Graduate Institute of Industrial Management
Southern Taiwan University
Application of a Fuzzy MCDM Model
to the Evaluation and Selection of
Smart Phones
Advisor:Prof. Chu, Ta-Chung
Student: Chen, Chih-kai
1
OUTLINE

INTRODUCTION

LITERATURE REVIEW

FUZZY SET THEORY

MODEL ESTABLISHMENT

NUMERICAL EXAMPLE

CONCLUSIONS
2
INTRODUCTION

Background and Motivation

Why We Apply FMCDM

Research Objectives

Research Framework
3
Background and Motivation

According to Gartner, International Research and Consulting:
1.
In the third quarter of 2010, worldwide mobile phone sales
volume of 417 million represented the growth of 35% over
the same period in 2009.
2.
In addition, smart phone sales volume is more substantial in
the third quarter of 2009 which grew 96%.
3.
The proportion of mobile phone sales volume has slightly
increased to 19.3%.
4



The prices of the mobile phones continue to lower and the
market tends to be saturated, the manufacturers cannot get as
high gross margin as they did in the past.
Therefore, manufacturers have begun to research and produce
smart phones.
The stereotype about the smart phones:
1. These devices are just for the businessmen.
2. It’s expensive.
3. Interfaces are too incomprehensible.
5
Why We Apply FMCDM


Evaluating smart phone many criteria (or factors) need to be
considered.
evaluating smart
phone
Quantitative
Qualitative
Hardware
Price
Brand Awareness
of Smart Phone
Operating Systems
Number of
Applications
Security
Different decision-makers also have different thoughts about
the weight of each criterion.
6
Research Objectives

The objectives of this study are listed as follows:
1.
Smart phone selection related to literature is investigated.
Criteria for selecting smart phones are analyzed.
A fuzzy MCDM approach is established.
A total relative area for ranking fuzzy numbers is suggested.
A numerical example is used to demonstrate the
computational process of the proposed model.
2.
3.
4.
5.
7
Research Framework
Chapter 1
Introduction
Chapter 4
Model Establishment
Chapter 2
Literature Review
Chapter 5
Numerical Example
Chapter 3
Fuzzy Set Theory
Chapter 6
Conclusion
8
LITERATURE REVIEW







Definition of Mobile Phone
What is a System of Smart Phone
The overview of the system development of the
smart phone Criteria Assessment
Related literatures on smart phones
Related Works on FMCDM
Fuzzy Number Ranking
Criteria Assessment
9
Definition of Mobile Phone
Feature Phones
Smart Phones
10
Feature Phones

Feature phone has its own mobile phone manufacturer’s
operating system (OS), which has a basic audio and video call
beside the additional features (e.g. taking photos, sending text
message, listening to music, etc.) but it is not allowed to install
or remove software (e.g. remove preset program or install GPS
software).

However, if the phone supports JAVA and BREW, it is able to
install applications.

The software being developed through the two systems is not
user-friendly.
11
Smart Phones

Su (2009) Smart phone is a phone equipped with function of
phone and PDA.

Yang (2009) Smart phones are developed because industrial
technology progresses and consumers require integrating
multiple requirements into one device.

Hsu (2004) In addition to the original function of voice
communications, a mobile phone should also be equipped with
an open operating system, and sufficient processing power,
allowing users to choose application freely and expand
multiple or limitless functions.
12
 According to the above-mentioned definitions of smart phones,
this study defined “Smart Phone” as follows:
1. Opening source operating system platform.
2.
Strong support on the third party's applications, and is allowed
to install or remove software freely.
3.
A strong hardware performance and faster processing power.
13
What is a System of Smart Phone

Alter (2002)
Operating System (OS)  “The system that controls the execution of all
other programs, communication with peripheral devices and use of
memory and resources.”

