Materials and Design 87 (2015) 949–959
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Materials and Design journal homepage: www.elsevier.com/locate/jmad
Comprehensive MULTIMOORA method with target-based attributes and integrated significant coefficients for materials selection in biomedical applications Arian Hafezalkotob a, Ashkan Hafezalkotob b,⁎ a b
Department of Mechanical Engineering, Islamic Azad University, South Tehran Branch, Tehran, Iran Department of Industrial Engineering, Islamic Azad University, South Tehran Branch, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 20 June 2015 Received in revised form 14 August 2015 Accepted 17 August 2015 Available online 24 August 2015 Keywords: Biomedical materials selection Multiple attribute decision making MULTIMOORA Target-based attributes Significant coefficients of attributes
a b s t r a c t Materials selection as a significant step in engineering design process can be effectively implemented with the aid of MADM methods. The traditional MADM approaches only focus on beneficial or non-beneficial attributes to choose the optimal material; however, the selection should be made considering target values of attributes in some cases. In spite of practical applications of the target-based decision-making specially in engineering design problems, few studies have focused on the field. In the present paper, we developed a target-based MULTIMOORA method through introducing a normalization technique for materials selection in biomedical applications. We utilized two combinations of subjective, objective, and inter-attribute correlation effect significant coefficients to give importance to attributes. The information entropy and standard deviation concepts were used to assign two objective significant coefficients. This assumption leads to two modes of solution. Each mode has three subordinate and a final rankings. The final ranks of the two modes were consolidated into an aggregate ranking. The final and aggregate rankings were obtained using the dominance theory. We discussed two real-world biomedical engineering problems including materials selection for femoral components of hip and knee joint prostheses. Our aggregate rankings for the practical cases were compared with the results of previous studies. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction At an estimate, more than 40,000 practical metallic alloys and a same number of nonmetallic materials like ceramics, composites, and polymers are commonly utilized in different industries [1]. Due to this enormous number of practical engineering materials and the fact that various materials are manufactured using diverse production techniques, the materials selection process can be considered as a complex task for a designer or engineer. If the process takes place unmethodically, many important materials may be disregarded. Thus, systemic mathematical methods are required to simplify making such selections. In this regard, multiple attribute decision making (MADM) techniques can be appropriately applied as effective structured tools for materials selection. By considering target-based attributes, an MADM method can be developed to an all-inclusive form. Traditional MADM techniques normally take account of beneficial (also known as profitable or positive) or nonbeneficial (also known as cost or negative) attributes in the decisionmaking process. However, the target-based MADM methods consider beneficial and non-beneficial attributes as well as those with target values [2]. Therefore, decision-making approaches that address target⁎ Corresponding author. E-mail address:
[email protected] (A. Hafezalkotob).
http://dx.doi.org/10.1016/j.matdes.2015.08.087 0264-1275/© 2015 Elsevier Ltd. All rights reserved.
based attributes along with beneficial and non-beneficial ones can help in careful selection of alternatives [3]. The target-based MADM techniques can also be employed in typical selection problems in which only beneficial and non-beneficial attributes exist. Importance of target-based attributes can be better realized in real-world engineering design problems. For example, in biomedical materials selection procedure, implant materials should possess almost identical properties to those of human body tissues [3]. Otherwise, the selection of a material with inappropriate properties can lead to severe irritations. In case of engineering problems in which electrical or thermal remarks should be noticed, thermal expansion is preferred to be limited to a certain value [4]. The multi-objective optimization on the basis of ratio analysis (MOORA) and its updated form MULTIMOORA method are effective and simple MADM techniques. In this paper, we made materials selection in biomedical applications using a comprehensive form of MULTIMOORA method. The proposed methodology was developed through considering a target-based normalization technique and integrated significant coefficients. The remainder of the present paper was organized as follows. A categorized literature review and explanation of the research gap were presented in Section 2. We introduced the traditional MULTIMOORA method in Section 3. The proposed approach, i.e., the target-based MULTIMOORA method, was described in Section 4 that includes the
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description of normalization technique and significant coefficients as well as the derivation of subordinate and final ranks. In Section 5, we discussed the materials selection process and two real-world problems in biomedical applications. Concluding remarks were mentioned in Section 6 to make a summary of the present study. 2. Literature review 2.1. Survey on applications of MADM methods in materials selection Many MADM methods have been used for materials selection, such as MOORA [5], the technique for order preference by similarity to ideal solution (TOPSIS) [6], compromise ranking also recognized as vlse kriterijumska optimizacija kompromisno resenje (VIKOR) [7], analytic hierarchy approach (AHP) [8], elimination and choice expressing the reality (ELECTRE) also known as outranking method [9], graph theory and matrix approach [10], gray relational analysis [11], quality function deployment [12], utility additive (UTA) [13], weighted property index [14], linear assignment [15], modified digital logic [16], and simplified fuzzy logic [17]. Two studies have focused on surveying the applications of MADM techniques in materials selection process [18,19]. 2.2. Survey on target-based MADM methods Some researchers have developed target-based versions of MADM methods. Jahan et al. [2] proposed a comprehensive form of VIKOR method with target-based attributes to select a suitable material for hip prosthesis. Bahraminasab and Jahan [20] employed the methodology of comprehensive VIKOR to implement a target-based decision-making concerning femoral component of knee replacement. Jahan et al. [4] suggested a target-based normalization technique to generate an inclusive extension of TOPSIS for application in electrical insulating materials selection problem. Jahan [21] proposed a mixed goal programming model considering target-based attributes for materials selection of hip implant and compared its results with those of comprehensive VIKOR method. Cavallini et al. [22] developed an integration of house of quality and the comprehensive VIKOR algorithm to choose suitable material for protective coating on an aluminum alloy substrate. Jahan and Edwards [23] developed a VIKOR approach that considers both target values of attributes and interval ratings. Jahan and Edwards [3] utilized a novel form of significant coefficients to develop target-based TOPSIS and VIKOR methods for materials selection in biomedical applications. Liu et al. [24] combined modified VIKOR and DEMATEL-based ANP techniques considering target values of attributes. Jahan and Edwards [25] classified thirty-one normalization techniques including multiple target-based norms and discussed the influence of the methods in ranking with focus on the materials selection. 