© A. Trabelsi et al., Published by EDP Sciences 2025

1 Introduction

Multidisciplinary teams strive to develop designs that comply with uncertainties coming from manufacturing, environmental, deterioration, degradation sources. In literature, studies on sustainable variability have a connection with robust design optimization (RDO) [1–3] and the reliability-based design optimization (RBDO) [4–6] approaches. This research is included in the RDO, especially, robustness for dynamic and static processes. The goal of the research is to determine the input parameter setting that desensitizes the process to the effects of the internal and external noise factores. Moreover, controlling the initiation of lattente effect because of cross interaction. The tolerance design stage is carried after to ensure a trade-off between the functional requirements, the deterioration effect/rate, and the overall cost of the product/process over the operating cycle time [7,8]. The deliverable is design, that accommodates operating factors so that the system response gets closer to the target value(s) with minimum variation in presence of background noise [9]. The Taguchi [10–12] technique − a groundbreaking piece of research in the realm of static and dynamic robust optimization design, involves two consecutive steps: (i) setting of the dispersion factors to minimize a quality loss function or maximize a signal-to-noise ratio, (ii) then, selection of the adjustment factors allowing to shift of the average output onto a process target. However, the Taguchi method can only answer mono-objective problems, moreover, it does not guarantee the same optimum at each stage unless the mean and the variance of the response are uncorrelated [13]. The research efforts to expand the Taguchi method's to address static and dynamic multi-objective processes are fuzzy and elusive to understand [14]. Therefore, different techniques such as the grey function, the desirability function and their hybrids [15], the principal component analysis techniques [15], the fuzzy logic [16], the meta-heuristic methods [17], the multiple regression models [18], and more recently, the frontier function [19] have been devised to bring viable alterntives to the Taguchi method for multi-objective processes. Even though these methods are effective in anserwing robust design optimization of static multi-objective processes, they are still in their infancy and to some extent uncapable of addressing dynamic multi-objective processes. Other methods such as the meta-heuristic and data mining techniques have shown success in this regard, but still, they failed to address issues such as producing the ideal mix of control elements in continuous space [20], for instance.

Dynamic multi-objective techniques for robust design optimization can be divided into three main categories, i.e., meta-heuristic [21,22], multiple attribute decision-making [23–26], and mathematical modeling [12,27,28]. Other strategies that take advantage of the quality loss function have also been developed to optimize both static and dynamic problems at the same time [29]. Dynamic robust design optimization consists of monitoring the output (Y) to a target T (min, max, or nominal) under the dynamic control of a signal factor (S). One solution is to determine the setting of the input parameters, which minimizes the average loss function L(Y,S,T). Formally, the dynamic model for multi-objective processes is stated as in equation (1):

$$Y_{js}=f_{js}\left(X,S_s\right)+e_{js}$$(1)

where X is a set of control factors, xi, i = (1,I). The response Y_js is the output j, j = (1,J) at the signal level s, s = (1,S). The fjs is the transfer function for the jth response while the signal factor S is set at the sth level. The e_js is the composed error for the Y_js process response.

In general, it is accepted that the signal factor(s) and the system output(s), Y_js have a linear relationship, as shown in equation (2):

$$Y_{js}=\beta \cdot S+{\epsilon }_{js}.$$(2)

In light of this constation, Taguchi proposes a two-step procedure to optimize dynamic mono-objective processes [12]: (i) maximization of the dynamic signal-to-noise ratio, (ii) then, adjustment of the slope (β); i.e., β = 0, 0 ≤ β ≤ ∞, and β = ∞ upon the objective function, i.e., Dynamic Smaller the Best (DSB), Dynamic Larger the Best (DLB), or Dynamic Nominal the Best (DNB), respectively. However, the two-step procedure is unpractical, as argued earlier since it is unpractical to find a set of control and dispersion parameters, which manage multi-objective responses, simultaneously. To solve the problem, one must first define a global objective function and then optimize it to bring each response variable as close to the target value as feasible. At each of the signal levels, the goal value should vary as little as possible [30].

In literature, many alternatives to the Taguchi dynamic multi-objective systems have been developed [31–33]. However, the frontier production approach has only been considered in a few research works. The article discusses the robust optimization design of dynamic systems while utilizing the stochastic frontier paradigm (SF). It attempts to expand on the ongoing research on the static Robust Design of Products and Processes using the Stochastic Frontier model (RDPP-SF) [34,35].

