APPLICATION OF ANALYTIC HIERARCHY PROCESS ON

CALCULATING THE WEIGHTS OF ECONOMIC MODEL

EVALUATION

Dai Wang, Dan Chang, Na Luo and Jingyi Xu

School of Economics and Management, Beijing Jiaotong University, No.3 Shangyuancun, Beijing, China

Keywords: Weight, Evaluation system, AHP, Adjust.

Abstract: With the combination of teaching resources and IT technology being more and more close, it has attracted

more attention on the issue of how to manage the quality of online teaching resources. This thesis has

constructed an evaluation system for the model library of the Economic Model Resource Platform, and

adjusted the weighted scales and the corresponding calculations based on both of the theory of the Analytic

Hierarchy Process (AHP to be brief) and the characteristics of the Economic Model Resource Platform

during the process of determining the weights of the evaluation system in order to make the final weights

more suitable for practical applications. This study helps achieve the purposes of monitoring the quality of

the economic models and promoting the optimization of the models. On the other hand, the calculation

method of determining the weights has provided a reference for the application of AHP.

1 INTRODUCTION

With the rapid development of information

technology, network technology gradually

penetrated into the educational applications.

However, with the rapid increase of the amount of

teaching resources on the Internet, how to effectively

improve the quality of teaching resources has gotten

more and more concentration. The foundation of this

thesis is the Economic Model Resource Platform.

This platform which works as a library of the

economic models based on the internet technology

contains 170 models up to now. And the platform is

an important teaching resource with the goals of

training the scientific and engineering thoughts of

the economics and management students and

training them to be the compound and creative

talents. With the increasing number of the models on

the platform, the quality of the models varies greatly,

and needs improving continuously. Therefore, the

management of the models on the platform is

particularly important.

Analytic Hierarchy Process (AHP to be brief) has

the characters of clear, simply, and with strong

systematicness (XIANG Qing, 1997). In recent

years, this method has been widely applied to

analyze and evaluate in many territories such as

social science, economy, education and others. The

examples of the applications come as the

constructing of the analysis method of meandering

river underground reservoir (Yue Dali et al., 2010),

the constructing of the evaluation system of the

emergency logistics plan(MA Li, 2010) , the

constructing of the quality evaluation system in the

hospital work (ZENG Wenting et al., 2010),

analyzing the leading industries in direct foreign

investments (Zhao Fu-hou, 2010), etc.

This thesis is based on AHP. The first step is to

construct the index system of model library

evaluation system, and the second step is to

determine the weights. In the process of the second

step, there is a combination of the theory of AHP

and the characteristics of the application of the

evaluation system. The method of constructing the

judgment matrixes has been adjusted, making it

more suitable with the actual situation. And it proves

to be correct by the validation afterwards.

2 CONSTRUCT THE INDEX

SYSTEM OF ECONOMIC

MODELS

This thesis uses the AHP method to construct the

index system of the economic model evaluation

108

Wang D., Chang D., Luo N. and Xu J..

APPLICATION OF ANALYTIC HIERARCHY PROCESS ON CALCULATING THE WEIGHTS OF ECONOMIC MODEL EVALUATION.

DOI: 10.5220/0003549201080116

In Proceedings of the 13th International Conference on Enterprise Information Systems (ICEIS-2011), pages 108-116

ISBN: 978-989-8425-55-3

 2011 SCITEPRESS (Science and Technology Publications, Lda.)

system (Rong RUAN et al., 2011). Based on the

basic principles of AHP, the index system is divided

into three levels, namely, the target layer, rule layer

and project layer. By consulting criteria of teaching

resources and combining the features of the

economic models, the target layer is divided into

content, technical, effectiveness and application.

Those are the first-level indexes. Then get the target

level indexes in the same way. After this, there are

11 indexes of the criterion level and 24 indexes in

the program level finally.

The index system is shown in Table 6.