2.3. Survey on development and applications of the MULTIMOORA method Brauers and Zavadskas [26] proposed the MOORA technique. Brauers and Zavadskas [27] developed the MOORA method into a general form known as the MULTIMOORA approach. The MOORA and MULTIMOORA methods have been commonly utilized in various applications. Baležentis et al. [28] considered personnel selection using MULTIMOORA technique under fuzzy environment for linguistic reasoning. Brauers and Zavadskas [29] employed the MULTIMOORA method to make a decision on a bank loan to purchase property. Baležentis and Zeng [30] presented an extended MULTIMOORA technique based upon interval-valued fuzzy numbers to choose a manager for R&D department of a telecommunication company. Datta et al. [31] developed the concept of the MULTIMOORA technique employing interval-valued gray numbers to generate an acceptable ranking order of candidate industrial robots. Dey et al. [32] suggested a MOORA based fuzzy multi-criteria decision making technique to arrange a ranking of supply chain strategies. Karande and Chakraborty [33] made a contribution to MOORA methodology utilizing
fuzzy values to rank enterprise resource planning systems. Mandal and Sarkar [34] recommended a fuzzy MOORA method to relieve the selection process of optimum intelligent manufacturing system. Karande and Chakraborty [35] employed the MOORA method for decision-making regarding the selection of appropriate supplier. Mishra et al. [36] suggested a decision model for supplier/partner selection in agile supply chain employing the fuzzy MULTIMOORA method. Deliktas and Ustun [37] integrated multi-choice conic goal programming and the fuzzy MULTIMOORA approach for student selection problem. Zavadskas et al. [38] used the concept of interval-valued intuitionistic fuzzy sets to develop the MULTIMOORA method for group decision making. Baležentis and Baležentis [39] presented a detailed survey on extensions and applications of the MULTIMOORA method. 2.4. Research gap To the best of the authors' knowledge, no study has considered target-based attributes to develop the MULTIMOORA method. We proposed an exponential normalization technique to insert the effect of attributes target values in derivation of the methodology. This formula is based on the exponential norms of the target-based VIKOR [2,20]. Subjective, objective, inter-attribute correlation effect significant coefficients were combined into two integrated forms. We utilized two concepts for considering objective significant coefficients including information entropy and standard deviation. Thus, the proposed methodology has two solution modes. The developed target-based MULTIMOORA method can be regarded as a generalized form of the approach that considers beneficial, non-beneficial, and target-based attributes. The suggested method can also be employed in the role of a traditional MADM technique in which no target-based attributes exist. Moreover, we utilized the dominance theory to generate the final rank of each solution mode and the aggregate ranking of the proposed methodology. We discussed two biomedical materials selection problems. The practical examples address materials selection of femoral components related to hip and knee joint prostheses. The results were compared with several studies that have employed other target-based methods to solve the biomedical problems. 3. The MULTIMOORA method The MOORA method consists of two components that are the ratio system and the reference point approach. However, the updated MOORA method, named MULTIMOORA, consists of three parts, i.e., the ratio system, the reference point approach, and the full multiplicative form. Similar to all MADM approaches, a decision matrix X is considered for the MULTIMOORA method. The arrays of the decision matrix, i.e., xij, denote the responses of alternative Ai to attribute ai called alternative ratings, i = 1, 2 …, m and j = 1, 2 …, n: X¼ xi j mn :
ð1Þ
The decision matrix is normalized to obtain comparable and dimensionless values named as normalized alternative ratings xij⁎. Normalization typically is a comparison between an alternative rating on a certain attribute, as a numerator, and a denominator that is a representative for all alternatives ratings on that attribute. Brauers and Zavadskas [26] recommended the following normalization ratio for the MULTIMOORA method: ," xi j
¼ xi j
m X
#1=2 x2i j
:
ð2Þ
i¼1
Relative significant coefficients of attributes, i.e., wj, can be considered in formulation of the MULTIMOORA method. Significant coefficients of attributes satisfy ∑nj = 1wj = 1.
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3.1. The ratio system The normalized ratings are multiplied by significant coefficients. The resultants are added for beneficial attributes and deducted for nonbeneficial attributes to obtain the assessment value of the ratio system yi [29]: yi ¼
g X
w j xi j −
j¼1
n X
w j xi j ;
ð3Þ
j¼gþ1
in which g indicates the number of beneficial attributes and (n − g) shows the number of non-beneficial attributes. The optimal alternative of the ratio system can be specified by listing the assessment values in descending order [31]: ARS ¼ fAi j maxi yi g:
ð4Þ
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Translating of ranks into a final ranking using the concept of the dominance theory may lead to circular reasoning. Readers can refer to [40] for details about the dominance theory. 4. The target-based MULTIMOORA with integrated significant coefficients 4.1. Target-based normalization In traditional MADM methods, normalization deals with ratings of alternatives on beneficial and non-beneficial attributes to obtain comparable values. However, the goal of achieving a certain target value of an attribute in some practical applications highlights the necessity of defining target-based normalization techniques [2]. The target (the goal or the most favorable) values for all attributes, i.e., Tj, j = 1, 2 …, n, can constitute the following set:
3.2. The reference point approach
T¼ T 1 ; T 2 ; …; T j ; …; T n :
This approach is established on the concept of maximal attribute reference point (MORP) and Tchebycheff min–max metric. The coordinate j of MORP vector is calculated as follows [26]:
Based on the norm of comprehensive VIKOR [2], we defined a normalization technique to consider target-based attributes for the MULTIMOORA method as follows:
rj ¼
maxi xi j ; mini xi j ;
j ≤g; jNg;
ð5Þ
−
in which g is the number of beneficial attributes. The deviation of normalized rating xij⁎ from reference point rj can be obtained as: di j ¼ r j −xi j :
ð6Þ
The assessment value of reference point approach can be determined as [29]: zi ¼ max w j di j : j
ð7Þ
The assessment values are listed in ascending order to produce the optimal alternative of the reference point approach [31]: ARP ¼ fAi j mini zi g:
ð8Þ
3.3. The full multiplicative form The third part of the MULTIMOORA approach is the full multiplicative form. In this technique, the allocation of significant coefficient as multiplier is meaningless. Instead, the significant coefficient should be considered as exponent [40]. The assessment value of the full multiplicative form can be obtained as follows: w j w j g n : ∏ j¼gþ1 xi j ui ¼ ∏ j¼1 xi j
ð9Þ
in which g denotes the number of beneficial attributes. Similar to the ratio system, the optimal alternative in this technique can also be calculated by finding the maximum assessment value: AM F ¼ fAi j maxi ui g:
ð10Þ
3.4. Final ranking of the MULTIMOORA method The subordinate ranks of Sections 3.1, 3.2, and 3.3 can be summarized into a final ranking, called the MULTIMOORA rank, by employing the theory of dominance. This theory is structured based on propositions such as dominance, being dominated, transitivity, and equability.
