Location Optimization of Multiple Distribution Centers under
Fuzzy Environment
Yong Wang
Ph.D. Student
School of Transportation Engineering, Southeast University
Nanjing, China, 210096
Email: yongwx6@uw.edu
Xiaolei Ma
Ph.D. Student
Department of Civil and Environmental Engineering, University of Washington, Seattle, WA, 981952700
Email: xiaolm@uw.edu
Yunteng Lao
Ph.D. Student
Department of Civil and Environmental Engineering, University of Washington, Seattle, WA, 981952700
Email: laoy@uw.edu
Yinhai Wang (Corresponding Author)
Professor
Box 352700
Department of Civil and Environmental Engineering, University of Washington
Seattle, WA 98195-2700
Tel: (206) 616-2696
Fax: (206) 543-1543
Email: yinhai@u.washington.edu
Haijun Mao
Professor
School of Transportation Engineering, Southeast University
Nanjing, China, 210096
Email: maohaijun@seu.edu.cn
Wang, Ma, Lao, Wang, Mao
1
ABSTRACT
Locating distribution centers optimally is a crucial and systematic task for logistic operators and
decision makers. Optimally located distribution centers can significantly improve the logistic
system’s efficiency and reduce its operational cost. However, it is not an easy task to optimize
distribution center locations and previous studies focused primarily on location optimization of a
single distribution center. With the growing logistics demands, multiple distribution centers
become necessary to meet customers’ requirements, but few studies have tackled the Multiple
Distribution Centers Location (MDCL) problems. This paper presents a comprehensive
algorithm to address the MDCL problem. Fuzzy integration and clustering approach using the
Improved Axiomatic Fuzzy Set (AFS) theory is developed for location clustering based on
multiple hierarchical evaluation criteria. Then, Technique for Order Preference by Similarity to
Ideal Solution (TOPSIS) is applied for evaluating and selecting the best candidate for each
cluster. Sensitivity analysis is also conducted to assess the influence of each criterion in the
location planning decision procedure. Results from a case study in Guiyang, China, reveals that
the approach developed in this study outperforms other similar algorithms for multiple
distribution center location selections. This new method may easily be extended to address
location planning of other types of facilities, including hospitals, fire stations, and schools.
Keywords: Multiple distribution centers; Location selection; Clustering algorithm; Axiomatic
Fuzzy Set; TOPSIS.
Wang, Ma, Lao, Wang, Mao
2
INTRODUCTION
Distribution centers play a vital role in transportation and logistics system. With the growing
urban freight movements, a single distribution center in a large region cannot efficiently
accommodate the growing demands from customers due to lengthened transit time and high
warehouse handling cost, especially for Just-In-Time and Cold Chain deliveries. Therefore,
systems with multiple distribution centers become highly desirable in practice and locating these
centers optimally is a critical issue faced by logistics system operators and transportation
planners (Yang et al. 2007). A method for optimal location selection of multiple distribution
centers helps balance production and consumption, which further improves the efficiency in
logistic systems, optimizes distribution system network, and alleviates urban traffic congestion.
However, developing such a method is not a straightforward task because many factors and
criteria need to take into account in the decision-making procedure (Sun et al. 2008; Syam 2002).
For example, distance to the shared production facility, local traffic pattern, economic condition
for each potential distribution center, all need to be considered. These factors require certain
domain knowledge to justify and should be adequately quantified and incorporated into the
multiple locations selection problem. It is worth mentioning that a general solution to the
problem of multiple distribution center locations selection (MDCL) also benefits the planning of
other types of multiple facility locations such as hospitals, fire stations, and schools.
Most previous studies focused on selecting single distribution center location rather than
simultaneously selecting multiple distribution center locations. The two most classical
quantitative frameworks for single facility location selection are the cost minimization approach
(Hansen et al. 1992; Lee 1993; Tyagi and Das 1995) and the profit maximization approach
(Hakimi and Kuo 1991). Yang et al. (2007) constructed a mathematical model to minimize the
total relevant cost, including setup cost, turnover cost, and transportation cost with each
customer’s demand. Sun et al. (2008) presented a bi-level mixed-integer programming model to
seek for the optimal distribution center location, where the upper-level model minimized the
planners’ cost, and the lower-level model minimized the customers’ cost. Hande et al. (2011)
also formulated a bi-level mixed-integer nonlinear programming model to deal with a
competitive facility location problem based on the game theory, which maximized the profit and
the attractiveness of each facility simultaneously.
