Haiqing Zhang

Aicha Sekhari
email: aicha.sekhari@univ-lyon2.fr

Yacine Ouzrout
email: yacine.ouzrout@univ-lyon2.fr

Abdelaziz Bouras
email: abdelaziz.bouras@qu.edu.qa

Deriving consistent pairwise comparison matrices in decision making methodologies based on linear programming method

Keywords: AHP, Fuzzy AHP, Inconsistency pairwise comparison matrix, Preserve information, Decision Making

Pairwise comparison matrix (PCM) with crisp or fuzzy elements should satisfy consistency requirements when it is used in analytic hierarchy process (AHP) or in fuzzy AHP methodologies. An algorithm has been presented to obtain a new modified consistent PCM for the corresponding inconsistent original one. The algorithm sets a linear programming problem based on all of the constraints. To obtain the optimum eigenvector of the middle value of the new PCM, segment tree is used to gradually approach the greatest lower bound of distance with the original PCM. As to obtain the lower value and upper value of the new PCM, a theory is proposed to reduce adding uncertainty factors and could maximum maintain the similarity with original PCM. The experiments for crisp elements show that the proposed approach can preserve more the original information than references. The experiments for fuzzy elements show that our method can effectively reduce inconsistency and obtain suitable modified fuzzy PCMs.

Introduction

Analytic Hierarchy Process (AHP) [START_REF] Saaty | The Analytic Hierarchy Process[END_REF][START_REF] Hq | Selection of optimal Product Lifecycle Management components based on AHP Methodologies[END_REF][START_REF] Hq | PLM Components Selection Based on a Maturity Assessment and AHP Methodology[END_REF] and Fuzzy AHP [START_REF] Torfi | Fuzzy AHP to determine the relative weights of evaluation criteria and Fuzzy TOPSIS to rank the alternatives[END_REF][START_REF] Yang | Prioritization of environmental issues in offshore oil and gas operations: A hybrid approach using fuzzy inference system and fuzzy analytic hierarchy process[END_REF] are multi criteria decisionmaking methodologies which are widely used in many real problems. The AHP expresses the relative importance of criteria by pairwise comparisons and converts the values of pairwise comparisons to priorities. However, AHP method has the inability to handle uncertain and imprecise situations due to the difficulty of mapping decision-makers' preferences into crisp value [START_REF] Deng | Multicriteria analysis with fuzzy pairwise comparison[END_REF]. Fuzzy sets theory can appropriately handle uncertainty and imprecision issues [31, 7, and 8]. It can better describe vague demands or capacities with tolerance than probability distributions. It is also a sophisticated technique to solve real problems when data is subject to imprecision. Fuzzy AHP methodology [START_REF] Torfi | Fuzzy AHP to determine the relative weights of evaluation criteria and Fuzzy TOPSIS to rank the alternatives[END_REF][START_REF] Yang | Prioritization of environmental issues in offshore oil and gas operations: A hybrid approach using fuzzy inference system and fuzzy analytic hierarchy process[END_REF] adopts triangular (or trapezoidal) membership functions to define fuzzy sets of decision-makers' preferences. It derives the fuzzy priorities of criteria from pairwise comparisons matrix with triangular (or trapezoidal) fuzzy elements to tackle the uncertainty and inaccurate issues in multicriteria decision-making process. To make sure the priorities of each criterion are accurate and sensible in either AHP or Fuzzy AHP, the consistency of Pairwise Comparison Matrix (PCM) with crisp or fuzzy elements must be achieved. For the PCMs which fail the consistency test, the decision-makers must redo the ratios. To be able to facile employ AHP or Fuzzy AHP, a strategy should be proposed to repair the inconsistency information of PCMs.

Some works [START_REF] Karapetrovic | A quality control approach to consistency paradoxes in AHP[END_REF][START_REF] Xu | A consistency improving method in the analytic hierarchy process[END_REF][START_REF] Cao | Modifying Inconsistent Comparison Matrix in Analytic Hierarchy Process: A Heuristic Approach[END_REF][START_REF] Chiclana | Cardinal Consistency of Reciprocal Preference Relations: A Characterization of Multiplicative Transitivity[END_REF] focus on studying inconsistency issues with crisp numbers. Karapetrovic and Rosenbloom [START_REF] Karapetrovic | A quality control approach to consistency paradoxes in AHP[END_REF] revise the single entry of a ratio's value till consistency of relative PCM (pairwise comparison matrix) matrix in an acceptable level. Xu and Wei [START_REF] Xu | A consistency improving method in the analytic hierarchy process[END_REF] preserve the initial ratios' value in PCM (pairwise comparison matrix) while obtaining satisfied consistency requirements. Cao et al. [START_REF] Cao | Modifying Inconsistent Comparison Matrix in Analytic Hierarchy Process: A Heuristic Approach[END_REF] develop a heuristic approach which can preserve more original comparison information compared with Xu and Wei [START_REF] Xu | A consistency improving method in the analytic hierarchy process[END_REF]. Chiclana et al. [START_REF] Chiclana | Cardinal Consistency of Reciprocal Preference Relations: A Characterization of Multiplicative Transitivity[END_REF] present functional equation to evaluate cardinal consistency of reciprocal preference relations and provide results toward the construction of consistent reciprocal preference relations. However, for these three works, the computing times are largely increased and the information of original matrix cannot be well preserved when the consistency requirement is increased. To find out an approximate PCM which is consistent and closest to the optimal one, we need to minimize the distance between inconsistent PCMs and their corresponding consistent PCMs.

