Conservative multi-objective optimization considering design robustness and tolerance: a quality engineering design approach

Abstract

Methods of multi-objective optimization are proposed to account for tolerance of design variable and variation in problem parameter. The post-optimization effort is initiated from deterministic Pareto-optimal solutions that were obtained from NSGA-II. The successive process to determine search directions and step sizes toward conservative multi-objective solutions was conducted by design of experiments to determine the worst design that had the highest constraint violation. The signal-to-noise (S/N) ratio was also employed to represent the robustness of constrained objective functions under parameter variation. Structural optimization was explored to accommodate both design tolerance and parameter variation and further apply S/N ratio in conservative multi-objective optimization.

1 Introduction

A deterministic optimum (Haftka and Gürdal 1992) is meaningful in the conceptual design process and could be further considered as a baseline toward detailed design. However, such a solution would be impractical in the case where the design problem is characterized by tolerance of the design variable (Lee and Park 2001) and/or variation in the problem parameter, since a deterministic optimum without considering either design uncertainty or safety factor often converges to a point that makes constraints active, thereby resulting in infeasible tolerance-treated designs.

Practical engineering design problems may include design tolerances and/or variations. Controllable design variables such as thickness, length, and shape values are expressed by their own lower and upper tolerances, and uncontrollable problem parameters such as material properties, allowable stress, and loading conditions have variations in their nominal values. The consideration of tolerances and variations in the design process is advantageous to enhance the design robustness and to accommodate the manufacturing and material handling process.

A robust optimization approach describes an objective function and constraints in terms of the design tolerance and variation (Lee and Park 2002). The robustness in the objective function means that the change of the objective function value with respect to the design tolerance should be insensitive as well as the minimization of the objective function value (Parkinson 1995). The robustness in the constraint implies that feasible designs should be guaranteed within the tolerance band of the design parameters (i.e., design variables and problem parameters). A conventional approach in the tolerance design has been explored by considering worst-case designs that exist within the specified tolerance band (Balling et al. 1986). In a previous study on the robust optimization, the robustness of the objective function was expressed in terms of its mean and standard deviation, which resulted in a multi-objective form, and the robust constraint functions were formulated by adding the absolute sensitivity of the constraint with respect to design tolerance to an original constraint function (Parkinson 1995).

A number of studies have been conducted to develop robust multi-objective design methodologies (Kunjur and Krishnamurty 1997; Li et al. 2005) and engineering applications (Doltsinis and Kang 2004; Forouraghi 2004) in the context of robust optimization. New recent developments in robust optimization have been recognized in the related literatures; the design robustness has been efficiently evaluated with the sensitivity information (Gunawan and Azarm 2005; Li et al. 2009; Ono and Nakayama 2009), the robust design approach that requires more expensive computational costs than a deterministic optimization method relies on the use of approximate meta-models such as response surface method and Kriging meta-model (Shimoyama et al. 2007; Sugimura et al. 2009), the emphasis on the concept of interval in evaluating the robust region has been explored (Li and Azaram 2008; Soares et al. 2009), and the adaptation of NSGA-II, a fast elitist non-dominated sorting genetic algorithm into the multi-objective robust optimization has been conducted (Qiang et al. 2010). However, the comparison among the existing robust engineering design methods has not been widely conducted in term of design conservativeness in the literature.

The present paper describes the conservative multi-objective optimization (CMO) method to account for both the tolerance in the design variable (controllable factor) and the variation in the problem parameter (uncontrollable factor). The paper also suggests the most efficient and practical design sequence of conservative robust multi-objective optimization method. The proposed method aims to accommodate design problems whose objective functions and constraints are convex in order to satisfy the local search capability; most of complex real-life engineering optimization problems are more than convex, but many of complex design problems are also practically handled with 2nd order polynomial based response surface method in the context of approximate meta-model based optimization strategy while artificial neural networks are actively being used in the approximate optimization society.

The post-optimization effort is initiated from the deterministic optimum as a baseline. The successive process to find the search directions and step sizes toward the final robust optimum is conducted by determining the worst design that had the highest level in the constraint violation. During the selection of the worst design, the orthogonal array in the context of the design of experiments (DOE) is used to reduce the constraint function evaluations, especially for the large-dimensionality problem. The analysis of means (ANOM) is adopted in the case where the variation in the problem parameter was considered. The measurement criterion to select the worst design is based on the degree of cumulative constraint violation. This post-optimization process is repeated until all the designs within the tolerance band satisfied the constraints. That is, the proposed method focuses on ensuring the constraint feasibility within tolerance bands of the problem parameters.