3 DETERMINE THE INITIAL

WEIGHTS USING AHP

3.1 Theory and Characteristics of AHP

AHP is proposed by the U.S. operations researcher

T.L Saaty (1980), which is a multi-objective

decision analysis method combined with qualitative

and quantitative analysis. It suits for the problem that

has complex structure and many decision criteria

that are difficult to quantify. The basic idea of AHP

is that the system is decomposed into different

elements according to the nature, decision-making or

evaluation of goals of the object. Then different

elements are arranged from high to low by the

linkages between elements.

AHP is not only a simple method for making

quantitative analysis by non-quantitative systems in

engineering event, but also an effective method to

make subjective judgments on the objective

description.

3.2 The Implementation Steps of AHP

The general process of AHP is, shown in Figure 1.

Analysis

Establish

Hierarchy Model

Construct

Judgment Matrix

Consistency

Test

One Order

Analysis

Sort of the

Total(All

Weights)

Value of

Comprehensive

Evaluation

Decision

Analysis

Conclusion

Yes

Figure 1: The general process of AHP.

1. Analyze the research questions.

2. Establish the hierarchy models based on the

results of the research questions.

3. Construct the judgment matrix.

4. Conduct consistency test, if the test failed, re-

analyze the problem or construct a new judgment

matrix, else make the single-pass order analysis.

5. Make the total sequencing analysis of all

weights.

6. Test the value of comprehensive evaluation, if

the decision can be accepted, draw conclusions, or

re-test the consistency.

3.3 Application of AHP to Get the

Weights of the Economic Model

Evaluation System

3.3.1 Construct Variables

The economic model evaluation system is divided

into three levels based on the target layer, rule layer

and project layer. Level one indexes include content,

technical, effectiveness and application; the

secondary indexes include integrity, accuracy and

ease of understanding and so on. Indexes have been

shown in Table 6.

Level one indexes will be set respectively

、

. The secondary indexes

corresponding to level one indexes will be set

respectively as

、

（

、

k are natural

numbers

），

so do the third level indexes.

Assume that the weights of the level one

indicators as



、



、



、



, and then:

01 (1,2,3)





 















(1)

(2)

Assume that the weights for the secondary indicators

are



( 1, 2,3; 1, 2,..., max( , , ))ij mnk

, and then:

max( , , )

0 1 ( 1,2,3)

mnk

iij













 



















(3)

(4)

(5)

APPLICATION OF ANALYTIC HIERARCHY PROCESS ON CALCULATING THE WEIGHTS OF ECONOMIC

MODEL EVALUATION

109

3.3.2 Determine the Weights of Evaluation

Indexes

(1) Construct judgment matrixes. On the same level,

the result of pair wise comparison can be generally

described by using "important", "slightly important",

"relatively important" and "very important". In AHP,

using

(, )

to express whether factors

is more

important than factor

y . If

(, ) 1fxy

, it indicates that

is more important than y . If

(, ) 1fxy

, it indicates

that

is less important than y . Only

when

(, ) 1fxy

, it indicates that

and y are

equally important. And it is commonly agreed

that

(, )

(,)

fxy



Numbers 1 to 9 are used to indicate the level of

importance of the indexes. The meanings of the

scales are in Table 1.

Table 1: The meaning of each scale in AHP.

Scale f(x,y) Definition Instructions

1 Equally important i is equally important with j

Somewhat

important

i is somewhat more

important than j

Relatively

important

i is relatively more

important than j

7 Very important i is more important than j

Absolutely

important

i is absolutely more

important than j

2 4 6 8 Between Between the two states

(2) Expert opinion to construct the Matrix method.

In the scoring process, the experts determine the

relative importance ratio of the indicators in the

same level based on their experience. The form of

collecting the data is questionnaire.

According to the evaluation system, we invited 55

experts to rate indexes in order to create a judgment

matrix by questionnaires. The experts are from three

research areas namely Department of Economics,

Department of Logistics and Department of

Information System. After the research, 50

questionnaires were recovered, and the recovery was

90.9%. Because each expert has a certain

understanding of bias and errors, we removed some

radical elements and do descriptive statistics when

using the original data. At last, we received 13

comparison matrixes.

Take the secondary indicators ‘integrity’ for

example. Its third-level indexes’ judgment matrix is

in Table 2.