Some works [6, 18, 25, and 36] are used to solve the inconsistency issues with fuzzy elements. Xu and Wang [START_REF] Xu | Eigenvector Method, Consistency Test and Inconsistency Repairing for an Incomplete Fuzzy Preference Relation[END_REF] repaire incomplete inconsistency fuzzy preference relation by finding out the unusual and false element until the consistency ratio was at a satisfied level. Leung and Cao [START_REF] Leung | On consistency and ranking of alternatives in fuzzy AHP[END_REF] propose a new definition of Fuzzy positive reciprocal matrix by setting deviation tolerances based on an idea of allowing inconsistent information. Zadeh and Bafandeh [START_REF] Zadeh | A New Method for Consistency Test in Fuzzy AHP[END_REF] further discuss Leung and Cao's work and proposed a new method of fuzzy consistency test by direct fuzzification of QR (Quick Response) algorithm which is one of the methods for the eigenvalues calculating of an arbitrary matrix. Cabrerizo et al. [START_REF] Cabrerizo | Managing the consensus in group decision making in an unbalanced fuzzy linguistic context with incomplete information[END_REF] adopt consistency measures in an iterative procedure to estimate the incomplete information, and the proposed consensus model supports the management of incomplete unbalanced fuzzy linguistic information. However, these works do not have standard parameters to criticize the reliability of their theory so far. Therefore, it is very important to prove the feasibility of a methodology that can reduce the inconsistency of the original matrix and preserve the original matrix's information as much as possible.

An algorithm is proposed to obtain the new modified PCM based on eigenvector method and linear programming method. The modified PCM is considered as a combination of the original inconsistent PCM and an adjustable consistent PCM. The algorithm aims to obtain the middle value, upper value, and lower value of adjustable matrix. The strategy is to find the optimum priority vector by solving a linear programming problem, and use eigenvector method to obtain the adjustable matrix based on the optimum priority vector. This paper is structured in the following way. Section 2 gives the basic concepts of the PCM with fuzzy and crisp elements, consistency indices, and parameters to judge the effectiveness of modified matrix. Section 3 gives the detail of the proposed algorithm. Section 4 and section 5 apply the algorithm to obtain the modified PCMs with crisp and fuzzy elements. Section 6 concludes our work.

Preliminaries

Pairwise Comparison matrixes with triangular fuzzy elements

Triangular fuzzy numbers are special cases of trapezoidal fuzzy numbers, which are widely used in fuzzy applications [START_REF] Torfi | Fuzzy AHP to determine the relative weights of evaluation criteria and Fuzzy TOPSIS to rank the alternatives[END_REF]. Triangular fuzzy number allows decision-makers to provide preferences in a fuzzy range that can keep certain unknown ratio [START_REF] Ramík | Inconsistency of Pairwise Comparison Matrix with Fuzzy Elements Based on Geometric Mean[END_REF]. Our work studies show how to obtain a consistent pairwise comparison matrix with triangular fuzzy elements when given an inconsistent one.

The pairwise comparison matrix with triangular fuzzy elements can be described in the following:

(2. Where, the operator is one of the operation rules of triangular fuzzy elements. This operation can be calculated by the following equation:

× m n A ! 11 11 11 1 1 1 1 1 1 ( , , ) ( , , ) A ( , , ) ( , ,
)

L M U L M U n n n L M U L M U m m m m n m n m n a a a a a a a a a a a a ! " # $ # % ae ö ç ÷ = ç ÷ ç ÷ è ø ! = ( , , ) L M U ij ij ij ij a a a a L ij a M ij a U ij a ! = ( , , ) L M U ij ij ij ij a a a a ! 1 1 1 = ( , , ) ji U M L ij ij ij a a a a A ! 1/ < < < < L M U ij ij ij σ a a a σ = [1/ , ], > 1 S σ σ σ A ! = = L M U ij ij ij a a a ! A [ ] ij n n a " = ! ! ! ik kj ij a a a " » ! When positive reciprocal matrix is crisp numbers A, consistent condition is ,{∀i, j, k|1≤i, j, k ≤n}.
However, this definition is too strict because it is unrealistic to reach perfect consistency of a PCM (crisp or fuzzy elements). Some works [START_REF] Aguar`on | The geometric consistency index: Approximated threshold[END_REF][START_REF] Gass | Singular value decomposition in AHP[END_REF][START_REF] Stein | The harmonic consistency index for the analytic hierarchy process[END_REF][START_REF] Saaty | Multi-criteria Decision Making-The Analytical Hierarchy Process[END_REF][START_REF] Leung | On consistency and ranking of alternatives in fuzzy AHP[END_REF][START_REF] Ramík | Inconsistency of Pairwise Comparison Matrix with Fuzzy Elements Based on Geometric Mean[END_REF][START_REF] Dubois | The role of fuzzy sets in decision sciences: Old techniques and new directions[END_REF][START_REF] Brunelli | A note on the article "Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean[END_REF] have developed consistency indices to accept a certain level of acceptable deviations. We will study consistency indices to decide whether the current PCM should be in an acceptable consistency level or not.

Consistency Indices for pairwise comparison matrix

Several consistency indices have been proposed for crisp numbers. For example: Geometric Consistency Index [START_REF] Aguar`on | The geometric consistency index: Approximated threshold[END_REF], singular value decomposition method [START_REF] Gass | Singular value decomposition in AHP[END_REF], harmonic consistency index [START_REF] Stein | The harmonic consistency index for the analytic hierarchy process[END_REF], etc. A large majority of indices have a good internal agreement, whereas few indices are outliers which have weaker agreement when compared with the other indices. Different consistency indices are different definition of consistency degree and there is still no evidence to show that which proposal is the best one. We use the most popular and oldest index of Satty'CR [START_REF] Saaty | Multi-criteria Decision Making-The Analytical Hierarchy Process[END_REF] to measure the consistency of crisp numbers.

The consistency index of CR [START_REF] Saaty | Multi-criteria Decision Making-The Analytical Hierarchy Process[END_REF] is defined as:

(2.2)
Where λmax is the principle eigenvalue of A. RI is random index which can be gotten by searching defined table. When the value of 0<CR<0.1, the consistency can be accepted.

