Fuzzy multivariate mean square error in equispaced pareto frontiers considering manufacturing process optimization problems

Abstract

This paper proposes a combined approach using the normal boundary intersection (NBI) and multivariate mean square error (MMSE) that is an alternative approach to outperform the traditional NBI driving to an equispaced Pareto Frontier in a low-dimension space with a considerable reduction in the number of iterations. The method participating in the evolutionary stage of creating a uniformly spread Pareto Frontier for a nonlinear multi-objective problem is the NBI using normalized objective functions allied to MMSE. In sequence, the fuzzy MMSE approach is utilized to determine the optimal point of the multi-objective optimization. For sake of comparison, the performance of arc homotopy length, global criterion method, and weighted sums were explored. To illustrate this proposal, a multivariate case of AISI H13 hardened steel-turning process is used. Experimental results indicate that the solution found by NBI-MMSE approach is a more appropriate Pareto frontier that surpassed all the competitors and also provides the best-compromised solution to set the machine input parameters. Further, this algorithm was also tested in benchmark functions to confirm the NBI-MMSE efficiency.

Appendix A: applications of other optimization methods

1.1 Weighted sum (WSUM)

This approach is tested since the weighted sum is perhaps the most natural approach to the MOP which minimizes a convex combination of the different objectives [47]; but at the same time, has serious drawbacks such as its inability to find an equispaced Pareto frontier. Thereby, the best algorithm of optimization which results in the Pareto frontier shown in Fig. 9 will be reconstructed to check the robustness of the algorithm optimization proposed. Essentially, different weights are attributed in the scalarized Eqs. (25) and (26) to obtain a single objective optimization as represented by Eq. (39).

$$\begin{gathered} {\text{Minimize}}\;\sum\limits_{{i=1}}^{n} {{w_i}{{\bar {f}}_1}(x)+(1 - } \,{w_i}){{\bar {f}}_2}(x) \hfill \\ {\text{subject}}\;{\text{to}}:\;x \in \Omega , \hfill \\ \end{gathered}$$

(39)

where ${w_i} \geq 0$and $\mathop \sum \nolimits_{{i=1}}^{n} {w_i}=0$. In sequence, the GRG algorithm is applied considering the objective functions in terms of MMSE and the experimental region constraint was the same as used in the NBI-MMSE approach of optimization. As a result, 21 optimum points are obtained as shown in Table 18 with the subsequent determination of the optimal point through the fuzzy decision maker.

1.2 Global criterion method (GCM)

Although the weighted sum method was popular due to its simplicity in implementation and usage to solve multi-objective problems, it could not reach the non-convex parts of the Pareto set. Therefore, as it is not known if the Pareto front a priori will be convex, the GCM is adopted which could reach the non-convex parts of the Pareto [48].

In this method, the distance between some reference points and the feasible objective regions is minimized. The decision maker has to select the reference point and the metric for measuring the distances [49]. In this way, the particular global criterion employed in this study was:

$$\begin{gathered} \mathop {{\text{Min}}}\limits_{{{\text{X}} \in \Omega }} {\text{ }}\bar {f}\left( {\mathbf{x}} \right)=\sum\limits_{{i=1}}^{m} {{w_i}} {\left[ {\frac{{{f_i}\left( {\mathbf{x}} \right) - {f^U}}}{{f_{i}^{U}}}} \right]^2}. \hfill \\ {\text{S.t.}}:{\text{ }}{{\mathbf{x}}^{\mathbf{T}}}{\mathbf{x}} \leq {\rho ^2} \hfill \\ \end{gathered}$$

(40)

The next step is to conduct the GRG algorithm in Eq. (40) using the same objective functions in terms of MMSE and the experimental region constraint of NBI-MMSE approach of optimization. As a result, 21 optimum points are obtained as shown in Table 19 with the subsequent determination of the optimal point through the fuzzy decision maker.

1.3 Arc homotopic length (AHL)

The algorithm is based on convex combinations of the objectives and homotopic continuation. The scalar objective function is introduced as

$$f(x,w)=(1 - w){f_1}(x)+w{f_2}(x),$$

(41)

where 0 ≤ w ≤ 1, and the same problem constraints apply. Standard homotopy starts at w = 0 and then steps w in some fashion, solving the successive problems to obtain a discrete sampling of the Pareto set and the Pareto front. A frequent criticism of this method is that there is no assured way to obtain a uniform sampling with it. because the parametrization of the Pareto front by w is usually a very nonlinear unknown map [50]. In the current method, the idea of intrinsic parametrization of the Pareto front using a discrete arc length was used.

Given a collection of points f_k = f(x_k), k = 0, 1, …, l + 1, the chord length of the polygon determined by these points was defined as ${S_l}=\sum\nolimits_{{k=0}}^{l} {\left\| {{f_{k+1}} - {f_k}} \right\|}$. Thus:

$$\gamma =\alpha \left\| {f({x_0}) - f({x_{l+1}})} \right\|/l$$

(42)

where the distance between the images of the minimizers times a factor α > 1 is an estimate of the total chord length of the Pareto front. Otherwise, γ ≠ 0 and it imposes the following equispaced constraint

$${\left\| {f(x) - {f_{{\text{prev}}}}} \right\|^2}={\gamma ^2},$$

(43)

where f_prev is a previous point in the homotopy process. Then, the AHL formulation can be stated as the following matrix resolution where it minimizes the scalarized function, subject to all constraints with w as an additional variable.

$$\begin{gathered} \mathop {{\text{Minimize}}}\limits_{{(x,w)}} :\quad (1 - w){f_1}(x)+w{f_2}(x) \hfill \\ {\text{Subject}}\;\,{\text{to}}:\quad g({\mathbf{x}}) \leq 0;\quad x \in D;\quad 0 \leq w \leq 1 \hfill \\ \quad \quad \;\quad \quad \quad {\left\| {f(x) - {f_{{\text{prev}}}}} \right\|^2}={\gamma ^2} \hfill \\ \end{gathered}$$

(44)

Next, the GRG algorithm is conducted in the system of Eq. (44) using the same objective functions in terms of MMSE. As a result, Table 20 shows the 21 optimum points with the subsequent determination of the optimal point through the fuzzy decision maker.