Design Improvement for Complex Systems with Uncertainty

: The uncertainty of the engineering system increases with its complexity, therefore, the tolerance to the uncertainty becomes important. Even under large variations of design parameters, the system performance should achieve the design goal in the design phase. Therefore, engineers are interested in how to turn a bad design into a good one with the least effort in the presence of uncertainty. To improve a bad design, we classify design parameters into key parameters and non-key parameters based on engineering knowledge, and then seek the maximum solution hyper-box which already includes non-key parameters of this bad design. The solution hyper-box on which all design points are good, that is, they achieve the design goal, provides target intervals for each parameter. The bad design can be turned into a good one by only moving its key parameters into their target intervals. In this paper, the PSO-Divide-Best method is proposed to seek the maximum solution hyper-box which is in compliance with the constraints. This proposed approach has a considerably high possibility to ﬁnd the globally maximum solution hyper-box that satisﬁes the constraints and can be used in complex systems with black-box performance functions. Finally, case studies show that the proposed approach outperforms the EPCP and IA-CES methods in the literature.


Introduction
With the rapid development of technology in recent years, the engineering system has become increasingly complex. The growing complexity of the system provides a new source of uncertainty. Uncertainty arises because some design parameters may be changed over the course of development, or they cannot yet exactly be specified [1][2][3]. The uncertainty appears as a deviation between the desired and realized parameter settings, which may bring in design failure, therefore it is essential to improve design under the uncertainty for complex systems.
Design points that achieve their design goals are called good, otherwise they are called bad. However, if there is already a bad design point, engineers are interested in how to turn it into a good design with the least effort. It can be fulfilled by modifying as few parameters from the existing bad design as possible.
Traditional optimization [4], sensitivity analysis [5,6], robust design optimization [7][8][9], and reliability-based design optimization [10][11][12] are some classical approaches to improve designs. Traditional optimization seeks an optimum design point in the design space, which does not take uncertainty into account. Sensitivity analysis computes the effect of variability of input design parameters on variability of output performance value. Unfortunately, sensitivity analysis does not provide information on how the parameters of a bad design need to be changed in order to turn it into a good design. Robust design optimization seeks a design point in a deterministic neighborhood with small output variation. However, robust design optimization only deals with the case where the variability of design parameters is given. If the variability of the design parameters is not Therefore, we focus on seeking for the maximum solution hyper-box which is in compliance with constraints. Fender et al. [28] extended the algorithm in [21] to account for constraints. However, the algorithm in [28] produces a hyper-box that may not have the largest size and may contain some bad designs, because the locations of good and bad designs are estimated by the Monte Carlo sampling. In this paper, the proposed PSO-Divide-Best method combines particle swarm optimization (PSO) [29] with the Divide-the-Best algorithm in [30], where the former is to evolve the solution hyper-box towards larger volume and the latter is to guarantee that the obtained solution hyper-box only includes good designs. The PSO-Divide-Best method is established based on black-box performance functions, so it is appropriate for most engineering problems.
The paper is organized as follows. The next section explains the motivation of this paper and gives a mathematical problem statement for optimizing the solution hyperbox with constraints. Section 3 introduces the Divide-the-Best algorithm and particle swarm optimization. Section 4 presents the proposed approach in detail. In Section 5, we apply the proposed approach to complex systems. In Section 6, we provide some concluding remarks.