(3) Calculating the weights and the maximum

eigenvalue of the third-level indexes of ‘integrity’.

Determine the matrix data (Row 2-5, Table 2) in

accordance with the instructions given afterwards.

The following is the calculation of the maximum

eigenvalue of the indexes; the results are in Table 3.

Calculation descriptions:

1/n

iij











(6)

Table 2: Judgment matrix of the level 3 indexes of ‘Integrality’.

Integrality Theoretical

source

Assumptions Theory to explain Analysis

Theoretical source 1 2 2 1

Assumptions 1/2 1 2 2

Theory to explain 1/2 1/2 1 1

Analysis 1 1/2 1 1

Table 3: Calculate the index weights and the maximum eigenvalue of the third-level indexes of ‘integrity’.

Scales A B C D Mi Wi (AW)i The largest eigenvalue Weights

1.00 2.00 2.00 1.00 1.414214 0.340657 2.043945 0.69628508 0.505198

0.50 1.00 2.00 2.00 1.189207 0.286458 1.575517 0.45131891 0.32746

0.50 0.50 1.00 1.00 0.707107 0.170329 0.510986 0.08703563 0.06315

1.00 0.50 1.00 1.00 0.840896 0.202556 0.708947 0.14360147 0.104192

Total

4.151424 1 1.3782411

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110

In this case n = 4.

ii i

WM M



(7)

In this case n = 4.

A is comprehensive judgment matrix. Each

number in the number i line in A multiplied by each

number of the corresponding values in the

column, and the summation of all products is

()

The calculation of the maximum eigenvalue:





max

AW n W





(8)

In this case, the maximum eigenvalue is equal to

1.3782411.

(4) Consistency Test. Because the matrix structure

made by the experts do not necessarily meets the

matrix consistency. Judge the matrix consistency

test in order to limit this kind of error. Take the

largest eigenvalue and n's relative error as the

consistency indicator of matrix. Denoted by:

max









(9)

( n equals to the order of matrix, also the number of

indexes in matrix.)

2n 

,the matrix is always exactly the same, it

means

0CI 

。

And when

2n 

,the ratio of the

matrix's consistency index and the average random

consistency index is random consistency ratio which

is denoted as

R<0.1C

, the judgment matrix has satisfactory

consistency and the calculated feature vector is

reliable. Otherwise the matrix the experts

constructed has larger error and is out of range. The

matrix needs to be re-adjusted until it gets the

satisfactory consistency.

After calculation, all comparison matrixes are

consistent, and the results are credible.

(5) Calculation of the total weight. The index system

is composed of the first-level indicators and

secondary indicators. Each first-level indicator

contains numbers of secondary-level indexes. As

described above, we calculate four first-level

indicators' weights, and analyze the consistency of

its matrix. And it is the same to the secondary-level

indexes.

Supposing that the indicator

i 's weight is equal

a ,its j secondary-level indicator's relative

weight is

b .Then this secondary-level indicator's

total weight is

ab

In order to verify whether the total weight also

has the satisfaction of consistency, we need to

calculate the total random consistency index, the

indicator is calculated as:

aCI

aRI





(10)

While:

a : the number i first-level indicator's weight.

CI : the number i first-level indicator's

consistency index value.

：

the number i first-level indicator's average

random consistency index values.

The final overall consistency test result is

0.028312 which is far less than 0.1.So it is consistent

with consistency. Evaluation indexes’ weights are in

Table 6, the unadjusted weights.

3.4 Existing Problems in Weights

According to the analysis of the system above, the

following questions are found:

1) There are nine scales 1 to 9 which bring heavy

interference to define the actual level of importance.

According to the statistics, in the valid

questionnaires, 89.93 percent of the scores are

between 1 and 4, and this is not accord with the

actual. And a few high points such as score nine or

eight will cause big effect to the result of the data.

The reason lies in that nine scales range too wide,

and in the economic model evaluation system, the

differences of importance between indexes of the

same layer do not need so big a range. At the same

time, because the differences between the graders

own subjective thoughts, the specific meanings of 1

to 9 are difficult to be defined.

