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CHEMICAL ENGINEERING TRANSACTIONS

VOL. 46, 2015

A publication of

The Italian Association
of Chemical Engineering
Online at www.aidic.it/cet

Guest Editors: Peiyu Ren, Yancang Li, Huiping Song
Copyright © 2015, AIDIC Servizi S.r.l.,
ISBN 978-88-95608-37-2; ISSN 2283-9216

Catastrophe Progression Method and Its Application to
Selection of Construction Scheme

Xianqi Zhang*, Xiaoshuang Fu, W eiwei Han, Luning Zhang

School of Water Conservancy, North China University of Water Resources and Electric Power, 450045, Zhengzhou, China.
zxqi@163.com

On the basis of the specific complexity, ambiguity and uncertainty of the selection of construction scheme for
multi-attribute decision making system, a new evaluation method based on catastrophe theory is proposed and
applied to the decision making for Construction Scheme. Firstly, the multi -layer assessment objects are
analyzed with this method and property values of evaluation indexes are quantified. Then, the ultimate
catastrophe function is obtained through different catastrophe models. In order to calculate, integrate and obtain
the total value of Catastrophe, the decision making can be carried out. The outcome from concrete applicatio n of
construction scheme optimization indicates that it is reasonable and has avoided the problem of calculating
coefficient of weight subjectively with clear thread and simple calculation.

1. Introduction

Selection of Construction Scheme is an important part of optimization in construction, which is restricted by
investment, construction period, construction quality, safety, environmental protection, energy policy and other
factors. Therefore, in order to achieve better economic efficiency and reduce energy consumption, it is very
necessary to compare different schemes in detail. Evaluation index system is large and complex, and the
interaction between the evaluation index is common due to the subjectivity of decision -makers and the
ambiguity of evaluation index, the interaction affects the decision making results (Zhu and Wu (2015), Li and
Zhang (2014)). In recent years, domestic and foreign scholars have carried on a lot of research on the
construction scheme optimization methods, and have achieved fruitful results. A lot of decision making methods
are proposed and applied, for example multi-criteria decision-making method (Vahid et al (2014), Daniel et al
(2014), Ke et al (2012)), simulation and goal programming (Karimi et al (2013), Panagiotis et al (2010), Yang
(2014)), TOPSIS method (Rodolfo and Renato (2014), Shu and Liu (2013), Raju and Kumar (2015)), Value
Engineering Method (Shu (2014)), Pattern recognition method (Ferreira et al (2014)), Fuzzy matter-element
evaluation method (Wang et al (2004)) and Grey method(Zhou and Xiao (2014), Lv and Cui (2002)). It seems
that either of these methods is legitimate, but there are also some problems in the mechanism of reflecting the
evaluation system (Edmundas et al (2015)). Catastrophe progression evaluation method comes from the
catastrophe model, which can be used to solve the multi-attribute decision making problem (Hidekazu et al
(2014), Lev and Alexander (2015)). The main characteristics of this method is that it carries out the
decomposition of multi-level conflicts on systematic evaluation overall goal at first. Then it brings forward
membership function based on catastrophe theory combined with fuzzy mathematics, carrying on integrated
quantitative calculations by the normalized formula. At last, it generates a comprehensive parameter and carries
out evaluation with the parameter.

2. Catastrophe progression method

Catastrophe theory was put forward by the French mathematician Rene Tom who stemmed from his book of
"Structural Stability and Morphogenesis" published in 1972. This book systematically described the catastrophe
theory (Antonio (2013)). It studied how nature and human society in a continuous gradient caused Catastrophe
or leaps and seeks a unified mathematical model to describe, predict and control the Catastrophe or leap as a
new science. Catastrophe theory is a mathematical tool to describe the system transition, giving the parameter
region when system is stable and unstable. Accordingly, it proved that when parameters verified, the situation of

DOI: 10.3303/CET1546127

Please cite this article as: Zhang X.Q., Fu X.S., Han W.W., Zhang L.N., 2015, Catastrophe progression method and its application to selection
of construction scheme, Chemical Engineering Transactions, 46, 757-762 DOI:10.3303/CET1546127

757

system would also change. When the parameters came through some certain specific locations, the state of
system will occur Catastrophe.
For a dynamic system, the potential function of the system can be described as :

),( UTFF  (1)

F is a collection of system state parameter it ,  ntttT 21 , ; U is a collection of system control
parameter

i
u ,  

n
uuuU 

21
, .

