03 - Forecasting
b. Round values to two decimals.
Initial Trend =
228  200
 9.33
3
Period
5
Actual
235
St-1 + Tt-1 = TAFt
228.00 + 9.33 = 237.33
TAFt + .3(At – TAFt) = St
237.33 + .3(235 – 237.33) = 236.63
Tt–1 + .2 (TAFt – TAFt–1 – Tt–1) = Tt
9.33
6
232
236.63 + 9.33 = 245.96
245.96 + .3(232 – 245.96) = 241.77
9.33 + .2(245.96 – 237.33 – 9.33) = 9.19
7
248
241.77 + 9.19 = 250.96
250.96 + .3(248 – 250.96) = 250.07
9.19 + .2(250.96 – 245.96 – 9.19) = 8.35
8
250
250.07 + 8.35 = 258.42
258.42 + .3(250 – 258.42) = 255.89
8.35 + .2(258.42 – 250.96 – 8.35) = 8.17
9
253
255.89 + 8.17 = 264.06
264.06 + .3(253 – 264.06) = 260.74
8.17 + .2(264.06 – 258.42 – 8.17) = 7.66
10
267
260.74 + 7.66 = 268.40
268.40 + .3(267 – 268.40) = 267.98
7.66 + .2(268.40 – 264.06 – 7.66) = 7.00
11
281
267.98 + 7.00 = 274.98
274.98 + .3(281 – 274.98) = 276.79
7.00 + .2(274.98 – 268.40 – 7.00) = 6.92
12
275
276.79 + 6.92 = 283.71
283.71 + .3(275 – 283.71) = 281.10
6.92 + .2(283.71 – 274.98 – 6.92) = 7.28
13
280
281.10 + 7.28 = 288.38
288.38 + .3(280 – 288.38) = 285.87
7.28 + .2(288.38 – 283.71 – 7.28) = 6.76
14
288
285.87 + 6.76 = 292.63
292.63 + .3(288 – 292.63) = 291.24
6.76 + .2(292.63 – 288.38 – 6.76) = 6.26
15
310
291.24 + 6.26 = 297.50
297.50 + .3(310 – 297.50) = 301.25
6.26 + .2(297.50 – 292.63 – 6.26) = 5.98
16
10.
301.25 + 5.98 = 307.23
The initial estimate of trend is based on the net change of 30 for the three periods from 1 to 4, for
an average of +10 units. Use  = .5 and  = .4. Round values to two decimals.
Initial trend = (240 – 210)/3 = 10.00
Period
Actual
1
210
Model
2
224
Development
3
229
4
240
Actual
Model Test
Next
Forecast
St + Tt = TAFt
TAFt + .5(At – TAFt) = St
Tt–1 + .4 (TAFt – TAFt–1 – Tt–1) = Tt
5
255
240.00 + 10.00 = 250.00 250.00 + .5(255 – 250.00) = 252.50 10.00
6
265
252.50 + 10.00 = 262.50 262.50 + .5(265 – 262.50) = 263.75 10.00 + .4(262.50 – 250.00 – 10.00) = 11.00
7
272
263.75 + 11.00 = 274.75 274.75 + .5(272 – 274.75) = 272.38 11.00 + .4(274.75 – 262.50 – 11.00) = 11.50
8
285
272.38 + 11.50 = 284.88 284.88 + .5(285 – 284.88) = 284.94 11.50 + .4(284.88 – 274.75 – 11.50) = 10.95
9
294
284.94 + 10.95 = 295.89 295.89 + .5(294 – 295.89) = 294.95 10.95 + .4(295.89 – 284.88 – 10.95) = 10.97
10
294.95 + 10.97 = 305.92
3-14
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Education.
Chapter 03 - Forecasting
11.
Yt = 70 + 5t
t = 0 (June of last year)
t = 1 (July of last year)
t = 7 (January of this year)
t = 8 (February of this year)
t = 9 (March of this year)
t = 19 (January of next year)
t = 20 (February of next year)
t = 21 (March of next year)
YJan. = 70 + (5)(19) = 165
YFeb. = 70 + (5)(20) = 170
YMar. = 70+ (5)(21) = 175
Forecast = Trend * Seasonal Relative (round to two decimals):
Month
January
12.
Trend * Seasonal Relative
165 * 1.10 = 181.50
February
170 * 1.02 = 173.40
March
175 * 0.95 = 166.25
The current quarter is Quarter 1 = t = 4. Quarter 1 from one year ago = t = 0.
Quarter 1 next year = t = 8.
Quarter
Value of t
Trend component, Ft
Quarter relative
Forecast
Next Year,
Q1
8
116.00
x 1.1 =
127.60
Next Year,
Q2
9
143.50
x 1.0 =
143.50
Next Year,
Q3
10
175.00
x 0.6 =
105.00
Next Year,
Q4
11
210.50
x 1.3 =
273.65
Two Years,
Q1
12
250.00
x 1.1 =
275.00
Trend component calculations: 𝐹𝑡 = 40 − 6.5𝑡 + 2𝑡 2 (round to two decimals):
𝐹8 = 40 − 6.5(8) + 2(82 ) = 116.00
𝐹9 = 40 − 6.5(9) + 2(92 ) = 143.50
𝐹10 = 40 − 6.5(10) + 2(102 ) = 175.00
𝐹11 = 40 − 6.5(11) + 2(112 ) = 210.50
𝐹12 = 40 − 6.5(12) + 2(122 ) = 250.00
3-15
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Education.
Chapter 03 - Forecasting
13.
Given:
Quarter
1
2
3
4
Year 1
2
6
2
5
Year 2
3
10
6
9
Year 3
7
18
8
15
Year 4
4
14
8
11
SA method (round season averages to three decimals and seasonal relatives to two decimals):
Quarter
1
2
3
4
Year 1
2
6
2
5
Year 2
3
10
6
9
Year 3
7
18
8
15
Year 4
4
14
8
11
Season
Average
4.000
12.000
6.000
10.000
8.000
Seasonal Relative
0.50 (4.000/8.000)
1.50 (12.000/8.000)
0.75 (6.000/8.000)
1.25 (10.000/8.000)
Overall
Average
Sum of Seasonal Relatives = 0.50 + 1.50 + 0.75 + 1.25 = 4.00
3-16
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Education.
Chapter 03 - Forecasting
14.
a. Centered Moving Average Method (round CMA to two decimals & Index to four
decimals):
Week
1
2
3
4
5
6
Day
Fri
Sales
149
Moving
Total
Centered
Moving
Average
Index
Sales/MA3
Sat
250
Sun
166
565
190.00
1.3275 Sat
0.8737
Fri
154
570
191.67
0.8035
Sat
255
575
190.33
Sun
162
571
189.67
1.3398 Sat
0.8541
Fri
152
569
191.33
0.7944
Sat
260
574
194.33
Sun
171
583
193.67
1.3379 Sat
0.8829
Fri
150
581
196.33
0.7640
Sat
268
589
197.00
Sun
173
591
200.00
1.3604 Sat
0.8650
Fri
159
600
201.67
0.7884
Sat
273
605
202.67
Sun
176
608
204.00
1.3470 Sat
0.8627
Fri
163
612
205.00
0.7951
Sat
276
615
207.33
1.3312 Sat
Sun
183
622
188.33
Seasonal relatives (round to two decimals):
x : Fri = 0.79; Sat = 1.34; Sun = 0.87
Sum of Seasonal Relatives = 0.79 + 1.34 + 0.87 = 3.00
3-17
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Education.
Chapter 03 - Forecasting
b. SA Method (round season averages to three decimals & seasonal relatives to two decimals).
Season
Friday
Saturday
Sunday
1
2
149 154
250 255
166 162
Week
3
4
5
152 150 159
260 268 273
171 173 176
Season Seasonal
6 Average Relative
163 154.500
0.79
(154.500/196.667)
276 263.667
1.34
(263.667/196.667)
183 171.833
0.87
(171.833/196.667)
196.667
Overall
Average
Sum of Seasonal Relatives = 0.79 + 1.34 + 0.87 = 3.00
c. In this problem, the two methods provide similar results because there are only 3 seasons;
therefore, the two methods are essentially averaging the same data. In addition, there is no
trend in the data.
15.
Given:
The restaurant is open 4 days. Thursday night accounts for 0.20 of the business. Friday night
accounts for 0.35 of the business. Saturday night accounts for 0.30 of the business.
Wednesday night: 1.00 – 0.20 – 0.35 – 0.30 = 0.15 (15.00%) of the business.
Seasonal relatives (round to two decimals):
Wednesday
= 0.15 x 4 = 0.60
Thursday
= 0.20 x 4 = 0.80
Friday
= 0.35 x 4 = 1.40
Saturday
= 0.30 x 4 = 1.20
Sum of Seasonal Relatives = 0.60 + 0.80 + 1.40 + 1.20 = 4.00
3-18
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Education.
Chapter 03 - Forecasting
16.
a. Centered Moving Average Method (round CMA to two decimals & Index to four decimals):
Day
1=1
2=2
3=3
4=4
5=5
6=6
7=7
(Data)
No.
Served
80
75
78
95
130
136
40
Moving
Total
Centered
Moving Average
Index
634
90.57
90.86
91.14
91.43
95/90.57 = 1.0489
130/90.86 = 1.4308
136/91.14 = 1.4922
40/91.43 = 0.4375
8=1
9=2
10 = 3
11 = 4
12 = 5
13 = 6
14 = 7
82
77
80
94
131
137
42
636
638
640
639
640
641
643
91.29
91.43
91.57
91.86
92.14
92.29
92.71
82/91.29 = 0.8982
77/91.43 = 0.8422
80/91.57 = 0.8736
94/91.86 = 1.0233
131/92.14 = 1.4217
137/92.29 = 1.4845
42/92.71 = 0.4530
15 = 1
16 = 2
17 = 3
18 = 4
19 = 5
20 = 6
21 = 7
84
78
83
96
135
140
44
645
646
649
651
655
658
660
93.00
93.57
94.00
94.29
94.71
95.29
96.00
84/93.00 = 0.9032
78/93.57 = 0.8336
83/94.00 = 0.8830
96/94.29 = 1.0181
135/94.71 = 1.4254
140/95.29 = 1.4692
44/96.00 = 0.4583
22 = 1
23 = 2
24 = 3
25 = 4
26 = 5
27 = 6
28 = 7
87
82
88
99
144
144
48
663
667
672
675
684
688
692
96.43
97.71
98.29
98.86
87/96.43 = 0.9022
82/97.71 = 0.8392
88/98.29 = 0.8953
99/98.86 = 1.0014
3-19
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Education.
Chapter 03 - Forecasting
Group and Average the Indexes to Derive Seasonal Relatives
1’s
2’s
3’s
4’s
5’s
6’s
7’s
1.0489
1.4308
1.4922
0.4375
0.8982
0.8422
0.8736
1.0233
1.4217
1.4845
0.4530
0.9032
0.8336
0.8830
1.0181
1.4254
1.4692
0.4583
0.9022
0.8392
0.8953
1.0014
2.7036
2.5150
2.6519
4.0917
4.2779
4.4459
1.3488
x : 0.90
0.84
0.88
1.02
1.43
1.48
0.45
Sum of Seasonal Relatives = 0.90 + 0.84 + 0.88 + 1.02 + 1.43 + 1.48 + 0.45 = 7.00
b. SA Method (round season averages to three decimals & seasonal relatives to two decimals):
Season
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
1
80
75
78
95
130
136
40
Week
Season
Seasonal
2
3
4 Average
Relative
82 84 87
83.250 0.89 (83.250/93.893)
77 78 82
78.000 0.83 (78.000/93.893)
80 83 88
82.250 0.88 (82.250/93.893)
94 96 99
96.000 1.02 (96.000/93.893)
131 135 144 135.000 1.44 (135.000/93.893)
137 140 144 139.250 1.48 (139.250/93.893)
42 44 48
43.500 0.46 (43.500/93.893)
93.893
Overall
Average
Sum of Seasonal Relatives = 0.89 + 0.83 + 0.88 + 1.02 + 1.44 + 1.48 + 0.46 = 7.00
3-20
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Education.
Chapter 03 - Forecasting
17.
Given:
Day
1
2
3
4
5
6
7
8
9
# Sold
36
38
42
44
48
49
50
49
52
Day
10
11
12
13
14
15
# Sold
48
52
55
54
56
57
a. The trend may be non-linear (although most students will view it as linear). Trend-adjusted
smoothing would have a slight edge over a linear trend line.
b. Yes. This would cause concern because you would not know the actual demand for the pain
reliever.
Sales
60
50
40
30
2
c.
4
6
8
Day
10
12
14
TAF
9 51.70
10 53.70
11 53.93
12 54.78
13 56.11
14 56.76
15 57.62
16 58.45
3-21
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Education.
Chapter 03 - Forecasting
Period
8
Actual
49
St-1 + Tt-1 = TAFt
50.00 (given)
TAFt + .3(At – TAFt) = St
50.00 + .3(49 – 50.00) = 49.70
Tt–1 + .3 (TAFt – TAFt–1 – Tt–1) = Tt
2.00 (given)
ei
-1.00
ei2
1.00
9
52
49.70 + 2.00 = 51.70
51.70 + .3(52 – 51.70) = 51.79
2.00 + .3(51.70 – 50.00 – 2.00) = 1.91
0.30
0.09
10
48
51.79 + 1.91 = 53.70
53.70 + .3(48 – 53.70) = 51.99
1.91 + .3(53.70 – 51.70 – 1.91) = 1.94
-5.70
32.49
11
52
51.99 + 1.94 = 53.93
53.93 + .3(52 – 53.93) = 53.35
1.94 + .3(53.93 – 53.70 – 1.94) = 1.43
-1.93
3.72
12
55
53.35 + 1.43 = 54.78
54.78 + .3(55 – 54.78) = 54.85
1.43 + .3(54.78 – 53.93 – 1.43) = 1.26
0.22
0.05
13
54
54.85 + 1.26 = 56.11
56.11 + .3(54 – 56.11) = 55.48
1.26 + .3(56.11 – 54.78 – 1.26) = 1.28
-2.11
4.45
14
56
55.48 + 1.28 = 56.76
56.76 + .3(56 – 56.76) = 56.53
1.28 + .3(56.76 – 56.11 – 1.28) = 1.09
-0.76
0.58
15
57
56.53 + 1.09 = 57.62
57.62 + .3(57 – 57.62) = 57.43
1.09 + .3(57.62 – 56.76 – 1.09) = 1.02
-0.62
0.38
16
57.43 + 1.02 = 58.45
𝑀𝑆𝐸 =
18.
42.76
8−1
Sum
= 6.11 (round two decimals)
a. As shown in the plot of Unit Sales, there appears to be a trend in Unit Sales.
Month
Jan
Units
Sold
640
Units
Sold
765
Index
0.90
Feb
648
0.80
Aug
805
1.15
Mar
630
0.70
Sep
840
1.20
Apr
761
0.94
Oct
828
1.20
May
735
0.89
Nov
840
1.25
Jun
850
1.00
Dec
800
1.25
Index Month
0.80
Jul
Unit Sales
900
800
700
600
500
Units
400
300
200
100
0
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
3-22
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Education.
42.76
Chapter 03 - Forecasting
b. Deseasonalize car sales: Units Sold / Index (round to two decimals)
Units
Sold
640
Index
0.80
Feb
648
0.80
Mar
630
0.70
900.00
Apr
761
0.94
May
735
0.89
Month
Jan
Jun
850
1.00
Deseasonalized Month
Jul
800.00
Aug
810.00
Units
Sold
765
Index Deseasonalized
0.90
850.00
805
1.15
700.00
Sep
840
1.20
700.00
809.57
Oct
828
1.20
690.00
825.84
Nov
840
1.25
672.00
850.00
Dec
800
1.25
640.00
c. Plotting the deseasonalized data on the same graph as the Units Sold data leads us to a
different conclusion than the conclusion in part a. There appears to be a downward trend in
sales.
Unit Sales & Deseasonalized Data
1000
900
800
700
600
500
400
300
200
100
0
Units
Deseasonalized
d. Part c indicated a downward trend in sales. We could forecast sales of the first three months
of the next year by fitting a monthly trend line to the deseasonalized values using t = 0 in
December of the previous year. Then, predict trend values for the first three months of next
year (t = 13, 14, 15). Finally, multiply each month’s trend value by the appropriate monthly
seasonal relative.
e. Based on the findings from the deseasonalized data, management should consider advertising
and sales promotions.
3-23
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Education.
Chapter 03 - Forecasting
19.
Deseasonalize the values, where:
Deseasonalized sales = (Actual sales) / (Seasonal relative) (round to two decimals):
Deseasonalized sales for quarter 1: 88/1.10 = 80.00
Deseasonalized sales for quarter 2: 99/0.99 = 100.00
Deseasonalized sales for quarter 3: 108/0.90 = 120.00
Deseasonalized sales for quarter 4: 141.4/1.01 = 140.00
There is a trend of +20 from previous quarter, hence the trend forecast would be 160 units.
Multiplying the trend forecast by the seasonal relative for quarter 1 yields a forecast for the first
quarter of next year: (140.00 + 20) * 1.10 = 176.00.
20.
t
11
Units
sold
147
Naïve
12
148
13
e
|e|
e2
Trend F
146
147
1
1
1
148
0
0
0
151
148
3
3
9
150
1
1
1
14
145
151
–6
6
36
152
–7
15
155
145
10
10 100
154
1
16
152
155
–3
3
9
156
–4
4 16
17
155
152
3
3
9
158
–3
3
9
18
157
155
2
2
4
160
–3
3
9
19
160
157
3
3
9
162
–2
2
4
20
165
160
5
5
25
164
1
1
1
Sum
18

