Fuzzy entropy-weighted MULTIMOORA method for materials selection

Abstract

The selection of proper material is a formative stage in engineering design and production process. Without a systematic approach, many practical engineering materials may be neglected for selection. Multiple attribute decision making techniques as efficient methodologies can aid decision-makers in finding the optimal option. As a robust multiple attribute decision making approach, the MULTIMOORA method can be usefully employed in materials selection process. In this study, we extend the MULTIMOORA approach based on Shannon entropy concept under fuzzy environment to solve a materials selection case that is a group multiple attribute decision making problem. In the present paper, subjective weights of attributes and the ratings of alternatives on attributes are assumed as fuzzy sets. The entropy concept is considered to assign objective relative importance to decision-making attributes. The fuzzy subjective weights of attributes that are based on decision-makers comments and the crisp objective weights of attributes obtained using the entropy concept are consolidated to generate the fuzzy integrated entropy weights. We evaluate a practical materials selection problem related to automotive industry. We perform a sensitivity analysis by altering aggregation and normalization techniques besides the types of weighting. The materials rankings of the example are compared with the related study.

Keywords

Multiple attribute decision making MULTIMOORA fuzzy logic Shannon entropy materials selection

1 Introduction

A huge number of practical metallic alloys and nonmetallic materials like polymers, ceramics, and composites are employed in industry [1]. Due to the variety of engineering materials and manufacturing processes, the material selection process is a challenging task for an engineer or designer. If the selection is made without using a structured method, many practical materials may be ignored.

Multiple attribute decision making (MADM) approaches can be exploited as useful tools for the process of materials selection. Each MADM approach has given assumptions and principles. So far, many MADM techniques have been utilized for the task of materials selection, such as the technique for order preference by similarity to ideal solution (TOPSIS) [2 –5], analytic hierarchy approach (AHP) [6, 7], analytic network process (ANP) [8, 9], compromise ranking also known as vlse kriterijumska optimizacija kompromisno resenje (VIKOR) [10, 11], different types of elimination and choice expressing the reality (ELECTRE) also known as outranking method [12 –14], preference ranking organization method for enrichment evaluation (PROMETHEE) [15, 16], graph theory and matrix approach [17], grey relational analysis [18, 19], various preference ranking-based techniques [20, 21], preference selection index [22], utility additive (UTA) [23], weighted property index [24], linear assignment [25], modified digital logic [26, 27], Z-transformation [28, 29], quality function deployment [30 –32], and multi-attributive border approximation area comparison (MABAC) [33]. The applications of MADM approaches in materials selection have been discussed in two studies [34, 35].

Many studies on materials selection using MADM techniques have rather complicated mathematical models. Thus, a straightforward MADM approach can aid decision-makers in selecting optimal material. The multi-objective optimization on the basis of ratio analysis (MOORA) technique introduced by Brauers and Zavadskas [36] is a simple but efficient method useful for materials selection. The MOORA method and its developed form called MULTIMOORA technique have been employed in various applications such as decision making in manufacturing environment [37], project management [38], supplier selection [39], selection of optimal software component [40], personnel selection in the mining industry [41], grinding circuit selection [42], and material selection [43 –46].

In this paper, we develop the MULTIMOORA technique employing Shannon entropy weight under fuzzy environment for application in a materials selection example that is a group multiple attribute decision making (GMADM) problem. Few studies have considered weights of attributes in the MOORA and MULTIMOORA models. Brauers and Zavadskas [36] stated that assigning significance to each attribute is possible, but they did not explain how the weights of attributes can be calculated. Özçelik et al. [47] used the fuzzy AHP technique to obtain the weights of attributes for the reference point approach of the MOORA method. El-Santawy [48] employed a new technique for considering weights in an extended version of the MOORA approach. Their suggested weights are different from those obtained based on Shannon entropy concept. Some researchers have utilized the fuzzy MOORA and MULTIMOORA techniques in industrial applications. Baležentis et al. [49] considered personnel selection based on computing with words and the fuzzy MULTIMOORA. Baležentis et al. [50] suggested the fuzzy MULTIMOORA method based on group decision-making (MULTIMOORA-FG) with an application to personnel selection. Karande and Chakraborty [51] proposed a fuzzy MOORA approach for enterprise resource planning (ERP) system selection. Mandal and Sarkar [52] conducted the selection of best intelligent manufacturing system employing the fuzzy MOORA approach. Dey et al. [53] suggested a MOORA based fuzzy multi-criteria decision making approach for supply chain strategy selection.

