Multi-criteria quality assessment of parts produced by single point incremental forming process: application of the ELECTRE method
- Open Access
- 16.12.2025
- ORIGINAL ARTICLE
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Abstract
1 Introduction
During the last years, the customization of the final products is progressively gaining a larger visibility. The paradigm of customization, indeed, can be approached according to several perspectives and, among the other, one is the geometrical complexity: more complex parts, especially for the automotive applications, generally lead to higher performance [1]. Such an improvement is even more evident if the complex part is made of a light material: light alloys [2], in fact, are universally recognized as the main solution to simultaneously reduce the vehicles’ mass, reduce the harmful emissions and bring toward a greener transportation system. Nevertheless, it should be underlined that most of the light alloys – aluminium and magnesium alloys above all – are characterized by a poor formability at room temperature which limits the achievable complexity, at least in a single step process.
Therefore, combining simultaneously two key aspects (complexity and reduced mass) is not trivial: the alloy has to be defined as well as the manufacturing route capable of overcoming the drawbacks related to the poor formability. Among the several innovative and flexible sheet metal forming processes able to accomplish such an ambitious goal [3], the Single Point Incremental Forming (SPIF) has promoted itself as a valuable candidate: the deformation progresses incrementally due to the local action of a rotating tool along a predefined path. Thanks to the local nature of the interaction between the tool and the blank, low forming forces are needed if compared with conventional sheet forming processes [4]. Besides its flexibility in processing different classes of materials, from metals [5‐7] to polymers [8] as well as composites [9], the main advantage of the SPIF comes probably from the forming mechanism that allows to achieve a formability limit remarkably higher than those found in conventional sheet metal stamping due to the localization of plastic deformation [10]. Additional systematic studies [11], based on the adoption of the numerical simulations, shed the light on the theory of distributed localized failure, the so called “noodle theory”, according to which the distribution of localized small deformations prolongs the speed of localization at one concentrated location, thus increasing the formability in the SPIF process. In light of this, the SPIF process has recently selected for the manufacturing of a complex benchmark geometry to fully demonstrate its huge potentialities [12]. Flexibility and versatility do not represent the only positive aspect of the SPIF process: the shorter time needed to design the process and the lower costs (the die is not strictly necessary) allow to reduce the environmental impact and represent two main advantages especially for small production scenarios [13].
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Due to its high flexibility and versatility, the SPIF process has been applied for the manufacturing of complex parts for automotive [9, 14], aerospace [15] and biomedical applications [16‐18].
Optimization plays a vital role in manufacturing processes, as it reduces waste, increases productivity, and improves overall system performance by analyzing and fine-tuning key parameters such as time, cost, quality, and resource consumption. Using mathematical algorithms and artificial intelligence, these methods provide optimal solutions to challenges such as production scheduling, resource allocation, and quality control, ultimately helping to reduce operating costs, increase competitiveness, and achieve sustainable production. The large number of variables involved in the SPIF process needs a systematic approach to properly optimize the quality of the formed part [19], sometimes focusing on demonstrators to better investigate the effect of the input parameters on the outcome of the process [20].
Numerous studies have been conducted by researchers on the optimization of the aforementioned process. Hussain et al. [21] focused on enhancement of profile accuracy in SPIF operation. This research work investigated forming parameters using response surface methodology (RSM) and analysis of variance (ANOVA), experimentally. They demonstrated that sheet thickness, wall angle, pitch size, and the interaction of sheet thickness with wall angle have a remarkable influence on the profile accuracy. Surface roughness is of great importance in the SPIF process. Accordingly, a comprehensive empirical study on the influences of SPIF forming conditions on surface roughness and formability has been carried out by Mulay et al. [22]. They investigated the effects of feed rate (f), pitch depth (p), tool diameter (d), and sheet thickness (ti) on surface roughness (Ra) and maximum forming angle (Ømax) in AA5052-H32 alloy. In their study, a mathematical model was developed using RSM-BBD. Based on their findings, ANOVA results revealed that pitch depth and tool diameter have a remarkable effect on surface roughness and formability (P < 0.0001), such that increasing pitch depth and decreasing tool diameter increased surface roughness, while increasing these parameters decreased the forming angle. Sbayti et al. [23] employed optimization techniques to improve quality of SPIF process in biomedical application. They concentrate on optimizing the SPIF main parameters for production of titanium dental plates. Using Box-Behnken experimental design and response surface methodology (RSM), numerical simulations were performed, and multi-objective genetic algorithms and Global Optimum Determination by Linking and Interchanging Kindred Evaluators algorithm were employed to find the optimal values. The optimization objectives included minimizing the sheet thickness, final depth, and also forming force. After production of specimen with the optimal