# Consistency Improvement Method of Pairwise Matrix Based on Consistency Ratio Decreasing Rate and Attribute Weighting Method Considered Decision Makers’ Levels in Analytic Hierarchy Process: Application to Hip Joint Prosthesis Material Selection

Authors: Won-Chol Yang; Hyon-Song Kang; Gyong-Su Ri; Jin-Sim Kim
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Analytic hierarchy process (AHP) is a well-known attribute weighting method in multiattribute decision-making. Its major requirement is to satisfy the consistency of pairwise matrix (PM). To solve this problem, we first propose a new consistency improvement method of PM based on consistency ratio (CR) decreasing rate. In this method, we calculate the CR decreasing rates of all the PMs reconstituted by replacing all elements of the PM with the lower and upper neighbouring 9-point scales and find the element with maximum CR decreasing rate, and then modify it to its lower or upper neighbouring scale. Second, we develop third-order approximate polynomial for random consistency index using least square method. It enables to determine the RI value according to the number of attributes without a numerical table. Third, we propose the final PM determining method and final attribute weighting method considered decision makers’ levels based on the CR values of the individual PMs in case several decision makers perform their own pairwise comparisons. We test the performances of the proposed and some previous consistency improvement methods with two numerical examples. The results demonstrate that the proposed method improves the consistency of PM better and faster with smaller amount of modification than that of the previous methods, while it modify the elements of the PM to 9-point scales, necessarily. We apply the proposed method to hip joint prosthesis material selection. The proposed methods may be widely used in practical applications of AHP.


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## Body

## 1. Introduction

Determining a reasonable attribute weights plays a vital role in multiattribute decision-making (MADM) and multiobjective optimization (MOO) because the decision-making and optimization results may differ according to the attribute weights. Analytic hierarchy process (AHP) is well-known attribute weighting method [1]. AHP determines the attribute weights based on pairwise comparison evaluation data for the pairs of attributes. The 9-point scales are used to transform the decision maker’s judgments into numerical quantities [2]. The essential feature of AHP is the pairwise comparisons between the attributes instead of the direct allocation of the weights [3]. The details of AHP have been described in the literature.AHP has been widely applied to calculate the subjective attribute weights in many practical MADM and MOO problems [4, 5]. Soni et al. [6] determined the criteria weights using AHP for material selection of reinforced sustainable composites by recycling waste plastics and agro-waste. Zhong et al. [7] constructed a cost evaluation system with 5 indices in the first level and 22 indices in the second level using AHP. Radulescu et al. [8] calculated the overall weights of the criteria as a linear combination of the individual weights obtained from the group AHP and extended entropy weighting method for evaluating the fourth wave of COVID-19 pandemic. Peng and Wu [9] determined the comprehensive weights of each index using AHP to score the index system for evaluating the benefit development of the offshore wind power after the cancellation of the public subsides. Wei et al. [10] constructed the evaluation model using AHP and fuzzy comprehensive evaluation. Mathew et al. [11] calculated the weights of the criteria using fuzzy AHP and determined the final ranking of the alternatives using spherical fuzzy TOPSIS. Rawa et al. [12] proposed an economical-technical-environmental operation for power networks with wind-solar-hydropower generation by using AHP and improved grey wolf algorithm. They used a weighted sum strategy using AHP to transform the multiobjective problem into a normalized single objective one. Okudan and Budayan [13] used fuzzy AHP for conducting the evaluation of project characteristics affecting risk occurrences in the construction projects. Dano [14] analysed the impacts of flash hazards using AHP and identified the most effective methods to reduce the flash flood impacts using expert’s opinions in Jeddah. Chen et al. [15] dealt with the uncertainty of the wind power, the load demand, and the multiobjective function by using fuzzy chance constraint programming and improved AHP.For group decision-making problems, hesitant fuzzy set, complex intuitionistic fuzzy set, and probabilistic hesitant fuzzy set have been introduced to handle and model the uncertainty and vagueness in decision-making very effectively, reflect the importance of different numerical values more clearly, elicit the decision makers’ knowledge, and develop more effective decision-making model. [16–18] Rani and Garg [16] proposed a novel algorithm for multiattribute group decision-making using complex intuitionistic fuzzy values. Jin et al. [17] proposed a decision-making model using probabilistic hesitant fuzzy preference relations for reflecting clearly the importance of different numerical values and eliciting the decision makers’ knowledge in the group decision-making problems. Liu et al. [18] calculated the probabilities of elements in the probabilistic hesitant fuzzy element and the probability of risk status by using two nonlinear programming models. Khan et al. [19] proposed a performance measure using an MADM method based on the complex T-spherical fuzzy power aggregation operators. Liu et al. [20] developed a novel correlation coefficient to measure the strength of the relationship between the hesitant fuzzy sets.Although AHP is a useful tool for attribute weighting, it has some drawbacks. One drawback is that it is difficult to conduct pairwise comparison in practical applications. Yang et al. [5] proposed a simplest questionnaire to conduct the pairwise comparison, easily and conventionally. Another drawback is that it is difficult to satisfy the consistency of pairwise matrix (PM) in practical applications. To determine the reasonable attribute weights using AHP, the consistency of PM must be satisfied. When it does not satisfy the consistency, it is need to repair the primary PM. In order to satisfy the consistency of the PM, some researchers proposed the reconstitution methods of the inconsistent PM. Girsang et al. [21] proposed ANTAHP method using ant optimization algorithm to reconstitute the inconsistent PM by minimizing the distance between the primary and modified PMs. Wu et al. [22] improved the inconsistency of the PM using marginal optimization method. The method is based to increase (or decrease) all elements by a fixed value, and it calculates the marginal effect of each modification. Zeshui and Cuiping [23] proposed a consistency improvement method based on auto-adaptive process. In the method, the element aij of the inconsistent PM A is replaced by bij = aijα (wi/wj)1−α, where w =  (w1,…,wi,…,wnT is the weight vector obtained from A. The generated matrix B = [bij] has a reduced CR. This process is repeated until the consistency is satisfied. Cao et al. [24] proposed a heuristic method to modify inconsistent PM. They decomposed the primary PM as the Hadamard product of the consistent PM and a reciprocal deviation matrix. They constituted a modified PM by convex combination of the reciprocal zero deviation matrices. They proposed auto-adaptive modification algorithm using such convex combinations. Yang et al. [25] modified the inconsistent PM by combining the particle swarm optimization and Taguchi method. Benítez et al. [26] proposed a linearization method to provide the closest consistent PM to the inconsistent PM by using orthogonal projection in a linear space.For consistency improvement, it needs to pay attention to guarantee a good balance between improvement of consistency and preservation of primary information. However, the previous methods are lacking in guaranteeing such balance. On the other hand, the elements of the PM are 9-point scales because the pairwise comparison is performed by means of 9-point scales {1/9, 1/8, …, 8, 9} in the conventional AHP. However, in the previous methods, the elements of the reconstituted PMs are no 9-point scales, and therefore, the PMs obtained from the previous methods are no inherent ones. To deal with this shortcoming, we propose a new consistency improvement method according to the following principles:(i)
The amount of consistency improvement of the PM should be as large as possible, and the deviation between the primary and reconstituted PMs and the number of the modified elements should be as small as possible.(ii)
The elements of the inconsistent PM should be replaced with the lower or upper neighbouring 9-point scales, and the elements of the reconstituted PM should be 9-point scales.When two or more decision makers take part in the pairwise comparison between the attributes, the PMs and the attribute weights may differ according to their knowledge and opinions. Therefore, it is necessary to constitute a final PM by synthesizing the individual PMs obtained from each decision makers and determine the final attribute weighting from the final PM. It needs to consider the decision makers’ levels to constitute the final PM. However, there is no reasonable objective method to determine the decision makers’ levels, while the previous methods are generally subjective ones. To overcome this shortcoming, we propose a new objective method to determine the decision makers’ levels based on the CR values of their PMs. The worse the consistency of the PM is, the more mistakes the decision maker’s judgment has, and therefore, we can regard that the CR value reflects the decision maker’s level.We propose a new consistency improvement method of pairwise matrix based on consistency ratio decreasing rate and attribute weighting method considered decision makers’ levels in analytic hierarchy process and apply the methods to hip joint prosthesis material selection.The novelties and advantages of the proposed methods are as follows:(i)
In the CR decreasing rate-based consistency improvement method of inconsistent PM, the elements of the inconsistent PM are modified to the adjacent 9-point scales, and all the elements of the reconstituted PM are the 9-point scales, not real numbers. This method improves the consistency more, better, and faster with smaller number of elements and smaller amount of modification, and it guarantees a very good balance between consistency improvement and information preservation of the primary PM.(ii)
The approximate formula is used to determine the RI value according to the number of attributes, not the numerical table for RI. It enables to test the consistency of PM without a numerical table for RI.(iii)
In the final attribute weighting method considered decision makers’ levels, the CR value is used as an objective measure that reflects the decision maker’s level. It is possible to determine the decision makers’ levels objectively, not subjectively. It enables to determine the attribute weights, more scientifically and reasonably. In this method, the elements of the final PM are also 9-point scales, and it enables to preserve the inherent characteristics of AHP.The rest of this paper is organized as follows: In Section2, we describe a new consistency improvement method of PM based on CR decreasing rate, a development method of approximate polynomial for RI according to number of attributes, and a constitution method of final PM and final attribute weighting method considered decision makers’ levels. In Section 3, we describe the numerical test results of the proposed method and its application to hip joint prosthesis material selection. In Section 4, we present the conclusions. In Appendix section, we describe the attribute weighting method using AHP with the simplest questionnaire, development method of approximate polynomial by least square method, and three well-known MADM methods (MADMs) such as simple additive weighting (SAW) method, technique for order preference by similarity to ideal solution (TOPSIS) method, and grey relational analysis (GRA) method used in this work.