Several important works focus on the consistency of pairwise comparison matrix with fuzzy elements. The first one is Leung and Cao [START_REF] Leung | On consistency and ranking of alternatives in fuzzy AHP[END_REF], who proposed a notion with consideration of a tolerance deviation. However, the notion is strongly related to Satty's CR and it has shortcomings to calculate consistency of pairwise comparison matrix with fuzzy elements [START_REF] Zadeh | A New Method for Consistency Test in Fuzzy AHP[END_REF]. The second one is Ramik and Jaroslav [START_REF] Ramík | Inconsistency of Pairwise Comparison Matrix with Fuzzy Elements Based on Geometric Mean[END_REF]'s work, which proposes a new consistency index NI to examine fuzzy elements based on the distance of the matrix to a special ratio matrix, and compare the properties with CR. This work has been further studied and used by several important works [START_REF] Aguar`on | The geometric consistency index: Approximated threshold[END_REF][START_REF] Dubois | The role of fuzzy sets in decision sciences: Old techniques and new directions[END_REF][START_REF] Brunelli | A note on the article "Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean[END_REF]. We have tested this work's performance which indicated that it can satisfy reasonable results with fuzzy elements, although it has some shortcomings [START_REF] Brunelli | A note on the article "Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean[END_REF].

The consistency index of NI [START_REF] Ramík | Inconsistency of Pairwise Comparison Matrix with Fuzzy Elements Based on Geometric Mean[END_REF] is defined as:

(2.3)
Where, is a normalization factor, the value of can be obtained from .

The third and the forth work are [START_REF] Bulut | Use of consistency index, decision prioritization and direct numerical inputs for generic Fuzzy-AHP modeling: A process model for shipping asset management[END_REF][START_REF] Dopazo | A Parametric Model for Determining Consensus Priority Vectors from Fuzzy Comparison Matrices[END_REF]'s study, they extend GCI (Geometric Consistency Index) [START_REF] Aguarón | The geometric consistency index: Approximated thresholds[END_REF] to CCI (centric consistency index) index to deal with PCM with triangle fuzzy elements. The consistency index CCI is defined as:

(2.4)
Where are the elements for fuzzy PCM, is priority vector derived by logarithmic least squares. When CCI( )=0, is considered as fully consistent. Thresholds remain identical with index GCI as CCI is a fuzzy extension of GCI. The thresholds are provided as: CCI=0.3147 for n=3, CCI=0.3526 for n=4 and CCI=0.370 for n>4 based on Aguaron's [START_REF] Aguarón | The geometric consistency index: Approximated thresholds[END_REF].

Parameters to judge the effectiveness of modified matrix

Fewer works describe the necessary parameters which can be used to measure the effectiveness of modified PCM B. Xu and Wei [START_REF] Xu | A consistency improving method in the analytic hierarchy process[END_REF] have given two parameters in the following:

(2.5) (2.6) σ and δ are used as the parameters of modificatory effectiveness. The authors [START_REF] Xu | A consistency improving method in the analytic hierarchy process[END_REF][START_REF] Cao | Modifying Inconsistent Comparison Matrix in Analytic Hierarchy Process: A Heuristic Approach[END_REF] argue that the modified matrix that preserves the most information of the original one must satisfy the condition:

. We extend these two parameters to be suitable to judge modificatory effectiveness of fuzzy elements, and the range is identical with σ and δ. σfuzzy and δfuzzy are:

! ! ! = ( , , ) ( , , ) = ( , , ) = ( , , ) L M U L M U ik kj ik ik ik kj kj kj L L M M U U L M U ij ik kj ik kj ik kj ij ij ij
a a a a a a a a a a a a a a a a a a

" " # # # » A ! ik kj ij a a a ! = max = (( -) ( -1)) (1 ) CR λ n n RI ! , = max{max{ ( ) -, ( )-,( )-} } , , ) σ σ L U L n n i j ij i j M M M U L U i j ij i j ij NI γ ω ω a ω ω a ω ω a ij n ! (1 £ £ σ n γ , , L M U i i i ω ω ω , , L M U ij ij ij a a a 2 (A) (2 ( 1)( 2)) (log(( ) 3) log( )) L M U ij ij ij L M U L M U i j i i i j j j CCI n n a a a w w w w w w < = - - + + ´- + + + + å A ( , , ) L M U ij ij ij a a a = ( ) i w w = A A , = max{ -}, , = 1,2, , ij ij i j δ b a i j n ! 2 1 1 ( ) n n ij ij i j b a n s = = = - åå 0 < < 2 & 0 < < 1 δ σ (2.7) (2.8)
Besides σ and δ, two parameters proposed by Xu and Wei [START_REF] Xu | A consistency improving method in the analytic hierarchy process[END_REF], a third parameter should be added, which is Condition of Order Preservation (COP) [START_REF] Bana E Costa | A Critical Analysis of the Eigenvalue Method Used to Derive Priorities in AHP[END_REF]. For example, suppose the original matrix A has alternatives (a1, a2, a3, a4, a5, a6, a7, a8), it has the relationship a1 dominates a2, a3 dominates a4, and the judgments indicate that the extent to which a1 dominates a2 is greater that the extent to which a3 dominates a4, then the priority vector ω should satisfy: ω(a1)> ω(a2) and ω(a3)> ω(a4) (preservation of order of preference), and ω(a1)/ω(a2)> ω(a3)/ ω(a4) (preservation of order of intensity of preference).

Deriving the optimum eigenvector by solving linear programming problem to obtain modified pairwise comparison matrix

Three conditions can guarantee a new modified consistent PCM B(or ) which has the closest and maximal similarity with the orginal inconsistent PCM A (or ):

1). The consistent value should be at an acceptable level based on different consistency indices;

2). The farthest distance between new PCM B(or ) and orginal PCM A (or ) should be as small as possible;

3). The obtained new matrix should have a strong similarity with the orginal matrix.

