1. Introduction

Comprehensively, much of the real industrial processes make use of several input variables (factors) at levels often unpredictable due to the instability displayed during operation in transient regime. The actual processes are very difficult to control, especially when it comes to numerous responses to be controlled.

Figure 1 shows in an illustrative way a real process where other factors that could directly influence the responses and interactions, called noise, were not considered. These noises can be related from the events of the environment where the process occurs, such as variations in the temperature of the medium, or events related to errors occurred by the operators. The use of complete second-order models to model processes should be restricted to only a certain interval specified by the levels presented for each of the factors analyzed. In the context of environmental processes, such as effluent decontamination in a treatment plant, the waste disposal parameters are defined according to country-specific standards and must be strictly followed. By using the effluent treatment plant as an example, it is practically impossible to maintain the constant input parameters such as the

individually, even knowing the existence of a high associated positive correlation. Thus, when the responses have a very high correlation, mainly positive, very common in processes that involve chemical reactions, it becomes impracticable to perform the modeling of the multiple objective functions in an independent way,

Normal Boundary Intersection Applied to Controllers in Environmental Controls

In multiobjective optimization problems, the assignment of convex combinations of weights to the multiple responses leads to the agglutination of the objective functions that represent each response through weighted sums, thus generating a Pareto border or surface. Pareto border or surface is therefore a set of optimal values for multiple features obtained from a list of viable optimal points, obviously within a region of viable space. This agglutination of functions can be performed according to some methods: weighted sum and global criterion method (GCM). Both allow the construction of the Pareto border with some constraints attributed to the convexity of the objective function presented in the region of the viable space where the boundary is constructed. When there is a non-convex region in a certain objective function to be analyzed, the Pareto boundary cannot detect optimal points

Analyzing Figure 2, it is possible to verify a Pareto frontier for two responses, where each of the points represents different operating conditions. However, there is a discontinuity indicating no convex region of both functions representing the responses. One way to solve this problem is to use the algorithm Normal Boundary Intersection (NBI) to construct the Pareto frontier. This algorithm is able to determine points along the boundary, even in non-convex regions of space. The NBI algorithm considers two fixed points of the frontier ("best of the best and worst of the worst") known as utopia and nadir respectively. Between these fixed points,

One of the great possibilities in using this algorithm as a transfer function in control processes is precisely the possibility of choosing a number of different process setups, which consequently lead to optimized responses between the utopia and nadir points, which may be the limits of specification of the particular disposal

leading to the wrong responses.

DOI: http://dx.doi.org/10.5772/intechopen.83662

all others that make up the border are distributed.

in this region.

Figure 2.

9

Pareto frontier showing discontinuity.

Figure 1. General process diagram.

incoming organic load, heavy metals, and turbidity, among others. In order to keep the process running at steady state, with possible variations of input, it is necessary that the levels of the controllable factors be adequate in order to keep the responses at the exit within the pre-established parameters. This adjustment of levels can be achieved through sensors connected to programmable logic controllers, which usually operate through a set point.

These types of equipment are microprocessor computers that perform the function of control through specific software. One of the major problems encountered in using this device can be attributed to numerous, generally correlated, input conditions that may occur throughout the operation. Even with controller performance due to the set point, it no longer considers the possibility of interactions between these input parameters, which may compromise the permanence of the steady state. When the noise source is not discovered for later quantification, instability in the process can lead to desired responses outside of the predefined standards, leading to losses, associated cost, and environmental damage. Therefore, the controllers currently applicable cannot consider this instability generated by the noise industrially.

In the environmental area, due to the large number of parameters that must be monitored and pre-established as waste disposal control standards in receiving bodies, it is very common to maintain a certain operation for numerous responses. Thus, it is fundamental that the process can be previously known, modeled, and later optimized through algorithms already fomented by the literature, allowing the implementation of robust multiobjective optimization from the polynomial that describes all the responses, factors, and levels of the process in detail.

The concept of multiobjective robust optimization can be described as the set of nonlinear constrained programming (NLP) methods and algorithms that are intended to simultaneously optimize the mean and variance of multiple process characteristics that are in a way correlated output quantities that are reasonably well modeled by complete quadratic models. However, in effluent treatment processes that have multiple output characteristics are generally correlated.

Any process can be defined through a quadratic polynomial, if it is properly constrained within certain predefined intervals. The original concept of "robust" process was introduced by Genichi Taguchi in 1980 [1]. To this concept we can associate the original idea of RPD (robust parameter design), applied to generic processes. The more "robust" the details of the process are known, the more accurately it can be modeled and optimized. Therefore, there are several situations in which the multiple means of responses must be optimized and the multiple variances associated with each of the responses individually, minimized. This routine can be performed in order to reduce the interference attributed to the noise and to maintain a more stable process. As already mentioned, independently of the innumerable responses to be analyzed to a process, they are easily analyzed

### Normal Boundary Intersection Applied to Controllers in Environmental Controls DOI: http://dx.doi.org/10.5772/intechopen.83662

individually, even knowing the existence of a high associated positive correlation. Thus, when the responses have a very high correlation, mainly positive, very common in processes that involve chemical reactions, it becomes impracticable to perform the modeling of the multiple objective functions in an independent way, leading to the wrong responses.

