INT J COMPUT COMMUN, ISSN 1841-9836
Vol.7 (2012), No. 3 (September), pp. 450-458

Data Consistency in Emergency Management

D. Ergu, G. Kou, Y. Peng, F. Li, Y. Shi

Daji Ergu
College of Electrical and Information Engineering
Southwest University for Nationalities
Chengdu, China, 610041
E-mail: ergudaji@163.com

Gang Kou, Yi Peng , Feixiong Li
School of Management and Economics
University of Electronic Science and Technology of China
Chengdu, China, 610054
E-mail: kougang@yahoo.com; pengyicd@gmail.com;
lifx@uestc.edu.cn

Yong Shi
1. Research Center on Fictitious Economy and Data Sciences
Chinese Academy of Sciences, Beijing 100190, China, and
2. College of Information Science & Technology
University of Nebraska at Omaha, Omaha, NE 68182, USA
E-mail: yshi@gucas.ac.cn

Abstract:
Timely response is extremely important in emergency management. However, car-
dinal inconsistent data may exist in a judgment matrix because of the limited exper-
tise, preference conflict as well as the complexity nature of the decision problems.
The existing inconsistent data processing models for positive reciprocal matrix either
are complicated or dependent on the priority weights, which will delay the decision
making process in emergency. In this paper, a geometric mean induced bias ma-
trix (GMIBM), which is only based on the original matrix A, is proposed to quickly
identify the most inconsistent data in the judgment matrix. The correctness and ef-
fectiveness of the proposed model are proved mathematically and illustrated by two
numerical examples. The results show that the proposed model not only preserves
most of the original information in matrix A, but also is faster than existing methods.
Keywords: cardinal inconsistency, positive reciprocal matrix, geometric mean in-
duced bias matrix (GMIBM), inconsistency identification

1 Introduction

Emergency management is an interdisciplinary field, and is in essence a complex multi-objective
optimization problem [1]. Multi-criteria decision making (MCDM) methods have therefore been ex-
tensively employed to study emergency management, for instance, the Ordered Weighted Averaging
(OWA) [2], Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) [3], Preference
Ranking Organization Method for Enrichment Evaluations (PROMETHEE) [4], Analytic Hierarchy Pro-
cess (AHP) ( [5], [6],and [7]), Analytic Network Process (ANP) ( [8], [9]), Decision Making Trial and
Evaluation Laboratory (DEMATEL) [10], Fusion Approach of MCDM (FAMCDM) methods [11] etc.
Among these MCDM methods, AHP and ANP are two of the most popular methods for studying emer-
gency management, and usually used to assess the emergency management performance, select the best
emergency response alternatives or emergency recovery alternatives and allocate reasonable relief re-
sources etc.

Copyright c⃝ 2006-2012 by CCC Publications



Data Consistency in Emergency Management 451

In the AHP/ANP, the tangible and intangible attributes or criteria are always measured by numerical
data through pairwise comparisons, and displayed in a judgment matrix whose numerical data are posi-
tive and reciprocal. The data of judgment matrix are usually provided by experts and collected through
questionnaire survey [12]. Therefore, the data may be inconsistent because of the limited expertise, pref-
erence conflict as well as the complexity nature of the decision problems etc ( [13], [14]). For instance,
assume there are three emergency response alternatives, A, B, and C, the ith expert thinks that alternative
A is preferred to B 2 times, and B is preferred to C 4 times, but alternative A is preferred to C 3 times
in stead of 8 times. This is called cardinal inconsistency, and for an inconsistent judgment, the incon-
sistent data should be identified and adjusted before it is used to make a valid decision. Therefore, the
inconsistent data processing issue in a judgment matrix has become a hot topic since the introduction
of the AHP/ANP. Currently, there are many methods for identifying and adjusting the inconsistent data,
for example, auto-adaptive algorithms in ( [14], [16]), absolute differences methods in ( [17], [18]), per-
turbation matrix method in [19] etc. However, the existing methods are either too complicated to delay
the speed of inconsistent data analysis or are difficult to preserve most of the original information in the
judgment matrix, or are extremely dependent on the priority weights. Therefore, it is necessary to pro-
pose a cardinal inconsistent data analysis model to effectively and simply identify the inconsistent data
in order to make a valid decision, which is independent to the priority weights while preserving most of
the original information in a judgment matrix. In an attempt to establish such models, the absolute differ-
ences of geometric mean matrix in [20], the induced bias matrix model (IBMM) in [21], were proposed
to identify the possible inconsistent data in a judgment matrix.

