New Decision Making Criterion for Multiobjective Optimization

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A New Decision Making Criterion for Multiobjective Optimization Problems Livia Pereira Lemos, Enrique Luis Lima, and José Carlos Pinto Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b04196 • Publication Date (Web): 21 Dec 2017 Downloaded from http://pubs.acs.org on December 26, 2017

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Industrial & Engineering Chemistry Research

A New Decision Making Criterion for Multiobjective Optimization Problems Lívia Pereira Lemosa,b, Enrique Luis Limaa and José Carlos Pintoa,* a

Programa de Engenharia Química / COPPE Universidade Federal do Rio de Janeiro Cidade Universitária, 68502 – Rio de Janeiro – 21941-972 RJ – Brasil b CENPES/PETROBRAS Avenida Horácio Macedo, 950 Cidade Universitária – Rio de Janeiro – 21949-900 RJ – Brasil

Abstract - This work presents a new decision making criterion for analysis of multiobjective optimization problems, based on the evaluation of the optimum solutions in the Pareto set. This new technique allows for discrimination among the solutions that constitute the Pareto set, so that a single alternative (the least sensitive one) is selected for real implementation. In order to do that, the effects of process uncertainties on the variable responses are analyzed individually, with the help of the process model, and used to compute a sensitivity index. The proposed technique was applied to three distinct problems and the obtained results showed that the method is able to discriminate the solutions successfully, allowing for appropriate selection of the best candidates for real implementation among the many candidates in the Pareto set.

Keywords: Multiobjective optimization, decision making, Pareto, sensitivity index.

*

Corresponding author - [email protected] - Tel:+55-21-39388337 - FAX:+55-21-39388300

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A New Decision Making Criterion for Multiobjective Optimization Problems Lívia Pereira Lemos Costaa,b, Enrique Luis Limaa and José Carlos Pintoa,* a

Programa de Engenharia Química / COPPE Universidade Federal do Rio de Janeiro Cidade Universitária, 68502 – Rio de Janeiro – 21941-972 RJ – Brasil b CENPES/PETROBRAS Avenida Horácio Macedo, 950 Cidade Universitária – Rio de Janeiro – 21949-900 RJ – Brasil

1. Introduction Multiobjective optimization problems are common in different areas and appear in the chemical engineering field with high frequency.1-11 For example, multiobjective optimization procedures (MOP's) have already been used for polymer synthesis,3 controller design5 and design of hydrogen processes,4 among many other examples. For this reason, MOP's have been widely studied lately. Generally speaking, MOP's with conflicting objectives can provide sets of equally efficient non-dominated optimal solutions for a particular multiobjective optimization problem. The optimal solutions constitute the Pareto set,12 whose image is called the Pareto front, and different numerical techniques can be used to calculate the Pareto set.13 However, from a practical point of view, it is always necessary to choose a particular solution from the Pareto set in order to implement the optimal solution in the real world. For this reason, it is necessary to establish a criterion for decision making and proper selection of a solution candidate from the Pareto set.14 In order to do that, most existing methods make use of heuristic procedures, being very dependent on the decision maker. Different decision making techniques can be found in the literature, although they have been generally grouped into two distinct classes: (i) the American School, based on the Utility Theory, where a utility function is used to discriminate different solutions in the Pareto set; (ii) and the French School, that makes use of outranking concepts, net flow and graphs to discriminate the solutions.14 The present work presents a new decision making criterion to solve conflicting multiobjective optimization problems and allow for real implementation of optimal solutions in some practical engineering problem. The proposed decision making criterion is based on the sensitivity of the solution that must be implemented, which seems appropriate when design and response variables are subject to uncertainties and experimental errors. Briefly, the proposed method disturbs the decision variables that constitute the solutions of the Pareto set, in accordance with a computational experimental plan, and ranks the solution candidates in terms of the distance between the *

Corresponding author - [email protected] - Tel:+55-21-39388337 - FAX:+55-21-39388300


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disturbed objectives and the original solutions. It is shown that the proposed criterion is able to produce efficient and coherent solutions for different multiobjective optimization problems, taking into account the existence of natural variable fluctuations in a systematic way. Illustrative examples are presented for polymer design, controller design and controller tuning.

