Extended MULTIMOORA method based on Shannon entropy weight for materials selection
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Abstract
Selection of appropriate material is a crucial step in engineering design and manufacturing process. Without a systematic technique, many useful engineering materials may be ignored for selection. The category of multiple attribute decision-making (MADM) methods is an effective set of structured techniques. Having uncomplicated assumptions and mathematics, the MULTIMOORA method as an MADM approach can be effectively utilized for materials selection. In this paper, we developed an extension of MULTIMOORA method based on Shannon entropy concept to tackle materials selection process. The entropy concept was considered to assign relative importance to decision-making attributes. The proposed model consists of two scenarios named the weighted and entropy-weighted MULTIMOORA methods. In the first scenario, subjective weight was considered in the formulation of the approach like most of conventional MADM methods. The general form of entropy weight that is a combination of subjective and objective weighting factors was employed for the second scenario. We examined two popular practical examples concerning materials selection to show the application of the suggested approach and to reveal the effect of entropy weights. Our results were compared with the earlier studies.
Keywords
Multiple attribute decision making MULTIMOORA Shannon entropy Materials selectionIntroduction
More than 40,000 practical metallic alloys and a same number of nonmetallic materials like polymers, ceramics, and composites are utilized in various industries (Farag 2002). Because of the considerable number, dissimilar production techniques, and different properties of engineering materials, the selection process of materials can be regarded as a complex undertaking for an engineer or designer. If the process takes place unsystematically, many significant materials may be neglected. Therefore, a structured mathematical approach is needed for materials selection.
MADM methods can be used as effective systematic tools for materials selection. Each MADM technique has specific assumptions and principles. A number of MADM methods have been utilized in the materials selection process by earlier researchers, like the technique for order preference by similarity to ideal solution (TOPSIS) (Bakhoum and Brown 2013; Das 2012; Huang et al. 2011; Jee and Kang 2000), analytic hierarchy approach (AHP) (Chauhan and Vaish 2013; Dweiri and Al-Oqla 2006), compromise ranking also known as vlse kriterijumska optimizacija kompromisno resenje (VIKOR) (Jahan and Edwards 2013b; Liu et al. 2013), diverse versions of elimination and choice expressing the reality (ELECTRE) also recognized as outranking method (Anojkumar et al. 2014; Chatterjee et al. 2009; Shanian and Savadogo 2009), preference ranking organization method for enrichment evaluation (PROMETHEE) (Jiao et al. 2011; Peng and Xiao 2013), graph theory and matrix approach (Rao 2006), gray relational analysis (Chan and Tong 2007; Zhao et al. 2012), various preference ranking-based techniques (Chatterjee and Chakraborty 2012; Maity et al. 2012), preference selection index (Maniya and Bhatt 2010), utility additive (UTA) (Athawale et al. 2011), weighted property index (Findik and Turan 2012), linear assignment (Jahan et al. 2010a), modified digital logic (Manshadi et al. 2007; Torrez et al. 2012), Z-transformation (Fayazbakhsh and Abedian 2010; Fayazbakhsh et al. 2009), and quality function deployment (Mayyas et al. 2011; Prasad 2013; Prasad and Chakraborty 2013). Two groups of researchers have reviewed the applications of MADM methods in materials selection (Jahan and Edwards 2013a; Jahan et al. 2010b).
Almost all the aforementioned methods have a key feature that is moderate to the extreme complexity of their mathematical models. Utilization of these techniques seems to be difficult, requiring advanced mathematical knowledge (Karande and Chakraborty 2012a). Accordingly, an undemanding MADM method can be a real blessing for decision makers. The multi-objective optimization on the basis of ratio analysis (MOORA) method proposed by Brauers and Zavadskas (2006) has uncomplicated mathematics. Therefore, it can be employed effortlessly and effectually for selection of materials. The MULTIMOORA method is a comprehensive form of the MOORA technique. As discussed by Brauers and Ginevičius (2010), because the final rank is generated by the integration of three subordinate ranks in the MULTIMOORA technique, its results can be more robust than traditional MADM methods in which a single rank is obtained. The MOORA and MULTIMOORA techniques have been used in different applications like decision making in manufacturing environment (Chakraborty 2011), robot selection (Datta et al. 2013), supplier selection (Farzamnia and Babolghani 2014; Karande and Chakraborty 2012b; Mishra et al. 2015), evaluating the risk of failure modes (Liu et al. 2014a), project selection (Rached-Paoli and Baunda 2014), selection of health-care waste treatment (Liu et al. 2014b), ranking of banks (Brauers et al. 2014), and student selection (Deliktas and Ustun 2015).