A central aim of the present study is to determine the search directions and step sizes toward the robust optimum, and use DOE (Sacks et al. 1989; Fowlkes and Creveling 1995; Montgomery 2001) to determine the worst design. The proposed method is also advantageous, since the robust design is obtained via the function evaluations of the constraints only; neither derivative-based sensitivity information nor statistical analysis is required. As an outline of the paper, the conservative method of robust multi-objective optimization is discussed with (1) only the tolerance of design variable, (2) both the tolerance of design variable and variation in the problem parameter, and (3) the use of the signal-to-noise (S/N) ratio, wherein the noise factor in the S/N ratio was considered as the variation in the problem parameter.

The proposed method of conservative multi-objective optimization following assumption and/or limitation: 1) objective functions and constraints should be convex in order to satisfy the local search capability, 2) for the assessment of design robustness and tolerance, the method requires a large number of function evaluations, especially in high dimensionality design problems, which can be computationally demanding, and 3) Uncontrollable problems parameters are expressed in form of discrete distribution in order to obtain their numerical values from DOE.

A mathematical function problem is first conducted to deal with the tolerance of the design variable. Subsequently, ten-bar truss and vibrating platform designs are explored to accommodate design tolerance and parameter variation, to further apply the S/N ratio in the context of CMO. In this study, a number of robust design sequences are compared to suggest the most efficient and conservative robust multi-objective optimization method.

2 Multi-objective genetic algorithm

The solution to a multi-objective problem (Park et al. 2009) is a set of design variable values, such that none of the objective functions can be further increased without a decrease in some of the remaining objective functions, i.e. every such value of a design variable is referred to as ‘Pareto-optimal’ (Mason et al. 1998). Pareto-optimal solutions have also been termed non-dominated. This name arises from the fact that no other solution is superior to them in all objectives. In other words, the non-dominated solutions are those solutions that cannot be simultaneously improved in all objectives. An evolutionary algorithm (EA) based, multi-objective optimization method has the ability to find multiple Pareto-optimal solutions in a single run as EAs work with a population of solutions. The present study employs NSGA-II as a multi-objective optimization tool (Deb et al. 2002; Deb and Gupta 2005). NSGA-II is known as a fast, non-dominated sorting algorithm for diversity preservation based on crowded comparison approach. In this study, NSGA-II is used to obtain robust multi-objective Pareto solutions as well as deterministic Pareto solutions. However, the issue on the spread measure of deterministic and/or robust Pareto solutions is not included.

3 Robust design approach

The first section discusses a robust optimization with the consideration of only the tolerance of design variables, and the second section deals with an approach, including both the tolerance of the design variable and variation in the problem parameter (Lee and Ahn 2006). The third section describes the robust optimization that utilizes the concept of the signal-to-noise (S/N) ratio to accommodate the objective function robustness.

3.1 Design tolerance

The deterministic Pareto optimal solutions are obtained from a formal multi-objective optimization method. A deterministic optimal design out of multi-objective Pareto solutions is denoted as $x_i^{D\ast }$, and could be positioned within the following tolerance region:

$$ \label{eq1} x_i^{D\ast } -T_i \le x_i^{D\ast } \le x_i^{D\ast } +T_i $$

(1)

where, T _i is a user-specified tolerance of the i-th design variable. It should be noted that some designs in the tolerance region are infeasible when the deterministic optimum is located at a point where constraints are active. Hence, the tolerance region needs to be moved into a feasible region. The first post-optimization process toward the robust optimum is to determine a new search direction, $S_i^{\left( q \right)}$ and a step size, α ^(q) based on the following equation:

$$ \label{eq2} x_i^{D\ast \left( {q+1} \right)} =x_i^{D\ast \left( q \right)} +\alpha ^{\left( q \right)}\cdot S_i^{\left( q \right)} . $$

(2)

where the notation, q is an iteration number. The search direction is determined by the following equation:

$$ \label{eq3} S_i^{\left( q \right)} =x_i^{D\ast \left( q \right)} -x_i^{worst\left( q \right)} . $$

(3)

where the search vector, $\vec{{S}}^{\left( q \right)}$is calculated by the difference between $\vec{{x}}^{D\ast \left( q \right)}$and $\vec{{x}}^{worst\left( q \right)}$. The notation, $\vec{{x}}^{worst\left( q \right)}$is a design variable vector whose degree of constraint violation is the highest within the tolerance region of interest. Such direction will make the deterministic optimum move into a new design point that is feasible with a marginal increase in the objective function value. Since the constraint functions used in the present study are convex, the most violated design often occurs at one of the vertices in the tolerance region. Thus, the design vector, $\vec{{x}}^{worst\left( q \right)}$is determined by (4).

$$ \label{eq4} x_i^{worst\left( q \right)} =x_i^{D\ast \left( q \right)} -T_i \qquad \mbox{or} \qquad x_i^{D\ast \left( q \right)} +T_i $$