To suit these conditions, we provide an adjustable PCM which can reduce the inconsistency of orginal PCM as much as possible. On the basis of the adjustable matrix, we propose the definition of the modified matrix Definition 2. The new modified PCM B is defined as a combination of the original PCM A and an adjustable matrix called B ' ,which is derived from A; the modified PCM B is defined in equation (3.1):

(3.1)
For fuzzy elements, the new modified PCM is defined as:

(3.2)
The new modified matrix is constructed by two parts based on definition 2: 1). One is the original matrix. The function of this matrix is to keep the original information and make sure the two matrixes is in an acceptable distance. 2). One is the adjustable matrix. The function of this matrix is to modify the inconsistency level to make sure the new modified matrix's consistency is based on consistency indexes.

One way to be able to obtain the adjustable PCM is by one element being is and one element of is , by making every element of closely resemble the elements of , which is , , and . Therefore, we go to find separately. After finding out , combine them together to get the new matrix .

Deriving the middle value of the adjustable matrix

Distance analysis between original matrix and adjustable matrix

The distance methods have logarithmic least square method (LSM) [START_REF] Jong | A statistical approach to Saaty's scaling methods for priorities[END_REF], eigenvector method [START_REF] Saaty | Decision-making with the AHP: Why is the principal eigenvector necessary[END_REF][START_REF] Bana E Costa | A Critical Analysis of the Eigenvalue Method Used to Derive Priorities in AHP[END_REF], and least squares method [START_REF] Almulhim | Deriving Weights from Group Fuzzy Pairwise Comparison Judgement Matrices[END_REF]. The optimum eigenvector should be as close as possible to the original eigenvector (derived from the original matrix). The adjustable matrix should strongly resemble the original matrix once the optimum eigenvector has been determined. Therefore, eigenvector can be used to calculate adjustable PCMs.

According to references [START_REF] Saaty | Ranking by the eigenvector versus other methods in the analytic hierarchy process[END_REF][START_REF] Saaty | Decision-making with the AHP: Why is the principal eigenvector necessary[END_REF], if matrix A is consistent, then we could find positive weights ω= {ω1, ω2, …, ωn} T which can satisfy such condition [START_REF] Kou | A New Consistency Test Index for the Data in the AHP/ANP[END_REF][START_REF] Saaty | Ranking by the eigenvector versus other methods in the analytic hierarchy process[END_REF][START_REF] Saaty | Decision-making with the AHP: Why is the principal eigenvector necessary[END_REF]:

. Therefore, if matrix A is close to consistent, then it must have .The obtained positive weights ω can be used to define the expected adjustable PCM.

On the basis of this idea, we want to make the fastest distance between and are the minimum value. The fastest distance between and is

! , , = max{ -} = max{ -, - , -}, , = 1,2, , L L M M U U fuzzy ij ij ij ij ij ij ij ij i j i j δ b a b a b a b a i j n " 2 2 2 1 1 1 1 1 1 ( max{( ( ) ), ( ( ) ), ( (
) )}) n n n n n n L L M M U U fuzzy ij ij ij ij ij ij i j i j i j b a b a b a n s = = = = = = = - - - åå åå åå B A B A ' 1- ( ) ij ij B a b b b = B ' 1- ' 1- ' 1- {(( ) ( ) ), (( ) ( ) ), (( ) ( ) )} L L M M ij ij ij ij U U ij ij B a b a b a b ! ! ! b b b b b b = ' B ' ' ' ' (( ) , ( ) , ( ) ) L M U ij ij ij ij b b b b = A ! ! ( , , ) L M U ij ij ij ij a a a a = ' B A ! ' ( ) L L ij ij b a ! ' ( ) M M ij ij b a ! ' ( ) U U ij ij b a ! ' ' '
( ) ,( ) ,( )

L M U ij ij ij b b b ' ' '
( ) ,( ) ,( )

L M U ij ij ij b b b ' B , , (1, 2,
..., )

ij i j a ij n w w = " Î ( / ) 0 i j ij a w w -» / i j w w ij a / i j
w w ij a defined as:

(3.3)
The issue of finding the optimal adjustable PCM with crisp elements can be portrayed as resolving the minimum value of function f, which can be expressed in equation (

Build Linear Programming Problem by defined inequalities of the middle value of the adjustable matrix

In this section, we aim to find out the least absolute worst distance by setting a more precise priority weights range and adding slack variables. The least absolute worst distance is used to achieve the middle value of the adjustable PCM. The middle value of from adjustable matrix should have a strong relationship with . is the optimum eigenvector matrix of . can be gotten in the following equation:

(3.6)
The optimum value can be gotten when the worst proportion is closest to . Eq.(3.6) has feasible solutions which means the following equation reaches to the minimum based value on Eq.(3.5):

(3.7)

We simply write the variable by introducing an additional variable Z= . Then, the problem is:

(3.8) (3.9)
Assume the value of Z is given, and then the constraint (3.8) can be rewritten as:

ij i j G A f a w ww = = - (A, ) min(max( / , / , / )) L L L M M M ij i j ij i j U U U ij i j G a a a w w w w w w w = - - - / i j w w ij a w M ij b ¢ B¢ M ij a M ! w M ij a M ij b ¢ / M M M ij i j b w w ¢ » M ij a ( , ) min(max( / )) M M M M M ij ij i j G a a w w w = - ( , ) M M ij G a w min Z . . s t | / | , , (1, 2,
..., )

M M M ij i j a Z ij n w w - £ " Î 1/ , , 1, 2,
...,

M M i j j n s w w s £ £ = ( / ) M M M ij i j Z a Z w w -£ - £ ( ) 0 M M M ij j i a Z !w w - - £ ( ) M M M i ij j a Z ! w w £ + / ( ) M M M j i ij a Z w w ³ 1 + ( ) 0 M M M i ij j a Z w w + - £ ( ) 0 M M M j ji i a Z w w + - £ (max{( ), }) 0 (
)

M M M i ij j M ji a Z a Z w w 1 - + - £ + (max{( ), }) 0 ( ) M M M j ji i M ij a Z a Z w w 1 - + - £ + , M M i j w w min X . . s t (max{( ), }) 0 ( ) M M M i i j j M ji a Z S a Z w w 1 - + - + = + (max{( ), }) 0 ( ) M M M j j i i M i j a Z T a Z w w 1 - + - + = + S T H = + 1 1 M X w + = 1/ , , 1

