Issue 
Int. J. Simul. Multidisci. Des. Optim.
Volume 13, 2022



Article Number  4  
Number of page(s)  6  
DOI  https://doi.org/10.1051/smdo/2021037  
Published online  06 January 2022 
Research Article
New multiobjective optimization algorithm using NBISASP approaches for mechanical structural problems
^{1}
LIMSAD, Department of Mathematics and computing, Hassan 2 University, Ain Chock Sciences Faculty, Km8 Route El Jadida, BOP 5366 Maarif, Casablanca, Morocco
^{2}
LM, Department of Physics, Hassan 2 University, Ain Chock Sciences Faculty, Km8 Route El Jadida, BOP 5366 Maarif, Casablanca, Morocco
^{*} email: selmoumen@yahoo.fr
Received:
26
January
2021
Accepted:
7
November
2021
Various engineering design problems are formulated as constrained multiobjective optimization problems. One of the relevant and popular methods that deals with these problems is the weighted method. However, the major inconvenience with its application is that it does not yield a well distributed set. In this study, the use of the Normal Boundary Intersection approach (NBI) is proposed, which is effective in obtaining an evenly distributed set of points in the Pareto set. Given an evenly distributed set of weights, it can be strictly shown that this approach is absolutely independent of the relative scales of the functions. Moreover, in order to ensure the convergence to the Global Pareto frontier, NBI approach has to be aligned with a global optimization method. Thus, the following paper suggests NBISimulated Annealing Simultaneous Perturbation method (NBISASP) as a new method for multiobjective optimization problems. The study shall test also the applicability of the NBISASP approach using different engineering multiobjective optimization problems and the findings shall be compared to a method of reference (NSGA). Results clearly demonstrate that the suggested method is more efficient when it comes to search ability and it provides a well distributed global Pareto Front.
Key words: Global optimization / hybrid method / simultaneous perturbation / simulated annealing / multiobjective optimization / normal boundary intersection approach / pareto front
© S. El Moumen and S. Ouhimmou, Published by EDP Sciences, 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Optimization in engineering has become a vital component with the growth of the capabilities of computers nowadays. In today's time, complicated calculations can be done in a very short time compared to years ago. Consequently, the use of numerical optimization has witnessed a dramatic increase. A significant amount of designing is usually performed intuitively. However, technical analyses involving numerical optimization can contribute a great deal to the amelioration of the designs see [1–4].
What characterizes any problem of engineering design is the presence of various objectives [5,6]. When all these objectives are taken into account, their conflicting nature finds it challenging to find an optimal solution to the problem. A set of solutions in which improving one objective can deteriorate the other ones is referred to as the Pareto set of nondominated solutions and approximating it contributes to understanding it in depth as well as assisting the process of decisionmaking. Two main methods for searching for the Pareto Front have been introduced in the last two decades: weighted method [7,8] and ϵconstraint method [7,8], Ehrgott and Gandibleux (2002), [9–12]. The issue with the Weighted method is that it does not have the ability to solve nonconvex objective space and it does not yield a welldistributed set. On the other hand, the ϵconstraint method deals with nonconvex space, but still does not yield a welldistributed set as well. This paper purports to propose the use of the Normal Boundary Intersection approach (NBI) [13] that allows us to handle nonconvex objective space and yields a well distributed set. In order to establish a transition to the global Pareto frontier, this paper is set to combine the NBI approach with a proposed global optimization method entitled Simulated Annealing Simultaneous Perturbation method (SASP). This method involves hybridization of the Simulated Annealing method and