In this paper, a geometric mean induced bias matrix (GMIBM), which is only based on the original
matrix A, is established to identify the most inconsistent data in a judgment matrix. Through observing
and adjusting the largest bias data in the induced bias matrix, the consistency ratio of the judgment
matrix can be quickly improved to make a fast decision for emergence management. Besides, a general
estimating formula of the mined cardinal inconsistent data of GMIBM is provided.

The remaining parts of this paper are organized as follows. The next section briefly describes the
cardinal inconsistency in a judgment matrix. The theorems, corollaries and identifying processes of
GMIBM are further proposed and presented in Section 3. Two numerical examples introduced in [21]
are used to test the proposed model in Section 4. Section 5 concludes the paper.

2 Cardinal Inconsistency

Let the judgment matrix be A = [ai j]n×n, where ai j>0 and ai j =1/ a ji for all i, j,and k, if ai j = aikak j
holds for all i, jandk, then matrix A is said to be perfectly cardinal consistency, otherwise, it is called
cardinal inconsistency. In practice, it is unrealistic to obtain a perfectly cardinal consistency of matrix
A, therefore AHP allows a certain level of cardinal inconsistency of the judgment matrix. To measure
the consistency of a judgment matrix and determine a certain acceptable level of inconsistency, Saaty
proposed a consistency index (CI), denoted as:

CI =
λmax − n

n − 1
(1)

where λmax is the maximum eigenvalue of matrix A, and n is the order of matrix A.
To define a unique consistency test index that does not rely on the order of judgment matrices, the

consistency index (CI) was extended and the consistency ratio (CR) method was further proposed by
Saaty [17],

CR =
CI
RI

(2)

where CI is the consistency index shown in equation (1) while RI is the average random index based on
Matrix Size, as shown in Table 1.



452 D. Ergu, G. Kou, Y. Peng, F. Li, Y. Shi

Table 1 The Average Random Index

n 1 2 3 4 5 6 7 8 9 10
RI 0 0 0.52 0.89 1.11 1.25 1.35 1.4 1.45 1.49

If the CR of a judgment matrix is less than or equal to 0.1 (CR≤0.1), indicating the inconsistency is
relatively small, the judgment matrix is said to be of acceptable inconsistency. If CR is greater than 0.1
(CR>0.1), the judgment matrix is of unacceptable cardinal inconsistency, and the decision makers are
asked to revise their judgments. To effectively identify the cardinal inconsistent data and improve the
consistency ratio of a judgment matrix, a geometric mean induced bias matrix (GMIBM) by Hadarmad
product is proposed in the following sections.

3 Geometric Mean Induced Bias Matrix (GMIBM)

3.1 Theorem of GMIBM

To identify the inconsistent data, a geometric mean induced bias matrix (for short GMIBM, here-
inafter), which is only based on the original judgment matrix and independent to the priority weights, is
established to amplify the most inconsistent data in this paper. Then, the most inconsistent data can be
identified by observing the largest data in the induced bias matrix. The related theorems and corollaries
are presented in this section.

Theorem 1. The geometric mean induced bias matrix (GMIBM) C should be a U matrix if judgment
matrix A is perfectly consistent, that is,

C = Ā ◦ AT =
(
ci j

)
= ( n

√√
n∏

k=1

aikak j · a ji) = U i f aikak j = ai j (3)

where Ā =
(
āi j

)
n×n
=

(
n

√
n∏

k=1
aikak j

)
n×n

represents an n-by-n geometric mean matrix composed of all

geometric mean of aikak j for all i, j and k, U =


1 · · · 1
...
...

...

1 · · · 1

, n denotes the order of A, AT represents the
transpose of matrix A. Symbol ′ ◦ ′ denotes Hadamard product (e.g. C = A ◦ B means ci j = ai jbi j for all
i and j ).