e | e | e2
1
1 1
–16
36 202
7 49
1
1
23 91
Round MAD & MSE to two decimals:
MAD :
36
 4.00
9
MAD :
MSE :
202
 25.25
9 1
MSE :
23
 2.30
10
91
 10.11
10  1
Linear trend provides forecasts with less average error and less average squared error.
3-24
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Education.
Chapter 03 - Forecasting
21.
1
68
66
2
2
4
(e/Demand)
F2
x 100 (%)
2.94% 66
2
75
68
7
7
49
9.33% 68
7
7
49
9.33%
3
70
72
–2
2
4
2.86% 70
0
0
0
0.00%
4
74
71
3
3
9
4.05% 72
2
2
4
2.70%
5
69
72
–3
3
9
4.35% 74
–5
5
25
7.25%
6
72
70
+2
2
4
2.78% 76
–4
4
16
5.56%
7
80
71
9
9
81
11.25% 78
2
2
4
2.50%
8
78
74
4
4
16
5.13% 80
–2
2
4
2.56%
32
176
24
106
32.84%
Period Demand
F1
e
e
e2
Sum
42.69%
e
e
e2
2
2
4
(e/Demand)
x 100 (%)
2.94%
a. MAD F1: 32/8 = 4.00 (round to two decimals)
MAD F2: 24/8 = 3.00 (F2 appears to be more accurate)
b. MSE F1: 176/(8-1) = 25.14
MSE F2: 106/(8-1) = 15.14 (F2 appears to be more accurate)
c. Either measure might be in use already, familiar to users, and have past values for
comparison. If control charts are used, MSE would be natural; if tracking signals are used,
MAD would be more natural.
d. MAPE calculations (round to two decimals):
MAPE (F1): 42.69%/8 = 5.34%
MAPE (F2): 32.84%/8 = 4.11%
Because 4.11% < 5.34%, F2 appears to be more accurate.
3-25
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Education.
Chapter 03 - Forecasting
22.
a. Compute MSE & MAD for each forecast method (round to two decimals). Round % to two
decimals.
Period
Demand
F1
1
770
771
-1
2
789
785
4
(e/Demand)
F2
x 100 (%)
1 1
0.13% 769
4 16
0.51% 787
1
1
(e/Demand)
x 100 (%)
1
0.13%
2
2
4
0.25%
3
794
790
4
4 16
2
2
4
0.25%
4
780
784
-4
4 16
-18
18
324
2.31%
5
768
770
-2
2
0.26% 774
0.52% 770
-6
6
36
0.78%
6
772
768
4
2
2
4
0.26%
7
760
761
-1
0.13% 759
0.52% 775
1
1
1
0.13%
8
775
771
4
4 16
0
0
0
0.00%
9
786
784
2
2
4
0.25% 788
0.25% 788
-2
2
4
0.25%
10
790
788
2
2
4
2
2
4
0.25%
Sum
12
-16
36
382
e e2
e
0.50% 792
0.51% 798
4
4 16
1
1
28 94
3.58%
e
e
e2
4.61%
MAD F1: 28/10 = 2.80
MAD F2: 36/10 = 3.60
MSE F1: 94/(10-1) = 10.44
MSE F2: 382/(10-1) = 42.44
F1 has both lower MAD and lower MSE so it seems better.
b. Compute MAPE for each forecast method (round to two decimals).
MAPE F1: 3.58%/10 = 0.36%
MAPE F2: 4.61%/10 = 0.46%
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Education.
Chapter 03 - Forecasting
c. Naïve Method Forecast
Naïve
Month Sales Forecast
1
770
e
|e|
e2
2
789
770
19
19
361
3
794
789
5
5
25
4
780
794
–14
14
196
5
768
780
–12
12
144
6
772
768
4
4
16
7
760
772
–12
12
144
8
775
760
15
15
225
9
786
775
11
11
121
10
790
786
4
4
16
11
790
At end of
Week 10