f ij ¼ e
ð11Þ
T j −xi j max max xi j ; T j − min min xi j ; T j
i
i
;
ð12Þ
In this paper, the exponential target-based normalization technique is employed in derivation of the assessment indices and significant coefficients of attributes. In addition to the role of normalization, Eq. (12) also measures the deviation of a rating from the target value that has been further discussed in Section 4.3.2. The target-based normalization technique, i.e., Eq. (12), was considered as an exponential formula to prevent insignificance of the assessment value of the target-based full multiplicative form that has been described in Section 4.3.3. Eq. (12) can also be utilized in traditional MULTIMOORA method in which only beneficial and non-beneficial attributes exist. Other forms of target-based normalization equations can be derived dependent on the type of MADM method. For instance, Jahan et al. [4] presented a linear target-based normalization technique for application in TOPSIS approach. 4.2. Significant coefficients of attributes 4.2.1. Subjective significant coefficient Relative importance of attributes obtained straight form decision makers opinions is called subjective significant coefficient. In the present paper, we show this significant coefficient as wsj . 4.2.2. Objective significant coefficient Two concepts including entropy and standard deviation were considered in this paper to allocate objective significant coefficient to attributes. Entropy concept has been broadly employed in social and physical sciences such as economics, language modeling, and spectral analysis. Shannon [41] developed the concept into information entropy to measure uncertainty in data. Information entropy can be effectively utilized in the process of decision making because it can evaluate existent contrasts between sets of data. To calculate objective significant coefficient through entropy, we used the technique presented by Zhang et al. [42] (named information entropy significant coefficient) that is slightly different from Shannon entropy significant coefficient. The following procedure should be
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used to determine the significant coefficient [42]. Initially, fij′ is introduced as follows: m X 0 1 þ f ij : f ij ¼ 1 þ f ij
fij′ is created from fij to avoid the insignificance of ln fij′ in Eq. (14). The information entropy measure is computed as: m X
0
0
f i j ln f i j :
ð14Þ
i¼1
Deviation degree Gj is defined as: G j ¼ 1−H j :
m X
ð13Þ
i¼1
Hj ¼ −
MADM methods. Inter-attribute correlation effect measure can be obtained as follows [3]:
ð15Þ
f i j− f j
w Hj ¼ G j
X n
G j:
ð16Þ
j¼1
Based on entropy concept, distributions with higher entropy represent more disorder, are smoother, are more probable, are less predictable, or assume less [43]. Thus, the set of ratings on a given attribute that is smoother shows higher information entropy H j , lower deviation degree Gj, and lower information entropy significant coefficient wjH. In statistics, standard deviation is a measure utilized to find the amount of variation of a data set. Generally, a standard deviation close to 0 shows that the set of data is near the mean point while a high standard deviation denotes a great spread of values. Standard deviation σj for application in the target-based MADM methods can be defined as follows [4]:
σj ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX m 2 u u f i j −f j t i¼1
m
;
ð17Þ
in which fj ¼
i¼1
ð20Þ
i¼1
in which k = 1, 2, …, n; moreover, fij or fik is the target-based normalized ratings obtained using Eq. (12) and f j or f k can be calculated through Eq. (18). Inter-attribute correlation effect significant coefficient is determined using Eq. (20) [3]: w cj ¼
n X
1−R jk
X n n X j¼1
1−R jk
! :
ð21Þ
k¼1
Based on Eq. (21), inter-attribute correlation effect significant coefficient wjc increases as inter-attribute correlation effect measure Rjk decreases. 4.2.4. Integration of the significant coefficients Subjective, objective, and inter-attribute correlation effect significant coefficients can be combined to generate the integrated form. Jahan et al. [44] proposed a formula for the integration of the three types of significant coefficients. Because of considering two concepts for objective significant coefficients, i.e., information entropy and standard deviation, two forms of integrated significant coefficients are obtained as follows: n 1 X 13 s H c 3 w s;H;c ¼ w w w w sj w Hj w cj ; j j j j
ð22Þ
j¼1 n 1 X 13 s σ c 3 ¼ w w w w sj w σj w cj ; w s;σ;c j j j j
ð23Þ
j¼1
in which wjs,H,c is the integration of subjective, information entropy, and inter-attribute correlation effect significant coefficients and wjs,σ,c indicates the combined form of subjective, standard deviation, and interattribute correlation effect significant coefficients. 4.3. Subordinate parts of the target-based MULTIMOORA method
m 1X f : m i¼1 i j
ð18Þ
The objective significant coefficient based on standard deviation is formulated as:
w σj ¼ σ j
f ik −f k
i¼1 ffi; R jk ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m 2 X 2 X f i j −f j f ik −f k
k¼1
Deviation degree Gj is higher if the value of information entropy Hj is smaller. The information entropy significant coefficient wjH is generated as:
X n
4.3.1. The target-based ratio system By considering Eq. (3), the assessment value of the target-based ratio system can be obtained as follows: Yi ¼
n X
w j f i j:
ð24Þ
j¼1
σ j:
ð19Þ
j¼1