a moderated method where the gradient is estimated by using the simultaneous perturbation. This paper endeavors to suggest NBISimulated Annealing Simultaneous Perturbation method (NBISASP) to ameliorate the efficiency of the optimization algorithms. Furthermore, despite the achievement of some improvements with regards to proposed algorithms of multiobjective design optimization, the complicated nature of design problems with conflicting objectives, wide solution space, and premature convergence to local optimum is considered as a drawback for the solution and computational cost of multiobjective design optimization problems. In order to compare the results, NSGAII shall be used as a method of references [6,8]. Then, a simulation of the results on various difficult test problems shall reveal that NBISASP outperforms NSGAII with regards to detecting a diverse set of solutions and converging near the true Paretooptimal set.
2 Multiobjective optimization problem
A multiobjective optimization problem is defined as follows:where f_{1}(x), f_{2}(x), ..., f_{k}(x) are the k objectives functions, x_{1}, x_{2}, ..., x_{n} are the n optimization parameters, and S ∈ ℝ^{n} is the solution or parameter space.
will be used to denote the individual minima of each respective objective function, and the utopian solution is defined as . As F^{*} simultaneously minimizes all objectives, it is an ideal solution that is rarely feasible. Figure 1 provides a visualization of the nomenclature.
Note that because F(x) is a vector, if the components of F(x) are involved in a competition, no unique solution can be found to this problem. Nevertheless, the concept of noninferiority (also called Pareto optimality) is necessary to characterize the objectives. A solution where improving one objective results in the deterioration of another is called a noninferiority solution. Obviously, a final design solution is preferred to be a member of the Pareto optimal set. Therefore, Pareto optimal solutions are also called nondominated or efficient solutions. Once a final solution is chosen from the set of Pareto optimal solutions, no other existing solution shall be better in all attributes. However, in case the solution does not belong to the Pareto optimal set, improvements can be made without deteriorating any objective, and, therefore, it is not a rational choice.
Fig. 1 Two dimensional problem with two objectives. 
3 The NBISASP method
3.1 Normal boundary intersection (NBI)
Normal Boundary Intersection is a method developed in 1998 by Das and Dennis. It aims to identify the Pareto front for a multiobjective optimization problem, it is proven that this method has succeeded in producing a uniform set of Pareto front points, and this gives NBI an advantage over the other methods used before, weighting method and ϵconstrain method.
Let and denote respectively the minimizer and minimum value of the f_{i} and let F^{*} denote the shadow minimum, i.e., the vector whose components are . Consider the shifted payoff matrix Φ whose i^{t}h column is . The Convex Hull of Individual Minima or CHIM is defined as the set of points that are convex combinations of the columns of Φ, i.e., {Φβ : β_{i} ≥ 0, ∑ _{i}β_{i} = 1 }.
For a two dimensional problem illustrated in Figure 2, CHIM is represented by segment AB.
The main purpose of NBI is to select an even spread of points on the CHIM (for example W in 2), and find the intersection point between the efficient frontier and a set of parallel normals resulting from the chosen set of points on the CHIM (C in 2). Taking into consideration a convex combination parameter vector β, and a normal direction n that points towards the origin, the point of intersection between the normal emanating from Φβ and the efficient frontier can be found by solving the following NBI_{β} subproblem:(1)where A is the set of feasible solution.
A solution to the subproblem NBI_{β} for different settings of β can generate various points on the efficient frontier. The β parameter's advantage is that an even spread of β parameters is related to an even spread of points on the CHIM. Thus, in order to ensure the convergence to the global Pareto frontier, a marriage of the NBI approach with a global optimization method is inevitable Figure 3.
