Introduction

It has been more than 40 years (Ashby, 2000) early works in material selection appeared; many methods have been proposed to analyze a big amount of data involved in the material selection process so as to obtain an appropriate result.

Various algorithms (techniques) have been developed, including Ashby’s method (Ashby, 2000; Ashby et al, 2004), Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), Vlse Kriterijumska Optimizacija Kompromisno Resenje (VIKOR), Multi Attribute Decision Making (MADM), Analytical Hierarchy Process (AHP), Simple Additive Weighted (SAW) method and Multi–Objective Optimization on the basis of Ratio Analysis (MOORA), etc (Zheng et al, 2021). Ashby’s method is difficult to be applied in cases which involve multiple criteria of selection (Ashby, 2000; Ashby et al, 2004; Zheng et al, 2021). Deshmukh et al employed the multi–objective optimization (MOO) techniques of TOPSIS and VIKOR to perform the material selection of the switching structure for RE-MEMS shunt capacitive switches (Deshmukh & Angira, 2019). However, there exist inherent problems of additive algorithms and subjective factors in the MADM, AHP, MOORA, TOPSIS and VIKOR due to their fatal scaling or normalization processes (Zheng et al, 2021).

Recently, a new probability-based multi-objective optimization method was developed (Zheng et al, 2021), attempting to solve the inherent problems of personal and subjective factors in the previous multi–objective optimization methods. The novel concept of preferable probability was introduced to reflect the preference degree of a candidate in the optimization where all performance utility indicators of candidates are divided into beneficial or unbeneficial types for the selection. Each performance utility indicator of a candidate contributes to a partial preferable probability quantitatively, and the total preferable probability of a candidate is the product of all partial preferable probabilities from the viewpoint of the probability theory, which is the overall and unique decisive index in the competitive selection process. The new multi-objective optimization method was also extended with the application of the multi-objective orthogonal test design method (OTDM) and the uniform test design method (UTDM), which results in appropriate achievements (Zheng et al, 2021).

In this paper, the new probability-based multi–objective optimization method is used to perform the optimal scheme in material engineering, which includes the selection of switching material of the RF-MEMS shunt capacitive switch, the optimization of the sintering parameters of natural hydroxyapatite and the optimal design of a connecting claw jig.

Brief introduction to the new multi–objective optimization method

In the new probability-based multi–objective optimization method (Zheng et al, 2021) beneficial utility index of material performance indicator contributes to a partial preferable probability in a positively linear manner, i.e.,

In Eq. (1), X_ij is the j^th beneficial utility index of the material performance indicator of the j^th candidate material; P_ij represents the partial preferable probability of the beneficial utility index X_ij; n is the total number of candidate materials in the material group involved; m is the total number of the utility indices of each candidate material in the group; ${\mathrm{\alpha }}_{\text{j}}$ is the normalized factor of the j_th utility index of the material performance indicator, ${\mathrm{\alpha }}_{\text{ij}}\text{=1/(n}\overline{{\text{X}}_{\text{j}}\text{)}}$ and Xj is the arithmetic mean value of the utility index of the material performance indicator in the material gropu involved.

Equivalently, the unbeneficial utility index of the material performance indicator contributes to a partial preferable probability in a negatively linear manner, i.e.,

In Eq. (3), X_jmax and X_jmin represent the maximum and minimum values of the utility indices X_j of the material performance indicator in the material group, respectively, and ${\mathrm{\beta }}_{\text{j}}$ is the normalized factor of the j^th utility indices of the material performance indicator, ${\mathrm{\beta }}_{\text{ij}}\text{=1/[n(}{\text{X}}_{\text{jmin}}\text{+}{\text{X}}_{\text{jmax}}\text{)-n}{\overline{\text{X}}}_{\text{j}}\text{].}$

Moreover, the total / comprehensive preferable probability of the i^th candidate material is the product of its partial preferable probability P_ij of each utility index of the material performance indicator in the overall selection due to the “simultaneous optimization” of the multi-objects in the viewpoint of probability theory (Zheng et al, 2021), i.e.,

The total preferable probability of a candidate is the uniquely decisive index in the overall selection process competitively, which transfers a multi–objective optimization problem (MOOP) into a single – objective optimization one. The main characteristic of the new probability-based multi–objective optimization is that the treatment for both beneficial utility index and unbeneficial utility index is equivalent and conformable, which is without any artificial or subjective scaling factors involved in the process.

Application of probability-based multi–objective optimization in material engineering

1) Multi–objective optimization in the material selection of RF‑MEMS shunt capacitive switches

Radio Frequency Micro Electro Mechanical Systems (RF-MEMS) is a promising technology for implementing passive devices in future wireless communication systems (Deshmukh & Angira, 2019). Switches have drawn more attention due to their frequent use in many cases in wireless communication systems. An RF-MEMS technology-based switch has low insertion loss, high isolation, high linearity and less power consumption (Deshmukh & Angira, 2019). Its shunt capacitive switch has two stable states i.e., up-state and down-state (Deshmukh & Angira, 2019). Power can flow from the input port to the output port in the switch upstate, while it stands at the off-state in its down-state (Deshmukh & Angira, 2019; Angira & Rangra, 2015a; Angira & Rangra, 2015b).