## 2. Methodology

### 2.1. CR Decreasing Rate-Based Consistency Improvement Method of Inconsistent PM

Let CR(A) be the CR value of the PM A=(aij)n×n.The main steps of a new consistency improvement method are as follows:Step 1. For each elementalm (l=1,n−1¯,m=l+1,n¯) in the upper triangular matrix of the PM A=(aij)n×n, reconstitute the PM Alm−=aij−n×n by replacing the element alm with the lower neighbouring scale hlm−, and then calculate its CR value CRAlm− and CR decreasing rate dCRrlm− as follows:(1)dCRrlm−=dCRlm−alm−hlm−+1/alm−1/hlm−,where(2)dCRlm−=CRA−CRAlm−,alm−=hlm−,aml−=1alm−,aij−=aij,i,j=1,n¯,i,j≠l,m.The lower and upper neighbouring scales of the 9-point scales {1/9, 1/8, …, 8, 9} are shown in Table1.Step 2. Find the maximum value of the CR decreasing ratedCRrr1s1− from {dCRrlm−; l=1,n−1¯,m=l+1,n¯} as follows:(3)dCRrr1s1−=max1≤l<m≤ndCRrlm−.Step 3. For each elementalm (l=1,n−1¯,m=l+1,n¯) in the upper triangular matrix of the PM A=(aij)n×n, reconstitute the PM Alm+=aij+n×n by replacing the element alm with the upper neighbouring scale hlm+, and then calculate its CR value CRAlm+ and CR decreasing rate dCRrlm+ as follows:(4)dCRrlm+=dCRlm+alm−hlm++1/alm−1/hlm+,where(5)dCRlm+=CRA−CRAlm+,alm+=hlm+,aml+=1alm+,aij+=aij,i,j=1,n¯,i,j≠l,m.Step 4. Find the maximum value of the CR decreasing ratedCRrr2s2+ from {dCRrlm+; l=1,n−1¯,m=l+1,n¯} as follows:(6)dCRrr2s2+=max1≤l<m≤ndCRrlm+.Step 5. IfdCRrr1s1−≥dCRrr2s2+, then reconstitute the PM B=bijn×n by replacing the element ar1s1 with the lower neighbouring scale hr1s1− in the PM A=(aij)n×n, where(7)br1s1=hr1s1−,bs1r1=1br1s1,bij=aij,i,j=1,n¯,i,j≠r1,s1.IfdCRrr1s1−<dCRrr2s2+, then reconstitute the PM B=bijn×n by replacing the element ar2s2 with the upper neighbouring scale hr2s2+ in the PM A=(aij)n×n, where(8)br2s2=hr2s2+,bs2r2=1br2s2,bij=aij,i,j=1,n¯,i,j≠r2,s2.Step 6. Calculate the CR value CR(B) of the reconstituted PM B=bijn×n.Step 7. If CR(B) > CR0, then A = B and go to Step 1.Step 8. If CR(B) ≤ CR0, then calculate the principal eigenvector v=v1,…,vj,…,vnT from the reconstituted PM B=bijn×n.Step 9. Calculate the attribute weightsw1,…,wj,…,wnT by normalizing v=v1,…,vj,…,vnT as follows:(9)wj=vj∑k=1nvk,j=1,n¯.Table 1
Lower and upper neighbouring scales of the 9-point scales.
9-Point scale1/91/81/71/61/51/41/31/2123456789Lower neighbouring scale1/91/91/81/71/61/51/41/31/212345678Upper neighbouring scale1/81/71/61/51/41/31/21234567899We call this method CR decreasing rate-based method.LetA = (aij)n×n and B = (bij)n×n be the primary inconsistent PM and reconstituted PM, respectively.To evaluate the performance of the consistency improvement method of the inconsistent PM, we use the following three measures:(i)
CR decreasing amount,(10)dCRA,B=CRB−CRA,where CR(A) and CR(B) are the CR values of the primary PM A and the reconstituted PM B, respectively.(ii)
Matrix deviation,(11)dMA,B=∑i=1n∑j=1naij−bij.(iii)
CR decreasing rate,(12)dCRrA,B=dCRA,BdMA,B.The greater the values ofdCR(A,B) and dCRr(A,B) are, the better the performance of the consistency improvement method is. The smaller the value of dM(A,B) is, the better the reconstituted PM preserves the information of the primary PM. The CR decreasing rate dCRr(A,B) becomes the major measure to evaluate the performance of the consistency improvement method of PM from among above three measures.


### 2.2. Development Method of Approximate Polynomial for RI According to the Number of Attributes

We developm-th order approximate polynomial for RI as the following form:(13)RI=RImn=a0+a1n…+am−1nm−1+amnm,with the data set {(n, RIn), n=3,15¯.}, where n is the number of attributes and RIn is the corresponding RI value (Table 2).Table 2
RI value according to the number of attributes [28].
n123456789101112131415RI0.00.00.580.901.121.241.321.411.451.491.511.531.561.571.59The MAE, MRE, and MSE of them-th order approximate polynomial are as follows:(14)MAEm=113∑n=315RIn−RImn,MREm=113∑n=315RIn−RImnRIn×100%,MSEm=113∑n=315RIn−RImn2.The main steps to develop the approximate polynomial for RI value according to the number of attributes are as follows:Step 1. Develop six approximate polynomialsRI = RIm(n); m=1,6¯ with the data {(n, RIn), n=3,15¯.} (Table 2).Step 2. Evaluate the MAE, MRE, and MSE of six approximate polynomialsRI = RIm(n); m=1,6¯.Step 3. Select the suitable approximate polynomialRI = RIr(n) with acceptable MAE, MRE, and MSE from among six approximate polynomials RI = RIm(n); m=1,6¯.