In the present paper, we extended the MULTIMOORA method using entropy weight based on Shannon information theory for application in materials selection. Our study is closely related to Karande and Chakraborty (2012a). They used the MOORA technique in the materials selection process of four practical cases. The novelties of our paper comparing the study of Karande and Chakraborty (2012a) are as follows: First, they did not calculate the final ranking of the MULTIMOORA method and only reported the three subordinate ranks. The third subordinate rank of the MULTIMOORA method, i.e., the full multiplicative form rank, was incorrectly called the MULTIMOORA ranking in their study. In this paper, we employed the dominance theory to integrate the three subordinate ranks into the final ranking, named the MULTIMOORA ranking. This aggregate final ranking is more robust than each of the subordinate ranks as stated by Brauers and Ginevičius (2010). Second, Karande and Chakraborty (2012a) did not utilize any relative significance for attributes. However, we used two forms of attributes weighting, i.e., subjective and the general Shannon entropy weights, to generate two solution modes named the weighted and entropy-weighted MULTIMOORA rankings, respectively. Third, Karande and Chakraborty (2012a) employed Voogd ratio (Voogd 1983) for normalization, whereas we utilized the original MULTIMOORA normalization equation that is the most robust option among various ratios as shown by Brauers and Zavadskas (2006). A few studies on assigning weights for the MOORA and MULTIMOORA techniques exist. Brauers and Zavadskas (2006) mentioned that giving importance to each attribute is possible, but they did not discuss on the specifications of these significance factors. Özçelik et al. (2014) assigned weight for the reference point approach of the MOORA method. In their study, the fuzzy analytic hierarchy process was utilized for the determination of significance coefficients of attributes. El-Santawy (2014) used a new form of entropy weight to develop the MOORA method. Derivation of their significance factors differs from Shannon entropy weight. In addition, they did not develop the MULTIMOORA method with their suggested weights. To the best of the authors’ knowledge, no study has been conducted on combination of Shannon entropy weight with MULTIMOORA technique. In our proposed approach, the general form of entropy weight was utilized that includes subjective and objective parts. The subjective significance coefficient is obtained directly from decision makers opinions. The objective part is calculated based on the entropy concept through analyzing the data regardless of decision makers’ comments. The general form of entropy weight improves the initial values of decision matrix and reliability of the ranking of alternatives obtained by the MULTIMOORA approach. We evaluated two practical examples in the field of materials selection. The results were compared with other studies that have considered these two problems. Eventually, concluding remarks were cited to make a summary of our work and to present an overview of the developed MULTIMOORA method and its application in materials selection.
The MULTIMOORA method
The ratio system
The reference point approach
The full multiplicative form
The assessment values of \( U_i^* \) differ from \( U_i^\prime \); however, the ranking calculated by both equations is analogous. Accordingly, to preserve a harmony between all parts of the MULTIMOORA method, we use Eq. (9) as the full multiplicative form representation.
The final ranking of the MULTIMOORA method based on the dominance theory
The dominance theory was employed as a tool for consolidation of subordinate rankings of the MULTIMOORA method (Brauers et al. 2011; Brauers and Zavadskas 2011, 2012). After the calculation of the subordinate ranks as above, they can be integrated into a final ranking, named the MULTIMOORA rank, based on the dominance theory. For a detailed explanation of the dominance theory, readers can refer to the study of Brauers and Zavadskas (2012).
Shannon entropy weight
Entropy concept has been widely employed in social and physical sciences. Economics, spectral analysis, and language modeling are a few typical practical applications of entropy. A mathematical theory of communication was proposed by Shannon (1948). Entropy evaluates the expected information content of a certain message. Entropy concept in information theory can be considered as a criterion for the degree of uncertainty represented by a discrete probability distribution.
Entropy idea can be effectively employed in the process of decision making, because it measures existent contrasts between sets of data and clarifies the average intrinsic information transferred to decision maker.