Malykhina (2007)
OS  “the heart of the smart phone, it determines a phone's features,
performance, security, and application installation.”
14
The overview development of the
smart phone

According to Gartner's statistics, the leading market
of smart phone operating system:
Symbian
29,480.1
3Q10
Market
share
36.6%
Android
20,500.0
25.5%
1,424.5
3.5%
iOS
Research In Motion
13,484.4
11,908.3
16.7%
14.8%
7,040.4
8,522.7
17.1%
20.7%
Microsoft Windows Mobile
22,247.9
2.8%
3,259.9
7.9%
1,697.1
1,214.8
2.1%
1.5%
1,918.5
612.5
4.7%
1.5%
80,532.6
100.0%
41,093.3
100.0%
Smart Phone Operating
Systems
Linux
Other OS
Total
3Q10
sales
3Q09
sales
18,314.8
3Q09
Market
share
44.6%
15
Related literatures on smart phones

Yang (2009), “ The Development Trend Analysis of
Smart Phone Industry”, he designs an expert questionnaire and interviews with 7 experts.




operating interface
entertainment platform
mobile business
specification of software and hardware.
16

Lin and Ye (2009), “ Operating System Battle in the
Ecosystem of Smart phone Industry”, they adopt concept of
Food Web to explain ecosystem of various smart phone OS.

They found out:
 device maker
 third-party application developer
are two key sources.
17
 Information Week (2007), “Survey of Emphasized Functions of
Smart phones towards 325 Experts in Related Field”.
The results demonstrated that security and easy integration with PC obtain
the first and second place respectively.

Gizmodo (2008), “Smart phone OS Comparison Chart”.
17 functions are listed in a table and the presence and absence of each
function are compared.
18

Many works investigated smart phone operating system as the
research object, however, these works cannot offer consumers a
method to evaluate and select smart phones.

Most importantly, when consumers choose to buy smart phones,
they will usually consider both hardware and operating system.

This thesis proposes a fuzzy multiple criteria decision making
approach to comprehensively consider criteria in hardware and
operating system in order to help consumers evaluate and select
smart phones.
19
Related literature on fuzzy MCDM

In 1970, Bellman and Zadeh introduced fuzzy set theory to
multi-criteria decision making, which involved fuzzy decision
analysis concepts and models for solving the problem of
uncertainty in decision-making.

Since then, the fuzzy multiple criteria decision making has
resulted in many researches.

Fuzzy numbers can be used to better describe suitability of
alternatives versus qualitative criteria under fuzzy multiple
criteria decision making environment.
20

Fuzzy MCDM is mainly divided into two
categories:
1. fuzzy multiple-objective decision-making:
It is mainly used in the "planning; design aspects ".
2. fuzzy multiple criteria (attributes) decision-making:
It is mainly used in the "assessment; selection aspects".
21

Chang et al. (2009) applied the fuzzy multi-criteria decision
making method in enterprise organization to establish the key
influential factors for the success of knowledge management.

Chou (2007) used fuzzy multiple criteria decision making
method to resolve the selection problem of transshipment
container port in marine transportation industry.
22
Fuzzy Number Ranking

In fuzzy multiple criteria decision making, the final evaluation
values are usually still fuzzy numbers.

A ranking method is needed to transform these final fuzzy
evaluation values into crisp values for decision making.

At present, there are many defuzzification methods which
have been investigated for ranking fuzzy numbers.
23

Some methods are briefly introduced as follows:

Liou and Wang (1992) introduced a total integral value generated by the
left and right integral values of a fuzzy number for ranking fuzzy numbers;
Chen and Hwang (1992) proposed ranking fuzzy numbers by preference
relations, the average of fuzzy number and degree, fuzzy rating, and
linguistic terms;
Abbasbandy and Hajjari (2009) presented a new approach for ranking
trapezoidal fuzzy numbers based on left and right spreads at some α-levels
of trapezoidal fuzzy numbers;
Farhadinia (2009) proposed a new approach to rank fuzzy numbers based
on the concept of lexicographical ordering in order to provide decision
makers algorithm in a simple and efficient way.