Based on Eq. (19), the set of ratings on a given attribute with higher variance leads to higher standard deviation significant coefficient wjσ. 4.2.3. Inter-attribute correlation effect significant coefficient This significant coefficient is established on correlation effects of attributes. The idea behind the significant coefficient is that when correlation of a specific attribute with another attribute is high, less importance should be considered for the correlated attributes [44]. Diakoulaki et al. [45] introduced the original inter-attribute correlation effect significant coefficient based on standard deviation approach. Jahan and Edwards [3] updated the significant coefficient for application in the target-based
Unlike Eq. (3) that has two terms for beneficial and non-beneficial attributes, Eq. (24) has been simplified to only one term because beneficial, non-beneficial, and target-based attributes are considered in the normalized ratings fij. wj in Eq. (24) is the significant coefficient of attributes and can appear in various forms as discussed previously in Section 4.2. The resultant optimal alternative of the target-based ratio system can be determined as follows: ATRS ¼ fAi j maxi Y i g:
ð25Þ
4.3.2. The target-based reference point approach The reference point rj of the original MULTIMOORA method, i.e., Eq. (5), is translated to 1 for the target-based MULTIMOORA method. We name the corresponding form of the reference point as normalized target value (that is equal to 1 for all attributes). Thus, the deviation
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of normalized rating fij from normalized target value, i.e., 1, is obtained as: Di j ¼ 1− f i j :
ð26Þ
Eq. (26) shows that greater value of fij leads to lower deviation D ij . By consideration this deviation and Eq. (7), the assessment value of the target-based reference point approach is specified as follows: Z i ¼ max w j Di j : j
4.3.3. The target-based full multiplicative form Similar to the target-based ratio system, the target-based full multiplicative form has one term. The assessment value of the full multiplicative form, i.e., Eq. (9), is transformed into Eq. (29) when target-based attributes are considered: w j n U i ¼ ∏ j¼1 f i j :
ð29Þ
The optimal alternative for the target-based full multiplicative form has the maximum assessment value:
ð27Þ ATM F ¼ fAi j maxi U i g:
Then, alternatives can be listed in ascending order based on the assessment values of Eq. (27) to find the optimal alternative of the target-based reference point approach as: ATRP ¼ fAi j mini Z i g:
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ð28Þ
ð30Þ
4.4. Final ranking of the target-based MULTIMOORA method By employing the dominance theory, the ranks obtained in Sections 4.3.1, 4.3.2, and 4.3.3 can be integrated into a final ranking.
Fig. 1. A step-by-step flowchart for materials selection process using the proposed methodology.
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for attributes are desired. For example, the elastic modulus of an implant material is an important target-based attribute. Significant variation between elastic modulus of implant material and that of the surrounding bone leads to stress concentration [21]. An illustrative description of the materials selection process with the proposed methodology has been presented in Section 5.1. We provided the solutions for two practical cases regarding materials selection for femoral components of hip and knee joint prostheses in Sections 5.1, 5.2 and 5.3, respectively. 5.1. Materials selection process using the proposed methodology A flowchart for materials selection process utilizing the target-based MULTIMOORA method with integrated significant coefficients was illustrated in Fig. 1. Because we have considered two types of integrated significant coefficients, i.e., wjs,H,c and wjs,σ,c, the methodology has two solution modes that are Mode 1: the target-based MULTIMOORA with wjs,H,c and Mode 2: the target-based MULTIMOORA with wjs,σ,c. The final ranks of the two modes were integrated into the aggregate ranking of the proposed methodology using the dominance theory. Fig. 2. Elements of a hip joint prosthesis.
5.2. Example 1: materials selection for femoral component of hip joint prosthesis This practical case deals with materials selection for femoral component of hip joint prosthesis using the proposed methodology. Other studies [2,3,21,46] have already solved this biomedical materials selection problem utilizing different methods. A hip prosthesis consists of three main elements: femoral component, acetabular cup, and acetabular interface. The femoral component is a stiff metal rod implanted into the hollowed-out femur. This component can be produced with either a built-in polished metal head or separately attached ceramic ball head. The acetabular cup is a polymeric or metal shell fixed to the ilium at the
5. Materials selection in biomedical applications using the proposed methodology Accelerated developments in manufacturing techniques have facilitated production of advanced materials. This issue has considerable significance in biomedical engineering because new materials with high quality and durability are required in the field. In biomaterials selection, besides beneficial and non-beneficial attributes, target values
Table 1 Decision matrix related to materials selection for femoral component of hip joint prosthesis (Example 1) [2]. Objective of design
Max
Max
Max
Max
Max
Max
Target value
Target value
Min
Material ID
Material name
Tissue tolerance
Corrosion resistance
Tensile strength (MPa)
Fatigue strength (MPa)
Toughness
Wear resistance
Elastic modulus (GPa)
Density (g/cm3)
Cost
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11