2) Traditional AHP method is applicable to the

systems with fewer indexes, and when they are used

to the economic model evaluation system, the big

workload will make raters be confused with the

concepts of indexes gradually.

3) Due to current mindset, in 90.1% of the

recovered questionnaires, the top right corner data of

judgment matrix are greater than 1. When some

experts do the rating, they default that the front

index is more important than the afterward indexes.

This kind of mindset leads to an obvious weight

stressed phenomenon in the same layer indexes,

namely the index Content’s weight is greater than

80%, which causes the weight of index Application

APPLICATION OF ANALYTIC HIERARCHY PROCESS ON CALCULATING THE WEIGHTS OF ECONOMIC

MODEL EVALUATION

111

can be ignored, or even the indexes of the secondary

layer and the third layer. The phenomenon does not

match the facts.

4 ADJUST THE WEIGHTS OF

THE EVALUATION SYSTEM

4.1 Adjustments

It has shown that the direct application of AHP does

not match the evaluation system exactly based on

the analysis of the evaluation system above. Because

of this, it is needed to adjust the ways of calculating

the weights based on the characteristics and the

actual applications of the Economic Model Resource

Platform and re-calculate them.

The specific adjustments of the application of

AHP are as follows.

1) Narrow the scope of the scales to three numbers.

Narrowing the scope of the scales is more suitable

for this study compared to the initial nine numbers.

2) Set the scales to -1, 0 and 1(YE Jun, WANG Lei,

2010). -1 is introduced into the process in order to

express the degree of the importance of the indexes

and reduce the fatigue caused by the large amount of

work.

The meaning of each scale is shown in Table 4.

Table 4: The meaning of each adjusted scale.

Scale f(x,y) Definition Interpretation

-1

To be less

important

i is less important

than j

To be equally

important

i is equally

important with j

To be more

important

i is more important

than j

3) Using the adjusted questionnaires to obtain new

data from the same respondents.

4.2 Calculating Weights

(1) Structure the Judgment Matrixes. A judgment

matrix is based on the relative importance of each

index, denoted by C. Still take the level three

indexes of the secondary index ‘Integrality’ for

example. The experts assess the relative importance

of the indexes as ‘Theory to explain’ >

‘Assumptions’ = ‘Theoretical source’ > ‘Analysis’.

And the initial judgment matrix is shown in Table 5.

Thereupon,

00 1 1

11 00

11 0 0

































(11)

(2) Calculate the optimal transfer matrix. Optimal

transfer matrix is set to be O. According to

()

ij ik kj

Occ







(12)

Get the optimal matrix as follows:

0 0.50 0.50 0

0.50 0 1.00 0.50

0.50 1.00 0 0.50

0 0.50 0.50 0

































(13)

(3) Get the consistent matrix. Set the consistent

matrix as K. According to

exp( )

ij ij

ko to get the

consistent matrix as follows:

1.00 1.65 0.61 1.00

0.61 1.00 0.37 0.61

1.65 2.72 1.00 1.65

1.00 1.65 0.61 1.00



























(14)

(4) Get the weights. Make use of the eigenvector to

get the weights. Set the weight matrix as

. Then









(15)

Table 5: Adjusted judgment matrix of the level 3 indexes

of ‘Integrality’.

Integrality

Theoretic

al source

Assumptions

Theory to

explain

Analysis

Theoretical

source

0 0 -1 1

Assumptions 0 0 -1 -1

Theory to

explain

1 1 0 0

Analysis -1 1 0 0

And the weight matrix is





0.235 0.143 0.387 0.235

P 

(16)

(5) Complete the calculation of the weights in the

evaluation system. Complete the calculation of the

ICEIS 2011 - 13th International Conference on Enterprise Information Systems

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Table 6: Economical model evaluation system and the weights.

