1 Introduction

1.1 Problems in robust design

Continuing improvement of product quality by reducing the effect of uncertain or uncontrollable factors on property responses of product quality is very important to sustaining development. Taguchi proposed a method entitled Taguchi's method in 1950s to conduct robust design for production process [1, 2], which aims to find a set of controllable input parameters (variables) to ensure the product quality being insensitive to uncertain or uncontrollable variables (noises). Furthermore, an approach of "signal–to–noise" ratio (SNR) was introduced to conduct the robust design in Taguchi method, which was used frequently [1, 2].

However, SNR and its expressions cast the two sub-responses of arithmetic value \(\overline{y}\) of a response and its dispersion γ into one new sole response, the optimization of the maximum of SNR does not equal to the simultaneous optimizations of the both minima of γ and \(\overline{y}\) approaching to its desirable target reasonably. Statisticians strongly criticized this point [3,4,5,6,7], and advised to consider both sub-responses of mean value \(\overline{y}\) and its dispersion γ with twin independent models individually. So, the concurrent optimizations of the both minima of γ and \(\overline{y}\) approaching to its desirable target should be done with separate models so as to get an effective robust optimization.

1.2 Brief introduction of robust design with probabilistic multi-objective optimization

Facing to above situation for robust design and keeping statisticians' tips in mind, a robustness evaluation with probabilistic multi–objective optimization (PMOO) was proposed [8]. PMOO was put forward from viewpoint of system theory to solve the intrinsic problems in the traditional (previous) multi–objective optimizations with subjective factors [8,9,10,11]. The novel concept of preferable probability was introduced to reveal the preference degree of performance utility indictor of a candidate scheme in the optimization. In PMOO, all performance utility indicators of alternative schemes were primarily divided into two kinds, i.e., beneficial or unbeneficial kinds in accordance with their roles or pre-required preferences in the optimization; every performance utility indicator of an alternative scheme contributes a partial preferable probability to the alternative scheme quantitatively; furthermore, the product of all partial preferable probabilities leads to the total preferable probability of the alternative scheme in the spirit of probability theory, which is the unique decisive index of the candidate scheme in the optimization process, and thus transfers the multi–objective optimization problem into a single–objective optimization one.

In the lights of probabilistic multi–objective optimization (PMOO) and the tips of statisticians that both sub-responses of mean value and its dispersion should be taken into account by using two individual models [8,9,10], the robust design with probabilistic multi–objective optimization (RDPMOO) can be conducted in principle.

In addition, sequential uniform design (algorithm) is a kind of sequential space filling design based on the idea of region compression, which was also be called sequential number theory optimization method (SNTO). Therefore, RDPMOO can be further combined with sequential uniform algorithm to formulate a hybrid of RDPMOO with SNTO to conduct deep optimization with subsequent steps for input variables.

Figure 1 illuminates the procedure of SNTO, D(1) and x(1) represent the domain and the optimum independent variables in the 1st step of sequential uniform design, D(2), D(3), x(2) and x(3), etc., indicate the subsequent corresponding quantities, respectively; the optimization domain around the provisional “optimum point” of the last step in the independent variable zone contracts step by step. This procedure is often used by practical researchers to find the global optimal point of black–box optimization problems. The core idea of this algorithm is to select and spread a point set with low deviation in each step of the compression sub–region [12, 13].

Fig. 1
figure 1

Illumination of the procedure of SNTO

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Recently, the hybrid of PMOO with SNTO was performed to conduct deep optimization subsequently [14], and the optimal scheme is with the highest total preferable probability comparatively in each step [14].

The procedure of hybrid of PMOO with SNTO is given here concisely.

Take a rectangle domain of D = [a, b] as the initial region for independent variable x, a point set with uniform distribution is determined in each step [12, 13], which is employed to assess the partial preferable probability and total preferable probability [14]. Additionally, the maximum value of total preferable probability Pi in each step can be appointed as the optimum point provisionally. Thus, a series of provisional optimization statuses is created from the sub-region during the domain compression processes. The hybrid of sequential uniform design with the PMOO is demonstrated as follows briefly [14].