In multiobjective optimization problems, the assignment of convex combinations of weights to the multiple responses leads to the agglutination of the objective functions that represent each response through weighted sums, thus generating a Pareto border or surface. Pareto border or surface is therefore a set of optimal values for multiple features obtained from a list of viable optimal points, obviously within a region of viable space. This agglutination of functions can be performed according to some methods: weighted sum and global criterion method (GCM). Both allow the construction of the Pareto border with some constraints attributed to the convexity of the objective function presented in the region of the viable space where the boundary is constructed. When there is a non-convex region in a certain objective function to be analyzed, the Pareto boundary cannot detect optimal points in this region.

Analyzing Figure 2, it is possible to verify a Pareto frontier for two responses, where each of the points represents different operating conditions. However, there is a discontinuity indicating no convex region of both functions representing the responses. One way to solve this problem is to use the algorithm Normal Boundary Intersection (NBI) to construct the Pareto frontier. This algorithm is able to determine points along the boundary, even in non-convex regions of space. The NBI algorithm considers two fixed points of the frontier ("best of the best and worst of the worst") known as utopia and nadir respectively. Between these fixed points, all others that make up the border are distributed.

One of the great possibilities in using this algorithm as a transfer function in control processes is precisely the possibility of choosing a number of different process setups, which consequently lead to optimized responses between the utopia and nadir points, which may be the limits of specification of the particular disposal

Figure 2. Pareto frontier showing discontinuity.

incoming organic load, heavy metals, and turbidity, among others. In order to keep the process running at steady state, with possible variations of input, it is necessary that the levels of the controllable factors be adequate in order to keep the responses at the exit within the pre-established parameters. This adjustment of levels can be achieved through sensors connected to programmable logic

describes all the responses, factors, and levels of the process in detail.

that have multiple output characteristics are generally correlated.

nonlinear constrained programming (NLP) methods and algorithms that are intended to simultaneously optimize the mean and variance of multiple process characteristics that are in a way correlated output quantities that are reasonably well modeled by complete quadratic models. However, in effluent treatment processes

The concept of multiobjective robust optimization can be described as the set of

Any process can be defined through a quadratic polynomial, if it is properly constrained within certain predefined intervals. The original concept of "robust" process was introduced by Genichi Taguchi in 1980 [1]. To this concept we can associate the original idea of RPD (robust parameter design), applied to generic processes. The more "robust" the details of the process are known, the more accurately it can be modeled and optimized. Therefore, there are several situations in which the multiple means of responses must be optimized and the multiple variances associated with each of the responses individually, minimized. This routine can be performed in order to reduce the interference attributed to the noise and to maintain a more stable process. As already mentioned, independently of the innumerable responses to be analyzed to a process, they are easily analyzed

These types of equipment are microprocessor computers that perform the function of control through specific software. One of the major problems encountered in using this device can be attributed to numerous, generally correlated, input conditions that may occur throughout the operation. Even with controller performance due to the set point, it no longer considers the possibility of interactions between these input parameters, which may compromise the permanence of the steady state. When the noise source is not discovered for later quantification, instability in the process can lead to desired responses outside of the predefined standards, leading to losses, associated cost, and environmental damage. Therefore, the controllers currently applicable cannot consider this instability generated by the noise industrially. In the environmental area, due to the large number of parameters that must be monitored and pre-established as waste disposal control standards in receiving bodies, it is very common to maintain a certain operation for numerous responses. Thus, it is fundamental that the process can be previously known, modeled, and later optimized through algorithms already fomented by the literature, allowing the implementation of robust multiobjective optimization from the polynomial that

controllers, which usually operate through a set point.

Figure 1.

8

General process diagram.

Biosensors for Environmental Monitoring

2.1 Stochastic response surface models

DOI: http://dx.doi.org/10.5772/intechopen.83662

a second-order response surface:

fields.

Figure 4. Counter graphic.

11

2.2 NBI algorithm

Yð Þ¼ x β<sup>0</sup> þ ∑

variables considered in the study, and ε is the error term.

of the independent variables, leads to better responses.

The response surface methodology (MSR) is a collection of mathematical and statistical techniques that allows modeling, analyzing, and optimizing problems whose response variables are influenced by many variables [2]. As mentioned earlier, there is great difficulty in knowing the behavior of independent and dependent variables in a process. Thus, the response surface allows the real approximation of the process from a quadratic model. The development through

a Taylor polynomial, truncated in the quadratic term, takes what we call

Normal Boundary Intersection Applied to Controllers in Environmental Controls

βixi þ ∑ k i¼1 βiix<sup>2</sup> <sup>i</sup> þ ∑ i < j

where β represents the coefficients of the model, k is the number of independent

The fact of using the response surface in a region close to high curvature of the model, presented according to local or global maxima or minima, according to convexity, does not effectively determine the best points or operation setups. However, what can be verified is a region of space that, depending on the levels of each

From the color gradient shown in Figure 4, it is possible to verify regions, delimited through the Cartesian axes representing the levels of each of the factors studied, leading to better responses. Thus, the construction of models through the surface response method becomes paramount for the application of later optimization algorithms. Among several optimization algorithms, the Normal Boundary Intersection (NBI) [3] has been used in several researches, in several different

The NBI algorithm is developed in terms of an array that we call the payoff matrix Φ , which represents the optimal values of the multiple objective functions

∑ βijxixj þ ε (1)

k i¼1

Figure 3. Pareto border for bi-objective problem.

parameter in an effluent treatment plant, for example. The polynomial (Figure 3), which represents the Pareto frontier, can be used as a transfer function in scaling of a possible dynamic process controller for multiobjectives.