The optimization of the performance of the switching structure involves many parameters (criteria), such as pull-in voltage, RF response (insertion loss and isolation), maximum displacement, thermal conductivity, etc (Deshmukh & Angira, 2019; Angira & Rangra, 2015a; Angira & Rangra, 2015b). Since many parameters are involved, it can be seen as a MOOP in the performance optimization of the switching material selection. Therefore, a MOOP can be used to decrease human effort since a large number of materials are available in practice, forming a material bank together with many manufacturing processes and selection attributes (Zheng et al, 2021).

Yang et al pointed out that if different normalization methods are applied, significant different results may be produced (Yang et al, 2021). Podviezko et al also stressed that different normalization of data applying to popular MCDM methods such as SAW or TOPSIS could lead to significant differences in the assessment (Podviezko & Podviezko, 2015). As a consequence, many researchers paid a lot of attention to the choice of the normalization type. However, it is still puzzling which normalization method is better and how to determine final results of material selection from different normalization algorithms.

A) Utility indices of the material performance indicators in the material selection of RF - MEMS shunt capacitive switches

In the study of Deshmukh & Angira (2019), the optimal objectives for this purpose are low pull–in voltage, low RF loss, high thermal conductivity and maximum displacement of the beam structure. As a result, the square root of Young’s modulus of the material E^0.5, the electrical resistive coefficient ${\mathrm{\rho }}_{\text{e}}$ $\textcolor{red}{}{\mathrm{\rho }}_{\text{e}}$, the thermal conductivity of the material $\mathrm{\lambda }$, the ratio of the fracture strength ${\mathrm{\sigma }}_{\text{f}}$ to Young’s modulus E of the material, ${\mathrm{\sigma }}_{\text{f}}\text{/E}$, are taken as the optimal utility indices of the material attribute indicators (Deshmukh & Angira, 2019).

B) Divisions of the utility indices in the material selection of RF - MEMS shunt capacitive switches

From analyzing the requirements of the optimizations of the bridge of RF-MEMS shunt capacitive switches, i.e., higher pull-in voltage (V_p), lower RF loss, higher thermal conductivity and the higher maximum displacement of the switch beam (Deshmukh & Angira, 2019), the utility indices of the square root of Young’s modulus of the material, E^0.5, the thermal conductivity of the material, $\mathrm{\lambda }$, the ratio of the fracture strength ${\mathrm{\sigma }}_{\text{f}}$, to Young’s modulus E of the material, ${\mathrm{\sigma }}_{\text{f}}\text{/E}$ belong to the beneficial type of the material performance index, while the electrical resistive coefficient, ${\mathrm{\rho }}_{\text{e}}$ belongs to the unbeneficial type of the material performance index in the assessment.

C) Assessment results

The values of the conventional material performance indicators for various materials are given in Table 1 (Deshmukh & Angira, 2019).

The partial preferable probabilities of the utility indices of E^0.5, $\mathrm{\lambda }$ and ${\mathrm{\rho }}_{\text{e}}$ and ${\mathrm{\sigma }}_{\text{f}}\text{/E}$ and the total preferable probabilities are assessed according to Equations (1) through (5), respectively, shown in Table 2. In addition, the ranking here by using the new probability-based multi–objective optimization method is given in Table 2 together with those of Vikor and Topsis from Ref. (Deshmukh & Angira, 2019) for comparison.

Table 1
Conventional material performance indicators for various materials

(Deshmukh & Angira, 2019)

Mat.	Young’s modulusE (GPa)	Electrical resistive coefficient re (Ω m) 10−8	Thermal conductivity l (W/m∙K)	Fracture strength sf (MPa)	(sf/E) ´103
Ni	193	6.99	90	345	1.7876
Au	70	2.44	315	220	3.1429
Al	70	2.82	204	47	0.6714
Ag	83	1.59	407	110	1.3253
Pt	168	10.5	73	125	0.7440
Cu	117	1.68	386	314	2.6838
Cr	279	12.9	90	370	1.3262
W	411	5.28	163	1725	4.1971
Co	209	6.24	69	675	3.2297
Fe	211	9.61	73	540	2.5592

Table 2
Partial preferable probabilities and total preferable probabilities for various materials for shunt capacitive switch optimization

Mat.	PE^0.5	Pre	Pl	Psf /E	Pt´104	Rank here	Rank Vikor	Rank Topsis
Ni	0.1073	0.0884	0.0481	0.0825	0.3766	6	6	6
Au	0.0646	0.1420	0.1684	0.1451	2.2423	3	1	1
Al	0.0646	0.1375	0.1091	0.0310	0.3005	7	4	4
Ag	0.0704	0.1520	0.2176	0.0612	1.4242	4	3	3
Pt	0.1001	0.0470	0.0390	0.0343	0.0631	10	8	9
Cu	0.0835	0.1510	0.2064	0.1239	3.2248	1	2	2
Cr	0.1290	0.0187	0.0481	0.0612	0.0712	9	10	10
W	0.1566	0.1085	0.0872	0.1937	2.8697	2	9	7
Co	0.1117	0.0972	0.0369	0.1490	0.5971	5	5	5
Fe	0.1122	0.0575	0.0390	0.1181	0.2975	8	7	8

It can be seen from Table 2 that the appropriate material from the new multi–objective optimization method is Cu, which is different from those of Vikor and Topsis from (Deshmukh & Angira, 2019); this is because of the inherent defects of personal and subjective factors in Vikor and Topsis (Deshmukh & Angira, 2019).

