**3. Results and Discussions**

Figure 2 presents the ranks of the near-*α* Ti alloys from the literature evaluated by the 12 MADMs. The ranks of the alloys represented as points in the figure by nature are discrete; thin dotted lines for a better visual effect connect the ranks assessed by each of the MADMs. Despite the unique mathematical aggregation procedures in various MADMs, the peaks and troughs of several MADMs somewhat coincide. For example, several MADMs assign similar ranks to WJZ-Ti-2, TKT-2, WJZ-Ti-1, PC-IMDF4, and KIMS-2 (green-shaded). Moreover, the rank assigned by various MADMs to most alloys differs significantly, for instance, as in the alloys designated as Ti-1100-5, IMI834-5 and JZ2-3 (pink shaded). Table 1 presents the Spearman rank (*Sρ*) that correlates ranks evaluated by the 12 MADMs. For example, the *Sρ* between CoCoSo and ROVM, MABAC and WEDBA, or MARCOS and TOPSIS is >0.95, which indicates strong correlations. On the contrary, *Sρ* between ARAS and SMART or TOPSIS and SMART is less than <0.3, which is expected owing to the distinct mathematical aggregation formulation in various MADMs. Out of the 66 combinations of MADM pairs, ~72% have rank correlations equal to or above 0.70, which elicits the robustness and validity of the ranking of the near-*α* Ti alloys. Therefore, it is imperative to consolidate the ranks obtained from various MADMs. Based on *Sρ* among all various combinations of MADMs, it is practical to consolidate the rankings evaluated by the 12 different MADM evaluations. Accordingly, the mean-based (arithmetic mean) rank consolidation of Figure 2 is shown in Figure 3. The ranks of the top ten data points are WJZ-Ti-2, WJZ-Ti-1, TKT-2, TA19-2, TKT-6, TKT-1, TA19-1, KIMS-2, IMI834-2, and PC-IMDF4 in that order.

**Figure 2.** The ranks of the near-*α* Ti alloys from the literature evaluated by the 12 multiple attribute decision making (MADM) methods. Several MADMs assign relatively similar ranks (green shaded) to WJZ-Ti-2, TKT-2, WJZ-Ti-1, PC-IMDF4, and KIMS-2, while Ti-1100-5, IMI834-5, and JZ2-3 are assigned diverse set of ranks (pink shaded).

**Table 1.** The Spearman rank (*Sρ*) correlation of the near-*α* Ti alloys ranks from the literature evaluated by the 12 multiple attribute decision-making (MADM) methods.


**Figure 3.** The arithmetic mean-based rank consolidation of the near-*α* Ti alloys from the literature evaluated by the 12 multiple attribute decision making (MADM) methods. The ranks of the top 10 data points are WJZ-Ti-2, WJZ-Ti-1, TKT-2, TA19-2, TKT-6, TKT-1, TA19-1, KIMS-2, IMI834-2, and PC-IMDF4 in that order.

Figure 4 is the score plot that presents the consolidated rank by PCA, of the near-*α* Ti alloys. It is the plot of the first two components (*PC1* and *PC2*), post-reduction of the data dimensionality (i.e., ranks from 12 MADMs) into a two-dimensional space. Table 2 presents the eigenvalues (and their proportion) that capture the variation of the distribution of each principal component. The first principal component (*PC1*) captures ~82% of the variation or scatter in the original data, while the second principal (*PC2*) describes ~17% of the variation. Since *PC1* captures nearly 82% of the variation in the initial 12 dimensions (sets of ranks), it approximates the rank of near-*α* Ti alloys. An imaginary reference line perpendicular to *PC1* traversing from left to right (−6 to 6) indicates the overall ranks of the near-*α* Ti alloys. The alloy grades WJZ-Ti, TKT-2, TA19, TKT-6, TKT-1, KIMS, IMI834, and PC top the list, followed by JZ1, JZ2, Ti-1100, and so on. The ranks of the top ten data points are WJZ-Ti-2, WJZ-Ti-1, TKT-2, TA19-2, TKT-6, TKT-1, TA19-1, KIMS-2, IMI834-2, and PC-IMDF4 in that order (the data points within the box in Figure 4), while certain variants of WJZ-Ti, JZ1, and JZ2 also appear promising (the data points close to the box). The top-ranked alloys by PCA-based consolidation are strikingly similar to the top-ranked alloys evaluated by mean-based consolidation. Specifically, the PCA-based consolidation refines the IMI834-2 (rank#9) and PC-IMDF4 (rank#9) assigned by mean-based consolidation to rank#9 and #10, respectively. Therefore, it is logical to label the score plot of PCA-based consolidated ranks as a 'rank chart'.