- Step 1 Normalization of the arrays of decision matrix (performance indices) to obtain the project outcomes p _{ ij }:$$ p_{ij} = \frac{{x_{ij} }}{{\sum_{i = 1}^m {x_{ij} } }} $$(11)
- Step 2 Computation of the entropy measure of project outcomes using the following equation:in which k = 1/ln(m).$$ E_j = - k\sum_{i = 1}^m {p_{ij} \ln p_{ij} ,} $$(12)
- Step 3 Defining the objective weight based on the entropy concept:$$ w_j = \frac{1 - E_j }{{\sum_{j = 1}^n {({1 - E_j })} }} $$(13)
- Step 4 Calculating the general form of the entropy weight, if the decision maker assigns subjective weight s _{ j }. By considering s _{ j }, Eq. (13) transforms into the following:in which subjective and objective weights (s _{ j } and w _{ j }) are combined to produce the general form of Shannon entropy weight \( w_j^* \).$$ w_j^* = \frac{s_j w_j }{{\sum_{j = 1}^n {s_j w_j } }}, $$(14)
The extended MULTIMOORA method based on Shannon entropy weight
In the initial paper on the MOORA method, Brauers and Zavadskas (2006) allocated a section for the importance given to an attribute. They mentioned that a significance coefficient can be considered to affix more importance to a specific attribute. Their weighted form of the MOORA method confines to general representation of the main formulas and no details concerning characteristics of significance coefficient have been cited. This concept was later updated to encompass all subsections of MULTIMOORA method (Brauers and Zavadskas 2010, 2011).
Significance coefficient can be subjective weight gained directly from the decision makers similar to the routine procedure of the majority of MADM methods. The coefficient can also be regarded as an objective factor like Shannon entropy weight. The inclusive significance coefficient is the combination of subjective and objective factors like the general Shannon entropy weight.
In the present paper, we designate two forms of weight as significance coefficient of attributes. If significance coefficient only consists of subjective weight s _{ j } earned from the decision makers, the resultant approach is named the weighted MULTIMOORA method. Application of the general Shannon entropy weight that is a combined subjective and objective significance coefficient leads to the so-called entropy-weighted MULTIMOORA technique. Based on Shannon entropy weight and the original MULTIMOORA approach, the following methodology is attained.
The extended ratio system
The extended reference point approach
The extended full multiplicative form
The final ranking of the extended MULTIMOORA method based on the dominance theory
By utilizing the dominance theory, we integrated the subordinate rankings into a final ranking.
Application of the extended MULTIMOORA method in materials selection
Karande and Chakraborty (2012a) utilized the MOORA technique to choose materials for different applications. However, they altered the original normalization ratio of the method, i.e., Eq. (2), into another form. They used Voogd ratio (Voogd 1983) as normalization formula that is \( x_{ij}^* = {{x_{ij} } / {\sum_{i = 1}^m {x_{ij} } }} \).
Brauers and Zavadskas (2006) established that among different choices for the denominator of the normalization ratio, the square root of the sum of each alternative performance index, \( [{\sum_{i = 1}^m {x_{ij}^2 } }]^{1/2} \), is the most robust option. Therefore, the results of the study of Karande and Chakraborty (2012a) may not be as robust as the original MULTIMOORA method. Thus, we do not verify our results with their outcomes.
In the following subsections, we calculated the weighted and entropy-weighted MULTIMOORA rankings for two material selection problems cited in the study of Karande and Chakraborty (2012a). Besides, we compared our results with the related studies on the field.