By solving the equations, we can get:

0




F
, 0




F
(2)

Get the critical point of system equilibrium state:

),(),,(),,(
2211 nn

UTUTUT 

Therefore, the center of the catastrophe theory is to study the process characteristics of the system by
studying the transformation between the critical points. If the element in the control parameters U is
not more than 4, then the function F is at most 7 mutated forms, namely Folding Catastrophe, Peak
point Catastrophe, Dovetail Catastrophe, and Butterfly Catastrophe. In the case of Peak point Catastrophe, the
catastrophe manifold and bifurcation are shown in Figure 1.

Figure 1: Manifold of Peak point Catastrophe

2.1 Basic model

Table 1: Catastrophe models and normalized equations

order type tendency function
dimension
of control
variable

normalized equation

1
Folding

Catastrophe
3

( )  V x x ax 1 ax a

2
Peak point

Catastrophe
4 2

( )   V x x ax bx 2 ax a ,
3

b
x b

3 Dovetail
Catastrophe

5 3 21 1 1
( )

5 3 2
   V x x ax bx cx 3


a

x a , 3bx b ,

4
c

x c

4 Butterfly
Catastrophe

6 4 3 21 1 1 1
( )

6 4 3 2
    V x x ax bx cx dx 4


a

x a , 3bx b ,
4

c
x c , 5dx d

The theoretical basis of catastrophe progression method is catastrophe theory, which uses topology theory and
singularity theory in dynamic system to build mathematical model. Then the process of qualitative change is

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described and forecast when something continuous interruptions happen in natural phenomena and society
activity. W hen state variable is one dimension, there are four Catastrophe models seen from table 1.
There are two kinds of variables in tendency function:(1) state variable x represents behavior state of the
system, (2) control variables (a, b, c, d)can be regarded as the factors influencing behavior state. It starts from
tendency function, analyses balance curved surface, singular point set and bifurcation set of different
Catastrophe models. Then, normalized equation and evaluation standard can be acquired.
Tendency function has two contradictions between state variable and control ones in the system. These
variables interact with each other. Hence, every state of system is the unification of state variable and control
ones, even of interactions with all control variables.

2.2 Evaluation standard
Facing various practical problems, there are three different kinds of criteria when applying Catastrophe theory to
fuzzy composite analysis and evaluation.
(1) Non-complementary criteria: effect of different control variables can’t be replaced. That is, they can’t
complement their shortcomings according to the criteria of min-max.
(2) Complementary criteria: they can make up their deficiency according to mean value.
(3) Complementary over threshold: they can complete after reaching some threshold which risk level can be
accepted. Only obeying above criteria can the requirements of bifurcation equation in Catastrophe theory be
satisfied.

3. Method of catastrophe progression to optimal scheme

3.1 Construct the system of evaluation indexes
According to the object of optimal Construction Scheme, decompose evaluation objects into some layers from
higher to lower. Above indexes are often abstract and not easy to quantify. After decomposition, more concrete
indexes can be obtained to quantify readily. When sub-index is calculated, the decomposition will discontinue.
Inverted tree hierarchical structure is set up through arraying all indexes. Requirement of Catastrophe
evaluation is that more important risk indexes is put frontier and less important ones is behind. In addition,
remark that some unimportant indexes should be removed and the layers had better not more than four.

3.2 Normalize decision making matrix
In order to avoid the effect of various dimensions, nondimensionalization of decision making matrix is the first
step. Every single index relates to corresponding fuzzy value. Evaluation indexes corresponding to standard
scheme belongs to fuzzy membership degree, which is called preferential membership. They are commo nly
normal. Based on these criteria, such criterion is named preferential membership. In fact, some characters are
better when they are larger, while others may be better when smaller. So, different equations are adapted to
different membership. However, there are many equations to calculate. To reflect its relativity sufficiently, this
paper employs following forms:
The better, the larger

/


 
m

ij ij ij

X X

(3)

The better, the smaller

(1 / ) / ( 1)


  
m

ij ij ij

j i

X X m

(4)

ij
 is preferential membership. m is the number of scheme.

Preferential membership decision making matrix
mn

R is set up.

1 2

1 11 21 1

2 12 22 2

1 2

 
 
 
 
 
 
 
 

mn m

n n n mn

M M M

R C

  

759

3.3 Determine the type of catastrophe
When the construction plan is respectively divided into 2, 3 and 4 layers, their corresponding Catastrophe type is
peak point, dovetail and butterfly.

3.4 Data integration and evaluation
According to the lowest hierarchy primitive data fuzzy membership of all indexes, decision scheme is selected.
Then, use the normalized equation of different Catastrophe systems to composite step by step until the highest
layer of construction plan.