parameters, comparing the final geometry with the target geometry by an optical measurement system showed that this method provides a robust solution for selecting the optimal parameters in this operation. Yan et al. [24] presented multi-stage tool path optimization in the SPIF process. In Their research, they focus on optimization of geometrical accuracy and reducing processing time in a two-stage SPIF strategy. At first, finite element analysis was employed with validation based on available experimental data. In the following, a statistical design of experiments approach was applied to optimize this two-stage technique. Based on their outcomes, the step size in the second stage had a remarkable influence on geometric accuracy, while the forming depth in the first stage had little effect on part quality. The geometric improvement was mainly observed in the base and wall regions, while the regions close to the clamping system had little improvement. After optimization, the average geometric deviation and forming time were decreased by, in turn, 25% and 55.56%, demonstrating the success of this two-stage strategy in improving process accuracy and speed, simultaneously. In another research work, Bhandari et al. [25] focused on optimization of forming parameter in SPIF process of A5052 Aluminum alloy. In their study, computer simulation was used to predict the findings and set the optimal parameters in the SPIF operation before conducting actual experiments. A 3D conical model was designed in Fusion 360 software and then simulated to determine the maximum forming depth before cracking. In the practical part, the influence of forming variables on the mentioned process was investigated by considering this depth and using grease as a lubricant to decrease friction. In the following, an optimization method was applied to reduce the cracking areas in the G-code. This optimization technique was able to effectively decrease the errors caused by the forming depth and provide a remarkable improvement in the final quality of the formed specimen. Mevada et al. [26] used optimization process in novel SPIF process to improve thinning and geometrical accuracy of Al alloys parts. This research is concerned with the effects of process variables such as tool diameter, pitch size, feed rate, and spindle speed on sheet thickness reduction and geometric accuracy in single point incremental forming process through numerical simulation. According to their reports, sheet thickness reduction is affected by tool diameter, feed rate, and spindle speed, so that increasing tool diameter and feed rate outcomes in thinner sheets. Geometric accuracy is also strongly influenced by tool diameter and feed rate, and increasing tool diameter increases geometric deviation. An inclusive study on the optimization of SPIF process to achieve better micro-hardness and surface roughness for Inconel 625 super-alloy specimens has been performed by Bishnoi and Chandna [27]. The effect of four variables, namely tool tip diameter (5–15 mm), pitch size (0.8–0.2 mm), tool spindle speed (500–1100 rpm), and wall angle (50–65 degrees), was analyzed using RSM. Their findings showed that tool tip and pitch diameters have the greatest impact on micro-hardness and surface roughness, respectively. Under optimal conditions (tool diameter 15 mm, pitch 0.4 mm, spindle speed 900 rpm, and angle 57.5 degrees), the desired values of micro-hardness (486.8 HV) and surface roughness (0.432 microns) were achieved. Bouhamed et al. [28] employed numerical optimization of single point incremental forming of steel matrix composites assisted by elastoplastic damage model and desirability-based RSM. This research study focused the formability and damage mechanisms in the SPIF process for ferritic steel matrix composites reinforced with TiB₂ ceramic particles. Concentrating on three key variables of sheet thickness, forming tool diameter, and cone wall angle, the influences of these factors on deformation mechanics and process performance were analyzed in detail. An inclusive computational method was employed to simulate and generate response surfaces to optimize the forming parameters. A multi-objective optimization framework was presented that allows achieving optimal conditions for maximum formability and minimum damage using the utility function. Based on their reports, the results of present study not only show the high potential of TiB₂-reinforced composites in advanced forming applications but also provide practical solutions for industries working with innovative composite materials.
The ELECTRE method is an acronym for “Elimination et Choice Translating to Reality. In this method, all options are evaluated using non-ranked comparisons, thereby eliminating ineffective options. The steps of the ELECTRE technique are based on a concordance set and a discordance set, which is why it is known as “outranking analysis”. The main goal of the ELECTRE method is to provide a framework for comparing options based on outranking relationships. Compared to simple additive weighting (SAW) or weighted sum model (WSM) methods, ELECTRE does not calculate a final score; instead, it seeks to identify a set of non-dominated options based on two types of relations: concordance and discordance. Therefore, its application in solving nonlinear problems with complex relationships, such as metal forming, is particularly important.
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According to the scenario described, it is clear that the SPIF process is regulated by several parameters and various indicators can be identified to evaluate the quality and soundness of the final part. Therefore, the present work focusses on the adoption of the Elimination and Choice Expressing Reality (ELECTRE) technique to identify several manufacturing scenarios (characterized by different combinations of spindle speed, step down and feed rate) to simultaneously optimize process indicator as the fracture depth, the forming load and the roughness of the final part (a truncated pyramid).
