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M. Zimmermann, Johannes Hoessle (2013)
Computing solution spaces for robust designInternational Journal for Numerical Methods in Engineering, 94
Craig Larman, V. Basili (2003)
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J. Fender, F. Duddeck, M. Zimmermann (2017)
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Stefan Erschen, F. Duddeck, M. Zimmermann (2015)
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M. Zimmermann, S. Königs, C. Niemeyer, J. Fender, Christian Zeherbauer, Renzo Vitale, M. Wahle (2017)
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Abstract Lack of knowledge or epistemic uncertainty in technical systems can be treated with so-called Solution Spaces. They are sets of good designs that reach by definition all design goals. Considering sets rather than one single design allows for unintended variations of component properties that are typical in the early stages of systems design. Box-shaped Solution Spaces can be expressed as the Cartesian product of permissible intervals for design variables. These intervals serve as independent target regions and can be interpreted as component requirements. Existing algorithms optimize the size of box-shaped Solution Spaces. Unfortunately, the size of the permissible intervals for crucial design variables is often not large enough to encompass all uncertainty and to ensure feasibility. A new approach is introduced where the design variables are divided into a set of early- and a set of late-decision variables. Early-decision variables are associated with permissible intervals on which they may assume any value to encompass uncertainty due to limited controllability. Late-decision variables are controllable and therefore associated with intervals where they can be adjusted to any specific value. The Cartesian product of these intervals is called a Solution-Compensation Space. It has the property that for all values of early-decision variables from their permissible intervals there exists at least one set of late-decision variable values from their intervals such that the resulting design reaches all design goals. The approach is applied to a design problem from vehicle driving dynamics. It is shown that the permissible intervals for the early-design variables can be increased significantly. 1. Introduction The early phase of engineering design is characterized by many uncertainties. The particular uncertainty relevant for this paper is caused by lack of knowledge about the final state of a complex product during development (see Zimmermann et al., 2017). Classical design approaches like incremental and iterative development (see Larman & Basili, 2003) seek a single design in order to reach all intended goals. Unfortunately, it is not guaranteed for this method to converge and the use of it may be ‘expensive, inefficient and vulnerable to uncertainty and variability’ (see Qureshi et al., 2015). In order to cope with those problems, set-based approaches can be used. These seek permissible intervals for each design variable rather than a single optimal solution, e.g. in the studies by Rocco et al. (2003) and Fung et al. (2005). Zimmermann & Von Hoessle (2013) propose Solution Spaces in order to find large box-shaped sets of good designs. A good design is defined here as a design that does not violate any constraints such as performance requirements. In this paper all requirements will be treated as mathematical constraints. Box-shaped Solution Spaces can be expressed as a product of intervals and thus allow to decouple the requirements on design variables. This enables a simultaneous and distributed component development and leads to an efficient design process that is also robust to variations. In Zimmermann & Von Hoessle (2013) a solution box is optimized with respect to the size measure μ and is required to have at least a specified fraction of good designs. Whether the required fraction is reached is tested by Monte Carlo sampling. Recently, new approaches were developed, which use classical optimization instead of a stochastic approach to derive the optimal box for general linear problems (see Erschen et al., 2015 and Fender et al., 2017). These