Material Selection Using Hybrid Grey Relation Analysis Approach Based on Weighted Entropy for Ranking: The Case of Helicopter Rotor Blade

E ngineering design relies highly on the selection of suitable materials. Because there are many engineering materials, selecting a suitable material for a product requires a systematic selection approach. This paper provides a hybrid strategy for choosing the best material for an engineering design to give the best performance at the lowest cost based on Ashby's performance indices. Then, it ranks the result by the grey relational approach integrated with the Weighted Entropy Method to choose the optimum material for the main rotor blade of a helicopter. Different materials used for manufacturing rotor blades, such as Aluminium alloys, titanium alloys, steel, composites, and wood, have been discussed. The performance indices chosen are stiffness, strength, and fracture toughness. The performance indices proved that the composite material has excellent structural strength, stiffness, and toughness. The result shows that CFRP is the best material for manufacturing helicopter rotors, while wood and steel were the best and cheapest when the design had to be economical.


INTRODUCTION
The Selection of materials is one of the most practical but challenging problems developers face since it is linked to process performance.Designers, engineers, and manufacturers continuously seek new and improved materials to enhance performance and lower the cost of the items to stay market competitive (Al-Mendwi, 2009; Mehmood et al., 2018).A helicopter, unlike a normal fixed-wing aircraft, uses rotary blades.A helicopter's wings, or blades, are part of a more extensive dynamic system called the rotor (Edwards and Davenport, 2006).The rotor is the significant component of the helicopter, consisting of blades attached to the center of the rotor.The main rotor blades achieve the vehicle's lift, which provides thrust and generates lift.Regarding design considerations, the rotor blade is considered a one-dimensional beam (Mishra et al., 2020).Fig. 1 shows the helicopter rotor blade profile.Selecting a material for the rotor blade frame is a critical component of the design process for a helicopter.Rotor blade frames can be constructed using a variety of materials.Metals and composites are the broad categories that cover these materials.Wood, aluminium, steel, titanium, fiber composites (glass, carbon, and aramid fibers), and other metals are used to make rotor blades.According to Balaji and his colleagues carbon epoxy and boron epoxy are recommended design considerations for the helicopter blade due to their good strength-to-weight ratio.They are also compared to Aluminium material using the ANSYS simulation tool This research outlines an integrated (hybrid) strategy for selecting the best material for an engineering design that provides the best performance at the lowest cost.The primary purpose is to identify the best candidate material for the rotor based on Ashby's performance indices and then rank the result by the grey relational approach based on the entropy weight method.The Entropy Weight Method computes the weights of criteria (performance indices).

MATERIALS SELECTION METHODOLOGY
Material selection is a crucial step in the design process and manufacturing.Generally, material selection aims to minimize cost while meeting customer requirements and performance goals (Rahim et al., 2020).

Short-Listed Materials and Relevant Attributes
Developing a lightweight rotor blade for a helicopter aims to improve mechanical attributes and reduce costs.Depending upon those basic parameters, material density, yield strength, Young's modulus, fracture toughness, and cost are relevant attributes.Different materials used for manufacturing rotor blades, such as Aluminium alloys, titanium alloys, steel, composites, and wood, have been discussed.The performance indices chosen are stiffness, strength, and fracture toughness.A list of materials meeting these requirements is shown in Table 1.The short-listed materials were then optimized using the performance indices.The performance weight was calculated using the entropy-weighted method, and the results were ranked using the grey relational analysis method.

Material Performance Index
The selection of rotor blade material depends mostly on structural strength and costeffectiveness (Mishra et al., 2020).Consequently, the material's mechanical attributes, including high strength, stiffness, and fracture toughness, are listed as the material requirement.
The strength-to-weight ratio of a material is among the most important requirements to consider when selecting the material in aero engineering applications (Mohammed, 2017).
The index that maximizes the ratio of strength to weight is as follows: for cost-effectiveness: The rotor blade structure must be rigid enough not to bend or buckle.Maximizing fracture stiffest index is as follows: for cost-effectiveness: The material's fracture toughness is another important attribute to consider.Increasing the fatigue life of the blades requires a material with high fracture toughness.The maximizing fracture toughness index is as follows: for cost-effectiveness: The optimum materials are evaluated using the indices.The material with the maximum M value is best fitted for rotor blade construction.Additional criteria, such as cost, will also be considered in the material selection.Supporting information is then gathered to ranking materials to a final choice, providing a close match between design requirements and material attributes.