0th Step: Initialization.

At moment t = 0, D(O) = D, a(O) = a and b(O) = b.

1st Step: Creation of an NT—net.

Number—theoretic method is employed to create a nt points Ƥ(t) which is distributed on D(t) = [a(t), b(t)] uniformly. Find the sampling scheme in the point set which gives the maximum value of total preferable probability Pi(x(t)) at moment t.

2nd Step: Evaluation of a novel approximate value.

Suppose x(t) ∈ Ģ(t) ∪ {x(t−1)} and M(t) such that M(t) = Pi(x(t)) ≤ Pi(y) for number of points with characteristic of nt-1 = nt = …, ∀y ∈ Ģ(t) ∪ {x(t−1)}, in which x(−1) is the empty set, x(t) and M(t) are the best approximations to x* and M provisionally.

3rd Step: Termination condition.

Suppose a parameter c(t) = (Max Pi(t−1)—Max Pi(t))/ Max Pi(t−1). If c(t) < δ, a pre-set small quantity, then x(t) and M(t) are acceptable; terminate algorithm. Otherwise, proceed to next step.

4th Step: Domain compression.

A new domain can be built, i.e., D(t+1) = [a(t+1), b(t+1)] can be set as follows: aj(t+1) = max (xj(t)—bcj(t), aj) and bj(t+1) = min (xj(t) + bcj(t), bj), in which b is a predefined compression ratio. Set t = t + 1. Go to Step 1.

According to Fang and Wang's experiences [12, 13], the assumption of n1 > n2 = n3 = … is used through the processing with b > 0 as constant.

Moreover, in our case, at kth step, Pi(x(k)) ≤ Pi(x(k−1)) in general for k > 2 only if n2 = n3 = … holds. Or else, one needs to examine the domain compression process again or stop the process of domain compression, and take the Pi(x(k−1)) and the corresponding x(k−1) as the temporary optimal results.

1.3 Problems in robust design with probabilistic multi-objective optimization

By using above procedures, a series of provisional optimization status is thus created from the process of sequential optimization of the hybrid of PMOO with SNTO, which raised a new problem of the rational option of optimization status in the optimization with multiple objectives specifically [14].

In fact, such kind of problem does not exist in the original sequential algorithm for the issues of optimization with single objective, in which the direct comparison of the response data for the single objective is accessible and convenient. However, the direct comparison of the response data for the issues of simultaneous optimization of multiple objectives is not possible since there is conflict of different objectives usually in the problem, which thus leads to the new problem of reasonable option from the provisional optimization statuses especially in multi–objective optimization.

Nevertheless, since the proposal of PMOO itself is to solve the problem of simultaneous optimization problem of multiple objectives with possible conflict among objectives, we could find an appropriate solution for this matter by means of total preferable probability in principle.

1.4 Summary

In the current paper, regulations of rational option of final optimum parameters of robust design with sequential uniform algorithm for optimization by means of PMOO are developed in Sect. 2, and examples of robust design with sequential uniform algorithm are given in details to illuminate the proposed procedure in Sect. 3.

2 Regulations of rational option of final optimum parameters in robust design with hybrid of PMOO and sequential uniform algorithm

The hybrid of PMOO with sequential uniform design was demonstrated [12,13,14], it can be used to improve the accuracy of approximate maximum by using discretization method. By employing sequential uniform algorithm for optimization, a series of provisional optimization statuses is thus produced in the subsequent steps, which induces a new problem of the option of final optimization status specifically in the MOO due to the appearances of values of the multiple objectives [14]. As was stated that there is no this kind of problem in the original sequential algorithm for the issues of optimization with single objective, since the comparison of the response data for the single objective can be done immediately and conveniently. However, the direct comparison of the response data for the issues of simultaneous optimization of multiple objectives is not possible due to the conflict of different objectives usually in the problem, which thus induces the new problem.