Most experimental research has adopted the design of experiment (DoE) methodology. Ideally, robust processes are laced with pure symmetrical random variation, solely, thus, they are under statistical control and reliable in time. However, nonrandom variation may initiate not only because of internal and external noises but also because of a violation of the premises of the DoE for industrial processes, i.e., the replication blocking and the “full randomization. In many scenarios such a hard-to-change factors, frequent clamping-unclamping operations, nonhomogeneity in raw materials, etc., full randomization is unpractical. The RDPP-SF method uses the stochastic frontier model to estimate composed errors, i.e., random and nonrandom components. The econometric studies, which link technical inefficiency with nonrandom variability among the Decision Making Units (DMUs), are the source of the SF paradigm that the RDPP-SF method. Thus, robustness in the RDPP-SF method to technical inefficiency.

The rest of paper unfolds as follows. Section 2 outlines the concept behind the stochastic frontier method. Section 3 presents the RDPP-SF method for static and dynamic multi-objective systems. In Section 4, a case study is proposed to demonstrate the employability of the RDPP-SF method for dynamic processes, hereinafter, called Dynamic Robust Design of Processes and Products using the Stochastic Frontier model (DRDPP-SF). The computer program, FRONTIER 4.1® is used to estimate the maximum likelihood of the parameters of the stochastic frontier model and tests statistical hypotheses about the functional form of the stochastic frontier, the robustness metric (γ-value), and the distributional forms of the non-random variation component. Section 5 is the conclusion, and it provides the scope and limitations of the RDPP-SF method for static and dynamic robust optimization design.

2 Stochastic frontier model

The stochastic frontier methodology is a widely used econometric technic for estimating the inefficiency of DMUs. The method can define the current state of technology in the industry and also measure the individual performance of the DMUs [36]. Initially, Aigner et al. [37], Meeusen and van Den Broeck [38], and Schmidt and Lovell [39] developed the SF model. Aigner et al. [37] devised the formal representation of a production frontier model is given in equation (3).

$$Y_i=f\left(x_i;\beta \right)+(v_i-u_i)$$(3)

where,

• Y_i and x_i are naturally logged variables representing the outputs and the inputs of the DMU_i (i = 1,N), respectively. The log transformation is employed to ensure the convexity of the stochastic production function as well as mitigate of the variability in the Y_is:

• β = [β₀β₁ … β_k] ^' is a (k + 1) column vector coefficients to be estimated;

• v_i represents the statistical noise distributed as $V_i\sim N\left(0,{\sigma }_v^2\right)$. In production processes, the v_i captures part-to-part variation, gauges repeatability, and the combined effects of unseen factors;

• u_i is a one-sided random term distributed as U_i ∼ N(0, σ_u²) and represents the technical inefficiency of the DMU, i. In production processes, the u_i stands for the variance brought on by noise factors such as environmental conditions, tool wear, material deviation from specifications, drift of parameter setting from the nominals because of time and/or manufacturing variation, lack of expertise/management performance, etc.

For output-oriented technical inefficiency, the ui term explains why a given DMUi cannot achive the feasible maximum output beyond random noise. As expressend in equation (1) and illustrated in Figure 1, the composed error term is the focus of the model. It allows for the conventional symmetrical random variation, $V_i^{iid}\sim N\left(0,{\sigma }_v^2\right)$, and a positive one-sided disturbance term, U_i ∼ iid|N(μ, σ²|, which represents the nonnatural variation occurring in a process.

Fig. 1 Output-oriented technical efficiency using the stochastic frontier method (SF).

3 Static and dynamic RDPP-SF method

3.1 RDPP-SF method for static processes

The static RDPP-SF method, which is devised by Trabelsi and Rezgui [34,35], is dedicated to estimating the inherent random and nonrandom variation in a signal-free process that is rife with internal and external noise sensitivity. The output-oriented production model, which is given in equation (3) is adopted by the RDPP-SF method. Table 1 shows how the static RDPP-SF technique and the econometric stochastic frontier model map out.

The procedural scheme of the RDPP-SF method for static and dynamic multi-objective problems is shown in Figure 2 and it involves four steps.