### 2.3. Constituting Method of Final PM Constituting Method and Final Attribute Weighting Method Considered Decision Makers’ Levels

LetM be the number of the decision makers.The main steps to constitute the final PM and determining the final attribute weights considered decision makers’ levels are as follows:Step 1. Constitute the simplest questionnaires byM decision makers and constitute M PMs {Am=aijmn×n; m=1,M¯ } from the simplest questionnaires, where Am=aijmn×n is the PM constituted from m-th decision maker’s questionnaire.Step 2. Test the consistency of the PMs.Step 3. Reconstitute the consistent PM by modifying the inconsistent PM using the CR decreasing rate-based consistency improvement method.Denote the consistent PMs asBm=bijmn×n; m=1,M¯.Step 4. Calculate the CR value CR(m) of the PM Bm=bijmn×n; m=1,M¯.Step 5. Calculate the inverse values of the CR values {ICR(m) = 1/CR(m); m=1,M¯.}.Step 6. Normalize the inverse values of the CR values {ICR(m); m=1,M¯} and determine the normalized values as the decision makers’ levels {hm; m=1,M¯}.(15)hm=ICRm∑j=1MICRj.Commonly, the greater the CR value is, the worse the consistency of the PM is. The worse the consistency of the PM is, the more mistakes the decision maker’s judgment has. Therefore, we can regard that the inverse value of the CR value of the PM reflects the decision maker’s level. This is why we assign the normalized inverse values of CR values to the decision makers’ levels.Step 7. Constitute the composite PMB0=bij0n×n as the geometric mean of the individual PMs {Bm=bijmn×n; m=1,M¯} as follows:(16)bij0=∏m=1Mbijmhm,i,j=1,n¯.The elements ofB0=bij0n×n may be no 9-point scales.Step 8. Constitute the final PMA0=aij0n×n by transforming the elements of B0=bij0n×n to the nearest neighbouring 9-point scales.The final PMA0=aij0n×n reflects not only the decision makers’ pairwise judgments but also their levels.Step 9. Determine the final attribute weightsw1,…,wj,…,wnT by normalizing the principal eigenvector v=v1,…,vj,…,vnT of the final PM A0=aij0n×n.Figure1 shows the flowchart of the proposed methods.Figure 1
Flowchart of the proposed methods.


## 2.1. CR Decreasing Rate-Based Consistency Improvement Method of Inconsistent PM

Let CR(A) be the CR value of the PM A=(aij)n×n.The main steps of a new consistency improvement method are as follows:Step 1. For each elementalm (l=1,n−1¯,m=l+1,n¯) in the upper triangular matrix of the PM A=(aij)n×n, reconstitute the PM Alm−=aij−n×n by replacing the element alm with the lower neighbouring scale hlm−, and then calculate its CR value CRAlm− and CR decreasing rate dCRrlm− as follows:(1)dCRrlm−=dCRlm−alm−hlm−+1/alm−1/hlm−,where(2)dCRlm−=CRA−CRAlm−,alm−=hlm−,aml−=1alm−,aij−=aij,i,j=1,n¯,i,j≠l,m.The lower and upper neighbouring scales of the 9-point scales {1/9, 1/8, …, 8, 9} are shown in Table1.Step 2. Find the maximum value of the CR decreasing ratedCRrr1s1− from {dCRrlm−; l=1,n−1¯,m=l+1,n¯} as follows:(3)dCRrr1s1−=max1≤l<m≤ndCRrlm−.Step 3. For each elementalm (l=1,n−1¯,m=l+1,n¯) in the upper triangular matrix of the PM A=(aij)n×n, reconstitute the PM Alm+=aij+n×n by replacing the element alm with the upper neighbouring scale hlm+, and then calculate its CR value CRAlm+ and CR decreasing rate dCRrlm+ as follows:(4)dCRrlm+=dCRlm+alm−hlm++1/alm−1/hlm+,where(5)dCRlm+=CRA−CRAlm+,alm+=hlm+,aml+=1alm+,aij+=aij,i,j=1,n¯,i,j≠l,m.Step 4. Find the maximum value of the CR decreasing ratedCRrr2s2+ from {dCRrlm+; l=1,n−1¯,m=l+1,n¯} as follows:(6)dCRrr2s2+=max1≤l<m≤ndCRrlm+.Step 5. IfdCRrr1s1−≥dCRrr2s2+, then reconstitute the PM B=bijn×n by replacing the element ar1s1 with the lower neighbouring scale hr1s1− in the PM A=(aij)n×n, where(7)br1s1=hr1s1−,bs1r1=1br1s1,bij=aij,i,j=1,n¯,i,j≠r1,s1.IfdCRrr1s1−<dCRrr2s2+, then reconstitute the PM B=bijn×n by replacing the element ar2s2 with the upper neighbouring scale hr2s2+ in the PM A=(aij)n×n, where(8)br2s2=hr2s2+,bs2r2=1br2s2,bij=aij,i,j=1,n¯,i,j≠r2,s2.Step 6. Calculate the CR value CR(B) of the reconstituted PM B=bijn×n.Step 7. If CR(B) > CR0, then A = B and go to Step 1.Step 8. If CR(B) ≤ CR0, then calculate the principal eigenvector v=v1,…,vj,…,vnT from the reconstituted PM B=bijn×n.Step 9. Calculate the attribute weightsw1,…,wj,…,wnT by normalizing v=v1,…,vj,…,vnT as follows:(9)wj=vj∑k=1nvk,j=1,n¯.Table 1
Lower and upper neighbouring scales of the 9-point scales.
9-Point scale1/91/81/71/61/51/41/31/2123456789Lower neighbouring scale1/91/91/81/71/61/51/41/31/212345678Upper neighbouring scale1/81/71/61/51/41/31/21234567899We call this method CR decreasing rate-based method.LetA = (aij)n×n and B = (bij)n×n be the primary inconsistent PM and reconstituted PM, respectively.To evaluate the performance of the consistency improvement method of the inconsistent PM, we use the following three measures:(i)
CR decreasing amount,(10)dCRA,B=CRB−CRA,where CR(A) and CR(B) are the CR values of the primary PM A and the reconstituted PM B, respectively.(ii)
Matrix deviation,(11)dMA,B=∑i=1n∑j=1naij−bij.(iii)
CR decreasing rate,(12)dCRrA,B=dCRA,BdMA,B.The greater the values ofdCR(A,B) and dCRr(A,B) are, the better the performance of the consistency improvement method is. The smaller the value of dM(A,B) is, the better the reconstituted PM preserves the information of the primary PM. The CR decreasing rate dCRr(A,B) becomes the major measure to evaluate the performance of the consistency improvement method of PM from among above three measures.


## 2.2. Development Method of Approximate Polynomial for RI According to the Number of Attributes

We developm-th order approximate polynomial for RI as the following form:(13)RI=RImn=a0+a1n…+am−1nm−1+amnm,with the data set {(n, RIn), n=3,15¯.}, where n is the number of attributes and RIn is the corresponding RI value (Table 2).Table 2
RI value according to the number of attributes [28].
n123456789101112131415RI0.00.00.580.901.121.241.321.411.451.491.511.531.561.571.59The MAE, MRE, and MSE of them-th order approximate polynomial are as follows:(14)MAEm=113∑n=315RIn−RImn,MREm=113∑n=315RIn−RImnRIn×100%,MSEm=113∑n=315RIn−RImn2.The main steps to develop the approximate polynomial for RI value according to the number of attributes are as follows:Step 1. Develop six approximate polynomialsRI = RIm(n); m=1,6¯ with the data {(n, RIn), n=3,15¯.} (Table 2).Step 2. Evaluate the MAE, MRE, and MSE of six approximate polynomialsRI = RIm(n); m=1,6¯.Step 3. Select the suitable approximate polynomialRI = RIr(n) with acceptable MAE, MRE, and MSE from among six approximate polynomials RI = RIm(n); m=1,6¯.


## 2.3. Constituting Method of Final PM Constituting Method and Final Attribute Weighting Method Considered Decision Makers’ Levels

LetM be the number of the decision makers.The main steps to constitute the final PM and determining the final attribute weights considered decision makers’ levels are as follows:Step 1. Constitute the simplest questionnaires byM decision makers and constitute M PMs {Am=aijmn×n; m=1,M¯ } from the simplest questionnaires, where Am=aijmn×n is the PM constituted from m-th decision maker’s questionnaire.Step 2. Test the consistency of the PMs.Step 3. Reconstitute the consistent PM by modifying the inconsistent PM using the CR decreasing rate-based consistency improvement method.Denote the consistent PMs asBm=bijmn×n; m=1,M¯.Step 4. Calculate the CR value CR(m) of the PM Bm=bijmn×n; m=1,M¯.Step 5. Calculate the inverse values of the CR values {ICR(m) = 1/CR(m); m=1,M¯.}.Step 6. Normalize the inverse values of the CR values {ICR(m); m=1,M¯} and determine the normalized values as the decision makers’ levels {hm; m=1,M¯}.(15)hm=ICRm∑j=1MICRj.Commonly, the greater the CR value is, the worse the consistency of the PM is. The worse the consistency of the PM is, the more mistakes the decision maker’s judgment has. Therefore, we can regard that the inverse value of the CR value of the PM reflects the decision maker’s level. This is why we assign the normalized inverse values of CR values to the decision makers’ levels.Step 7. Constitute the composite PMB0=bij0n×n as the geometric mean of the individual PMs {Bm=bijmn×n; m=1,M¯} as follows:(16)bij0=∏m=1Mbijmhm,i,j=1,n¯.The elements ofB0=bij0n×n may be no 9-point scales.Step 8. Constitute the final PMA0=aij0n×n by transforming the elements of B0=bij0n×n to the nearest neighbouring 9-point scales.The final PMA0=aij0n×n reflects not only the decision makers’ pairwise judgments but also their levels.Step 9. Determine the final attribute weightsw1,…,wj,…,wnT by normalizing the principal eigenvector v=v1,…,vj,…,vnT of the final PM A0=aij0n×n.Figure1 shows the flowchart of the proposed methods.Figure 1
Flowchart of the proposed methods.