Find feasible solutions to solve the linear programming problem

The aim of this section is to find the feasible solutions of ((3.18)-(3.23)) and determine the greatest lower bound of Z. The problem can be described as storing intervals of [0, Zmax], analyzing the corresponding X value, and finding the greatest lower of Z that makes X=0, which can be solved by segment tree. The segment tree is special for storing intervals. The built time is O(n log n) for n intervals, and it uses O(n log n) storage. The reason we adapted to the segment tree is because: the segments can be stored in any arbitrary manner, it can easily be adapted to counting queries, and it helps us to query the number of segments that contain a given point. The steps to solve inequalities ((3.18)-(3.23)) are as follows:

Step 1 sets the initial value. Assume the accuracy level is ξ; let the initial value of be , and ; let Zmax=Z= , and Zmin=Z=0.

Step 2 builds a segment tree by using interval [0, Z]. Let P1, P2, P3, P4 be the list of distinct interval endpoints. We separate intervals into two parts in every division and terminate this process till the value of the interval is less than the accuracy level (ξ). Then obtain the X value in problem ((3.18)-(3.23)) by setting the current Z value. If X=0, then the next Z value is equal to the lower bound of the current node. The calculation steps will end till it reaches to endpoints (P1; P2; P3; P4) based on accuracy level ξ.

Step 3 selects the greatest lower value of Z when X=0, inputs Z value to inequalities ((3.18)-(3.23)), then outputs , finally, adopts Eq.(3.6) to achieve .

Deriving the upper and lower value of the adjustable matrix

This section focuses on obtaining the value of and . The modified matrix is a combination of the adjustable matrix and original matrix based on definition 2; then the adjustable matrix should have the minimum fuzziness and maximum preservation of the original matrix's pattern. If is minimum fuzziness, then fuzziness of will mostly come from , and will be more similar with . In fact, the minimum fuzziness of could reduce uncertainty factors of . If could maximally preserve the pattern of , then the combination of and could reach to the most potential of similarity with . We propose theorem 1 to obtain the value of and based on the above theory.

Theorem 1. The optimal solution vector is: , then , i=1,2,…,n. Set CL,CU are arbitrary positive constants. Define the value of in the following formulas:

(3.24)
Where and (3.25)

1 1 M w = M w M i w M i w 1 M ii a = M i w 1/ 1 ( ) n M n ij j a = Õ 1 1 M w = ( , ) M M ij G a w ( , ) M M i j w w M ij b ¢ L ij b ¢ U ij b ¢ B¢ B A B A B¢ B B¢ A B¢ A A L ij b ¢ U ij b ¢ M i w ' ( ) M M i ij M j B w w = , L U i i w w . L M i L i C w w = 1/ 1 1 1/ 1 1 ( ) min{ , } (
)

n n M n M ij ij j j L n n L L n ij ij j j a a C a a = = = = = å Õ å Õ . U M i U i C w w =

An Algorithm to obtain Modified PCM

The adjustable matrix is derived by combining section 3.1 and 3.2. The expected modified PCM can be gotten based on definition 2. A proposed algorithm is to conclude the steps of how to obtain the modified consistency PCM. Algorithm 1: Find modified PCM

Numerical illustration and comparison with crisp numbers

First illustration

We run the experiments by software Matlab (R2009a) on a personal computer with Intel(R) Core(TM) 2.2 GHZ and 4 G RAM. First we test crisp numbers by using algorithm 1, and then compare with references. Consider the following inconsistency PCM with crisp numbers in [START_REF] Saaty | Fundamentals of decision making and priority theory with the analytic hierarchy process[END_REF]:

(4.1)
For PCM A, , CR=0.2151>0.1, the principal eigenvector is: ω= (0.3208 0.1395 0.0348 0.1285 0.2374 0.1391) T .

The value of CR is much more than 0.1. Therefore, we will adopt algorithm 1 to obtain the new consistency PCM.

Demonstration of Algorithm 1:

1/ 1 1 1/ 1 1 ( ) max{ , } (
)

n n M n M ij ij j j U n n U U n ij ij j j a a C a a = = = = = å Õ å Õ L ij b ¢ U ij b ¢ L L L ij i j b w w ¢ = U U U ij i j b w w ¢ = M i w A ! L M U i i i w w w £ £ min L M i i C w w = max U M i i C w w = , , L M U ij ij ij a a a L ij a M ij a M ij a U ij a 1/ 1 1 min 1/ 1 1 ( ) min{ , } (
)

n n M n M ij ij j j n n L L n ij ij j j a a C a a = = = = = å Õ å Õ 1/ 1 1 max 1/ 1 1 ( ) max{ , } (
)

n n M n M ij ij j j n n U U n ij ij j j a a C a a = = = = = å Õ å Õ A ! A ! B ! M i w M ij b ¢ L ij b ¢ U ij b ¢ B ! ¢ L ij b ¢ M ij b ¢ U ij b ¢ B ! M i w 1 1 1 4 3 1 3 4; 1 7 3 1; 4 5 1 1 1 1 1 1 1 A 1 ;1 5 1 1 ; 3 7 5 5 6 3 3 1 1 1 5 5 1 1 3; 1 6 3 1 3 4 3 ae ö ç ÷ ç ÷ ç ÷ = ç ÷ ç ÷ ç ÷ ç ÷ è ø max = 7.

λ

Input the initial value of , initial value of ξ, Zmax, and Zmin 1). The original matrix is , as

shown here, crisp number is a specific case of a fuzzy number in our model. Set .

2). The acceptable precise degree is ξ=0.1. Here, we adopt 0.1 as an example.

3). The initial value of Zmax= 5.7655 which is obtained by Eq.(3.7), Zmin=0, and the initial Z=5.7655. Algorithm 1 has a solution for the first Z value. To better demonstrate, we start from the second iteration.

Begin second time:

Step1. (Zmax-Zmin)> ξ; then go to 2;

Step2.