Fig. 2 An illustrative integrated design. 
Fig. 3 The SASP organizational chart. 
Fig. 4 Flow chart of the NBI algorithm for generating Pareto optimal sets. 
3.2 The global optimization method (SASP)
The SASP method offers a good compromise between exploration and exploitation, it is based on two main ideas, the first idea comes from the fact that the simulated annealing algorithm e.g. [14–18], Kirkpatrick et al. (1983), [18] can easily escape from a local optimum, and the second idea is that gradient descent methods converge rapidly towards the local minimum.
This method presents two phases that are executed alternately. In the first phase a local search is run using a descent method where the gradient is estimated by using the simultaneous perturbation [19], from a starting point . It will converge (presumably) to a minimum (local) . And to deviate from this local solution, the second phase runs using some iterations of the simulated annealing algorithm to find the next best point. Then we take this point as a starting point and start a new search with the gradient method. This procedure continues until convergence and finally we find the overall solution [15].
3.3 The NBISASP algorithm
Flow chart of the NBI algorithm for generating Pareto optimal sets is briefly described in the following Figure 4.
4 Numerical example
In this section, we shall introduce a solution for three problems in structural mechanics [20–26]. The application of these solutions will be done using both methods. First using the NBISASP approach, and then a close comparison to the NSGA reference approach will be made to check which one will have more efficiency.
4.1 The two bar truss design problem
As can be seen in Figure 5, the truss carries a specific load without elastic failure. Therefore, the design objectives involve first and primarily reducing the volume of the structure (which equals less fabrication costs). Secondly, the next step is to reduce or minimize the stresses in each of the two members AC and BC. For this reason, any design problem involves a twoobjective optimization problem for three variables: vertical distance y between B and C (metres), length x_{1} of AC (metres), and length x_{2} of BC in (metres).
And the mathematical description of the problem is as follows(2)
Fig. 5 The twobar truss design problem. 
Fig. 6 The IBeam design example. 
Fig. 7 The disc brake design optimization problem. 
4.2 The IBeam design problem
The purpose of multiobjective optimization for the I Beam design (see Fig. 6), is to find the optimal compromise between the dimensions of the concrete IBeam and geometric and strength constraints [27–29]. The two objective functions are simultaneous minimization of f_{1} cross sectional area of the IBeam, and f_{2} the static deflection of the IBeam taking into consideration the orthogonal and cross sectional forces P = 600KN and Q = 50kN.
The problem definition is written below(3)
4.3 The disc brake design problem
One of the pertinent structural multiobjective optimization problems is the optimization problem of the disk brake design. It is mentioned here to exemplify the effects of the algorithm NBISASP compared to other algorithms without improvements for solving multiobjective design optimization problems such as NSGAII.
The design has as an objective to minimize the mass of the brake as well as the stopping time [5,30,31] The four variables are defined for the disc brake mathematical optimization model as x_{1}, inner radius of the discs; x_{2}, outer radius of the discs; x_{3}, engaging force; and x_{4}, number of the friction surface Figure 7. The objective functions and constraints of the disc brake design optimization model are:(4)
5 Results
In the tree problem, the nonlinear constraints can cause difficulties in finding the Pareto solutions. As shown in Figures 8–10 it is clearly that the NBI, based on the hybrid method, have exclude the nonPareto and local Pareto points, compared with the result obtained by the NSGAII algorithm. Also all the figure can be shown that our approach guarantees uniform distribution of the Pareto solutions.