Stainless steel 316 Stainless steel 317 Stainless steel 321 Stainless steel 347 Co–Cr alloy (castable) Co–Cr alloy (wrought) Pure titanium Ti–6Al–4V Epoxy—70% glass Epoxy—63% carbon Epoxy—62% aramid
10 9 9 9 10 10 8 8 7 7 7
7 7 7 7 9 9 10 10 7 7 7
517 630 610 650 655 896 550 985 680 560 430
350 415 410 430 425 600 315 490 200 170 130
8 10 10 10 2 10 7 7 3 3 3
8 8.5 8 8.4 10 10 8 8.3 7 7.5 7.5
200 200 200 200 238 242 110 124 22 56 29
8 8 7.9 8 8.3 9.1 4.5 4.4 2.1 1.6 1.4
1 1.1 1.1 1.2 3.7 4 1.7 1.9 3 10 5
Table 2 Normalized decision matrix for Example 1. Material ID
Tissue tolerance
Corrosion resistance
Tensile strength
Fatigue strength
Toughness
Wear resistance
Elastic modulus
Density
Cost
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11
1 0.717 0.717 0.717 1 1 0.513 0.513 0.368 0.368 0.368
0.368 0.368 0.368 0.368 0.717 0.717 1 1 0.368 0.368 0.368
0.430 0.527 0.509 0.547 0.552 0.852 0.457 1 0.577 0.465 0.368
0.587 0.675 0.667 0.696 0.689 1 0.545 0.791 0.427 0.401 0.368
0.779 1 1 1 0.368 1 0.687 0.687 0.417 0.417 0.417
0.513 0.607 0.513 0.587 1.000 1 0.513 0.567 0.368 0.435 0.435
0.442 0.442 0.442 0.442 0.374 0.368 0.656 0.617 0.966 0.832 0.936
0.465 0.465 0.471 0.465 0.447 0.403 0.732 0.742 1 0.937 0.913
1 0.989 0.989 0.978 0.741 0.717 0.925 0.905 0.801 0.368 0.641
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Table 3 Information entropy and standard deviation measures for Example 1. Measure
Tissue tolerance
Corrosion resistance
Tensile strength
Fatigue strength
Toughness
Wear resistance
Elastic modulus
Density
Cost
Hj σj
2.387 0.244
2.385 0.251
2.392 0.180
2.392 0.177
2.387 0.255
2.390 0.203
2.389 0.215
2.389 0.219
2.392 0.187
beneficial. The material ratings on tissue tolerance, corrosion resistance, toughness, wear resistance, and cost collected in Table 1 are the quantified values of the related linguistic variables. The material ratings of Table 1 can be normalized by employing Eq. (12) as shown in Table 2. The measures of information entropy and standard deviation for the problem listed in Table 3 are computed using Eqs. (14) and (17), respectively. Table 4 exhibits the arrays of inter-attribute correlation effect measure obtained through Eq. (20). Significant coefficients based on information entropy, standard deviation, inter-attribute correlation effect, and the two types of the integrated significant coefficients are obtained using Eqs. (16), (19), (21), (22), and (23), respectively. All significant coefficients were shown in Table 5. The assessment values related to the three subordinate parts of the proposed method for this practical case are obtained by considering
acetabulum, i.e., the hip socket. The acetabular interface is placed between the femoral component and the acetabular cup and may be manufactured with various material combinations, i.e., metal on metal, metal on polymer, and ceramic on ceramic, to decrease wear generated by friction [3]. Fig. 2 shows the typical structure and parts of a hip joint prosthesis. The femoral component is commonly made up of cobalt chrome or titanium, but previously also of stainless steel. Decision matrix for the materials selection problem is generated from ratings of eleven potential materials (m = 11) on nine attributes (n = 9) as presented in Table 1 [2]. Elasticity modulus and density are the target-based properties with goal values of 14 GPa and 2.1 g/cm3, respectively. The target values are determined near the real values of the femur properties. Tissue tolerance, corrosion resistance, fatigue strength, toughness, and wear resistance are the beneficial attributes. Cost is the only attribute regarded as nonTable 4 Inter-attribute correlation effect measure for Example 1. Attribute
Tissue tolerance
Corrosion resistance
Tensile strength
Fatigue strength
Toughness
Wear resistance
Elastic modulus
Density
Cost
Tissue tolerance Corrosion resistance Tensile strength Fatigue strength Toughness Wear resistance Elastic modulus Density Cost
1 0.072 0.158 0.697 0.465 0.770 −0.901 −0.912 0.376
0.072 1 0.584 0.435 −0.057 0.384 −0.166 −0.045 0.098
0.158 0.584 1 0.737 0.243 0.419 −0.254 −0.156 0.092
0.697 0.435 0.737 1 0.661 0.763 −0.798 −0.755 0.365
0.465 −0.057 0.243 0.661 1 0.200 −0.686 −0.721 0.645
0.770 0.384 0.419 0.763 0.200 1 −0.709 −0.678 −0.017
−0.901 −0.166 −0.254 −0.798 −0.686 −0.709 1 0.982 −0.517
−0.912 −0.045 −0.156 −0.755 −0.721 −0.678 0.982 1 −0.553
0.376 0.098 0.092 0.365 0.645 −0.017 −0.517 −0.553 1
Table 5 Significant coefficients of attributes for Example 1. Significant coefficient
Tissue tolerance
Corrosion resistance
Tensile strength
Fatigue strength
Toughness
Wear resistance
Elastic modulus
Density
Cost
wjs wjH wjσ wjc wjs,H,c wjs,σ,c
0.200 0.111 0.126 0.105 0.194 0.214
0.200 0.111 0.130 0.096 0.179 0.203
0.080 0.111 0.093 0.089 0.066 0.054
0.120 0.111 0.092 0.085 0.095 0.076
0.080 0.111 0.132 0.104 0.077 0.089
0.080 0.111 0.105 0.099 0.074 0.067
0.080 0.111 0.111 0.159 0.118 0.115
0.080 0.111 0.113 0.156 0.116 0.114
0.080 0.111 0.097 0.108 0.081 0.068
Table 6 The assessment values and rankings for Example 1. Material ID
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11
Target-based MULTIMOORA with integrated significant coefficient (Mode 1)
Target-based MULTIMOORA with integrated significant coefficient (Mode 2)
TRSa
TRS
TRPa
TMFa
Final rank
Ass.a
#a
Ass.
#
Ass.
#
0.629 0.612 0.604 0.613 0.682 0.773 0.689 0.747 0.570 0.507 0.529
5 7 8 6 4 1 3 2 9 11 10
0.113 0.113 0.113 0.113 0.074 0.075 0.095 0.095 0.123 0.123 0.123
5 5 5 5 1 2 3 3 9 9 9
0.583 0.579 0.571 0.580 0.641 0.726 0.664 0.724 0.520 0.473 0.488
5 7 8 6 4 1 3 2 9 11 10
5 7 8 6 4 1 3 2 9 11 10
TRP
TMF
Final rank
Ass.