**Figure 4.** Score plot by principal component analysis (PCA) of the ordinal data, i.e., PCA-based rank consolidation of the near-*α* Ti alloys evaluated by the 12 MADM methods. The top-ranked alloy variants evaluated by PCA-based consolidation are strikingly similar to the top-ranked alloy variants evaluated by the mean-based consolidation. Specifically, the PCA-based consolidation refines the IMI834-2 (rank#9) and PC-IMDF4 (rank#9) assigned by the mean-based consolidation to rank#9 and #10, respectively.

**Table 2.** The eigenvalues and their proportion by the principal component analysis (PCA) of the ranks of the near-*α* Ti alloys from the literature by the 12 multiple attribute decision making (MADM) methods.


For deeper insight into the rank chart (Figure 4) of near-*α* Ti alloys, Figure 5a–d presents the score plots through the lens of various categories. Here, the region of interest (green box) corresponds to the top 10 alloy variants. Key inferences from the figures are as follows: (i) majority (seven out of 10) of the data points in the area of interest have aluminum equivalent to 8 (Figure 5a), (ii) all of the data points in the region of interest have a bimodal matrix, i.e., primary *α* + transformed *β* (Figure 5b), (iii) among the top ten data points, five (WJZ-Ti-1, TKT-2, TKT-1, TA19-2, and TA19-1) have no precipitates; one of them, WJZ-Ti-2, has precipitates Ti3Al in *α*p-1 (inside primary *α*); one of them (TKT-6) has germanide precipitates; one has silicide precipitates (KIMS-2—Hf in silicide and no Zr); and two (IMI834-2 and PC-IMDF4) have no information regarding the precipitates (Figure 5c) based on the chemistry, thermomechanical processing and the thermal treatments, these two variants would highly likely have Ti3Al and silicides; and lastly (iv) among the top 10 data points, five have no precipitates, four of them have nanocrystalline precipitates, and one has no information about any precipitate (Figure 5d). These analyses suggest guidelines for developing next-generation commercial near-*α* Ti alloys. The alloy design strategy for near-*α* Ti alloys for high-temperature applications with a combination of high *YS*, high *UTS*, and high *%EL* has two distinct options: (i) a combination of the aluminum equivalent to 8 and a bimodal matrix (primary *α* + transformed *β*) with no precipitates, (ii) a combination of the aluminum equivalent to 8, bimodal matrix, and nanocrystalline Ti3Al or germanide or silicide (no Zr, but Hf, as the silicides containing Hf, do not reduce ductility, however, Hf provides solid solution strengthening [3]) precipitates in *α*.

**Figure 5.** Score plots by the principal component analysis (PCA) of the ordinal data, i.e., PCA-based rank consolidation of the near-*α* Ti alloys evaluated by the 12 MADM methods through the lens of, i.e., categorized based on (**a**) aluminum equivalent, (**b**) matrix, (**c**) precipitates, and (**d**) precipitate size. The region of interest (green box) shows the top-ranked ten alloy variants.

In this investigation, we compile, evaluate, sort, and select near-*α* Ti alloys in the current literature for high-temperature applications in aeroengines, driven by decision science, by integrating MADM and principal component analysis (PCA). The evaluation provided valuable insight into potential existing materials ('research alloys') to focus on further research and development for commercialization. Among the top-ranked ten alloy variants (WJZ-Ti-2, WJZ-Ti-1, TKT-2, TA19-2, TKT-6, TKT-1, TA19-1, KIMS-2, IMI834-2, and PC-IMDF4), seven variants belong to the six 'research grade' alloys (WJZ-Ti, TKT-2, TKT-6, TKT-1, KIMS, and PC), while the data point IMI834-2 is a variant of an existing commercial alloy IMI834. Thus, all of these alloys appear to be strong contenders for large-scale development and testing. Additionally, in the future, newly discovered novel high-temperature Ti alloys (conventional and high-entropy alloys) can be included in the list and evaluated to assess their relative position in the rank chart and infer their potential to replace existing materials. In the near future, we plan to expand the decision science driven material selection by including several other relevant mechanical properties as

they become available. Lastly, this effort (i) validates the decision science driven MADM coupled with PCA for sorting, ranking, and material selection, (ii) weeds out the alloys that need not be pursued further with time-consuming experimental studies to generate data on additional attributes that are required for use for the intended application/s, and (iii) provide directions for advancing alloys that are under development or suggest some critical improvements for possible newer alloys by providing metallurgical perspectives. Developing a methodology that applies decision science principles to compile and sort a relatively large literature data, select or identify top-ranked alloys, unearth metallurgical patterns, and recommend guidelines for developing next-generation commercial near-*α* Ti alloys for aeroengines is the novelty of the investigation.