Example 1: Material selection for flywheel
Candidate materials and their properties for Example 1 (Jee and Kang 2000)
Materials | Properties | ||||
---|---|---|---|---|---|
Fatigue strength (Mpa) | Fracture toughness (Mpa·m^{1/2}) | Density (g/cm^{3}) | Price/mass (10^{3} US$/t) | Fragmentability | |
300 M | 800 | 68.9 | 8 | 4.2 | 3 (poor) |
2024-T3 | 140 | 38 | 2.82 | 2.1 | 3 (poor) |
7050-T73651 | 220 | 35.4 | 2.82 | 2.1 | 3 (poor) |
Ti–6Al–4V | 515 | 123 | 5 | 10.5 | 3 (poor) |
E glass–epoxy FRP | 140 | 20 | 2 | 2.735 | 9 (excellent) |
S glass–epoxy FRP | 330 | 50 | 2 | 4.095 | 9 (excellent) |
Carbon–epoxy FRP | 700 | 35 | 2 | 35.47 | 7 (fairly good) |
Kevlar 29–epoxy FRP | 340 | 40 | 1 | 11 | 7 (fairly good) |
Kevlar 49–epoxy FRP | 900 | 50 | 1 | 25 | 7 (fairly good) |
Boron–epoxy FRP | 1000 | 46 | 2 | 315 | 5 (good) |
Decision matrix for Example 1 (Jee and Kang 2000)
Materials | Beneficial and non-beneficial attributes | |||
---|---|---|---|---|
MAX | MAX | MIN | MAX | |
Specific strength (kN·m/kg) | Specific toughness (kPa·m^{1/2}/kg/m^{3}) | Price/mass (10^{3} US$/t) | Fragmentability | |
300 M | 100 | 8.613 | 4.2 | 3 |
2024-T3 | 49.645 | 13.475 | 2.1 | 3 |
7050-T73651 | 78.014 | 12.553 | 2.1 | 3 |
Ti–6Al–4V | 108.879 | 26.004 | 10.5 | 3 |
E glass–epoxy FRP | 70 | 10 | 2.735 | 9 |
S glass–epoxy FRP | 165 | 25 | 4.095 | 9 |
Carbon–epoxy FRP | 440.252 | 22.013 | 35.47 | 7 |
Kevlar 29–epoxy FRP | 242.857 | 28.571 | 11 | 7 |
Kevlar 49–epoxy FRP | 616.438 | 34.247 | 25 | 7 |
Boron–epoxy FRP | 500 | 23 | 315 | 5 |
Normalized decision matrix for Example 1
Materials | Beneficial and non-beneficial attributes | |||
---|---|---|---|---|
MAX | MAX | MIN | MAX | |
Specific strength | Specific toughness | Price/mass | Fragmentability | |
300 M | 0.103 | 0.124 | 0.013 | 0.156 |
2024-T3 | 0.051 | 0.194 | 0.007 | 0.156 |
7050-T73651 | 0.080 | 0.181 | 0.007 | 0.156 |
Ti–6Al–4V | 0.112 | 0.375 | 0.033 | 0.156 |
E glass–epoxy FRP | 0.072 | 0.144 | 0.009 | 0.468 |
S glass–epoxy FRP | 0.170 | 0.360 | 0.013 | 0.468 |
Carbon–epoxy FRP | 0.453 | 0.317 | 0.111 | 0.364 |
Kevlar 29–epoxy FRP | 0.250 | 0.412 | 0.035 | 0.364 |
Kevlar 49–epoxy FRP | 0.634 | 0.493 | 0.079 | 0.364 |
Boron–epoxy FRP | 0.514 | 0.331 | 0.989 | 0.260 |
Entropy measure and weighting factors for Example 1 (Jee and Kang 2000)
Entropy and weights | Beneficial and non-beneficial attributes | |||
---|---|---|---|---|
MAX | MAX | MIN | MAX | |
Specific strength | Specific toughness | Price/mass | Fragmentability | |
s _{ j } | 0.4 | 0.3 | 0.2 | 0.1 |
E _{ j } | 0.861 | 0.963 | 0.415 | 0.960 |
w _{ j } | 0.174 | 0.047 | 0.730 | 0.050 |
\( w_j^* \) | 0.296 | 0.060 | 0.623 | 0.021 |
Assessment values and rankings of the weighted MULTIMOORA method for Example 1
Materials | Assessment values | Rankings | |||||
---|---|---|---|---|---|---|---|
\( y_i^w \) | \( z_i^w \) | \( U_i^w \) | \( y_i^w \) | \( z_i^w \) | \( U_i^w \) | Final | |
Rank | Rank | Rank | Rank | ||||
300 M | 0.091 | 0.212 | 0.425 | 10 | 7 | 9 | 9 |
2024-T3 | 0.093 | 0.233 | 0.422 | 9 | 10 | 10 | 10 |
7050-T73651 | 0.101 | 0.222 | 0.495 | 8 | 8 | 6 | 7 |
Ti–6Al–4V | 0.166 | 0.209 | 0.510 | 5 | 6 | 5 | 5 |
E glass–epoxy FRP | 0.117 | 0.225 | 0.469 | 7 | 9 | 8 | 8 |
S glass–epoxy FRP | 0.220 | 0.186 | 0.802 | 4 | 4 | 2 | 4 |