Motivation
A voltage divider shown in Figure 1 is considered. The terminal voltage V ab is expressed as V ab = 10R/(R + R a ) and the resistance as R in = R + R a . and R in ¼ R þ R a : Let us assume that V o ¼ 10 V and that R ¼ 50 V and R a ¼ 70 V: We can evaluate how V ab and R in are affected by uncertainty in R and R a . For example, if R a ¼ 70 V and R ¼ 50^10 V we obtain that V ab will vary between 5.40 and 6.36 V and 110 # R in # 130 V: Additionally, if R a ¼ 70^30 V; then V ab will vary between 4 and 7.14 V and 80 # R in # 140 V: Now suppose that specifications require that V ab between 4.5 and 5.5 V and R in between 80 and 120 V. Fig. 3 shows the feasible region (region delimited by dotted lines) [9]. How should we select R and R a in order to satisfy both specifications?
In general, robust design deals with non-linear optimisation and it is first assumed that y(x) is a continuous function [5]. The proposed approach, based on CES, does not require this assumption.
A second normal assumption is that x must vary in a continuous way within a so-called experimental region X. For the approach developed, the region simply consists of lower and upper bounds on the design parameters. In a technological context it is common to use an expression about a relation y(x) like 'the relation is valid on this range' [4]. In the voltage divider example, we are looking for the maximum deviation for R and R a such as to satisfy the specification. Fig. 4 shows the theoretical tolerance region for R and R a . R o and R ao are the centre values, whereas DR and DR a are the maximum deviations to be determined. We will assume that with any parameter the range of variation is symmetrical about a central or nominal value.

Feasible solution set and approximate desc
Suppose the area shown in Fig. 5(a) is the fea that is any pair (R, R a ) in that zone sa specifications.
An exact description of the feasible solution [10] or feasible operating region [11] (Fig. 5(a)) i not simple, since it may be a very complex set. the FSS could be limited by non-linear function reason, approximate descriptions are often looke simply shaped sets like boxes containing (outer Fig. 5(b)) or contained in (inner bounding, Fig. 5 the set of interest. In particular we are looking (Fig. 5(c) and (d)): we can guarantee that whate R a ) is selected in those boxes, it will compl specifications. The maximum ranges of possible of the feasible values are the sizes (along co-ord of the axis-aligned box of minimum volume cont [10]. The ranges for each variable define th tolerance region [9].
Several references report on methods to d FSS. For example, Milanese et al. [10] present te obtain MIB. Such methods can be applied in the the FSS is limited by linear functions. The   Good designs fulfilling the design goals should satisfy: 80 Ω ≤ R in ≤ 120 Ω.
The complete solution space defined by Expression (1) is shown in Figure 2 as a grey region. Now, a design with R = 43 Ω and R a = 60 Ω is considered. It violates Equation (1a) and therefore is a bad design. To improve this bad design with the least effort, three scenarios are considered: (a) Figure 2a shows a solution hyper-box with the maximum volume. Both R and R a are changed so as to achieve the design goals. (b) Figure 2b shows a solution hyper-box that includes R a = 60 Ω. To achieve the design goals, only R needs to be modified. The solution hyper-box is obtained by requiring a minimum safety margin of ±2 Ω for R a . This is essential, because R a is subject to uncertainty and cannot be controlled exactly. The realized lower safety margin is larger than the required minimum safety margin, thus more tolerance to uncertainty is provided. (c) Figure 2c shows a solution hyper-box that includes R = 43 Ω. To realize the design goals, only R a needs to be modified. The same minimum safety margin as in Scenario (b) is required. The solution hyper-box of Scenario (a) is much larger than those of Scenarios (b) and (c). In this sense, the solution hyper-box of Scenario (a) is the most robust one. However, Scenario (a) requires redesigning two design parameters, while Scenarios (b) and (c) only require redesigning one design parameter. If there is already a bad design, Scenarios (b) and (c) are recommended, because they only change one parameter. Otherwise, Scenario (a) is more appropriate, because it provides more tolerance to the uncertainty. To choose among Scenarios (b) and (c), two different strategies are possible. If there is no information on which design parameter is easier to redesign, the scenario with the maximum solution hyper-box is chosen, because it provides a larger target region and is easier to reach. This would be Scenario (a) with R as the key parameter. As alternative approach, engineers can take advantage of their engineering knowledge and choose Scenario (b) where R a is regarded as the key parameter, if they know that design parameter R a is easier to redesign than R.
This procedure can described by seeking the maximum solution hyper-boxes under the constraint that certain parameters should be included with a specified safety margin.