20
96 1,248
Round MSE, MAD, TS, & control limits to two decimals:
𝑀𝑆𝐸 =
1,248
9−1
= 156.00
𝑇𝑟𝑎𝑐𝑘𝑖𝑛𝑔 𝑆𝑖𝑔𝑛𝑎𝑙 =
20
10.67
𝑀𝐴𝐷 =
96
9
= 10.67
= +1.87
Control Limits for Naïve Method: 0  2 156 = 0  24.98 [in control because + 1.87 falls
within the range of -24.98 to +24.98].
It appears that the naïve forecast is in control because its tracking signal at the end of Week
10 is within the limits. However, the MAD and MSE values for the naïve method are much
higher than the MAD and MSE values for the other two forecasting methods (refer to the
table below). Therefore, the naïve forecasting method does not appear to be performing as
well as the other two forecasting methods.
Method
F1
F2
Naïve
MAD
2.80
3.60
10.67
MSE
10.44
42.44
156.00
3-27
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill
Education.
Chapter 03 - Forecasting
23.
a.
850
Insurance
needed ($000)
0
10
20
30
40
Current Age of Head of Household (years)
b. y = 850 – 0.1(30) = 847. Thus, a 30-year old will need $847,000 of life insurance.
24.
a. Let x1 = weight in lb.
x2 = distance in miles
y
y = $0.10x1 + $0.15x2 + $10
= delivery charge
b. y = $0.10(40) + $0.15(26) + $10 = $17.90 (round to two decimals)
25.
a.
X
Y
Price Sales
X*Y
X2
Y2
6.00
200
1,200.00
36.00
40,000
6.50
190
1,235.00
42.25
36,100
6.75
188
1,269.00
45.56
35,344
7.00
180
1,260.00
49.00
32,400
7.25
170
1,232.50
52.56
28,900
7.50
162
1,215.00
56.25
26,244
8.00
160
1,280.00
64.00
25,600
8.25
155
1,278.75
68.06
24,025
8.50
156
1,326.00
72.25
24,336
8.75
148
1,295.00
76.56
21,904
9.00
140
1,260.00
81.00
19,600
9.25
133
1,230.25
92.75 1,982 15,081.50
85.56
729.05
17,689
332,142
Round b & a to two decimals:
n  xy   x  y (12)(15,081.50)  (92.75)(1,982)