Carbon–epoxy FRP | 0.290 | 0.072 | 0.724 | 2 | 2 | 4 | 2 |
Kevlar 29–epoxy FRP | 0.253 | 0.154 | 0.779 | 3 | 3 | 3 | 3 |
Kevlar 49–epoxy FRP | 0.422 | 0.014 | 1.014 | 1 | 1 | 1 | 1 |
Boron–epoxy FRP | 0.133 | 0.197 | 0.482 | 6 | 5 | 7 | 6 |
Assessment values and rankings of the entropy-weighted MULTIMOORA method for Example 1
Materials | Assessment values | Rankings | |||||
---|---|---|---|---|---|---|---|
\( y_i^{ew} \) | \( z_i^{ew} \) | \( U_i^{ew} \) | \( y_i^{ew} \) | \( z_i^{ew} \) | \( U_i^{ew} \) | Final | |
Rank | Rank | Rank | Rank | ||||
300 M | 0.033 | 0.157 | 6.404 | 8 | 6 | 5 | 8 |
2024-T3 | 0.026 | 0.173 | 8.230 | 9 | 9 | 3 | 9 |
7050-T73651 | 0.034 | 0.164 | 9.370 | 7 | 7 | 1 | 6 |
Ti–6Al–4V | 0.038 | 0.155 | 3.965 | 5 | 5 | 8 | 5 |
E glass–epoxy FRP | 0.035 | 0.167 | 7.774 | 6 | 8 | 4 | 7 |
S glass–epoxy FRP | 0.074 | 0.138 | 8.233 | 4 | 4 | 2 | 4 |
Carbon–epoxy FRP | 0.092 | 0.065 | 2.834 | 2 | 2 | 9 | 2 |
Kevlar 29–epoxy FRP | 0.085 | 0.114 | 5.003 | 3 | 3 | 6 | 3 |
Kevlar 49–epoxy FRP | 0.176 | 0.045 | 3.997 | 1 | 1 | 7 | 1 |
Boron–epoxy FRP | −0.438 | 0.612 | 0.752 | 10 | 10 | 10 | 10 |
Comparison between the materials ranks of the proposed model and other methods for Example 1
Materials | Methods | |||||
---|---|---|---|---|---|---|
Weighted MULTIMOORA | Entropy-weighted MULTIMOORA | TOPSIS (Jee and Kang 2000) | ELECTRE (Chatterjee et al. 2009) | VIKOR (Chatterjee et al. 2009) | Linear assignment (Jahan et al. 2010a) | |
300 M | 9 | 8 | 5 | 10 | 9 | 7 |
2024-T3 | 10 | 9 | 9 | 9 | 10 | 10 |
7050-T73651 | 7 | 6 | 7 | 8 | 8 | 8 |
Ti–6Al–4V | 5 | 5 | 6 | 6 | 6 | 6 |
E glass–epoxy FRP | 8 | 7 | 8 | 7 | 7 | 9 |
S glass–epoxy FRP | 4 | 4 | 3 | 3 | 5 | 5 |
Carbon–epoxy FRP | 2 | 2 | 4 | 2 | 2 | 3 |
Kevlar 29–epoxy FRP | 3 | 3 | 2 | 4 | 4 | 4 |
Kevlar 49–epoxy FRP | 1 | 1 | 1 | 1 | 1 | 1 |
Boron–epoxy FRP | 6 | 10 | 10 | 5 | 3 | 2 |
Example 2: Material selection for cryogenic storage tank
Decision matrix for Example 2 (Manshadi et al. 2007)
Materials | Beneficial and non-beneficial attributes | ||||||
---|---|---|---|---|---|---|---|
MAX | MAX | MAX | MIN | MIN | MIN | MIN | |
Toughness index (MPa) | Yield strength (MPa) | Elastic modulus (GPa) | Density (g/cm^{3}) | Thermal expansion (10^{−6}/°C) | Thermal conductivity (cal/s/cm/°C) | Specific heat (cal/g/°C) | |
Al 2024-T6 | 75.5 | 420 | 74.2 | 2.80 | 21.4 | 0.370 | 0.16 |
Al 5052-O | 95 | 91 | 70 | 2.68 | 22.1 | 0.330 | 0.16 |
SS 301-FH | 770 | 1365 | 189 | 7.90 | 16.9 | 0.040 | 0.08 |
SS 310-3AH | 187 | 1120 | 210 | 7.90 | 14.4 | 0.030 | 0.08 |
Ti–6Al–4V | 179 | 875 | 112 | 4.43 | 9.4 | 0.016 | 0.09 |
Inconel 718 | 239 | 1190 | 217 | 8.51 | 11.5 | 0.310 | 0.07 |
70Cu–30Zn | 273 | 200 | 112 | 8.53 | 19.9 | 0.290 | 0.06 |
Normalized decision matrix for Example 2
Materials | Beneficial and non-beneficial attributes | ||||||
---|---|---|---|---|---|---|---|
MAX | MAX | MAX | MIN | MIN | MIN | MIN | |
Toughness index | Yield strength | Elastic modulus | Density | Thermal expansion | Thermal conductivity | Specific heat | |
Al 2024-T6 | 0.084 | 0.179 | 0.184 | 0.160 | 0.472 | 0.565 | 0.564 |
Al 5052-O | 0.106 | 0.039 | 0.174 | 0.154 | 0.487 | 0.504 | 0.564 |
SS 301-FH | 0.858 | 0.581 | 0.469 | 0.453 | 0.373 | 0.061 | 0.282 |
SS 310-3AH | 0.208 | 0.477 | 0.521 | 0.453 | 0.318 | 0.046 | 0.282 |
