A Decision Framework for Safe Pressure Vessel Material Selection: Integrating k-means Clustering, Multi-Criteria Analysis, and Decision-Makers' Perceptions

Ortiz, Guilherme O.; Dwek, Mauricio; Bolfarini, Claudemiro; Staffa, Lucas H.

doi:10.1590/1980-5373-MR-2025-0698

Abstract

Standard tools for dealing with complex material selection cases involving multi-criteria optimization strategies, e.g., multiple constraints and conflicting objectives, can sometimes be challenging. The material selection case study for a safe pressure vessel is a typical multiple-constraints problem. When addressed using standard tools, the material selection may be occasionally conditioned by half the maximum length of any crack the vessel wall might contain ( $a_{c}$ ) without catastrophic failure under the state stress acting in the vessel, a parameter that is not always readily available. Therefore, a combined approach using unsupervised machine learning (k-means clustering) and TOPSIS, with different relative weighting vectors and no initial constraints, was employed. Applying TOPSIS with a subjective weighting vector to the best-ranked cluster identified Nickel superalloys (Ni-Cr, Ni-Fe-Cr, Ni-Mo, Ni-Cr-Co-Mo) as top candidates for safe pressure vessels, underscoring the role of decision-makers' perceptions in material selection, even when aided by machine learning.

Keywords:
Machine Learning; Material Selection; k-means; Ashby

1. Introduction

With advances in technology, the availability and development of new materials have increased. As a result, a need arose to define systematic methods for material selection. To this end, material property charts and merit indices emerged as tools to aid material selection, aiming to maximize performance¹ and, when combined with digital tools²,³, can help reduce the complexity of several material selection problems.

However, in some cases, it is necessary to address multiple conflicting constraints and objectives, which can significantly complicate the material selection process. The standard tools available to deal with these multi-criteria optimization challenges, such as the ‘min–max’ resolution method, penalty functions, and exchange constants, are sometimes recognized as limited⁴.

Pressure vessels are equipment that contain fluids under internal or external pressure, which differs from atmospheric pressure. When a pressure vessel needs to be moved, for instance, liquid-natural gas containers, the equipment must be lightweight and simultaneously ensure safety⁴.

The materials selection for a light and safe pressure vessel, for instance, is a classic case of multiple constraints, where the vessel can fail due to yielding ( $σ_{y}$ ) or fast fracture ( $K_{I c}$ ) since they are designed to yield or leak before their failure. These two criteria lead to two different merit indices: $M_{1} = \frac{ρ}{σ_{y}}$ and $M_{2} = \frac{ρ}{K_{I c}}$ ⁴. Both these merit indices were derived from two objective-functions: $m_{1} = 2 π r^{3} Δ p \frac{ρ}{σ_{y}}$ and $m_{2} = 2 π r^{3} Δ p \sqrt{π a_{c}} \frac{ρ}{K_{I c}}$ , where $2 r$ is the pressure vessel diameter, $p$ is the pressure difference across the vessel wall, $ρ$ is the density (or the specific density: kg/m3 or kgf/m3) and $a_{c}$ is the half of the maximum length of any crack that the wall might contain before catastrophic failure.

When the coupling equations that link the merit indices are derived from m₁=m₂: $M_{2} = \frac{1}{\sqrt{π a_{c}}} M_{1}$ , it is possible to see that the candidate material for a light and safe pressure vessel depends on the $a_{c}$ . According to this approach and for $a_{c}$ ≤ 5 mm, titanium alloys, aluminum alloys, and stainless steels, for instance, are good candidates for light-pressure vessels⁴.

Here, we employed a different approach than usual in selecting materials for a safe pressure vessel. An unsupervised machine-learning method, k-means clustering⁴, was explored in a broader context, combined with TOPSIS (Technique for Order Preference by Similarity to Ideal Solution), without imposing initial constraints and using any merit indices. For that purpose, the dataset will be partitioned into k user-defined clusters. Each cluster will have a centroid, i.e., a set of coordinates located at the center of each cluster⁵. The chronological sequence of the steps is illustrated in the Figure S1 (Supplementary Material).

k-means has applications ranging from data mining to image compression⁶; in the medical field for diagnosing heart disease⁷; in the field of computer science for anomaly detection in network monitoring data⁸ identification of rock types with various microstructures and transport properties⁹.