Proof: If the judgment matrix satisfies the perfect consistency condition, namely, aikak j = ai j holds for
all i, j and k. Since ai j =1/a ji, we have

ci j =
n

√√
n∏

k=1

aikak j · a ji = n
√√

n∏
k=1

ai j · a ji = n
√

ai jn · a ji = ai ja ji = 1 (4)

Therefore, all entries in matrix C are ones if the matrix is perfectly consistent, and matrix C is a U matrix,
whose entries are ones. �

To simply compute the GMIBM and easily understand the theorem of GMIBM, the Theorem 1 is
transformed to following theorem.



Data Consistency in Emergency Management 453

Theorem 2. The geometric mean induced bias matrix (GMIBM) C should be a U matrix if judgment
matrix A is perfectly consistent, that is,

C = L × R ◦ AT =
(
ci j

)
= ( n

√√
n∏

k=1

aik · n
√√

n∏
k=1

ak j · a ji) = U i f aikak j = ai j (5)

where L =
(

n

√
n∏

k=1
aik

)
n×1

represents an n-by-one column matrix composed of geometric mean of rows

in matrix A, while R =
(

n

√
n∏

k=1
ak j

)
1×n

denotes an one-by-n row matrix composed of geometric mean of

columns in matrix A.

Corollary 3. The geometric mean induced bias matrix (GMIBM) C should be as close as possible to a
U matrix if judgment matrix A is approximately consistent.

Corollary 4. There must be some inconsistent data in the geometric mean induced bias matrix (GMIBM)
C deviating far away from one if the judgment matrix is inconsistent.

Based on Corollary 4, we can identify the most inconsistent data in matrix A by observing the largest
data deviating from 1 in the geometric mean induced bias matrix (GMIBM) C. Details of inconsistency
identification processes are presented below.

3.2 Inconsistency Identification and Adjustment Processes of GMIBM

To propose the inconsistency identification and adjustment processes of GMIBM based on above
Theorems and Corollaries, the aforementioned n-by-n judgment matrix A = [ai j]n×n is used in the fol-
lowing. The processes of inconsistent data analysis and adjustment of GMIBM include two major steps,
inconsistency identification and inconsistency adjustment:

Step I: Inconsistency Identification
Step 1: Compute a column matrix L and a row matrix R, which are composed of geometric means

of rows and columns respectively.

L =

A =



a11 · · · a1i · · · a1 j · · · a1n
...

...
...

ai1 · · · aii · · · ai j · · · ain
...

...

a j1 · · · a ji a j j
... a jn

an1 · · · ani · · · an j · · · ann



n

√
n∏

k=1
a1k

n

√
n∏

k=1
aik

n

√
n∏

k=1
a jk

n

√
n∏

k=1
ank

R = n
√

n∏
k=1

ak1, n
√

n∏
k=1

aki, n
√

n∏
k=1

ak j, n
√

n∏
k=1

akn

.

where 
L =

(
n

√
n∏

k=1
a1k, · · · n

√
n∏

k=1
aik, · · · n

√
n∏

k=1
a jk, · · · n

√
n∏

k=1
ank

)T
R =

(
n

√
n∏

k=1
ak1, · · · n

√
n∏

k=1
aki, · · · n

√
n∏

k=1
ak j, · · · n

√
n∏

k=1
akn

) (6)



454 D. Ergu, G. Kou, Y. Peng, F. Li, Y. Shi

Step 2: Compute geometric mean matrix by formula

Ā = L × R (7)

We can obtain the geometric mean matrix Ā. The computing processes are shown below:

Ā = L × R =



n

√
n∏

k=1
a1k
...

n

√
n∏

k=1
aik
...

n

√
n∏

k=1
a jk
...

n

√
n∏

k=1
ank


n×1

×
(

n

√
n∏

k=1
ak1, · · · n

√
n∏

k=1
aki, · · · n

√
n∏

k=1
ak j, · · · n

√
n∏

k=1
ank

)
1×n

=



n

√
n∏

k=1
a1kak1 · · · n

√
n∏

k=1
a1kaki · · · n

√
n∏

k=1
a1kak j · · · n

√
n∏

k=1
a1kakn

... · · ·
...