Remark 2.

1).The value of ξ is a stopping sign. If current value of ξ is less than 0.1, stop the calculation process.

2).The value of X can determine how to change current Z value. If X=0, it means there is a solution for problem (3.18)-(3.23), then save the value of vector ω and set Zmax=Z; else X=1, it means there is not a solution for problem (3.18)-(3.23), then set Zmin= lower bound of the current interval, Z= upper bound of the current interval.

3). It needs a seven-time iteration to find out the final optimum vector of inconsistency PCM A. The detailed information of each parameter is shown in Table 1. Let ωi be the obtained vector of each feasible solution. ω5 is the optimum vector for the new consistency PCM B when β=0.4 in definition 2.

The new obtained consistency PCM B is: Three consistencies of PCM B are obtained under different precise degrees. To evaluate the effective ness of obtained PCM B, CR (consistency index), δ (Eq.2.5), σ (Eq.2.6), iteration times (mark "II" in Table 2), and COP (Condition of Order Preservation) are shown in Table 2.

The consistency value of the original A is 0.2151, whereas the consistency of new obtained PCM B is around 0.03. The effective parameters δ (less than 2) and σ (less than 1) are both at an acceptable level. The aim to calculate COP is to make sure the obtained PCM could preserve of the order of preference and preserve the order of intensity of preference of original PCM A. COP tests every pair of weights and the proportion among every pairs' weights. The iteration times are less than 14.

Comparison with References

Xu and Wei [START_REF] Xu | A consistency improving method in the analytic hierarchy process[END_REF] defined the original matrix A (the elements are aij) that can be replaced by the new matrix B (the elements are bij), which is showed in following equation:

(4.2)
Where α is the positive value which is less than, but approaching to 1.

Cao et al. [START_REF] Cao | Modifying Inconsistent Comparison Matrix in Analytic Hierarchy Process: A Heuristic Approach[END_REF] proposed a equation to obtain the new matrix B, which is showed the following:

(4.3)
Where is the symbol of Hadamard product; for example, means , and is the modified deviation matrix, which is showed in the following equation:

(4.4) A ! A = ( , , ) L M U ij ij ij a a a ! = = = L M U ij ij ij a a a A
ae ö ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ è ø 1- = ( / ) α α ij ij i j b a ω ω ' = ( / ) ij i j b ω ω D ! ! A=BoC , , 1, , ij ij ij a b c i j n ! = ´ " = ' D ' ' = [ ] = ij D d γD γ DI Å (1-)
Where D is the deviation matrix, DI is a zero deviation matrix when [dij]=1. The value of γ is between 0 and 1.

α and γ has different meaning for two papers; but, the two parameters should be as close to 1 as possible. In two papers, they mentioned that α=γ=0.98 is the most suitable value to get the optimal new matrix. We compare with two references in two situations: one is the required Critical Ratio (CR) less than 0.1 (Table 3); the other one is the required Critical Ratio (CR) close to 0 (Table 3).

Algorithm 1 can reach a CR value lower than reference [START_REF] Xu | A consistency improving method in the analytic hierarchy process[END_REF] and [START_REF] Cao | Modifying Inconsistent Comparison Matrix in Analytic Hierarchy Process: A Heuristic Approach[END_REF] when CR is approaching to 0.1, and at the same time, the method could achieve lower values of δ and σ in short iterative times. We also rank the priority weight ω derived from A. The ranking results are same in two similar weights, which mean the priority weight ω which derived from our method is acceptable. The iterative times for reference [START_REF] Karapetrovic | A quality control approach to consistency paradoxes in AHP[END_REF] and [START_REF] Xu | A consistency improving method in the analytic hierarchy process[END_REF] are quite large, 820 and 1619 when CR is approaching to 0, which will cost the running time. The value of δ and σ in these two references is far away from the acceptable value, which means the priority weight is not acceptable at all; But, the iterative times of our method is less than 10 times, the value of δ is less than 4, and the value of σ is less than 2, which means our result is acceptable to some extent. On the basis of these results, we go to discover the difference of COP parameter in our method and two references. Cao et al. is closer to the original matrix's vector, but the gap among Cao, Xu, et al. and our method is very narrow (less than 1).The three methods have almost the same effectiveness in terms of the COP parameter. Thus, our method is also acceptable in this parameter.

In conclusion, Algorithm 1 can preserve more original information and obtain more consistent new matrix in short iterative times for PCMs with crisp elements.

Numerical illustration and comparison with fuzzy numbers

First analysis for pairwise comparison matrix with fuzzy elements

To judge the effectiveness of Algorithm 1 for PCMs with fuzzy elements, we analyze several parameters including the inconsistency index NI [START_REF] Ramík | Inconsistency of Pairwise Comparison Matrix with Fuzzy Elements Based on Geometric Mean[END_REF], consistency index CCI [START_REF] Dopazo | A Parametric Model for Determining Consensus Priority Vectors from Fuzzy Comparison Matrices[END_REF][START_REF] Jong | A statistical approach to Saaty's scaling methods for priorities[END_REF], δfuzzy, σfuzzy, COP, iteration times and running time. There is no evidence showing the thresholds of NI. The thresholds for CCI are provided as: CCI=0.3147 for n=3, CCI=0.3526 for n=4 and CCI=0.370 for n>4 based on Aguaron's [START_REF] Aguarón | The geometric consistency index: Approximated thresholds[END_REF]. Considering the following PCM with fuzzy elements: 4. In Table 4, the inconsistency index NI has been reduced by 0.1 and the consistency index CCI, is around 0.1, which is much less than 0.3147 and indicates the modified matrix can reach to consistent requirement; δfuzzy satisfies 0<δfuzzy<2 and σfuzzy satis-

1 1 1 1 1 1 1 1 1 (1,1,1) ( , , ) ( , , ) ( , , ) 8 7 6 6 5 4 6 5 4 (6, 7,8) (1,1,1) (4,5, 6) (2,3, 4) A= 1 1 1 1 (4,5, 6) ( , , ) (1,1,1) ( ,1,1) 6 5 4 2 1 1 1 (4,5, 6) ( , , ) (1,1, 2) (1,1,1) 4 3 2 ae ö ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ è ø
A NI(A) 0.6704 = fies 0<σfuzzy<1, which shows the obtained new matrix is within acceptable distance with ; the value of COP which is gotten from priority vector of can preserve order of preference and order of intensity preference, which present can maintain the pat-tern of (similarity). Meanwhile, algorithm 1 has high convergence speed based on the less iteration times and running time. The gap among these four matrixes is quite narrow, is the best choose when we select the smallest consistency index.

