Operations and Supply Chain Management The Core

58 OPERATIONS AND SUPPLY CHAIN MANAGEMENT
where
Y t = Dependent variable computed by the equation
y = The actual dependent variable data point (used below)
a = Y t intercept
b = Slope of the line
t = Time period
The least squares method tries to fit the line to the data that minimizes the sum of the squares of the
vertical distance between each data point and its corresponding point on the line. If a straight line is
drawn through the general area of the points, the difference between the point and the line is y − Y.
Exhibit 3.6 shows these differences. The sum of the squares of the differences between the plotted
data points and the line points is
​(​y​ 1 ​ − ​Y​ 1 ​)​ 2 ​ + ​(​y​ 2 ​ − ​Y​ 2 ​)​ 2 ​ + ⋯ +​ (​y​ 12 ​ − ​Y​ 12 ​)​ 2 ​
The best line to use is the one that minimizes this total.
As before, the straight line equation is
​Y​ t ​ = a + bt​
In the least squares method, the equations for a and b are
​b = ​ Σty − N​ t ¯
__________ ​ ⋅ ​ y ¯ ​
​ [3.8]
Σ​t 2 − N t ¯ ​
2 ​
​a = ​ y ¯ ​ − b​ t ¯ ​ [3.9]
exhibit 3.6
Least Squares Regression Line
Excel:
Forecasting
Sales
$5,000
4,500
4,000
3,500
3,000
2,500
2,000
1,500
1,000
500
Y 1
y 1
Y 2
y 2 y 3
Y 3
Y 4
y 4
y 5
Y 5
Y 6
y 6
y 7
Y 7
Y 8 Y 9
y 8
y 9
y 10
Y 10
Y 11
y 11
y 12
Y 12
0
1
2 3 4 5 6 7 8 9 10 11 12
Quarters