A Method Based on Grey Correlation Clustering for Improving the Hierarchy ofAnalytic Hierarchy Process

Di Zhao, Hongfu Liu, Hongyi LiLMIB, School of Mathematics System Science

Beihang UniversityBeijing 100191, P.R. China

zdbuaa2009@yahoo.cn

Abstract—As the AHP is a relatively crude method ofsorting, this paper is intended for improving the constructionof AHP hierarchical structure model, which clusters the layeraccording to the grey cluster of the grey system, in order toreduce the inconsistency of judging matrix and the number ofjudging matrix. At the same time, the improvement can notonly simplified the hierarchical structure, but also reduce thenumber of the comparisons, to make the problem easier.

Keywords-Grey Correlation Clustering; AHP; Judgment Ma-trix;

I. INTRODUCTION

Analytical Hierarchy Process(AHP), as a combinationof qualitative and quantitative analysis of multi-objectivedecision analysis methods, which was proposed by ProfessorSaaty TL in the early 1970s( [1], [2]), has been widelyapplied in various fields, but also exposed much to be de-sired. Therefore, scholars from various countries are seekingfor more comprehensive approach. However, among theseimprovements, most scholars have put forward one way orother algorithms, in terms of especially for the definitiveor linear AHP. But also there is no shortage of ProfessorWang Yingming in solving the least-squares sequencingbreakthrough [7]. Some academics have also proposed AHMmethod according to AHP( [5], [6]). AHM seems better thanAHP, but in terms of applications in the real world, AHP isused much more widely than AHM. In today’s integrated useof multi-model, many articles are also based on the ideologyof AHP, using fuzzy matrix( [8], [9]) and main componentmethods [10] to improve the original results. Combininggrey method with AHP( [11], [12]) is not new. They arebased on different aspects from my paper. All of the methodsmentioned are based on the original AHP, whose results arealmost the same with the original AHP. The improvementis that these methods enhance the accuracy more or less.However, this paper aims at the better hierarchical structurerather than accuracy.

I believe that the main reason for the popularity ofAHP method is its ability to bring the results of subjectivejudgments principled, rather than eliminate the impact ofthe subjective factors on the model. In some cases, it isvery difficult to avoid the subjective judgments. People

use different methods to meet the rank preservation of thesituation, and get the weight of each factor, which does notdiffer greatly. Because of subjective factors in the mixedmodel, continuously improving the precision of meaningis not very efficient. There are two reasons for this point.First, people at different times set different weights of thefactors. The second is to determine matrix relies mainly onthe experience of experts. So, the idea of this paper is theimprovement in the modeling, which in a simple way is thescientific criterion for layer.

In the nine calibration scale method proposed by Saaty,the proposed criteria level is best not more than 9 elements,because the comparison matrix is too large, and it is difficultto reach a satisfactory consistency. Sometimes, it is difficultfor people to have a clear conclusion on the importanceof two things, even for some people to put forward thethree standard degree method algorithm. Whether it is 9scale method or other scale methods [13], if a number offactors are related of high levels, judgments made by peoplecould result in a large degree of error. Therefore, on theestablishment of the hierarchical model, it is necessary forfactors to cluster among the guidelines layer. There are manyclustering methods, such as the reachability matrix, statistics,in accordance with the distance of the cluster analysis, andprincipal component of the data classified by dimensionalityreduction. However, the above-mentioned methods requirethe premise of large amounts of data, if the data is notenough, it is difficult to find a suitable classification criteria.In this paper, gray clustering evaluation, aiming at the objectthrough a small number of observations, classifies the dataappropriately.

AHP (Analytic Hierarchy Process for short AHP) isproposed by Professor Saaty (U.S.) in the early 1970s,which is a quantitative analysis of qualitative issues. It isa simple, flexible and practical method of multi-criteriadecision-making.Its character is by making an orderly hi-erarchy structure according to a complicated issue to let itbe clear. Based on the objective reality and experts’ advice,we can get the relative importance between two differentfactors among the same level. Then, the use of mathematicalcalculations reflects the relative importance of elements at

2010 IEEE/WIC/ACM International Conference on Web Intelligence and Intelligent Agent Technology

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DOI 10.1109/WI-IAT.2010.164

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each level. Through all levels of the total order among allelements, calculate the relative weight and sort.

Gray system theory (GST or GS), is proposed by Pro-fessor Deng Julong in 1982. Different from ”random un-certainty” probability statistics and ”cognitive uncertainty”fuzzy mathematics, the research objects of gray systemtheory are systems with uncertainty, such as ”part of theinformation known and partial information unknown” or”small samples” or ”poor information”. Through generationand development of ”some” known information, we couldbetter understand the real world, and realize the running be-havior of system and evolution of the correct understandingand description.