(4)

Equation (4) demonstrates that the worst design during the direction finding process is chosen as either the smaller (a minus tolerance, − T _i ) or larger (a plus tolerance, + T _i ) design value that is expressed in terms of the deterministic optimum and its tolerances. A total of 2ⁿ vertices (design points) exist in the tolerance region, where n is the number of design variables. The low dimensionality optimization problem would additionally require the small number of function evaluations to determine $\vec{{x}}^{worst\left( q \right)}$. When a higher dimensionality (i.e., many of design variables are involved), larger scale (i.e., many of constraints are involved) design problem is considered, DOE is an efficient approach to reduce the number of constraint function evaluations. The present study employs the well-known orthogonal array for savings in computational costs (Sacks et al. 1989; Fowlkes and Creveling 1995). The computational savings of orthogonal array compared to full factorial design become increased as the number of design variables and problem parameters gets large. The analysis of means (ANOM) is also used to accommodate factor effects under the use of orthogonal array. Especially, a two-level orthogonal array is appropriate, since a tolerance-treated design point consists of two ends values of the smaller (a minus tolerance, Level-1) and larger (a plus tolerance, Level-2) ones. Once the search direction is determined, an instantaneous step size, α ^(q) is calculated based on the worst design and constraint violation, as follows:

$$ \label{eq5} \alpha^{\left( q \right)}=\left| {x_i^{worst\left( q \right)} -x_i^{g_j \_active\left( q \right)} } \right|. $$

(5)

In the above expression, $x_i^{g_j \_active\left( q \right)}$ is an active design that produces the constraint function value of g _j  = 0, and such constraint is infeasible to $\vec{{x}}^{worst\left( q \right)}$. That is, (5) allows the worst design to move up to an active constraint position with the marginal increase in the objective function value. This step size can be obtained by identifying violated constraints through an orthogonal array under the use of convex functions. Using (3) to (5), a new design, $x_i^{D\ast \left( {q+1} \right)}$ is obtained, as shown in (2). Such process is repeated until all the designs (i.e., all the vertex points) within the tolerance band satisfied the constraints.

3.2 Parameter variation

Design robustness can be reinforced by including the variation of the problem parameters, such as material properties, applied loading, and boundary conditions. This concept is more practical in design under manufacturing uncertainties. The present study utilizes the orthogonal array to accommodate discrete levels in both the design variable and the problem parameter. After finding the deterministic optimum as previously mentioned, the orthogonal array table needs to be constructed by locating levels of design variables (controllable factors) at inner arrays and positioning levels of the problem parameters (uncontrollable factors) at outer arrays. The current approach also uses (2) to (5) to determine the robust optimum, however the distinction from the previous approach is to include the variation in the problem parameter, which is why problem parameters are considered uncontrollable factors. In this case, the variation of the problem parameter would be pre-specified by the user, as well. The present study uses the two-level array for each problem parameter (Sundaresan et al. 1995); for example Level-1 is 5% or 10% less than the nominal value of a given problem parameter, and Level-2 is 5% or 10% greater. It should be noted that the nominal problem parameter values in the outer array are fixed, while design values in the inner arrays are sequentially changed during the robust optimization process. The measurement function in the orthogonal array table is M _k , the degree of cumulative constraint violation calculated via ANOM, as follows:

$$ \label{eq6} M_k =\frac{1}{L}\sum\limits_{j=1}^J {M_{jk} } =\frac{1}{L}\sum\limits_{j=1}^J {g_j \left( {x_i ,p_l } \right)_k } $$

(6)

where, p _l denotes the l-th problem parameter (l = 1,...,L). A notation, J is the number of constraints (j = 1,...,J), and K is the number of computational experiments in orthogonal array (k = 1,...,K). The analysis of means (ANOM) facilitates to determine the meaningful level of a design variable by calculating the mean value of all levels in design variable. The result of ANOM is called ‘the factor effect’ in the context of DOE. From (6) utilizing the ANOM, the combination of levels in design variable that generates the highest value of the degree of cumulative constraint violation is a set of the worst design at the current iteration as shown in (4). Such combination of levels is to be used to obtain the search direction of (3) and step size of (5) toward the robust design.