To the best of authors’ knowledge, no study has introduced a model based on Shannon entropy weights of attributes and the fuzzy MULTIMOORA approach. In our methodology, the integrated entropy weight is proposed that comprises subjective and objective parts. The subjective weight is determined based on decision-makers comments as a fuzzy number. The objective weight is obtained using entropy concept as a crisp value based on the data of decision matrix disregarding the opinions of decision makers. The ratings of alternatives on attributes are also considered as fuzzy sets.

The remainder of the paper is structured as follows. We explain the original MULTIMOORA method and its constituents in Section 2. The derivation of integrated entropy weight based on Shannon information theory is described in Section 3. Fuzzy set theory is briefly introduced in Section 4. We describe the preparation of decision matrix and weights for solution as a step-by-step approach in Section 5. The proposed fuzzy entropy-weighted MULTIMOORA method is discussed in Section 6. We evaluate a real-world decision-making problem related to materials selection of an automobile component using the proposed method in Section 7. A discussion of application of the proposed method consists of a sensitivity analysis and a comparison with a related study is performed in Section 8. Finally, a summary of the study and prospective directions for development of the proposed methodology are given inSection 9.

2 The MULTIMOORA method

Brauers and Zavadskas [36] presented the MOORA method as a straightforward MADM technique. Their suggested technique has two parts: the ratio system and the reference point approach. Brauers and Zavadskas [38] developed the MOORA method using the full multiplicative form and the dominance theory to find the final ranking. This updated method called MULTIMOORA starts with a decision matrix X. Each element of decision matrix x_ij indicates the rating of alternative i on attribute j, i = 1, 2, … m and j = 1, 2, …, n. Decision matrix X is as follows: $X = [x_{ij}]_{m \times n} .$ (1)

The next step is the normalization of decision matrix. For normalization, each rating of an alternative on an attribute is divided by a representative for all ratings on that attribute. Different denominators can be used for normalization ratio. Brauers and Zavadskas [36] utilized the square root of the sum of squares of ratings per attribute as the denominator. Therefore, the MULTIMOORA method normalization ratio is: $x_{ij}^{*} = x_{ij} / - {[\sum_{i = 1}^{m} x_{ij}^{2}]}^{1 / 2},$ (2) in which $x_{ij}^{*}$ is the normalized rating of alternative i on attribute j. Brauers and Zavadskas [36] proved that Equation (2) is the most robust normalization ratio for the MULTIMOORA technique.

2.1 The ratio system

The first part of the MULTIMOORA method is the ratio system. To obtain the assessment value, the normalized ratings are added for beneficial attribute or subtracted for non-beneficial attribute as follows [1]: $y_{i}^{*} = \sum_{j = 1}^{g} x_{ij}^{*} - \sum_{j = g + 1}^{n} x_{ij}^{*},$ (3) in which g is the number of beneficial attributes and (n - g) denotes the number of non-beneficial attributes. $y_{i}^{*}$ indicates the ratio system assessment value of alternative i. The optimal alternative based on the ratio system obtains the highest value of $y_{i}^{*}$ as follows [54]: $A_{Rs}^{*} = {A_{i} | max_{i} y_{i}^{*}} .$ (4)

2.2 The reference point approach

This technique starts with finding the maximal objective reference point (MORP). Co-ordinate j of MORP vector is determined as [36]: $r_{j} = {\begin{matrix} max_{i} x_{ij}^{*}, & j \leq g, \\ min_{i} x_{ij}^{*}, & j > g . \end{matrix}$ (5)

To evaluate the distance between alternatives and the reference point, the Tchebycheff Min-Max metric is selected [36]. The deviation of a normalized rating from the reference point r_j, can be obtained as $(r_{j} - x_{ij}^{*})$ . The assessment value of the reference point approach is determined by finding the maximum deviation from the related reference point among all attributes, as follows: $z_{i}^{*} = max_{j} | r_{j} - x_{ij}^{*} | .$ (6)

The optimal alternative based on the reference point approach has the minimum value of Equation (6) among all alternatives [54]: $A_{RP}^{*} = {A_{i} | min_{i} z_{i}^{*}} .$ (7)

2.3 The full multiplicative form

The full multiplicative form is the third part of the MULTIMOORA method proposed by Brauers and Zavadskas [38]: $U_{i}^{'} = \frac{\prod_{j = 1}^{g} x_{ij}}{\prod_{j = g + 1}^{n} x_{ij}} .$ (8)

The product of ratings of alternative i on beneficial attributes is divided by the product of ratings of alternative i on non-beneficial attributes to form Equation (8).