approaches ensure that each design inside the box is good. Unfortunately, the size of the permissible intervals derived by the Solution Space approach is often not sufficiently large for crucial design variables to compensate for the uncertainties during the development process. Hence, solution-compensation spaces are introduced in this paper, which allow to compute larger intervals for early-decision variables. Early-decision variables refer to one of two different types of design variables, which can be identified in many industrial development processes: Early-decision variables underlie large uncertainty and have to be bounded during the early phases of the development process since they have strong influence on the system performance (e.g. the design of a suspension). When computing solution-compensation spaces, these variables are associated with permissible intervals on which they may assume any value. Late-decision variables are specified in late stages of the development process since they are associated with parts, which are easy to adjust (e.g. the tuning parameters of a control system or the design of an anti-roll bar). When computing solution-compensation spaces, these variables are associated with intervals on which they have to be able to assume any value. The Cartesian product of the early-decision variable intervals and the late-decision variable intervals is called a solution-compensation space. Solution-compensation spaces serve to increase the size of the permissible intervals for early-decision variables requiring additional conditions for late-decision variables. The objective of this paper is to introduce an algorithm to compute these solution-compensation spaces. The paper is organized as follows: in Section 2, the search for solution-compensation spaces is motivated by a simple example problem. A general problem statement is introduced in Section 3. Section 4 explains in detail how the proposed algorithm works. In Section 5, an eight-dimensional real-world problem is solved. Section 6 discusses the capabilities and the numerical performance of the algorithm. 2. Description of the underlying problem 2.1. Seeking a box-shaped Solution Space In order to illustrate the basic idea of computing solution-compensation spaces, a two-dimensional example problem is shown in Fig. 1a. The three straight lines forming the grey triangle visualize the constraints. The triangle depicts the area of all good designs, where all requirements are satisfied, called the complete Solution Space. The box-shaped Solution Space allows decoupling of the parameters. Fig. 1. View largeDownload slide (a) Box-shaped Solution Space (small rectangle) inside the complete Solution Space (triangle) and the solution-compensation space (large rectangle with dashed lines). Dashed/solid lines indicate that the respective axis is associated with a late/early decision variable. (b) A realized value for the early-decision variable |$x_a^*$| and the resulting late-decision variable permissible interval |$\Omega_b^* $|. Fig. 1. View largeDownload slide (a) Box-shaped Solution Space (small rectangle) inside the complete Solution Space (triangle) and the solution-compensation space (large rectangle with dashed lines). Dashed/solid lines indicate that the respective axis is associated with a late/early decision variable. (b) A realized value for the early-decision variable |$x_a^*$| and the resulting late-decision variable permissible interval |$\Omega_b^* $|. 2.2. Loss of Solution Space For the two-dimensional example problem the derived complete Solution Space as well as the box-shaped Solution Space are depicted in Fig. 1a. The design space Ωds is represented by the grey dashed line. Inside of the complete Solution Space the box-shaped Solution Space is shown. Comparing the box-shaped Solution Space with the complete Solution Space shows that the box only covers a small part of the complete Solution Space. This represents a significant loss of good designs. 