Taking into account the whole analysis, it can be deduced that the NBI SPSA solutions are better then NSGAII with regards to both the closeness to the true optimum and their spread for all test problems employed in the study. The NBI SPSA does not face any challenges in order to reach an effective spread of Paretooptimal solutions for constrained multiobjective optimization. The results yielded for engineering design problems sufficiently show that the NBI SPSA method can come up with efficient, uniformly distributed, nearcomplete and near optimal Paretooptimal solutions for multiobjective optimization.
Fig. 8 The solutions obtained by NBISASP and NSGAII on twobar truss design problem. 
Fig. 9 The solutions obtained by NBISASP and NSGAII on beam design example. 
Fig. 10 The solutions obtained by NBISASP and NSGAII on beam design example. 
6 Conclusions
This study has proposed the Normal Boundary Intersection approach (NBI) based on the SASP method to solve multiobjective design problems. This method is capable of dealing with nonconvex objective space, as well as it yields a well distributed Pareto front. The resulting solution given by the suggested approach for the two bar truss design problem, the IBeam design problem and the disk brake design problem is favorable in contrast to the result given by NSGAII. Furthermore, the aforementioned results propose that the system of optimization suggested is more efficient and practical for solving realworld application problems. For upcoming research, more practical tests shall be done on the algorithm to show its application. Thus, future modifications and improvements of the algorithm are still possible.
References
 D. Bassir, F.X. Irisarri, J.F. Maire, N. Carrere, Incorporating industrial constraints for multiobjective optimization of composite laminates using a GA, Int. J. Simul. Multidisci. Des. Optim. 2, 101–106 (2008) [CrossRef] [EDP Sciences] [Google Scholar]
 K. Deb, Current trends in evolutionary multiobjective optimization, Int. J. Simul. Multidisci. Des. Optim. 1, 1–8 (2007) [CrossRef] [EDP Sciences] [Google Scholar]
 R. El Maani, S. Elouardi, B. Radi, A. El Hami, Multiobjective aerodynamic shape optimization of NACA0012 airfoil based mesh morphing, Int. J. Simul. Multidisci. Des. Optim. 11, 1–10 (2020) [CrossRef] [EDP Sciences] [Google Scholar]
 A. Tchvagha Zeine, N. El hami, S. Ouhimmou, R. Ellaia, A. Elhami, Multiobjective optimization of trusses using Backtracking Search Algorithm, Incertitudes et fiabilité des systèmes multiphysiques 1, 1–10 (2017) [Google Scholar]
 M. Duran Toksari, A heuristic approach to find the global optimum of function, J. Comput. Appl. Math. 209, 160–166 (2007) [CrossRef] [MathSciNet] [Google Scholar]
 M. Ehrgott, X. Gandibleux, Multiobjective combinatorial optimization–theory, methodology and applications, in Multiple Criteria Optimization–State of the Art Annotated Bibliographic Surveys, edited by M. Ehrgott and X. Gandibleux. sInternational Series in Operations Research and Management Science (Springer, Boston, MA, 2003), vol 52, pp. 369–444 [Google Scholar]
 V. Chankong, Y.Y. Haimes, Multiobjective Decision Making Theory and Methodology (NorthHolland, New York, 1983) [Google Scholar]
 K. Miettinen, Nonlinear Multiobjective Optimization ( Kluwer, Boston, 1999) [Google Scholar]
 D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (AddisonWesley Pub. Co., 1989) [Google Scholar]
 C.R. Houck, J. Joines, M. Kay, A genetic algorithm for function optimization: A matlab implementation, Technical Report NCSUIETR9509, North Carolina State University, Raleigh, NC, 1995 [Google Scholar]
 R.E. de Castro, A Genetic Algorithm for Multiobjective Structural Optimization”, IV Simposio Mineiro de Mecanica Computacional (2000) 219–226 [Google Scholar]
 I.G. Tsoulos, Modifications of real code genetic algorithm for global optimization, Appl. Math. Comput. 203, 598–607 (2008) [Google Scholar]
 I. Das, J.E. Dennis, Normal boundary intersection, a new methode for generating the pareto surface in nonlinear multicreteria optimization problems, SIAM J. Optim. 3, 631–657 (1998) [CrossRef] [MathSciNet] [Google Scholar]