The ELECTRE method is superior to other Multi-Attribute Decision Making (MADM) methods in many problems due to its flexibility in managing uncertainty and weighting, as well as the ability to make accurate pairwise comparisons. Accordingly, in the present research work, an attempt has been made to select the best experiment by considering different scenarios and determining the importance of each indicator. The simultaneous evaluation of the three indicators by determining the desired importance weight for each of them helped to select the most appropriate values of the process parameters, and this type of comprehensive study has not been done for the SPIF process so far.
2 Material and methodology
2.1 The alloy under investigation
The SPIF tests were carried out on 150 mm square samples extracted from a 1 mm thick AZ31-B sheet purchased in the H24 temper. The mechanical properties of the alloy, as well as its chemical composition, are reported in Table 1.
Table 1
Mechanical properties and chemical composition of the AZ31-B alloy
Mechanical properties | Ultimate Tensile Strength [MPa] | Yield Stress [MPa] | ||||||||
283 ~ 294 | 223 ~ 232 | |||||||||
Chemical composition | Al | Zn | Mn | Ni | Fe | Cu | Si | Ca | Others | Mg |
2.5 ~ 3.5 | 0.7 ~ 1.3 | 0.2 ~ 1.0 | < 0.005 | < 0.005 | < 0.06 | < 0.05 | < 0.04 | 0.30 | Balance | |
2.2 Experimental SPIF tests
Experimental SPIF tests were carried out on a Kondia HS1000 3 axis milling machine equipped with a Fidia numerical control. An overview of the test chamber is reported in Fig. 1a, where also the thermographic camera is evident for the temperature monitoring during SPIF operations.
Fig. 1
Experimental set up for the SPIF tests: (a) overview of the Kondia 3 axis and (b) detail of the fixture for the blank clamping
Figure 1b shows a detailed view of the clamping fixture that, in turn, was bolted on a steel structure equipped with a Kistler 9257B table-type dynamometer: loads exerted by the punch during its local action on the blank were monitored and collected every 0.1 s (frequency of 10 Hz) using a DaqBoard® 505 data acquisition card managed by the DaqView® 9.0.0 software.
Before carrying out the SPIF tests, the oxide (naturally developed) on the specimens’ surface was mechanically removed using a grain abrasive sponge. An ordered grid of 3 mm circles was printed on a thick paper and transferred to the Mg surface once grinded. Finally, the blank was clamped and the surface experiencing the contact with the punch tip covered by an abundant quantity of Houghton TD-52 lubricant, specific for metal forming operations and characterized by a light-colored appearance due to the presence of chlorine-based agents to improve the extreme pressure behavior.
The target geometry was a pyramidal frustum characterized by a circular generatrix (radius of 80 mm) and an initial wall angle inclination of 45 degrees and a final depth of 43 mm, as shown in Fig. 2a. The toolpath was designed according to a Z-level contouring (assuming that the Z direction is the vertical one) while considering the bidirectional countering (the tool changed its direction in each plane).
Fig. 2
SPIF part: (a) Schematic representation of the target geometry (dimensions: mm) (b) Example of surface roughness measurement (detailed view of the stylus in contact with the inner surface of the formed part)
During each test, the temperature was acquired using the thermographic camera IRBIS ImageIR 3300, setting a frequency of 1 image every 10 s: in such a way, the temperature could be continuously monitored in the proximity of the tool/surface contact without affecting the process.
As for the load data, the acquired forces were processed in the Matlab environment to get rid of the noise and neglect the component along the x and y axis since the predominant force in ISF is developed in the axial direction of the tool [29].
At the end of each test, post forming properties of the samples were analyzed in terms of: (i) final height and, for parts where premature fracture occurred, height at fracture, (ii) principal strains by measuring the deformed circle and calculating the main components in the blank plane (Ɛ₁, Ɛ2), (iii) the average surface roughness, along a distance of 8 mm in three different sections of the same specimen, by means of a Mitutoyo Surftest SV-2000 profilometer (Fig. 2b reports also a detailed view of the stylus while in contact with the inner surface of the formed part).
SPIF tests were designed according to a central composite design plan. Since the formability of the sheets was supposed to be enhanced because of elevated temperatures caused by frictional heat, Spindle Speed (S), Step Down (Δz) and Feed Rate (F) were chosen as input variables (factors). In fact, frictional stress exerted at the tool-sheet interface is made of two components: a meridional component due to the step down movement of the tool and a circumferential component due to the feed rate combined with the spindle speed [30]. The CCD plan was built up combining a reduced factorial plan (8 designs), one central point (that was replicated three times) and the star points extending beyond the factorial design and characterized by a value of α equal to 1.682. The interval ranges of the CCD plan, along with the indication of the star points, are enlisted in Table 2.