2.3. Underlying idea of solution-compensation spaces It is assumed that the Solution Space interval for the early-decision parameter xa is too small and shall be enlarged. This is made possible, by changing the character of xb, rather than treating it as an uncertain variable that may assume any value within some interval, it is considered as a variable that can be adjusted arbitrarily well to any desired value from an assigned interval. Its final value will be determined in a later stage, in particular after xa was chosen. One possibility to derive the early-decision variable interval for xa is to project the complete Solution Space onto the xa axis. The late-decision variable interval for xb is chosen to be its entire design interval. In the two-dimensional example (see Fig. 1a) the solution-compensation space is represented by the blue line. As can be seen for all values in the xa interval there exists at least one value in xb ∈ Ωb such that the resulting design reaches all design goals. Figure 1b shows an example of the resulting late-decision variable interval |$\Omega _{b}^{\ast }$| for a chosen early-decision variable value |$x_{a}^{\ast }\in \Omega _{a}$|. A typical iterative design process consists of a component design phase, in which the design is chosen and iteratively improved (see Fig. 2). The classical Solution Space approach extends the component design phase by a system design phase in which permissible intervals for all variables are derived (see Fig. 3). This enables the development of a single design, which satisfies all design goals without any iterative steps. As depicted in Fig. 4 the solution-compensation space approach adds a third design step called compensation phase. In the system design phase a set of early- and late-decision variables needs to be determined. Then the increased permissible intervals for the early-design variables xa are derived. The increased intervals for xa will grant more flexibility during the development process. This accounts for the lack-of-knowledge situation in early development stages. In the component design phase all early-decision variables xa are chosen. In the compensation phase, all late-decision variable intervals are computed and their values xb are chosen such that in combination with the early-decision variables a good design is generated. Fig. 2. View largeDownload slide Iterative design process according to Larman & Basili (2003). Fig. 2. View largeDownload slide Iterative design process according to Larman & Basili (2003). Fig. 3. View largeDownload slide Approach using system and component design of classical Solution Spaces according to Zimmermann & Von Hoessle (2013). Fig. 3. View largeDownload slide Approach using system and component design of classical Solution Spaces according to Zimmermann & Von Hoessle (2013). Fig. 4. View largeDownload slide Extension to a three-step design approach for sequential development using solution-compensation spaces. Fig. 4. View largeDownload slide Extension to a three-step design approach for sequential development using solution-compensation spaces. 3. Computing solution-compensation spaces 3.1. General problem statement Design points or designs are represented by the vector \begin{align} x=(x_{a},x_{b}) \end{align} (3.1) with xa = (xa, 1, xa, 2, …, xa, p) and xb = (xb, 1, xb,2, …, xb, q), where p and q are the total numbers of early- and late-decision variables, respectively. The index a indicates an early-decision variable whereas the index b indicates a late-decision variable. The set of all possible design points x is called design space Ωds, with Ωds =Ωds, a ×Ωds, b and xa ∈Ωds, a, xb ∈Ωds, b. The response of the system at x is given by \begin{align} z=f(x_{a}, x_{b}) \quad \quad f: \mathbb{R}^{p+q} \to \mathbb{R}^{m}, \end{align} (3.2) where f is the performance function. A classical optimization problem reads \begin{align} \mathop{\textrm{maximize}}_{{x}}(\varphi({x})) \quad \quad \varphi : \mathbb{R}^{m} \to \mathbb{R} \quad \quad x\in \Omega_{ds}, \end{align} (3.3) with φ being an appropriate objective function. By contrast the Solution Space approach does not seek a single design point but a set of designs described by the lower |${x_{i}^{l}}$| and the upper |${x_{i}^{u}}$| bounds for each variable \begin{align} \Omega = I_{a,1} \times ... \times I_{a,p} \times I_{b,1}\times ... \times I_{b,q} \subset \mathbb{R}^{p+q} \end{align} (3.4) with |$I_{i}=\big[{x_{i}^{l}},{x_{i}^{u}}\big]$|, where |${x_{i}^{l}}$| and |${x_{i}^{u}}$| denote the lower and upper bound of xi, respectively. This box-shaped Solution Space is the solution of the following optimization problem (see Zimmermann & Von Hoessle, 2013): \begin{align} \mathop{\textrm{maximize}}_{{\Omega}\subseteq\Omega_{ds}} \mu({\Omega}) \end{align} (3.5a) \begin{align} s.t. \quad f(x)\leqslant f_{c},\quad \forall x\in \Omega, \end{align} (3.5b) where μ(Ω) is typically the volume of the set Ω and fc is a threshold value for the performance criteria. The solution is the box maximizing μ(Ω) while every design that is part of the box satisfies all constraints. Solution-compensation space are computed similarly to classical Solution Spaces, i.e. a box-shaped set of design points is derived, described by the lower and upper bounds; the difference is that only early-decision variables xa and its associated Solution Space are optimization variables. A solution-compensation space is the solution of the following optimization problem: \begin{align} \mathop{\textrm{maximize}}_{\Omega_{a}\subseteq\Omega_{ds,a}} \mu(\Omega_{a}) \end{align} (3.6a) \begin{align} s.t. \quad \forall x_{a} \in \Omega_{a}, \quad \exists x_{b} \in \Omega_{ds,b},\quad f(x_{a}, x_{b}) \leqslant f_{c}. \end{align} (3.6b) Note that only Ωa = Ia, 1 × Ia, 2 × ... × Ia, p is a degree of freedom, whereas Ωb =Ωds, b is fixed. For every design that is part of Ωa there exists at least one set of values for the late-decision variables xb such that all constraints are satisfied. 3.2. The linear problem This paper proposes an algorithm for linear systems with \begin{align} z= Fx + c =Ax_{a} + Bx_{b} + c \quad \quad x = (x_{a}, x_{b}) \quad F=(A, B) \quad A\in \mathbb{R}^{m\times p} \quad B\in \mathbb{R}^{m\times q} \quad c\in \mathbb{R}^{m}. \end{align} (3.7) In addition, the design space Ω has to be box-shaped and hence convex \begin{align} \Omega = I_{ds,1}\times ... \times I_{ds,\,p+q} \subset \mathbb{R}^{p+q}, \end{align} (3.8) with |$I_{ds,i}=\big[x^{l}_{ds,i}, x^{u}_{ds,i}\big]$|, where |$x^{l}_{ds,i}$| and |$x^{u}_{ds,i}$| denote the lower and upper bound of the design space, respectively. Similarly to expression (3.6), box-shaped solution-compensation spaces for linear system responses for the early-decision variables xa are sought. In order to derive these intervals, the following optimization problem is solved: \begin{align} \mathop{\textrm{maximize}}_{\Omega_{a}\subseteq\Omega_{ds,a}} \mu({\Omega_{a}}) \end{align} (3.9a) \begin{align} s.t. \quad \forall x_{a} \in \Omega_{a},\quad \exists x_{b} \in \Omega_{ds,b},\quad Ax_{a} + Bx_{b} \leqslant f_{c}. \end{align} (3.9b) Note that the constant c is included in fc. 4. An algorithm to compute solution-compensation spaces The idea of the algorithm is to modify the constraints (3.6b) of the initial problem statement such that the late-decision variables xb are eliminated from the expression. Late-decision variables xb are eliminated by projecting the complete Solution Space into the design space of the early-decision variables Ωds, a. This is accomplished in four steps: (1) Seek the intersections of all constraint hyper-planes. (2) Determine which of the intersections satisfy all constraints and are part of the design space. (3) Project the intersection points onto Ωds, a. (4) Determine the convex hull of the projected intersection points. In the following, these four steps are explained in detail. Considering the design space boundaries as linear constraints, the linear problem statement (3.6b) can be rewritten as \begin{align} \mathop{\textrm{maximize}}_{{\Omega_{a}}} \mu({\Omega_{a}}) \end{align} (4.1a) \begin{align} s.t. \quad \forall x_{a} \in \Omega_{a},\quad \exists x_{b},\quad Gx \leqslant g_{c}, \end{align} (4.1b) \begin{align} G = \begin{bmatrix} A \quad B\\-I\\I \end{bmatrix}\in \mathbb{R}^{(2p+2q+m)\times (p+q)}\quad \quad g_{c} = \begin{bmatrix} f_{c}\\-x^{l}_{ds}\\x^{u}_{ds} \end{bmatrix}\in \mathbb{R}^{(2p+2q+m)}, \end{align} (4.1c) where |$x^{l}_{ds} = \{x_{ds,i}^{l}\}$| and |$x^{u} = \{x_{ds,i}^{u}\}$| with i = 1, …, p + q. (1) In order to compute the intersections of all constraint hyper-planes Λc, the following procedure is applied: construct all possible sub-matrices Gk of G by removing