Step 1 (data preparation): Define the DoE strategy and assign factor levels. Different formattings are acceptable, even though, a ±3σ coding is preferred. The reason is that the three sigma quality level is adopted by major Design For Six Sigmas (DFSS) procedures. By analogy to the econometric model, every combination of the factor levels (run) in the designed experiment, is seen as a decision-making unit (DMU), which makes use of the resources x_i (process inputs). The process responses, Y_jer are scaled and translated depending on whether a nominalization, maximization, or minimization target is sought. The outputs of the types smaller the best (STB) and nominal the best (NTB) should be transformed and scaled beforehand since the stochastic function initially aims to produce the maximum output for each combination of the inputs (x_i).

Equation (4) states the transformation for the NTB outputs.

$$Y_i=\exp \left[-abs\left(y_i-y_T\right)\right].$$(4)

For the STB outputs, the transformation is given in equation (5).

$$Y_i=\frac{1}{y_i}.$$(5)

The original interval of the raw data for the STB and NTB cases is recovered using equation (6).

$$Y_{scaled}=\left(U_b-L_b\right)\frac{Y_i-Y_{\min }}{Y_{Max}-Y_{\min }},$$(6)

where y_i, y_T, Y_i, Y_scaled, U_b, and L_b represent, in that order, the actual output (not transformed), the target value, the transformed output (not scaled), the transformed and scaled output, and the upper and lower bounds of the original data interval.

Step 2 (data analysis): The input-output function is used to account for both the interaction and the individual effects of the input parameters, x_i. The following hypotheses are tested at a 95% confidence level for each output (Y_jer) using the FRONTIER 4.1^® program.

• (H₀: β_k = 0 vs. H₁: β_k ≠ 0). The goal is to test the statistical significance of the two-way interactions among the process parameters.

• (H₀: γ = 0 vs. H₁: γ > 0). The test checks, for each Y_jer, the statistical significance of non-random variations in a process, which could be brought on by both internal and external noises. In any case, the test establishes whether there is confounding between the stochastic frontier (SF) and the average line model (RSM).

• The test (H₀: U_i∼half normal vs. H₁: U_i ∼ truncated normal distribution) ascertains the distributional form of the non-random variation component. The RDPP-SF method for static and dynamic processes assumes that the u_i(s) are following a half-normal distribution as suggested in Aigner et al. [37].

Step 3 (constitute the uncertainty arrays for each Y_jer): The components of the composed error term (u_i and v_i) are estimated using the test on the γ-value (test b in step 2). For each Y_jer, the uncertainty array is created as follows. If γ ≥ 95% then the v_i ≈ 0, and the u_i(s) estimates compose the uncertainty array (ε_i). The variations in the process is entirely non-random. If γ ≤ 5% then the u_i ≈ 0 and the process is experiencing pure random unit-to-unit variations, solely. The uncertainty array (ε_i) is then composed of the v_i(s) estimates at each run. The average line model (RSM) is confounding with the SF model in this case. If 5% ≤ γ ≤ 95%, the two types of variations-random unit-to-unit and nonrandom, are there and should be accounted for. The uncertainty array (ε_i) is composed of the estimates (ε_i = v_i − u_i) for each run. In the FRONTIER4.1^® program, the individual inefficiency (exp(–u)) is used to calculate the u_i(s) terms for each run.

Step 4 (determination of the robust design solution): The robust optimization design solution correlates with the run in the DoE layout, which adds up to a minimum (Σ abs(ε_i)) across the Y_jer(s). Trabelsi and Rezgui [34,35] have compiled applications of the RDPP-SF method for static processes.

Fig. 2 Procedural scheme of the RDPP-SF method for static and dynamic processes.

3.2 RDPP-SF method for dynamic processes

In dynamic multi-objective processes, the control factors should be set at suitable levels so that the signal-output relationship is insensitive to internal and external noise. The dynamic DRDPP-SF process, which still adheres to the functional scheme depicted in Figure 2, modifies steps 1 and 3.

Step 1 revised (data preparation): In dynamic processes, each output, Y_jer, is correlated to a signal factor level (S = s). The designed experiment is formatted so that it reflects the combination levels of the control factors, x_ei (e = 1, E and i = 1, I), the replication, r (r = 1, R), and the signal set, S (s = 1, S). The apdated DoE layout formatting for a dynamic output, Y_jer, at a signal level, S = s is displayed in Table 2.

Step 3 revised (determination of the robust design solution): The uncertainty array for evry output, Y_jer, and signal level, (S = s) is composed is composed using the estimate of the γ-value (refer to step 3 above). The global uncertainty array is obtained by totaling the individual uncertainty scores for the outputs Y_jer over the S^J factorial combinations across the E executions (runs) of the designed experiment runs. The level combination of the input factors (x_ie, i = 1, I, and e = 1, E) and the signal level (S = s), giving the minimum global uncertainty score corresponds to the robust solution.