Ti–6Al–4V | 0.199 | 0.372 | 0.278 | 0.254 | 0.207 | 0.024 | 0.317 |
Inconel 718 | 0.266 | 0.506 | 0.538 | 0.488 | 0.254 | 0.473 | 0.247 |
70Cu–30Zn | 0.304 | 0.085 | 0.278 | 0.489 | 0.439 | 0.443 | 0.211 |
Entropy measure and weighting factors for Example 2
Entropy and weights | Beneficial and non-beneficial attributes | ||||||
---|---|---|---|---|---|---|---|
MAX | MAX | MAX | MIN | MIN | MIN | MIN | |
Toughness index | Yield strength | Elastic modulus | Density | Thermal expansion | Thermal conductivity | Specific heat | |
s _{ j } | 0.28 | 0.14 | 0.05 | 0.24 | 0.19 | 0.05 | 0.05 |
E _{ j } | 0.855 | 0.879 | 0.954 | 0.953 | 0.979 | 0.819 | 0.964 |
w _{ j } | 0.404 | 0.338 | 0.127 | 0.132 | 0.058 | 0.505 | 0.101 |
\( w_j^* \) | 0.571 | 0.238 | 0.032 | 0.159 | 0.055 | 0.127 | 0.026 |
Assessment values and rankings of the weighted MULTIMOORA method for Example 2
Materials | Assessment values | Rankings | |||||
---|---|---|---|---|---|---|---|
\( y_i^w \) | \( z_i^w \) | \( U_i^w \) | \( y_i^w \) | \( z_i^w \) | \( U_i^w \) | Final | |
Rank | Rank | Rank | Rank | ||||
Al 2024-T6 | −0.127 | 0.217 | 0.684 | 6 | 7 | 6 | 6 |
Al 5052-O | −0.139 | 0.210 | 0.593 | 7 | 6 | 7 | 7 |
SS 301-FH | 0.148 | 0.072 | 1.528 | 1 | 1 | 1 | 1 |
SS 310-3AH | −0.034 | 0.182 | 1.051 | 4 | 4 | 3 | 4 |
Ti–6Al–4V | 0.004 | 0.184 | 1.243 | 2 | 5 | 2 | 2 |
Inconel 718 | −0.029 | 0.166 | 1.045 | 3 | 3 | 4 | 3 |
70Cu–30Zn | −0.122 | 0.155 | 0.744 | 5 | 2 | 5 | 5 |
Assessment values and rankings of the entropy-weighted MULTIMOORA method for Example 2
Materials | Assessment values | Rankings | |||||
---|---|---|---|---|---|---|---|
\( y_i^{ew} \) | \( z_i^{ew} \) | \( U_i^{ew} \) | \( y_i^{ew} \) | \( z_i^{ew} \) | \( U_i^{ew} \) | Final | |
Rank | Rank | Rank | Rank | ||||
Al 2024-T6 | −0.042 | 0.441 | 0.233 | 6 | 7 | 6 | 6 |
Al 5052-O | −0.055 | 0.429 | 0.188 | 7 | 6 | 7 | 7 |
SS 301-FH | 0.535 | 0.048 | 1.388 | 1 | 1 | 1 | 1 |
SS 310-3AH | 0.146 | 0.370 | 0.620 | 3 | 4 | 3 | 3 |
Ti–6Al–4V | 0.148 | 0.376 | 0.677 | 2 | 5 | 2 | 2 |
Inconel 718 | 0.132 | 0.337 | 0.540 | 4 | 3 | 4 | 4 |
70Cu–30Zn | 0.039 | 0.316 | 0.366 | 5 | 2 | 5 | 5 |
Comparison between the materials ranks of the proposed model and other methods for Example 2
Materials | Methods | |||||||
---|---|---|---|---|---|---|---|---|
Weighted MULTIMOORA | Entropy-weighted MULTIMOORA | The method of Manshadi et al. (2007) | WPM (Manshadi et al. 2007) | GTMA (Rao 2006) | AHP-TOPSIS (Rao and Davim 2008) | Fuzzy logic (Khabbaz et al. 2009) | Z-transformation (Fayazbakhsh et al. 2009) | |
Al 2024-T6 | 6 | 6 | 5 | 5 | 6 | 5 | 6 | 6 |
Al 5052-O | 7 | 7 | 7 | 6 | 7 | 6 | 7 | 7 |
SS 301-FH | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
SS 310-3AH | 4 | 3 | 4 | 4 | 3 | 4 | 4 | 4 |
Ti–6Al–4V | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
Inconel 718 | 3 | 4 | 3 | 3 | 4 | 3 | 3 | 3 |
70Cu–30Zn | 5 | 5 | 6 | 7 | 5 | 7 | 5 | 5 |
Conclusion