Another formula for the assessment value of the full multiplicative form can be established using the normalized ratings of decision matrix as $U_{i}^{*} = \frac{\prod_{j = 1}^{g} x_{ij}^{*}}{\prod_{j = g + 1}^{n} x_{ij}^{*}} .$ (9)

Although the values of $U_{i}^{*}$ differ from $U_{i}^{'}$ , their resultant rankings are identical. We utilize Equation (9) as the full multiplicative form assessment value to maintain consistency by using the normalized ratings of decision matrix in all subordinate parts of the MULTIMOORA method.

The optimal alternative based on the full multiplicative form has the maximum of $U_{i}^{*}$ among all alternatives: $A_{MF}^{*} = {A_{i} | max_{i} U_{i}^{*}} .$ (10)

2.4 The final ranking of the MULTIMOORA method based on the dominance theory

Brauers and Zavadskas [38] utilized dominance theory to integrate the rankings of the three subordinate parts of the MULTIMOORA method into a final rank list. For details about the dominance theory, readers can refer to the study of Brauers and Zavadskas [55].

3 Integrated Shannon entropy weight

Shannon [56] suggested information entropy in his famous paper “A Mathematical Theory of Communication”. Information entropy evaluates the expected information content of a certain message. The degree of uncertainty in information can be measured using the entropy concept.

Information entropy idea can regulate decision making process because it is able to measure existent contrasts between sets of data and thus clarify the intrinsic information for decision maker.

The following procedure should be employed to determine integrated weights through Shannon entropy [57]:

Normalize decision matrix to obtain the project outcomes of attribute j, p_ij: $p_{ij} = x_{ij} / - \sum_{i = 1}^{m} x_{ij} .$ (11)

Calculate the entropy measure of the project outcomes as follows: $E_{j} = \frac{1}{ln (m)} \sum_{i = 1}^{m} p_{ij} ln p_{ij} .$ (12)

Obtain objective weights as: $w_{j}^{o} = \frac{1 - E_{j}}{\sum_{j = 1}^{n} (1 - E_{j})} .$ (13)

Calculate the integrated entropy weight as follows: $w_{j}^{*} = \frac{w_{j}^{s} w_{j}^{o}}{\sum_{j = 1}^{n} w_{j}^{s} w_{j}^{o}},$ (14)

in which subjective weight

w_{j}^{s}

and objective weight

w_{j}^{o}

are combined to generate integrated weight

w_{j}^{*}

4 Fuzzy set theory

Fuzzy set theory was introduced in 1965 by Zadeh [58]. The theory has been employed in many fields from control theory to artificial intelligence. Fuzzy set theory can be used to convert linguistic variables to numerical variables in decision making process. The definition of fuzzy sets were used to develop fuzzy multi-criteria decision making (FMCDM) methodology by Bellman and Zadeh [59] to increase precision in allocating weights of attributes and the ratings of alternatives on attributes.

A fuzzy set is defined through a membership function, that allocates a degree of membership in the interval [0, 1] to each element of the set. Fuzzy numbers have different forms. The most common are triangular and trapezoidal fuzzy numbers. We use trapezoidal fuzzy numbers to map the linguistic variables to fuzzy sets in the present paper. Figure 1 represents a positive trapezoidal fuzzy number $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4})$ .