rows such that – rank(Gk) = p + q and – Gk is quadratic. Then, construct a gc, k for each Gk by removing the same rows from gc and solve the linear equations \begin{align} G_{k} x_{k} = g_{c,k}. \end{align} (4.2) This leads to a maximum number of |$\left ({2p+2q+m}\atop{p+q} \right )$| linear equations, which have to be solved. The set of all basic solutions found is denoted as Λc. Note that these solutions are not necessarily feasible. (2) The set of all feasible vertices Λ is denoted as \begin{align} \Lambda=\{ x\in \Lambda_{c} |\: Gx \leqslant g_{c}\}. \end{align} (4.3) (3) In order to project all feasible vertices Λ, a simple projection operator is used \begin{align} p: \mathbb{R}^{p+q} \rightarrow \mathbb{R}^{p} \quad \quad p(x)=\big[\mathbf{I \quad\; 0}\big] \begin{bmatrix}\! x_{1}\\ \vdots \\ x_{p} \\ x_{p+1}\\ \vdots \\ x_{p+q} \end{bmatrix} = \begin{bmatrix} x_{1} \\ \vdots \\ x_{p} \!\end{bmatrix} \end{align} (4.4) \begin{align} \Lambda_{p} = \{p(x)|x\in \Lambda\}. \end{align} (4.5) (4) In the last step, the convex hull of Λp is determined \begin{align} \Omega_{c,a}= \textrm{conv}\{\Lambda_{p}\} = \{x|\: \tilde{A}x\leqslant \tilde{f_{c}}\}. \end{align} (4.6) In order to compute the convex hull, the quick hull algorithm developed by Barber et al. (1996) is used. The resulting polytope is called the projected early-decision variable space. After the late-decision variables xb are eliminated from the problem statement it can be written as a classical Solution Space problem \begin{align} \mathop{\textrm{maximize}}_{\Omega_{a}\subseteq\Omega_{ds,a}} \mu(\Omega_{a}) \end{align} (4.7a) \begin{align} s.t. \quad \forall x_{a} \in \Omega_{a},\quad \tilde{A}x_{a} \leqslant \tilde{f_{c}}. \end{align} (4.7b) At this point, the stochastic Solution Space algorithm (e.g. Zimmermann & Von Hoessle, 2013) as well as the corner tracking approach (e.g. Erschen et al., 2015) can be used to find the box-shaped Solution Space. 5. High-dimensional driving dynamics problem 5.1. Problem description In order to simulate the vehicle behaviour, a modified version of the two-track model by Heissing et al. (2011) is used and the output is linearized by a linear regression model. Three driving manoeuvres are simulated and assessed with respect to costumer relevant properties: Quasi-steady state cornering (QSSC; see N.N. 2012): the vehicle follows a circular trajectory with a specified radius, while the velocity is increased so slowly, such that it can be estimated as being constant at any time step. The velocity is increased up to a point where the vehicle can no longer follow its specified trajectory. Ramp steering (RAST): the vehicle maintains a constant velocity while cornering with an increasing steer angle until a certain lateral acceleration is reached. Sine with dwell (SWD, see International Organization for Standardization, 2016): the vehicle performs a lane-change manoeuvre with maximal lateral acceleration. The linearized vehicle performance measures z = Fx + c used in this application are shown in Table 1. All design goals are met when the performance measures are between their lower and upper bound, i.e. |$Fx\leqslant f_{c}$| (see Section 3). The input of the system x consists of eight variables, which can be divided into a set of four early-decision variables and four late-decision variables. Further details on the input variables are listed in Table 2. For the design of the tyres and the axles in an early-development phase, a large Solution Space is sought. Hence, the corresponding variables are considered early-decision variables. The anti-roll bars as well as the bump stops can be specified later, since these parts can be adjusted with less effort; it is easy to change the design even in the later stages of the development process, so the corresponding variables can be categorized as late-decision variables. Classical Solution Spaces and solution-compensation spaces are identified in Section 5.2. Table 1. Vehicle performance measures and the associated requirements represented by lower and/or upper bounds Perf. measure Lower bound Upper bound Unit Description, manoeuvre zα 0.65 |$^{\circ }/\frac{m}{s^{2}}$| Self-steering gradient, QSSC |$z_{a_{y}}$| 9.0 m/s2 Maximum lateral