 19.53
n  x 2  ( x) 2
(12)(729.05)  (92.75) 2
 y  b  x 1,982  (19.53)(92.75)
a

 316.12
n
12
b
3-28
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill
Education.
Chapter 03 - Forecasting
Y = 316.12 – 19.53X
Actual data are represented by circles.
Predicted values are represented by pluses.
200
+
190
+
+
180
+
sales
+
170
+
160
+
+
150
+
+
140
+
+
130
6
7
8
9
price
Round r to four decimals:
r
𝑟=
n( xy)  ( x)(  y )
n(  x )  (  x ) 2 n(  y 2 )  (  y ) 2
2
(12)(15,081.50) − (92.75)(1,982)
√(12)(729.05) − (92.75)2 √(12)(332,142) − (1,982)2
= −0.9854
b. r = –0.9854 implies a strong, negative relationship between price and demand.
r2 = (–0.9854) 2 = 0.9700. It appears that 97.00% of the variation in sales can be accounted for
by the price of our product. This indicates that price is a good predictor of sales.
3-29
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill
Education.
Chapter 03 - Forecasting
26.
a.
100


y

  









50
0
10
20
30
x
40
50
b.
t
1
2
3
4
5
6
7
8
9
10
11
12
13
x
15
25
40
32
51
47
30
18
14
15
22
24
33
366
y
74
80
84
81
96
95
83
78
70
72
85
88
90
1,07
6
x*y
1,110
2,000
3,360
2,592
4,896
4,465
2,490
1,404
980
1,080
1,870
2,112
2,970
x2
225
625
1,600
1,024
2,601
2,209
900
324
196
225
484
576
1,089
y2
5,476
6,400
7,056
6,561
9,216
9,025
6,889
6,084
4,900
5,184
7,225
7,744
8,100
31,329
12,078
89,860
Round b & a to two decimals:
n  xy   x  y (13)(31,329)  (366)(1,076)

 0.58
n  x 2  ( x) 2
(13)(12,078)  (366) 2
 y  b  x 1,076  (0.58)(366)
a

 66.44
n
13
b
3-30
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Education.
Chapter 03 - Forecasting
c.
r
r
n( xy)  ( x)( y )
n(  x 2 )  (  x ) 2 n(  y 2 )  (  y ) 2
(13)(31, ,329)  (366)(1,076)
(13)(12,078)  (366) 2 (13)(89,860)  (1,076) 2
 0.8691 (round to 4 decimals)
r 2  (0.8691) 2  0.7553 (round to 4 decimals)
Approximately 75.53% of the variation in the dependent variable is explained by the
independent variable.
d. y = 66.44 + 0.58 (41) = 90.22 (round to two decimals)
27.
a. & b.
X
1.6
Y
10
X*Y
16.00
X2
2.56
Y2
100
2
1.3
8
10.40
1.69
64
3
1.8
11
3.24
121
4
2.0
12
19.80
24.00
4.00
144
5
2.2
12
26.40
4.84
144
2.56
81
Period
1
6
1.6
9
14.40
7
1.5
8
12.00
2.25
64
8
1.3
7
9.10
1.69
49
9
1.7
10
17.00
2.89
100
10
1.2
6
7.20
1.44
36
11
1.9
11
20.90
3.61
121
12
1.4
8
11.20
1.96
64
13
1.7
10
17.00
2.89
100
1.6
9
14.40
2.56
81
22.8
131
219.80
38.18
1,269
14
r
r
n( X iYi )  ( X i )( Yi )
n( X i2 )  ( X i ) 2 n( Yi 2 )  ( Yi ) 2
(14)( 219.80)  (22.8)(131)
(14)(38.18)  (22.8) 2 (14)(1,269)  (131) 2
 0.9592 (round to four decimals)
Because the value of r is close to +1, there is a strong positive relationship between these
variables.
3-31
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Education.
Chapter 03 - Forecasting
Round b & a to two decimals:
n  xy   x  y (14)( 219.80)  (22.8)(131)