24

In this research, we suggest total relative area to rank the final
fuzzy numbers, and it is developed based on the concept of
Chen’s (1985) maximizing set and minimizing set which is
one of the most frequently used methods for the problems
under fuzzy MCDM environment.
25
Criteria Assessment
Evaluation criteria
Nature Literature source
Qualitative
brand awareness of
smart phone
C1
Benefit
This Study
brand awareness of
OS
C2
Benefit
J.D. Power(2010)
Designs
C3
Benefit
J.D. Power(2010)
Security
C4
Benefit
Jakajima (2008)
Operability
C5
Benefit
Jakajima (2008)
Entertainment
C6
Benefit
Su (2009)
Execution efficiency
C7
Benefit
Gizmodo (2009)
26
Quantitative
Screen size
C8
Benefit
This Study
Camera resolution
C9
Benefit
This Study
C10
Benefit
Lin and Ye (2009)
Price of applications
C11
Cost
Distimo (2010)
Price
C12
Cost
This Study
Weight
C13
Cost
This Study
Number of
applications
27
FUZZY SET THEORY





Fuzzy Sets
Fuzzy Numbers
α-cut
Arithmetic Operations on Fuzzy Numbers
Linguistic Values
28
Fuzzy Set
The fuzzy set A can be expressed as:
A  {( x, f A ( x)) | x U }
(3.1)
where U is the universe of discourse, x is an element in U,
A is a fuzzy set in U, f  x  is the membership function of A
at x. The larger f  x  , the stronger the grade of
membership for x in A.
A
A
29
Fuzzy Numbers

A real fuzzy number A is described as any fuzzy subset of the real
line R with membership function f A which possesses the following
properties (Dubois and Prade, 1978):
(a)
(b)
(c)
(d)
(e)
(f)
f A is a continuous mapping from R to [0,1];
f A ( x)  0, x  (, a] ;
f A is strictly increasing on [a ,b];
f A ( x)  1, x  b, c;
f A is strictly decreasing on [c ,d];
f A ( x)  0, x  [d , ) ;
where, A can be denoted as a , b , c , d  .
30

The membership function f of the fuzzy number A can also be
expressed as:
A
 f AL ( x), a  x  b,

bxc
1,
f A ( x)   R
 f A ( x), c  x  d ,
0,
otherwise,

(3.2)
where f AL ( x) and f AR ( x) are left and right membership functions of
A,respectively.
31
α-cut

The α-cuts of fuzzy number A can be defined as:
A  x | f A  x    ,  0, 1
(3.3)
where A is a non-empty bounded closed interval contained in R
and can be denoted by A  [ A , A ], where Al and Au are its lower
and upper bounds, respectively.
l
u
32
Arithmetic Operations on
Fuzzy Numbers

Given fuzzy numbers A and B, A, B  R , the α-cuts of A and B are
A  [ A , A ] and B  [B , B ], respectively. By interval arithmetic,
some main operations of A and B can be expressed as follows
(Kaufmann and Gupta, 1991):
l
u
l
u

 A  B
 [ Al  Bl , Au  Bu ]
(3.4)

 A  B
 [ Al  Bu , Au  Bl ]
(3.5)

 A  B
 [ Al  Bl , Au  Bu ]
(3.6)

 A () B 

 A  r 

Al Au
[  ,  ]
Bu Bl

(3.7)

 Al  r , Au  r , r  R 
(3.8)
33
Linguistic Variable

According to Zadeh (1975), the concept of linguistic variable is
very useful in dealing with situations which are complex to be
reasonably described by conventional quantitative expressions.
A1=(0,0,0.2)=Unimportant
A2=(0.1,0.3,0.5)=
Between Unimportant and Important
A3=(0.3,0.5,0.7)=Important
A4=(0.5,0.7,0.9)=Very important
A5=(0.8,1,1)=Absolutely important
Figure 3.1. Linguistic Values and Fuzzy
Numbers for Degree of Importance
34
MODEL DEVELOPMENT
Average Importance Weights
Aggregate Ratings of Alternatives versus
Qualitative Criteria
 Normalize Values of Alternatives versus Quantitative
Criteria
 Aggregate the Ratings and Weights
 Rank Fuzzy Numbers