MDCL problem needs to take into account a wealth of influential factors such as resource
availability and local economic conditions. These factors are traditionally measured by human
perception, and are considered fairly difficult to process by computers in an automatic fashion.
To tackle the vagueness of the selection criteria, Fuzzy set theory was used to evaluate the
numerous attributes of potential facilities’ locations, and has been utilized to handle ambiguous
scenarios in the decision-making procedure by many researchers ((12), (13), (14), (15)). One of
the main merits for fuzzy set theory is that it is capable of not only considering uncertainty in
human’s cognitive process, but also evaluating different subjective attributes by linguistic terms
((16), (17)). Fuzzy set theory was then developed and applied in the facility location selection
process, since valuation criteria for each alternative location are often defined in linguistic terms,
e.g. “Low”, “Medium”, ”High”, ”Poor”, “Fair”, “Good”, etc. (13). This suggests that Fuzzy set
theory is well suitable to transform these linguistic attributes into fuzzy relations, fuzzy numbers,
and fuzzy inference systems (18). Thus, there are many facility location selection studies
Wang, Ma, Lao, Wang, Mao
3
employed fuzzy set theory to interpret evaluation criteria, such as labor costs, resource
availability, and freight regulation (19).
Based on fuzzy set theory, numerous systematic analysis methods are adopted into the
potential location ranking process. Chou proposed an integrated short-term and long-term
multiple-criteria decision-making approach to solve location selection problems (20). Wey and
Chang presented a hybrid analytical hierarchy process (AHP) and data envelopment analysis
(DEA) location ranking method (21). Yu et al. (22) ranked and evaluated the potential location
plans for multiple urban transit hubs by using an AHP-based framework and fuzzy set theory.
Among all location ranking algorithms, the TOPSIS algorithm proposed by Negi is most widely
used and has been improved by many researchers (23). Chu presented a fuzzy TOPSIS model
under a group of decisions to solve the facility location selection problem (19). Awasthi et al. (15)
also used the fuzzy TOPSIS approach to evaluate and select the best urban distribution center.
However, these methods can only be used to select single facility location, and most of them are
hard to expend to tackle the MDCL problem.
To fill this research gap and provide a potential solution for the MDCL problem, a novel
approach for multiple distribution centers location selection is proposed. The remaining part of
this paper is organized as follows: Comprehensive evaluation index system is first established by
using fuzzy integration and clustering algorithms, which groups several potential locations into
different clusters. Then fuzzy TOPSIS method is applied for selecting the alternative distribution
centers location within each cluster. Next, a case study of MDCL in Guiyang, China is presented
to demonstrate the effectiveness of our method, followed by a comparison among different
relevant algorithms. Last and but not least, a sensitivity analysis is conducted to examine impacts
incurred by changing selection criteria weights. Finally, conclusions are given at the end of this
paper.
METHDOLOGY
The study methodology contains three parts: 1) major criteria and sub-criteria are established to
form a hierarchical analysis structure; 2) the linguistics variables used for location evaluation are
then defined and transformed into fuzzy numbers for further assessments; and 3) a location
planning framework for urban multiple distribution centers is finally developed based on the
fuzzificated evaluation criteria.
Hierarchical analysis structure for Location Planning
The location selection criteria are used for evaluating potential urban distribution centers location.
As shown in Figure 1, six major criteria ( Ci ) and sixteen sub-criteria ( Cij ) are finalized to
determine the locations for implementing urban distribution centers. These criteria are obtained
from previous studies (e.g. Chen, 2001; Heizer and Render, 2004; Stevenson 2005) and are
considered important after the discussions with several transportation experts and members in the
local transportation agencies.