2. Theoretical Framework 2.1. Multiobjective Optimization It is common to deal with optimization problems that involve multiple objectives (in many cases, conflicting objectives) that can be related to economic, social or environmental factors.12 A typical multiobjective optimization problem can be described in the form:

min = ⋯ ⋯ ∈

subject to ≤ 0, = 1, … ,

ℎ = 0, = 1, … ,

(1)

where is the set that contains the k objectives ( ⋯ ⋯ ) that must be minimized simultaneously, are the inequality constraints and ℎ are the equality constraints. Traditionally, multiobjective problems can be solved with techniques that combine the multiple competing objectives into a single objective function.15 However, it is also well established that this traditional approach is inadequate to solve a large number of real practical problems, when grouping of very different objectives into a singular mathematical function seems artificial and when one is interested in analyzing the pros-and-cons of a particular optimal solution.2 In general, when dealing with conflicting objectives, the multiobjective optimization problem does not allow for computation of a single solution that minimizes all the objectives simultaneously, but for computation of a set of solutions that are equally efficient and constitute the Pareto set. For the solution candidates in the Pareto set (or non-dominated optimal solutions) it is not possible to reduce one particular objective function without sacrificing at least one of the remaining objectives.16 Although many distinct numerical methods can be used to generate the Pareto set, the use of non-deterministic algorithms, such as the Monte Carlo, particle swarm and genetic algorithms, has been preferred in the literature for this computation.3,16-24 In the present work, particle swarm technique25,26 has been implemented and used to obtain the Pareto fronts and the reader is encouraged to refer to these works for implementation details. The number of particles and the number of iterations adopted in each example have been chosen by trial, aiming to achieve a well-explored Pareto set. Also, in this work, for multiobjective purposes, the particle swarm algorithm has been computed with the specified number of particles and iterations for each objective.



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2.2. The Decision Making Problem The final solution of a conflicting multiobjective optimization problem comprises a set of solution candidates that constitute the Pareto set. From a practical point of view, the proper selection of the solution candidate that must be implemented constitutes a very important step. This is particularly true in engineering problems, as the optimization scheme aims at finding a suitable process operation condition, a particular process formulation or an appropriate tuning for the optimal controller. In this case, the simple presentation of a set of solution candidates does not suffice, as a single solution must be implemented in the real world problem. Therefore, it is necessary to select one of the solution candidates in the Pareto set that must be used in the real implementation12 with the help of a decision making criterion. Distinct methods have been proposed for decision making in the literature.27,28 Some of them are based on weighted sums or weighted products,29 hierarchical processes,30 the Promethee,31 Electre32 and multiattribute Utility Theory33 and robustness aspects.34-37 Most of these methods rely on heuristics and parameters that must be defined by the analyst, such as the weights used in weighted sums, rendering the decision process very dependent on the mood of the decision maker. Decision making procedures have been applied in different areas, including finances,38 energetic planning,29 transports,39 medicine40 and marketing41. Decision making procedures have also been used in chemical engineering problems.2,42,43 Nevertheless, even when the multiobjective formalism is used to provide the Pareto set, decision making in multiobjective chemical engineering problems has been usually achieved through combination of the multiple objectives into a single objective function, with help of weighted sums. Thus, the use of decision making criteria in multiobjective chemical engineering problems is still an area to be explored. In this sense, Wang and Rangaiah44 compared 10 methods with different objective weightages for selecting an optimal solution from the Pareto-optimal front in 12 benchmark mathematical problems and 13 chemical engineering problems. Particularly, some methods for decision making criterion based on sensitivity or uncertainty analysis can be found in the literature.45,46 Generally, these works stablish a key criterion, such as an economic criterion, and test the solutions in the Pareto for a large number of uncertain scenarios. The present work suggests a different approach in which there is no key criterion, but an index, which considers all the objectives and their uncertainties, computed for several uncertain scenarios. Alternatively, some works propose an optimization problem based on a norm, such as a conditional-value-at-risk norm or a distance from a utopia point, to find a compromise solution from the Pareto set. This approach is done without the need to calculate the entire Pareto set.47-49 However, for problems in which the sensitivity to uncertainties is fundamental, the new proposed approach could be very useful, since sensitivity is directly considered, although it is computationally expensive.

