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# Assignment #2 - Solution Winter 2010

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University of Waterloo

Management Sciences

MSCI 432

Binyamin Mantin

Fall

Description

MSCI 432.001/633: Production and Service Operations Management, Winter 2010
Assignment # 2
Due by Friday, February 12, 2010, at noon time. Drop your solutions in the MSci432/633 slot of the metal box
located in front of CPH 2367 (2nd floor, where CPH building meets E2 building). No late submissions.
Individual submissions. You must indicate whether you have collaborated/cooperated on this assignment with
other students. Also indicate whether you are 432.001 or 633.
As a rule, type your solution for all text-dominated so lutions. Save paper, submit double-sided and please no
fancy marginalia.
1. [6 marks] The number of students spotted at the Grad House in each of the last six weeks has been:
t 1 2 3 4 5 619 0
Students t 83 10695 91 110 108
a. Use a two-week moving average to forecast the number of students in each of weeks 7 and 10. Also
compute the MSE based on the # of students and forecasts for period 1 to 6. To initialize assume
x-1x =000 (i.e., your prediction for the first period is 100).
b. A consultant suggests the use of simple (single) exponential smoothing with α=0.1. Use such a
procedure to develop forecasts of # of students in weeks 7 and 10. Initialize with F=100. Alst
compute MSE and MAD.
c. Without redoing all the computations, what would be the forecast of students in week 7 using
exponential smoothing if F wert 90 instead of 100?
a. [2 marks]
xt,2 xt xt1/2
and
at xt,2 x t,t at
Now using these three formulas, we obtain,
110108 /2 109
x6,7 x6,10 a6
Now developing the whole table;
t 1 2 3 4 5 619 0
xˆt1,t 100 91.5 94.5 100.5 93 100.5 109 109 109 109
xt 83 10695 91 110 108
x - 1714.50.5 9.5 17 7.5
t xt1,t
(xt- xt1,t 289.10.25 0.25 90.25 289.00 56.25
The MSE = the sum of the last row divided by 6 = 155.83.
b. [2 marks] In simple exponential smoothing, we have;
aˆt xt 1 at1 here 0.1
xˆ aˆ
t,t t
Using these formula, the following table is developed.
T 1 2 3 4 5 61 870 xt1,t 100 98.3 99.1 98.7 97.9 99.1 100 100
xt 83 106 95 91 110 108
xt-xˆt1,t 17 7.7 4.1 7.7 12.1 8.9
2
(xt-xt1,t 289.00 59.29 16.81 59.29 146.41 79.21
MAD = 9.6, and MSE = 108.34, which is better than moving average.
c. [2 marks]
x6,7 a6 x6 1 a5
x6 1 x 5 1 a4
2
x6 x5 a4
Continuing in this manner, we get
6 6
New x6,7Old x6,7 new a0 1 old a0
6
new a0 oldaˆ0
0.9690100 5.3
New x6,7 old x6,75.3
100.05.3
94.7
2. [6 marks] Union Gas reports the following (normalized) figures for gas consumption by households in
a metropolitan area of Ontario:
Jan. Feb. Mar. Apr. May June July Aug. Sept.Oct. Nov. Dec.
200179619617310575 39 13 20 37 73 108191
200282721719711088 51 27 23 41 40 107172
200291621819312099 59 33 37 59 95 128201
20228
a. Using a simple (single) exponential smoothing model (with α=0.1), determine the forecast for
consumption in (1) February 2010 and (2) June 2012.
b. Do the same using the Winters seasonal smoothing model with α=0.2, β=0.05, and δ=0.1. Use the first
two years of data for initialization purposes.
c. Plot the demand data and the two sets of forecasts. (i) Does there appear to be a point of abrupt change in
the demand pattern? (ii) Assess the performance of the Winters model briefly for the different months
(without calculating any error measures). (iii) Which method is preferred?
a. Assuming initial forecast is 196, we have the following predictions as in the last column of the table
below.
i. The forecast for February, 2010, is 124.2 ii. Single exponential smoothing assumes no trend or seasonal factors. Thus, the forecast for
June, 2012 is also 124.2.
b. Refer to the table below to see the initialization using the first two years and updating thereafter.
i. The forecast for February, 2010, is 247.8
ii. The forecast for June, 2012 is (161.81+1.86*29)*0.53=114.98, since June 2012 is 29 months
away from January 2010, and we are using S and G from that month and the best factor available
for June.
Lin. exponential
Demand Reg. Ct Average Ct Winter smoothing
Use first two years (24 obs.) for initial
Jan‐07 1 196 127.49 1.54 estimates (linear reg.) 196
Feb‐07 2 196 125.56 1.56 slope ‐1.93391 196
Mar‐07 3 173 123.62 1.40 intercept 129.4239 196
Apr‐07 4 105 121.69 0.86 193.7
May‐07 5 75 119.75 0.63 184.8
Jun‐07 6 39 117.82 0.33 173.8
Jul‐07 7 13 115.89 0.11 160.4
Aug‐07 8 20 113.95 0.18 145.6
Sep‐07 9 37 112.02 0.33 133.1
Oct‐07 10 73 110.08 0.66 123.5
Nov‐07 11 108 108.15 1.00 118.4
Dec‐07 12 191 106.22 1.80 1.67 1.43 117.4
Jan‐08 13 227 104.28 2.18 1.87 1.60 124.7
Feb‐08 14 217 102.35 2.12 1.76 1.51 135.0
Mar‐08 15 197 100.42 1.96 1.41 1.21 143.2
Apr‐08 16 110 98.48 1.12 0.87 0.75 148.5
May‐08 17 88 96.55 0.91 0.62 0.53 144.7
Jun‐08 18 51 94.61 0.54 0.33 0.28 139.0
Jul‐08 19 27 92.68 0.29 0.23 0.20 alpha= 0.2 130.2
Aug‐08 20 23 90.75 0.25 0.29 0.25 beta= 0.05 119.9
Sep‐08 21 41 88.81 0.46 0.56 0.48 delta= 0.1 110.2
Oct‐08 22 40 86.88 0.46 0.73 0.63 103.3
Nov‐08 23 107 84.94 1.26 1.53 1.31 G S F 97.0
Dec‐08 24 172 83.01 2.07 2.12 1.82 ‐1.93 83.01 115.92

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