An Overview of Grey Relational Analysis Method
The GRA method was developed based on the grey system theory (Vatansever and Akgűl, 2018).The grey relational grade, often known as the GRG, can represent the degree of the relationship between multiple responses (Hammood, 2021); better solutions will have a higher GRG.As a result, GRG can be utilized as an evaluation index for problems with multiple objectives; GRA is used to determine a priority ranking for all possible design phases (Zhang et al., 2022).This theory is widely used in various research fields because of its advantages in evaluating complex systems with several linked indicators.This concept has been demonstrated to aid in processing uncertain, incomplete, or inaccurate data (Maidin et al., 2022;Wu et al., 2018).GRA is often used to measure financial performance, logistic performance, and process optimization (Patil et Additionally, the cost attribute index, where the smaller the better, can be normalized as follows (Wu et al., 2018): where   is the value of performance indices for each material and   is the linear scale standardized matrix (Wu et al., 2018).

Derivation of the Reference Sequence
After the grey relational sequence was generated, a reference sequence, X0, with values equal to 1, was defined and compared to the generated sequence.The following is the reference sequence (Maidin et al., 2022): where 0 is the reference sequence value.
The matrix can be written as (Wu et al., 2018): where  is the reference sequence matrix.

Calculating the Grey Relational Coefficient
The grey relational coefficient shows a degree of grey relation between the reference sequence and experiment sequence that can be computed using the equations below (Wu et al., 2018): where   is the grey relational coefficient of the ℎ index of the ℎ alternative.

Grey Relational Grade
The grey relational grade is distributed between zero and one (Sarraf and Nejad, 2020) Grey relational quality is obtained by using the formula below: where   is the weight assigned to the attribute j.The total weight assigned to the attributes is unity (Zhang et al., 2022)

An Overview of Entropy Weighting Method
The Entropy technique is used to assign weights to the requirements.It is an essential information-weighting method that eliminates the effects of personal factors on variable weighting.It is widely applied and has several uses in engineering and other industries where   is a numerical number indicating the alternative's performance.The second step is the normalization of the decision matrix (performance indices) as follows (Zhu et al., 2020): The third step is to calculate the entropy (Zhu et al., 2020) The fourth step is to calculate the objective weight value:

APPLICATION AND RESULTS
This research aims to determine the optimal material for designing helicopter rotor blades.The design must be stiff, strong, and tough while light and cheap for a better design.Regarding Ashby's method, the importance of benefit and non-benefit attributes in the design is essential for identifying the differences between attributes when generating the material indices.The goal is always to enhance the value of the benefits characteristic and minimize that of the non-benefit attribute.Among the considered attributes, density and cost are classified as non-benefit features, whereas the remaining attributes are benefit attributes.The following material indices are maximized or minimized based on what needs to be maximized or minimized.The following indices confirm that a given design's component performs at an optimum level:  Young's modulus versus density  1/2 /  Young's modulus versus cost  1/2 /  Yield strength versus density  2/3 /  Yield strength versus cost  2/3 /  Fracture toughness versus density 1/  Fracture toughness versus cost 1/

Performance Evaluation without Cost Criteria
The study's application is divided into three sections.The first includes calculating the rotor blade performance indices.The Entropy Weight Method is used in the second step to determine the indices' weights.In the last step, the performance indices were ranked using Grey Relational Analysis to select the optimal material.

The Entropy Weight Method to Determine the Weights of the Indices
The entropy-weighted method was used to determine each alternative's performance weight by applying Eq.s, (14-17) to the performance values in Table 2, which is written as a decision matrix.Table 3 shows the normalization procedure using Eqn.15.The following procedure is for finding the entropy value using Eqn.16, as in Table 4.The last step is calculating the weight using Eq. ( 17), as shown in Table 5, which illustrates the weights of the indices.Examining these values reveals that the criteria weights are quite close, as shown in Fig. 3.

Performance ranking utilizing grey relational analysis
Table 6 shows the normalization matrix derived from Eq. ( 7), where the more significant, the better (or benefit attributes).The grey relational coefficient is determined by calculating the deviation after normalization.Table 7 illustrates the deviation of the reference sequence value derived using the Eq.s (9, 10), and Table 8 shows the grey relation coefficient and grade derived using the Eq.s (11,12).Ranking the results obtained, the best material alternative, as shown in Fig. 4, according to the GRA method, is CFRP.