acceleration, QSSC |$z_{F_{z}}$| 550.0 N Minimum of the vertical tyre forces at the maximum lateral acceleration, QSSC |$z_{z_{FA}}$| 10.0 mm Vertical displacement of the front part of the car body at a specified lateral acceleration, RAST |$z_{z_{RA}}$| 10.0 mm Vertical displacement of the rear part of the car body at a specified lateral acceleration, RAST zΦ 0.06 – Roll angle of the body while cornering with a specified lateral acceleration, QSSC zC 3.50 – Maximum of the amplitude frequency response of the vehicle body when passing a one sided road bump zδ 4.5 – Maximum steering angle factor before loss of stability, SWD Perf. measure Lower bound Upper bound Unit Description, manoeuvre zα 0.65 |$^{\circ }/\frac{m}{s^{2}}$| Self-steering gradient, QSSC |$z_{a_{y}}$| 9.0 m/s2 Maximum lateral acceleration, QSSC |$z_{F_{z}}$| 550.0 N Minimum of the vertical tyre forces at the maximum lateral acceleration, QSSC |$z_{z_{FA}}$| 10.0 mm Vertical displacement of the front part of the car body at a specified lateral acceleration, RAST |$z_{z_{RA}}$| 10.0 mm Vertical displacement of the rear part of the car body at a specified lateral acceleration, RAST zΦ 0.06 – Roll angle of the body while cornering with a specified lateral acceleration, QSSC zC 3.50 – Maximum of the amplitude frequency response of the vehicle body when passing a one sided road bump zδ 4.5 – Maximum steering angle factor before loss of stability, SWD View Large Table 1. Vehicle performance measures and the associated requirements represented by lower and/or upper bounds Perf. measure Lower bound Upper bound Unit Description, manoeuvre zα 0.65 |$^{\circ }/\frac{m}{s^{2}}$| Self-steering gradient, QSSC |$z_{a_{y}}$| 9.0 m/s2 Maximum lateral acceleration, QSSC |$z_{F_{z}}$| 550.0 N Minimum of the vertical tyre forces at the maximum lateral acceleration, QSSC |$z_{z_{FA}}$| 10.0 mm Vertical displacement of the front part of the car body at a specified lateral acceleration, RAST |$z_{z_{RA}}$| 10.0 mm Vertical displacement of the rear part of the car body at a specified lateral acceleration, RAST zΦ 0.06 – Roll angle of the body while cornering with a specified lateral acceleration, QSSC zC 3.50 – Maximum of the amplitude frequency response of the vehicle body when passing a one sided road bump zδ 4.5 – Maximum steering angle factor before loss of stability, SWD Perf. measure Lower bound Upper bound Unit Description, manoeuvre zα 0.65 |$^{\circ }/\frac{m}{s^{2}}$| Self-steering gradient, QSSC |$z_{a_{y}}$| 9.0 m/s2 Maximum lateral acceleration, QSSC |$z_{F_{z}}$| 550.0 N Minimum of the vertical tyre forces at the maximum lateral acceleration, QSSC |$z_{z_{FA}}$| 10.0 mm Vertical displacement of the front part of the car body at a specified lateral acceleration, RAST |$z_{z_{RA}}$| 10.0 mm Vertical displacement of the rear part of the car body at a specified lateral acceleration, RAST zΦ 0.06 – Roll angle of the body while cornering with a specified lateral acceleration, QSSC zC 3.50 – Maximum of the amplitude frequency response of the vehicle body when passing a one sided road bump zδ 4.5 – Maximum steering angle factor before loss of stability, SWD View Large Table 2. Early- and late-decision variables Variable Type Unit Part Description μmax, X μmax, Y Early – tyre Maximum friction coefficient in the tyre longitudinal/transverse axle Zrc, FA Zrc, RA Early m axle Roll centre height at the front/rear axle. carb, FA carb, RA Late N/mm anti-roll bar Stiffness of the anti-roll bar at the front/rear axle. cbs, FA cbs, RA Late N/mm bump stop Stiffness of the bump stop at the front/rear axle. Variable Type Unit Part Description μmax, X μmax, Y Early – tyre Maximum friction coefficient in the tyre longitudinal/transverse axle Zrc, FA Zrc, RA Early m axle Roll centre height at the front/rear axle. carb, FA carb, RA Late N/mm anti-roll bar Stiffness of the anti-roll bar at the front/rear axle. cbs, FA cbs, RA Late N/mm bump stop Stiffness of the bump stop at the front/rear axle. View Large Table 2. Early- and late-decision variables Variable Type Unit Part Description μmax, X μmax, Y Early – tyre Maximum friction coefficient in the tyre longitudinal/transverse axle Zrc, FA Zrc, RA Early m axle Roll centre height at the front/rear