## 3. Results and Discussion

### 3.1. Numerical Test Results of the Proposed Consistency Improvement Method

We test the performance of proposed consistency improvement method of PM by applying it to two examples and compare with the previous methods.Example 1.
The primary PM [21, 22] is as follows:(17)153766131415113533151/71331634615131516113141718161313311215161612144211516351675511247586621,CR=0.169087.
The reconstituted PM using the ANTAHP method [21] is as follows:(18)14.22.47.45.85.412.413.414.21134.42.62.41/416.812.4316.83.44.24.414.217.414.416.8112.813.4171915.812.613.42.8111.81516.815.412.414.23.41.811516.82.4414.47551123.46.84.296.86.821,CR=0.099838.
The reconstituted PMs using Wu method [22] are as follows:
In case ofd = 0.1 (where d is the fixed modifying amount),(19)14.32.377535.30.45450.3030232561103333432.30.23260.15870434783.003300031674.35.30.23260.129870.232560232560149251076920.14710.11490.188680434780434780270272299910.23260.14930.18868043478023256330031300110.23260.14932.20024299201886867981429924299210.38463.300363012429928703266979669792.60011,CR=0.099955.
In case ofd = 0.01,(20)14.282.287.725.285.280.43860.30490.2336410.33674.282.282.280.233600.15920.43862.9716.723.724.325.280.23360.129530.233640.1488110.43860.30490.14840.11470.189390.43860.268822.2810.78120.23360.14880.189390.43860.231483.279811.280110.23360.14882.284.28080.189396.73854.28084.280810.38023.27986.28144.28088.71846.72046.72042.63021,CR=0.099825.
In case ofd = 0.001,(21)14.2822.2827.7185.2825.2820.43820.30470.2335410.33644.2822.2822.2820.23350.15920.438212.972716.7183.7184.3285.2820.23350.129570.233540.1488510.43820.30470.14830.11470.189320.438210.268962.282110.780.23350.14890.189320.438210.231053.28191.282110.23350.14892.28214.28270.189326.74314.28274.282710.38023.28196.28144.28278.71846.71596.71592.63021,CR=0.099997.
The reconstituted PM using Xu and Wei method [23] is as follows:(22)14.5242.3397.5235.8885.6860.4250.2920.22110.3264.5162.6712.5800.2220.1470.4273.06716.7493.4604.1884.1550.2490.1330.2210.14810.3730.2870.1340.1040.1700.3740.2892.68110.5610.1970.1470.1760.3880.2343.4791.78410.2040.1532.3544.4970.2417.4795.0734.89910.5013.4196.7864.0249.6246.7836.5511.9961,CR=0.096964.
The reconstituted PM using Cao method [24] is as follows:(23)14.444122.36827.67435.85595.60790.42010.29680.225210.32104.42242.61753.53920.22680.14860.42233.115116.91493.53514.27744.51050.24870.13030.22610.144610.38050.29270.13130.10300.17080.38200.28292.627810.57220.19600.14440.17830.39380.23383.41661.747810.20550.14942.38044.40990.22177.61365.10124.866110.50043.36896.72934.02159.71306.92356.69491.99841,CR=0.107316.
The reconstituted PM using the proposed method is as follows (CR0 = 0.1.):(24)1537661314151135331517133163411517151611314171816131331121516161314421151635175511247586621,CR=0.087973.
Table3 shows the comparison results between some previous and proposed consistency improvement methods. The CR, dCR, and dCRr are, respectively, 0.087973, 0.081114, and 0.013905 by modifying only two elements with the dM of 5.833333 using the proposed method. The CR, dCR, and dCRr are, respectively, 0.096964, 0.072123, and 0.004792 by modifying 56 elements with the dM of 15.050557 using the previous Cao’s method.Table 3
Comparison results between some consistency improvement methods about Example1.
Performance measureANT-AHP [21]Wu et al. [22]Zeshui and Cuiping [23]Cao et al. [24]Proposed methodd = 0.1d = 0.01d = 0.001CR0.0998380.0999550.0998250.0999970.0969640.1073160.087973dCR0.0692480.0691300.0692620.0690900.0721230.0617710.081114dM14.84297419.45700020.00900019.95600015.05055716.6055575.833333dCRr0.0046650.0035530.00346150.0034620.0047920.0037200.013905Number of modified elements4654565656562Example 2.
The primary inconsistent PM [26] is as follows:(25)1131511423311221333521456511214114124315413112131611311313131512131,CR=0.112090.
The reconstituted PM using the Benítez method [26] is as follows:(26)10.5260.1540.7940.4711.7381.171.90210.2931.5100.8963.3062.2256.4873.41115.1493.05511.287.5901.2600.6620.19410.5932.1901.4742.1231.1160.3271.68513.6912.4840.5750.3020.0890.4570.27110.6730.8550.4490.1320.6780.4031.4861,CR=0.099677.
The reconstituted PMs using the proposed method are as follows:
In case of CR0 = 0.01,(27)113151141131122133521426511214114114112413211316113111131511211,CR=0.009261.
In case of CR0 = 0.005,(28)113151141131122133521426511214113114112313311316113111131511311,CR=0.0004667.
In case of CR0 = 0.001,(29)113151131131123133521526511315113113112313311316113111131511311,CR=0.000772.
Table4 shows the comparison result between Benítez method [26] and the proposed method.
The CR,dCR, and dCRr are, respectively, 0.000772, 0.111318, and 0.005214 by modifying 22 elements with the dM of 21.35 using the proposed method (CR0 = 0.001). The CR, dCR, and dCRr are, respectively, 0.000007693, 0.112082, and 0.003194 by modifying 42 elements with the dM of 35.096667 using the previous Benítez’s method.
As can be seen in Tables3 and 4, the proposed method is much better than the previous ones from the viewpoints of CR, dCR, dM, dCRr, and the number of modified elements.Table 4
Comparison result between Benítez method and the proposed method about Example2.
Performance measureBenítez method [26]Proposed methodCR0 = 0.005CR0 = 0.01CR0 = 0.005CR0 = 0.001CR0.0000076930.0092610.0046670.000772dCR0.1120820.102830.107420.111318dM35.09666715.46718.0521.35dCRr0.0031940.0066480.00595140.005214Number of modified elements42141622