3. The Proposed Approach The decision making criterion proposed here is intended to eliminate weights and other similar heuristic parameters from the decision making process, in order to minimize the sensitivity of the expected performance to undesired (but unavoidable) fluctuations of 4 ACS Paragon Plus Environment

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the optimizing variables and to make the final selection of the optimal solution less dependent on the decision maker. It is well-known in the industry that measured process variables are subject to uncertainties that can be caused by natural fluctuations of the process conditions and by the real measuring performances of process instruments. Besides, variable values calculated with mathematical process models are also subject to model uncertainties, as models cannot represent all aspects of the real analyzed process and model parameters obtained with parameter estimation procedures are also uncertain to some extent due to the unavoidable experimental errors.50 For these reasons, the formulation of decision making criteria that take into consideration the sensitivity of the optimum candidate solutions in the Pareto set to uncertainties is appealing. Thus, it is assumed here that a good candidate solution in the Pareto set must depend weakly on small fluctuations of the operation conditions (or be less sensitive), in order to minimize the effect of uncertainties on the obtained performances during the real implementation. Based on the previous remarks, the proposed multiobjective optimization procedure comprises the following two fundamental steps: 1. Step 1 – calculation of the Pareto set; 2. Step 2 – classification of the solution candidates in terms of the sensitivity. In order to perform the second step, optimizing variables (possibly including design variables and model parameters) must be disturbed in accordance with a previously defined experimental design (taking into consideration the confidence regions of expected fluctuations) and the models must be re-evaluated, so that new objective function values can be obtained for each solution candidate. As a consequence, the analyst can evaluate the effect of real process uncertainties on the performances of the solution candidates of the Pareto set. If for simplicity the analyst assumes that fluctuations follow the normal distribution and that the confidence level must be equal to 95%, then confidence regions are approximately equal to ± 2 σ , where σ is the standard deviation. The confidence region determine the range in which each optimizing variable will be disturbed during the experimental design. In chemical engineering problems, the standard deviations for the variables can be computed from a historical sample. In the case of parameters, the standard deviations can be obtained from the parameter estimation step. In the present manuscript, the strategy for experimental design used to disturb the optimizing variables was the latin hypercube design (LHD).51,52 This technique aims to fill the search space so that all dimensions of the vector space are represented. Briefly, LHD was implemented as follows: assuming the computation of npert disturbances for each point of the Pareto set, the confidence regions were uniformly divided into npert values for each variable and these individual values were randomly combined to generate an uncorrelated numerical grid where sensitivities were computed, as in a standard factorial experimental design. Therefore, all analyzed optimizing variables are perturbed and analyzed simultaneously. After computation of the objective function values in the proposed numerical grid, it becomes possible to define a distance (Dk) between the disturbed results and the original Pareto front values in the form



=

,-./ "0

!," , $ = 1, … , %&'()*+

!," = min 1

,78 0

2 ,", − 4, 5 , = 1, … , %&'()*+ 6

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(2)

(3)

where %+9 is the number of pursued objectives; %&'()*+ is the number of candidate solutions in the Pareto set; %(! is the number of points in the proposed numerical grid;