Problem Statement
Let x = (x 1 , x 2 , . . . , x m ) represent the design point or design, which is composed of m design parameters. The output performance at x is given by where f (x) is the performance function. The performance is sufficient by satisfying the performance criterion where f c is a predefined threshold value. Designs or design points are called good if their performance satisfies (3), otherwise they are called bad. The lower and upper bounds of the design parameters, x l and x u , define the continuous design space. The region consisting of all good design points is the complete solution space.
To obtain interval boundaries of each parameter that are independent of other parameters, hyper-boxes are considered. The hyper-box D = D(x low , x up ) is expressed by ) denote the bounds of the hyper-box, that is, We see that the first m components of x bound are the lower bound x low and the last m ones are the upper bound x up . The volume of the hyper-box D(x low , x up ) is thus given by A hyper-box which only includes good design points is called a solution hyper-box. The design parameter which can be modified easily with low modification cost is called the key parameter. The design parameter which is hard to be modified is called the non-key parameter.
As mentioned in Section 2.1, to turn a bad design into a good one with the least effort, a solution hyper-box which already includes non-key parameters of the bad design is sought. Formally, the problem we focus on is formulated as the following constrained optimization problem: where k is the number value that corresponds to the dimension of a non-key parameter, K is the set of the indices of non-key parameters, and x low ck and x up ck are the lower and upper limit values, respectively. The constraints x low k ≤ x low ck and x up k ≥ x up ck ensure that the non-key parameter is included in the obtained hyper-box with a specified safety margin.
It can be seen from (7) that the aim of this paper is to seek for the largest solution hyper-box satisfying the constraints. The proposed problem is fundamentally different from robust design optimization in that it is to seek the intervals of a permissible design range, and interval boundaries are used as degrees of freedom rather than design parameters.
If the explicit expressions of f (x) and min x∈D(x low ,x up ) f (x) are analytically known, the problem (7) can be solved by classical optimization methods. However, the performance function f (x) is not analytically known in most engineering problems, thus min x∈D(x low ,x up ) f (x) cannot be explicitly expressed by x low and x up . Therefore, it is necessary to propose an approach which aims at black-box performance function.

Divide-the-Best Algorithm
The problem (7) in Section 2.2 involves the following box-constrained global optimization sub-problem: where . . , m} is an mdimensional hyper-box (hyper-rectangle) and f (x) is a performance function.
Numerous algorithms have been proposed (see, e.g., [27,[30][31][32][33][34][35][36][37][38][39]) for solving the problem (8), under the assumption that the performance function f (x) satisfies the Lipschitz condition over the hyper-box with an unknown Lipschitz constant. The Divide-the-Best approach is the most well-known partitioning-based one. Particularly, the Divide-the-Best method in [30] has very promising performance, which uses the multiple estimates of the Lipschitz constant and an efficient diagonal partition. Therefore, the Divide-the-Best method in [30] is adopted in this paper to solve the problem (8). Figure 3 shows the flow Mathematics 2021, 9,1173 6 of 20 chart of the Divide-the-Best algorithm in [30]. Given a vector q k of the method parameters, at each iteration k, the admissible region D(x low , x up ) is adaptively partitioned into a collection {D k i } of the finite number of robust subsets D k i .

Estimation of the Lipschitz Constant
Step 2 Calculation of Characteristics

Stopping Criterion k > Kmax
Step 3 The Promising Hyper-box and New Trials Step 4 Diagonal Partition Figure 3. Flow chart of Divide-the-Best algorithm in [30].
More precisely, in Step 1, several possible Lipschitz constants are chosen from a set of values varying from zero to infinity, denoted byL. Then, the "merit" (called characteristic) Figure 3). The hyper-box over which characteristic is the minimum is called the "promising hyper-box". It has higher possibility to find the global minimizer within the "promising hyper-box" D k t (see Step 3). Subsequently, the new sample points are obtained from the old ones by adding and subtracting the two-thirds-side length of the longest edge of the "promising hyper-box". Then, the evaluation of performance function at the new sample point is performed. Finally, in Step 4, the "promising hyper-box" is subdivided by an efficient diagonal partition strategy for performing the next iteration (see [30] for more details). Naturally, more than one "promising hyper-box" can be partitioned at every iteration. The stopping criterion is that the number of iteration reaches the pre-defined maximal allowed number. The evaluation of the performance function at a point is referred to as a trial.
To better demonstrate how the Divide-the-Best algorithm performs, its first three iterations on a two-dimensional function in [30] are shown in Figure 4.  The Divide-the-Best algorithm in [30] balances global and local search in a more sophisticated way and provides a faster convergence to the global minimizers of difficult multi-extremal black-box performance functions.