Wang, Ma, Lao, Wang, Mao
4
Geological condition C11
Natural
environment C1
Hydrology condition C12
Landform condition C13
Road facilities C21
Transportation
C2
Accessibility C22
Connectivity to Multi-modal transport C23
Business
environment C3
Proximity to Customers C31
The degree of close to logistics center C32
Resource availability C41
Candidate land
C4
Sustainable freight regulations C42
Land costs C43
Water supply C51
Supply condition
C5
Power supply C52
Gas supply C53
Environmental
impact C6
Drainage C61
Solid castoff disposal C62
l
s
er
nt
ce
n
io
ut
ib
tr
is
D
e
pl
ti
ul
M
FIGURE 1 Hierarchical Analysis Structure for Locations Selection of Multiple Distribution
Centers
These experts gave their linguistic evaluations on each sub-criterion. Based on these
evaluations, each potential location can be evaluated and graded using linguistic terms by the
experts. The scores of an upper level criterion can then be calculated from these sub-criteria
evaluations.
The above sub-criteria structure can be divided into two categories: cost criteria and benefit
criteria. The criterion C43 , criterion C61 and criterion C62 fall into the cost criteria, where an
alternative location with higher cost is less likely to be selected as the final location. The
remaining sub-criteria are grouped as benefit criteria, where an alternative location with higher
benefit criteria is more likely to become the final location.
Linguistic Variables Fuzzification and Related definitions
Linguistic Variables Fuzzification
As the first step, natural languages are converted into numerical inputs using fuzzy set theory.
Adopted in this study is the triangle fuzzy transformation, in which a triangular fuzzy number is
Wang, Ma, Lao, Wang, Mao
5
represented as a triplet n = (n1 , n2 , n3 ) (15, 25), and the membership function µn ( x) can be
calculated based on the triangular fuzzy number n as follows:
0,
 x−n
1

,
 n2 − n1
µ n ( x) = 
 n3 − x ,
 n3 − n2

0,
x ≤ n1 ,
n1 ≤ x ≤ n2 ,
(1)
n2 ≤ x ≤ n3 ,
x ≥ n3
Where n1 , n2 , n3 are real numbers, x at n2 gives the maximal grade of µn ( x) , and x at n1
and n3 gives the minimal grade of µn ( x) .
In the fuzzy set theory, each linguistic term is transformed into a fuzzy number. Due to the
uncertain nature of the MDCL problem, criteria ratings and alternative ratings are used as the
linguistic variables. In the current study, as shown in Table 1, we apply a scale of 1-9 (VL-VH)
to rate the criteria and (VP-VG) to rate the alternative locations. Table 1 presents the linguistic
variables and fuzzy numbers for each criterion and alternative.
TABLE 1 Linguistic Terms for Criteria and Alternative Ratings
Linguistic term
Abbreviation
Fuzzy numbers
Very Low (Very Poor)
VL (VP)
(0.56，0.62，0.65)
B.VL&L
(B.VP&P)
(0.62，0.65，0.68)
L (P)
(0.65，0.68，0.73)
B.L&M
(B.P&F)
(0.72，0.75，0.78)
M (F)
(0.75，0.78，0.83)
B.M&H
(B.F&G)
(0.80，0.83，0.90)
H (G)
(0.85，0.88，0.93)
B.H&VH
(B.G&VG)
(0.90，0.93，0.96)
VH (VG)
(0.95，0.98，1)
Between Very Low and Low
(Between Very Poor and Poor)
Low (Poor)
Between Low and Medium
(Between Poor and Fair)
Medium (Fair)
Between Medium and High
(Between Fair and Good)
High (Good)
Between High and Very High
(Between Good and Very Good)
Very High (Very Good)
Related Definitions
To address the MDCL selection problem, several related definitions are needed and presented as
follows.