axle. carb, FA carb, RA Late N/mm anti-roll bar Stiffness of the anti-roll bar at the front/rear axle. cbs, FA cbs, RA Late N/mm bump stop Stiffness of the bump stop at the front/rear axle. Variable Type Unit Part Description μmax, X μmax, Y Early – tyre Maximum friction coefficient in the tyre longitudinal/transverse axle Zrc, FA Zrc, RA Early m axle Roll centre height at the front/rear axle. carb, FA carb, RA Late N/mm anti-roll bar Stiffness of the anti-roll bar at the front/rear axle. cbs, FA cbs, RA Late N/mm bump stop Stiffness of the bump stop at the front/rear axle. View Large 5.2. Results Figure 5(a)–(d) shows a good nominal design for the vehicle considered. Each coloured dot represents one design point, where the input value of each dimension not shown in the plot is equal to the nominal design value shown in the other diagrams. The nominal design values are marked with a bold black cross. If a dot is green the design fulfils all goals. If the dot has a different colour the design misses at least one goal. The colour of each dot indicates which goal is not met. The different performance measures are explained in Table 1. For reference, the corner-tracking algorithm (see Erschen et al., 2015) is used to optimize an interval for each variable such that the resulting hyper-box includes only good designs and maximizes the volume. The result is shown in Fig. 5(e)–(h). For each dimension not shown in the respective plot, a random value (determined by Monte Carlo sampling) from the box interval shown in the other diagramsis chosen. Thus, the coloured design point borders become fuzzy. Unfortunately, box-shaped Solution Spaces that were computed using state of the art algorithms are too small for application. In particular, μmax, Y is difficult to adjust during the tyre design process, therefore, a larger interval is required to encompass the deviation between desired and realized design variable value. In order to derive larger intervals for the early-decision variables (see Table 2), the solution-compensation space is computed. When using the solution-compensation space algorithm explained in Section 4, the projected early-decision variable space is obtained, which is shown in Fig. 5(i)–(j). The dashed black box in Fig. 5(k)–(l) shows the compensation space for the late-decision variables in which they have to be able to assume any value. For easy comparison, the box-shaped Solution Space for the early-decision variables is shown. Compared to Fig. 5(e)–(h), the area of green design points has increased dramatically. For each green design point in Fig. 5(i)–(j) there exists at least one combination of late-decision variables in Fig. 5(k)–(l) such that all design goals are reached. The final result is shown in Fig. 5(m)–(p). The black box shows the Solution Space of the early-decision variables. The dashed black box shows the compensation space for the late-decision variables. The Solution Space volume is 240 times larger compared to the classical Solution Space approach for the variables μmax, Y, μmax, X, Zrc, FA and Zrc, RA. Fig. 5. View largeDownload slide (a)–(d) Nominal design (black cross) and cross-sections of the complete Solution Space (monochrome area surrounding the black cross). (e)–(h) Classical box-shaped Solution Space (black box, all design variables are early). (i)–(j) Box-shaped Solution Space (black box) and the projected early-decision variable space (monochrome area surrounding the black box). |$\Omega_{ds,b} $| is shown as a dashed rectangle. (m)–(p) solution-compensation space (black box) and the projected early-decision variable space (monochrome area surrounding the black box). |$\Omega_{ds,b} $| is shown as a dashed rectangle. Fig. 5. View largeDownload slide (a)–(d) Nominal design (black cross) and cross-sections of the complete Solution Space (monochrome area surrounding the black cross). (e)–(h) Classical box-shaped Solution Space (black box, all design variables are early). (i)–(j) Box-shaped Solution Space (black box) and the projected early-decision variable space (monochrome area surrounding the black box). |$\Omega_{ds,b} $| is shown as a dashed rectangle. (m)–(p) solution-compensation space (black box) and the projected early-decision variable space (monochrome area surrounding the black box). |$\Omega_{ds,b} $| is shown as a dashed rectangle. 