,", is the value of the objective function after perturbation : of the candidate solution $ in the Pareto set; 4, is the value of the objective function for the candidate solution in the Pareto set; 6 is the characteristic variance of the objective function ; !," represents the minimum distance between the calculated objectives after introduction of disturbance : in the candidate solution $ and the objectives belonging to the Pareto front; and represents the sum of the least distances between the perturbed solutions from the candidate solution k and the original solutions of the Pareto set. values can be used to quantify the sensitivity of the candidate $ to perturbations of the optimizing variables. It is assumed here that the analyst is interested in obtaining very low values, in order to minimize the sensitivity of obtained performances to modification of the optimizing variables. Obviously, the distance can be defined using a different vector norm, although this is not pursued here on purpose, as any useful distance function can be used to replace Equations (2-3) without any loss of generality. Similarly, 6 values can be regarded as suitable weighting parameters, although this interpretation is avoided here, as 6 clearly represents the natural variability (or accepted variability) of the analyzed objectives in the proposed approach and, therefore, does not depend on the analyst, but on the particular analyzed problem. Besides, in order to explore the Pareto front conveniently, it is important to use as many particles and iterations as possible during the computation of the Pareto set, although the use of small number of particles and iterations during the first step does not reduce the generality of the proposed approach.

4. Numerical Applications 4.1. Example 1 - A Polymer Design Problem Example 1 proposes the multiobjective formulation of a polymer design problem, as investigated by Latado et al.53. The original reference analyzes the formulation of polypropylene resins with specified performance properties. This problem is very common in the polymer industry, as several conflicting resin properties must be satisfied to attend clients' requirements. Latado et al.53 developed semi-empirical models to allow for prediction of melt index (;), flexural modulus (?@ ) and Izod 6 ACS Paragon Plus Environment

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impact resistance at 23ºC ( values reflects the decrease of the degree of isotacticity of the resin, leading to poorer mechanical performance. ?@ and =

jU 1 ) _lm = ) _Nm 1 + kU > 1 + 20>

(5)

The selected performance criteria used for optimization and tuning of the PID controller were the minimization of the integral absolute error (IAE), of integral time absolute error (ITAE) and of integral squared error (ISE). Thus, the proposed multiobjective optimization problem is formulated in three dimensions, where the controller parameters j, kn and ko are the decision variables (=, = and =B, respectively) and IAE, ITAE and ISE are the pursued objectives ( , and B , respectively). In Example 2, the particle swarm optimization was performed using 1000 particles and 5 iterations and the decision making step included 100 perturbations for each point of the Pareto set. It took approximately 60 minutes to solve this problem on an Intel Core 2 Duo PC (3.00 GHz, 8 Gb RAM). This example compares the performances of different tuning methods of PID controllers with the proposed methodology. As the analyzed objective functions do not present a clear physical meaning, accepted variances of , and B were assumed to be equal to 2.8, 53.5 and 1.3, respectively, leading to similar contributions to calculation of . Similarly, absolute errors of jU and kU were assumed to be equal to 0.2 and 4, respectively. ISE tuning, computed for comparison, was obtained from heuristic optimization using swarm algorithm for a servo control scenario. A Pareto set of around 30 non-dominated solutions was obtained in this example and their values varied from approximately 57 until 1847. This example was performed several times and the final results were very similar. Tables 2 and 3 and Figures 3 and 4 summarize the results obtained in Example 4, when a unit step is introduced into the set point of the controlled variable, with different PID controller tunings. The system behavior can be observed when the real process is identical to the reference model used to tune the controller parameters (Table 2, Figure 3) and when it is different from the model used during the tuning step (Table 3, Figure 4). In this last case, the process is represented by Equation (6), while the model used for tuning the controller parameters is presented in Equation (5). For simplicity, the least sensitive result obtained by the proposed methodology was called Multi 1 and the most sensitive one was called Multi 2.

iU > =

1.2 ) _Nm 1 + 24>

(6)

Table 2. Parameters and performance criteria of different tuning methods, without model mismatch. X1 X2 X3 f1 f2 f3 IAE ITAE ISE j kn ko ZN ISE IMC Multi 1 (least sensitive)