Table 2
CCD plan for the SPIF tests: interval ranges and star points
Input Variable | Upper bound | Lower bound | Star Points |
|---|---|---|---|
Spindle Speed, S [rpm] | 5000 | 3000 | 2318/5682 |
Step Down, Δz [mm] | 0.5 | 0.2 | 0.1/0.6 |
Feed Rate, F [mm/min] | 500 | 300 | 232/568 |
2.3 The ELECTRE methodology
As mentioned earlier, the Elimination and Choice Expressing Reality (ELECTRE) method, or approximate dominance, is one of the remarkable multi-criteria decision-making methods. Various articles provide examples of the application of this method, including project ranking, facility planning, factory site selection, strategic planning, and more. In this method, all options are evaluated using non-ranking comparisons, thereby eliminating ineffective options [31]. All these stages are based on a concordance and discordance matrixes, which is why this method is also known as concordance analysis [32]. The ELECTRE method has various versions. This technique possesses the following characteristics: (a) It is classified as a compensatory method, (b) Independence of attributes is not required, and (c) Qualitative attributes are transformed into quantitative attributes [33‐35]. The steps of the ELECTRE method are shown in Figs. 3 and 4. The ELECTRE is an effective technique that offers several advantages. This method has a logical and clear structure that makes the decision-making process understandable and provides high flexibility, allowing for the modification of criteria and weights based on specific needs [36]. Additionally, ELECTRE pays special attention to the preferences and priorities of individuals, enabling a deep analysis of options and the identification of their strengths and weaknesses [37]. One of the important advantages of this method is that it does not require independence between indicators, meaning that dependent criteria can also be considered. This method simplifies the decision-making process by reducing complexity and making it easier to compare options based on various criteria. Furthermore, ELECTRE facilitates collective decision-making by taking into account the opinions of group members and ultimately ensures increased transparency and credibility in the decision-making process. ELECTRE presents outcomes as a set of ranks instead of providing a linear ranking of options [38]. This allows decision-makers to identify potentially suitable options and discard undesirable ones. Overall, this method helps improve the quality of decisions and facilitates the decision-making process.
Fig. 3
A comprehensive flowchart of ELECTRE Method (part I)
Fig. 4
A comprehensive flowchart of ELECTRE Method (part II)
3 Results and discussion
3.1 The SPIF tests
An overview of the formed samples after the 17 tests is reported in Fig. 5.
Fig. 5
SPIF tests: overview of the formed samples
Thanks to the sensors equipping the Kondia HS1000 milling machine, along with the availability of the thermal camera, several quantities were monitored during each test while some other could be calculated at the end of the test, as detailed in Sect. 2.2. All the output quantities are enlisted in Table 3.
Table 3
Results from the SPIF tests
Run | Input variables | Output variables | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
S | Δz | F | Test duration [h] | Hmax [mm] | Hfracture [mm] | Fmax [N] | Safe | Tmax [°C] | Ɛ₁,max | Ra [µm] | |
01 | 4000 | 0.35 | 400 | 1.7 | 42.7 | 36.4 | 760.63 | no | 252.7 | 0.80 | 0.402 |
02 | 4000 | 0.35 | 400 | 1.7 | 42.7 | 36.5 | 676.16 | no | 261.6 | 0.86 | 0.327 |
03 | 4000 | 0.35 | 400 | 1.7 | 42.7 | 37.1 | 740.03 | no | 274 | 0.81 | 0.306 |
04 | 3000 | 0.2 | 300 | 2.8 | 28 | 24.5 | 751.65 | no | 253.4 | 0.65 | 0.243 |
05 | 3000 | 0.2 | 500 | 2.2 | 34.7 | 30.6 | 792.18 | no | 221.7 | 0.84 | 0.382 |
06 | 4000 | 0.35 | 232 | 2.7 | 42.7 | 36.5 | 685.09 | no | 259.2 | 0.79 | 0.498 |
07 | 4000 | 0.35 | 568 | 1.3 | 42.7 | 42.7 | 728.06 | yes | 230.2 | 0.86 | 0.354 |
08 | 3000 | 0.5 | 300 | 1.4 | 43 | 43 | 927.87 | yes | 194 | 0.84 | 0.544 |
09 | 3000 | 0.5 | 500 | 0.2 | 6.6 | 6.6 | 985.73 | no | 158.9 | 0.31 | 0.803 |
10 | 5000 | 0.2 | 300 | 2.8 | 43 | 39.6 | 607.6 | no | 276 | 0.87 | 1.303 |
11 | 5000 | 0.2 | 500 | 2.5 | 43 | 43 | 549.54 | yes | 302.3 | 0.90 | 0.576 |
12 | 5000 | 0.5 | 300 | 1.5 | 43 | 37.58 | 662.97 | no | 300.4 | 0.90 | 1.723 |