II. THE THEORETICAL BACKGROUND OF THE NEWMODEL

Proposition 1 The action sequence is

𝑋𝑖 =(𝑥𝑖(1), 𝑥𝑖(2), ⋅ ⋅ ⋅ , 𝑥𝑖(𝑛)

)Denote

(𝑥𝑖(1)− 𝑥𝑖(1), 𝑥𝑖(2)− 𝑥𝑖(1), ⋅ ⋅ ⋅ , 𝑥𝑖(𝑛)− 𝑥𝑖(1))

as 𝑋𝑖 − 𝑥𝑖(1).

Let 𝑠𝑖 =∫ 𝑛

1

(𝑋𝑖 − 𝑥𝑖(1)) 𝑑𝑡.

Then we have i when 𝑋𝑖 is an ascending sequence 𝑠𝑖 ≥ 0.ii when 𝑋𝑖 is a descending sequence 𝑠𝑖 ≤ 0.iii when 𝑋𝑖 is an oscillating sequence, the

sign of 𝑠𝑖 varies.

Definition 1 Denote action sequence 𝑋𝑖 =(𝑥𝑖(1), 𝑥𝑖(2), ⋅ ⋅ ⋅ , 𝑥𝑖(𝑛)

), 𝐷 is the sequence functor,

and 𝑥𝑖(𝑘), 𝐷 = 𝑥𝑖(𝑘) − 𝑥𝑖(1), 𝑘 = 1, 2, ⋅ ⋅ ⋅ , 𝑛. we call𝐷 is starting point of zero-based functor, 𝑋𝑖𝐷 is startingpoint of zero-based elements, Get

(𝑥0𝑖 (1), 𝑥

0𝑖 (2), ⋅ ⋅ ⋅ , 𝑥0

𝑖 (𝑛))

Proposition 2 Denote the action sequence

𝑋𝑖 =(𝑥𝑖(1), 𝑥𝑖(2), ⋅ ⋅ ⋅ , 𝑥𝑖(𝑛)

)𝑋𝑗 =

(𝑥𝑗(1), 𝑥𝑗(2), ⋅ ⋅ ⋅ , 𝑥𝑗(𝑛)

)Whose starting point of zero-based elements, as follows

𝑋𝑖 =(𝑥0𝑖 (1), 𝑥

0𝑖 (2), ⋅ ⋅ ⋅ , 𝑥0

𝑖 (𝑛))

𝑋𝑗 =(𝑥0𝑗 (1), 𝑥

0𝑗 (2), ⋅ ⋅ ⋅ , 𝑥0

𝑗 (𝑛))

Definition 2 The sum of the distance between the variousobservational data is called the length of 𝑋𝑖 [4].

Definition 3 Denote the length of 𝑋0 and 𝑋𝑖 is the same,𝑠0, 𝑠1, as the definition1 follows, then

𝜀0𝑖 =1 + ∣𝑠0∣+ ∣𝑠𝑖∣

1 + ∣𝑠0∣+ ∣𝑠𝑖∣+ ∣𝑠𝑖 − 𝑠0∣

is the degree of Gray absolute correlation between 𝑋0 and𝑋𝑖, short for Absolute correlation [4].

Theorem 1i 0 < 𝜀0𝑖 < 1;ii 𝜀0𝑖 is only related with the geometrical shape of 𝑋0

and 𝑋𝑖, which has nothing to do with the relative positionin their space. In another word, translating cannot changethe value of Absolute correlation;iii Any two sequences are not absolutely independent,thatmeans that 𝜀0𝑖 is always above zero;iv the more the degree of similarity on Geometry between𝑋0 and 𝑋𝑖, the larger 𝜀0𝑖;v when 𝑋0 and 𝑋𝑖 is parallel, or 𝑋0

𝑖 swing around 𝑋00 ,

and the area that 𝑋0𝑖 is above 𝑋0

0 and the area that 𝑋0𝑖 is

below 𝑋00 , equals, 𝜀0𝑖 = 1;

vi when any element in 𝑋0 or 𝑋𝑖 changes, 𝜀0𝑖 changes;vii the length of 𝑋0 or 𝑋𝑖 changes, 𝜀0𝑖 changes;viii 𝜀𝑖𝑖 = 1; ix 𝜀𝑖𝑖 = 𝜀𝑖0 [4].

Definition 4 Denote there exists 𝑛 Observing ObjectseveryObserving Objects has 𝑚 Feature datawe can get thesequences as follows:

𝑋1 = (𝑥1(1), 𝑥1(2), ⋅ ⋅ ⋅ , 𝑥1(𝑛))

𝑋2 = (𝑥2(1), 𝑥2(2), ⋅ ⋅ ⋅ , 𝑥2(𝑛))

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅𝑋𝑚 = (𝑥𝑚(1), 𝑥𝑚(2), ⋅ ⋅ ⋅ , 𝑥𝑚(𝑛))

For all 𝑖 ≤ 𝑗, 𝑖, 𝑗 = 1, 2, ⋅ ⋅ ⋅ ,𝑚, calculate the absolutecorrelation between 𝑋𝑖 and 𝑋𝑗 , 𝜀𝑖𝑗 . By the property, justget upper triangular matrix,

𝐴 =

⎡⎢⎢⎢⎣𝜀11 𝜀12 ⋅ ⋅ ⋅ 𝜀1𝑚

𝜀22 ⋅ ⋅ ⋅ 𝜀2𝑚. . .

