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

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.


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 di er according to the attribute weights. Analytic hierarchy process (AHP) is wellknown attribute weighting method [1]. AHP determines the attribute weights based on pairwise comparison evaluation data for the pairs of attributes. e 9-point scales are used to transform the decision maker's judgments into numerical quantities [2]. e essential feature of AHP is the pairwise comparisons between the attributes instead of the direct allocation of the weights [3]. e 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 agrowaste. Zhong et al. [7] constructed a cost evaluation system with 5 indices in the rst 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. ey 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][17][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. e 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 a ij of the inconsistent PM A is replaced by b ij � a ij α (w i /w j ) 1−α , where w � ((w 1 , . . . , w i , . . . , w n ) T is the weight vector obtained from A. e generated matrix B � [b ij ] has a reduced CR. is process is repeated until the consistency is satisfied. Cao et al. [24] proposed a heuristic method to modify inconsistent PM. ey decomposed the primary PM as the Hadamard product of the consistent PM and a reciprocal deviation matrix. ey constituted a modified PM by convex combination of the reciprocal zero deviation matrices. ey 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) e 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) e elements of the inconsistent PM should be replaced with the lower or upper neighbouring 9point 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. erefore, 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. e 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. e 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. is 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) e 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.
e rest of this paper is organized as follows: In Section 2, 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.

CR Decreasing Rate-Based Consistency Improvement
Method of Inconsistent PM. Let CR(A) be the CR value of the PM A � (a ij ) n×n . e main steps of a new consistency improvement method are as follows: Step 1. For each element a lm (l � 1, n − 1, m � l + 1, n) in the upper triangular matrix of the PM A � (a ij ) n×n , reconstitute the PM A − lm � (a − ij ) n×n by replacing the element a lm with the lower neighbouring scale h − lm , and then calculate its CR value CR(A − lm ) and CR decreasing rate dC Rr − lm as follows: where (2) e lower and upper neighbouring scales of the 9-point scales {1/9, 1/8, . . ., 8, 9} are shown in Table 1.
Step 2. Find the maximum value of the CR decreasing rate dC Rr − r 1 s 1 from {dC Rr − lm ; l � 1, n − 1, m � l + 1, n} as follows: Step 3. For each element a lm (l � 1, n − 1, m � l + 1, n) in the upper triangular matrix of the PM A � (a ij ) n×n , reconstitute the PM A + lm � (a + ij ) n×n by replacing the element a lm with the upper neighbouring scale h + lm , and then calculate its CR value CR(A + lm ) and CR decreasing rate dC Rr + lm as follows: where Step 4. Find the maximum value of the CR decreasing rate dC Rr + r 2 s 2 from {dC Rr + lm ; l � 1, n − 1, m � l + 1, n} as follows: Step 5. If dC Rr − r 1 s 1 ≥ dC Rr + r 2 s 2 , then reconstitute the PM B � (b ij ) n×n by replacing the element a r 1 s 1 with the Mathematical Problems in Engineering If dC Rr − r 1 s 1 < dC Rr + r 2 s 2 , then reconstitute the PM B � (b ij ) n×n by replacing the element a r 2 s 2 with the upper Step 6. Calculate the CR value CR(B) of the reconstituted PM B � (b ij ) n×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 Step 9. Calculate the attribute weights (w 1 , . . . , w j , . . . , w n ) T by normalizing (v � v 1 , . . . , v j , . . . , v n ) T as follows: We call this method CR decreasing rate-based method. Let A � (a ij ) n×n and B � (b ij ) 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:  (A,B) is, the better the reconstituted PM preserves the information of the primary PM. e 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.

Development Method of Approximate Polynomial for RI
According to the Number of Attributes. We develop m-th order approximate polynomial for RI as the following form: with the data set {(n, RI n ), n � 3, 15.}, where n is the number of attributes and RI n is the corresponding RI value ( e main steps to develop the approximate polynomial for RI value according to the number of attributes are as follows: Step 1. Develop six approximate polynomials RI � RI m (n); m � 1, 6 with the data {(n, RI n ), n � 3, 15.} ( Table 2).
Step 3. Select the suitable approximate polynomial RI � RI r (n) with acceptable MAE, MRE, and MSE from among six approximate polynomials RI � RI m (n); m � 1, 6.