13 | 5000 | 0.5 | 500 | 1 | 43 | 43 | 729.77 | yes | 281.1 | 0.89 | 0.682 |
14 | 2318 | 0.35 | 400 | 0.2 | 4.55 | 4.55 | 732.85 | no | 89.9 | 0.23 | 1.942 |
15 | 5682 | 0.35 | 400 | 1.7 | 42.7 | 37.73 | 667.61 | no | 303.3 | 0.88 | 0.868 |
16 | 4000 | 0.1 | 400 | 0.6 | 3.4 | 3.4 | 568.54 | no | 87.7 | 0.20 | 2.091 |
17 | 4000 | 0.6 | 400 | 0.9 | 37.2 | 28.6 | 838.32 | no | 189.2 | 0.83 | 0.59 |
Reported data suggest that, for each of the tested conditions, two final heights were recorded since the moment at which the crack appeared was not always the same moment when the test was stopped due to the leakage of the lubricant. Therefore, the quantity Hmax refers to the final height at which the test was stopped, whereas Hfracture is the height at which the rupture could be clearly distinguished: an example is reported in Fig. 6a. On the contrary, there were only two cases in which Hfracture and Hmax were coincident: (i) a sound part was obtained (see Fig. 6b) or (ii) rupture occurred in the very first stages so the test could be immediately stopped (see Test #09, Test #14 and Test #16 in Table 3).
Fig. 6
SPIF tests: (a) fractured part (Test #06) in the corner region and (b) sound part (Test #08)
The availability of the thermal camera during each test allowed to draw some interesting considerations. At first, as shown in Fig. 7a, the maximum temperature was always located in the contact region between the blank and the tool, as a further evidence of the local nature of the incremental forming process. Moreover, by plotting the evolution of the recorded maximum value throughout the whole test (see Fig. 7b), it is evident that the first stages were characterized by a progressive increase due to the combination of the frictional heat and the portion that was accumulated by the material. Although the contact area changed continuously (according to the tool path), the blank tended to experience a continuously higher temperature as a consequence of the heat generated in the previous stages. Nevertheless, after a certain amount of time (around 3000 s in Fig. 7b), the temperature settles around a quite stable value resulting from an achieved equilibrium between the generated heat and the quantity dissipated by the tool, the fixture and exchange with the surrounding environment. Figure 7b, in addition, reports the comparison between three replications both in terms of forces and in terms of temperature acquisition of the same operative condition (Test #01, Test #02, Test #03), demonstrating the repeatability of the SPIF process. Comparing the evolution of forces and maximum temperatures, it can be noticed that following an initial transient phase, the force reaches its maximum value at the onset of the temperature increase. Subsequently, the force decreases, with the peak temperature coinciding with the minimum force value. Then, the force exhibits a slight increase, while the temperature undergoes a slight decrease. These results align with the material’s temperature-dependent behaviour, namely an inverse correlation in which higher temperatures are associated with lower forming forces [39].
Fig. 7
Temperature monitoring during the SPIF tests: (a) thermographic image (Test #01); (b) SPIF process temperature (dotted markers) and force (solid lines) acquisition for Test #01, Test #02 and Test #03
Due to the large number of monitored outputs, some preliminary statistical analyses were inferred from the correlation matrix shown in Fig. 8. The plot allows to understand the input/output linear association, expressed by the Pearson product-moment correlation coefficients ranging from 1 (perfectly positively correlated) to −1 (perfectly negatively correlated) and passing through 0 (absence of any correlation). For example, the matrix highlights that the spindle speed (S) has a strong positive correlation with most of the considered outputs whereas it shows a strong negative correlation with the maximum force and almost no correlation with the final roughness. The feed rate (F), as expected, has a strong correlation with the total duration of the test but it results weakly correlated with the other response variable. It is also reasonable to see the positive correlation between the step down (Δz) and the maximum load (\(\:{\text{F}}_{\text{max}})\), as well as its negative correlation with the total time of the test.