4.059 4.464 3.618

2.279 6.845 6.200

9.386 10.37 7.895

13.03 9.469 9.408

152.5 90.83 98.01

8.269 6.710 6.915

3.376

8.052

5.301

8.573

54.07

7.034



Multi 2 (most sensitive)

4.346

6.711

8.597

9.092

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75.27

6.723

Table 3. Parameters and performance criteria of different tuning methods, with model mismatch. X1 X2 X3 f1 f2 f3 IAE ITAE ISE j kn ko ZN ISE IMC Multi 1 (least sensitive) Multi 2 (most sensitive)

4.059 4.464 3.618

2.279 6.845 6.200

9.386 10.37 7.895

13.66 10.05 10.33

168.1 117.6 127.9

8.625 6.767 6.983

3.376

8.052

5.301

9.316

83.63

7.049

4.346

6.711

8.597

9.742

102.6

6.79

According to Tables 2 and 3, the least sensitive solution from the Pareto set is characterized by low IAE and ITAE values, without great loss of the ISE value. On the other hand, although the most sensitive solution presents the smallest ISE value, justifying its presence in the Pareto optimal set, it leads to high IAE and ITAE values, when compared with the least sensitive solution. The results presented in Tables 2 and 3 and in Figures 3 and 4 make clear that the use of the least sensitive solution obtained with the proposed methodology leads to the best performance, when compared to the performances obtained after use of the remaining tuning methods. With the least sensitive solution, the controlled variable reaches its new set point with small overshoot and short stabilization time. Analysis of responses obtained with solutions Multi 1 and Multi 2 clearly show that different solutions in the Pareto set can lead to very different performances, especially in the presence of model mismatch.

Figure 3. Control system performance in Example 4 without model mismatch. SP (---), ZN ( ), ISE ( ), IMC ( ), Multi 1 ( ), Multi 2 ( )


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Figure 4. Control system performance in Example 4 with model mismatch. SP (---), ZN ( ), ISE ( ), IMC ( ), Multi 1 ( ), Multi 2 ( ) In the present example, the proposed method was able to distinguish the solutions in the Pareto set, indicating that the least sensitive solution was indeed able to provide the best control performance. Therefore, it seems clear that it is very important to rely on a proper decision making criterion to adequately distinguish the quality of the solutions of the Pareto set.

4.3. Example 3 - A MIMO Controller Tuning Problem Example 3 proposes the multiobjective formulation of a MIMO controller tuning problem, using a multivariable predictive controller as illustration. The multivariable predictive control (MPC) comprises a class of algorithms that generate a sequence of actions for the manipulated variables, aiming to optimize the future behavior of a process, as predicted by a model.62 Although the multivariable predictive control had been originally developed to satisfy the requirements of energy generation and petroleum refining segments, this technique has been widely used in different industrial areas, such as the chemical, food, automotive and aerospace fields.63 Multivariable predictive control applications involve basically two steps. The first step consists in the identification of a mathematical model that is able to represent the process, while the second step comprises the tuning of an appropriate control algorithm that must be used to optimize the future behavior of the process, using the identified model as reference.64 Therefore, if a linear model is used as reference, control actions can be calculated to optimize the process performance as described in Equation (7):

Problem P1

J min rst . $ + 1 − tu$ + 1 v ∙ Γ ∙ st . $ + 1 − tu$ + 1 v + ∆x J $ ∙ Λ ∙ ∆x$ z

∆q

| $

tu$ + 1 = tu ∗ $ + 1 + > ∙ ∆x$ +

subject to

x }~ ≤ x$ ≤ x } ∆x }~ ≤ ∆x$ ≤ ∆x } t }~ ≤ tu $ + 1 ≤ t }

(7)



Page 14 of 26

where t . contains the reference trajectory for controlled variables during the prediction horizon (HP), tu is the model predicted trajectory for controlled variables, tu ∗ are the predicted values for open loop controlled variables, ∆x are the changes applied to the manipulated variables during the control horizon (HC), > represents the process dynamic

K./L/ ./ | $ = t L" matrix and − tu L" is a deviation that aims to correct model mismatches caused by dynamic disturbances that are not considered by the model or sources of noise associated with the instrumentation system. Γ and Λ are weight matrices

for controlled and manipulated variables, represented here in the form of Equation (8). For matrix Λ, the weight factors are called suppression factors.