Although several robust multicriteria decision-making method (MCDM) methods have recently been applied to material selection problems¹⁰, for instance, in flexible polymeric heat pipes¹¹, knee implant femoral components¹², and cryogenic storage tanks¹³, the strategy here is to use the k-means clustering method combined with TOPSIS for selecting the best material for safe pressure vessels, aiming to overcome challenges that are, sometimes, inherent in most conventional methods and standards tools for this case, such as (i) multiple constraints; (ii) conflicting objectives, (iii) establishment of merit indices; and (iv) initial requirements imposed during the screening stage.

In our approach, merit indices were not used, and no initial quantitative constraints were applied, just the “must not fail by yielding and fast fracture” criteria. Additionally, the k-means method enables the simultaneous handling of multiple properties in a multivariable approach. Furthermore, this study aims to investigate how different weighting philosophies influence material selection.

2. Methods

The prospective materials include all those available in the Ansys® Granta Selector 2024 R1 catalog¹⁴, excluding liquids and gases. The physical and mechanical properties of interest used in this material selection process were yield strength (σ_y), fracture toughness (K_Ic), and density (ρ). Since such properties are described in a range, each one was considered at its minimum and maximum values. In this way, for each material analyzed, six columns of physical-mechanical properties were considered: σ_y,min; σ_y,max; K_Ic,min; K_Ic,max; ρ_min and ρ_max. The design requirements for this material selection study are presented in Table 1.

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Table 1
Design requirements for a safe pressure vessel.

From then on, the material selection process followed the flowchart described in the Supplementary Material (Figure S2), in which each step is explained in detail below:

Establishment of relative weighting vectors: Four weighting vectors were employed here (Table 2);

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Table 2
Relative weighting vectors used.
1. ¹
  Subjective Weighting Vector: based on the author's perceptions of the relative importance of each property in the performance of a large and safe pressure vessel. In practice, fracture toughness is often considered more critical than yield strength for ensuring safety for large pressure vessels¹⁵. Therefore, K_Ic,min was considered the most important mechanical property, as the material is more likely to fail at the lower bound of a property’s value range;
2. ²
  Analytic Hierarchy Process (AHP): AHP is a multi-criteria decision-making weighting methodology that transforms qualitative judgments into quantitative weights and can therefore be regarded as a structured subjectivity approach. Here, the pairwise comparison framework enforces explicit trade-offs rather than ad hoc weight assignments. The methodology used to obtain the AHP Weighting Vector followed Saaty's study¹⁶ regarding the axiomatic foundation of the AHP, through paired comparison (scaling criteria). The corresponding Saaty matrix used to structure the AHP weighting is provided in the Supplementary Material (Table S1).
3. ³
  Objective Weighting Vector: This vector was based on the concept of entropy proposed by Shannon and Weaver¹⁷,¹⁸, in which the relative weight of each property is related to the degree of disorder/diversification of its values. This is a purely objective approach, without the need to collect the subjective perceptions of the decision-makers¹⁹. Information entropy is a quantitative measure of the randomness of an information system . The more informative or discriminative a given property column is, the lower its entropy and, consequently, the higher the weighting it receives, as it more effectively distinguishes candidate materials¹⁴.
4. ⁴
  Equal Weighting Vector: an arbitrary weighting method whereby all properties are assigned the same relative importance.
Data standardization: the data were standardized using the Z-score methodology ( $\frac{X - m e a n (X)}{s t d (X)}$ )²⁰ to ensure uniformity, scale invariance, and enable meaningful comparison between properties. Here, X is the property value, mean(X) is the column mean, and std(X) is the column standard deviation. The specific steps in our methodology where data standardization was applied are indicated in the supplementary file (Figure S2). This standardization is an essential preprocessing step in data clustering and data mining, particularly in methods that partition datasets into groups of patterns, improving the performance of the k-means algorithm by transforming attributes with different dynamic ranges into a common scale²¹.
Defining k-value: an inherent difficulty when using k-means is determining the value of k. Some strategies can be employed to address this, such as the Elbow Method⁶ and the Silhouette Score. Since the result of the Elbow Method is a curve showing the sum of squared errors for each cluster, the value of k may be chosen subjectively by the decision-maker¹⁰. To avoid such subjectivity, the Silhouette Score ( $\frac{b - a}{max (a, b)})$ is an interesting option for determining the value of k, where a is the average intracluster distance, i.e., the average distance of each data point to the other instances within the same cluster to which it is assigned; b is the average intercluster distance, i.e., the average distance of each data point to the instances in the nearest neighboring cluster. In this sense, with the data standardized, the Silhouette Score Method was used to determine the k-value. For that, a Python script was employed utilizing an open-source and free library from Scikit-learn: sklearn.metrics.silhouette_score²². The Silhouette Score was evaluated across 3 to 15 clusters. The number of clusters that yielded the highest Silhouette Score was used as the k-value. For comparison, the elbow rule was also applied to determine the k value. The corresponding curves are provided in the Supplementary Material (Figure S3)
Clusterization k-means: this step followed the flowchart in Figure S2. After determining the k-value, a Python script employed an open-source and free library from Scikit-learn: sklearn.cluster.KMeans²³. At this point, each material will be assigned to a specific cluster. For each run, a 6-dimensional data space was projected onto a 3-dimensional space using Principal Component Analysis (PCA) for visualization purposes. The contribution of each component to explaining the PCA dataset is provided in the Supplementary Material (Figure S4).
TOPSIS among the clusters: TOPSIS is a well-established multicriteria decision-making method (MCDM) that has been widely adopted in engineering material-selection studies and does not require the definition of concordance/discordance thresholds. Furthermore, unlike other MCMD methods, the present study was not intended to identify a compromise solution among conflicting stakeholders, but rather to rank materials according to predefined weighting vectors. Since the proposed methodology requires repeated ranking of clusters and candidate materials generated by the k-means algorithm, TOPSIS was considered a suitable, efficient, and transparent ranking strategy herein for material selection. In this way, the clusters were ranked, with the average value of each cluster subjected to a method for material selection aided by decision-making theory via TOPSIS. The goal was to determine the surviving cluster based on the selection objectives. The TOPSIS method was conducted according to Jee and Kan¹⁷, using the same standardization method as described previously (ii). The clusters were ranked according to the relative closeness ( $C_{i}^{+}$ )¹⁷. The cluster with the highest $C_{i}^{+}$ is considered ideal. To control the number of code runs and avoid overwhelming documentation, we decided that steps (iv) and (v) would be completed only when the number of materials in the best-ranked cluster was less than 20. If the number of materials in the best-ranked cluster exceeds 20, the process returns to step (iv) only for those materials. The threshold of 20 materials was adopted as a pragmatic stopping criterion rather than as an optimized parameter.
TOPSIS inside the best-ranked cluster: when the number of materials available in the best-ranked cluster is fewer than 20, TOPSIS is applied among these materials. The material with the highest $C_{i}^{+}$ is considered ideal.
Seek documentation: finally, the highest-ranked materials are documented for the final decision.