...𝜀𝑚𝑚

⎤⎥⎥⎥⎦in 𝜀𝑖𝑖 = 1, 𝑖 = 1, 2, ⋅ ⋅ ⋅ ,𝑚 [4].

Definition 5 Denote critical point threshold 𝑟 ∈ [0, 1], when𝜀𝑖𝑗 ≥ 𝑟(𝑖 ∕= 𝑗) , we take 𝑋𝑖 and 𝑋𝑗 as the same class.Characteristic variable, under the critical point threshold 𝑟is called Grey Correlation Clustering [4].

III. APPLICATION INSTANCE

One of applications using AHP is to choose the rightperson for the right place. It’s clear that using AHP tosolve this multi-index question is exactly right. And ev-eryone one knows that the first step is to build a sci-entific hierarchical structure. To evaluate employees’ jobrequirements,15 indicators are proposed by the judges: (1)application image, (2)academic ability, (3)pleasant, (4)self-confidence, (5)sagacity, (6)honesty, (7)salesmanship, (8)ex-perience, (9) enthusiasm, (10) ambition, (11) appearance,

157

Figure 1. Hierarchical structure diagram

(12) understanding, (13) potential, (14) communicative com-petence, (15) adaptability [4]. We, however, do not know therelationship of these indicators. That means we put these 15indicators on the same level, a huge level, and we assume4 candidates. Here comes the structure (See Figure.1).

We can see that we have to build 19 matrixes includingone 15-order matrix. Take the 15-order matrix into account,there exist two problems. First, Satty did not give the R.I.of a 15-order matrix; second so difficult it is to build such asatisfactory matrix in term of the consistency. And whenit comes to the number of comparisons in the matrixes,195(14*13/2+15*4*3/2) comparisons must be made.

So, we take the Grey Correlation Clustering into accountin order to simplify this problem. First, we need some littleinformation about these 15 indicators to clarify the relation-ship among these indicators. Scoring of these indicators tomake quantitative approach, 9 employees obtained scores ofindividual indicators are in the following table(See Table I).

For all 𝑖 ≤ 𝑗, 𝑖, 𝑗 = 1, 2, ⋅ ⋅ ⋅ , 15, calculate the absolutecorrelation between 𝑋𝑖 and 𝑋𝑗 , 𝜀𝑖𝑗 . By the propertyjust getupper triangular matrix(See Table II).

Set 𝑟 = 0.80, we start from the first line followed byinspection, select out more than 0.80𝜀𝑖𝑗 , that is 𝜀1,3 =0.88, 𝜀1,11 = 0.90, 𝜀1,12 = 0.88, 𝜀1,13 = 0.80, 𝜀2,8 =0.99 𝜀3,11 = 0.80, 𝜀3,13 = 0.90, 𝜀6,11 = 0.84, 𝜀6,12 =0.86, 𝜀6,14 = 0.81 𝜀7,10 = 0.83, 𝜀7,15 = 0.89, 𝜀9,10 =0.81, 𝜀10,15 = 0.92, 𝜀11,12 = 0.97 Thus we can see:𝑋3, 𝑋11, 𝑋12, 𝑋13 should be in the same class with 𝑋1.𝑋8 and 𝑋2 are in the same class. 𝑋11, 𝑋13, 𝑋3 are inthe same class. 𝑋11, 𝑋12, 𝑋14, 𝑋6 are in the same class.𝑋10, 𝑋15, 𝑋7 are in the same class. 𝑋10, 𝑋9 are in the sameclass. 𝑋15 and 𝑋10 are in the same class. 𝑋12 and 𝑋10 arein the same class.

Take the smallest label indicators as representative ofvarious types, put the elements which class is the same with𝑋6. 𝑋6, 𝑋14, 𝑋11, 𝑋12 into the same class which 𝑋1 is in.Put 𝑋9 and 𝑋10 in the class which 𝑋7 is in. 𝑋 − 4 isis regarded as a single class. Get 15 indicators of cluster-ing: {𝑋1, 𝑋3, 𝑋6, 𝑋11, 𝑋12, 𝑋13, 𝑋14}, {𝑋2, 𝑋8}, {𝑋4},{𝑋5}, {𝑋7, 𝑋9, 𝑋10, 𝑋15}

From the above five categories are 𝐴2 Internal quality, 𝐴2

ability, 𝐴3 confidence, 𝐴4 sagacity, 𝐴5 External character.Using the above classification, the establishments of hier-

Figure 2. New Hierarchical structure diagram

archical structure diagram(See Fig.1).Again, we see the new structure. Only 9 matrixes are

needed, which are small matrixes, 4 or 5-order matrixes.And the number of comparisons reduce to 40.