35
Model Development

Dt decision makers, Dt , t  1,2,..., l
 Ai candidate of smart phones, Ai , i  1,2,..., m
 C j selected criteria, C j , j  1,2,..., n
 In model development process, criteria are
categorized
into three groups:
 Benefit qualitative criteria: C j , j  1,..., g
 Benefit quantitative criteria: C j , j  g  1,..., h
 Cost quantitative criteria: C j , j  h  1,..., n
36
Average Importance Weights
Assume w jt  (a jt , b jt , c jt ) , w jt  R , j  1,..., n , t  1,..., l ,
1
w j   (w j1  w j 2  ...  w jl )
l
(4.1)
1 l
1 l
1 l
where, a j   a jt , b j   b jt , c j   c jt .
l t 1
l t 1
l t 1
w jt represents the weight assigned by each decision maker for each criterion.
w j represents the average importance weight of each criterion.
37
Aggregate Ratings of Alternatives
versus Qualitative Criteria
Assume xijt  (dijt , eijt , fijt ) , i  1,..., m, j  1,..., g , t  1,..., l
1
xij   ( xij1  xij 2  ...  xijl )
l
(4.2)
1 l
1 l
1 l
where dij   dijt , eij   eijt , fij   fijt
l t 1
l t 1
l t 1
xijt denotes ratings assigned by each decision maker for each alternative versus each
qualitative criterion.
xij denotes averaged ratings of each alternative versus each qualitative criterion.
38
Normalize Values of Alternatives
versus Quantitative Criteria
the value of an alternative Ai , i  1,2,..., m,
versus a benefit quantitative criterion j, j  g  1,..., h, and
cost quantitative criterion j , j  h  1,..., n .
 y ij  (oij , pij , qij ) is
rij denotes the normalized value of y ij
rij  (
oij pij qij
, * , * ) , qij*  max qij , j  B
*
qij qij qij
(4.3)
oij oij oij
rij  ( * , * , * ) , o*  min o , j  C
ij
ij
qij pij oij

For calculation convenience, assume r
ij
 (aij , bij , cij ) , j  g  1,..., n .
39
The membership function of the final fuzzy evaluation value, Ti , i  1,..., n of each
alternative can be developed as follows:
g
Ti   w j  xij 

j 1


h

j  g 1


w j  xij 
n

j h1
wj  xij ,
(4.4)
The membership functions are developed as:
wj  [(b j  a j )  a j ,(b j  c j )  c j ] ,
(4.5)
xij  [(eij  dij )  dij ,(eij  fij )  fij ] .
(4.6)
40
From Eqs. (4.5) and (4.6), we can develop Eqs. (4.7) and (4.8) as follows:
wj  xij  [(b j  a j )(eij  dij ) 2  (dij (b j  a j )  a j (eij  dij ))  a j dij ,
(b j  c j )(eij  fij ) 2  ( fij (b j  c j )  c j (eij  fij ))  c j fij ] .
(4.7)
n
 w j  xij
j 1


n
n
n
j 1
j 1
n
n
j 1
j 1
 [ (b j  a j )(eij  dij )   (dij (b j  a j )  a j (eij  dij ))   a j dij ,
2
j 1
n
 (b j  c j )(eij  fij )   ( fij (b j  c j )  c j (eij  fij )))   c j fij ] .
j1
2
(4.8)
41

When applying Eq. (4.8) to Eq.(4.4), three equations are developed:
g
 wj

j 1
g
g
g
j 1
j 1
 xij  [ (b j  a j )(eij  dij )   (dij (b j  a j )  a j (eij  dij ))   a j dij ,

2
j 1
g
 (b j  c j )(eij  fij )
g
g
j 1
j 1
  ( fij (b j  c j )  c j ( eij  fij ))   c j fij ] .
2
j1
(4.9)
h

j  g 1
wj
 xij
h

[
j  g 1
h


j h1
wj  xij  [
(b j  c j )(eij  fij ) 
2
j g 1
n
(b j  a j )(eij  dij ) 
n

j h1
n

2

j  g 1
h

j  g 1
(b j  a j )(eij  dij ) 2 
 (b j  c j )(eij  fij )
jh1
h
2
(dij (b j  a j )  a j (eij  dij )) 
( fij (b j  c j )  c j (eij  fij )) 
n