Particle Swarm Optimization (PSO)
Particle swarm optimization (PSO) is a population-based meta-heuristics technique [29,40]. The population consists of potential solutions, called particles, which are a metaphor of birds in bird flocking. The particles are randomly initialized and then freely fly across the multidimensional search space. In social science context, a PSO system combines a socialonly model and a cognition-only model [41]. The social-only component suggests that particles ignore their own experience and adjust their behavior according to the successful beliefs of particles in the neighborhood. On the other hand, the cognition-only component treats individuals as isolated beings. One advantage of PSO is that it often locates near optimal solutions significantly more quickly than evolutionary optimization [42].
Each particle is equivalent to a candidate solution of the problem (7). The particle moves according to an adjusted velocity, which is based on the corresponding particle's experience and the experience of its companions. The velocity of the ith particle is modified under the following equation in the PSO algorithm: where t is the current iteration number, v t i is the velocity of the ith particle at iteration t, x bound,t i is the position of the ith particle at iteration t, pbest t i is the best position of the particle i until iteration t, gbest t is the best position among all particles until iteration Iter max t is the inertia weight at iteration t, Iter max is the allowed maximum iteration number, c 1 and c 2 are weight factors, and r 1 and r 2 are random numbers between 0 and 1. Naturally, according to Equation (5), x bound,t i represents the bounds of the ith hyper-box at iteration t.
It is worth mentioning that the second term of Equation (9) represents the cognitive part of PSO where the particle changes its velocity based on its own thinking and memory. The third term represents the social part of PSO where the particle changes its velocity based on the social-psychological adaptation of knowledge.
Each particle moves from the current position to the next one by the modified velocity in (9) using the following equation: (10)

The Proposed Particle Swarm Optimization Divide-the-Best Algorithm
The particle swarm optimization algorithm does not rely on mathematical properties for application. The Divide-the-Best algorithm in [30] is an efficient algorithm to calculate the global minimum of a black-box performance function over a hyper-box. Therefore, an innovative approach which combines the particle swarm optimization algorithm with the Divide-the-Best algorithm is proposed to solve the problem (7), referred to as the particle swarm optimization divide-the-best algorithm (PSO-Divide-Best).
Specifically, particle swarm optimization drives the evolution toward increasing the volume of hyper-box. The Divide-the-Best algorithm solves the optimization sub-problem (8), and thus ensures that the obtained hyper-box only includes good designs.
The PSO-Divide-Best algorithm is illustrated in Algorithm 1. Initially, it generates randomly a group of particles satisfying constraints. During the process of optimization, at iteration t, firstly, the velocity v t+1 i and position x bound,t+1 i of the ith particle are updated according to Equations (9) and (10), respectively. Then, if the conditions x low ck , and ∀k ∈ K are not satisfied, the re-updating is performed. Otherwise, the value of min x∈D(x low,t+1 is evaluated by the Divide-the-Best algorithm in Figure 3 and denoted by f t+1 min,i . If the constraint f t+1 min,i ≥ f c is met, we calculate the corresponding volume according to Equation (6), otherwise the particle is re-updated. Next, pbest t+1 i and gbest t+1 are updated. The ith particle velocity v t+1 i is updated according to Equation (9). If a particle violates the velocity limits, set its velocity equal to the limit.

2.1.2
The position of each particle x bound,t+1 i is modified by Equation (10). If a particle violates its position limits in any dimension, set its position equal to the limit.

2.1.4
If the constraints x low,t+1 are not satisfied, go to Step 2.1.1.

2.1.5
Calculate the value of min x∈D(x low,t+1 Figure 3 with parameter setting K max = 100 ln 2m , denoted by f t+1 min,i .

2.1.6
If the constraint f t+1 min,i ≥ f c is met, then calculate the corresponding volume by Equation (6), namely u t+1 Update gbest t+1 as the one with the maximum volume among all pbest t+1 i , i = 1, . . . , N.

