Emergency is a typical complex system because of its diversity, randomness, ... Based on complexity measure, the specific analysis of the cause for system.
Available online at www.sciencedirect.com
Systems Engineering Procedia 5 (2012) 289 – 294
International Symposium on Engineering Emergency Management 2011
A Model of Complexity Measurement for Emergency Decision Support System Yunhua WANGa,b,*, Shihong CHENa , Huiyan KEc a
School of Computer, Wuhan University, Wuhan ,430072, China School of Computer Science and Technology, Wuhan University of Technology, Wuhanˈ 430070, China c Soil and Water Conservation Monitoring Centre, Department of Water Resources of Hubei Province, Wuhanˈ 430071, China b
Abstract Emergency decision-making is the core of emergency management, and directly determines success or failure of emergency disposal activities. This paper introduced the information entropy in the complexity theory. With the combination of qualitative and quantitative methods, the paper proposed a model of evaluation and decision-making for complexity measurement based on information entropy. The model can effectively choose the best decisions in decision-making programs that have been developed and evaluation system. And an example shows its value. © 2012 2011Published Published Elsevier Selection and peer-review under responsibility Desheng byby Elsevier Ltd.Ltd. Selection and peer-review under responsibility of DeshengofDash Wu. Dash Wu Open access under CC BY-NC-ND license. Keywords: Complexity measurement; Information entr opy; Information analysis; Decision support System
1. Introduction Emergency is a typical complex system because of its diversity, randomness, sudden, disorderly and other features. The complexity of emergency refers to the states which are difficult to understand, describe, predict and control, but from the perspective of information theory, it is the state of the system that is expected to take quantity of information. The greater of the degree of complexity, indicates the more uncertainty and unpredictability of the states of incident, and need obtain more information to understand it [1-3]. The complexity characteristics of emergency determine the complexity of decision-making. And the level of understanding the emergency and the program of decision-making will affect its handling and control effects. Generally, there are two ways that respond to decision-making complexity of emergency. One the hand, it is as possible as to minimize or eliminate complexity from the point of management. The strategy is to simplify emergencies in order to improve their manageability, such as the status of the incident process, reducing the amount of resources. On the other hand, trying to understand and measure the complexity from the perspective of theoretical methods. That is to say, according to qualitative and quantitative description and analysis of system complexity, the purpose of effective choice system behaviour will be to achieve. Measurement is the basis for management, and the measure of system complexity is the basis for the studying quantitatively complexity. Based on complexity measure, the specific analysis of the cause for system complexity can quantitatively evaluate different disposal options in order to optimize emergency decision-making [4-6]. The key of emergency decision-making is that how to develop and optimize decision-making program. And the key of program optimization is choice of rational target weight with a certain amount of subjective and arbitrary [7]. 2211-3819 © 2012 Published by Elsevier Ltd. Selection and peer-review under responsibility of Desheng Dash Wu. Open access under CC BY-NC-ND license. doi:10.1016/j.sepro.2012.04.045
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How to better determine the target weight and eliminate subjectivity and optimize decision-making from a number of alternatives is main research aim of this paper. With of complexity measurement based on method of complexity theory, the paper proposed a model of evaluation and decision-making for complexity measurement based on information entropy, which can be used to describe quantitatively the complexity of target of emergency decisionmaking and evaluation. 2. Complexity Measurement In information theory, entropy reflects the degree of disorder and solves the problem of complexity measure as a measure of the scale of complex information. An indicator in the decision-making programs carry the more information that the index the greater the role of decision-making. The smaller the entropy value, the smaller the degree of disorder in system [2]. 2.1. Definition of Information Entropy Let discrete random variable X with n possible values ( x1 , x2 L xn ) , and their probability are respectively ( p1 , p2 L pn ) , then entropy of X is defined the following [8-10]: n