Second analysis for pairwise comparison matrix with fuzzy elements

The second analysis is developed in a company manufacturing hard disk drives for storage and home entertainment. This company mainly organizes internal storage in Ayutthaya and Pathumtani of Thailand. To better handle sustainability of a plant in Pathumtani, we define three main criteria: economic, environmental, social. Then define different sub-criteria for these three main criteria. To make a believable decision of which criteria play fundamental roles in sustainability, decision makers give the fuzzy PCMs. Algorithm 1 analyzes these PCMs and provide the closest consistency PCMs for inconsistency ones. Twenty-eight fuzzy matrixes are inconsistency matrixes, and three of them are selected in Table 4. (1 2 3 4;0.5 1 1.5 0.5; 0.3 0.5 1 2; 0.25 1.5 0.5 1) (1 2.5 4 4; 0.55 1 2 1.5; 0.5 0.8 1 2.5; 0.3 2 0.8 1) (1 3 4.5 5; 0.67 1 2.5 2; 1.5 1.2 1 3; 0.5 1.5 0.8 1) NI: 0.2599;CCI:0.7880 Algorithm 1 is adopted to obtain the modified PCMs. Table 5 displays the results. The results show the new modified PCMs have good consistency level because of all NI value less than 0.1 and all CCI value less than 0.3147. Meanwhile, the new modified PCMs are closest to original PCMs because δfuzzy and δfuzzy are in an acceptable range, and the COP value shows that the new modified PCMs preserve order of preference and order of intensity preference. Ten PCMs are studied in Figure 1. The results show that Algorithm 1 could effectively reduce the inconsistency of original PCM (reduced the value of NI and CCI) and preserve the information of original PCM. Algorithm 1 can also approach to its limits faster than [START_REF] Xu | A consistency improving method in the analytic hierarchy process[END_REF][START_REF] Cao | Modifying Inconsistent Comparison Matrix in Analytic Hierarchy Process: A Heuristic Approach[END_REF]. The maximum iteration times for Algorithm 1 is less than twenty, even when ξ=0.0001, while the maximum iteration times for [START_REF] Cao | Modifying Inconsistent Comparison Matrix in Analytic Hierarchy Process: A Heuristic Approach[END_REF] and [START_REF] Xu | A consistency improving method in the analytic hierarchy process[END_REF] are 1619 and 820 respectively. The convergence rate of Algorithm 1 has low growth rate. Figure 2 shows the results.

Conclusions

Algorithm 1 has been proposed to derive a consistent PCM with crisp or fuzzy elements from an inconsistent one. The presented approach is tested in several examples. The experiments results in Table 3 conclude the effectiveness of Algorithm 1 by comparing with Cao et al. [START_REF] Cao | Modifying Inconsistent Comparison Matrix in Analytic Hierarchy Process: A Heuristic Approach[END_REF] and Xu and Wei [START_REF] Xu | A consistency improving method in the analytic hierarchy process[END_REF] based on PCMs with crisp elements. Algorithm 1 can retain more original information and achieve lower value of δ and σ when both α [4] and γ [5] approach to 1. Tables 4 and 5 summarize the effectiveness of Algorithm 1 for PCMs with fuzzy elements. The modified PCMs with fuzzy elements can maintain original information and reach to acceptable consistency level as well.

Two effective criteria δ and σ show how the distance between modified PCM and original PCM is acceptable for both crisp and fuzzy elements. A new effective criterion COP reflects that the modified PCM resembles the original PCM (crisp and fuzzy elements). In conclusion, this approach could enhance the quality of vague and inaccurate data for decision makers, and also can better handle the inconsistency problems of AHP and Fuzzy AHP.

However, β in Algorithm 1 has a different meaning with α and γ. The true meaning of β is the proportion of the original matrix to an adjustable matrix. The approximate prefect range of β is [0.25, 0.8] based on experiments' results, and there is no clue to which value is the best one for β as the inconsistency level of PCMs are differs. In future research, it would be interesting to figure out the exact prefect range of β. Meanwhile, efforts should be made to test Algorithm 1 for more cases.

  3.4): (3.4) This equation (3.4) is extended into equation (3.5) to solve adjustable PCM with fuzzy elements: (3.5) Eq.(3.4) and Eq.(3.5) can reach the absolute minimum value (lowest point) when even the worst situation of proportion of is closest to . The idea to obtain the optimum positive eigenvector ω ( ) is to find out all possible constraints and solve feasible solutions by selecting a suitable linear programming pattern in section 3.1.2.