 6.16
n  x 2  ( x) 2
(14)(38.18)  (22.8) 2
 y  b  x 131  (6.16)( 22.8)
a

 0.67
n
14
b
c. Y = –0.67 + (6.16)(2) = 11.65 (round to two decimals)
Therefore, we expect about 12 mowers to be sold in the first week of August.
28.
Round MAD & Tracking Signal to two decimals:
t
Period
1
A
Demand
129
F
Predicted
124
e
Error
5
|e|
5
Cum.
Error
5
2
194
200
–6
6
–1
3
156
150
6
6
5
4
91
94
–3
3
2
5
85
80
5
5
7
5.00*
1.40***
6
132
140
–8
8
–1
5.90**
–0.17
7
126
128
–2
2
–3
4.73
–0.63
8
126
124
2
2
–1
3.91
–0.26
9
95
100
–5
5
–6
4.24
–1.42
10
149
150
–1
1
–7
3.27
–2.14
11
98
94
4
4
–3
3.49
–0.86
12
85
80
5
5
2
3.94
0.51
13
137
140
–3
3
–1
3.66
–0.27
14
134
128
6
6
5
4.36
1.15
MADt
Tracking
Signal
*Initial MAD = Sum of Cumulative |e| [1 through 5]/5 = 25/5 = 5.00
**Updated MADs [6 through 14]: MADt = MADt–1+  (| e |t – MAD t–1)
e.g., MAD6 = MAD5 + .1(| e |6 – MAD5) = 5.00 + .3(8 – 5.00) = 5.90
***Tracking Signal = Cumulative Error/MADt = 7/5.00 = 1.40
3-32
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Education.
Chapter 03 - Forecasting
5
Tracking Signal Data
4
4
3
2
1
0
TS
0
-1
-2
-3
-4
-4
-5
5
6
7
8
9
10
11
12
13
14
Because all tracking signal values are within the limits, the forecast method is not exhibiting bias.
29.
Refer to data in Problem 22 (shown below).
Period
Demand
F1
e
1
770
771
2
789
785
3
794
790
4
780
784
5
768
770
6
772
768
7
760
761
8
775
771
9
786
784
10
790
788
-1
4
4
-4
-2
4
-1
4
2
2
12
Sum
e
1
4
4
4
2
4
1
4
2
2
28
e2
(e/Demand)
F2
x 100 (%)
1
0.13% 769
16
16
16
4
16
1
16
4
4
94
0.51%
0.50%
0.51%
0.26%
0.52%
0.13%
0.52%
0.25%
0.25%
3.58%
787
792
798
774
770
759
775
788
788
e
1
2
2
-18
-6
2
1
0
-2
2
-16
e
e2
1
1
2
4
2
4
18 324
6 36
2
4
1
1
0
0
2
4
2
4
36 382
(e/Demand)
x 100 (%)
0.13%
0.25%
0.25%
2.31%
0.78%
0.26%
0.13%
0.00%
0.25%
0.25%
4.61%
MAD F1: 28/10 = 2.80
MAD F2: 36/10 = 3.60
MSE F1: 94/(10-1) = 10.44
MSE F2: 382/(10-1) = 42.44
3-33
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill
Education.
Chapter 03 - Forecasting
a. Tracking signal using cumulative error for months 1 to 10:
F1: 12/2.80 = +4.29
F2: -16/3.60 = -4.44
Both are slightly beyond the limits of ± 4.
Because +4.29 > 4, the forecasting method 1 is biased. Using F1, we are underestimating
demand.
Because –4.44 < –4, the forecasting method 2 is biased. Using F2, we are overestimating
demand.
b. Compute 2s limits for errors of each forecast method (round to two decimals).
Control limits are 0  2 MSE :
#1: 0  2 10.44  0  6.46
8.00
F1: 2s Limits for Errors
6.46
6.00
4.00
2.00
0.00
0.00
e
-2.00
-4.00
-6.00
-6.46
-8.00
1
2
3
4
5
6
7
8
9
10
Because all errors are within these limits, forecast method F1 is in control.
3-34
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Education.
Chapter 03 - Forecasting
0  2 42.44  0  13.03
#2
F2: 2s Limits for Errors
15.00
13.03
10.00
5.00
0.00
0.00
e
-5.00
-10.00
-13.03
-15.00
-20.00
1
2
3
4
5
6
7
8
9
10
Because the error for month 4 is below the lower control limit, forecast method F2 is not in
control.
3-35
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill
Education.
Chapter 03 - Forecasting
30.
a. Round MAD & Tracking Signal values to two decimals:
t
e
Cum.
Month Error | e | Error MADt
-8
8
-8
1
-2
2
-10
2
Tracking
Signal
3
4
4
-6
4
7
7
1
5
9
9
10
6
5
5
15
7
0
0
15
8
-3
3
12
9
-9
9
3
10
-4
4
-1
11
1
1
0
4.73*
0.00
12
6
6
6
4.86**
1.23
13
8
8
14
5.17
2.71
14
4
4
18
5.05
3.56
15
1
1
19
4.65
4.09***
16
-2
2
17
4.39
3.87
17
-4
4
13
4.35
2.99
18
-8
8
5
4.72
1.06
19
-5
5
0
4.75
0.00
20
-1
1
-1
4.38
-0.23
*Initial MAD = Sum of Cumulative |e| [1 through 11]/11 = 52/11 = 4.73
**Updated MADs [11 through 20]: MADt = MADt–1+  (| e |t – MAD t–1)
e.g., MAD12 = MAD11 + .1(| e |12 – MAD11) = 4.73 + .1(6 – 4.73) = 4.86
***Tracking Signal = Cumulative Error/MADt = 19/4.65 = 4.09
Assuming limits of ±4, the tracking signal in month 15 is outside the limits. The forecast
method is exhibiting bias.
3-36
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Education.
Chapter 03 - Forecasting
b.
Month
1
Error
–8
Error2
64
2
–2
4
3
4
16
4
7
49
5
9
81
6
5
25
7
0
0
8
–3
9
9
–9
81
10
–4
16
Sum
345
MSE 
 e2
345

 38.33
n  1 10  1
Control limits = 0 ± 2√38.33= 0 ± 12.38
2s Limits for Errors for 11-20
15.00
12.38
10.00
5.00
0.00
0.00
e
-5.00
-10.00
-12.38
-15.00
11
12
13
14
15
16
17
18
19
20
The errors may be cyclical, suggesting that there may be a cyclical component in demand
that is being overlooked in the forecast.
3-37
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill
Education.
Chapter 03 - Forecasting
31.
a.
Linear regression model:
1
Y
40.2
t*Y
40.20
t2
1
2
44.5
89.00
4
3
4
48.0
52.3
144.00
209.20
9
16
5
55.8
279.00
25
6
57.1
342.60
36
7
62.4
436.80
49
8
69.0
552.00
64
9
45
73.7
663.30
81
t
503.0 2,756.10 285
b
n  tY   t  Y 9(2,756.10)  45(503.0)

 4.02
n  t 2  ( t ) 2
9(285)  (45) 2
a
 Y  b  t 503.0  4.02(45)

