Page 318 - Solid Waste Analysis and Minimization a Systems Approach

296 BENCHMARKING AND EVALUATION

The following mathematics (in matrix form) were applied to calculate the confidence
intervals multivariable regression models:

−
1 −
ˆ y − t s x′ (X X ) x < μ < ˆ y + t s x′ ( ′ XX ) x
′
1
0 α /2 0 0 Yx ,x 0 α /2 0 0 0
10 20
ˆ
where y 0 = predicted mean annual solid waste generation
t α /2 = t value at the specified confidence level and n – k – 1 degrees of freedom
s = unbiased estimate of residual mean square
x = condition vector of independent variable values for prediction
0
X = matrix of x values that give rise to the response y i
i

The performance parameters were established using a t α/2 = 3. A value of 3 cor-
responds to approximately 3σ above or below the mean annual waste generation
tonnage. A performance parameter of 3σ was chosen to balance the two types of
errors associated with statistical quality control, type I and type II. Type I error is
the risk of a point falling beyond performance parameters, indicating an out of
control condition when no assignable cause is present (Montgomery, 1997). Type
II error is the risk of a point falling between the performance parameters when the
process is really out of control. By using a 3σ the probability of type I error is 0.27
percent; that is an incorrect out of control signal will be generated only 27 out of
10,000 data points. A 3σ limit is typically used in statistical quality control and
was applied to this research. The performance parameters for the 20 waste groups
involved a similar statistical basis used with the x (x-bar) chart. The x-bar chart is
commonly used in manufacturing environments to track and control the quality of
products or services.
The following list displays the three values that were calculated to establish the per-
formance parameters:

1 The expected or mean solid waste quantity waste
2 Upper performance parameter
3 Lower performance parameter

The latter two were calculated from the previous equation. The benefits of this
method are: ease of calculations, ease of programming, ability to benchmark, stan-
dardization, objectivity, and confidentiality. The confidentiality aspect of the method
involves its usage. This method is applicable to Internet systems and may be easily
available from a Web site for private use by companies to quickly and privately eval-
uate their waste generation performance. The performance parameters identify high
and low waste generators based on research specific data. The performance parame-
ters developed will aid manufacturing and service companies in evaluating their solid
waste generation in comparison with the waste group standards established from this
research. The performance parameters determine when a company is out of control in