  Take the reciprocal of constraint (3.12); then the new constraint (3.13)-(3.15) and can be gotten. (3.13) (3.14) (3.15) Combine constraint (3.11) and constraint (3.15) and remove one of the inequalities, and then the new inequality can be gotten: (3.16) Analogously, a similar inequality can be gotten: (3.17) Next, we add slack variable S, T, and objective function H=S+T, to change the constraints (3.16) and (3.17) to equality constraints (3.18) and (3.19). Then we specify constraints, propose formulas to calculate , and add additional stopping parameter X. Then constraints correspond to the following linear programming problem:

  In constraint (3.18), we define to normalize vector . If the stopping parameter X=0, then it means constraint (3.18) has a solution, and the problem (3.18)-(3.23) has a feasible solution; the values of is the optimal solution. If the stopping parameter is X=1 which contradicts with constraint (3.21), then the problem (3.18)-(3.23) is inconsistent; then the values of is not the optimal solution. 2). The equalities (3.18)-(3.23) can be solved by Simplex algorithm [32]. The main idea of this algorithm is to walk along edges of the polytope to find out extreme points with lower and lower objective values till the minimum value is reached or an unbounded edge is visited. If the extreme point is reached, then the problem (3.18)-(3.23) have feasible solutions. 3). If Z=Z ' can make problem (3.18)-(3.23) have a feasible solution, then it must have Z≥Z ' that can also make the problem (3.18)-(3.23) have a feasible solution. The linear programming problem has the optimum solution when inequalities ((3.18)-(3.23)) reach to the greatest lower bound of Z. 4). If i=j, then , then constraint (3.18) can be always be satisfied. Analogous, constraint (3.20) is always satisfied. Then the equalities (3.18)-(3.23) can be used to find adjustable PCM B ' for original PCM A.

(4. 5 )

 5 For matrix , , CCI=0.3746. It is an inconsistent PCM with fuzzy elements. We adopt Algorithm 1 to obtain the modified new matrix under different parameters. The results are shown in Table

  ĺ =0.1, ĺ =0.01, ĺ =0.001, ĺ =0.0001 Number of Iterations

Algorithm1. Find Modified _Matrix(Zmax,

  Zmin, ξ, )

	Input: Zmax, Zmin, ξ, original matrix ,Z=Zmax
	Output: Zmax, Zmin, optimal matrix
	BEGIN
	1.	If (Zmax-Zmin<ξ)&(X=0)
	Then
	The current	is the desired optimal solution of weight
	vector.
	Adopt Eq.(3.6) to get	.
	Use Theorem1 to obtain	and	.
	Adjustable PCM	= (	,	,	).
	Obtain Modified PCM	by Definition 2.
	Stop.
	2.	Else Search segment tree
		While (Z-Zmin)> ξ
		Z=(( Zmax -Zmin)/2+ Zmin);
	Input Z into problem (3.18)-(3.23) to output the value
	of X
		If X=0
	Save the new value of	, and Zmax=Z
		Else
		Z=upper bound of the current interval;
		Zmin=lower bound of the current interval;
	END

Table 1 Parameter

 1

			values for each iterative time
	Itera-
	tive	Zmax	Zmin	Z	ξ	X ω
	time
	1	5.7655	0	5.7655	5.7655	0 ω1

Table 2

 2 Effective analysis for algorithm 1 based on five parameters

	PCM	CR	δ	σ	COP	II
	A	0.2151	0	0		-
	B(ξ=0.1)	0.0325	1.6602 0.9916	Keep	7
	B(ξ=0.01)	0.0325	1.6294 0.9868	Keep	11
	B(ξ=0.001) 0.0324	1.6291 1.0308	Keep	14

Table 3

 3 Effective analysis of Algorithm 1 by comparing with references based on parameters

	CR≤0.1

Table 4 .

 4

			Effective Analysis for algorithm 1 based on nine parameters
	β=0.5	NI	CCI	δfuzzy	σfuzzy	COP	Iteration times	Running time(seconds)
	ξ=0.1	0.5816	0.0928	1.8924	0.8656	keep	7	0.021
	ξ=0.01	0.5798	0.0928	1.8682	0.8639	keep	10	0.027
	ξ=0.001	0.5806	0.0928	1.8787	0.8628	keep	14	0.037
	β=0.6	NI	CCI	δfuzzy	σfuzzy	COP	Iteration times	Running time(seconds)
	ξ=0.1	0.5980	0.1338	1.4713	0.7185	keep	7	0.016
	ξ=0.01	0.5965	0.1338	1.4530	0.7170	keep	10	0.03
	ξ=0.001	0.5971	0.1338	1.4610	0.7161	keep	14	0.035

Table 4 Inconsistency

 4

	(1 0.35 3.3 0.42; 0.4 1 0.9 0.55;
	1.55 1.15 1 0.85; 2.4 2.25 1.5 1)
	NI: 0
	PCMs with fuzzy elements
	(1 1.5 0.5;0.67 1 0.55; 1.5 1.8 1)
	(1 2 0.67;0.7 1 0.67;1.5 1.8 1)
	(1 2.5 1;0.8 1 1;2 2 1)
	NI: 0.1635; CCI: 0.2835;
	(1 0.25 2.5 0.35;0.3 1 0.6 0.3;
	1.25 0.8 1 0.6; 2 1.8 0.9 1)
	(1 0.3 3 0.4;0.35 1 0.8 0.5;
	1.5 0.95 1 0.8; 2.2 2 1 1)

.1372; CCI: 1.7524;

Table 5

 5 Modified PCMs with fuzzy elements (1.0000 1.3784 0.8130; 0.7266 1.0000 0.6855; 1.1282 1.4545 1.0000) (1.0000 1.4460 0.8376; 0.7650 1.0000 0.7132; 1.1957 1.4833 1.0000) (1.0000 1.5290 0.9241; 0.8052 1.0000 0.7956; 1.3322 1.5474 1.0000) NI: 0.0895; CCI: 0.0235; δfuzzy:0.3130; δfuzzy:0.4270 (1 0.7163 1.5830 0.7193; 0.6418 1 0.9503 0.6326; 0.8891 0.8444 1 0.7031; 1.2491 1.3140 1.1822 1) (1 0.7707 1.6859 0.7380; 0.6599 1 1.0254 0.7135; 0.9314 0.8981 1 0.7493; 1.3040 1.4015 1.2481 1) (1 0.8067 1.7315 0.7643; 0.6873 1 1.0609 0.7498; 0.9424 0.9524 1 0.7803; 1.3115 1.4218 1.3785 1) Fig. 1. Consistency analysis for obtained new PCMs with fuzzy elements Fig. 2. Convergence rate for Algorithm 1

		A
	A
	ω	B
		B(ξ=0.1,β=0.5)
	B