#
Ass.
#
Ass.
#
0.632 0.609 0.602 0.609 0.686 0.775 0.696 0.743 0.558 0.503 0.523
5 7 8 6 4 1 3 2 9 11 10
0.128 0.128 0.128 0.128 0.072 0.072 0.104 0.104 0.136 0.136 0.136
5 5 5 5 1 2 3 3 9 9 9
0.585 0.575 0.568 0.576 0.643 0.728 0.671 0.720 0.509 0.469 0.482
5 7 8 6 4 1 3 2 9 11 10
5 7 8 6 4 1 3 2 9 11 10
a The utilized abbreviations are “TRS”: the target-based ratio system, “TRP”: the target-based reference point approach, “TMF”: the target-based full multiplicative form, “Ass.”: assessment value, and “#”: rank.
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Table 7 The rankings of the proposed methodology and the other approaches for Example 1. Material ID
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11
Material name
Stainless steel 316 Stainless steel 317 Stainless steel 321 Stainless steel 347 Co–Cr alloy (castable) Co–Cr alloy (wrought) Pure titanium Ti–6Al–4V Epoxy—70% glass Epoxy—63% carbon Epoxy—62% aramid
Our rankings
The rankings of other MADM methods
Target-based MULTIMOORA with integrated significant coefficient (Mode 1)
Target-based MULTIMOORA with integrated significant coefficient (Mode 2)
Aggregate ranking of the proposed methodology
Target-based TOPSIS with integrated significant coefficient (λ = 0.6) [3]
Target-based VIKOR with subjective significant coefficient [2]
Target-based VIKOR with integrated significant coefficient (λ = 0.6) [3]
Target-based LOP with subjective significant coefficient[2]
Target-based GP with subjective significant coefficient [21]
5 7 8 6 4 1 3 2 9 11 10
5 7 8 6 4 1 3 2 9 11 10
5 7 8 6 4 1 3 2 9 11 10
5 7 8 6 4 1 3 2 9 11 10
5 7 8 6 2 1 4 3 9 11 10
5 7 8 6 2 1 4 3 9 11 10
4 5 7 6 8 2 3 1 9 11 10
7 5 6 4 9 2 3 1 8 11 10
Fig. 3. Correlation between the aggregate ranking of the proposed method and the ranks of other approaches for Example 1.
Eqs. (24), (27), and (29) as well as the two integrated significant coefficients, i.e., Eqs. (22) and (23). Subordinate rankings are obtained based on Eq. (25), (28), and (30). These are then consolidated into a final ranking on the basis of the dominance theory. The assessment values, the subordinate ranks, and the final rankings of the two solution modes for Example 1 were collected in Table 6. The ranks were shown in bold format in Table 6. The subordinate and final rankings of the two modes have one-to-one correspondence. Except the TRP approach, the subordinate and final ranks of the two modes show a same optimal material. That is, based on Eqs. (25), (28), (30), and the dominance theory, (ATRS⁎)1,2 = (ATRP⁎)1,2 = (ATMF⁎)1,2 = (AFinal⁎)1,2 = M6, i.e., Co–Cr alloy (wrought). The indices of these terms display the number of the modes. The final ranks of the two modes were integrated into an aggregate ranking utilizing the dominance theory. Table 7 shows a comparison between the aggregate ranking for Example 1 and the ranks of other studies that have considered this biomedical problem. The comparison between the aggregate ranking and the rank of the target-based TOPSIS with integrated significant coefficient (λ = 0.6) [3] shows exact correspondence. Besides, our aggregate ranking and the ranks of the target-based VIKOR with subjective significant coefficient [2] and the target-based VIKOR with integrated significant coefficient (λ = 0.6) [3] have close agreement. We utilized Spearman rank correlation coefficient to specify connection between the aggregate ranking of the proposed methodology and the ranks of the other methods presented in Table 7. This coefficient proposed by Spearman [47] is a real number between − 1 and 1. The value of 1 exhibits exact correspondence of compared ranks while the value of −1 shows complete opposition. Fig. 3 demonstrates Spearman
rank correlation coefficients for Example 1. Because the target-based TOPSIS with integrated significant coefficient (λ = 0.6) [3] has exact agreement with our aggregate ranking, the top value of Spearman coefficient, i.e., 1, goes to this approach. The target-based GP with subjective significant coefficient [21] has the bottom value of Spearman coefficient value, i.e., 0.80, that shows the lowest agreement with our aggregate ranking.
Fig. 4. Elements of a knee joint prosthesis.