Constituting Method of Final PM Constituting Method and Final Attribute Weighting Method Considered Decision
Makers' Levels. Let M be the number of the decision makers.
e 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 by M decision makers and constitute M PMs ij ) n×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 as Commonly, the greater the CR value is, the worse the consistency of the PM is. e worse the consistency of the PM is, the more mistakes the decision maker's judgment has. erefore, we can regard that the inverse value of the CR value of the PM reflects the decision maker's level. is is why we assign the normalized inverse values of CR values to the decision makers' levels.
Step 7. Constitute the composite PM e elements of B (0) � (b (0) ij ) n×n may be no 9-point scales.
Step 8. Constitute the final PM ij ) n×n to the nearest neighbouring 9-point scales.
e final PM A (0) � (a (0) ij ) n×n reflects not only the decision makers' pairwise judgments but also their levels.
Step 9. Determine the final attribute weights Figure 1 shows the flowchart of the proposed methods.

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.
e primary PM [21,22] is as follows: e reconstituted PM using the proposed method is as follows (CR 0 � 0.1.): e reconstituted PMs using the proposed method are as follows: In case of CR 0 � 0.01, In case of CR 0 � 0.005, In case of CR 0 � 0.001,  As can be seen in Tables 3 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.

Approximate Polynomial for Random Consistency Index (RI) According to the Number of Attributes.
We develop approximate polynomial for RI according to the number of attributes by using the proposed method in Section 3.2. Table 5 and Figures 2 and 3 show the performance test results of approximate polynomials for RI with different orders.
From Table 5 and Figures 2 and 3, we can know that third-order approximate polynomial is appropriate from the viewpoints of the performance and complexity.
(30) Table 6 shows the performance test result of the above third-order approximate polynomial.
As the result, the approximate formula of CR is as follows: 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) Table 7 shows the performance test result of the previous approximate formula (32).
From Tables 6 and 7, we know that the proposed thirdorder approximate polynomial (30) has much better performance than the previous approximation formula (32).
First, we conduct the correlation analysis between the properties of hip joint prosthesis materials. Table 9 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. erefore, 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. Table 10 shows the decision matrix.       As shown in Table 10, 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.
e material attribute weights are determined by using the AHP with simplest questionnaire by three decision makers. Table 11 shows the simplest pairwise comparison questionnaire completed by the first decision maker.

Alternative material
From Table 12, the maximum CR decreasing rate is 0.0077709 (CR � 0.10632) when we replace the element a 67 � 3 in the primary PM with the lower neighbouring scale 2.
We next reconstitute the PM by replacing all elements of A (1) with the upper neighbouring scales and calculate the CR values and CR decreasing rates (Table 13).
From Table 13, the maximum CR decreasing rate is 0.010371 (CR � 0.10449) when we replace the element a 37 � 1/5 in the primary PM with the upper neighbouring scale 1/4. As 0.0077709 < 0.010371, we decide to modify the element a 37 � 1/5 of the primary PM to the upper neighbouring scale 1/4. e reconstituted PM is as follows: As CR � 0.10449 > 0.1, we should modify the PM B (1) 1 until CR < 0.1 again.
In the next iteration, we first reconstitute the PM B (1) 1 by replacing all elements with the lower neighbouring scales and calculate the CR values and CR decreasing rates (Table 14).
From Table 14, the maximum CR decreasing rate is 0.0071198 (CR � 0.096187) when we replace the element a 67 � 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 (Table 15).
From Table 15, the maximum CR decreasing rate is 0.010142 (CR � 0.093506) when we replace the element a 37 � 1/4 with the upper neighbouring scale 1/3. As 0.0071198 < 0.010142, we decide to modify the element a 37 � 1/4 to the upper neighbouring scale 1/3. e reconstituted PM is as follows: As CR � 0.093506 < 0.1, the reconstituted PM B (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:  Tissue tolerance  1  1  5  5  3  5  5  Corrosion resistance  1  5  5  3  5  5  Tensile strength  1  5  5  3  5  Fatigue strength  1  3  3  5  Relative toughness  1  3  5  Relative wear  resistance  1  3 Cost 1 It indicates that the value is the maximum. I have no objection to remove the Bold type.
14 Mathematical Problems in Engineering By transforming the elements of B (0) � (b (0) ij ) n×n to the nearest neighbouring 9-point scales, the final PM is as follows: 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. Table 16 and Figure 4 show the comprehensive scores of the alternatives from three MADMs. Table 17 shows the ranking of the comprehensive scores of the alternatives from three MADMs.
From Table 17, we know that the final ranking of the alternative materials is as follows: erefore, 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.