### 3.2. Approximate Polynomial for Random Consistency Index (RI) According to the Number of Attributes

We develop approximate polynomial forRI according to the number of attributes by using the proposed method in Section 3.2.Table5 and Figures 2 and 3 show the performance test results of approximate polynomials for RI with different orders.Table 5
Performance test results of approximate polynomials for RI with different orders.
mMean absolute errorMaximum of absolute errorsMean relative error (%)Maximum of relative errors (%)Mean-squared error (%)Maximum of squared errors (%)10.111010.3363710.75657.9950.0183120.1131520.0468060.117914.383720.330.00312640.01390330.0166330.0389211.53165.40920.000403140.001514940.00788260.02280.66341.72739.4729⋅10−50.0005198350.00563810.0147330.421491.11614.8904⋅10−50.0002170560.00492970.012720.39410.963653.9756⋅10−50.0001618Figure 2
Performance test results of approximate polynomials for RI with different orders: (a)m = 1, (b) m = 2, (c) m = 3, (d) m = 4, (e) m = 5, and (f) m = 6.
(a)(b)(c)(d)(e)(f)Figure 3
MAE, MRE, and MSE values of the approximate polynomials for RI according to different orders: (a) MAE, (b) MRE, and (c) MSE.
(a)(b)(c)From Table5 and Figures 2 and 3, we can know that third-order approximate polynomial is appropriate from the viewpoints of the performance and complexity.Third-order approximate polynomial for RI is as follows:(30)RI=0.001311n3−0.045332n2+0.533262n−0.615824.Table6 shows the performance test result of the above third-order approximate polynomial.Table 6
Performance test result of the third-order approximate polynomial for RI (equation (21)).
nRICalculated RIAbsolute errorRelative error (%)Squared error30.580.611370.0313745.40920.000984340.90.875820.0241762.68620.0005844751.121.08110.0389213.47510.001514961.241.2350.0049950.402822.495⋅10−571.321.34550.025471.92950.000648781.411.42030.010340.733310.0001069191.451.46750.0174831.20570.00030564101.491.49480.00476520.319812.2707⋅10−5111.511.51015.4945⋅10−50.00363873.019⋅10−9121.531.52120.00878120.573947.711⋅10−5131.561.53610.0238761.53050.00057007141.571.56260.00736260.468965.4208⋅10−5151.591.60860.0186261.17150.00034694Mean0.0166331.53160.00040314Max0.0389215.40920.0015149As the result, the approximate formula of CR is as follows:(31)CR=CIRI=λmax−nn−1RI=λmax−nn−10.00311n3−0.045332n2+0.533262n−0.615824.By using (30) and (31), we can directly calculate the RI value and CR value without the numerical table for RI values according to the number of attributes and evaluate the consistency of the constituted PM rapidly.On the other hand, we test the performance of the following previous approximate formula for RI: [4].(32)RI=1.98n−2n.Table7 shows the performance test result of the previous approximate formula (32).From Tables6 and 7, we know that the proposed third-order approximate polynomial (30) has much better performance than the previous approximation formula (32).Table 7
Performance test result of the previous approximate formula for RI (equation (23)).
nRICalculated RIAbsolute errorRelative error (%)Squared error30.580.660.0813.7930.006440.90.990.09100.008151.121.1880.0686.07140.00462461.241.320.086.45160.006471.321.41430.0942867.14290.008889881.411.4850.0755.31910.00562591.451.540.096.20690.0081101.491.5840.0946.30870.008836111.511.620.117.28480.0121121.531.650.127.84310.0144131.561.67540.115387.39640.013314141.571.69710.127148.09830.016165151.591.7160.1267.92450.015876Mean0.0976787.68010.00991Max0.1271413.7930.016165


### 3.3. Application to Hip Joint Prosthesis Material Selection

We use the proposed consistency improvement method of PM based on CR decreasing rate to hip joint prosthesis material selection [27].The alternative materials for hip joint prosthesis are, respectively, stainless steels 316 (A1), stainless steels 317 (A2), stainless steels 321 (A3), stainless steels 347 (A4), Co-Cr alloys-cast alloy (A5), Co-Cr alloys-wrought alloy (A6), unalloyed titanium (A7), Ti-6Al-4V (A8), composites (fabric reinforced)-epoxy-70% glass (A9), composites (fabric reinforced)-epoxy-63% carbon (A10), and composites (fabric reinforced)-epoxy-62% aramid (A11). The material attributes are, respectively, tissue tolerance (TT), corrosion resistance (CR), tensile strength (TS) (MPa), fatigue strength (FS) (MPa), relative toughness (RT), relative wear resistance (RWR), elastic modulus (EM) (GPa), specific gravity (SG) (g/cc), and cost (C). Table8 shows some hip joint prosthesis materials and their properties. [27].Table 8
Alternative materials and their attribute values [27].
Alternative materialTTCRTSFSRTRWREMSGCA11075173508820081A297630415108.520081.1A3976104101082007.91.1A497650430108.420081.2A51096554252102388.33.7A610989660010102429.14A7810550315781104.51.7A881098549078.31244.41.9A97768020037222.13A107756017037.5561.610A117743013037.5291.45First, we conduct the correlation analysis between the properties of hip joint prosthesis materials.Table9 shows the correlation coefficients between properties of hip joint prosthesis materials. As shown in Table 9, the elastic modulus and specific gravity have the high correlation coefficients with the tissue tolerance, fatigue strength, and relative wear resistance.Table 9
Correlation coefficients between properties of hip joint prosthesis materials.
TTCRTSFSRTRWREMSGCTT1.0000.1610.2290.7800.5300.7920.9660.961−0.479CR1.0000.5700.476−0.0080.4840.1750.101−0.127TS1.0000.7290.2870.4650.3000.268−0.157FS1.0000.6810.7920.8480.832−0.492RT1.0000.2510.6240.688−0.624RWR1.0000.8150.744−0.129EM1.0000.985−0.472SG1.000−0.576C1.000In this table, the bold values indicate higher correlation coefficients (ñ>0.8) than others.Therefore, we remove two properties such as EM (elastic modulus) and SG (specific gravity) from the consideration.Consequently, we constitute the decision matrix with seven attributes: TT, CR, TS, FS, RT, RWR, and C. Table10 shows the decision matrix.Table 10
Decision matrix.
Alternative materialsTTCRTSFSRTRWRCA1107517350881A297630415108.51.1A3976104101081.1A497650430108.41.2A51096554252103.7A610989660010104A7810550315781.7A881098549078.31.9A977680200373A107756017037.510A117743013037.55As shown in Table10, there are 11 alternative materials with 7 attributes. Excepting C (cost), six attributes are benefit attributes among seven attributes, while C (cost) is cost attribute.The material attribute weights are determined by using the AHP with simplest questionnaire by three decision makers. Table11 shows the simplest pairwise comparison questionnaire completed by the first decision maker.Table 11
Simplest pairwise comparison questionnaire completed by the first decision maker.
Tissue toleranceCorrosion resistanceTensile strengthFatigue strengthRelative toughnessRelative wear resistanceCostTissue tolerance1155355Corrosion resistance155355Tensile strength15535Fatigue strength1335Relative toughness135Relative wear resistance13Cost1It indicates that the value is the maximum. I have no objection to remove the Bold type.From Table11, the first decision maker’s PM is as follows:(33)A1=115535511553551515115151315151551133513135313515153131313151551515131.The consistency test result of the PMA(1) is as follows:(34)λmax=7.913828,CI=0.152305,CR=0115382>01.Therefore, we should modify the PMA(1) until CR < 0.1 using the proposed consistency improvement method.In the first iteration, we first reconstitute the PM by replacing all elements ofA(1) with the lower neighbouring scales and calculate CR values and CR decreasing rates (Table 12).Table 12
CR decreasing rates by replacing all elements with the lower neighbouring scales.
dCRrpq−2345671−0.004171−0.00775670.00381980.00423790.00012511−0.00286972−0.00775670.00381980.00423790.00012511−0.00286973−0.00229490.00080871−0.0031613−0.0102734−0.00776960.00616790.006202950.000354390.001863460.0077709It indicates that the value is the maximum. I have no objection to remove the Bold type.From Table12, the maximum CR decreasing rate is 0.0077709 (CR = 0.10632) when we replace the element a67 = 3 in the primary PM with the lower neighbouring scale 2.We next reconstitute the PM by replacing all elements ofA(1) with the upper neighbouring scales and calculate the CR values and CR decreasing rates (Table 13).Table 13
CR decreasing rates by replacing all elements with the upper neighbouring scales.
dCRrpq+23451−0.0041710.0051298−0.004292−0.0058214−0.0010120.001327620.0051298−0.004292−0.0058214−0.0010120.001327630.0014299−0.00219980.00073210.01037140.0063837−0.0075791−0.00650445−0.0025883−0.00258986−0.0090527It indicates that the value is the maximum. I have no objection to remove the Bold type.From Table13, the maximum CR decreasing rate is 0.010371 (CR = 0.10449) when we replace the element a37 = 1/5 in the primary PM with the upper neighbouring scale 1/4.As 0.0077709 < 0.010371, we decide to modify the elementa37 = 1/5 of the primary PM to the upper neighbouring scale 1/4. The reconstituted PM is as follows:(35)B11=115535511553551515115151314151551133513135313515153131313151541515131,CI=010449>01.As CR = 0.10449 > 0.1, we should modify the PMB11 until CR < 0.1 again.In the next iteration, we first reconstitute the PMB11 by replacing all elements with the lower neighbouring scales and calculate the CR values and CR decreasing rates (Table 14).Table 14
CR decreasing rates by replacing all elements with the lower neighbouring scales.
dCRrpq−2345671−0.0042129−0.0072570.00379170.00423495.7868e − 05−0.00326522−0.0072570.00379170.00423495.7868e − 05−0.00326523−0.00263550.00050903−0.0036426−0.0103714−0.00782550.00626760.005737150.000316940.001497560.0071198From Table14, the maximum CR decreasing rate is 0.0071198 (CR = 0.096187) when we replace the element a67 = 3 with the lower neighbouring scale 2.We next reconstitute the PM by replacing all elements with the upper neighbouring scales and calculate the CR values and CR decreasing rates (Table15).Table 15
CR decreasing rates by replacing all elements with the upper neighbouring scales.
dCRrpq+2345671−0.00421290.0047588−0.0042749−0.0058397−0.000965720.001654920.0047588−0.0042749−0.0058397−0.000965720.001654930.0018419−0.00181350.00137290.01014240.0064268−0.0076851−0.00607285−0.0025853−0.00226066−0.0084573From Table15, the maximum CR decreasing rate is 0.010142 (CR = 0.093506) when we replace the element a37 = 1/4 with the upper neighbouring scale 1/3. As 0.0071198 < 0.010142, we decide to modify the element a37 = 1/4 to the upper neighbouring scale 1/3. The reconstituted PM is as follows:(36)B1=115535511553551515115151313151551133513135313515153131313151531515131,CR1=0.093506.<0.1.As CR = 0.093506 < 0.1, the reconstituted PMB(1) satisfies the consistency.By the similar way, the reconstituted PMs based on the PMs constituted by the second and third decision makers are as follows:(37)B2=11/375355315535717171131511151531133513135313515151131313151711515131,CR2=0.071796<0.1,B3=13755751317557517171131511315153113331515531531717113151115153131311,CR3=0.076189<0.1.Based on the CR values {0.093506, 0.071796, 0.076189}, three decision makers’ levels are determined as follows:(38)h1=0.28331,h2=0.36898,h3=0.34771.By using (31), the composite PM is as follows:(39)B0=109769636355358315620651023615620653583156206566090157150177921028842020732530499950202346721033333341863027909027909531358314186301779201779213651033333027909120475020176652.00020.238870.238870.48841.By transforming the elements ofB0=bij0n×n to the nearest neighbouring 9-point scales, the final PM is as follows:(40)A0=116546511654661616114151212151541133414145314416162131412151621414121.The CR value of the final PMA(0) is CR(0) = 0.064623, and therefore the final PM satisfies the consistency.The final attribute weights calculated from the final PMA(0) are as follows:(41)0.31665,0.32205,0.031323,0.093804,0.14673,0.048864,0.040575.With these attribute weights, we can select the best hip joint prosthesis material using the well-known three MADMs such as SAW, TOPSIS, and GRA.Table16 and Figure 4 show the comprehensive scores of the alternatives from three MADMs.Table 16
Comprehensive scores of the alternatives from three MADMs.
MADMsAlternative materialsA1A2A3A4A5A6A7A8A9A10A11SAW0.5320.4910.4800.4930.6800.8740.6170.6800.0780.0420.049TOPSIS0.5080.4390.4370.4390.6770.7970.6080.6210.0800.0450.059GRA0.6400.5740.5700.5750.7010.8470.6520.6940.3600.3440.348Figure 4
Comprehensive scores of the alternatives from three MADMs.Table17 shows the ranking of the comprehensive scores of the alternatives from three MADMs.Table 17
Ranking of the alternatives using three MADMs.
MADMsAlternative materialsA1A2A3A4A5A6A7A8A9A10A11SAW5786214391110TOPSIS5786214391110GRA5786214391110Final ranks5786214391110From Table17, we know that the final ranking of the alternative materials is as follows:(42)A6>A5>A8>A7>A1>A4>A2>A3>A9>A11>A10.Therefore, we can select A6 (Co-Cr alloys-wrought alloy) as the best hip joint prosthesis material, and the next are A5 (Co-Cr alloys-cast alloy), A8 (Ti-6Al-4V), A7 (unalloyed titanium), and so on.