 35.79
n
9
Forecasts for periods 10 through 14 using Linear Trend are (round to two decimals):
Y10 = 35.79 + (4.02)(10) = 75.99
Y11 = 35.79 + (4.02)(11) = 80.01
Y12 = 35.79 + (4.02)(12) = 84.03
Y13 = 35.79 + (4.02)(13) = 88.05
Y14 = 35.79 + (4.02)(14) = 92.07
3-38
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill
Education.
Chapter 03 - Forecasting
b.
Prepare a control chart using 2s limits.
Year Sales
t
Y
40.2
1
t*Y
40.20
t2 F
e
e2
1 39.81 -0.39 0.15
2
44.5
89.00
4 43.83 -0.67 0.45
3
4
48.0
52.3
144.00
209.20
9 47.85 -0.15 0.02
16 51.87 -0.43 0.18
5
55.8
279.00
25 55.89 0.09
0.01
6
57.1
342.60
36 59.91 2.81
7.90
7
62.4
436.80
49 63.93 1.53
2.34
8
69.0
552.00
64 67.95 -1.05 1.10
9
45
73.7
663.30
81 39.81 -1.73 2.99
503.0 2,756.10 285
15.1
4
 e 2 15.14

 1.89
n 1 9 1
2s control limits are 0  2 1.89  0  2.74
MSE 
c.
Year
t
10
Sales
Y
77.2
Forecast
F
75.99
Error
e
1.21
11
82.1
80.01
2.09
12
87.8
84.03
3.77
13
90.6
88.05
2.55
14
98.9
92.07
6.83
3-39
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Education.
Chapter 03 - Forecasting
2s Limits for Errors for 10-14
8.00
6.00
4.00
2.74
e
2.00
0.00
0.00
-2.00
-2.74
10
11
12
13
14
-4.00
The forecast method is not in control. Two of the five errors are outside of the limits. In an
actual situation, the error in year 12 would have triggered an examination of forecast
performance.
3-40
Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill
Education.
Chapter 03 - Forecasting
32.
a.
Period
1
Actual
37
Forecast
1
36
Forecast
2
36
e1
+1
e2
+1
2
39
38
37
+1
3
37
40
38
4
39
42
38
5
45
46
6
49
7
47
8
e12
e22
e1
e2
1
1
1
1
+2
1
4
1
2
–3
–1
9
1
3
1
–3
+1
9
1
3
1
41
–1
+4
1
16
1
4
46
52
+3
–3
9
9
3
3
46
47
1
0
1
0
1
0
49
48
48
1
+1
1
1
1
1
9
51
52
52
–1
–1
1
1
1
1
10
54
55
53
–1
+1
1
1
1
1
–2
+4
34
35
16
15
MSE 1 
34
 3.78
10  1
MAD 1 
16
 1.60
10
MSE 2 
MAD 2 
35
 3.89
10  1
15
 1.50
10
The analyst is indifferent between the two alternatives because both forecasting methods have
MADs that are approximately equal (MAD1 = 1.60, MAD2 = 1.50), and MSEs that are also
approximately equal (MSE1 = 3.78, MSE2 = 3.89).
b. The errors for Forecast 1 cycle (+1, +1, –3, –3, –1, +3, +1,+1, –1, –1), although all are within
2s control limits. The errors for Forecast 2 (+1, +2, –1, +1, +4, –3, 0, +1, –1, +1) do not
appear to cycle, but the error of +4 is just beyond the 2s control limits for Forecast 2.
Forecast 1: 2s control limits are 0 ± 2√3.78 = 0 ± 3.89
Forecast 2: 2s control limits are 0 ± 2√3.89 = 0 ± 3.94
While Forecast 1 has a small negative bias (slight overestimation), Forecast 2 has a small
positive bias (slight underestimation).
MFE1 = -2/10 = –0.20. MFE2 = +4/10 = +0.40. MFE = the Mean Forecast Error.
3-41
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Chapter 03 - Forecasting
33.
t
Period
1
A
(Sales)
15
F
A–F Cumulative
(Forecast) (Error)
Error
Error Error Error2 MAD
15
0
0
0
0
0
0.00
2
21
20
1
1
1
1
1
0.05
2.00
3
23
25
–2
–1
2
3
4
1.00
–1.00
4
30
30
0
–1
0
3
0
0.75
–1.33
5
32
35
–3
–4
3
6
9
1.20
–3.33
6
38
40
–2
–6
2
8
4
1.33
–4.51
7
42
45
–3
–9
3
11
9
1.57
–5.73
8
47
50
–3
–12
3
14
9
1.75
–6.86
TS
0.00
Note: MAD is not updated and smoothed.
MSE 
 e2
36

 5.14
n 1 8 1
2s control limits are 0 ± 2√5.14 = 0 ± 4.53
All errors fall within the 2s control limits; however, there is a bias in the forecast method as seen
in the tracking signal measures that keep getting more negative. In addition, if we set the tracking
signal limits at ± 4, then the tracking signals in periods 6 – 8 would fall outside the limits. In
conclusion, the forecast method is not performing adequately—it is exhibiting bias.
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Chapter 03 - Forecasting
34.
t
Period
1
A
(sales)
14
T = 10 + 5t F = T * S
Cumulative
T
Forecast Error
Error
Error Error Error2 MAD
13.50
0.50
0.50
0.50
0.50
0.25 0.50
15
19.00
1.00
1.50
1.00
1.50
1.00 0.75
20
2
20
3
24
25
26.25
-2.25
-0.75
2.25
3.75
5.06
1.25
-0.60
4
31
30
33.00
-2.00
-2.75
2.00
5.75
4.00
1.44
-1.91
5
31
35
31.50
-0.50
-3.25
0.50
6.25
0.25
1.25
-2.60
6
37
40
38.00
-1.00
-4.25
1.00
7.25
1.00
1.21
-3.51
7
43
45
47.25
-4.25
-8.50
4.25
11.50
18.06
1.64
-5.18
8
48
50
55.00
-7.00
-15.50
7.00
18.50
49.00
2.31
-6.71
9
52
55
49.50
2.50
-13.00
2.50
21.00
6.25
2.33
-5.58
Note: MAD is not updated and smoothed.
MSE 
 e 2 84.87