j h1
n

j h1

j  g 1
a j dij ,
h

j  g 1
(dij (b j  a j )  a j (eij  dij )) 
( fij (b j  c j )  c j (eij  fij )) 
h
c j fij ] .
n

j  h1
(4.10)
a j dij ,
n

j h1
c j fij ] .
(4.11)
42
g
Oi1   a j dij
To simplify equation, assume:
j 1
g
Ai1   (b j  a j )(eij  dij )
j 1
Ai 2 
Ai 3 

j  g 1

j  h 1
(b j  a j )(eij  dij )
(b j  a j )(eij  dij )
g
Bi1    dij (b j  a j )  a j (eij  dij ) 
j 1
Bi 2 
Bi 3 
h

j  g 1
n

j  h 1
Ci1   (b j  c j )(eij  fij )
j 1
h
n
g
Ci 2 
Ci 3 
Oi 3 
h

j  g 1
n

j  h 1
(b j  c j )(eij  fij )
(b j  c j )(eij  fij )
 dij (b j  a j )  a j (eij  dij )  Di 3 


j 1
h

j  g 1
n

j  h 1
 fij (b j  c j )  c j (eij  fij ) 



j  g 1
a j dij
n

j  h 1
a j dij
g
Pi1   b j eij
j 1
Pi 2 
g
Di1    fij (b j  c j )  c j (eij  fij ) 
 dij (b j  a j )  a j (eij  dij )  D 

 i2
Oi 2 
h
Pi 3 
h

j  g 1
b j eij
n

j  h 1
b j eij
g
Qi1   c j fij
j 1
 fij (b j  c j )  c j (eij  fij ) 


Qi 2 
Qi 3 
h

j  g 1
c j fij
n

j  h 1
c j fij
43

By applying the above equations, Eqs. (4.9)-(4.11) can be
arranged as Eqs. (4.12)-(4.14) as follows:
g
 wj  xij  [ Ai1 2  Bi1  Oi1, Ci1 2  Di1  Qi1] .
(4.12)
j 1
h

wj  xij  [ Ai 2 2  Bi 2  Oi 2 , Ci 2 2  Di 2  Qi 2 ] .
(4.13)
wj  xij  [ Ai 3 2  Bi 3  Oi 3 , Ci 3 2  Di 3  Qi 3 ] .
(4.14)
j  g 1
n

j  h 1
44
Applying Eqs.(4.12)-(4.14) to Eq.(4.4) to produce Eq.(4.15):
Ti  [( Ai1  Ai 2  Ci 3 ) 2  ( Bi1  Bi 2  Di 3 )  (Oi1  Oi 2  Qi3 ) ,
(Ci1  Ci 2  Ai 3 )  ( Di1  Di 2  Bi 3 )  (Qi1  Qi 2  Oi3 )] .
2
Assume:
(4.15)
Ii1  Ai1  Ai 2  Ci 3
J i1  Bi1  Bi 2  Di 3
Ii 2  Ci1  Ci 2  Ai 3
J i 2  Di1  Di 2  Bi 3
Qi  Oi1  Oi 2  Qi3
Yi  Pi1  Pi 2  Pi 3
Zi  Qi1  Qi 2  Oi 3
45
By applying the above equations, Eqs. (4.15) can be arranged as
Eqs. (4.16) and (4.17) as follows:
Ii1 2  J i1  Qi  x  0
(4.16)
Ii 2 2  Ji 2  Zi  x  0
(4.17)
The left and right membership function of Ti can be obtained as
shown in Eq. (4.18) and (4.19) as follows:
a
1
2
f
 J i1   J i12  4 Ii1  x  Qi 

 ,Q  x  Y ;
x 

i
i
Ti
2 Ii1
f
 J i 2   J i 2  4 I i 2  x  Zi 

 ,Y  x  Z
x 

i
i
Ti
2 Ii 2
L
2
a
R
(4.18)
1
2
(4.19)
46
Rank Fuzzy Numbers

In this research, we applied total relative area to rank the final
fuzzy numbers, and it is developed based on the concept of
Chen’s (1985) maximizing set and minimizing set.
47