Fig. 8
Preliminary statistical investigation: the correlation matrix
Despite the usefulness of the correlation matrix, it does not provide any information on cause-and-effect relationships between the defined variables; therefore, the preliminary analysis was completed by the Smoothing Spline ANOVA (SS-ANOVA) model, which is a statistical modeling algorithm based on a function decomposition similar to the classical analysis of variance (ANOVA) decomposition. The advantage lies in the direct interpretability of the results, since the percentage of the single input contribution to the global variance can be easily quantified. The overall main effect stacked bar chart shown in Fig. 9, that is basically a graphical representation of the effect table, confirms the predominant effect of the spindle speed on most of the response quantities with a particular incidence on the maximum temperature. On the other hand, the step down shows its main influence on the load necessary to deform the blank and on the time needed to complete the test.
Fig. 9
Preliminary statistical analysis: the overall effect chart
Among the several outputs, the attention was subsequently focused on three of them, namely the height at fracture (Hfracture), the maximum load recorded during the test (Fmax) and the average roughness (Ra) of the deformed surface. The three chosen outputs were considered the most prominent since related to the final quality of part (in terms of final height and quality of the final surface) and also to the process (the maximum load).
3.2 Preliminary regression analysis
Once evaluated the effect of the inputs on the three main outputs, a regression analysis was initially conducted, separately for each input variable. Figure 10 shows the results for the Hfracture output.
Fig. 10
Regression analysis of the Hfracture index in terms of: (a) spindle speed S, (b) step down Δz and (c) feed rate F
The separated regression analysis was then formalized according to the following Eq.
$$\begin{array}{c}H_{fracture}=7.944+0.006\ast S+4.94E-6\ast{(S-4000)}^2\\+1.4E-9\ast{(S-4000)}^3-3.05E-12\ast{(S-4000)}^4\end{array}$$
(1)
$$\begin{array}{c}H_{fracture}=46.41-38.14\ast\Delta z+182.54\ast{(\Delta z-0.35)}^2\\+1416.67\ast{(\Delta z-0.35)}^3-7290.13\ast{(\Delta z-0.35)}^4\end{array}$$
(2)
$$\begin{array}{c}H_{fracture}=47.01+0.05\ast F+0.001\ast{(F-400)}^2\\+2.49E-6\ast{(F-400)}^3-1.35E-8\ast{(F-400)}^4\end{array}$$
(3)
Subsequently, by combining the three relations, the regression equation of the \(\:{\text{H}}_{\text{fracture}}\) as a function of the three inputs could be determined.
$$\:\begin{array}{c}H_\text{fracture}=0.802+0.074\text{*}\left(\frac{S-4000}{2000}\right)\\-0.154\ast\left(\frac{\Delta z-0.35}{0.25}\right)-0.236\ast\left(\frac{F-400}{200}\right)\end{array}$$
(4)
The graphical visualization of the regression equation is shown by means of the contour plots in Fig. 11.
Fig. 11
Contour plot of the Hfracture output considering the combination of (a) spindle speed/step down, (b) spindle speed/feed rate and (c) step down/feed rate
The contour plot, as expected and in accordance with the literature, suggested that higher regimes of spindle speed, in combination with average values of the step down and the feed rate (Fig. 11a and b), increased the maximum height at fracture as a direct consequence of the more severe heating generated by the contact between the tool and the blank. There’s also a wide region, close to a step between 0.3 and 0.4, where larger height at fracture could be reached irrespective of the feed rate (Fig. 11c).
Similarly, the regression equations considering only one input per time were determined for the Fmax index, as shown in Fig. 12.
Fig. 12
Regression analysis of the Fmax index in terms of (a) spindle speed S, (b) step down Δz, and (c) feed rate F
The analytical relationship between the Fmax index and each single input is reported by Eqs. 5–7.
$$\:\begin{array}{c}F_{max}=1373.27-0.16\ast S+6.00E-5\ast\left(S-4000\right)^2\\{{+5.14E-8\ast\left(S-4000\right)}^3-2.29E-11\ast\left(S-4000\right)}^4\end{array}\\$$
(5)
$$\begin{array}{c}F_{max}=543.26+484.75\ast\Delta z+2724.01{\ast\left(\Delta z-0.35\right)}^2\\{{+876.89\ast\left(\Delta z-0.35\right)}^3-46013.69\ast\left(\Delta z-0.35\right)}^4\end{array}\\$$
(6)
$$\begin{array}{c}\:{\text{F}}_\text{max}\text{=657.13+0.14*F+0.006*}\left(\text{F-400}\right)^2\\{{\text{-3.31E-7*}\left(\text{F-400}\right)}^\text{3}\text{-2.24E-7*}\left(\text{F-400}\right)}^\text{4}\end{array}\\$$
(7)
The combination of the three regression functions led to the definition of a single equation including the three contributions (see Eq. 8).
$$\begin{array}{c}\:{{F}}_{max}{=729.68-148.96*}\left(\frac{{S-4000}}{{2000}}\right)\\{+129.71*}\left(\frac{{\Delta z-0.35}}{{0.25}}\right){+26.284*}\left(\frac{{F-400}}{{200}}\right)\end{array}\\$$
(8)
The contour plots in Fig. 13 allow to visualize the effect of each pair of input variables on the maximum load \(\:{\text{F}}_{\text{max}}\).