 10.61
n 1 9 1
2s control limits are 0 ± 2√10.61 = 0 ± 6.51
The error in Period 8 is outside the 2s control limits. In addition, there is a bias in the forecast
method as seen in the tracking signal measures that keep getting more negative (except in
Period 9). In addition, if we set the tracking signal limits at ± 4, then the tracking signals in
periods 7 – 9 would fall outside the limits. In conclusion, the forecast method is not
performing adequately. It is not in control and is exhibiting bias.
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TS
1.00
2.00
Chapter 03 - Forecasting
Case: M & L Manufacturing
1.
2.
The potential benefit of using a formalized approach to forecasting is that it will be easier to
utilize the computer and easier to quantify the information. A less formalized approach is more
likely to utilize personal intuition. For small forecasting problems, intuition may involve personal
bias, which may be reflected in the forecast. As the forecasting problem gets larger, it will be
impossible to rely solely on a less formalized approach because a person’s intuition will be
unable to process the large quantity of information.
Product 1
Plotting the data for Product 1 reveals a linear pattern with the exception of demand in week 7.
Demand in week 7 is unusually high and does not fit the linear trend pattern of the remaining
data. Thus, the demand for the 7th week is considered an outlier. There are different ways of
dealing with outliers. A simple and intuitive way is to replace the demand for the week in
question with the average demand from the previous week and the next week in the time-series.
Therefore in this case, the demand of 90 in week 7 will be replaced with 71.5 = [(67 + 76)/2].
t
1
2
3
4
5
6
7
8
9
10
11
12
13
14
105
Y
50
54
57
60
64
67
71.5
76
79
82
85
87
92
96
1,020.5
t*Y
50.00
108.00
171.00
240.00
t2
1
4
9
16
320.00
25
402.00
36
500.50
49
608.00
64
711.00
81
820.00 100
935.00 121
1,044.00 144
1,196.00 169
1,344.00 196
8,449.50 1,015
Round b & a to two decimals:
b
n  tY   t  Y 14(8,449.50)  105(1,020.5)

 3.50
n  t 2  ( t ) 2
14(1,015)  (105) 2
a
 Y  b  t 1,020.5  3.50(105)

 46.64
n
14
Y = 46.64 + 3.50t
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Education.
Chapter 03 - Forecasting
The next four forecasts (t = 15, 16, 17, 18) are:
Period
15
Forecast (T = 46.64 + 3.50t)
T = 46.64 + 3.50(15) = 99.14
16
T = 46.64 + 3.50(16) = 102.64
17
T = 46.64 + 3.50(17) = 106.14
18
T = 46.64 + 3.50(18) = 109.64
Product 2
Plotting the data for Product 2 yields a more complex pattern: There is a spike once every four weeks; the
values between the spikes are fairly close to each other. In addition, the data appear to be increasing at the
rate of about one unit per week. An intuitive approach would be to use the average of the three nonspike
periods plus 1.0 to predict the next three nonspike periods. Doing so for the data up to period 15 yields a
very small average forecast error (MAD = 0.54). Given the fact that we have only two data points
following the last spike, a reasonable forecast might be to use the last three period average plus 1.0 (i.e.,
43.33 to predict orders for period 15, and use the average of the values for periods 13 and 14 plus 1.0 (i.e.,
43.5 + 1.0 = 44.5) as a forecast for periods 17 and 18.
The values of the spikes also seem to be increasing. The initial increase was 1.0 and the second increase
was 2.0. A naive forecast here would be 49 + 2 = 51. However, the average increase was 1.5. Using that
would yield a value of 50.50. One might even be tempted to project an increase of 3.0, although either of
the others seems more justifiable. Still, the fact that there is a limited amount of data makes this forecast
more risky.
Hence, the forecasts are:
Period
15
Forecast
43.33
16
50.50
17
44.50
18
44.50
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Education.
Chapter 03 - Forecasting
Case: Highline Financial Services, Ltd.
Aligning data by quarters, we can see (in the tables and in the figures) that demand for service A is
increasing, demand for service B is decreasing, and demand for service C is mixed. Note, though, that
total annual demand for service C has changed only slightly.
A
Year
1
2
Change
Quarter
1
2
3
4
60
45 100 75
72
51 112 85
+12 +6 +12 +10
Forecast
84
57
124
B
95
Quarter
1
2
3
95
85
92
85
75 85
-10 -10 -7
75
65
72
C
4
65
50
-15
Quarter
1
2
3
93
90 110
102 75 110
+9 -15
0
4
90
100
+10
35
121
110
60
110
Freddie should be concerned about service B, because that has declined for every quarter.
Forecasts were made using a simple naïve (additive) approach. An argument could be made for using a
multiplicative approach (i.e., basing the forecast on the percentage change from one year to the next
instead of the actual change).
Service B
Service A
120
80
Series1
60
Series2
40
20
0
1
2
3
4
Demand
Demand
100
100
90
80
70
60
50
40
30
20
10
0
Year 1
Year 2
1
2
3
4
Quarter
Quarter
Service C
120
Demand
100
80
Series1
60
Series2
40
20
0
1
2
3
4
Quarter
3-46
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Education.
Chapter 03 - Forecasting
Enrichment Module: Additional Methods for Evaluating Forecast Accuracy
The major problem in determining which forecast accuracy measure to use is that there is no universally
accepted accuracy measure. In Chapter 3, several different accuracy measures are covered. To develop a
better understanding of the forecast accuracy measures, first we must understand the nature of the forecast
errors. There are two types of forecast errors.
The first type of error is called the forecast bias, where the direction of the error is the primary
consideration. If the value of the error is negative, then we can conclude that the forecasting method
overestimated sales or demand. If the value of the error is positive, then we can conclude that the
forecasting method underestimated sales or demand because in calculating the error term, we always
subtract the forecasted value from the actual value. Below are three forecast accuracy measures to assess
forecast bias:
1.
Mean Forecast Error (MFE)
2.
Tracking Signal
3.
Control Charts
When we sum the error terms, if there is no bias, positive and negative error terms will cancel each other
out, and the MFE will be zero. As was pointed out above, negative MFE is an indication of
overestimation, and positive MFE is an indication of underestimation. However, if the positive and
negative error values tend to cancel each other out and the MFE or Tracking Signal value is zero or near
zero, then we can conclude that the forecasting method does not result in bias (underestimation or
overestimation). Even if there is no significant bias, it is possible that the forecasting method results in too
much overall variation from the actual values. We call measurement of this type of variation overall
forecast accuracy.
In measuring forecast bias, we cannot measure the overall accuracy of the forecasting method because the
positive and negative errors will cancel each other out. In measuring the overall accuracy, there is never
an error value with a negative sign. There are different overall accuracy measures. In general, in
determining the overall forecast accuracy, we either square the error terms or take the absolute value of
the error terms. The goal of the overall accuracy is to determine how well the forecasting method
estimates the actual values without evaluating forecast bias. In Chapter 3, we covered numerous ways of
evaluating the overall accuracy of forecasts including:
1.
Mean Absolute Deviation (MAD)
2.
Mean Squared Error (MSE)
3.
Standard Error of Estimate
To be able to assess both the overall accuracy and forecast bias, an analyst probably should utilize at least
one method from each category. In the next section, we will discuss two additional methods for
evaluating forecast accuracy.
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Chapter 03 - Forecasting
Warning on MSE: The utilization of MSE as a criterion in determining the accuracy of forecasts has
some drawbacks. One of the drawbacks is that in many cases it is not appropriate to compare MSE values
obtained from different forecasting models because different methods use different ways of obtaining the
forecasted values. Thus, comparison of methods using a single criterion such as MSE becomes
questionable. Relative measures use percentages in determining the accuracy of forecasts. Because of the
limitation of MSE mentioned above, analysts may want to use multiple measures to evaluate the accuracy
of forecasting methods. In addition, multiple accuracy measures may be needed to evaluate both forecast
bias and the overall forecast accuracy. Relative measures are generally considered desirable because
percentages are easy to interpret.
The relative forecast accuracy measures that we will discuss in the remaining portion of this section are:
1.
Mean Percentage Error (MPE)
2.
Mean Absolute Percentage Error (MAPE)
MPE measures the forecast bias while MAPE measures overall forecast accuracy. As with any other
forecast bias measure, when calculating MPE, negative and positive error terms offset each other.
Therefore, for a given time-series data, MPE  MAPE.
Before stating the equations for MPE and MAPE, first we need to define Percentage Error. The
Percentage Error (PE) for a given time-series data measures the percentage points deviation of the
forecasted value from the actual value. The equations for PE in period i, MPE, and MAPE are given
below in equations 1, 2, and 3 respectively.
 A  Fi 
(100 )
PEi   i