In the present paper, we extended MULTIMOORA method using entropy weight based on the Shannon information theory to solve materials selection problem. The extended model has two scenarios called the weighted and entropy-weighted MULTIMOORA methods. To attach relative importance to attributes, subjective weight was considered in the first scenario whereas the combined subjective and objective weights were used in the second scenario. Subjective weight is obtained straight from decision makers’ comments based on their knowledge of materials and their experiences of the engineering design process. However, objective weight is calculated using entropy idea. The two forms of weighting factor can be integrated to produce the general form of Shannon entropy weight. Each of the two scenarios has three subordinate parts. To integrate the subordinate rankings, the dominance theory was exploited. Two practical materials selection examples were discussed to show the effect of the entropy weight on MULTIMOORA ranking. Moreover, the final rankings of the examples were compared with those of other methods.
The comparison between our final ranks and other studies demonstrates close correspondences, especially over the best rank or the optimal material. Spearman rank correlation coefficients obtained for the two examples show that the correlation between the ranks of the weighted MULTIMOORA method and the most of the earlier studies is higher than that of the entropy-weighted MULTIMOORA method. This fact is due to considering subjective weights in the models of the most of the references. Because of readily comprehensible mathematical derivation, the model based on MULTIMOORA method and the entropy concept gives an efficient means for decision making in the field of materials selection. Another strong point of our model is that our final rankings that were calculated by the consolidation of three subordinate ranks are more robust than those of other studies in which a single rank has been reported. The proposed model may have practical limitations in some real-world applications. The data of decision matrix may be presented as uncertain values. In this regard, new developments of the model are required based on fuzzy, interval, green, or other uncertain numbers dependent of the type of vagueness of the data. Moreover, our suggested methodology is to be developed for the case studies in which target-based attributes exist in the decision-making process, such as biomaterials selection problems. If a large number of alternatives and attributes exist in the decision matrix for a practical case, the manual calculation may be exhausting. Thus, the algorithm of this study can be computerized for such cases.
As future research, the extended MULTIMOORA approach can be considered for application in many case studies other than materials selection problem. For instance, decision making over the selection of optimal manufacturing process and the evaluation of failure modes risks can be done using the proposed model. In the field of materials selection, only two typical practical examples were presented in this paper. Other real-world materials selection problems with a number of various alternatives and attributes can be considered. The final rankings of the proposed model for the two examples were compared with a few approaches. The comparison of the present paper results with other MADM methods or expert systems seems to be interesting. As different extensions of the MULTIMOORA method, other concepts for assigning relative importance of attributes can be utilized.
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