Using Chen’s maximizing set and minimizing set on the given
sets of examples, we can see three rankings for the same
alternatives.
Table 4.1 Chen’s maximizing set and minimizing set ranking outcomes
Example
A1
A2
A3
A4
Ranking
1
(3, 5, 7)
(4, 5, 51/8)
(2, 3, 5)
(8, 9, 10)
A1=A2
2
(3, 5, 7)
(4, 5, 51/8)
(2, 3, 5)
(6, 7, 8)
A1<A2
3
(3, 5, 7)
(4, 5, 51/8)
(2, 3, 5)
(10, 11, 12)
A1>A2
To resolve the inconsistency problem above, modification was
made to Chen’s ranking method.
48
Definition 1
The maximizing set M is a fuzzy subset with fM as
f
 xRi1  xmin k
) , xmin  xRi1  xmax ,
(
x

   xmax  xmin
M
0, otherwise .

(4.20)
The minimizing set N is a fuzzy subset with fN as
f
 xLi1  xmax k
) , xmin  xLi1  xmax ,
(
x

   xmin  xmax
N
0, otherwise ,

Where xmin  infx S , xmax  sup S , S  i1 Si
x
is set to 1.
n
(4.22)
, Si  {x f Ai ( x)  0} , usually
k
49
S ( xLi1 )  Yi  xLi1 
Yi
 f V  x  dx
xLi1
S ( xRi 2 ) 
xRi 2
L
ij
S ( xLi 2 ) 
xLi 2
 f
Yi

 x  dx   xLi 2  Yi
Vij

R
 f V  x  dx
Qi
L
Figure 4.1. Total Relative Area

Zi
S ( xRi1 ) 
ij
ST ( Ai ) 

 xL  xmax
 i2
 xmin  xmax

 f V  x  dx
xRi1
1
( S ( xRi1 )  S ( xLi1 )  S ( xRi 2 )  S ( xLi 2 )) , i  1 ~ n .
4
R
ij
50




By applying the new method, we can see that the rankings have
changed and are now consistent.
Table 4.4 Total relative area ranking outcomes
Example
A1
A2
A3
A4
Ranking
1
(3, 5, 7)
(4, 5, 51/8)
(2, 3, 5)
(8, 9, 10)
A1>A2
2
(3, 5, 7)
(4, 5, 51/8)
(2, 3, 5)
(6, 7, 8)
A1>A2
3
(3, 5, 7)
(4, 5, 51/8)
(2, 3, 5)
(10, 11, 12)
A1>A2
51


Compare the Chen’s (2010) maximizing area and minimizing area and the
total relative area method by changing values of A2.
Using Chen’s maximizing area and minimizing area on the given sets of
examples, we can see three rankings for the same alternatives as shown in
Table 4.6.
Table 4.6 Maximizing area and minimizing area ranking outcomes

Example
A1
A2
A3
A4
Ranking
1
(3, 5, 7)
(41/20, 5, 55/8)
(2, 3, 5)
(8, 9, 10)
A1=A2
2
(3, 5, 7)
(41/20, 5, 55/8)
(2, 3, 5)
(6, 7, 8)
A1>A2
3
(3, 5, 7)
(41/20, 5, 55/8)
(2, 3, 5)
(10, 11, 12)
A1<A2
The Chen’s (2010) method have the inconsistency problem.
52

By applying the total relative area method, we can see that the
rankings have changed and are now consistent as shown in
Table 4.8.
Table 4.8 Total relative area ranking outcomes
Example
A1
A2
A3
A4
Ranking
1
(3, 5, 7)
(41/20, 5, 55/8)
(2, 3, 5)
(8, 9, 10)
A1>A2
2
(3, 5, 7)
(41/20, 5, 55/8)
(2, 3, 5)
(6, 7, 8)
A1>A2
3
(3, 5, 7)
(41/20, 5, 55/8)
(2, 3, 5)
(10, 11, 12)
A1>A2
53