...

n

√
n∏

k=1
aikak1

... n
√

n∏
k=1

aikaki · · · n
√

n∏
k=1

aikak j · · · n
√

n∏
k=1

aikakn
...

...
...

...

n

√
n∏

k=1
a jkak1 · · · n

√
n∏

k=1
a jkaki · · · n

√
n∏

k=1
a jkak j · · · n

√
n∏

k=1
a jkakn

...
...

...
...

n

√
n∏

k=1
ankak1 · · · n

√
n∏

k=1
ankaki · · · n

√
n∏

k=1
ankak j · · · n

√
n∏

k=1
ankakn


n×n

.

=

 n
√√

n∏
k=1

aikak j


n×n

where L is an n-by-one column matrix, which is composed of all geometric means of rows while R is
a one-by-n row matrix R composed of all geometric means of columns, as shown in formula (6) and the
two edges of matrix A in Step 1. The geometric mean matrix Ā can be easily computed by multiplying
L to R.

Step 3: Compute geometric mean induced bias matrix (GMIBM) C by formula,

C = Ā ◦ AT =
(
ci j

)
=

(
āi j · a ji

)
= ( n

√√
n∏

k=1

aikak j · a ji) (8)

Step 4: Identify the data with the largest value, denoted as cmaxi j , deviating from 1 in matrix C, then
the corresponding ai j is regarded as the most inconsistent data in matrix A. If there are other data, say
cmn, cpq, whose values are also deviating far away from 1, then their corresponding data in matrix A,
amn, apq, can also be considered as the possible inconsistent elements. Once the inconsistent data are



Data Consistency in Emergency Management 455

identified, the following two steps are proposed to adjust the inconsistent data.

Step II: Inconsistency Adjustment:
Step 1: Estimate the value of identified inconsistent data by formula

ãi j =n−2

√√√ n∏
k=1,,i, j

aikak j =n−2
√√

āni j
a2i j
= āi j

(
āi j
ai j

) 2
n−2

(9)

where ãi j denotes the estimated value of the most inconsistent data ai j while āi j is the geometric mean
value located at the ith row and the jth column of geometric mean matrix Ā.

Step 2: Test the consistency of the revised matrix A by replacing the inconsistent data with the
estimated values.

To summarize, the processes of dealing with inconsistency by GMIBM include two major steps, that
is, inconsistency identification and inconsistency adjustment. The first two steps in Step I are used to
show the specific procedure of computing geometric mean matrix Ā. For simplicity, one can directly
use the latter two steps as the sub-steps of Step I to identify the most inconsistent data. To verify the
effectiveness and accuracy of GMIBM, in the following, two numerical examples introduced in [20] are
used to illustrate the proposed model.

4 Illustrative Examples

To test the effectiveness and correctness of the proposed GMIBM, and illustrate the processes of
the proposed specific inconsistency identification and adjustment by numerical examples, two numerical
examples in [21] are used in this paper.

Example 1 The Example 1 used in [21], which was firstly introduced in [20], is a 4 × 4 inconsistent
pair-wise comparison matrix A with CR=0.173>0.1.

A =


1 1/9 3 1/5

9 1 5 2

1/3 1/5 1 1/2

5 1/2 2 1


Apply the GMIBM to this matrix:

Step I: Inconsistency Identification
Step 1: Compute the column matrix L and the row matrix R by formula (6),

L =
(

0.5081 3.0801 0.4273 1.4953
)T
,R =

(
1.968 0.3247 2.3403 0.6687

)
Step 2: Compute geometric mean matrix by formula (7),

Ā = L × R

=


1 0.165 1.1892 0.3398

6.0615 1 7.2084 2.0598

0.8409 0.1387 1 0.2857

2.9428 0.4855 3.4996 1





456 D. Ergu, G. Kou, Y. Peng, F. Li, Y. Shi

Step 3: Compute geometric mean induced bias matrix (GMIBM) C by formula (8) ,

C = Ā◦AT

=


1 1.4848 0.3964 1.6990

0.6735 1 1.4417 1.0299

2.5227 0.6936 1 0.5715

0.5886 0.9710 1.7498 1


Step 4: Identify the largest value cmaxi j in matrix C. Here c

max
i j = c

max
31 = 2.5227, deviating from 1

in matrix C, then the corresponding element a31 in matrix A is regarded as the most inconsistent ele-
ment,indicating that it is smaller than its average values.