 R. Aboulaich, R. Ellaia, S. El Moumen, The meanvarianceCVaR model for portfolio optimization Modeling using a MultiObjective Approach based on a hybrid method, Math. Model. Nat. Phenom. 7, 93–98 (2010) [Google Scholar]
 S. El Moumen, R. Ellaia, R. Aboulaich, A new hybrid method for solving global optimization problem, Appl. Math. Comput. 218, 3265–3276 (2011) [Google Scholar]
 S. Kirpatrick, C.D. Gelatt, M.P. Vecchi, Optimization by simulated annealing, Science 220, 671–680 (1983) [CrossRef] [MathSciNet] [Google Scholar]
 N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, Equations of state calculations by fast computing machines, J. Chem. Phys. 21, 1087–1091 (1953) [CrossRef] [Google Scholar]
 C.R. Reeves, Modern Heuristic Techniques for Combinatorial Problems (John Wiley and Sons, New York, NY, 1993) [Google Scholar]
 J.C. Spall, Multivariate stochastic approximation using a simultaneous perturbation gradient approximation, IEEE Trans. Autom. Control 37, 332–341 (1992) [CrossRef] [Google Scholar]
 B. Azvine, G.R. Tomlinson, R. Wynne, Use of active constrainedlayer damping for controlling resonant vibration, Smart Mater. Struct. 4, 1–6 (1995) [CrossRef] [Google Scholar]
 M.J. Lam, D.J. Inman, W.R. Saunders, Variations of hybrid damping, in L.P. Davis (ed.), Smart Structures & Materials 1998: Passive Damping and Isolation, edited by L.P. Davis (SPIE, Bellingham, USA, 1998), Vol. 3327, pp. 32–43 [CrossRef] [Google Scholar]
 S. El Moumen, R. Ellaia, R. Aboulaich, New hybrid algorithm for multiobjective structural optimization, in Proceedings of2013 International Conference on Industrial Engineering and Systems Management (IESM), (2013), pp. 1–5 [Google Scholar]
 Q. Yuan, Z. He, H. Leng, A hybrid genetic algorithm for a class of global optimization problems with box constraints, Appl. Math. Comput. 197, 924–929 (2008) [MathSciNet] [Google Scholar]
 J. Zhang et al., An effective multiagent evolutionary algorithm integrating a novel roulette inversion operator for engineering optimization, Appl. Math. Comput. 211, 392–416 (2009) [MathSciNet] [Google Scholar]
 J. Schuurmans, J.A. Rossiter, Robust predictive control using tight sets of predicted states, IEE Proc. Control Theory Appl. 147, 13–18 (2000) [CrossRef] [Google Scholar]
 M. Janga Reddy, D. Nagesh Kumar, An efficient multiobjective optimization algorithm based on swarm intelligence for engineering design, Eng. Optim. 39, 49–68 (2007) [Google Scholar]
 K. Deb, Optimal design of a welded beam via genetic algorithms, AIAA J. 29, 2013–2015 (1991) [CrossRef] [Google Scholar]
 B. Yang, Y. Yeun, W. Ruy, Managing approximation models in multiobjective optimization, Struct Multidiscip Optim. 24, 141–156 (2002) [CrossRef] [Google Scholar]
 T. Erfani, S.V. Utyuzhnikov, B. Kolo, A modified directed search domain algorithm for multiobjective engineering and design optimization, Struct. Multidiscip. Optim. 48, 1129–1141 (2013) [CrossRef] [MathSciNet] [Google Scholar]
 B. Raphael, I.F.C. Smith, A direct stochastic algorithm for global search, Appl. Math. Comput. 146, 729–758 (2003) [MathSciNet] [Google Scholar]
 W. Gong, Z. Cai, L. Zhu, An efficient multiobjective differential evolution algorithm for engineering design, Struct. Multidisc. Optim. 38, 137–157 (2009) [CrossRef] [Google Scholar]
Cite this article as: Samira El Moumen, Siham Ouhimmou New multiobjective optimization algorithm using NBISASP approaches for mechanical structural problems, Int. J. Simul. Multidisci. Des. Optim. 13, 4 (2022)
All Figures
Fig. 1 Two dimensional problem with two objectives. 

In the text 
Fig. 2 An illustrative integrated design. 

In the text 
Fig. 3 The SASP organizational chart. 

In the text 
Fig. 4 Flow chart of the NBI algorithm for generating Pareto optimal sets. 

In the text 
Fig. 5 The twobar truss design problem. 

In the text 
Fig. 6 The IBeam design example. 

In the text 
Fig. 7 The disc brake design optimization problem. 

In the text 
Fig. 8 The solutions obtained by NBISASP and NSGAII on twobar truss design problem. 

In the text 
Fig. 9 The solutions obtained by NBISASP and NSGAII on beam design example. 

In the text 
Fig. 10 The solutions obtained by NBISASP and NSGAII on beam design example. 

In the text 
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