3.3 Use of the S/N ratio

The S/N ratio is an important measure of the design quality under the variation of the problem parameter (i.e., noise factor). In the present study of CMO, two well-known S/N ratios such as smaller-the-better (STB) and larger-the-better (LTB) are adopted (Fowlkes and Creveling 1995), since they can represent the minimization and/or maximization of objective function(s), respectively. For the S/N ratio based multi-objective function formulation, each of multiple objective functions is expressed as follows:

$$\begin{array}{lll} \mbox{Maximize} \quad S \mathord{\left/ {\vphantom {S {N_m }}} \right. \kern-\nulldelimiterspace} {N_m }\nonumber\\ &&\quad =-10\log_{10} \left[ {\frac{1}{n}\sum\limits_{i=1}^n {Y_i^2 } } \right]_m , \quad m=1,...,\mathit{MOBJ} \end{array}$$

(7)

$$\begin{array}{rll} Y_i &=&f\left( {x,p\pm \Delta p} \right)\nonumber\\ &&+R\sum\limits_{j=1}^J {\max <0,g_j \left( {x,p\pm \Delta p} \right)>^2}\nonumber\\ && \mbox{for}\;\mbox{constrained}\;\mbox{minimization} \end{array}$$

(8)

$$\begin{array}{rll} Y_i &=&\frac{1}{f\left( {x,p\pm \Delta p} \right)} \nonumber\\ &&+R\sum\limits_{j=1}^J {\max <0,g_j \left( {x,p\pm \Delta p} \right)>^2} \nonumber\\ && \mbox{for}\;\mbox{constrained}\;\mbox{maximization} \end{array}$$

(9)

where MOBJ is the number of objective functions and n is the number of problem parameters for the variation. f is an objective function and g is an inequality constraint formulated by exterior penalty function method with a penalty parameter R. Equation (7) implies that the robustness, that is, the variation of problem parameters as noise factors is included in both objective functions and constraints. The proposed method using the S/N ratio is implemented such that the NSGA-II based maximization of (7) is conducted until all the feasible designs are obtained, starting with Pareto-optimal solutions as the initial designs.

The present study proposes the post-deterministic optimization-based CMO considering design tolerance and parameter variation, using the S/N ratio, and sequentially applying both. The first method obtains conservative designs located within constraint-feasible regions, and the second method accommodates the robustness of the objective function(s) with use of the S/N ratio, for which constraints are formulated via the exterior penalty function method. A design that violates the tolerance-considered constraints is removed during the S/N ratio, so that the proposed CMO method identifies constraint-feasible designs that maximize the S/N ratio. The CMO by the S/N ratio generates more robust solutions since its initial designs are solutions already obtained by considering the design tolerance and parameter variation. Therefore, the final solution is a conservative design that satisfies both constraint feasibility and robust characteristics of objective functions, which matches all of the robustness of the constrained multi-objective optimization.

4 Proposed CMO methods

The present study suggests a total of three major design sequences to account for the design tolerance (T), parameter variation (V), and S/N ratio (S/N) of the constrained multi-objective functions in the context of the CMO. Once the deterministic multi-objective optimization has been conducted, all of deterministic Pareto solutions are participated as initial designs to subsequently find CMO solutions. That is, the proposed CMO methods are post-optimization efforts starting from the deterministic Pareto solutions.

• 0) ‘D-P’: Run NSGA-II to find deterministic Pareto solutions using unconstrained/constrained multiple objectives, where GA based constrained multi-objective optimization is formulated via a well-known exterior penalty function method. (Deterministic Pareto solutions)

• 1) ‘D-T-P’: Conduct the DOE based process to find feasible multi-objective Pareto solutions considering design tolerance only. (Pareto solutions with design Tolerance)

• 2) ‘D-TV-P’: Conduct the DOE based process to find feasible multi-objective Pareto solutions considering both design tolerance and parameter variation. (Pareto solutions with both design Tolerance and parameter Variation)

• 3) ‘D-TV-S/N-P’: Using solutions of D-TV-P as initial designs, find feasible solutions using the S/N ratio of each constrained objective functions. (Pareto solutions with both design Tolerance and parameter Variation first and S/N ratio applied next) The S/N ratio considers problem parameters as uncontrollable noise factors. Since the S/N ratio is formulated via the constrained objective function(s), a design solution that violates any of the constraints is excluded from the feasible Pareto solutions during the run of NSGA-II.

Besides the aforementioned three major design sequences of ‘D-T-P’, ‘D-TV-P’ and ‘D-TV-S/N-P’, additional versions of conservative design methods are further proposed to identify the most conservative optimization method. Two more design sequences are ‘D-T-S/N-P’ and ‘D-S/N-T-P’, which are variants of ‘D-TV-S/N-P’.

• 4) ‘D-T-S/N-P’: Using solutions of D-T-P as initial designs, find feasible solutions using the S/N ratio of each constrained objective functions. (Pareto solutions with design Tolerance first and S/N ratio applied next)

• 5) ‘D-S/N-T-P’: This is a reverse sequence of ‘D-T-S/N-P’. (Pareto solutions with the S/N ratio applied first and the design Tolerance considered next)

In the later part of Section 6, the implication of each design sequence is to be discussed in terms of design conservativeness—through the implementation of proposed design sequences, the most conservative optimization method is to be achieved in the context of quality engineering design.