Fig. 13
Contour plot of the Fmax index considering the combination of input variables as (a) spindle speed-step down, (b) spindle speed-feed rate, (c) step down-feed rate
Also in this case, the contour plot confirmed what expected: increasing the spindle speed led to higher temperature regimes which, in turns, reduced the strength of the material thus needing lower forces to impart the needed deformation. At the same time, lower speed rate and lower step down led to a less severe deformation which resulted in a lower maximum load.
Eventually, the same analysis was carried out on the third output (i.e. the final roughness Ra), whose representation is reported in Fig. 14.
Fig. 14
Regression analysis of the Ra index in terms of (a) spindle speed S, (b) step down Δz, and (c) feed rate F
Equations 9–11 describe the influence of the single input on the output.
$$\begin{array}{c}{\text{R}}_\text{a}\text{=-1.83+0.0006*S+5.48E-8}{\text{*}\left(\text{S-4000}\right)}^\text{2}\\{{\text{-3.25E-10*}\left(\text{S-4000}\right)}^\text{3}\text{+7.464E-14*}\left(\text{S-4000}\right)}^\text{4}\end{array}\\\:$$
(9)
$$\begin{array}{c}\:{{R}}_{a}{=-0.49+3.31*\Delta z+1.68*}\left({\Delta z-0.35}\right)^2\\{{{-101.05*}\left({\Delta z-0.35}\right)}^{3}{+144.47*}\left({\Delta z-0.35}\right)}^{4}\end{array}\\$$
(10)
$$\:\begin{array}{c}{\text{R}}_\text{a}\text{=1.899-0.0024*F-1.343E-5}{\text{*}\left(\text{F-400}\right)}^\text{2}\\{{\text{+7.05E-8*}\left(\text{F-400}\right)}^\text{3}\text{-1.60E-10*}\left(\text{F-400}\right)}^\text{4}\end{array}\\$$
(11)
On the other hand, Eq. 12 summarizes the influence of the three parameters simultaneously on the mentioned output.
$$\begin{array}{c}\:{\text{R}}_{a}{=0.802+0.074*}\left(\frac{{S-4000}}{{2000}}\right)\\{+0.154*}\left(\frac{{\Delta z-0.35}}{{0.25}}\right){+0.236*}\left(\frac{{F-400}}{{200}}\right)\end{array}\\$$
(12)
The graphical representation of the input influence on the surface roughness is clearly described by the contour plots in Fig. 15.
Fig. 15
Contour plot of the Surface roughness index considering the combination of input variables as (a) spindle speed-step down, (b) spindle speed-feed rate, (c) step down-feed rate
According to the 2D plots, there’s a wide portion of the design space (in the proximity of a spindle speed value of 4000 RPM) where the quality of the formed surface could be improved regardless of the feed rate regime (see Fig. 15b). Moreover, it can also be seen that even larger values of the step down, but in combination with proper levels of the feed rate, allow the enhancement of the quality of the formed part.
3.3 Evaluation of different operative scenarios using the ELECTRE technique
As described in Sect. 2.3, the ELECTRE method is a multi-criteria decision-making approach that operates through pairwise comparison of alternatives. In this method, after defining the criteria and determining their weights, a decision matrix is formed. Then, for each pair of options, two matrices of agreement and disagreement are calculated. If the agreement is high enough and the disagreement is low enough, a preference relationship is created between the options. Finally, by forming the overall matrix of preference relationships, superior options are identified and poor options are eliminated. This method, which has a non-compensatory approach, allows for the comparison of qualitative and quantitative criteria and is suitable for problems with multiple and conflicting criteria. However, its sensitivity to the determination of weights and thresholds is considered one of its limitations.
In the following, the aim is to extract superior experiments using the ELECTRE technique. More in details, the scope of the adopted technique is to identify possible optimal scenarios while simultaneously considering the three objective functions: maximization of the height at fracture (i.e. the final value as close as possible to the desired one), minimization of the forming load and minimization of the final surface roughness (to increase the overall quality of the final part). In fact, despite the three functions can be satisfied simultaneously, the optimal set of input parameters may change if different weights are assigned to each single objective. Table 4, in fact, enlists five different possible scenarios (indicated as “Model”) including: (i) Model 2 and Model 4 for which the maximization of the final height is more prominent than the other two objectives, (ii) Model 1 and Model 5 where the final quality is considered as predominant over the other two aspects and (iii) Model 3 in which all the objectives have the same weight.