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Table 8 Decision matrix related to materials selection for femoral component of knee joint prosthesis (Example 2) [20]. Objective of design
Target value
Max
Target value
Max
Max
Max
Max
Material ID
Material name
Density (g/cm3)
Tensile strength (MPA)
Elastic modulus (GPA)
Elongation (%)
Corrosion resistance
Wear resistance
Osseointegration
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
Stainless steel 316L (annealed) Stainless steel 316L (cold-worked) Co–Cr alloy (wrought) Co–Cr alloy (castable) Pure titanium Ti–6Al–4V Ti–6Al–7Nb (wrought) Ti–6Al–7Nb (hot-forged) NiTi (SMA) Porous NiTi (SMA)
8.0 8.0 9.1 8.3 4.5 4.4 4.5 4.5 6.5 4.3
517 862 896 655 550 985 900 1050 1240 1000
200 200 240 240 100 112 113 110 48 15
40 12 20 20 54 12 10 13 12 12
0.67 0.67 0.75 0.75 0.96 0.96 0.96 0.96 0.96 0.75
0.59 0.75 0.87 0.87 0.59 0.67 0.67 0.67 0.96 0.96
0.59 0.59 0.67 0.67 0.75 0.75 0.75 0.75 0.50 0.96
Table 9 Normalized decision matrix for Example 2. Material ID
Density
Tensile strength
Elastic modulus
Elongation
Corrosion resistance
Wear resistance
Osseointegration
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
0.425 0.425 0.368 0.409 0.665 0.670 0.663 0.663 0.515 0.682
0.368 0.593 0.621 0.445 0.385 0.703 0.625 0.769 1 0.718
0.441 0.441 0.370 0.370 0.688 0.653 0.651 0.659 0.867 0.996
0.727 0.385 0.462 0.462 1 0.385 0.368 0.389 0.385 0.385
0.368 0.368 0.485 0.485 1 1 1 1 1 0.485
0.368 0.567 0.784 0.784 0.368 0.457 0.457 0.457 1 1
0.447 0.447 0.532 0.532 0.633 0.633 0.633 0.633 0.368 1
5.3. Example 2: materials selection for femoral component of knee joint prosthesis In this example, a biomedical problem regarding finding the optimal material for femoral component of knee joint prosthesis is discussed. This problem has already been solved using the other MADM methods [20,23]. Knee implants are produced to replace the elements of a knee joint that have been damaged. A knee prosthesis normally includes three main elements: femoral component, tibial tray, and tibial insert as illustrated in Fig. 4. The femoral component usually cemented or pressed in the place is a substitution for the distal femur. The currently used metals for application in femoral component of knee joint prosthesis are stainless steel, Co–Cr alloys, as well as titanium and its alloys. However, some materials like NiTi shape memory alloys (SMA), including dense and porous NiTi, can also be prospective options
for the application. Decision matrix for the engineering materials selection problem is formed from ratings of ten candidate materials (m = 10) on seven attributes (n = 7) as shown in Table 8 [20]. Density and elasticity modulus are the target-based properties with goal values of 1.3 g/cm3 and 16 GPa, respectively. The target values are determined near the real values of the distal femur properties. Other attributes including tensile strength, elongation, corrosion resistance, wear resistance, and osseointegration are beneficial. The material ratings on corrosion resistance, wear resistance, and osseointegration listed in Table 8 are the quantified values of the related linguistic variables. Readers can refer to [20] to gather original data of the materials selection problem. The material ratings of Table 8 are normalized to generate dimensionless values of Table 9. The measures of information entropy and standard deviation for the problem were gathered in Table 10. Table 11 shows the
Table 10 Information entropy and standard deviation measures for Example 2. Measure
Density
Tensile strength
Elasticity modulus
Elongation
Corrosion resistance
Wear resistance
Osseointegration
Hj σj
2.299 0.125
2.296 0.182
2.295 0.200
2.294 0.197
2.289 0.284
2.292 0.235
2.297 0.165
Table 11 Inter-attribute correlation effect measure for Example 2. Attribute
Density
Tensile strength
Elasticity modulus
Elongation
Corrosion resistance
Wear resistance
Osseointegration
Density Tensile strength Elasticity modulus Elongation Corrosion resistance Wear resistance Osseointegration
1 0.260 0.745 0.013 0.711 −0.217 0.661
0.260 1 0.594 −0.687 0.431 0.556 0.019
0.745 0.594 1 −0.129 0.480 0.373 0.564
0.013 −0.687 −0.129 1 0.023 −0.473 −0.084
0.711 0.431 0.480 0.023 1 −0.240 0.038
−0.217 0.556 0.373 −0.473 −0.240 1 0.165
0.661 0.019 0.564 −0.084 0.038 0.165 1
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Table 12 Significant coefficients of attributes for Example 2. Significant coefficient
Density
Tensile strength
Elastic modulus
Elongation
Corrosion resistance
Wear resistance
Osseointegration
wsj wH j wσj wcj ws,H,c j ws,σ,c j
0.061 0.143 0.090 0.111 0.047 0.028
0.105 0.143 0.131 0.140 0.102 0.088
0.136 0.143 0.144 0.098 0.092 0.087
0.105 0.143 0.142 0.213 0.155 0.144
0.179 0.142 0.205 0.133 0.163 0.220
0.229 0.143 0.169 0.170 0.268 0.299
0.185 0.143 0.119 0.135 0.173 0.135
Table 13 The assessment values and rankings for Example 2. Material ID
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
Target-based MULTIMOORA with integrated significant coefficient (Mode 1)
Target-based MULTIMOORA with integrated significant coefficient (Mode 2)
TRSa
TRS
TRPa
TMFa
Final rank
Ass.a
#a
Ass.
#
Ass.
#
0.447 0.470 0.567 0.551 0.660 0.618 0.607 0.626 0.761 0.777
10 9 7 8 3 5 6 4 2 1
0.169 0.116 0.084 0.084 0.169 0.146 0.146 0.146 0.109 0.095
9 5 1 1 9 6 6 6 4 3
0.433 0.463 0.549 0.534 0.608 0.588 0.577 0.595 0.694 0.727
10 9 7 8 3 5 6 4 2 1
10 9 7 8 3 5 6 4 2 1
TRP
TMF
Final rank
Ass.
#
Ass.
#
Ass.
#
0.438 0.468 0.576 0.562 0.671 0.634 0.625 0.641 0.801 0.764
10 9 7 8 3 5 6 4 1 2
0.189 0.139 0.113 0.113 0.189 0.162 0.162 0.162 0.089 0.113
9 5 2 2 9 6 6 6 1 2
0.425 0.460 0.557 0.543 0.614 0.599 0.589 0.605 0.739 0.714
10 9 7 8 3 5 6 4 1 2
10 9 7 8 3 5 6 4 1 2
a The utilized abbreviations are “TRS”: the target-based ratio system, “TRP”: the target-based reference point approach, “TMF”: the target-based full multiplicative form, “Ass.”: assessment value, and “#”: rank.