5 Design applications

Consider the following mathematical optimization problem referred to as SRN (Deb et al. 2001):

$$\begin{array}{rll} \mbox{Minimize}\quad f_1 \left( {x_1 ,x_2 } \right)&=&\left( {x_1 -2} \right)^2+\left( {x_2 -1} \right)^2+2 \nonumber\\ [4pt] \mbox{Minimize}\quad f_2 \left( {x_1 ,x_2 } \right)&=&9x_1 -\left( {x_2 -1} \right)^2 \nonumber\\ [4pt] \mbox{Subject\thinspace to} \quad g_1 \left( {x_1 ,x_2 } \right)&=&x_1^2 +x_2^2 -225\le 0 \nonumber\\ [4pt] \quad g_2 \left( {x_1 ,x_2 } \right)&=&x_1 -3x_2 +10\le 0 \nonumber\\ [4pt] -20.0&\le& x_1 ,x_2 \le 20.0. \end{array}$$

(10)

The SRN is tested first, in order to identify multi-objective, Pareto-optimal solutions considering only the design tolerance. That is, this example problem discusses how much tolerance-treated Pareto solutions are shifted from deterministic Pareto solutions when the design tolerance is of concern.

As an example of a larger dimensionality-problem, the ten-bar truss optimization in Fig. 1 is explored. The design objective is to find the cross-sectional areas of truss members, X _i (i = 1, 10) by minimizing both the total weight of a structure W(X _i ) and the tip deflection δ(X _i ) subjected to constraints on stress, σ _j (X _i ) (j = 1, 10) (Haftka and Gürdal 1992). The optimization statement is written as follows:

$$\begin{array}{lll} {\mbox{Minimize}} \qquad{W\left( {X_i } \right)} \nonumber\\ {\mbox{Minimize}} \qquad{\delta \left( {X_i } \right)} \nonumber\\ {\mbox{Subject}\,\mbox{to}} \qquad\sigma_j \left( {X_i } \right)\le 25, \nonumber\\ \qquad\qquad\qquad\left( {j=1,2,3,4,5,6,7,8,10} \right) \quad{\mbox{unit:}\,\left[ {\mbox{ksi}} \right]} \nonumber\\ \qquad\qquad\qquad{\sigma_j \left( {X_i } \right)\le 75,\left( {j=9} \right)} \qquad\qquad{ \mbox{unit:}\,\left[ {\mbox{ksi}} \right]} \nonumber\\ \qquad\qquad\qquad{0.1\le X_i \le 20.0} \qquad\qquad\qquad{ \;\mbox{unit:}\,\left[ {\mbox{in}} \right].} \nonumber\\ \end{array}$$

(11)

For another structural design problem, a pinned-pinned sandwich beam supported with a vibrating motor on top is considered, as shown in Fig. 2 (Parkinson 2005). The objective is to determine five sizing design variables by minimizing the total cost of the sandwich beam structure and maximizing its natural frequency subjected to a number of constraints. The formal statement of the multi-objective optimization is written as follows:

$$\begin{array}{rll} \mbox{Minimize}\quad cost&=&2b\left[ {c_1 d_1 +c_2 \left( {d_2 -d_1 } \right)+c_3 \left( {d_3 -d_2 } \right)} \right] \nonumber\\ \mbox{Maximize}\quad f&=&\left( {\frac{\pi }{2L^2}} \right)\left( {\frac{EI}{\mu }} \right)^{0.5} \nonumber\\ \mbox{Subject}\,\mbox{to} \quad g_1 &=&\mu L-2800\le 0 \nonumber\\ g_2 &=&d_1 -d_2 \le 0 \nonumber\\ g_3 &=&d_2 -d_1 -0.15\le 0 \nonumber\\ g_4 &=& d_2 -d_3 \le 0 \nonumber\\ g_5 &=& d_3 -d_2 -0.01\le 0 \nonumber\\ 0.05&\le& d_1 \le 0.5,\;0.2\le d_2 \le 0.5,\nonumber\\ \;0.2&\le& d_3 \le 0.6 \nonumber\\ 0.35&\le& b\le 0.5,\;3.0\le L\le 6.0 \nonumber\\ [-15pt] \end{array}$$

(12)

Ten-bar planar truss and vibrating platform problems are used for the CMO, considering both design tolerance and parameter variation and further applying the S/N ratio. The problem parameters in the ten-bar truss design are the material density and two applied loads. The material density, Young’s modulus, and cost coefficients, c _i are considered as problem parameters in the vibrating platform problem.