Γ 0, × , Γ= ⋮ 0 , × ,

Λ 0, × , Λ= ⋮ 0 , × ,

0, × , Γ

⋮ 0, × ,

0, × , Λ

⋮ 0, × ,

⋯

0, × ,

⋯

0, × ,

⋯

⋱ ⋯

⋯ ⋱ ⋯

γ 0, × , 0 Γ = ⋮ ⋮ 0 ΓU

λ 0, × , 0 Λ = ⋮ ⋮ 0 Λ_

0 ⋮ 0

0 ⋮ 0

⋯ ⋯ ⋱ ⋯

⋯ ⋯ ⋱ ⋯

0 0 ⋮ , = 1, … , V4 ,

0 0 , = 0, … , V − 1 ⋮ ,

(8)

At each sampling time k, the optimization problem defined in Equation (7) must be solved. Although the MPC calculates a set of M actions ∆x$ during the control horizon, only the first control action ∆$ is in fact implemented. It must be noted that the objective function that must be minimized involves simultaneously deviations from set point values and the magnitudes of the manipulated variables. Thus, both Γ and Λ exert great influence on the controller results. Detailed description of theoretical and implementation issues related to multivariable predictive controllers can be found in standard textbooks.65 For proper tuning of the multivariable predictive controller, it is necessary to arbitrate some controller parameters, including weights Γ and Λ. This can pose some practical implementation problems, as thoroughly discussed in this field.66-69 In order to tackle this problem, the proposed multiobjective optimization methodology can be used to design the weights Γ and Λ with help of Equation (9).

Problem P2

min , , … , , , , , … , , ¡ ,

(9)

subject to


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=

=

£,"

. 2 , − ¢, 5 , = 1,2, … , %

0S £,"

0S

2Δ, 5 , = 1,2, … , %

. where * %': represents the simulation time (control horizon), , is the reference value for controlled variable at time , ¢, is the predicted value for controlled variable at . time , Δ, is the movement value for manipulated variable at time . The , , ¢, and Δ, values can be obtained as responses at the end of Problem P1, when all implemented movements and controlled variables are calculated as functions of the controller parameters. Thus, during the resolution of Problem P2 (external layer), Problem P1 must be solved several times (internal layer). Problem P2 is solved here in accordance with the proposed methodology, while Problem P1 is solved with a standard quadratic Newton algorithm. The process analyzed here is a Shell Benchmark, comprising a heavy oil distillation column, as described by Prett and Morari70 and realized by Vlachos et al.71, Vlachos et al.72 and Chiou and Zafiriou73. The fundamental problem is composed by two controlled variables (top and side compositions) and two manipulated variables (top and side output flowrates). The 2 x 2 process model can be represented by Equation (10), while the plant can be represented by Equation (11), both in the Laplace domain:

4.05 ) _Pm ; > ; >

1 ; = ¥ ¦ = 50> +_Qm ; > ; >

5.39 ) 50> + 1

1.77 ) _Qm 60> + 1 5.72 ) _Mm 60> + 1

4.05 + 2.11∆ ) _Pm 4 > 4 >

50> + 1 4 = ¥ ¦= 4 > 4 >

5.39 + 3.29∆ ) _Qm 50> + 1

1.77 + 0.39∆ ) _Qm 60> + 1 5.72 + 0.57∆ ) _Mm 60> + 1

(10)

(11)

The difference between plant and model reflects the model uncertainties and is important for proper tuning of the controller performance. In Equation (11), −1 ≤ ∆ ≤ 1 $ = 1, 2 . The adopted constraints for manipulated variables, controlled variables and control actions are shown in Equation (12). In this example, the control horizon (1), the prediction horizon (30) and the sampling period (5) were kept constant and neglected as possible tuning parameters.