Membership function $μ_{\tilde{A}} (x)$ is determined as follows [60]: $μ_{\tilde{A}} (x) = {\begin{matrix} 0, & x < a_{1}, \\ \frac{x - a_{1}}{a_{2} - a_{1}}, & a_{1} \leq x \leq a_{2}, \\ 1, & a_{2} \leq x \leq a_{3}, \\ \frac{x - a_{4}}{a_{3} - a_{4}}, & a_{3} \leq x \leq a_{4}, \\ 0, & x > a_{4} . \end{matrix}$ (15)

The basic mathematical operations for two positive trapezoidal fuzzy numbers $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4})$ , $\tilde{B} = (b_{1}, b_{2}, b_{3}, b_{4})$ , and a positive real number r are as follows [60]: $\tilde{A} \oplus \tilde{B} = (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3}, a_{4} + b_{4}),$ (16) $\tilde{A} ⊖ \tilde{B} = (a_{1} - b_{4}, a_{2} - b_{3}, a_{3} - b_{2}, a_{4} - b_{1}),$ (17) $\tilde{A} \otimes \tilde{B} ≅ (a_{1} b_{1}, a_{2} b_{2}, a_{3} b_{3}, a_{4} b_{4}),$ (18) $\tilde{A} ø \tilde{B} ≅ (a_{1} / b_{4}, a_{2} / b_{3}, a_{3} / b_{2}, a_{4} / b_{1}),$ (19) $\tilde{A} \otimes r = (a_{1} r, a_{2} r, a_{3} r, a_{4} r) .$ (20)

The distance between two trapezoidal fuzzy numbers $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4})$ and $\tilde{B} = (b_{1}, b_{2}, b_{3}, b_{4})$ can be calculated utilizing the vertex method as [61]: $d (\tilde{A}, \tilde{B}) = \sqrt{\frac{1}{6} [{(a_{1} - b_{1})}^{2} + 2 {(a_{2} - b_{2})}^{2} + 2 {(a_{3} - b_{3})}^{2} + {(a_{4} - b_{4})}^{2}]}$ (21)

Various defuzzification techniques exist for transforming a fuzzy number into a crisp value. The centroid-based defuzzified value for a trapezoidal fuzzy number $\tilde{A} = (a_{1}, a_{2}, a_{3}, a_{4})$ is [62]:

$\begin{matrix} \bar{A} \\ = \frac{1}{3} [a_{1} + a_{2} + a_{3} + a_{4} - \frac{a_{4} a_{3} - a_{1} a_{2}}{(a_{4} + a_{3}) - (a_{1} + a_{2})}] . \end{matrix}$ (22)

5 The preparation of decision matrix and weights for solution

We develop an extension of the MULTIMOORA method using Shannon entropy weight under fuzzy environment to address a GMADM problem. The following sets exist in a GMADM problem:

m possible alternatives A = {A₁, A₂, …, A_m},

n decision attributes a = {a₁, a₂, …, a_n},

h decision makers called D = {D₁, D₂, …, D_h}.

We describe the overall procedure of the preparation phase as a step-by-step approach.

5.1 Initial decision matrix

Two sets of relevant linguistic variables are required to obtain subjective weights of attributes and the ratings of alternatives assigned by decision makers. The corresponding fuzzy ratings of alternatives and fuzzy subjective weights of attributes considered by decision maker k are ${\tilde{x}}_{ijk} = (x_{ijk 1}, x_{ijk 2}, x_{ijk 3}, x_{ijk 4})$ and ${\tilde{w}}_{jk}^{s} = (w_{jk 1}^{s}, w_{jk 2}^{s}, w_{jk 3}^{s}, w_{jk 4}^{s})$ .

5.2 Aggregation (fuzzy decision matrix)

The aggregated subjective weights vector is ${\tilde{W}}^{s} = [{\tilde{w}}_{j}^{s}]_{1 \times n} .$ (23)

The aggregated fuzzy subjective weight ${\tilde{w}}_{j}^{s}$ of each attribute is obtained as [60]: $\begin{matrix} {\tilde{w}}_{j}^{s} = (w_{j 1}^{s}, w_{j 2}^{s}, w_{j 3}^{s}, w_{j 4}^{s}) \\ = (\frac{1}{h} \sum_{k = 1}^{h} w_{jk 1}^{s}, \frac{1}{h} \sum_{k = 1}^{h} w_{jk 2}^{s}, \frac{1}{h} \sum_{k = 1}^{h} w_{jk 3}^{s}, \frac{1}{h} \sum_{k = 1}^{h} w_{jk 4}^{s}) . \end{matrix}$ (24)

