The optimum seeking method of Pavement maintenance based on interval fuzzy soft sets

Wu Congliang Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 5, Issue 8, (Part - 2) August 2015, pp.20-23
www.ijera.com 20 | P a g e
The optimum seeking method of Pavement maintenance based on
interval fuzzy soft sets
Wu Congliang1
, Yang Xiaoye2
School of Traffic and transportation1
, School of civil engineering ＆architecture2
, Chongqing Jiaotong University,
Chongqing, China
Abstract
For pressed for time or limited funding, Policy-makers may not know the accurate value of evaluation indexes,
and only know the interval-number of the evaluation index, in determining the order of pavement maintenance.
Aiming at this situation, the decision model, making decisions based on the scope of the evaluation index
values, of pavement maintenance is established. Model adopts the analytic hierarchy process (AHP) to
determine the weight of evaluation index, and sorts through the decision-making method of interval fuzzy soft
sets. At last, the same sort result with other methods was obtained by an example to prove the feasibility of the
model.
Key Words: pavement maintenance, interval-number, soft sets, decision model
I. INTRODUCTION
The road performance will gradually deteriorate
as it bears the effect of the traffic loads and
environmental factors, so it is important to
maintenance and repair the road timely and effective.
In the case of restricted budgets, the priorities of road
maintenance should be considered. Sorting, widely
used, is one of the most important methods in the
pavement management system around the world [1]
.
But, at present, the majority of sorting method is based
on the situation of the evaluation index attribute value
is known, does not take into account the case of only
knowing the interval-number of the evaluation index[2]
.
And, sometimes it is very difficult to know the exact
value of the index attribute. Meanwhile, time is short,
it is not necessary to investigate all roads within the
road network in detail. So, to select the most in need of
maintenance project by knowing the interval-number
of the evaluation index, which can achieve the result
that to save time, energy and money.
II. To establish the interval-number
decision model
Assuming that the number of pavement need
maintenance or reconstruction is m, the alternative
project sets h= { h1, h2,···, hi }, i∈m, m= {1, 2,···, m},
m≥2; there are n indicators reflecting pavement
performance, so the decision attribute set ε= {ε1, ε2,···,
εj}, j∈n, n= {1, 2,···, n}, n≥2; corresponding index εj,
the attribute value of project hi is interval-number［aij
-
,
aij
+
］, aij
-
is lower limit value, aij
+
is ceiling value, in
which i=1,2,···,m; j=1,2,···,n. The decision-making
information matrix constructed by original data is A =
(aij) m×n, aij=［aij
-
, aij
+
］.
2.1 decision matrix standardized treatment
Efficiency type attribute: pavement condition
index PCI, lateral force coefficient SFC, etc; cost
attribute: roughness index IRI, pavement crack rate,
etc. The model utilize the Interval-numbers decision
matrix standardization method in literature[3]
to
standardize the two kinds of attribute:
RESEARCH ARTICLE OPEN ACCESS

Efficiency attribute index was calculated by the
following equation:
bij
-
=
aij
−−min 1<𝑖<𝑚 (aij
−
)
max 1<𝑖<𝑚 (aij
+
)−min 1<𝑖<𝑚 (aij
−)
bij
+
=
aij
+−min 1<𝑖<𝑚 (aij
−
)
+
−)
··············equation (1)
Cost type attribute index according to the following
equation:
bij
-
=
+
)−aij
+
+
−)
bij
-
=
+
)−aij
−
+
−)
···············equation (2)
j=1,2,···,n。
After normalization of attributes, decision–
making information matrix composed of the original
data A=(aij)m×n is converted into standardized matrix
B=(bij)m×n，bij=［bij
-
, bij
+
］.
2.2 To determine decision attribute weights[4]
Determine the attribute weights Through the
analytic hierarchy process (AHP) is (w1,w2,···,wn),
and wj
n
j=1 =1,wj≥0, so the interval Numbers decision
evaluation matrix is R=（rij）m×n，rij=［rij
-
, rij
+
］.
R= （ bij ） * wj= ［ bij
-
* wj, bij
+
* wj ］
=
b11 ∗ w1
b21 ∗ w1
···
bm1 ∗ w1
b12 ∗ w2
b22 ∗ w2
···
bm2 ∗ w2
···
···
···
···
b1n ∗ wn
b2n ∗ wn
···
bmn ∗ wn
2.3 The fuzzy soft set decision method
2.3.1 The theory
Definition 1[5]
U is the initial field, E is the
parameter set. Sequence of (F, E) is called soft set if
and only if F is a mapping of power set from E to set U,
namely F: E→P (U), P (U) is the power set of U.
Definition 2[6]
U is the initial field, E is a set of
parameters, §(U) said all of fuzzy subset collection on
U. Make A∈E, the sequence of (F, A) is known as a
basic fuzzy soft set on U, F is a mapping, F: A→
§(U).
In short, a fuzzy soft set is Parameter set
composed of fuzzy subsets on field U. if ε ∈A. F(ε)
can be regarded as A fuzzy soft set of the ε
approximation of fuzzy set (F, A).
2.3.2 Decision-making method
Firstly, according to the decision method in
literature[7]
, to build basic fuzzy soft set (F, E). As
table 1:
Tabe1. The tabular form of basic fuzzy Soft Set (F,E)
U ε1 ε2 ··· εn
h1 ［r11
-
, r11
+
］［r12
-
, r12
+
］ ··· ［r1n
-
, r1n
+
］
h2 ［r21
-
, r21
+
］［r22
-
, r22
+
］ ··· ［r2n
-
, r2n
+
］
··· ··· ··· ··· ···
hm ［rm1
-
, rm1
+
］［ rm2
-
,
rm2
+
］
··· ［ rmn
-
,
rmn
+
］
Table {h1, h2,···, hi} is field, namely all
alternatives of the multiple attribute decision making
problems; ｛ε1,ε2,···, εj｝as the parameter set, namely
all decision attribute of the multiple attribute decision
making problems; rij said attribute value of
decision-making objects (alternatives) hi about the
parameter (decision attribute) εj.
Secondly, according to the data in table 4,
calculate choice value Ci of decision-making object hi
of the basic fuzzy soft set (F, E). Option value
calculated by the next equation:
Ci=［ui, vi］=［ rij
−m
j=1 , rij
+m
j=1 ］ ········Equation(3)
Where rij means fuzzy comprehensive evaluation
value of object hi about parameter εj, m said the
number of parameters.
Finally, according to the choice value of all
decision-making object, calculated decision values ri
of decision object (alternatives) hi (∀ hi∈U). Decision
value ri were calculated by the next equation:
riz ( ui − uj + (vi − vj))hj∈U ·············Equation(4)
so the object have maximum decision value ri should
be maintained at the earliest.
III. example
The feasibility and effectiveness of basic fuzzy
soft set multiple attribute decision making method be