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0 ≤ ≤ 200 = 1, 2

0 ≤ ≤ 200 = 1, 2

−200 ≤ ∆ ≤ 200 = 1, 2

(12)

In this example, 2000 particles and 5 iterations were used to obtain the Pareto set through particle swarm optimization. Then, each point of the Pareto set was subjected to 10 perturbations in order to perform the decision making step. Unlike the previous examples, these numbers allow only for fair exploration of the search region and were smaller than used previously because of the higher computational costs of individual iterations. In spite of that, as shown in the next section, obtained numerical results can be regarded as appropriate. It took approximately 300 minutes to solve this problem on an Intel Core 2 Duo PC (3.00 GHz, 8 Gb RAM). According to the proposed formulation, four parameters must be estimated: and in Γ; and in Λ. Variance values were considered equal to 1 for all objective functions and the model mismatch was defined in accordance with Equation (11). The ranges in which the optimizing variables have been disturbed were equal to ± 0.1 for Γ and ± 1 for Λ. Table 4 shows the obtained tuning parameters and the objective function values for some solutions of the Pareto set, ranked in terms of the sensitivity criterion.

fy1

fy2

fu1

fu2

1 (least sensitive)

4.134

5.000

0.6577

0.2856

236.3

748.3

2.592

4.831

4.39E+01

16

0.5919

4.269

0.4029

0.3127

249.5

628.9

2.641

6.444

5.43E+01

95

4.509

3.405

0.5585

0.4611

235.7

553.2

3.701

10.69

7.03E+01

1001

4.807

4.475

0.4427

0.4380

286.2

597.8

1.985

6.080

1.23E+02

3050

1.592

2.061

0.2411

0.7934

202.7

489.0

7.060

18.97

2.40E+02

6400

0.0758

0.6088

1.330

4.411

160.9

467.6

39.98

72.25

6.55E+02

9900

4.599

3.734

0.1162

0.0111

336.4

25305.6

0.1039

0.0345

1.00E+03

10560

4.157

4.677

0.0258

0.0446

6549.4

1738.9

0.0352

0.0426

1.65E+03

10845

3.801

3.639

0.2296

1.767

497.19

467.6

6.994

16.36

1.06E+04

10950

0.3812

3.122

0.1683

1.028

320.8

483.0

7.718

13.23

1.05E+05

11190

3.198

1.699

3.792

0.0356

192.8

2262.1

7.130

15.50

5.64E+05

11355

4.458

3.912

0.0335

1.858

8999.8

457.3

5.820

12.90

1.26E+06

11940 (most sensitive)

3.459

1.148

0.0531

3.739

82995.3

259.8

153.4

197.09

3.18E+06

Table 4. Some solutions of the Pareto set in Example 3.