## 3.1. Numerical Test Results of the Proposed Consistency Improvement Method

We test the performance of proposed consistency improvement method of PM by applying it to two examples and compare with the previous methods.Example 1.
The primary PM [21, 22] is as follows:(17)153766131415113533151/71331634615131516113141718161313311215161612144211516351675511247586621,CR=0.169087.
The reconstituted PM using the ANTAHP method [21] is as follows:(18)14.22.47.45.85.412.413.414.21134.42.62.41/416.812.4316.83.44.24.414.217.414.416.8112.813.4171915.812.613.42.8111.81516.815.412.414.23.41.811516.82.4414.47551123.46.84.296.86.821,CR=0.099838.
The reconstituted PMs using Wu method [22] are as follows:
In case ofd = 0.1 (where d is the fixed modifying amount),(19)14.32.377535.30.45450.3030232561103333432.30.23260.15870434783.003300031674.35.30.23260.129870.232560232560149251076920.14710.11490.188680434780434780270272299910.23260.14930.18868043478023256330031300110.23260.14932.20024299201886867981429924299210.38463.300363012429928703266979669792.60011,CR=0.099955.
In case ofd = 0.01,(20)14.282.287.725.285.280.43860.30490.2336410.33674.282.282.280.233600.15920.43862.9716.723.724.325.280.23360.129530.233640.1488110.43860.30490.14840.11470.189390.43860.268822.2810.78120.23360.14880.189390.43860.231483.279811.280110.23360.14882.284.28080.189396.73854.28084.280810.38023.27986.28144.28088.71846.72046.72042.63021,CR=0.099825.
In case ofd = 0.001,(21)14.2822.2827.7185.2825.2820.43820.30470.2335410.33644.2822.2822.2820.23350.15920.438212.972716.7183.7184.3285.2820.23350.129570.233540.1488510.43820.30470.14830.11470.189320.438210.268962.282110.780.23350.14890.189320.438210.231053.28191.282110.23350.14892.28214.28270.189326.74314.28274.282710.38023.28196.28144.28278.71846.71596.71592.63021,CR=0.099997.
The reconstituted PM using Xu and Wei method [23] is as follows:(22)14.5242.3397.5235.8885.6860.4250.2920.22110.3264.5162.6712.5800.2220.1470.4273.06716.7493.4604.1884.1550.2490.1330.2210.14810.3730.2870.1340.1040.1700.3740.2892.68110.5610.1970.1470.1760.3880.2343.4791.78410.2040.1532.3544.4970.2417.4795.0734.89910.5013.4196.7864.0249.6246.7836.5511.9961,CR=0.096964.
The reconstituted PM using Cao method [24] is as follows:(23)14.444122.36827.67435.85595.60790.42010.29680.225210.32104.42242.61753.53920.22680.14860.42233.115116.91493.53514.27744.51050.24870.13030.22610.144610.38050.29270.13130.10300.17080.38200.28292.627810.57220.19600.14440.17830.39380.23383.41661.747810.20550.14942.38044.40990.22177.61365.10124.866110.50043.36896.72934.02159.71306.92356.69491.99841,CR=0.107316.
The reconstituted PM using the proposed method is as follows (CR0 = 0.1.):(24)1537661314151135331517133163411517151611314171816131331121516161314421151635175511247586621,CR=0.087973.
Table3 shows the comparison results between some previous and proposed consistency improvement methods. The CR, dCR, and dCRr are, respectively, 0.087973, 0.081114, and 0.013905 by modifying only two elements with the dM of 5.833333 using the proposed method. The CR, dCR, and dCRr are, respectively, 0.096964, 0.072123, and 0.004792 by modifying 56 elements with the dM of 15.050557 using the previous Cao’s method.Table 3
Comparison results between some consistency improvement methods about Example1.
Performance measureANT-AHP [21]Wu et al. [22]Zeshui and Cuiping [23]Cao et al. [24]Proposed methodd = 0.1d = 0.01d = 0.001CR0.0998380.0999550.0998250.0999970.0969640.1073160.087973dCR0.0692480.0691300.0692620.0690900.0721230.0617710.081114dM14.84297419.45700020.00900019.95600015.05055716.6055575.833333dCRr0.0046650.0035530.00346150.0034620.0047920.0037200.013905Number of modified elements4654565656562Example 2.
The primary inconsistent PM [26] is as follows:(25)1131511423311221333521456511214114124315413112131611311313131512131,CR=0.112090.
The reconstituted PM using the Benítez method [26] is as follows:(26)10.5260.1540.7940.4711.7381.171.90210.2931.5100.8963.3062.2256.4873.41115.1493.05511.287.5901.2600.6620.19410.5932.1901.4742.1231.1160.3271.68513.6912.4840.5750.3020.0890.4570.27110.6730.8550.4490.1320.6780.4031.4861,CR=0.099677.
The reconstituted PMs using the proposed method are as follows:
In case of CR0 = 0.01,(27)113151141131122133521426511214114114112413211316113111131511211,CR=0.009261.
In case of CR0 = 0.005,(28)113151141131122133521426511214113114112313311316113111131511311,CR=0.0004667.
In case of CR0 = 0.001,(29)113151131131123133521526511315113113112313311316113111131511311,CR=0.000772.
Table4 shows the comparison result between Benítez method [26] and the proposed method.
The CR,dCR, and dCRr are, respectively, 0.000772, 0.111318, and 0.005214 by modifying 22 elements with the dM of 21.35 using the proposed method (CR0 = 0.001). The CR, dCR, and dCRr are, respectively, 0.000007693, 0.112082, and 0.003194 by modifying 42 elements with the dM of 35.096667 using the previous Benítez’s method.
As can be seen in Tables3 and 4, the proposed method is much better than the previous ones from the viewpoints of CR, dCR, dM, dCRr, and the number of modified elements.Table 4
Comparison result between Benítez method and the proposed method about Example2.
Performance measureBenítez method [26]Proposed methodCR0 = 0.005CR0 = 0.01CR0 = 0.005CR0 = 0.001CR0.0000076930.0092610.0046670.000772dCR0.1120820.102830.107420.111318dM35.09666715.46718.0521.35dCRr0.0031940.0066480.00595140.005214Number of modified elements42141622