Stopping criteria:
If t < Iter max , set t = t + 1 and go to Step 2.1. Finally, to verify whether the best particle satisfies the constraints, the Divide-the-Best algorithm with more iterative times is performed. If the constraints are satisfied, PSO-Divide-Best outputs the best particle. Otherwise, it goes back to Step 2.
PSO is derivative-free and global search [43]. The Divide-the-Best algorithm in [30] can converge to the global minimum with any degree of accuracy provided there are enough iterations. Therefore, the proposed PSO-Divide-Best method in this paper has the following advantages: (1) PSO-Divide-Best has great possibility to reach the globally maximum solution hyperbox satisfying the constraints.
(2) Due to the discrete nature of the trial points in the Divide-the-Best algorithm, the PSO-Divide-Best method can be applied to both analytically known and black-box performance functions. (3) PSO-Divide-Best guarantees that any point selected within the obtained hyper-box is a good design provided that the performance function is continuous.
In most engineering problems, the performance function is continuous whether it is analytically known or black-box. Therefore, the PSO-Divide-Best approach has strong applicability in engineering problems.

Case Studies
A stochastic approach based on Monte Carlo sample is discussed in [28]. This approach consists of two phases: exploration phase and consolidation phase. The purpose of the exploration phase is to identify a solution box as large as possible. The consolidation phase includes an algorithm which shrinks the hyper-box such that it contains only good designs. Therefore, we denote this method by "EPCP" hereafter (an abbreviation for "exploration phase and consolidation phase"). Besides, a method in [24] which combines interval arithmetic with cellular evolutionary strategies is denoted by "IA-CES" hereafter.
To compare the proposed PSO-Divide-Best method with the EPCP and IA-CES ones, two cases are considered. The first case is the vehicle structure design problem which has been studied by Fender et al. in [28]. The second case is the power-shift steering transmission control system (PSSTCS) with a price of approximately 500,000 USD.
In the proposed PSO-Divide-Best method, the weight factors c 1 and c 2 are set to 2, the maximum number of iterations is 2000, ω max = 0.9, ω min = 0.4, and the number of particles N is set to 160.
All experiments were performed in MATLAB R2016b on a windows platform with Intel Core i7-4790 CPU 3.60 GHz, 16 GB RAM.