Fig. 1

Ten-bar planar truss

Fig. 2

Vibrating platform

6 Results and discussion

6.1 Mathematical function problem

The design tolerance is assumed to be 10% of a deterministic optimum. There are a total of 200 deterministic Pareto-optimal solutions obtained by NSGA-II, wherein the population size is 200 and the number of GA generations is 500.

The post-optimization process toward the robust multi-objective Pareto solutions is explained using one of the deterministic optimal solutions, $\vec{{x}}=\left[ {-2.50,2.50} \right]^T$, referred to as “Point-a” in Fig. 3. If the design tolerance is assumed to be ±10% about the nominal value, it is expected that some shaded designs in the tolerance region, referred to as “Band-A” are infeasible. The next procedure is to successively find search directions and step sizes toward the robust optimum using (2). To evaluate the most violated design, a DOE based orthogonal array is used as shown in Table 1. Since each of the design variables has two-level tolerance limits, a total of four constraint function evaluations are required to calculate the degree of constraint violations. From the first iterative use of DOE, the maximum constraint violation value has been determined to occur at $\vec{{x}}=\left[ {-2.25,2.25} \right]^T$; this is a necessary point to calculate the search direction. Using (3), the search direction is obtained, as follows:

$$\begin{array}{rll} S_i^{\left( 1 \right)} &=&x_i^{D\ast \left( 1 \right)} -x_i^{worst\left( 1 \right)} \nonumber\\ [6pt] &=&\left[ {{\begin{array}{*{20}c} {-2.50} \hfill \\ {2.50} \hfill \\ \end{array} }} \right]-\left[ {{\begin{array}{*{20}c} {-2.25} \hfill \\ {2.25} \hfill \\ \end{array} }} \right]=\left[ {{\begin{array}{*{20}c} {-0.25} \hfill \\ {0.25} \hfill \\ \end{array} }} \right] \end{array}$$

(13)

Fig. 3

One of deterministic Pareto solutions and its tolerance band of mathematical problem

Table 1 Initial orthogonal array for mathematical problem

A step size, α ⁽¹⁾ is obtained by (5) as follows:

$$ \label{eq14} \alpha^{\left( 1 \right)}=\sqrt {\left. {\left( {-0.25} \right)^2+\left( {0.25} \right)^2} \right)} =0.3536. $$

(14)

The normalization of the search direction using α ⁽¹⁾ yields $S_i^{\left( 1 \right)} =\left[ {-0.7071,0.7071} \right]^T$. In this case, $x_i^{D\ast \left( 1 \right)} =x_i^{g_j \_active\left( 1 \right)}$ since the deterministic optimum converges to the point where a constraint, g ₂ is active. Such step size allows the worst (i.e., the most violated) design to move from $x_i^{worst\left( 1 \right)}$ to $x_i^{g_j \_active\left( 1 \right)}$, consequently introducing the change in the tolerance region from “Band-A” to “Band-B,” as shown in Fig. 4. The successive results to obtain the final robust optimum are summarized in Table 2. In the first step, the design solution $\vec{{x}}=\left[ {-2.50,2.50} \right]^T$is the deterministic optimum to be used as the initial design for the post-optimization process. Based on the iterative use of (2)–(5) with the updated DOE, the design solution in the third step, $\vec{{x}}=\left[ {-3.025,3.025} \right]^T$ is the robust optimum whose tolerance region is successfully located within the feasible designs.

Fig. 4

Search direction and step size

Table 2 Process of design with tolerance

The above DOE-based post-optimization process is repeated for all of the deterministic Pareto-optimal solutions in order to obtain tolerance-treated designs. The remaining 199 Pareto solutions can also be changed to tolerance-treated designs (i.e., ‘D-T-P’) by the same procedure, and they are shown in Fig. 5. Among a total of 200 Pareto solutions, only 41 designs are shifted into other positions, with an overall shrinking of the Pareto fronts. Therefore, the 159 deterministic Pareto-optimal solutions are within the 10% tolerance limit.

Fig. 5

Tolerance-treated solutions of mathematical problem

6.2 Ten-bar planar truss design

The design tolerance is assumed to be 5% or 10% of the deterministic optimum. Using NSGA-II, a set of 200 deterministic Pareto-optimal solutions is explored in a population of 200 individuals over 500 generations.

The post-optimization process toward the robust multi-objective Pareto solutions is explained using one of the deterministic optimal solutions. The NSGA-II based deterministic multi-objective optimization is first conducted to obtain the deterministic Pareto-optimal solutions under nominal values of the problem parameters, and post-optimization is subsequently performed with tolerances on the design parameters; the deterministic optimum and its two-level tolerance on each design parameter are shown in Table 3. Since this design example includes the variation in the problem parameter and the tolerance of the design variable, the orthogonal array table with both inner and outer arrays is necessary, as shown in Table 4, wherein the degree of cumulative constraint violation is used as a measurement function to compute ANOM using a total of 48 function evaluations. A number of ANOMs, that is, the sensitivity of the degree of constraint violation with respect to the design variable, are shown in Fig. 6. As a result, the higher level of each design variable represents more constraint violations, and this set of designs should be moved into the feasible region by computing the search direction and appropriate step size using (3) and (5).