4 The DRDPP-SF method: an illustrative example

A simulated dataset, which is retrieved from Chang and Chen [20] is used to demonstrate the employability of the DRDPP-SF method for dynamic processes. The DoE plan is an L18 OA, which is replicated once. The process has four control factors x_i, (i = 1, 4); each at three discrete levels (1, 2, and 3). The process outputs Y_jer (j = 1, 3, e = 1, 18, and r = 1, 2) are of DLB, DNB, and DSB types, respectively. The signal factor has two levels, s = 0.1, and s = 0.2. Table 3 shows the L18 dataset. The DRDPP-SF is executed in the same manner as the static RDPP-SF, except for steps 1 and 3, which are revised in the following section.

Step 1 revised: The dataset is templated as in Table 2 for each signal level (S = s). Because the rationale behind the stochastic frontier function (SF) drew on the maximization of the production level, the dynamic outputs Y_2er (DNB) and Y_3er (DSB) are transformed and mapped to their original bounds using equations (4), (5), and (6). The target value for the output, Y_2er, is decided to be the average of the raw data, i.e., Ȳ₂= 0.987 at s = 0.1 and s = 0.2. Table 4 shows the prepared dataset with s = 0.1 and the original and transformed outputs (Y_2er) and (Y_3er).

Step 2 (data analysis): The x_i and Y_jer values given in Table 4 should be logged before the FRONTIER4.1^® program starts processing. The Neperian log transformation is applied to eliminate extrema, correct skewness, and assure convexity of the stochastic frontier. The DRDPP-SF method considers two input-output models: the Translog, a quadratic regression in log(x_i), which takes interactions into account, and the Cobb Douglas, a multiple linear regression in log(x_i), which does not. The hypotheses tests (H₀: CD vs TL) in Table 5 show the Cobb-Douglas model is the best fit for Y_1er, and Y_3er when the signal levels are set at s = 0.1 or s = 0.2, indistinctly. This is because the LR statistics (13.23, 14.50, −49.10, and 7.20, respectively) are smaller than the critical value, ${\chi }_{10}^2=18.31$. However, a TL model provides a superior fit for Y_2er at signal levels, s = 0.1 and s = 0.2 (LR 18.96, and 41.48, respectively). In the remainder, we apply the TL model to all outputs. For the robustness index (γ_estimate), Table 5 indicates that the test (H₀: γ = 0 vs H₁: γ > 0) is statistically significant at a 95% confidence interval regardless of the signal levels, i.e., the Log Ratios are higher than the critical value ${\chi }_c^2={\chi }_1^2\left(10\%\right)=\mathrm{2.71.}$ Therefore, the RSM model does not confound with the stochastic frontier for all outputs Y_1er, Y_2er, and Y_3er.

The regressional models for the process responses, Y_1er, Y_2er, and Y_3er are provided in Table 6.

Step 3 revised (constitute the uncertainty arrays for the outputs Y_1er, Y_2er, and Y_3er): Using ε_i = (v_i − u_i) for each run, i, the uncertainty arrays for the dynamic outputs, Y_1er, Y_2er, and Y_3er at the signal levels s = 0.1 and s = 0.2 are obtained. Table 7 shows the uncertainty scores (ε_i) as generated by the FRONTIER^® 4.1 program.

There are six uncertainty arrays in total since the uncertainty arrays are arranged based on the signal level for each output. The global error at each run, i, of the L18 layout is calculated by totting the uncertainty scores over the factorial combinations of the signal levels for the outputs Y_1er, Y_2er, and Y_3er as shown in Table 8. It represents the number. of applications from the set of {Y_1er, Y_2er, and Y_3er} to the set of two signal levels {s = 0.1 and s = 0.2}, i.e., 2³ combinations. Because the purpose is to determine the control and signal factor settings, which produce the least amount of variation in magnitude rather than direction, the absolute values of the uncertainty scores are considered.