## 3.2. Approximate Polynomial for Random Consistency Index (RI) According to the Number of Attributes

We develop approximate polynomial forRI according to the number of attributes by using the proposed method in Section 3.2.Table5 and Figures 2 and 3 show the performance test results of approximate polynomials for RI with different orders.Table 5
Performance test results of approximate polynomials for RI with different orders.
mMean absolute errorMaximum of absolute errorsMean relative error (%)Maximum of relative errors (%)Mean-squared error (%)Maximum of squared errors (%)10.111010.3363710.75657.9950.0183120.1131520.0468060.117914.383720.330.00312640.01390330.0166330.0389211.53165.40920.000403140.001514940.00788260.02280.66341.72739.4729⋅10−50.0005198350.00563810.0147330.421491.11614.8904⋅10−50.0002170560.00492970.012720.39410.963653.9756⋅10−50.0001618Figure 2
Performance test results of approximate polynomials for RI with different orders: (a)m = 1, (b) m = 2, (c) m = 3, (d) m = 4, (e) m = 5, and (f) m = 6.
(a)(b)(c)(d)(e)(f)Figure 3
MAE, MRE, and MSE values of the approximate polynomials for RI according to different orders: (a) MAE, (b) MRE, and (c) MSE.
(a)(b)(c)From Table5 and Figures 2 and 3, we can know that third-order approximate polynomial is appropriate from the viewpoints of the performance and complexity.Third-order approximate polynomial for RI is as follows:(30)RI=0.001311n3−0.045332n2+0.533262n−0.615824.Table6 shows the performance test result of the above third-order approximate polynomial.Table 6
Performance test result of the third-order approximate polynomial for RI (equation (21)).
nRICalculated RIAbsolute errorRelative error (%)Squared error30.580.611370.0313745.40920.000984340.90.875820.0241762.68620.0005844751.121.08110.0389213.47510.001514961.241.2350.0049950.402822.495⋅10−571.321.34550.025471.92950.000648781.411.42030.010340.733310.0001069191.451.46750.0174831.20570.00030564101.491.49480.00476520.319812.2707⋅10−5111.511.51015.4945⋅10−50.00363873.019⋅10−9121.531.52120.00878120.573947.711⋅10−5131.561.53610.0238761.53050.00057007141.571.56260.00736260.468965.4208⋅10−5151.591.60860.0186261.17150.00034694Mean0.0166331.53160.00040314Max0.0389215.40920.0015149As the result, the approximate formula of CR is as follows:(31)CR=CIRI=λmax−nn−1RI=λmax−nn−10.00311n3−0.045332n2+0.533262n−0.615824.By using (30) and (31), we can directly calculate the RI value and CR value without the numerical table for RI values according to the number of attributes and evaluate the consistency of the constituted PM rapidly.On the other hand, we test the performance of the following previous approximate formula for RI: [4].(32)RI=1.98n−2n.Table7 shows the performance test result of the previous approximate formula (32).From Tables6 and 7, we know that the proposed third-order approximate polynomial (30) has much better performance than the previous approximation formula (32).Table 7
Performance test result of the previous approximate formula for RI (equation (23)).
nRICalculated RIAbsolute errorRelative error (%)Squared error30.580.660.0813.7930.006440.90.990.09100.008151.121.1880.0686.07140.00462461.241.320.086.45160.006471.321.41430.0942867.14290.008889881.411.4850.0755.31910.00562591.451.540.096.20690.0081101.491.5840.0946.30870.008836111.511.620.117.28480.0121121.531.650.127.84310.0144131.561.67540.115387.39640.013314141.571.69710.127148.09830.016165151.591.7160.1267.92450.015876Mean0.0976787.68010.00991Max0.1271413.7930.016165