Keep in mind that the proposal of PMOO itself is to solve the problem of simultaneous optimization of multiple objectives with possible conflict among objectives, thus the PMOO might be employed in turn to assess the property of optimization statuses reasonably by means of total preferable probability comparatively, therefore the regulations of the assessment can be put forward in the light of PMOO itself conveniently.

The regulations of rational option of final optimum parameters from a series of provisional optimization statuses with sequential uniform algorithm for optimization by means of PMOO are demonstrated in Fig. 2, which is demonstrated as follows.

Fig. 2
figure 2

Illumination of the rational option of parameters in robust design of PMOO hybrid with SNTO

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(1) Formation of “special point set”

During the sequential uniform algorithm for optimization process, a series of provisional “optimization status” is produced from each step, thus a series of specific values of independent variables corresponding to these provisional “optimization statuses” appears, which forms a “special point set”.

(2) Evaluation of preferable probabilities of the “special point set”

Evaluations of partial preferable probabilities and total preferable probabilities of the “special point set” could be conducted by using PMOO once more.

(3) Rational option of final optimum parameters

The rational option of final optimum parameters can be obtained by means of the total preferable probability in the evaluations within this special point set comparatively. The final optimum status is with the highest total preferable probability.

3 Application examples

3.1 A robust design of an electronic circuit for electric source

The robust design with sequential uniform algorithm for optimization is taken as an example to illuminate the assessment first.

(1) The problem of robust design of an electronic circuit for electric source

Zhao once raised a robust design problem of an electronic circuit for electric source [15]. The electronic circuit is used to exchange the alternating current (AC) of 110 V into a direct current (DC) of 115V. This is a typical issue of ″target value being the best″ problem, which is quite significant in product design [15,16,17,18,19]. There are two input parameters to be optimized, i.e., the electric resistance parameter A (in Ω) and the magnification of transistor parameter B. Eleven experiments were conducted, the experimental results are shown in Table 1.

Table 1 Design of the experiment of electric source and its results
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Now, the problem is to seek a set of optimum designs for input parameters A and B to guarantee the electronic circuit for electric source with robustness under condition of parameters A suffering possible ± 10% fluctuations and parameter B bearing ± 50% uncertainties, respectively.

(2) Discretization treatment by using uniform design and sequential uniform algorithm for optimization

A regressed expression of response Y (V) vs input parameters A (Ω) and B can be obtained from the experimental data in Table 1 for subsequent use [12,13,14], which is

$$ Y \, = \, 139.9016 \, {-} \, 0.62344*B \, {-} \, 0.00163*A^{2} + \, 0.000314*B^{2} + \, 0.003953*A*B + \, 3.19*10^{ - 6} *A^{3} + \, 4.34*10^{ - 8} *B^{3} {-} \, 4.6*10^{ - 6} *A^{2} *B \, {-}1.5*10^{ - 6} *A*B^{2} , R^{2} = \, 0.9989. $$
(1)

Successively, the uniform design table U*17(175) is employed to perform the sequential uniform algorithm for optimization [12,13,14]. The design and assessment are presented in Table 2.

Table 2 Design of sequential uniform algorithm with U*17(175) and assessment result
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In Table 2, a response ε is used to reveal the discrepancy of the average value \(\overline{Y}\) of output voltage Y from its target value Y0 = 115 V, which is with the characteristic of the smaller the better, i.e., an unbeneficial indicator; while σ = [(∂Y/∂A⋅δA)2 + (∂Y/∂B⋅δB)2]0.5 reflects the uncertainty of the output voltage response Y due to the uncertainties of parameters A ± 10% suffering fluctuations and parameter B bearing ± 50% uncertainties, of which the assessment is conducted according to the error transfer function [16]; while γ = (σ2 + ε2)0.5 indicates the actual averaged deviation of the output voltage response Y from its target value Y0 = 115 V in accordance with the mean squared error (MSE) criterion of Lin and Tu [20], which is attributed to an unbeneficial indicator, too.

Actually, the essence of the robust design of this problem is to make minimum of both ε and γ simultaneously [14].