#### **4. Summary and Conclusions**

We compiled, evaluated, ranked, and selected near-*α* Ti alloys in the current literature for high-temperature applications in aeroengines, driven by decision science by integrating MADM and principal component analysis (PCA). A combination of 12 MADM methods ranked a list of 105 alloy variants based on the thermomechanical processing (TMP) conditions of 19 different near-*α* Ti alloys. PCA consolidated the ranks from various MADMs and identified ten top-ranked alloy variants for the intended application/s. The ten top-ranked alloy variants are WJZ-Ti-2, WJZ-Ti-1, TKT-2, TA19-2, TKT-6, TKT-1, TA19-1, KIMS-2, IMI834-2, and PC-IMDF4 in that order and they correspond to the following eight alloys: Ti-6.7Al-1.9Sn-3.9Zr-4.6Mo-0.96W-0.23Si, Ti-4.8Al-2.2Sn-4.1Zr-2Mo-1.1Ge, Ti-6.6Al-1.75Sn-4.12Zr-1.91Mo-0.32W-0.1Si, Ti-4.9Al-2.3Sn-4.1Zr-2Mo-0.1Si-0.8Ge, Ti-4.8Al-2.3Sn-4.2Zr-2Mo, Ti-6.5Al-3Sn-4Hf-0.2Nb-0.4Mo-0.4Si-0.1B, Ti-5.8Al-4Sn-3.5Zr-0.7Mo-0.35Si-0.7Nb-0.06C, and Ti-6Al-3.5Sn-4.5Zr-2.0Ta-0.7Nb-0.5Mo-0.4Si. The top-ranked alloys evaluated by PCA-based consolidation are strikingly similar to the top-ranked alloys evaluated by mean-based consolidation. The top-ranked alloys suggest the following metallurgical characteristics: bimodal matrix (primary *α* + transformed *β*), aluminum equivalent preferably up to 8, and nanocrystalline precipitates of Ti3Al, germanides, or silicides. The analyses driven by decision science made metallurgical sense. It provides guidelines for developing next-generation commercial near-*α* Ti alloys. The alloy design strategy for near-*α* Ti alloys for high-temperature applications with a combination of high *YS*, high *UTS*, and high *%EL* has two distinct options: (i) a combination of the aluminum equivalent to 8 and a bimodal matrix with no precipitates, or (ii) a combination of the aluminum equivalent to 8, bimodal matrix, and nanocrystalline Ti3Al or germanide or silicide (not Zr, but Hf, as the silicides containing Hf do not reduce ductility, to the contrary, Hf provides solid solution strengthening) precipitates in *α*. Thus, novel alloys could be developed based on these directions for the future. A similar analysis could include data from newer exotic experimental materials, such as composites, for compressor parts.

**Author Contributions:** Conceptualization, T.V.J. and R.C.; Methodology, T.V.J.; Software, T.V.J.; Validation, R.C. and T.V.J.; Formal Analysis, T.V.J. and R.C.; Investigation, T.V.J. and R.C.; Data Curation, R.C.; Writing—Original Draft Preparation, T.V.J.; Writing—Review & Editing, T.V.J. and R.C.; Visualization, T.V.J.; Supervision, R.C. and T.V.J.; Project Administration, R.C. and T.V.J.; Funding Acquisition, R.C. and T.V.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** Weldaloy Specialty Forgings (Research and Development Account# 8860.00) Institute of Advanced Vehicle Systems (grant# 052349), University of Michigan-Dearborn.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** No applicable.

**Data Availability Statement:** The data used in the analyses is presented in Tables A1 and A2.

**Acknowledgments:** The author R. Canumalla thanks the Weldaloy Specialty Forgings management for all their support; the author T. V. Jayaraman, thanks the Department of Mechanical Engineering, College of Engineering and Computer Science, University of Michigan-Dearborn for all their support. **Conflicts of Interest:** The authors declare no conflict of interest.