The aggregated matrix for fuzzy alternatives ratings, i.e., the fuzzy decision matrix, is $\tilde{X} = [{\tilde{x}}_{ij}]_{m \times n} .$ (25)

The aggregated fuzzy ratings ${\tilde{x}}_{ij}$ of alternatives on each attribute are calculated using the following formula [60]: $\begin{matrix} {\tilde{x}}_{ij} = (x_{ij 1}, x_{ij 2}, x_{ij 3}, x_{ij 4}) \\ = (\frac{1}{h} \sum_{k = 1}^{h} x_{ijk 1}, \frac{1}{h} \sum_{k = 1}^{h} x_{ijk 2}, \frac{1}{h} \sum_{k = 1}^{h} x_{ijk 3}, \frac{1}{h} \sum_{k = 1}^{h} x_{ijk 4}) . \end{matrix}$ (26)

5.3 Normalization

The normalized fuzzy decision matrix can be defined as: ${\tilde{X}}^{*} = [{\tilde{x}}_{ij}^{*}]_{m \times n} .$ (27)

Normalization can be performed based on the study of Brauers et al. [63] as follows:

$\begin{matrix} {\tilde{x}}_{ij}^{*} = (x_{ij 1}^{*}, x_{ij 2}^{*}, x_{ij 3}^{*}, x_{ij 4}^{*}) \\ = (\frac{x_{ij 1}}{\sqrt{\sum_{i = 1}^{m} x_{ij 1}^{2}}}, \frac{x_{ij 2}}{\sqrt{\sum_{i = 1}^{m} x_{ij 2}^{2}}}, \frac{x_{ij 3}}{\sqrt{\sum_{i = 1}^{m} x_{ij 3}^{2}}}, \frac{x_{ij 4}}{\sqrt{\sum_{i = 1}^{m} x_{ij 4}^{2}}}) \end{matrix}$ (28)

5.4 Fuzzy integrated entropy weight

The integrated entropy weight in fuzzy environment can be calculated based on the followingsteps:

5.4.1 Computation of crisp objective weight

To determine objective weights, first the decision matrix should be defuzzified. The defuzzified decision matrix is shown as: $\bar{X} = [{\bar{x}}_{ij}]_{m \times n} .$ (29)

By employing Equation (22), the defuzzified ratings, i.e., crisp ratings, can be calculated as:

$\begin{matrix} {\bar{x}}_{ij} & = & \frac{1}{3} [x_{ij 1} + x_{ij 2} + x_{ij 3} + x_{ij 4} \\ - \frac{x_{ij 4} x_{ij 3} - x_{ij 1} x_{ij 2}}{(x_{ij 4} + x_{ij 3}) - (x_{ij 1} + x_{ij 2})}] . \end{matrix}$ (30)

Project outcomes are computed by entering defuzzified rating ${\bar{x}}_{ij}$ in Equation (11). Then, entropy measures and crisp objective weights are obtained employing Equations (12) and (13).

5.4.2 Integration of crisp objective weight and fuzzy subjective weight

To generate fuzzy integrated entropy weights, crisp objective and fuzzy subjective weights can be combined. Thus, the fuzzy form of Equation (14), is determined as: ${\tilde{w}}_{j}^{*} = ({\tilde{w}}_{j}^{s} \otimes w_{j}^{o}) ø \sum_{j = 1}^{n} ({\tilde{w}}_{j}^{s} \otimes w_{j}^{o}) .$ (31)

6 The proposed fuzzy entropy-weighted MULTIMOORA method

The subordinate and final rankings of the developed methodology are obtained based on the concept of fuzzy logic and entropy theory as follows:

6.1 The fuzzy entropy-weighted ratio system

By considering the crisp ratio system, Equation (3), and fuzzy integrated entropy weights, Equation (31), we have: ${\tilde{y}}_{i}^{ew} = \sum_{j = 1}^{g} ({\tilde{w}}_{j}^{*} \otimes {\tilde{x}}_{ij}^{*}) ⊖ \sum_{j = g + 1}^{n} ({\tilde{w}}_{j}^{*} \otimes {\tilde{x}}_{ij}^{*}) .$ (32) in which ${\tilde{y}}_{i}^{ew}$ denotes the assessment value of the fuzzy entropy-weighted ratio system. To rank the assessment values, the fuzzy values of ${\tilde{y}}_{i}^{ew}$ are defuzzified into ${\bar{y}}_{i}^{ew}$ . The resultant optimal alternative can be identified as follows: $A_{FERS}^{*} = {A_{i} | max_{i} {\bar{y}}_{i}^{ew}} .$ (33)