6. Discussion Numerical complexity and properties of the optimization problems. The challenge is to compute solution-compensation spaces for high dimensions. The algorithm introduced in Section 4 produces a result for any linear problem within a high-dimensional space. It is divided into four steps: (1) The upper bound of the binomial coefficient |$\left ({n} \atop{k}\right )$| can be estimated with |$\left (\frac{n e}{k}\right )^{k}$|, where n = 2p + 2q + m and k = 2p + 2q. This leads to an exponential growth of the computational effort. (2) The computational complexity to determine, which of the intersections fulfil all constraints is O(s), where s is the number of intersections. (3) The computational complexity to project all good design points is O(sg), where sg is the number of good intersection points. (4) According to Barber et al. (1996), the computational complexity of the quick hull algorithm for a space with dimensionality higher than three is O(nfr/r). Where n is the number of input points, r is the number of processed points and fr is the maximum number of facets created by r vertices. The computationally most expensive part is step 1. In the industrial example (see Section 5) an eight-dimensional problem with eight performance constraints is examined. Therefore, |$\left ({24} \atop{8}\right )\approx 7.3\times 10^{5}$| linear equations have to be solved to compute all vertices. For higher dimensional problems, the number of linear systems becomes very expensive. In this case, the introduced algorithm is not suitable any more. Gain of Solution Space. According to Erschen et al. (2017) using the volume as a measure is suitable, since it correlates directly with the number of good designs. Providing larger permissible intervals for the input variables is advantageous during the development process since it provides significantly increased flexibility and allows for increased robustness against unexpected changes. For the high-dimensional driving dynamics application introduced in Section 5, it is shown that the volume of the solution for four out of eight decision variables can be increased by 240 times compared to the classical Solution Space approach. 7. Conclusion A new approach to compute solution-compensation spaces was presented. A solution-compensation space is the Cartesian product of (1) permissible intervals for so-called early-decision variables and (2) intervals for so-called late-decision variables, that can be adjusted to any value of their respective interval. The approach projects high-dimensional complete Solution Spaces for linear objective functions onto a subset of input dimensions. Then a box-shaped Solution Space is sought in the projected space. The approach maximizes the size of permissible intervals for early-decision variables. The approach is applicable for a sequential development process where a subset of variables can assume any value in a given design interval. In future research work, new algorithms will be developed in order to improve the performance and thus enable solution-compensation spaces for higher dimensional problems. Additional stochastic algorithms will be developed in order to solve problems with non-linear output functions. References Barber , C. B. , Dobkin , D. P. & Huhdanpaa , H. ( 1996 ) The quickhull algorithm for convex hulls . ACM Trans. Math. Software , 4 , 469 -- 483 . Google Scholar Crossref Search ADS Erschen , S. , Duddeck , F. & Zimmermann , M. ( 2015 ) Robust design using classical optimization-computation of solution spaces with application to chassis design . Proc. Appl. Math. Mech. , 15 , 565 -- 566 . Google Scholar Crossref Search ADS Erschen , S. , Duddeck , F. , Gerdts , M. & Zimmermann , M. ( 2017 ) On the optimal decomposition of high-dimensional solution spaces of complex systems . ASCE-ASME J. Risk Uncertainty Eng. Syst., Part B: Mech. Eng. (submitted) . Fender , J. , Duddeck , F. & Zimmermann , M. 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IMA Journal of Management Mathematics – Oxford University Press
Published: May 17, 2019
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