3. Results and Discussions

Below is a step-by-step description of how k-means clustering is applied using the subjective weighting vector. The initial determination of the k-value, as executed in stage (iv), yielded the following Silhouette Score profile. Figure 1 summarizes the Silhouette Score of all runs. The k-value for the first run was four (4) (see Figure 1a), as it demonstrated the highest Silhouette Score.

Figure 1
Silhouette Score curves summarizing (a) 1^st, (b) 2^nd, and (c) the 3^rd runs.

Subsequently, steps (v) and (vi) were initiated. The average properties of each cluster are listed in Table 3. Cluster 1 achieved the highest $C_{i}^{+}$ ; however, the number of materials is higher than 20. Therefore, a second algorithm execution was decided upon.

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Table 3
Average properties of each cluster (1st run).

Dimensionality reduction of the six-dimensional feature space was carried out using Principal Component Analysis (PCA). The data were projected onto the first three principal components, and the resulting three-dimensional (3D) space for the 1^st run is presented in Figure 2a, enabling visualization of the clustering process.

Figure 2
3D space for the a) 1^st run (99.4% of the total variance explained), b) 2^nd run (98.5% of the total variance explained), and c) 3^rd run (97.9% of the total variance explained).

In the second run, stage (iv) was applied again within Cluster 1. The second run's K-value was five (5) (see Figure 1b), which demonstrated the highest Silhouette Score. The 3D space for the 2nd run is presented in Figure 2b.

Subsequently, steps (v) and (vi) were initiated again. The average properties of each cluster are listed in Table 4. It is noted that in the 2nd run, cluster 3 achieved the highest $C_{i}^{+}$ , however, the number of materials remains higher than 20. Therefore, a third and final algorithm execution was decided upon.

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Table 4
Average properties of each cluster (2nd run).

In the 3^rd run, stage (iv) was applied again within Cluster 3, generating the following Silhouette Score profile, as shown in Figure 1c. The K-value for the third run was three (3), as it demonstrated the highest Silhouette Score.