In fact, the evaluation result of the new probability-based method for multi–objective optimization in material selection is no need to equal to those of other previous approaches exactly due to their involvements of personal or other subjective coefficients.

2) Optimization of sintering parameters of natural hydroxyapatite

Abifarin conducted the optimization of hydroxyapatite (HAp) mechanical characteristics using Taguchi grey relational analysis design which includes hardness and compressive strength (Abifarin, 2021). Three levels of sintering temperature and two levels of compaction pressure are employed during sintering (Abifarin, 2021). The design and the results are shown in Table 3. Again, the probability-based multi-objective optimization is used to conduct the assessment with hardness and compressive strength as the beneficial type index. The evaluation results are shown in Table 4.

Table 4
Evaluation results of HАp

No	Pressure	Temperature °C	Hardness	Compressive Strength
1	0	900	0.54	0.39
2	0	1000	0.838	0.58
3	0	1100	0.940	0.84
4	5	900	0.656	0.34
5	5	1000	0.929	0.5
6	5	1100	1.103	0.69

Table 4
Evaluation results of HАp

	Partial preferable probability		Total
No	Hardness	Strength	Pt*102	Rank
1	0.1079	0.1168	1.2596	6
2	0.1674	0.1737	2.9069	3
3	0.1878	0.2515	4.7225	1
4	0.1311	0.1018	1.3340	5
5	0.1856	0.1497	2.7781	4
6	0.2203	0.2066	4.5518	2

Table 4 indicates that the optimal sintering parameters of natural hydroxyapatite are at 1100°C and 0 compaction pressure.

3) Optimal design of a connecting claw jig

Yan et al conducted the multi-objective optimal design of a connecting claw jig with ANSYS Workbench finite element analysis software (Yan et al, 2021). The maximum equivalent stress (MPa) Y₁, the weight (kg) Y₂, the minimum safety factor Y₃ and the maximum deformation (mm) Y₄ of the claw jig are taken as the optimization objectives, while the thickness of the substrate FD₁ (mm) x₁, the angle of the connecting claw A1 (°) x₂, the thickness of the connecting claw FD₂ (mm) x₃, and the outside diameter of the jig base R₁ (mm) x₄ are taken as the input variables.

After the simulation and the analysis, three candidate schemes with good objective functions are selected by the system, as shown in Table 5. The object Y₃ is a beneficial type index, while Y₁, Y₂ and Y.₄are all unbeneficial type indexes. The evaluation results are shown in Table 6.

Table 6 shows that the optimal scheme is scheme No 1.

Table 5
Three candidate schemes of the connecting claw jig

Original scheme	x1 (mm)	x2 (°)	x3 (mm)	x4 (mm)	Y1 (MPa)	Y2 (kg)	Y3	Y4 (mm)
	56	79	38	32.5	151.22	6.176	1.6532	1.171
1	54.125	73.13	35.414	31.051	128.42	5.615	1.9467	0.923
2	48.125	73.77	37.982	32.395	143.31	5.577	1.7444	1.043
3	46.625	76.38	39.908	30.715	161.48	5.620	1.5482	1.375

Table 6
Evaluation results of the connecting claw jig

No.	Partial preferable probability				Total
	Y1	Y2	Y3	Y4	Pt*100	Rank
1	0.3700	0.3327	0.3716	0.3870	1.7697	1
2	0.3358	0.3349	0.3329	0.3532	1.3229	2
3	0.2942	0.3324	0.2955	0.2598	0.7507	3

Conclusion

The application of the new probability-based multi–objective optimization method in dealing with three optimal problems of material engineering has shown that: the appropriate material (Cu) is successfully selected, which meets the requirements of the optimizations of the bridge of RF ‑ MEMS shunt capacitive switches; the optimal sintering parameters of natural hydroxyapatite are at 1100°C and 0 compaction pressure; and the optimal scheme of the connecting claw jig is scheme No 1. The main feature of the new probability-based multi–objective optimization method is that the treatment is equivalent and conformable for both the beneficial utility index and the unbeneficial utility index, without any artificial or subjective scaling factors involved in the process.

References

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Yan, Y., Fu, N., Zhang, X., Wang, C., Sun, J. & Lu, L. 2021. Research on the Multi-Objective Optimization Design of Connecting Claw Jig. International Journal of Steel Structures, 21(6), pp.1911-1920. Available at: https://doi.org/10.1007/s13296-021-00542-6.

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Author notes

mszhengok@aliyun.com

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