Vehicle Structure Design Problem
The vehicle structure design problem in [28] consists of two structural components. The design parameters are the two deformation forces x 1 and x 2 . The performance functions are as follows: where x = (x 1 , x 2 ), m = 2000 kg is the mass, a c = 32 g is a critical threshold level, v 0 = 15.6 m/s is the speed, and µ 1c = µ 2c = 0.3 m are the limits of the deformation measures. The design goals are achieved if f i (x) ≥ 0 for all i = 1, 2, 3. A design with x 1 = 275 kN and x 2 = 450 kN is considered in [28]. It violates Equation (11a) and therefore is a bad design. To improve this bad design with the least effort, three scenarios are performed. In Scenario (a), both x 1 and x 2 are key parameters, that is, both F 1 and F 2 need to be modified. In Scenario (b), only x 1 is the key parameter. More precisely, x 2 is included in the solution hyper-box that is obtained by requiring a minimum safety margin of ±25 kN. In Scenario (c), only x 2 is the key parameter.
Since each performance function in Equation (11) is monotone, the exact solution is obtained by the penalty method [44], as shown in Table 1. The results in [28] obtained by the EPCP method are listed in Table 1. Due to the stochastic nature of the PSO, as adopted in [24], we ran the PSO-Divide-Best algorithm 20 times for each scenario, and the best solutions among the 20 runs are shown in Table 1. The results obtained by the IA-CES method are also shown in Table 1 relative error is given by ϕ low i = ∆x low i /x low 0,i . Besides, the relative error for the upper boundaries of the hyper-box are defined in a similar way. The row entitled "volume" is the volume of hyper-box, and the row entitled "error" is the relative error between the volume of hyper-box computed numerically and the exact one.     Table 1 shows that the hyper-boxes obtained by the PSO-Divide-Best method are nearly identical to the exact ones, while the hyper-boxes obtained by the EPCP and IA-CES approaches have a high deviation from the exact ones. Besides, the hyper-boxes obtained by the PSO-Divide-Best method are significantly larger than those obtained by the EPCP and IA-CES methods.
Furthermore, a visualization of the results of the EPCP, IA-CES, and PSO-Divide-Best methods in Table 1 is shown in Figures 5-7, respectively, where the grey region is the complete solution space. Figures 5-7 also give some examples of how to turn the bad design into a good one.
In Scenarios (a) and (b), the blue hyper-boxes exceed the grey region. This implies that the hyper-boxes obtained by the EPCP method contain some bad designs. Based on the hyper-box obtained by the EPCP method, a bad design may fail to be turned into a good one (see Figure 5a). Consequently, the EPCP method should be used with great caution. The yellow hyper-boxes obtained by the IA-CES method all lie within the grey region; however, they are not the largest ones. This implies that target intervals for each parameter determined by the IA-CES method are relatively shorter. Therefore, the good design point which is determined by the IA-CES method has relatively poor robustness against unintended variations. The green hyper-boxes obtained by the PSO-Divide-Best method all locate within grey region, therefore they only include good designs. Actually, as long as the key parameters of the bad design are moved into their target intervals defined by our method, the bad design is turned into a good one. Taking Scenario (b) as an example, according to the solution hyper-box obtained by the PSO-Divide-Best method shown in Table 1, the bad design is turned into a good one by changing   proposed in terms of expounding the GO operators representing a standby structure in any place. Next, the multifunction characteristics, multi-state units and closed-loop feedback link are explained, and a new exact algorithm with shared signals is proposed. In addition, the new availability assessment method is explained in detail by conducting a system analysis, developing the GO model, determining the test units, collecting the availability data of the test units and evaluating the system availability's lower confidence limit. Based on the results, the process is formulated. Then, the ECSoPSST is taken as an example to evaluate the lower confidence limit of its system availability as determined by the new availability assessment method. Finally, to verify the advantages and rationality of the new availability assessment method, the availability assessment result and evaluation efficiency are compared with those obtained by the regular Monte Carlo method and the regular exact GO algorithm with shared signals. Furthermore, the coverage rate of the system availability's lower confidence limit by the new availability assessment method is compared with the nominal significance level. The comparison results show that this method has the following obvious advantages:

The Power-Shift Steering Transmission Control System (PSSTCS)
1) The availability assessment method discussed in this paper uses the GO model as the system availability model, and thus, it can connect the system structure, functions, and system characteristics directly and closely. The method is also easy to check. Moreover, it can solve the availability assessment modeling for a noncoherent system and avoid the influence of the experience of engineer(s) in availability modeling. 2) The availability assessment method uses the GO operation to obtain the system availability, and thus it can avoid an influence of sampling on the availability assessment result. Moreover, it can obtain a stable availability assessment result with higher efficiency. The PSSTCS is a non-monotonic coherent system which consists of 86 components. The design goal is that the system reliability should be higher than 0.9900. However, the system reliability function is not analytically known, so it is evaluated by the goal-oriented (GO) reliability assessment method in [45]. Now, we consider a design shown in Table A1 in Appendix A. Its system reliability is 0.9856; therefore, it is insufficient. To improve this bad design with the least effort, firstly, the engineers classify design parameters into key parameters and non-key ones according to modification difficulty degree. The key parameters (components) are marked with red color in Table A1. Secondly, we search the maximum solution hyper-box which already includes non-key parameters of this bad design. More precisely, in this PSSTCS case, the optimization problem (7) is formulated as follows: where R low = (R low 1 , . . . , R low 86 ), R up = (R  3,4,5,10,11,12,13,15,16,17,18,22,23,26,28,30,31,32,35,36,37,38,39,42,43,45,51,55,56,57,59,63,65,66,68,69,71,72,73,82, 83}, f c = 0.9900 is the reliability threshold, and f (R) is the system reliability function which is evaluated by the GO method in [45]. Note that the volume here is replaced by the log-volume (logarithmic transformation of the volume) in order to calculate conveniently.
The IA-CES method is established based on analytically known function; therefore, it is not applicable to this complex system with black-box reliability function. The obtained hyper-boxes of the problem (12) by the EPCP and PSO-Divide-Best methods are listed in Table A3 in Appendix A. We see that the lower bounds obtained by the PSO-Divide-Best method are almost always smaller than those obtained by the EPCP method, and the upper bounds obtained by the PSO-Divide-Best method are almost always larger than those obtained by the EPCP method. This implies that the PSO-Divide-Best method provides much wider target intervals for most design parameters. Therefore, the PSO-Divide-Best method has stronger robustness against uncertainty.
Besides, the log-volume of the obtained hyper-box is listed in Table 2. We see that the log-volume of hyper-box obtained by the PSO-Divide-Best method is much larger than that by the EPCP method. This is further reflected that the PSO-Divide-Best method is more robust against variations.
The hyper-box provides target intervals for each design parameter; therefore, the bad design can be turned into a good design by only moving its key parameters into their target intervals. Particularly, the good design for which the key parameters are located at the midpoints of their target intervals may be the most representative one. It provides the maximum robustness if the variation of design parameter is the same on both sides of a nominal value. These representative good designs obtained by the EPCP and PSO-Divide-Best methods are listed in Table A1. To verify whether these good designs satisfy the design goal, their system reliabilities were evaluated by the GO method, as also shown in Table A1. We can see these two good designs indeed achieve the design goal. Therefore, the system reliability for the PSSTCS can change from insufficient to sufficient by only modifying the reliabilities for key components according to Table A1.
To illustrate whether the obtained hyper-boxes meet the design goal, we use the Latin hypercube sampling to choose n designs from each hyper-box, and then calculate the rate of good designs as follows: where #(·) is the indicator function, i.e., #( Figure 10 shows the rates of good designs under different sample sizes. We see that the rates of good designs of the PSO-Divide-Best method are all 1, while those of the EPCP method are all below 1. This implies that the hyper-box obtained by the PSO-Divide-Best method only includes good designs, while those obtained by the EPCP method includes some bad designs. Therefore, the EPCP method should be used cautiously, because it may fail to turn a bad design into a good one. However, the PSO-Divide-Best method is valid as it ensures any design within the obtained hyper-box is good.

Conclusions
To improve a bad design with comparatively little effort in the presence of uncertainty, rather than changing all design parameters of this bad design, this paper only modifies its key parameters. To this end, the maximum solution hyper-box which already includes non-key parameters of a current bad design is sought. The solution hyper-box provides target intervals for each design parameter. A current bad design can be turned into a good one by only moving its key parameters into their target intervals. The volume of the solution hyper-box should be as large as possible for providing stronger robustness against unintended variations.
The PSO-Divide-Best algorithm combines the PSO and the Divide-the-Best algorithms [30] to seek a solution hyper-box which has the maximum volume and satisfies all the constraints. The case studies show the solution hyper-boxes obtained by the PSO-Divide-Best method only include good designs, and they are much larger than those obtained by the EPCP and IA-CES methods. This implies that a good design determined by the EPCP method may have stronger robustness against uncertainty. Therefore, our method is better than the EPCP and IA-CES methods.
Since the Divide-the-Best algorithm only evaluates the performance function at trial points, the PSO-Divide-Best method has strong application and can be applied to complex systems with black-box performance functions. As long as the performance function is continuous, our method can provide a solution hyper-box that is guaranteed to include only good designs. Therefore, a bad design can be turned into a good one provided that its key parameters are moved into the target intervals obtained by our method. This implies that our method is valid. Our method converges to the globally maximum solution hyper-box with very great probability. However, its convergence speed becomes slow as the number of design parameters increases. Therefore, the scalability of our method is relatively poor.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript:

PSO
Particle Swarm Optimization PSSTCS Power-Shift Steering Transmission Control System PSO-Divide-Best Particle Swarm Optimization Divide-the-Best algorithm EPCP Approach in [28] including exploration phase and consolidation phase IA-CES Method in [24] which combines interval arithmetic with cellular evolutionary strategies Appendix A