Subsequently, steps (v) and (vi) were initiated again. The average properties of each cluster are listed in Table 5. In the 3^rd run, cluster 2 achieved the highest $, C_{i}^{+}$ , with 16 materials, i.e., 0.43% of the initial candidates survived. With fewer than 20, the process proceeded to the final step (vii). The 3D space for the 3rd run is presented in Figure 2c.

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Table 5
Average properties of each cluster (3rd run).

Now, step (vii) was applied in this best-ranked cluster. Table 6 presents the final result using the Subjective Weighting Vector.

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Table 6
TOPSIS on the best-ranked cluster after the 3^rd run using the Subjective Weighting Vector.

The final TOPSIS results for the best-ranked cluster, using AHP (AHP-TOPSIS), equal and objective (entropy-based) weighting vectors, are presented in Tables 7, 8, and 9, respectively.

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Table 7
AHP-TOPSIS on the best-ranked cluster after the 3^rd run.

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Table 8
TOPSIS on the best-ranked cluster after the 3^rd run using the Equal Weighting Vector.

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Table 9
TOPSIS on the best-ranked cluster after the 3^rd run using the Objective Weighting Vector.

Table 10 summarizes the iterative k-means/TOPSIS workflow for all weighting strategies. All weighting strategies required three iterations to reduce the number of surviving materials below the predefined threshold of 20, except the entropy-based weighting, which required only two iterations. In addition, our purely subjective weighting vector and AHP weighting approaches yielded the same number of surviving materials at each iteration.

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Table 10
Summary of the iterative k-means/TOPSIS workflow for all weighting strategies. The values indicate how many candidate materials remain in the highest-ranked cluster after each k-means/TOPSIS iteration.

4. Seek Documentation and Discussions

The subjective and AHP weighting vectors (see Tables 6 and 7) yielded similar and consistent results based on Ni-base superalloys, which are suitable and widely used for thick-walled pressure vessels. These alloys are characterized by high strength, good creep resistance, and excellent oxidation and corrosion resistance, making them ideal for service in extreme pressure and temperature environments. In addition, they can form a protective oxide layer at high temperatures (passive film), which hinders further degradation²⁴. Hastelloy and Inconel Ni-base superalloys have already been used in large and small transportable vessels, static vessels, and cryogenic pressure vessels²⁵.

The equal weighting vector (see Table 8) also yielded consistent results based on Ti-based alloys, which have already been employed in manufacturing pressure vessels²⁶,²⁷. Although titanium alloys offer a high strength-to-weight ratio and good general corrosion resistance in many environments, Ti-based alloys can undergo hydrogen-assisted cracking or stress corrosion cracking (SCC) under certain conditions¹⁵. In contrast, Ni-based superalloys operate safely at high temperatures and are reliable for welding in critical vessel applications. They retain significant strength, maintain toughness, and resist stress-corrosion cracking in highly aggressive media, an essential feature for ensuring vessel safety in environments under pressure and containing corrosive chemicals.

The objective weighting vector (see Table 9) yielded results that, although some are incoherent, include others related to synthetic fibers that could theoretically be used with polymers to fabricate lightweight composite pressure vessels²⁸. Although less common, such vessels have been manufactured using composite materials, including carbon and glass fibers. However, concerns remain regarding the bonding interfaces between fiber multilayers, which may lead to delamination and, consequently, severe accidents in thick-walled vessels operating in high-pressure environments²⁹. Moreover, it is recognized that no appropriate criteria have been established for assessing overall failure in thick-walled composite pressure vessels, due to limited experimental validation, insufficient design standards, and a lack of rigorous quality control during manufacturing²⁹,³⁰.

In this sense, the subjective-TOPSIS and AHP-TOPSIS approaches yielded similar rankings, both identifying Ni-based superalloys as the preferred candidates. This agreement indicates that the ranking method managed to moderate variations in subjective judgment and reinforces K_Ic,min as the dominant criterion when safety outweighs weight reduction.

In contrast, the equal weighting vector produced a different outcome, favoring titanium alloys. When all properties were assigned identical importance, density became as influential as fracture toughness and yield strength. Consequently, materials exhibiting an attractive balance between mechanical performance and low density became more competitive.

The entropy-based weighting vector produced the most divergent ranking. Because entropy weights are based on statistical dispersion, fracture toughness received low importance, while yield strength dominated the ranking. As a result, several materials with limited suitability for thick-walled pressure vessels ranked highly. The results, therefore, suggest that purely objective weighting schemes may not always adequately capture the engineering significance of critical failure mechanisms in safety-driven material-selection problems.