Performance Evaluation with Cost Criteria
The rotor blade performance indices with cost attributes are given in Table 9.The indices are calculated using the values of the individual properties in Table 1 by applying Eq.s (2, 4, and 6) for cost-effectiveness.The cheapest material in each category has a maximum index value.Wood and Steel have maximum values according to the stiffness, toughness and strength indices.As a result, it is understood from the decision matrix that wood and Steel a good material alternative.Fig. 5 illustrates the result obtained from Table 9.The performance weight was calculated.Table 10 demonstrates the weights of the indices with cost-effectiveness.The performance weight of each alternative is determined by applying Eq.s (15-17) to the performance values in Table 9, which is written as a decision matrix as described previously.Examining these values reveals that the toughness and strength criteria weights are highest compared to the stiffness, as shown in Fig. 6.   9 is considered maximized indices.The normalization matrix provided in Table 11 is calculated using Eq.7; the larger it is the better, as described previously.The grey relation coefficient is determined by calculating the deviation; Table 12 illustrates the deviation value of the reference sequence derived using the Eqs.(9, 10), whereas Table 13 shows the grey relation coefficient and grey grade derived using the Eq.s (11,12).Ranking the results obtained by the GRA method according to the GRA method, the cheapest material alternative is wood, followed by steel, as shown in Fig. 7.

CONCLUSIONS
The helicopter rotor blade material was successfully selected using the integrated material selection strategy for choosing the best material for an engineering design based on the performance indices.Different materials used for manufacturing rotor blades, such as aluminum alloys, titanium alloys, steel, composites, and wood, have been discussed.The performance indices chosen are stiffness, strength, and fracture toughness.The performance weight was calculated using the entropy-weighted method, and the results were ranked using the grey relational analysis method.
Based on the results obtained, we come to the following significant conclusions:  The performance indices proved that the composite material has excellent structural strength, stiffness, and toughness. The method used in this research found that CFRP is the best material for manufacturing helicopter rotors without cost consideration.The results are logical and reasonable because CFRP has low density and high strength but is also very expensive. Wood, followed by steel, was the best and cheapest material when the design had to be economical. Steel has more than 4 times the density of the CFRP and less strength.But it is cheaper than CFRP many times. Wood is low in density, strength, and cost, but its manufacturing process is complicated and unproductive.

Figure 1 .
Figure 1.Helicopter blade profile (Bagheinia and Ghassemi, 2018) (Balaji et al., 2016).(Mishra et al., 2020) propose a technique for analyzing the vibration of a rotor blade using FEM methods, which is achieved by designing a rotor blade using Composite materials (Glass Epoxy 2024).Generally, many methods are available to make the material selection (Emovon and Oghenenyerovwho,2020).Design engineers and decision-makers utilize a variety of approaches to select the best material from several alternatives.Identifying the objective, constructing the selection criteria, defining the suitable options, and final selection are the four stages of the selection process (Erzaij and Bidan, 2016).(Patil et al., 2017) illustrate using a Fuzzy Analytical Hierarchical Approach in parallel with Grey Relation Analysis to choose the most suitable automobile.(Radhi and Burhan,2022; Hasan and Jaber, 2023; Erzaij and Bidan, 2016; Zakeri et al., 2023) intended to examine the appropriateness of various MC-DM strategies.In (Wu et al., 2018), A hybrid systematic evaluation model was suggested, which integrates grey relational analysis (GRA) with the analytic hierarchy process (AHP) and a unique entropy-based method to derive the objective weighting of indices.(Unal et al., 2018; Lee et al., 2020) illustrated the material selection procedure; they accomplished the study with the use of a design performance index together with Ashby charts.

Figure 2 .
Figure 2. Performance indices and their values for each alternative.

Figure 3 .
Figure 3. Performance indices weight according to entropy weighted method.

Figure 4 .
Figure 4. Weighted grey relational coefficients and grad results

Figure 5 .Figure 6 .
Figure 5. Cost performance indices and their values for each alternative

Table 1 .
Material short-list and their properties.

al., 2017; Al-Taie, and Doos, 2023).
The following are the procedures involved in the traditional grey relational analysis (

Table 2
shows the rotor blade performance indices.The indices are calculated using the values of the individual properties in Table1 by applying Eqs.(1, 3, and 5) without costeffectiveness.Composites, Aluminum alloys, Titanium alloys, steel, and wood are the five alternative materials.It indicates that each material's index is different.The best material in each category has the most significant index value.CFRP has the highest values according to the stiffness, toughness, and strength indices.It is understood from the decision matrix that CFRP is a good material alternative.Fig.2illustrates the result obtained from Table2.

Table 2 .
Performance indices and their values for each alternative.

Table 3 .
The normalized matrix for indices

Table 4 .
Entropy value for indices

Table 5 .
Performance indices weight according to entropy weighted method.

Table 6 .
Performance indices normalization for benefit attributes.

Table 8 .
Weighted grey relational coefficients and grad results

Table 9 .
Cost performance indices and their values for each alternative

Table 13 .
Weighted grey relational coefficients and grad results.
Figure 7. Weighted grey relational coefficients and grad results.