Question

Production Management:

Given the following table, forecast for the production of shirt using least square method for 2019, 2020, and 2021.Show your soluction.

YEAR	PRODUCTION
2012	343.5 PIECES
2013	381.2 PIECES
2014	437.5 PIECES
2015	323.6 PIECES
2016	231.4 PIECES
2017	443.5 PIECES
2018	525.2 PIECES

Answer 1

Here response y = Production

and x = (year - origin) / Interval

year 2015 is the origin as it is in the middle

Constructing the table again with new column x we get,

Year	Production (y)	x
2012	343.5 PIECES	-3
2013	381.2 PIECES	-2
2014	437.5 PIECES	-1
2015	323.6 PIECES	0
2016	231.4 PIECES	1
2017	443.5 PIECES	2
2018	525.2 PIECES	3

Now creating the linear regression line between x and y we get,

$\hat \beta_1 = \frac{S_{xy}}{S_{xx}} = \frac{\sum_{i = 1}^{7}x_i y_i - n \bar x \bar y}{\sum_{i = 1}^{7}x_i^2 - n \bar x^2} = \frac{463.6 - 7 * 0 * 383.7}{28 - 7 * 0^2} = 16.557$

and,

$\hat \beta_0 = \bar y - \hat \beta_1 \bar x = 383.7 - 16.557 * 0 = 383.7$

So the regression line becomes, y_i = 383.7 + 16.557 x_i

Now for year 2019, 2020 and 2021, x will be 4, 5 and 6 respectively.

Now for year 2019 (x = 4) the forecasted value of Production (y) will be,

$y = 383.7 + 16.557 * 4 = 449.928$

Now for year 2020 (x = 5) the forecasted value of Production (y) will be,

$y = 383.7 + 16.557 * 5 = 466.485$

Now for year 2021 (x = 6) the forecasted value of Production (y) will be,

$y = 383.7 + 16.557 * 6 = 483.042$

**If your answer does not match please comment.