6.2 The fuzzy entropy-weighted reference point approach

By using Equation (5), co-ordinate j of the fuzzy maximal objective reference point (FMORP) vector is determined as [60]: ${\tilde{r}}_{j} = {\begin{matrix} (max_{i} x_{ij 1}^{*}, max_{i} x_{ij 2}^{*}, max_{i} x_{ij 3}^{*}, max_{i} x_{ij 4}^{*}), & j \leq g, \\ (min_{i} x_{ij 1}^{*}, min_{i} x_{ij 2}^{*}, min_{i} x_{ij 3}^{*}, min_{i} x_{ij 4}^{*}), & j > g . \end{matrix}$ (34)

Then, the deviation $d [({\tilde{w}}_{j}^{*} \otimes {\tilde{r}}_{j}), ({\tilde{w}}_{j}^{*} \otimes {\tilde{x}}_{ij}^{*})]$ is calculated considering Equation (21). The assessment value of the fuzzy entropy-weighted reference point approach is obtained as follows: $z_{i}^{ew} = max_{j} d [({\tilde{w}}_{j}^{*} \otimes {\tilde{r}}_{j}), ({\tilde{w}}_{j}^{*} \otimes {\tilde{x}}_{ij}^{*})] .$ (35)

The optimal alternative based on this approach has the minimum value of Equation (35): $A_{FERP}^{*} = {A_{i} | min_{i} z_{i}^{ew}} .$ (36)

6.3 The fuzzy entropy-weighted full multiplicative form

The assessment values of this subordinate part can be obtained by employing the formula of the crisp full multiplicative form, Equation (9), and fuzzy integrated entropy weights, Equation (31). Unlike the fuzzy entropy-weighted ratio system and reference point approach, the weights cannot be considered as coefficients in the fuzzy entropy-weighted full multiplicative form because such assumption is meaningless. Instead, the weights should be allocated as exponents in this method [55]: ${\tilde{U}}_{i}^{ew} = \prod_{j = 1}^{g} {({\tilde{x}}_{ij}^{*})}^{{\tilde{w}}_{j}^{*}} ⊘ \prod_{j = g + 1}^{n} {({\tilde{x}}_{ij}^{*})}^{{\tilde{w}}_{j}^{*}} .$ (37)

Because all elements of ${\tilde{x}}_{ij}^{*}$ and ${\tilde{w}}_{j}^{*}$ are between 0 and 1, it can be shown that:

$\begin{matrix} {({\tilde{x}}_{ij}^{*})}^{{\tilde{w}}_{j}^{*}} & = & ({({\tilde{x}}_{ij 1}^{*})}^{w_{j 4}^{*}}, {({\tilde{x}}_{ij 2}^{*})}^{w_{j 3}^{*}}, \\ {({\tilde{x}}_{ij 3}^{*})}^{w_{j 2}^{*}}, {({\tilde{x}}_{ij 4}^{*})}^{w_{j 1}^{*}}) . \end{matrix}$ (38)

Similar to Section 6.1, ${\tilde{U}}_{i}^{ew}$ are defuzzified into ${\bar{U}}_{i}^{ew}$ . The resultant optimal alternative is: $A_{FEMF}^{*} = {A_{i} | max_{i} {\bar{U}}_{i}^{ew}} .$ (39)

6.4 The final ranking of the proposed method based on the dominance theory

By employing the dominance theory, we consolidate the rankings of Sections 6.1, 6.2, and 6.3 into a final ranking.

7 Application of the proposed method in materials selection process

In this section, we utilize the proposed method to solve a materials selection problem in automotive industry. We employ the case study of Girubha and Vinodh [64]. They used the fuzzy VIKOR method to tackle the materials selection problem.

Girubha and Vinodh [64] conducted a study on materials selection for an automobile component, an instrument panel. The company have a group of 5 (h = 5) decision makers that are responsible for materials selection.