Step II: Inconsistency Adjustment
Step 1: Estimate the possible proper value of a31 using the estimating formula (9)

ã31 =
4−2

√√
ā431
a231
=

√
0.84094

(1/3)2
= 2.1213 ≈ 2

Step 2: Test the consistency of the revised matrix A by replacing the inconsistent elements a31 and
a13 with the estimated values 2 and 1/2. The revised matrix passed with CR=0.0028<0.1.

The identified inconsistent data and its estimated value are the same as the ones in [19] and [20], but
the proposed method is faster to find the inconsistent element and estimate the values.

Example 2 The second example in [20] is a 4 × 4 inconsistent pair-wise comparison matrix A with
CR=1.0242>0.1.

A =


1 2 4 18
1
2 1 2 4
1
4

1
2 1 2

8 14
1
2 1


Apply the GMIBM to this matrix:

Step I: Inconsistency Identification
Step 1: Compute the column matrix R and the row matrix L by formula (6),

L =
(

1 1.4142 0.7071 1
)T
, R =

(
1 0.7071 1.4142 1

)
Step 2: Compute geometric mean matrix by formula (7),

Ā = L × R

=


1 0.7071 1.4142 1

1.4142 1 2 1.4142

0.7071 0.5 1 0.7071

1 0.7071 1.4142 1





Data Consistency in Emergency Management 457

Step 3: Compute geometric mean induced bias matrix (GMIBM) C by formula (8),

C = Ā◦AT

=


1 0.3536 0.3536 8

2.2828 1 1 0.3536

2.2828 1 1 0.3536

0.125 2.8284 2.8284 1


Step 4: Identify the largest value cmaxi j in matrix C. Here c

max
i j = c

max
14 = 8, deviating from 1 in matrix

C, then the corresponding element a14 in matrix A is regarded as the most inconsistent element, indicat-
ing that it is smaller than its average values.

Step II: Inconsistency Adjustment
Step 1: Estimate the possible proper value of a14 using the estimating formula (9),

ã14 = ā14

(
ā14
a14

) 2
4−2
= 1 ·

(
1

1/8

)
= 8

Step 2: Test the consistency of the revised matrix A by replacing the inconsistent elements a14 and
a41with the estimated values 8 and 1/8. The revised matrix passed with CR=0<0.1.

The identified inconsistent data and the estimated value are the same as the ones in [20], but the
proposed method is faster to find the inconsistent data and estimate the values.

5 Conclusions

In this paper, we proposed a geometric mean induced bias matrix (GMIBM), which is only based
on the original matrix and independent to the way of deriving the priority weights, to identify the cardi-
nal inconsistent data in the judgment matrix. The inconsistent data identification process includes two
major steps, namely, inconsistency identification and inconsistency adjustment. The inconsistent data
can be easily and quickly identified by observing the data with the largest value(s) deviating from 1 in
the induced bias matrix C. Besides, the identified data can be estimated by the estimated formula. Two
examples are used to illustrate the proposed model. The results show that the proposed model is easier
and faster to identify and adjust the inconsistent data than existing models.

Acknowledgments
This research has been partially supported by grants from Academic Degree Programs Construction

at Southwest University for Nationalities (#2012XWD-S1201), and grants from the National Natural
Science Foundation of China (#70901015 and #70921061), the Fundamental Research Funds for the
Central Universities and Program for New Century Excellent Talents in University (NCET-10-0293).
No additional external funding received for this study. The funders had no role in study design, data
collection and analysis, decision to publish, or preparation of the manuscript.

Bibliography

[1] Tufekci, S., Wallace, W.A, The emerging area of emergency management and engineering,IEEE
Transactions on Engineering Management, 1998, 45(2):103-105, 1998.