Table 14 The rankings of the proposed methodology and the other approaches for Example 2. Material ID Material name
Our rankings Target-based MULTIMOORA with integrated significant coefficient (Mode 1)
M1 M2 M3 M4 M5 M6 M7 M8 M9 M10
Stainless steel 316L (annealed) 10 Stainless steel 316L (cold-worked) 9 Co–Cr alloy (wrought) 7 Co–Cr alloy (castable) 8 Pure titanium 3 Ti–6Al–4V 5 Ti–6Al–7Nb (wrought) 6 Ti–6Al–7Nb (hot-forged) 4 NiTi (SMA) 2 Porous NiTi (SMA) 1
The rankings of other MADM methods Target-based MULTIMOORA with integrated significant coefficient (Mode 2)
Aggregate ranking of the proposed methodology
Target-based TOPSIS with subjective significant coefficient [23]
Target-based VIKOR with subjective significant coefficient (λ = 0.5) [20]
Interval target-based VIKOR with subjective significant coefficient [23]
10 9 7 8 3 5 6 4 1 2
10 9 7 8 3 5 6 4 1 1
10 9 7 8 6 4 5 3 2 1
8 7 9 10 6 4 5 3 2 1
10 8 3 4 9 6 7 5 2 1
arrays of inter-attribute correlation effect measure. Subjective, objective, and inter-attribute correlation effect significant coefficients as well as the two types of the integrated significant coefficients were listed in Table 12.
The assessment values, the subordinate rankings, and the final rankings of the two solution modes for Example 2 were listed in Table 13. Except the TRP approach, subordinate and final ranks of Mode 1 show the optimal material as (ATRS⁎)1 = (ATMF⁎)1 = (AFinal⁎)1 =
Fig. 5. Correlation between the aggregate ranking of the proposed method and the ranks of other approaches for Example 2.
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M10, i.e., porous NiTi (SMA). However, subordinate and final ranks of Mode 2 show the optimal material as (ATRS⁎)2 = (ATRP⁎)2 = (ATMF⁎)2 = (AFinal⁎)2 = M9, i.e., NiTi (SMA). The aggregate ranking for Example 2 and the ranks of the other studies that have considered this biomedical problem were also listed in Table 14. Based on Table 14, the comparison between the aggregate ranking and the rank of the target-based TOPSIS [23] shows close correspondence. Fig. 5 demonstrates Spearman rank correlation coefficients for Example 2. The top value 0.93, is for the target-based TOPSIS with subjective significant coefficient [23]. The interval target-based VIKOR with subjective significant coefficient [23], by 0.58, has the lowest agreement with our aggregate ranking. 6. Conclusion In biomedical engineering design problems, decision making concerning materials selection should often be implemented in view of target values of attributes. Therefore, structured target-based methods are required to solve such problems. In this paper, we extended the MULTIMOORA method using target-based attributes. We suggested an exponential target-based normalization technique to generate a comprehensive MULTIMOORA approach. The combinations of subjective, objective, and inter-attribute correlation effect significant coefficients were utilized in formulation of the proposed method to give relative importance to attributes. We considered information entropy and standard deviation concepts to assign objective significant coefficients. We obtained the final rank of each solution mode and the aggregate ranking of the proposed methodology employing the dominance theory. Two engineering decision-making problems in biomedical applications were presented. These biomedical problems deal with materials selection for femoral components of hip and knee joint prostheses. The aggregate rankings of the proposed methodology for the practical biomedical cases were compared with those of earlier studies. As further researches, other MADM methods can be developed into comprehensive forms considering the target-based attributes. Moreover, the proposed target-based MULTIMOORA method can be employed in other practical selection problems. References [1] M.M. Farag, Quantitative methods of materials selection, in: M. Kutz (Ed.), Handbook of Materials Selection, John Wiley & Sons, Canada 2002, pp. 3–24. [2] A. Jahan, F. Mustapha, M.Y. Ismail, S. Sapuan, M. Bahraminasab, A comprehensive VIKOR method for material selection, Mater. Des. 32 (2011) 1215–1221. [3] A. Jahan, K.L. Edwards, Weighting of dependent and target-based criteria for optimal decision-making in materials selection process: biomedical applications, Mater. Des. 49 (2013) 1000–1008. [4] A. Jahan, M. Bahraminasab, K. Edwards, A target-based normalization technique for materials selection, Mater. Des. 35 (2012) 647–654. [5] P. Karande, S. Chakraborty, Application of multi-objective optimization on the basis of ratio analysis (MOORA) method for materials selection, Mater. Des. 37 (2012) 317–324. [6] E. Celik, M. Gul, A.T. Gumus, A.F. Guneri, A fuzzy TOPSIS approach based on trapezoidal numbers to material selection problem, J. Inf. Technol. Appl. Manag. 19 (2012) 19–30. [7] H.-C. Liu, L.-X. Mao, Z.-Y. Zhang, P. Li, Induced aggregation operators in the VIKOR method and its application in material selection, Appl. Math. Model. 37 (2013) 6325–6338. [8] D. Das, Selection of materials in engineering design using Ashby's chart and AHPTOPSIS(Bachelor Dissertation) Jadavpur University, Kolkata, India, 2012. [9] A. Shanian, A.S. Milani, C. Carson, R.C. Abeyaratne, A new application of ELECTRE III and revised Simos' procedure for group material selection under weighting uncertainty, Knowl.-Based Syst. 21 (2008) 709–720. [10] R.V. Rao, A material selection model using graph theory and matrix approach, Mater. Sci. Eng. A 431 (2006) 248–255. [11] J.W.K. Chan, T.K.L. Tong, Multi-criteria material selections and end-of-life product strategy: grey relational analysis approach, Mater. Des. 28 (2007) 1539–1546. [12] K. Prasad, S. Chakraborty, A quality function deployment-based model for materials selection, Mater. Des. 49 (2013) 525–535. [13] V.M. Athawale, R. Kumar, S. Chakraborty, Decision making for material selection using the UTA method, Int. J. Adv. Manuf. Technol. 57 (2011) 11–22.
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