E( X )
H ( p1 , p2 , L , pn )
¦ [ pi log pi ]
(1)
i 1
n
In the formula, pi t 0 ˈ ¦ pi =1 If X denotes a system, xi and pi respectively denote n possible states and its i 1
occurrence probability in the system, E(X) is information entropy which is the amount of information required for describing the system X. E(X) also describes the degree of uncertainty of the system X. The greater the entropy, the system is more uncertainly, the greater the complexity. Information entropy described equation (1) has the following features: ķWhen only one is 1 and the others are 0 in pi ͑͝ the information entropy of the system is minimal. Namely E(X)=0, X is a completely deterministic system. ĸWhen the states of system are the same probability distribution ( pi
1/ n ), the information entropy of the
system is maximal and E(X)=logn. Because of X is the most uncertainly, the system is in the most complex state. ĹThe uncertainty, information entropy, and complexity of system will increase if pi changes by equalization. 2.2. Definition of Information Entropy Weight According to the concept and features, entropy combines the inherent information of programs to be selected in the multi-attribute decision-making evaluation with quantitative information of decision-makers’ subjective experience, which can be more essential to measure the complexity of programs and can create a model of multiattribute decision-making evaluation based on entropy weight. It is supposed that there are n programs to be evaluated, and an index system consists of m evaluation indexes, which can indicate by assessment matrix [Y ] ( yij ) num that normalized is [ X ] ( xij ) num . Then the entropy of the j-th index can be defined as follow: n
ej
k ¦ [ fij ln fij ] i 1
(2)
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Yunhua Wang et al. / Systems Engineering Procedia 5 (2012) 289 – 294 n
In the equation (2), f ij
¦x
xij
ij
ˈk
1 ln n ˄if fij =0ˈ fij ln fij =0, the j-th index of entropy weight
j 1
can be defined as follow. m
hj
1 ej m ¦ ej
(3)
j 1
3. Model of Emergency Decision-making Evaluation indexes of decision-making programs in emergency are often very complex. Therefore, in order to establish a more effective evaluation model, the indexes are divided into two levels that integrate to measure the complexity of the Decision Support System. 3.1. Establishing Original Data Matrix Assuming an emergency has formulated m decision-making programs, which established set of indexes U. The all evaluation indexes as a system that has l evaluation indexes of primary level (subsystem).Namely the system is U i {U1 , U 2 , L , U l } where there are ni evaluation indexes of secondary level in the i-th subsystem Ui
X
{U i1 , U i 2 , L , U ik } . Then the original data matrix of evaluation indexes is following:
ª x (1) º « (2) » «x » « M» « (n) » ¬x ¼
ª x (1)11 « (2) « x 11 « M « (m) ¬« x 11
x (1)1n1 L x (2)1n1 L M L (m) x 1n1 L
L L L L
In the above matrix, x
(k ) ij
x (1) l1 L x (2) l1 L M L (m) x l1 L
x (1) lnl º » x (2) lnl » M » » x ( m )lnl ¼»
indicates the assessed value of the k-th decision-making program in the j-th
secondary level of the i-th subsystem. 3.2. Standardization of Matrix In the multi-index comprehensive evaluation, it involves two basic variables which are the actual value of evaluation indexes and the evaluation value of each index. Each index represents different physical meaning, therefore, the dimension is different. Because of the different degree of non-public caused by the eliminated dimensional and dimensionless unit, non-dimensional treatment for the evaluation indexes is very important before decision-making. Maximized indexes and minimized indexes are mainly ways in the system of evaluation indexes, and are formalized as follow:
y ( k )ij
x ( k )ij max{ x ( k )ij }
(1 d i d m)
y ( k )ij
min{ x ( k )ij } x ( k )ij
(1 d i d m)
In the above two equations, y
(k ) ij
is the index value of non-dimension standardization.
To calculate the entropy of indexes and entropy weight in system, it is need to normalize the standardized matrix. The normalized formula is as follow:
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Yunhua Wang et al. / Systems Engineering Procedia 5 (2012) 289 – 294 m
y ( k )ij
Pij
¦y
(k ) ij
i 1
3.3. Computing Complexity Take U i for instance, the standardized and normalized matrix is as follow:
Pi
ª p (1)i1 « (2) « p i1 « « M « p ( m )i1 ¬
Where p
(k ) ij
p (1)i 2
L
p (2)i 2
L
M
L L
p
(m) i2
p (1)ini º » p (2)ini » » M » p ( m )ini »¼
is the j-th evaluation index of secondary level in the i-th subsystem of the k-th decision-making
program. Using equation (2) and (3) to get the complexity of the j-th evaluation index of secondary level in the i-th subsystem, that is, entropy value eij and entropy weight hij . Then the entropy vector of each subsystem is as follow.