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. e main conclusions are as follows: (i) e 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) e approximate polynomial for RI enables to calculate the RI value according to the number of attributes without the numerical table for RI. (iii) e 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.
e 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.

MADMs
Alternative materials  A1  A2  A3  A4  A5  A6  A7  A8  A9  A10  A11  SAW  5  7  8  6  2  1  4  3  9  11  10  TOPSIS  5  7  8  6  2  1  4  3  9  11  10  GRA  5  7  8  6  2  1  4  3  9  11  10  Final ranks  5  7  8  6  2  1  4  3  9  11 10 First row First Coloumn  e main steps for determining the attribute weights using AHP with simplest questionnaire are as follows: [5] Step 1. Record the pairwise comparison judgment values for the importance of the attributes in the simplest questionnaire. In the simplest questionnaire, the attributes C 1 , . . ., C j , . . ., C n are listed in the same order at the first row and first column, respectively. e cell with slash "\" () is called recording cell. Figure 5 shows the recording method of the pairwise comparison judgment value. e recording cell cell(i, j) corresponds to i-th attribute C i at the first column and j-th attribute C j at the first row (i � 1, n − 1, j � i + 1, n). Compare i-th attribute C i at the first column and j-th attribute C j at the first row facing the recording cell cell(i,j). If i-th attribute C i at the first column is more important over j-th attribute C j at the first row, then record the judgment value a in the left area of the recording cell as . If j-th attribute C j at the first row is more important over i-th attribute C i at the first column, then record the judgment value a in the right area of the recording cell as . If two attributes C i and C j have an equal importance, then record the number 1 in the arbitrary area. Table 18shows the recording value according to the verbal judgment. By the similar way, record all the recording cells in the simplest questionnaire.
Step 2. Constitute the PM A � (a ij ) n×n based on the complete questionnaire. e recording cell cell(i,j) corresponds to the element a ij of the upper triangular matrix in the PM A. If the state of cell(i,j) is , then a ij is a (a ij � a). If the state of cell(i, j) is , then a ij is 1/a (a ij � 1/a). From the states of all the recording cells, all the elements of the upper triangular matrix are determined in A. e lower triangular elements are always the positive reciprocal of the upper triangular elements (a ji � 1/a ij ). All the principal diagonal elements are equal to 1. (a ii � 1; i � 1, n) Step 3. Determine the principal eigenvector v � (v 1 , . . . , v j , . . . , v n ) T of A � (a ij ) n×n using the eigenvector method. e principal eigenvector v � (v 1 , . . . , v j , . . . , v n ) T can be determined from the following equation: where λ max is the maximum eigenvalue of A and v is the corresponding eigenvector.
Step 4. Test the consistency of A using the consistency ratio (CR): where CI is the consistency index of A. It is calculated as follows: RI is the random consistency index (RI) ( Table 2). If CR < CR 0 , then the PM satisfies the consistency. Otherwise, the PM should be modified. CR 0 is the consistency threshold value. Usually, CR 0 is 0.1.
Step 5. Calculate the attribute weights w 1 . . ., w j . . ., w n by normalizing the principal eigenvector v � (v 1 , . . . , v j , . . . , v n ) T of the consistent PM A as follows: A.2. Development method of approximate polynomial by least square method. To develop approximate polynomial P m (x) � a 0 + a 1 x + . . . + a m−1 x m−1 + a m x m with a set of data (x n , y n ), n � l, 2, . . ., N, least square method could be used [29,30]. It requires to determine the coefficients a 0 , a 1 , . . ., a m−1 , a m to minimize the following function: e error may be reduced by increasing the polynomial order. However, for the high-order polynomial, the curve shows oscillatory behaviour. erefore, the low-order polynomial with acceptable error is used commonly