Ranking


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Table 4 shows that solutions in the Pareto set containing very different tuning parameters can present very similar sensitivity indexes. This is the case of the solutions classified as 1, 16 and 95. Considering the high values that D k index can assume in this problem (3.18·106) and the large number of solutions that belong to the Pareto set (11940), it can be said that the sensitivity indexes of solutions 1, 16 and 95 are very similar, although the estimated values for , , and parameters are very different. This illustrates the difficulty to estimate these tuning parameters appropriately. Another interesting aspect is the large observed range of values in the Pareto set, which emphasizes the very different sensitivity characteristics of the solutions. When a set point change is imposed on the controlled variables (decreasing from 100 to 95 for y1 and increasing from 100 to 110 for y2), the multivariable predictive controller was tuned with the least sensitive solution obtained with the proposed methodology. The results were compared with the ones obtained when all parameters were assumed to be equal to 1 (Figure 5) and with the ones obtained with the most sensitive solution (Figure 6). Figure 5 shows that the proposed methodology was efficient and able to select a proper tuning solution from the Pareto set for real implementation. Particularly, the least sensitive solution led to smooth transitions to the new set points, as desired. Figure 6 illustrates the inadequate behavior obtained when the tuning parameters were selected from the most sensitive solution of the Pareto set. Once more, the proposed method was able to distinguish the solutions of the Pareto set, indicating that the least sensitive solution was indeed able to provide the best control performance. Figures 7 and 8 show similar results in presence of model mismatch (Δ = 1 and Δ = −1), where it becomes even more evident that the least sensitive solution leads to better controller performance than the other solutions, with smooth transitions to the new set points, as desired. Figure 8 illustrates the very poor performance of the most sensitive solution of the Pareto set, making clear that the proposed method was able to distinguish the solutions of the Pareto set and indicating again that the least sensitive solution was indeed able to provide the best control performance. In order to illustrate the controller tuning degradation as the sensitivity index decreases, Figures 9 and 10 show the controller performances for different solutions of the Pareto set. Figures 9 and 10 confirm the superiority of the least sensitive solutions when compared to the most sensitive solutions, according to the decision making criterion defined in the present work. This is particularly true when model mismatch is present, as one might already expect.



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Figure 5. Performance of the multivariable predictive controller in Example 3. Initial tuning with all parameters equal to 1 ( ) and least sensitive tuning obtained with the proposed methodology ( --- ), without model mismatch.

Figure 6. Performance of the multivariable predictive controller in Example 3. Most 18 ACS Paragon Plus Environment

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sensitive tuning obtained with the proposed methodology ( mismatch.

), without model

Figure 7. Performance of the multivariable predictive controller in Example 3. Initial tuning with all parameters equal to 1 ( ) and least sensitive tuning obtained with the proposed methodology ( --- ), with model mismatch.



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Figure 8. Performance of the multivariable predictive controller in Example 3. Most sensitive tuning obtained with the proposed methodology ( ), with model mismatch.

Figure 9. Performance of the multivariable predictive controller in Example 3 with different tuning parameters in the Pareto set, without model mismatch. (---) Ranking 1, (---) Ranking 16, (---) Ranking 1001, (---) Ranking 9900, (---) Ranking 10845, (---) Ranking 11190. 20 ACS Paragon Plus Environment

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Figure 10. Performance of the multivariable predictive controller in Example 3 with different tuning parameters in the Pareto set, with model mismatch. (---) Ranking 1, (---) Ranking 16, (---) Ranking 1001, (---) Ranking 9900, (---) Ranking 10845, (---) Ranking 11190.

5. Conclusions A new decision making criterion for analysis of multiobjective optimization problems was proposed, based on the evaluation of the sensitivity of the optimum solutions in the Pareto set. This new technique allows for discrimination among the solutions that constitute the Pareto set, so that a single alternative (the least sensitive one) must be selected for real implementation, minimizing the sensitivity of the expected performance to undesired (but unavoidable) fluctuations of the optimizing variables and making the final selection of the optimal solution less dependent on the decision maker. In order to do that, the effects of process uncertainties on the variable responses are analyzed individually, with the help of the process model, and used to compute a sensitivity index. The proposed technique was applied to three distinct problems and the obtained results showed that the method is able to discriminate the solutions successfully, allowing for appropriate selection of the best candidates for real implementation among the many candidates in the Pareto set. Particularly, the proposed method was able to provide guidelines for preparation of polymer materials, when applied to a multiobjective polymer design problem, and to provide well-tuned process controllers, when applied to multiobjective controller designs.



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6. Acknowledgments The authors thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brasil), FAPERJ (Fundação Carlos Chagas Filho de Apoio à Pesquisa do Estado do Rio de Janeiro) and PETROBRAS for scholarships and financial support.

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New Decision Making Criterion for Multiobjective Optimization

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