Hi
(i 1, 2, L , l )
(hi1 , hi 2 , L , hini )
According to the additive property of entropy, the complexity of the i-th evaluation index of primary level is established. ni
Zi
l
¦ (1 eis )
ni
¦¦ (1 e
is
s 1
)
i 1 s 1
The index weight based on entropy may only represent local circumstances and be difference from actual situation sometimes. To solve the problem, the experience, i.e. expert judgment, is introduced for correction. Establishing comprehensive weight combined with entropy weight and expert judgment, which is as far as possible truly reflect the actual situation and provide more accurate decision-making support for decision-makers. The formula of Comprehensive weight
Zic
is as follow:
l
Zic Zi vi
¦Z v
i i
(i 1, 2, L , l )
i 1
Where vi is the weight value of expert judgment. Finally, using the comprehensive evaluation algorithm of AHP, the comprehensive evaluation value of k-th program can be formulated as follow: l
i
i 1
Where i
ni
¦¦ Z ch
F (k )
ij
p ( k )ij
j
1, 2, L , l ; j 1, 2, L , ni ; k 1, 2, L , m
4. Case Study
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Yunhua Wang et al. / Systems Engineering Procedia 5 (2012) 289 – 294
It is supposed that a large-scale landslide have occurred somewhere, and decision-makers had developed five rescue plan, which primary and secondary evaluation indexes are in the table 1. Table 1 Evaluation indexes of emergency decision-making. Feasibility (A)
Timeliness(B)
Risk (C)
Flexibility (D)
Economy (A1)
Personnel scheduling (B1)
Environment (C1)
Personnel (D1)
Technology (A2)
Equipment scheduling (B2)
Rescue (C2)
Equipment (D2)
Management (A3)
Rescue rate (B3)
Trapped (C3)
Rescue (D3)
Table 2 is the original data which is the average score of some experts’ judgment for each evaluation index. Table 2 Data of evaluation indexes Primary level
A
B
C
D
Secondary level
A1
A2
A3
B1
B2
B3
C1
C2
C3
D1
D2
D3
Program1
0.86
0.87
0.96
0.90
0.80
0.75
0.87
0.77
0.65
0.81
0.75
0.65
Program2
0.67
0.75
0.88
0.90
0.60
0.75
0.64
0.83
0.75
0.75
0.75
0.84
Program3
0.78
0.92
0.75
0.95
0.50
0.65
0.82
0.87
0.95
0.89
0.74
0.75
Prog.4
0.96
0.85
0.95
0.97
0.70
0.87
0.91
0.98
0.60
0.84
0.85
0.65
Prog.5
0.92
0.96
0.85
0.87
0.70
0.78
0.95
0.94
0.60
0.78
0.65
0.76
Using formula (4), (5) and (6) to calculate the standardized matrixes of all primary indexes:
P1i
ª 0.11 «0.17 « « 0.18 « « 0.22 «¬ 0.23
P2i
ª 0.21 « 0.18 « «0.22 « « 0.23 «¬0.20
0.20 0.20 º 0.18 0.21»» 0.22 0.18» P3i » 0.19 0.20 » 0.20 0.20 »¼ 0.25 0.21º 0.19 0.20 »» 0.14 0.16 » P4i » 0.20 0.22 » 0.22 0.22 »¼
ª0.19 «0.16 « « 0.23 « « 0.21 «¬0.22
0.18 0.19 º 0.18 0.20 »» 0.21 0.28 » » 0.20 0.18 » 0.24 0.17 »¼
ª 0.21 «0.19 « «0.20 « «0.22 «¬0.20
0.19 0.17 º 0.21 0.22 »» 0.21 0.20 » » 0.22 0.19 » 0.16 0.20 »¼
Then according to equation (2) and (3), the complexity of each secondary index is as follow: H1
(0.1803, 0.3607, 0.4590), H 2
(0.0558, 0.7386, 0.2026)
H3
(0.8179, 0.1450, 0.0371), H 4
(0.2222, 0.3889, 0.3889)
According W
to
equation
(9)
to
get
the
value
of
entropy
weight
of
primary
indexes:
(0.0194, 0.0487, 0.9090, 0.0072)
It is supposed that the reference weight established by experts is V
(0.30, 0.20, 0.35, 0.15) , and using equation
(10), comprehensive weight is W c (0.0174, 0.0291, 0.9503, 0.0032) At last, according to equation (11), the comprehensive evaluation value of reliability for each sample of rescue system can be calculated as shown in table 3.
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Yunhua Wang et al. / Systems Engineering Procedia 5 (2012) 289 – 294
Table 3 Comprehensive evaluation values Program
Evaluation value
Rank
Prog.1
0.8023
4
Prog.2
0.8104
5
Prog.3
0.8833
2
Prog. 4
0.8945
1
Prog5
0.8721
3
Table 3 showed that program 4 is the best among the all decision-making programs. 5. Conclusion The choice of emergency decision-making program directly relates to minimize disaster losses and social impact during the disposal process. Because the traditional optimization emergency programs rely solely on the decisionmakers and experts to obtain based on data of evaluation indexes, it is too subjective to response objective complexity of emergency. Using Information entropy method to give weight of each level in the evaluation index system, and calculate the complexity of primary and secondary indexes, and draw comprehensive weight of evaluation index through the revise of the index weight which is evaluated by experts, it is effective to avoid weighting subjectively in the system of emergency decision-making evaluation. Experimental analysis showed that this measurement method of complexity can choose optimal decision-making programs and point out the lack of the others.
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