Table 3 Initial values of a Pareto solution used in ten-bar truss design

Table 4 DOE result of a Pareto solution used in ten-bar truss design

Fig. 6

Degree of constraint violation in ten-bar truss

The CMO requires two steps to determine its robust optimum, starting from the deterministic optimum, as shown in Table 5. At every step, ANOM should be evaluated using a newly obtained set of design variables (i.e., new vertex points). After the robust optimization, it is necessary to make sure that all the vertices near the robust optimum point satisfy the design constraints. As a result, the orthogonal array-based ANOM is efficient for a certain level of tolerance bands in terms of computational costs. The solution is now evaluated in terms of conservativeness and/or robustness. The worst case is as follows; two sets of factor levels associated with the maximum constraint violation are Level(x _i ) = [1, 2, 1, 1, 1, 2, 1, 1, 1, 1] and Level(p _l ) = [1, 2, 2], as shown in the 2nd step of Table 5, and the maximum stress is σ ₉ = 72.4 kpsi at the 9-th truss member, which means that the solution with the design tolerance and parameter variation satisfies the robustness. The worst cases of the ‘D-P’ and ‘D-TV-P’ are shown in Table 6. A design trade-off exists for the solution of ‘D-TV-P’, as compared with ‘D-P’, wherein the weight is increased 8% and the deflection is decreased 5%. That is, the method of ‘D-TV-P’ trades product performance for design robustness.

Table 5 Design process of a Pareto solution with tolerance and variation

Table 6 Comparison of worst case Pareto design

The tolerance-treated design of ‘D-T-P’ results in 163 designs being shifted into the feasible region, as shown in Fig. 7. The solutions of ‘D-TV-P’ shifted and shrank more than ‘D-T-P’. The Pareto curve of ‘D-TV-P’ is also oriented in the clock-wise direction, due to the inclusion of the parameter variation. The parameter variation means a change in the environmental condition. In this case, another constraint-feasible optimal design should be obtained, since the problem parameter values are changed. That is, the consideration of the parameter variation would generate differently shifted Pareto solutions from the original deterministic Pareto solutions. For the case with the S/N ratio, as shown in Fig. 8, solutions of ‘D-TV-S/N-P’ are more conservative (i.e., the Pareto curve is shrank even more). It is now necessary to look at the variation of the proposed design sequence by comparing ‘D-T-S/N-P’ and ‘D-TV-S/N-P,’ as shown in Fig. 9. Both results are nearly the same, since the parameter variation (V) has already been included in the S/N ratio, such that the consideration of the parameter variation (V) in ‘D-TV-S/N-P’ should be redundant. Thus, the method of ‘D-T-S/N-P’ is recommended instead of ‘D-TV-S/N-P’.

Fig. 7

Tolerance and variation in ten-bar truss

Fig. 8

Tolerance, variation and S/N ratio in ten-bar truss

Fig. 9

Effect of parameter variation in ten-bar truss

It is realized that the method of ‘D-T-S/N-P’ is currently the most conservative design sequence. Another alternative design sequence of ‘D-S/N-T-P’ is compared with ‘D-T-S/N-P,’ as shown in Fig. 10, wherein solutions of ‘D-S/N-T-P’ are marginally better in terms of the constraint feasibility and conservativeness. The method of ‘D-S/N-T-P’ developed in the present study is exactly compatible with a well-known design stage of the ‘(deterministic) parameter design → robust design → tolerance design’ in the context of quality engineering design. That is, the present study emphasizes that from a number of test results on the design sequence as shown in Figs. 7, 8, 9 and 10, ‘D-S/N-T-P’ in Fig. 10 is the most conservative design method that can accommodate the constraint feasibility considering design tolerance, parameter variation, and eventually secure the design robustness.

Fig. 10

Difference of design sequence between tolerance and S/N ratio in ten-bar truss

Additionally, the use of design of experiments is now examined by comparing results of the orthogonal array (OA) with those of full factorial (FF) designs. For each DOE step, an orthogonal array with 12 inner experiments and 4 outer experiments requires a total of 48 constraint function evaluations, while FF requires a total of 8,192 calculations since there are 2¹⁰ = 1,024 inner arrays and 2³ = 8 outer arrays. Tolerance/variation-treated Pareto solutions obtained from OA and FF are presented in Fig. 11. A total of 57 designs are different between OA and FFD, and this number, 57 out of 200 Pareto solutions is considered quite large. Such difference may come from the strong nonlinearity of a truss whose stress constraints cannot be expressed with a convex function. Since the use of an orthogonal array is known as valid for small-scale problems, other advanced DOE methods should be devised to handle the high-nonlinearity, large-scale design.