The assessment results for partial preferable probabilities of ε and γ, as well as total preferable probability for each sampling scheme are presented in Table 2. The ranking result reveals that the sampling scheme No. 15 has the highest total preferable probability provisionally, so the sampling scheme No. 15 can be used as the primary optimum status at the first glance. Here actually the entire partial preferable probability is the total preferable probability fortunately.

(3) Sequential uniform algorithm for subsequent deep optimization

Sequential uniform algorithm for subsequent deep optimization around scheme 15 of Table 2 is performed by employing uniform design table U*17(175) repeatedly.

Table 3 represents the consequences of sequential uniform algorithm for subsequent deep optimization. If a pre-set value for δ is assumed as 2%, then the sequential uniform algorithm for subsequent optimization can be seemingly stopped at step k = 7. But here, we take 10 steps, so as to collect more data to conduct our analysis of the comparative study.

Table 3 Results of subsequent optimization with sequential uniform design table U*17(175) at each step
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Table 3 indicates that totally there appear 11 provisional candidate optimum statuses, of course the primary optimum status No. 15 of Table 2 and the 10 candidate optimum statuses of the sequential uniform algorithm are all involved.

However, the forthright comparison of the response data for the issues of concurrent optimization of multi-objective is not possible since the tendencies of different objectives usually are complex in the problem, which therefore leads to a novel problem for the selection of final optimization status.

Nevertheless, these 11 provisional candidate statuses of optimization could build a 'special set' of the possible option of the robust design issue, and the corresponding input parameters (variables) form a "special point set". Therefore, PMOO can be reasonably employed to perform an integral (whole) comparison within this "special point set" again naturally.

(4) Rational option in the subsequent deep optimization

Employing PMOO to the "special point set" of last section, the reasonable selection of final optimum status can be thus gained comparatively in manner of maximum value of total preferable probability within this special point set. Table 4 presents the assessment results of the "special point set".

Table 4 Assessment consequences on the special point set
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Table 4 reveals that at the step 9 of sequential algorithm the output response exhibits comparatively the highest total preferable probability Pt, which is tightly followed by that of the step 8. Thus, the final optimum status is at the optimum point of the step 9; the averaged value \(\overline{Y}\) of output response is 115.003 V with averaged deviation γ = 0.060 V from the target value Y0 = 115 V correspondingly. Clearly, the optimum status of the step 9 at parameters of A″ = 368.735 Ohm and B″ = 793.118 is near the test result of experiment No. 15 of Table 2, while the latter has the specific parameters of A′ = 370.588 Ohm and B′ = 782.353.

This result indicates that the optimum parameters of A″ = 368.735 Ohm and B″ = 793.118 are reasonable cooperation for the output DC voltage with target value Y0 of 115 V in conditions of the fluctuations of parameter A suffering ± 10% and parameter B bearing ± 50%.

(5) Discussion

From above robust design and analysis, it can be seen that the final optimized parameters for electric resistance and magnification of transistor are A″ = 368.735 Ohm and B″ = 793.118, which is not far from the test result of experiment No. 15 with the specific parameters of A′ = 370.588 Ohm and B′ = 782.353 in Table 1. This result indicates a rational cooperation due to the fluctuations of parameter A suffering ± 10% and parameter B bearing ± 50%. However, it differs from Zhao's results of A* = 350 Ω and B* = 260 by using visualization without detailed robust analysis [15]. It implies the necessary of providing sufficient argument for robust design.

Therefore, the relevant analysis of robust design with hybrid of PMOO and sequential uniform algorithm can be used to perform the evaluation of alternative candidate in the design.

3.2 Rational determination of the Box’s robust problem

Box once took two hypothetical samples to show the difficulty of using signal –noise–ratio (SNR) to distinguish the comparative preference of their consequences [21]. His problems are with four observed values in the circumstances of the smaller the better, respectively, which were,

Hypothetical Example 1, the observed values of the are: 0, 0, 4, 4, which leads to \(\overline{Y} = 2\), σ = 2; MSE = 8, SNR = – 9.03.