The global uncertainty of 0.986 (line 2 in Tab. 8), for instance, is obtained when summing up the absolute errors in run 2 while setting Y_1er at s = s₂ = 0.2 (ε₂ =|0.058|), Y_2er at s = s₂ = 0.2 (ε₂ = |0.600|), and Y_3er at s = s₁ = 0.1 (ε₂ = |–0.328|). According to Table 8, when the signal factor is set at s = 0.1 for Y_1er and Y_2er and s = 0.2 for Y_3er, respectively, the minimum global error of 0.032 is met. This corresponds to run 1 (x1 = x2 = x3 = x4 = 1), which is deemed as a robust optimization solution. Another workable solution is run 17 (x1 = 3, x2 = 2, x3 = 1, and x4 = 3), which has a global uncertainty score of 0.040. In certain operational scenarios, it is often advised to consider the same signal sitting for all outputs. In this scenario, run 7 (x1 = 3, x2 = 1, x3 = 2, and x4 = 1) with a signal setting of s = 0.1 for Y_1er, Y_2er, and Y_3er is the proper alternative. The global uncertainty error is 0.091.

5 Conclusion

The RDPP-SF approach for dynamic systems is covered in this article. The method's novelty may be seen in how the stochastic frontier composed error is decomposed, i.e., random and nonrandom components. So, the control of dynamic processes, which are vulnerable to environmental noise, unseen control factors, and violation of the DoE premises such as the randomization, replication, and blocking principles is affordable. A major application of the static and dynamic RDPP-SF method is estimatimg the design and manufacturing process maturity. This is performed using the test on the γ-value (H0: γ = 0 vs. H1: γ > 0). So, at a 5% level, for instance, if γ ≤ 5%, the process is mature since it only experiences random variance and may be tested for long-term statistical process control. If the γ-value falls between 5% and 95%, then, the sensitivity is marginal, and both natural and non-natural sources of variation are accountable for. If the γ ≥ 95%, the nonrandom sources of variation are predominating, meaning the process is immature and should be improved/redesigned. The presence of non-random variation is the main cause of short-term drift and bias in a process. The additional advantages and limitations of the DRDPP-SF are outlined below.

1. The RSM optimization technique, which uses the statistical Ordinary Least Square method, yields a constant uncertainty estimate. On the other hand, the stochastic frontier model provides a variable (and often better) estimate of the uncertainty; i.e., each pairing of the input and signal parameters has its uncertainty estimate.

2. The stochastic frontier model is parametric, uses flexible functional models, can estimate the standard errors, and makes use of the hypotheses tests to ascertain the statistical significance of the non-natural variation using the Maximum Likelihood Method. In that regard, it is superior to other frontier-based methods such as Data Envelopment Analysis [19], which incorporates noise as part of the efficiency score.

3. The proposed DRDPP-SF method can provide the link between robustness and the signal-to-noise metric, which is used in many robust design and reliability methods. The estimate of the γ-value in the DRDPP-SF method can be used to do this. Furthermore, the method holds the most promise for engineering fields, such as reliability, versatility, resilience, adaptability, and flexibility.

4. The DRDPP-SF method uses a Translog as a transfer function to account for the interactions among the control factors. Nevertheless, more research is still needed to figure out other functional forms that are more suitable to manufacturing processes.

5. The DRDPP-SF technique has considered robust optimization design of processes but in discrete space. Research using Latin hypercube sampling and ANN-GA algorithm is launched to address optimization in continuous space.

Funding

This research received no external funding.

Conflicts of interest

The authors declare no potential conflicts of interest concerning the research, authorship, and/or publication of this article.

Data availability statement

The data that support the findings of this study are available from the corresponding second author (Mohamed-Ali Rezgui) upon reasonable request.

Author contribution statement

Conceptualization, A.T. and M.A.R.; Methodology, A.T. and M.A.R.; Software, A.T., M.A. and A.D.; Validation, A.T. and A.D.; Formal Analysis, A.T. and M.A.R.; Investigation, A.T. and M.A.R.; Resources, A.T. and M.A.R.; Data Curation, A.T., M.A.R.; Writing − Original Draft Preparation, A.T. and M.A.R.; Writing − Review & Editing, A.T., M.A.R. and J.H.; Visualization, A.T., M.A.R. and H.J.; Supervision, A.T. and M.A. and H.J.; Project Administration, M.A.R. and H.J.; Funding Acquisition, H.J. All authors have read and agreed to the published version of the manuscript.

References

All Tables

All Figures

	Fig. 1 Output-oriented technical efficiency using the stochastic frontier method (SF).
In the text

	Fig. 2 Procedural scheme of the RDPP-SF method for static and dynamic processes.
In the text