On this basis, build the judgment matrix, as follows:⎡⎢⎢⎢⎢⎢⎢⎣𝐴1 𝐴2 𝐴3 𝐴4 𝐴5

1 1 1 4 11 1 2 4 11 1

2 1 5 514

14

15 1 1

31 1 1

3 3 1

⎤⎥⎥⎥⎥⎥⎥⎦Get the weight of every element, 𝐴1 Internal quality, 𝐴2

ability, 𝐴3 confidence, 𝐴4 sagacity, 𝐴5 External character(0.2374, 0.2371, 0.2696, 0.0750, 0.1446), indicating that theassessment pays more attention to internal quality, compe-tence and self-confidence. We may use this as the standardfor the assessment of all participating personnel.

IV. CONCLUSION

In this application instance, the improved method is notonly the guidelines layers from 15 down to five-order matrix,but also streamline the layers significantly reduce the numberof matrix from 19 to 9, and the number of comparisons from195 to 40, thereby reducing the computational. This papercombine the Grey Correlation Clustering and AHP, thussimplifying the judgment matrix. Empowerment in the multi-index has shown a definite advantage. The most importantis that the improved method can reduce the number ofcomparisons in the judgment matrixes.

ACKNOWLEDGMENT

The project is supported by the National Natural ScienceFoundation of China (Grant No.60831001), the Projectsin the National Science & Technology Pillar Program ofChina(Grant No.2008BADA4B03) and National Natural Sci-ence Foundation of China(Grant No.10771011).

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Table IINDIVIDUAL INDICATORS

indicator observer 1 2 3 4 5 6 7 8 9application image 𝑋1 6 9 7 5 6 7 9 9 9academic ability 𝑋2 2 5 3 8 8 7 8 9 7pleasant 𝑋3 5 8 6 5 8 6 8 8 8self-confidence 𝑋4 8 10 9 6 4 8 8 9 8sagacity 𝑋5 7 9 8 5 4 7 8 9 8honesty 𝑋6 8 9 9 9 9 10 8 8 8salesmanship 𝑋7 8 10 7 2 2 5 8 8 5experience 𝑋8 3 5 4 8 8 9 10 10 9enthusiasm 𝑋9 8 9 9 4 5 6 8 9 8ambition 𝑋10 9 9 9 5 5 5 10 10 9appearance 𝑋11 7 10 8 6 8 7 9 9 9understanding 𝑋12 7 8 8 8 8 8 8 9 8potential 𝑋13 5 8 6 7 8 6 9 9 8communicative competence 𝑋14 7 8 8 6 7 6 8 7 8adaptability 𝑋15 10 10 10 5 7 6 10 10 10

Table IIABSOLUTE CORRELATION BETWEEN DIFFERENT INDIVIDUAL INDICATORS

𝑋1 𝑋2 𝑋3 𝑋4 𝑋5 𝑋6 𝑋7 𝑋8 𝑋9 𝑋10 𝑋11 𝑋12 𝑋13 𝑋14 𝑋15

𝑋1 1.00 0.66 0.88 0.52 0.58 0.77 0.51 0.66 0.51 0.51 0.90 0.88 0.80 0.67 0.51𝑋2 1.00 0.72 0.51 0.53 0.59 0.50 0.99 0.51 0.51 0.63 0.62 0.77 0.55 0.51𝑋3 1.00 0.51 0.56 0.70 0.51 0.72 0.51 0.51 0.80 0.78 0.90 0.63 0.51𝑋4 1.00 0.56 0.53 0.58 0.51 0.69 0.62 0.52 0.52 0.51 0.54 0.60𝑋5 1.00 0.65 0.51 0.53 0.53 0.52 0.61 0.61 0.55 0.75 0.52𝑋6 1.00 0.51 0.59 0.52 0.52 0.84 0.86 0.66 0.81 0.51𝑋7 1.00 0.50 0.70 0.83 0.51 0.51 0.51 0.51 0.89𝑋8 1.00 0.51 0.51 0.63 0.62 0.77 0.55 0.51𝑋9 1.00 0.81 0.52 0.52 0.51 0.53 0.76𝑋10 1.00 0.51 0.51 0.51 0.52 0.92𝑋11 1.00 0.97 0.74 0.71 0.51𝑋12 1.00 0.73 0.72 0.51𝑋13 1.00 0.60 0.51𝑋14 1.00 0.52𝑋15 1.00

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