5. Conclusions

This study employed an unsupervised machine learning approach, specifically a combined k-means clustering method and the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), without any initial constraints, to aid in the material selection for a safe pressure vessel. The adoption of the entire Ansys® Granta Selector 2024 R1 catalog as initial candidates reduces confirmation bias in the screening stage.

This is a typical case involving multiple conflicting constraints, which can be conditioned by half the maximum length of any crack in the vessel wall ( $a_{c}$ ), a parameter that is not always accessible. To this end, four weighting vectors were employed: (i) a subjective weighting vector, (ii) a structure subjective weighting vector (AHP), (iii) an equal weighting vector, and (iv) an objective weighting vector based on the concept of entropy proposed by Shannon and Weaver. Approach (iv) is considered purely objective, as it disregards the decision-makers' perceptions. The choice of a relative weighting vector proved to be the most critical factor in material selection within a multi-criteria optimization context.

Using the subjective or AHP weighting vectors and conducting three code runs, the nickel-based superalloys (Ni-Cr, Ni-Fe-Cr, Ni-Mo, and Ni-Cr-Co-Mo) were identified as prominent candidates for the manufacture of safe pressure vessels. This outcome aligns with the historical use of these materials and is considered the most coherent outcome for the application under study. This reflects that, even when supported by machine learning techniques, decision-makers’ perceptions in material selection remain imperative.

This approach offers a credible alternative for material selection in the specific case studied, with potential for extrapolation to other contexts involving multi-criteria optimization strategies.

A formal sensitivity analysis of the weighting vectors was not included in the present study. The objective of this work was to compare representative weighting philosophies and evaluate the proposed decision-making framework, rather than to quantify the robustness of the rankings under weight perturbations. Future studies should investigate the sensitivity of the proposed framework to different weighting scenarios in order to further assess the stability and robustness of the resulting rankings.

5. Acknowledgments

This study was financed by the Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES) - Finance Code 001 and by the Modernization of Undergraduate Education PMG 2018984450P, supported by CAPES (88887.302130/2018-00) and by the Fulbright Commission in Brazil, in cooperation with the U.S. Embassy in Brasília, Brazil.

6. Data Availability

The Python script used here is from the free, open-source Scikit-learn library. The developed code is included in a public repository. See here: https://github.com/guiooz/kmeans-topsis.

Supplementary Material

The following online material is available for this article:

Figure S1

Sequence of the k-means algorithm operation.

Figure S2

Sequence of the algorithm operation used.

Figure S3

Elbow curves compared with the silhouette score for (a) the 1st run, (b) the 2nd run, and (c) the 3rd run using the subjective weighting vector. The number of clusters (K) selected subjectively based on the elbow curve: K = 7 for the first run, K = 5 for the second run, and K = 7 for the third run.

Figure S4

a) Contribution of each component to explaining the PCA dataset for the 1st run using the subjective weighting vector. b) Contribution of each component to explaining the PCA dataset for the 2nd run using the subjective weighting vector.

Table S1

Saaty’s pairwise comparison matrix employed to derive the AHP weighting vector.

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*

e-mail: lucas.staffa@ufscar.br

Associate Editor:

Hugo Sandim

Editor-in-Chief:

Luiz Antonio Pessan.

Publication Dates

Publication in this collection
21 Sept 2026
Date of issue
2026

History

Received
28 Oct 2025
Reviewed
01 Aug 2026
Accepted
13 Aug 2026

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Boundary conditions
Function	A spherical vessel with radius r that supports a pressure p safely.
Constraints	None
Objective	Must not fail by yielding and fast fracture, i.e., maximize K_Ic and σ_y
Free variable	Choice of the material

Properties	Subjective Weighting Vector (1)	AHP Weighting Vector (2)	Objective Weighting Vector (3)	Equals Weighting Vector (4)
K_{Ic, min}	0.300	0.36	0.0883	0.166
K_{Ic, max}	0.150	0.17	0.0838	0.166
σ_y,min	0.150	0.16	0.273	0.166
σ_y,max	0.150	0.14	0.252	0.166
ρ_min^-1	0.150	0.09	0.155	0.166
ρ_max^-1	0.100	0.06	0.148	0.166