A
i


(1)
n
MPE 
 PE
i
i 1
( 2)
n
n
MAPE 
 PE
i 1
n
i
(3)
where:
Ai is the actual value from period i.
Fi is the forecasted (estimated) value from period i.
Both MPE and MAPE are more intuitive and easier to understand and interpret than most of the other
measures because 4.00% has far more meaning to the user than MSE of 224.00 or Tracking Signal value
of 3.50.
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Education.
Chapter 03 - Forecasting
Problems for the Enrichment Module
Problem 1
An analyst must decide between two different forecasting techniques for weekly sales of bicycles: a linear
trend equation and the naïve approach. The linear trend equation is: Yˆi  12  2 X i , and it was developed
using data from periods 1 through 10. Based on the data from periods 11 through 20, calculate the MPE
and MAPE. Based on the values of MPE and MAPE, comment on which of the two methods has the
greater overall accuracy. Compare the two methods in terms of the forecast bias.
t
11
Units Sold
25
t
16
Units Sold
39
12
28
17
48
13
34
18
50
14
40
19
47
15
44
20
54
Problem 2
In solving problem 20 in the textbook, we calculated both MAD and MSE values. In this exercise, we are
going to use the same data and information and calculate MPE and MAPE values. The revised problem is
stated as follows:
Two different forecasting techniques (F1 and F2) were used to forecast demand for cases of bottled water.
Actual demand and the two sets of forecasts are as follows:
a.
b.
t
1
Actual Demand
68
Forecasted demand 1
(F1)
66
Forecasted demand 2
(F2)
66
2
75
68
68
3
70
72
70
4
74
71
72
5
69
72
74
6
72
70
76
7
80
71
78
8
78
74
80
Compute MPE for both sets of forecasts. Which of the two forecasting methods has a higher forecast
bias? Explain.
Compute the MAPE for the two sets of forecasts. Which of the two forecasting methods provides
higher overall accuracy with this data set?
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Chapter 03 - Forecasting
Solutions to Enrichment Module Problems
Solution to Problem 1 (round % to two decimals)
Percentage Error Calculations using the Naïve Method
Period
Actual
Forecast
Error
11
25
–
–
12
28
25
13
34
14
PE i
PE i
3
–
10.71%*
–
10.71%
28
6
17.65%
17.65%
40
34
6
15.00%
15.00%
15
44
40
4
9.09%
9.09%
16
39
44
–5
-12.82%
12.82%
17
48
39
9
18.75%
18.75%
18
50
48
2
4.00%
4.00%
19
47
50
–3
-6.38%
6.38%
20
54
47
7
12.96%
12.96%
Sum
68.96%
107.36%
𝐴𝑖 − 𝐹𝑖
∗ 𝑃𝐸𝑖 = (
) 100
𝐴𝑖
28 − 25
𝑃𝐸12 = (
) 100 = 10.71%
28
68.96%
 7.66%
9
107.36%
MAPE 
 11.93%
9
MPE 
3-50
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Chapter 03 - Forecasting
Percentage Error Calculations using the Linear Trend Equation
PE i
PE i
Period
Actual
Forecast
Error
11
25
34
–9
-36.00%*
36.00%
12
28
36
–8
-28.57%
28.57%
13
34
38
–4
-11.76%
11.76%
14
40
40
0
0.00%
0.00%
15
44
42
2
4.55%
4.55%
16
39
44
–5
-12.82%
12.82%
17
48
46
2
4.17%
4.17%
18
50
48
2
4.00%
4.00%
19
47
50
–3
-6.38%
6.38%
20
54
52
2
3.70%
3.70%
Sum
-79.11%
111.95%
𝐴𝑖 − 𝐹𝑖
∗ 𝑃𝐸𝑖 = (
) 100
𝐴𝑖
25 − 34
𝑃𝐸11 = (
) 100 = −36.00%
25
 79.11%
 7.91%
10
111.95%
MAPE 
 11.20%
10
MPE 
Note: We had 10 periods for which we had both actual demand and forecasts.
a. MPE & Forecast Bias:
Naïve method: Because the MPE is positive for the naïve forecasting method (7.66%), it is
underestimating sales by 7.66%.
Linear trend method: Because the MPE is negative (–7.91%) for the linear trend method, the
trend equation is overestimating sales by 7.91%.
Conclusion: The linear trend method has higher bias.
b. MAPE & Overall Accuracy
Naïve method: The MAPE 11.93%
Linear trend method: The MAPE is 11.20%.
Conclusion: The linear trend method has higher overall accuracy.
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Chapter 03 - Forecasting
To decide between the two methods, we have to compare the consequences and the cost of
overestimation with the consequences and the cost of underestimation. The cost of overestimation
involves the cost of carrying excess inventories, while the cost of underestimation includes the
cost of shortages, backordering, and lost sales.
If the cost of underestimation is less than the cost of overestimation, then the naïve method
should be selected.
If the cost of overestimation is less than the cost of underestimation, then the linear trend method
should be used.
Solution to Problem 2 (round % to two decimals)
a. Forecast Errors and Percentage Errors Using the First Forecasting Method
Period
ei
PE i
PE i
1
2
2.94%
2.94%
2
7
9.33%
9.33%
3
–2
-2.86%
2.86%
4
3
4.05%
4.05%
5
–3
-4.35%
4.35%
6
2
2.78%
2.78%
7
9
11.25%
11.25%
8
4
5.13%
5.13%
Sum
28.27%
42.69%
28.27%
 3.53%
8
42.69%
MAPE 
 5.34%
8
MPE 
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Chapter 03 - Forecasting
b. Forecast Errors and Percentage Errors Using the Second Forecasting Method
Period
ei
PE i
PE i
1
2
2.94%
2.94%
2
7
9.33%
9.33%
3
–2
0.00%
0.00%
4
3
2.70%
2.70%
5
–3
-7.25%
7.25%
6
2
-5.56%
5.56%
7
9
2.50%
2.50%
8
4
-2.56%
2.56%
Sum
2.10%
32.84%
2.10%
 0.26%
8
32.84%
MAPE 
 4.11%
8
MPE 
We recommend the second forecasting method for two reasons:
1. The second forecasting method has less forecast bias because (MPE2 = 0.26%) < (MPE1 =
3.53%).
2. The second forecasting method results in higher overall forecast accuracy because (MAPE2 =
4.11%) < (MAPE1 = 5.34%).
3-53
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