verified by the example of road maintenance decision
problem in the paper [8]
.
The survey data of asphalt pavement using state
in a region as table 2, sorting the 5 need maintenance
road.
Tab.2 The data of road states
road SFC Crack rate/% IRI
1 25.8 3.7 2.85
2 20-25 0.5-2 3-4
3 25-30 1-3 3-5
4 18-22 1-3 4-6
5 20-25 3-5 2-4
Decision-making information matrix A can be
obtained through the table 2:
A=
25.8,25.8
20,25
25,30
18,22
20,25
3.7,3.7
0.5,2
1,2
1,3
3,5
2.85,2.85
3,4
3,4
3,5
2,4
3.1 Standardization
Crack rate and IRI belongs to the Cost type
indicator, standardizing with equation(2), SFC
belongs to benefit index, standardizing with
equation(1). So the normalized interval Numbers
decision matrix B can be calculated:
B=
0.20,0.24
0.16,0.23
0.20,0.28
0.14,0.20
0.16,0.23
0.06,0.17
0.11,0.61
0.07,0.61
0.07,0.61
0.04,0.20
0.20,0.29
0.14,0.27
0.11,0.27
0.09,0.20
0.14,0.41
3.2 To determine the weights
Determined through the analytic hierarchy process
(AHP), the attribute weights of each indicator is
w1=0.5，w2=0.2，w3=0.3
So the decision-making evaluation matrix R is:
0.100,0.120
0.080,0.115
0.100,0.140
0.070,0.100
0.080,0.115
0.012,0.034
0.024,0.122
0.014,0.122
0.014,0.122
0.008,0.040
0.060,0.087
0.042,0.081
0.033,0.081
0.027,0.060
0.042,0.123
3.3 build fuzzy soft set and make decisions
The five alternatives as filed, the three attributes
as parameter set, basic fuzzy soft set (F, E) can be set
up, such as table 3:
Table 3 basic fuzzy soft set (F, E)
U ε1 ε2 ε3
h1 0.100,0.120 0.012,0.034 0.060,0.087
h2 0.080,0.115 0.024,0.122 0.042,0.081
h3 0.100,0.140 0.014,0.122 0.033,0.081
h4 0.070,0.100 0.014,0.122 0.027,0.060
h5 0.080,0.115 0.008,0.040 0.042,0.123
In the table, filed is the five alternatives, namely
U= {h1, h2, h3, h4, h5,}; Parameter set E is the four
decision attribute, namely E= {ε1, ε2, ε3}, ε1 said SFC,
ε2 said Crack rate, ε3 said IRI.
So according to the data table 6, calculate by the
equation (3), the choice value Ci of the five
alternatives hi (∀ hi ∈U) are:
C1= 0.172,0.241 ， C2= 0.146,0.308 ，
C3= 0.147,0.343 ， C4= 0.111,0.282 ，
C5= 0.130,0.278
According to the choice value of the five roads,
calculate by the formula (4), the decision values ri of
the 5 alternatives hi (∀ hi∈U) are:
r1=-0.093 ， r2=0.112 ， r3=0.292 ， r4=-0.193 ，
r5=-0.118
According to the above decision value size: r4 < r5 < r1
< r2 < r3, so the order of road maintenance plan is:
h3→h2→h1→h5→h4. That is to say, under the
condition of the limited funding, road 3 should be
maintained at the earliest.
IV. Conclusions
An interval number multiple attribute decision
making model has been established through applying
with the analytic hierarchy process (AHP) and interval
fuzzy soft set decision method. The model can
determine the order of maintenance when decision
makers only know the scope of the attribute value and
do not know the specific value. This paper obtained
the same sequence of maintenance with the literature

[8]
, but the multiple attribute decision making method
in the literature [8]
need complex computation, this one
is easier. The solution to the problem of multiple
attribute decision making method to avoid the human
subjectivity and randomness is put forward in this
essay, and the results is more objective. The model
provides a reference for the maintenance or rebuilding
project decisions.
References
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Pavement Maintenance Management and
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Traffic Press, Beijing, China.
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[3] HU Mingli, FAN Chengxian, SHI Kaiquan.
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matrix [J]. Journal of computer science, 40
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[4] LIAO Hongqiang, QIU Yong, YANG Xia,
WANG Xinggang, GE Renwei. Study on
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[5] Molodtsov D, Softset Theory-first Results[J].
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[6] P.K. Maji, R. Biswas, A.R. Roy. Fuzzy Soft
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[7] YANG Xibei, Tsau YOUNG Lin, YANG
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[8] WANG Chao-hui, WANG Xuan-cang, MA
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