	Unity	Cluster 0	Cluster 1	Cluster 2	Cluster 3
K_{Ic, min}	MPa.m^1/2	14.4708	109.162	0.007	28.165
K_{Ic, max}	MPa.m^1/2	19.2102	139.689	0.011	39.372
σ_y,min	MPa	152.418	423.407	0.190	1522.310
σ_y,max	MPa	184.649	567.568	0.237	1785.060
ρ_min	kg.m^-3	3037.95	9352.08	31.700	8166.59
ρ_max	kg.m^-3	3128.78	9478.04	36.003	8306.09
$C_{i}^{+}$		0.11	0.61	0.29	0.4356
Amount		2731	596	33	384
			Selected

	Unity	Cluster 0	Cluster 1	Cluster 2	Cluster 3	Cluster 4
K_{Ic, min}	MPa.m^1/2	98.726	90.195	91.139	252.364	72.067
K_{Ic, max}	MPa.m^1/2	119.962	120.052	126.969	284.97	86.167
σ_y,min	MPa	432.11	783.325	266.859	305.197	790.593
σ_y,max	MPa	599.35	938.470	339.549	409.288	890.630
ρ_min	kg.m^-3	16591	8124.500	8011.51	8285.15	4526.67
ρ_max	kg.m^-3	16783	8238.340	8122.94	8421.82	4585.19
$C_{i}^{+}$		0.191	0.387	0.202	0.620	0.44
Amount		100	151	252	66	27
					Selected

	Unity	Cluster 0	Cluster 1	Cluster 2
K_{Ic, min}	MPa.m^1/2	224.933	230.514	325.875
K_{Ic, max}	MPa.m^1/2	264.000	257.857	363.938
σ_y,min	MPa	512.733	211.286	316.062
σ_y,max	MPa	673.933	304.114	425.000
ρ_min	kg.m^-3	7714.000	8411.71	8543.75
ρ_max	kg.m^-3	7851.330	8544.57	8688.12
$C_{i}^{+}$		0.475	0.0679	0.589
Amount		15	35	16
				Selected

	$C_{i}^{+}$	Ranking
Nickel-Fe-Cr alloy, INCOLOY 800, annealed	0.688	1
Nickel-chromium alloy, INCONEL 686, annealed	0.542	2
Nickel-chromium alloy, HASTELLOY G, solution-treated	0.532	3
Nickel-molybdenum alloy, HASTELLOY B-2, plate	0.478	4
Nickel-Cr-Co-Mo alloy, INCONEL 617, annealed	0.461	5
Nickel-Mo-Cr alloy, HASTELLOY C-276	0.439	6
Nickel-chromium alloy, INCONEL 600, annealed	0.430	7
Nickel-chromium alloy, INCONEL 690, annealed	0.412	8
Nickel-molybdenum alloy, HASTELLOY B-2, sheet	0.399	9
Nickel-Mo-Cr alloy, HASTELLOY C-4	0.394	10
High alloy steel, Kovar, annealed	0.382	11
Nickel-Mo-Cr alloy, HASTELLOY S	0.350	12
Nickel-chromium alloy, HASTELLOY G-3, solution treated	0.345	13
Nickel-Mo-Cr alloy, HASTELLOY C-22, annealed	0.289	14
Nickel-chromium alloy, HAYNES 230, annealed	0.256	15
Nickel-Mo-Cr alloy, HASTELLOY N	0.186	16

	$C_{i}^{+}$	Ranking
Titanium, alpha-beta alloy, Ti-6Al-4V, cast	0.828	1
Titanium, alpha-beta alloy, Ti-6Al-4V, annealed	0.602	2
Titanium, alpha-beta alloy, Ti-6Al-4V ELI, annealed	0.601	3
Titanium, alpha-beta alloy, Ti-5Al-2.5Sn-0.5Fe, annealed	0.176	4

	$C_{i}^{+}$	Ranking
Silicon carbide (w)	0.618	1
Silica (25–35-micron monofilament, f)	0.616	2
Asbestos (tremolite)(f)	0.540	3
Carbon fibers, high strength (5 micron,f)	0.508	4
Glass, S grade (10-micron monofilament, f)	0.347	5
Asbestos (white)(f)	0.271	6
Asbestos (blue)(f)	0.232	7
Glass, C grade (10 micron monofilament,f)	0.201	8
Borsic (SiC/B/S 100-150 micron,f)	0.177	9

Weighting strategy	Iterations			Final number of materials
Weighting strategy	1st	2nd	3rd	Final number of materials
Subjective Weighting Vector (1)	596	66	16	16
AHP Weighting Vector (2)	596	66	16	16
Objective Weighting Vector (3)	384	9	-	9
Equals Weighting Vector (4)	596	27	4	4