[2] Shan, L., Zhao, Z.P, Evaluation on emergency management system in supply chain based on
trapezoidal fuzzy order weighted average (FOWA) operator, Applied mechanics and materials,34-
35:1170-1174, 2010.



458 D. Ergu, G. Kou, Y. Peng, F. Li, Y. Shi

[3] Chen, Y., Li, K., Xu, H.Y., Xu, H.Y, A DEA-TOPSIS method for multiple criteria decision analysis
in emergency management,Journal of Systems Science and Systems Engineering, 18(4):489-507,
2009.

[4] Zhang, K., Kluck, C., Achari, G, A Comparative Approach for Ranking Contaminated Sites Based on
the Risk Assessment Paradigm Using Fuzzy PROMETHEE, Environmental Management, 44(5):952-
967, 2009.

[5] Tie, Y.B. Tang, C, Zhou, C.H. The application of AHP to emergency response capability assess-
ment in urban disaster, Journal of Geological Hazards and Environment Preservation,16(4):433-437,
2005.

[6] Ergu, D., Kou, G., Peng, Y., Shi, Y., Shi, Y, The Analytic Hierarchy Process: Task Schedul-
ing and Resource Allocation in Cloud Computing Environment,Journal of Supercomputing, DOI:
10.1007/s11227-011-0625-1, 2011.

[7] Wu, W.; Kou, G.; Peng, Y.; Ergu, D., Improved AHP-group decision making for investment
strategy selection, Technological and Economic Development of Economy, 18(2), 2012. DOI:
10.3846/20294913.2012.680520

[8] Ergu, D., Kou, G., Shi, Y., Shi, Y., Analytic Network Process in Risk Assessment and Decision
Analysis, Computers & Operations Research, 2011. DOI: 10.1016/j.cor.2011.03.005.

[9] Levy J.K., Taji K., Group decision support for hazards planning and emergency management: A
Group Analytic Network Process (GANP) approach, Mathematical and Computer Modelling,46(7-
8): 906-917, 2007.

[10] Zhou Q., Huang W., Zhang Y., Identifying critical success factors in emergency management using
a fuzzy DEMATEL method, Safety Science, 49(2):243-252, 2011.

[11] Peng, Y., Kou, G., Wang, G., and Shi, Y.,FAMCDM: A Fusion Approach of MCDM
Methods to Rank Multiclass Classification Algorithms, Omega, 39(6): 677-689, 2011,
DOI:10.1016/j.omega.2011.01.009

[12] Ergu, D., Kou, G., Questionnaire Design Improvement and Missing Item Scores Estimation for
Rapid and Efficient Decision Making, Annals of Operations Research, DOI: 10.1007/s10479-011-
0922-3, 2011.

[13] Filip,F.G., D.A. Donciulescu, Cr.I. Filip.,Towards intelligent real-time Decision Support Systems
for industrial milieu,Studies in Informatics and Control , 11 (4):303-311, 2001.

[14] Filip F.G., Decision support and control for large-scale complex systems, Annual Reviews in Con-
trol, 32(1):61-70,2008.

[15] Xu.Z, Wei.C ., A consistency improving method in the analytic hierarchy process, European Journal
of Operational Research, 116:443–449, 1999.

[16] Cao, D., Leung, L.C., Law, J.S., Modifying inconsistent comparison matrix in analytic hierarchy
process: A heuristic approach, Decision Support Systems ,44:944–953, 2008.

[17] Saaty, T.L., The Analytical Hierarchy Process, New York: McGraw-Hill, 1980.
[18] Saaty, T.L., How to Make a Decision: The Analytic Hierarchy Process, Interfaces, 24:19-43, 1994.
[19] Saaty, T.L., Decision-making with the AHP: Why is the principal eigenvector necessary, European

Journal of Operational Research, 145(1): 85-91, 2003.
[20] Yang, Y.Q., Study on adjustment method for the inconsistency of the judgment matrix in AHP,

Operations research and management science, 8(3):12-16,1999 (in Chinese).
[21] Ergu, D., Kou, G., Peng, Y., Shi, Y., A Simple Method to Improve the Consistency Ratio of the Pair-

wise Comparison Matrix in ANP, European Journal of Operational Research, 213(1):246–259, 2011.