Table 4
Response variable weights
Output variable | Hfracture [mm] | Fmax [N] | Ra [µm] |
|---|---|---|---|
Obj. Function | Maximize | Minimize | Minimize |
Model 1 | 0.2 | 0.3 | 0.5 |
Model 2 | 0.5 | 0.2 | 0.3 |
Model 3 | 0.33 | 0.33 | 0.33 |
Model 4 | 0.5 | 0.1 | 0.4 |
Model 5 | 0.4 | 0.1 | 0.5 |
Results from the application of the ELECTRE methodology are reported in Table 5: according to the weights distribution listed in Table 4, the optimal working conditions are ranked as first (highlighted by the light grey cells).
Table 5
Ranking of practical experiments based on ELECTERE
Model 1 finds its optimum in the combination of parameters characterizing the Test #04. In fact, a limited step down (0.2 mm) leads to the reduction of the average roughness in the formed surface, thus satisfying the objective characterized by the largest weight. At the same time, due to the smaller weights of the other two functions, low values of spindle speed (3000 RPM) and feed rate (300 mm/min) lead to a ruptured part (Hfracture is lower than 43 mm) and quite high value of the forming load, probably due to low temperature reached during the deformation stages.
Model 2 and Model 4 – both characterized by a predominance in the maximization of the final height over the other two objectives – are satisfied by the combination of high spindle speed and low step down (i.e. the Test #11). A high rotation speed, in fact, increases the quantity of heat generated by the friction, the temperature increases accordingly and the enhanced material formability can be exploited to achieve the final desired height. Moreover, especially in the case of Model 4, the maximization of the final quality of the formed surface (weight of correspondent function equal to 0.4) imposes a limited value of the step down (0.2 mm).
An interesting consideration comes from the fact that the Test #07 is identified as the optimal combination of process parameters for three different scenarios. In fact, Model 4 and Model 5 share comparable values of the weights assigned to the objectives related to the final height and the average roughness (besides the same weight on the maximum load function). Consequently, they share also the same optimal combination of process parameters. Moreover, Test #07 represents also an optimal solution for the Model 3 for which it is more important that all the outputs are simultaneously balanced in terms of quality, without one excelling on the others (according to a common principle of “everywhere good, but nowhere excellent”).
4 Conclusions
In the present work, the ELECTRE decision-support tool for multi-criteria evaluation and prioritization has been applied to the SPIF process.
The first step of the analysis regarded the manufacturing of a benchmark component – a truncated pyramid – while changing the main process parameters (feed rate, spindle speed and step down) according to a CCD plan. All the tests were analyzed in terms of several output indicators, among which the attention was mainly focused on the depth at fracture, the surface roughness (more related to the overall quality of the formed part) and the maximum load recorded during the deformation (more related to the process).
The preliminary analysis, based on the SS-ANOVA methodology, highlighted the well-known main effects of the process parameters on the defined indicators: higher spindle speed allows the achievement of the final depth, lower step down results in a smoother surface (lower average roughness).
Nevertheless, the complexity of the process, mainly due to the number of inputs and outputs involved, represented the ideal condition to exploit the potentialities of the ELECTRE technique. In fact, being a support-decision tool based on the outranking of alternatives and thanks to its capability of managing several criteria even conflicting, it was possible to outline several scenarios of simultaneous optimization of all the output indicators, each characterized by a different combination of the process parameters resulting from the assignment of different weight to the single output indicator. It was demonstrated that when the minimization of the both forming load and the average surface roughness is privileged, the correspondent superior combination of process parameters do not allow to obtain a sound component (depth a fracture remarkably lower than 43 mm). In the opposite case, when the maximization of the final height is predominant over the other two indicators, the decision tool moves the spot on the increase of the spindle speed (and the consequent enhancement of the material formability due to the increase in the heat generated by friction). The presented results have demonstrated the capability of the ELECTRE technique to identify, needing a limited number of experimental conditions, different optimal scenarios while simultaneously optimizing all the output indicators.
Acknowledgements
Financed by the European Union - NextGenerationEU (National Sustainable Mobility Center CN00000023, Italian Ministry of University and Research Decree n. 1033–17/06/2022, Spoke 11 - Innovative Materials & Lightweighting). The opinions expressed are those of the authors only and should not be considered as representative of the European Union or the European Commission’s official position. Neither the European Union nor the European Commission can be held responsible for them.
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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