Four (m = 4) candidate materials are considered as alternatives for the problem that are styrene maleic anhydride (SMA), polycarbonate (PC), polypropylene (PP), and acrylonitrile butadiene styrene (ABS). The following eight (n = 8) materials properties are considered as attributes for the problem:

MTL: Maximum temperature limit

RCY: Recyclability

TSS: Tensile strength

WGT: Weight

THC: Thermal conductivity

ELG: Elongation

CST: Cost

TXL: Toxicity level

The first three attributes, i.e., MTL, RCY, TSS are beneficial and the others are non-beneficial.

The algorithm of the proposed method according to the steps mentioned in Sections 5 and 6 for the materials selection problem is as follows:

Step 1: Initial decision matrix

Linguistic terms and corresponding fuzzy numbers for subjective weights of materials properties and ratings of materials on their properties are allocated as shown in Tables 1 and 2, respectively. The subjective weights and ratings have been collected from the decision makers in the form of linguistic terms as shown in Tables 3 and 5, respectively. The linguistic terms related to the subjective weights and ratings are transformed into fuzzy sets as displayed in Tables 4 and 6, respectively.

Step 2: Aggregation (fuzzy decision matrix)

The aggregated matrix for subjective weights of attributes and material ratings are computed employing Equations (24) and (26) respectively as demonstrated in Tables 7 and 8. Table 8 represents the fuzzy decision matrix for the materials selection problem.

Step 3: Normalization

The ratings of the fuzzy decision matrix are normalized utilizing Equation (28) to generate the normal values shown in Table 9.

Step 4: Fuzzy integrated entropy weight

To obtain fuzzy integrated entropy weight several preliminary phases should be passed. First, fuzzy material ratings are converted to defuzzified values applying Equation (30). Project outcomes, entropy measures, and objective weights for the problem are computed using (Equations 11–13), respectively. The values of entropy measures and objective weights are gathered in Table 10. In addition, Table 10 denotes fuzzy integrated entropy weights calculated utilizing Equation (31) that is a combination of crisp objective and fuzzy subjective weights of the materialproperties.

Step 5: Solution

The assessment values of the subordinates parts of the proposed method for this case study can be obtained by considering Equations (32), (35), and (37) as listed in Tables 11–13, respectively. The optimal material based on the three sub-techniques identically is SMA, i.e., $A_{FERS}^{*} = A_{FERP}^{*} = A_{FEMF}^{*} = SMA$ . Table 14 displays the final ranking of materials for the problem based on the dominance theory. The optimal material based on the proposed fuzzy entropy-weighted MULTIMOORA method is SMA and the final ranking is [SMA, PC, PP, ABS].

8 Discussion

8.1 Sensitivity analysis

We conduct a sensitivity analysis by changing aggregation and normalization techniques as well as type of attributes weights. Different aggregation methods exist for GMADM problems. Additionally, various normalization techniques exist for fuzzy MADM problems. In the previous sections, only one type of aggregation and normalization technique is considered. For the current sensitivity analysis, we use other types of aggregation and normalization techniques to generate final rankings.

Equations (24) and (26) are considered as Type 1 of aggregation technique. Aggregation technique Type 2 (for subjective weights and alternatives ratings) is defined as:

$\begin{matrix} {\tilde{w}}_{j}^{s} & = & (w_{j 1}^{s}, w_{j 2}^{s}, w_{j 3}^{s}, w_{j 4}^{s}) \\ = & (min_{k} {w_{jk 1}^{s}}, \frac{1}{h} \sum_{k = 1}^{h} w_{jk 2}^{s}, \\ \frac{1}{h} \sum_{k = 1}^{h} w_{jk 3}^{s}, max_{k} {w_{jk 4}^{s}}), \\ {\tilde{x}}_{ij} & = & (x_{ij 1}, x_{ij 2}, x_{ij 3}, x_{ij 4}) \\ = & (min_{k} {x_{ijk 1}}, \frac{1}{h} \sum_{k = 1}^{h} x_{ijk 2}, \\ \frac{1}{h} \sum_{k = 1}^{h} x_{ijk 3}, max_{k} {x_{ijk 4}}) . \end{matrix}$ (40)

