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**preview**shows pages 1-2. to view the full**6 pages of the document.**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 solutions. 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 6 7 8 9 10

Students xt 83 106 95 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-1=x0=100 (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 Ft=100. Also

compute MSE and MAD.

c. Without redoing all the computations, what would be the forecast of students in week 7 using

exponential smoothing if Ft were 90 instead of 100?

a. [2 marks]

axxa

xxx

ttttt

ttt

ˆˆ

and

ˆ2/

,2,

12,

Now using these three formulas, we obtain,

1092/108110

ˆˆˆ 610,67,6 axx

Now developing the whole table;

t 1 2 3 4 5 6 7 8 9 10

xtt

ˆ,1 100 91.5 94.5 100.5 93 100.5 109 109 109 109

xt 83 106 95 91 110 108

xt - xtt

ˆ,1 17 14.5 0.5 9.5 17 7.5

(xt - xtt

ˆ,1)2 289.00 210.25 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;

ax

axa

ttt

ttt ˆˆ 1.0 here

ˆ

1

ˆ

,

1

Using these formula, the following table is developed.

T 1 2 3 4 5 6 7 8 9 10

Only pages 1-2 are available for preview. Some parts have been intentionally blurred.

xtt

ˆ,1 100 98.3 99.1 98.7 97.9 99.1 100 100

xt 83 106 95 91 110 108

xt - xtt

ˆ,1 17 7.7 4.1 7.7 12.1 8.9

(xt - xtt

ˆ,1)2 289.00 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]

axx

axx

axax

ˆ

11

ˆ

11 ˆ

1

ˆˆ

4

2

56

456

5667,6

Continuing in this manner, we get

7.94 3.5 0.100 3.5

ˆ

old

ˆ

New 3.510090

9.0

ˆ

old

ˆ

new1

ˆ

old 1

ˆ

new 1

ˆ

Old

ˆ

New

7,67,6

6

00

6

0

6

0

6

7,67,6

xx

aa

aaxx

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.

2007 196 196 173 105 75 39 13 20 37 73 108 191

2008 227 217 197 110 88 51 27 23 41 40 107 172

2009 216 218 193 120 99 59 33 37 59 95 128 201

2010 228

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

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