4. Case study

The total distance of some channel in the project from south to north is 1840m. The channel is trapezoid section,
whose bottom is from 13m to 18m wide and longitudinal shrinking slope is 1/28000. Its first side slope is 1/2~1/3
in the bottom and external slope is 1/1.5~1/1.2. First levee crest is 5.0m wide. Channel slope plate is 10cm thick.
Bottom channel is 8cm thick.

4.1 Construct the system of decision making evaluation indexes
On the basis of practical technical level in China construction and Catastrophe evaluation theory, the system of
decision making evaluation indexes is constructed. Criteria affecting project construction contain basic demand,
technical demand, and social profit and so on. Experts give some score to the se quantitative indexes which
seen as table 2.

Table 2: Construct the system of decision making evaluation indexes

Object Criteria indexes
Attribute of scheme

Scheme Ⅰ Scheme Ⅱ Scheme Ⅲ

Optimal
scheme of

construction

B1 basic
demand

C1cost/10000yuan 4568 4700 4380

C2duration/month 15 17 16

C3quality/% 85 82 80

B2 technical

C4degree of hard/% 80 85 78

C5technical
extension/%

80 90 75

C6management
range/%

78 85 70

B3 social
profit

C7social model/% 75 80 72

C8environmental/% 80 82 78

4.2 Normalize the decision making matrix
According to the equation (3) and (4), normalized decision making matrix is constructed as follows:

1 2 3

0.331 0.348 0.321

0.344 0.323 0.333

0.344 0.332 0.324

0.335 0.325 0.340

0.327 0.367 0.306

0.336 0.318 0.350

0.330 0.352 0.317

0.333 0.342 0.325

 
 
 
 
 
 
 
 
 
 
 
 
 
 

M M M

R C

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4.3 Catastrophe method to integrate and select schemes
For scheme Ⅰ, integrate the data on the basis of Catastrophe progression evaluation method. Indexes C1, C2
and C3 form dovetail Catastrophe model based on the complementary criteria,

1/ 2 1/3 1/ 4 1/ 2 1/3 1/ 4

1 1 2 3
( ) / 3 (0.331 0.344 0.344 ) 0.6807      B C C C .

Indexes C4, C5, C6 form dovetail Catastrophe model based on the complementary criteria,
Indexes C7, C8 form peak point Catastrophe model based on the complementary criteria,

1/ 2 1/3 1/ 2 1/3 1/ 4

3 7 8
( ) / 2 (0.335 0.327 0.333 ) 0.6341     B C C .

Indexes B1, B2, B3 form dovetail Catastrophe model on the basis of min-max criteria,

1/ 2 1/3 1/ 4 1/ 2 1/3 1/ 4

1 1 2 3
min( , , ) min(0.6801 , 0.6757 , 0.6341 ) 0.8250  A B B B .

Based on the similar theory, A2＝0.8235 for scheme Ⅱ and A3＝0.8194 for scheme Ⅲ.
A1＞A2＞A3 so scheme Ⅰ is the optimal plan.

5. Conclusions

Construct Catastrophe model step by step with the method of Catastrophe progression evaluation. Intergrades
all the indexes based on the inner interaction in the normalized equation. Sort indexes in accordance to intrinsic
logical relationship to the importance. This can make up the shortcoming of static evaluation method to select
plans recently used and reduce subjectivity of assigning weight. In this way, subjective decision making caused
by uncertainty can be avoided. Catastrophe progression method is applied to select plans through determine,
quantifying and integrating the evaluation model. Evaluation tends to the truth and the result is reasonabl e.
Compared with other methods, Catastrophe evaluation has easy calculation, clear thread and accurate result.
Note that there is still something subjectivity in the process of sorting indexes. In addition, normalized equation is
integrated. Final composite value is higher and the difference among evaluation values is smaller. Hence, visual
effect on evaluation is worse and there is still something to improve in the later research.

Acknowledgments

This work is financially supported by NSFC-Henan Provincial People's Government Joint Fund of Personnel
Training (No. U1304511), Henan Provincial Natural Science Foundation (No. 132300410020, No.
112300410035), Collaborative Innovation Center of W ater Resources Efficient Utilization and Protection
Engineering, Henan Province, Program for Science & Technology Innovation Talents in Universities of Henan
Province (No. 15HASTIT049), and Program for Innovative Research Team (in Science and Technology) in
University of Henan Province (No. 14IRTSTHN028). Our gratitude is also extended to reviewers for their efforts
in reviewing the manuscript and their very encouraging, insightful and constructive comments.

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