Equation (28) is considered as Type 1 of normalization technique. Normalization technique Type 2 can be obtained as:

$\begin{matrix} {\tilde{x}}_{ij}^{*} & = & (x_{ij 1}^{*}, x_{ij 2}^{*}, x_{ij 3}^{*}, x_{ij 4}^{*}) \\ = & {\begin{matrix} (\frac{x_{ij 1}}{x_{j^{4}}^{+}}, \frac{x_{ij 2}}{x_{j^{4}}^{+}}, \frac{x_{ij 3}}{x_{j^{4}}^{+}}, \frac{x_{ij 4}}{x_{j^{4}}^{+}}), & j \leq g, \\ (\frac{x_{j 1}^{-}}{x_{ij 4}}, \frac{x_{j 1}^{-}}{x_{ij 3}}, \frac{x_{j 1}^{-}}{x_{ij 2}}, \frac{x_{j 1}^{-}}{x_{ij 1}}), & j > g, \end{matrix} \end{matrix}$ (41) in which $x_{j 4}^{+}$ and $x_{j 1}^{-}$ are defined as: ${\begin{matrix} x_{j 4}^{+} = max_{i} {x_{ij 4}}, & j \leq g, \\ x_{j 1}^{-} = min_{i} {x_{ij 1}}, & j > g . \end{matrix}$ (42)

Table 15 shows eight different cases and the related final rankings based on two types of aggregation and normalization techniques using two weighting methods. In Table 15, the term “weighted” denotes the fuzzy subjective weights of attributes and the term “entropy-weighted” represents the fuzzy integrated entropy weights.

Cases 1 and 2 show an identical ranking [SMA, PC, PP, ABS]. The ranking for Cases 3, 4, 7, and 8 is [PP, ABS, SMA, PC]. Cases 5 and 6 gain the ranking [SMA, PC, ABS, PP]. Table 15 demonstrates 3 different sets of rankings; therefore, it implies that the aggregation and normalization techniques and the weighting systems may affect the rankings.

Therefore, in the real-world applications, selection of appropriate aggregation and normalization techniques and weighting systems is important. The suitable adoption of the solution procedure depends on decision makers knowledge and experience.

8.2 Comparison with the fuzzy VIKOR method

Using the fuzzy VIKOR method, the final ranking obtained in the study of Girubha and Vinodh [64] is [PP, ABS, SMA, PC]. Their assumptions concerning the aggregation and normalization techniques and the weighting systems are identical to Case 7 of our study that has the final ranking [PP, ABS, SMA, PC]. Therefore, if we employ the same assumptions as Girubha and Vinodh [64] within the proposed methodology, the resultant final ranking would be similar to that of the fuzzy VIKOR method [59].

9 Conclusion

In this paper, we introduced an extension of the MULTIMOORA technique employing the integrated Shannon entropy weight under fuzzy environment to address an engineering material selection problem. Fuzzy subjective weights of material properties are gained from decision-makers dependent on their knowledge and experiences of materials and their properties. Whereas, crisp objective weights of materials properties are computed based on entropy idea. The two forms of weight can be combined to provide the fuzzy integrated entropy weight. Due to uncomplicated mathematical computation, the presented fuzzy entropy-weighted MULTIMOORA approach helps engineers in making a robust selection of suitable material. A real-world example related to materials selection of an automobile component was considered in the present paper. By changing aggregation and normalization techniques as well as type of attributes weights, a sensitivity analysis is conducted that consists of eight cases. Moreover, the final rankings of the example are compared with the results of the fuzzy VIKOR method [59]. The comparison demonstrates exact correspondence between the results of several cases of the proposed technique and the fuzzy VIKOR method [59].

As future research directions, different types of aggregation and normalization techniques can be considered for the fuzzy MULTIMOORA method under group decision-making environment. Therefore, the sensitivity analysis discussed in this paper can be extended. Such comprehensive analysis would better reveal the effects of aggregation and normalization techniques on final results. For objective weighting of attributes, other approaches like standard deviation and inter-attribute correlation effect measure can be employed. The comparison of the differences between the results based on such weights and the outcomes of this study is interesting. Moreover, the proposed methodology of this paper that is based on fuzzy integrated entropy weights can be utilized for other uncertain real-world selection problems.

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