Hypothetical Example 2, the observed values of the are: 1, 2, 3, 4, which leads to \(\overline{Y} = 2.5\), σ = 1.1180; MSE = 7.5, SNR = – 8.75.

Then, Box's problem is which example is better. Obviously, Example 1 had smaller mean value, but Example 2 had smaller variance, smaller mean square error (MSE) and hence smaller quadratic loss around 0 and the more desirable value of SNR [21]. So, Box queried which example showing a preferable result.

In his opining, the Example 1 exhibits preferable to Example 2 with two response values of 0, which represents perfection, whereas Example 2 had none such kind of value.

In fact there are two respects to consider Box's robust problem in detail: 1) the optimum status is only ″the smaller the better″ for the average response value of \(\overline{Y}\) (original Box’s problem); 2) there is a target (or expected, desirable) value of Y0 = 0 for the average response value of \(\overline{Y}\). Let’s discuss them here in more details.

(1) The optimum status being ″the smaller the better″ for the average response \(\overline{Y}\)

Under this condition, the probabilistic robustness assessment could only take both mean value \(\overline{Y}\) and variance σ as dual individual response objectives which follow the suggestion of statisticians by using two separate models, its rationality is logically appropriate.

As to this problem, in spirit of PMOO both the mean value \(\overline{Y}\) and variance σ all belong to unbeneficial type of attributes. Therefore the partial preferable probabilities of the mean value and the variance and the total preferable probability of Hypothetical Example 1 are 0.5556, 0.3583 and 0.1991, respectively; while the partial preferable probabilities of the mean value and the variance and the total preferable probability of Hypothetical Example 2 are 0.4444, 0.6417 and 0.2852, individually.

Therefore, the Example 2 is superior to Example 1 in spirit of PMOO due to its larger total preferable probability.

(2) The optimum status being a ″target (or expected, desirable) value″ of Y0 = 0 for the average response value of \(\overline{Y}\)

Under this condition, the probabilistic robustness assessment could take both discrepancy ε of average value of \(\overline{Y}\) from target value of Y0 (ε =|\(\overline{Y}\)-Y0|) and averaged deviation γ of actual response value of Y from its target value Y0 as two individual response objectives, in which γ = (ε2 + σ2)0.5 in accordance with Lin and Tu [20], ε expresses the discrepancy of the average value of output voltage \(\overline{Y}\) from its target value Y0 [20]. Hypothetical Example 1 indicates its \(\overline{Y} = 2\), σ = 2, and \( \curlyvee= \sqrt 8\); while Hypothetical Example 2 indicates its \(\overline{Y} = 2.5\), σ = 1.1180, \( \curlyvee= \sqrt {7.5}\).

The evaluation results are presented in Table 5. In this condition, Hypothetical Example 1 gets the higher total preferable probability and ranked first, so, Hypothetical Example 1 is the prefer result.

Table 5 Evaluation of Box's problem under condition of target value Y0 = 0
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4 Concluding remarks

From above discussions, following conclusions can be obtained:

  1. (1)

    Continuing improvement of product quality can be effectively evaluated by using rational option of optimum parameters of robust design of PMOO hybrid with sequential uniform algorithm;

  2. (2)

    Sequential uniform design with probabilistic multi-objective optimization is employed in the robust design successively. Rational option of final optimum parameters can be obtained by means of the total preferable probability in the evaluation within the special point set, which consists of the provisional ″optimum statuses″ of the sequential uniform design (algorithm) for subsequent deep optimization. The final optimum status is with the highest total preferable probability within the special point set comparatively;

  3. (3)

    Robust design of product especially for case of ″target value being the best″ differs from the evaluations of ″the smaller the better″ and ″the bigger the better″ by means of probabilistic approach significantly. In condition of ″target value being the best″, both discrepancy of average value \(\overline{Y}\) from target value Y0 (ε =|\(\overline{Y}\)-Y0|) and averaged deviation γ of actual response value of Y from target value Y0 are employed as two individual optimal response objectives simultaneously.

  4. (4)

    Further exploration and digging in this area is still needed in future research due to its special importance.