## 3.3. Application to Hip Joint Prosthesis Material Selection

We use the proposed consistency improvement method of PM based on CR decreasing rate to hip joint prosthesis material selection [27].The alternative materials for hip joint prosthesis are, respectively, stainless steels 316 (A1), stainless steels 317 (A2), stainless steels 321 (A3), stainless steels 347 (A4), Co-Cr alloys-cast alloy (A5), Co-Cr alloys-wrought alloy (A6), unalloyed titanium (A7), Ti-6Al-4V (A8), composites (fabric reinforced)-epoxy-70% glass (A9), composites (fabric reinforced)-epoxy-63% carbon (A10), and composites (fabric reinforced)-epoxy-62% aramid (A11). The material attributes are, respectively, tissue tolerance (TT), corrosion resistance (CR), tensile strength (TS) (MPa), fatigue strength (FS) (MPa), relative toughness (RT), relative wear resistance (RWR), elastic modulus (EM) (GPa), specific gravity (SG) (g/cc), and cost (C). Table8 shows some hip joint prosthesis materials and their properties. [27].Table 8
Alternative materials and their attribute values [27].
Alternative materialTTCRTSFSRTRWREMSGCA11075173508820081A297630415108.520081.1A3976104101082007.91.1A497650430108.420081.2A51096554252102388.33.7A610989660010102429.14A7810550315781104.51.7A881098549078.31244.41.9A97768020037222.13A107756017037.5561.610A117743013037.5291.45First, we conduct the correlation analysis between the properties of hip joint prosthesis materials.Table9 shows the correlation coefficients between properties of hip joint prosthesis materials. As shown in Table 9, the elastic modulus and specific gravity have the high correlation coefficients with the tissue tolerance, fatigue strength, and relative wear resistance.Table 9
Correlation coefficients between properties of hip joint prosthesis materials.
TTCRTSFSRTRWREMSGCTT1.0000.1610.2290.7800.5300.7920.9660.961−0.479CR1.0000.5700.476−0.0080.4840.1750.101−0.127TS1.0000.7290.2870.4650.3000.268−0.157FS1.0000.6810.7920.8480.832−0.492RT1.0000.2510.6240.688−0.624RWR1.0000.8150.744−0.129EM1.0000.985−0.472SG1.000−0.576C1.000In this table, the bold values indicate higher correlation coefficients (ñ>0.8) than others.Therefore, we remove two properties such as EM (elastic modulus) and SG (specific gravity) from the consideration.Consequently, we constitute the decision matrix with seven attributes: TT, CR, TS, FS, RT, RWR, and C. Table10 shows the decision matrix.Table 10
Decision matrix.
Alternative materialsTTCRTSFSRTRWRCA1107517350881A297630415108.51.1A3976104101081.1A497650430108.41.2A51096554252103.7A610989660010104A7810550315781.7A881098549078.31.9A977680200373A107756017037.510A117743013037.55As shown in Table10, there are 11 alternative materials with 7 attributes. Excepting C (cost), six attributes are benefit attributes among seven attributes, while C (cost) is cost attribute.The material attribute weights are determined by using the AHP with simplest questionnaire by three decision makers. Table11 shows the simplest pairwise comparison questionnaire completed by the first decision maker.Table 11
Simplest pairwise comparison questionnaire completed by the first decision maker.
Tissue toleranceCorrosion resistanceTensile strengthFatigue strengthRelative toughnessRelative wear resistanceCostTissue tolerance1155355Corrosion resistance155355Tensile strength15535Fatigue strength1335Relative toughness135Relative wear resistance13Cost1It indicates that the value is the maximum. I have no objection to remove the Bold type.From Table11, the first decision maker’s PM is as follows:(33)A1=115535511553551515115151315151551133513135313515153131313151551515131.The consistency test result of the PMA(1) is as follows:(34)λmax=7.913828,CI=0.152305,CR=0115382>01.Therefore, we should modify the PMA(1) until CR < 0.1 using the proposed consistency improvement method.In the first iteration, we first reconstitute the PM by replacing all elements ofA(1) with the lower neighbouring scales and calculate CR values and CR decreasing rates (Table 12).Table 12
CR decreasing rates by replacing all elements with the lower neighbouring scales.
dCRrpq−2345671−0.004171−0.00775670.00381980.00423790.00012511−0.00286972−0.00775670.00381980.00423790.00012511−0.00286973−0.00229490.00080871−0.0031613−0.0102734−0.00776960.00616790.006202950.000354390.001863460.0077709It indicates that the value is the maximum. I have no objection to remove the Bold type.From Table12, the maximum CR decreasing rate is 0.0077709 (CR = 0.10632) when we replace the element a67 = 3 in the primary PM with the lower neighbouring scale 2.We next reconstitute the PM by replacing all elements ofA(1) with the upper neighbouring scales and calculate the CR values and CR decreasing rates (Table 13).Table 13
CR decreasing rates by replacing all elements with the upper neighbouring scales.
dCRrpq+23451−0.0041710.0051298−0.004292−0.0058214−0.0010120.001327620.0051298−0.004292−0.0058214−0.0010120.001327630.0014299−0.00219980.00073210.01037140.0063837−0.0075791−0.00650445−0.0025883−0.00258986−0.0090527It indicates that the value is the maximum. I have no objection to remove the Bold type.From Table13, the maximum CR decreasing rate is 0.010371 (CR = 0.10449) when we replace the element a37 = 1/5 in the primary PM with the upper neighbouring scale 1/4.As 0.0077709 < 0.010371, we decide to modify the elementa37 = 1/5 of the primary PM to the upper neighbouring scale 1/4. The reconstituted PM is as follows:(35)B11=115535511553551515115151314151551133513135313515153131313151541515131,CI=010449>01.As CR = 0.10449 > 0.1, we should modify the PMB11 until CR < 0.1 again.In the next iteration, we first reconstitute the PMB11 by replacing all elements with the lower neighbouring scales and calculate the CR values and CR decreasing rates (Table 14).Table 14
CR decreasing rates by replacing all elements with the lower neighbouring scales.
dCRrpq−2345671−0.0042129−0.0072570.00379170.00423495.7868e − 05−0.00326522−0.0072570.00379170.00423495.7868e − 05−0.00326523−0.00263550.00050903−0.0036426−0.0103714−0.00782550.00626760.005737150.000316940.001497560.0071198From Table14, the maximum CR decreasing rate is 0.0071198 (CR = 0.096187) when we replace the element a67 = 3 with the lower neighbouring scale 2.We next reconstitute the PM by replacing all elements with the upper neighbouring scales and calculate the CR values and CR decreasing rates (Table15).Table 15
CR decreasing rates by replacing all elements with the upper neighbouring scales.
dCRrpq+2345671−0.00421290.0047588−0.0042749−0.0058397−0.000965720.001654920.0047588−0.0042749−0.0058397−0.000965720.001654930.0018419−0.00181350.00137290.01014240.0064268−0.0076851−0.00607285−0.0025853−0.00226066−0.0084573From Table15, the maximum CR decreasing rate is 0.010142 (CR = 0.093506) when we replace the element a37 = 1/4 with the upper neighbouring scale 1/3. As 0.0071198 < 0.010142, we decide to modify the element a37 = 1/4 to the upper neighbouring scale 1/3. The reconstituted PM is as follows:(36)B1=115535511553551515115151313151551133513135313515153131313151531515131,CR1=0.093506.<0.1.As CR = 0.093506 < 0.1, the reconstituted PMB(1) satisfies the consistency.By the similar way, the reconstituted PMs based on the PMs constituted by the second and third decision makers are as follows:(37)B2=11/375355315535717171131511151531133513135313515151131313151711515131,CR2=0.071796<0.1,B3=13755751317557517171131511315153113331515531531717113151115153131311,CR3=0.076189<0.1.Based on the CR values {0.093506, 0.071796, 0.076189}, three decision makers’ levels are determined as follows:(38)h1=0.28331,h2=0.36898,h3=0.34771.By using (31), the composite PM is as follows:(39)B0=109769636355358315620651023615620653583156206566090157150177921028842020732530499950202346721033333341863027909027909531358314186301779201779213651033333027909120475020176652.00020.238870.238870.48841.By transforming the elements ofB0=bij0n×n to the nearest neighbouring 9-point scales, the final PM is as follows:(40)A0=116546511654661616114151212151541133414145314416162131412151621414121.The CR value of the final PMA(0) is CR(0) = 0.064623, and therefore the final PM satisfies the consistency.The final attribute weights calculated from the final PMA(0) are as follows:(41)0.31665,0.32205,0.031323,0.093804,0.14673,0.048864,0.040575.With these attribute weights, we can select the best hip joint prosthesis material using the well-known three MADMs such as SAW, TOPSIS, and GRA.Table16 and Figure 4 show the comprehensive scores of the alternatives from three MADMs.Table 16
Comprehensive scores of the alternatives from three MADMs.
MADMsAlternative materialsA1A2A3A4A5A6A7A8A9A10A11SAW0.5320.4910.4800.4930.6800.8740.6170.6800.0780.0420.049TOPSIS0.5080.4390.4370.4390.6770.7970.6080.6210.0800.0450.059GRA0.6400.5740.5700.5750.7010.8470.6520.6940.3600.3440.348Figure 4
Comprehensive scores of the alternatives from three MADMs.Table17 shows the ranking of the comprehensive scores of the alternatives from three MADMs.Table 17
Ranking of the alternatives using three MADMs.
MADMsAlternative materialsA1A2A3A4A5A6A7A8A9A10A11SAW5786214391110TOPSIS5786214391110GRA5786214391110Final ranks5786214391110From Table17, we know that the final ranking of the alternative materials is as follows:(42)A6>A5>A8>A7>A1>A4>A2>A3>A9>A11>A10.Therefore, we can select A6 (Co-Cr alloys-wrought alloy) as the best hip joint prosthesis material, and the next are A5 (Co-Cr alloys-cast alloy), A8 (Ti-6Al-4V), A7 (unalloyed titanium), and so on.


## 4. Conclusions

We proposed a new consistency improvement method of inconsistent PM based on CR decreasing rate, developed an approximate polynomial for RI according to the number of attributes, and proposed a method to determine final PM and final attribute weights considered decision makers’ levels. We compared the performances of the proposed and previous consistency improvement methods with two numerical examples, and then applied the proposed methods to determine material attribute weights in hip joint prosthesis material selection.The main conclusions are as follows:(i)
The CR decreasing rate-based consistency improvement method of PM improves the consistency better with much smaller modification amounts and guarantees a very good balance between improvement of consistency and preservation of primary information.(ii)
The approximate polynomial for RI enables to calculate the RI value according to the number of attributes without the numerical table for RI.(iii)
The proposed method for constituting the final PM and determining the final attribute weights enables to reflect the decision makers’ levels well, and it is useful for group AHP.The limitation of this work is that we do not deal with the uncertainty of the pairwise comparison judgments. To handle the fuzziness and vagueness effectively, elicit the decision makers’ knowledge and develop more effective decision-making model; future work needs to introduce the hesitant fuzzy set, complex intuitionistic fuzzy set, and so on.



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