A Target layer

Unadjusted

weights

Adjusted

weights

B Rule layer

Unadjusted

weights

Adjusted

weights

C Project layer

Unadjusted

weights

Adjusted

weights

1 Content 0.870 0.276

11 Integrity 0.503 0.124

111 Theoretical source 0.234 0.029

112 Assumptions 0.244 0.017

113 Theory to explain 0.009 0.048

114 Analysis 0.016 0.047

12 Accuracy 0.238 0.089

121 Interpretation accuracy 0.073 0.038

122 The accuracy of

understanding

0.033 0.018

123 Whether to seize the

essence

0.132 0.027

13 Understand-

ability

0.125 0.063

131 Example 0.046 0.018

132 Analogy to explain 0.075 0.009

133 Whether has

interpretation of the

academic term

0.004 0.036

2 Technical 0.080 0.455

21 Operability 0.077 0.256

211 Whether the output of

experimental operation

0.068 0.148

212 Whether the result is

that the process

0.005 0.053

213 Whether has input data

validation

0.004 0.053

22 Friendly

interface

0.002 0.067

221 Tips range of input data 0.002 0.028

222 Sample data 0 0.016

223 The reasonable of the

control

0 0.022

23 Experiment to

explain

0.002 0.131

231 Experiment description 0.002 0.041

232 Interpretation of results 0 0.057

233 Whether is the steps to

explain logical

0 0.030

3 Effectiveness 0.030 0.102

31 Intuitive 0.025 0.027

311 Text 0 0.015

312 Graphic description 0.016 0.008

313 Other visual

presentation methods

0.008 0.004

32 Data Validation 0.075 0.075

321 The length of time to

get result

0 0.054

322 Error rate 0.001 0.018

4 Application 0.030 0.167

41 Case relevance 0.025 0.096 0.025 0.096

42 Case Study 0.001 0.035 0.001 0.035

43 Questions 0 0.035 0 0.035

APPLICATION OF ANALYTIC HIERARCHY PROCESS ON CALCULATING THE WEIGHTS OF ECONOMIC

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entire weights in the evaluation system according to

the methods above. Set the weight of index i as

a ,

its secondary index j’s weight as

b , and the final

weight of the secondary weight is

ab

4.3 The Evaluation System and the

Weights after the Adjustments

According to the methods above, the economical

model evaluation system and the weights are in the

table 6, the adjusted weights.

5 VALIDATION OF THE

EVALUATION INDEXES

5.1 The Basis of the Validation

To verify that the adjusted index’s weights are more

suitable for the economic model evaluation system,

it is needed to test the two sets of weights.

According to the experience of statistics, when

the size of the sample is not less than 30，it is large

sample, and the samples’ average can eliminate

personal biases and errors. In this case, 60 users of

the Economic Model Resource Platform were

selected for the investigation. And they were

randomly divided into group A and group B, 30

people in each group to make the results of the

validation more general and reliable (E.L.Lehman,

2010).

5.2 The Process of the Validation

1) Sampling. Randomly select 20 models from the

170 models in the model library of the Economic

Model Resource Platform, number them 1-20.

2) Select the respondents. Select 60 users of the

Economic Model Resource Platform randomly and

divided them into group A and group B, according

to the methods above.

3) Rating. Let the users of group A rate the models

0-20 according to the practical usage of the models

with 10 being the highest and 0 being the lowest. In

this case, it has been the rules that the scores of 8-10

are level A, 5-8 are level B, and the rest are level C.

4) Scoring. Request the users of group B score the

indexes of the 20 models.

5) Data analysis.

Firstly, the data of Group A were statistically

analyzed. Work out the average score of each model

and rate the models in the method above.

Secondly, put the scores of each index from

group B into the two sets of the index systems to get

two total scores for each model. And take the

average scores of all the raters as the final scores of

the models using two evaluation systems.

Finally, compare the three sets of data, as shown

in Table 7.

Notes:

Horizontal axis: Model Numbers.

Vertical axis: Scores.

Grey areas: Levels from Group A.

Triangle marks: Model scores rated by Group B

using the evaluation system unadjusted.

Quadrate markers: Model scores rated by Group B

using the evaluation system after adjusted.

Table 7: The results of the validation.