Using Liou and Wang method on the given sets of examples, we can see
three rankings for the same alternatives as shown in Table 4.10.
Table 4.10 The Liou and Wang method ranking outcomes
Example
A1
A2
A3
A4
Ranking
1
(3, 5, 7)
(41/20, 5, 55/8)
(2, 3, 5)
(8, 9, 10)
A1>A2
2
(3, 5, 7)
(41/20, 5, 55/8)
(2, 3, 5)
(6, 7, 8)
A1>A2
3
(3, 5, 7)
(41/20, 5, 55/8)
(2, 3, 5)
(10, 11, 12)
A1>A2

According to Table 4.10, the rankings are consistent; therefore, the
feasibility of this model can be demonstrated.
54
 xRi1 is
developed as follows:
xRi1  xmin
xmax  xmin
1



=  J i 2   J i22 +4I i 2 xRi1  Zi  2  2I i 2








xRi1
1
2
  xmax  xmin   J i 2   xmax  xmin    2I i 2 xmin    xmax  xmin   J i 2   xmax  xmin   +4I i 2  xmin  Zi 


=
2Ii 2
2
(4.29)
55
 xLi1 is
developed as follows:
xLi1  xmax
xmin  xmax
xLi1 =
  xmin  xmax   J i1 
1



=  J i1 +  J i21 +4I i1 xLi1  Qi  2  2I i1








 xmin  xmax    2Ii1xmax  +  xmin  xmax   J i1   xmin 
xmax   +4I i1  xmin  Qi 
2
1
2

2Ii1
(4.30)
56
 xRi 2 is
developed as follows:
xRi 2  xmin
xmax  xmin
1



=  J i1 +  J i21 +4I i1 xRi 2  Qi  2  2I i1








1
2
  xmax  xmin   J i1   xmax  xmin   2Ii1xmin  +  xmax  xmin  J i1   xmax  xmin  +4I i1  xmin  Qi 


xRi 2 =
2Ii1
2
(4.33)
57
 xLi 2
is developed as follows:
xLi 2  xmax
xmin  xmax
xLi 2 
  xmin  xmax   J i 2 
1



=  J i 2   J i22 +4I i 2 xLi 2  Zi  2  2I i 2








 xmin  xmax    2Ii 2 xmax    xmin  xmax   J i 2   xmin 
2Ii 2
xmax   +4I i 2  xmin  Zi 
2
1
2

(4.34)
58

By equations(4.29)，the S  xRi1  area, denoted as:
 
S xRi1 

 J i 2 Zi  xRi1
2 Ii 2

J i32
12 Ii22


3
2
1  2

J i 2  4 Ii 2 xRi1  Zi


12 Ii 2
(4.31)
xRi1 =equations(4.29)
59

By equations(4.30)，the
 
S xLi1  Yi  xLi1 

J i1 Yi  xLi1
2 Ii1

  area, denoted as:
S xLi1
3


3
1  2
1  2
2

J

4
I
Y

Q

J  4Ii1 xLi1  Qi  2
i1  i
i 
2  i1
2  i1

12 Ii1
12 Ii1
(4.32)
xLi1 =equations(4.30)
60

By equations(4.33)，the S  xRi 2  area, denoted as:
 
S xRi 2 

J i1 xRi 2  Qi
2 Ii1

3
2
J i31
1  2
J  4 Ii1 xRi 2  Qi

2  i1
 12 I 2
12 Ii1
i1


(4.35)
xRi 2 =equations(4.33)
61

By equations(4.34)，the S  xLi 2  area, denoted as:
 
S xLi 2 

 J i 2 xLi 2  Yi
2Ii 2

1


3
2
3
2
1
 J i22 +4I i 2 xL  Zi
 J i22 +4I i 2 Yi  Z i 

i2


12 Ii22 
12 Ii22 
 xL2  xL  xmax  Yi   Yi xmax  
i2

  i2
 xmin  xmax 


(4.36)
xLi 2 =equations(4.34)
62

By equations (4.31),(4.32),(4.35)and(4.36), the total
area, denoted as:
ST ( Ai ) 
S ( xRi1 )  S ( xLi1 )  S ( xRi 2 )  S ( xLi 2 )
4
,i 1 ~ n .
(4.37)
63
64