Fig. 11

Comparison between orthogonal array and full factorial design

6.3 Vibrating platform design

Tolerance limits of 5% or 10% about a deterministic optimum are used in this design problem. For the parameter variation, the material density $\uprho$, Young’s modulus E, and one of cost coefficients, c₁ are considered as 3-level problem parameters such as $\uprho =$ [95, 100, 105 kg/m³], E = 1.44, 1.60, 1.76 GPa], and c₁ = [450, 500, 550$/m³], respectively. Other parameters are fixed as c₂ = 150$/m³ and c₃ = 800$/m³. The NSGA-II identifies a total of 200 Pareto solutions using 200 individuals in a population over 500 generations. A total of 174 Pareto solutions are shifted by the post-optimization design process. Two cases with 5% and 10% tolerance and variation intervals are compared in Fig. 12. Since the infeasible region is positioned in the upper (right hand) side of the Pareto-fronts, tolerance/variation-treated Pareto solutions move to the lower (left hand) side of the Pareto-fronts. In the case of ‘D-TV-P’, the active constraint, g₄ becomes critical to the constraint violation, wherein g ₄ = (d ₂ − d ₃) >0 (i.e., violated) such that the tolerance/variation-treated design of ‘D-TV-P’ tries to make d ₃ increase. As a result, the frequency is increased so that its tolerance-treated solutions are moved to the left hand side. As expected, solutions with 10% tolerance are shifted more than those with 5% tolerance.

Fig. 12

Tolerance and variation intervals in vibrating platform

The design sequences of ‘D-TV-P’, ‘D-TV-S/N-P’ are compared in Fig. 13; the additional use of S/N is more conservative. For the comparison between ‘D-T-S/N-P’ and ‘D-S/N-T-P’ in the vibrating platform design as shown in Fig. 14, the method of ‘D-S/N-T-P’ shows better performance in terms of design conservativeness; which is the same as the ten-bar planar truss design and represents the design sequence of the parameter design → robust design → tolerance design.

Fig. 13

Effect of S/N ratio in vibrating platform

Fig. 14

Difference of design sequence in vibrating platform

Each design process, such as ‘T’, ‘TV’, and ‘S/N’ requires a number of objective and constraint function evaluations to reach conservative and robust Pareto solutions. For a mathematical function problem considering only design tolerance, a single deterministic Pareto solution requires two more post-optimization steps to determine the tolerance-feasible solution, as explained with Table 2. Throughout this study, each design process of ‘T’, ‘TV’ or ‘S/N’ takes approximately two or three steps in order for each Pareto solution to move into a more conservative region; for example, the method of ‘D-S/N-T-P’ roughly needs four to six more steps, starting from the deterministic Pareto solution. Thus, the total number of function evaluations required to shift the deterministic Pareto curve into the conservative region is dependent upon how many designs participate in constructing a Pareto surface. Using two structural design problems, a number of design sequences are explored to identify the most efficient and conservative design method to satisfy the robustness of the Pareto-optimal solutions in the context of multi-objective optimization. The present study suggests the method of ‘D-S/N-T-P’ as the best design sequence to meet the conservativeness and robustness, and the method is exactly compatible with the design stage of ‘parameter design → robust design → tolerance design’ in quality engineering design.

7 Concluding remarks

CMO methods considering tolerance of design variable and variation in the problem parameter and using the S/N ratio are proposed in the present study. The proposed methods aim to accommodate design problems whose objective function and constraints are expressed with convex functions in order to satisfy the local search capability. Uncontrollable problems parameters are expressed in form of the discrete distribution in order to obtain their numerical values from DOE. The deterministic multi-objective Pareto-optimal solutions obtained from NSGA-II are baselines toward robust optimal designs. CMO designs satisfy constraints by successively moving the tolerance region into a feasible space. As a result, a robust optimum is conserved with a marginal increase in the objective function value. The paper proposes the benefit of the DOE method to evaluate the measurement of the constraint violation. Such method reduces the constraint function evaluation by employing a traditional orthogonal array-based ANOM. The present study also suggests a method of the objection function robustness using the S/N ratio, wherein the noise factor in the S/N ratio is considered as the variation in the problem parameter. Subsequently, robust designs are obtained based on the tolerance/variation-treated solutions. The present study suggests the method of ‘D-S/N-T-P’ as the best design sequence to meet the conservativeness and robustness. The suggested method is compatible with the design stage of ‘parameter design → robust design → tolerance design’ in the context of quality engineering design.