Model Number 1 2 3 4 5 6 7 8 9 10

Rating by Group A

A B B C C A C B B C

Scoring by Group B(unadjusted)

9.82 2.85 9.43 0.78 7.99 9.58 7.78 9.65 8.87 8.96

Scoring by Group B(adjusted)

9.76 5.48 6.42 0.61 2.97 8.90 2.44 7.87 7.54 3.54

Model Number 11 12 13 14 15 16 17 18 19 20

Rating by Group A

C C B C A A B C B C

Scoring by Group B(unadjusted)

1.54 8.99 2.01 7.45 9.46 9.67 9.06 8.79 1.13 7.06

Scoring by Group B(adjusted)

1.08 3.25 8.50 2.48 9.08 8.16 6.35 4.70 6.78 2.35

The results of the matching are shown in Figure 2.

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Figure 2: The matching Results of the Validation.

5.3 Analysis of the Validation

As is shown above, the number of matched models

between the rating from group A and the scores

gotten by the evaluation system unadjusted is 6

(respectively, model 1, model 4, model 6, model 11,

model 15 and model 16 ), 30% of the total number

of the samples. According to the data, the scores

from the unadjusted evaluation system are extreme,

and most of them are between 7-10 points. Observe

the weight of each index, it can be find that

“Theoretical source”, “Assumptions”,”

Interpretation accuracy ”,“Whether to seize the

essence ”,“Whether the output of experimental

operation” and “Analogy to explain” occupy the

larger share, weighing 82.6% in the whole

evaluation. So they have too much excessive traction

to the final score, resulting in relatively extreme

scores of models.

However, the number of matched models

between the rating from group A and the scores

made by the evaluation system adjusted is 19, 95%

of the total number of the samples. Therefore, the

weights made by the adjusted evaluation system are

more applicable and objective, mainly embodied in

the following three aspects.

Firstly, they solve the unsuitable problem of the

weights made by the unadjusted evaluation system.

Reduce the scales to - 1, 0 and 1 can indicate the

level of importance among indexes more clearly.

Meanwhile, the reduction of the number of scales

eases the workload of respondents and alleviates the

scoring fatigue, thus enhancing the effectiveness of

the initial matrixes. In addition, adjusting scales

solutes the difficulty of discriminating the large

amount of scales.

Secondly, accord with the actual importance of

the indexes. By the weights of the final indexes,

"Experimental operation is output or not" occupies

the largest share, as 14.8%. Followed is "Case

correlation" which occupies as 9.6%. The Economic

Model Resource Platform takes IT as the core, and

focuses on the applications of the models, so the

maneuverability and correlation of the cases indeed

weigh a larger proportion in evaluating the quality of

the models. And the weights of other indexes are

also accord with the actual situation.

Thirdly, be helpful with practical applications.

This economic model evaluation system can show

clearly the total scores of the models and typically

each score of the indexes. It not only evaluates the

models, but also indicates the low grade models’

weak more intuitively, which helps build a clear

target for the administrator to improve the quality of

the models.

6 CONCLUSIONS

There are numerals ways to get the weights of

indexes in an evaluation system. And this thesis gets

the weights by adjusting the process of AHP based

on the theory of AHP and the characteristics of the

Economic Model Resource Platform. This method

not only makes the meaning of the scales more

directly which results in reducing the workload of

scoring and relieving the fatigue of scoring, but also

gets a result more suitable to the practical

applications. The method in this thesis can be used

not only on the Economic Model Resource Platform,

but also provides some references to other similar

studies.

APPLICATION OF ANALYTIC HIERARCHY PROCESS ON CALCULATING THE WEIGHTS OF ECONOMIC

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However, although the way of getting the

weights is adjusted, it is unavoidable that the data

from the experts are some kind of subjective.

Therefore, it is important to choose the proper

experts, both from the experts of the corresponding

subjects and the users of the Economic Model

Resource Platform in order to make the weights of

the indexes more reasonable and practicable. The

users of the Economic Model Resource Platform can

rate the models according to this evaluation, and

when the number of the raters is large enough, we

can get the evaluation of the models.

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