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Fig. 3 Response Surface and its Contour Plot.
On the theory and practice of RSM, it is assumed that the
mean response (η) is related to values of the process variables (ξ1, ξ2, …, ξk) by an fitted unknown mathematical function f [3]. The
functional relationship between the mean response and k process
variables can be written as η = f (ξ), if ξ denotes a column vector with
elements ξ1, ξ2, …, ξk. Estimation of such surfaces, and hence
identification of near optimal setting for process variables is an
important practical issue with interesting theoretical aspects.
The procedure begins with any types of experimental
designs around the current operating condition. A sequence of first
order model and line searches are conventionally justified on the basis
that such a plane would be fitted well as a local approximation to the
true process response. The estimated coefficients of multiple regression
models are usually determined by the method of least squares. A
sequence of run is carried out by moving in the direction of steepest
descent with the predetermined step length. In contrast to this other
algorithmic processes search the system approximation via the
systematic searches or the measurement of the response in the design
points. When curvature is detected, another factorial experimental
design is conducted. This is used either to estimate the position of the
optimum or the systematic searches to specify a new direction of
steepest descent or the new design point with the better yields.
In this study, the mixed integer linear constrained response
surface optimization model (MI-LCRSOM) is deployed to set up a
relationship of the linear constrained responses and both types of
influential process variables. Originally, linear programs are problems
that can be expressed in canonical form:
46
Minimize C T X
Subject to AX ≥ B
And X ≥ 0
where X represents the vector of process variables (to be
determined), C and B are vectors of (known) coefficients and A is a
(known) matrix of coefficients of problem constraints. The expression
to be maximized or minimized is called the objective function (C T X in
this case). The constraints Ax ≥ B specify a convex polytype over
which the objective function is to be optimized. In this problem, some
of the unknown variables are required to be integers. The problem is
then called a mixed integer linear programming (MILP) problem.
Sequential procedures of MI-LCRSOM are repeated. A factorial
experiment design is use to investigate the optimal responses of process
of interest. When the model is formulated, analysis of variance
(ANOVA) is applied to find statistically significant process variables
and determine the most effective levels. Regression analysis is used to
fit a relationship equation of the response and its factor. A restriction of
process variables is also considered as the constraints of the process. A
mathematical programming is use to find the optimal levels in each
process variable via a generalized reduced gradient algorithm that can
bring the suit levels.
3.2 Mixed Integer Linear Constrained Response Surface
Optimization Model (MI-LCRSOM)
In order to optimize the response of meandering that might
be influenced by several process variables various sequential procedures
via statistic tools are then used. One among those is the multiple
regression analysis. It is used to determine the relationship between the
influential process variable of x’s and the dependent process variable or
response of y that is modeled as a linear or nonlinear model. Multiple
regression fits a linear relationship between the value of x’s and the
corresponding conditional mean of y and has been use to describe the
linear phenomena.
To minimize the variance of the unbiased estimators of the
coefficients, multiple regression analysis played and important role in
the development of regression analysis, with a greater emphasis on
issues of design and inference. An aim of regression analysis is to