Wang, Ma, Lao, Wang, Mao
6
Definition 1. Notations for the parameters are defined as follows:
D{Di i = 1, 2,  , m ' } denotes the decision makers who evaluate the alternative distribution
center locations;
C 1{Ct1 t = 1, 2, , r} denotes the major criteria for location selection of multiple distribution
centers;
C 2 {Ct2 t = 1, 2, , k} denotes the sub-criteria for location selection of multiple distribution
centers;
A{ Ai i = 1, 2, , n} is the potential location i of urban distribution centers;
=
Ah' { Ahi' i 1,=
2, , n ; h 1, 2, , c} is the number of potential locations within cluster
'
=
Wut2l (u 1,=
2,, m' ; t 1, 2, , r )
h;
denotes the fuzzy numbers for sub-criterion l of major criterion t
by decision maker u ;
=
Wut2' (u 1,=
2,  , m ' ; t ' 1, 2,  , k )
denotes the fuzzy numbers for sub-criterion t by decision
maker u ;
2
=
X uhit
, m' ; h 1,=
, n' ; t ' 1, 2, , k ) denotes the fuzzy numbers for alternative
1, 2,=
2,, c; i 1, 2,=
' (u
location i under sub-criterion t ' by decision maker u within cluster h ;
=
xujt2 l (u 1, 2,=
, m' ; j 1,=
2, , n; t 1, 2, , r ) denotes the fuzzy numbers for alternative location j
under sub-criterion l of major criterion t by decision maker u ;
p '{ ph' h = 1, 2, , c ' } represents each initial cluster, and c ' is the total number of initial clusters.
p{ ph h = 1, 2, , c} represents each final cluster, and c is the total number of final clusters.
Definition
2. X {x1 , x2 , , xn } ⊆ R n denotes the sample set of potential urban distribution centers
=
location; µ = {µ1 , µ2 , , µr } is the attribute set of X ; xtj = µt ( x j ) ， t = 1, 2, , r ， j = 1, 2, , n ， where
xtj denotes the membership function value of the sample x j with attribute µt ; The attribute µt
can be further separated as mt ,1 , mt ,2 , , mt , st , thereby, the attribute set µ can be expressed as
µ =M = {m1,1 , m1,2 , , m1,s1 , m2,1 , m2,2 , , m2,s2 ,  , mr ,1 , mr ,2 , , mr ,sr } , which is defined as the fuzzy attribute (concept)
set.
Definition 3. Let X , M be sets, 2 M is the power set of M , τ：X × X → 2 M , if τ satisfies the
following conditions (26),
AX 1：∀( x1 , x2 ) ∈ X × X ,τ ( x1 , x2 ) ⊆ τ ( x1 , x1 )
AX 2：∀( x1 , x 2 ),( x2 , x3 ) ∈ X × X ，τ ( x1 , x2 ) ∩ τ ( x2 , x3 ) ⊆ τ ( x1 , x3 )
Wang, Ma, Lao, Wang, Mao
7
( M ,τ , X ) is called an Axiomatic Fuzzy Set (AFS) structure, X is called a sample set, M is
called an attribute (concept) set, τ is called structure.
Definition 4. Let R be the binary relation of the sample set X (27), when x, y ∈ X , x ≠ y, R
satisfies the following conditions,
(1) If ( x, y ) ∈ R , then ( x, x) ∈ R ;
(2) If ( x, x) ∉ R and ( y, y ) ∈ R , then ( y, x) ∈ R ;
(3) If ( x, y ) ∈ R and ( y, z ) ∈ R , then ( x, z ) ∈ R ;
(4) If ( x, x) ∈ R and ( y, y ) ∈ R , then ( x, y ) ∈ R or ( y, x) ∈ R ;
R is called sub-preference relation, the corresponding concept of sub-preference relation is
called simple concept, otherwise R is called complex concept.
Definition 5. Let m be the simple fuzzy concept, and m ∈τ ( x, y ) , ρ m：X → R + =[0, ∞) (28),
(1) ρ m ( x) =
0 ⇔ ( x, x) ≠ Rm , x ∈ X ;
(2) ( x, y ) ∈ Rm ⇒ ρ m ( x) ≥ ρ m ( y ), x, y ∈ X ;
ρ m is called membership function of simple fuzzy concept m .
Definition
6.
Let
η(x j )
be
the
simple
fuzzy
concept
set,
that
is,
η ( x=
or st ;j 1, 2, , n} , let ζ A be the fuzzy description of each
2, , r; s ' 1 or  =
j )={mt , s ' |t 1,=
j
sample Aj , then,
ζ Aj =argmin{ ∏ mt , s' | mt , s' ∈η ( x j ),=t 1, 2, , r;=
s ' 1 or  or st ;=j 1, 2, , n} .
Definition 7. Let X , M be two sets, and let ( M ,τ , x) be an AFS structure (29), A ⊆ X , B ⊆ M ,
then B(=
A) { y | y ∈ ρ m ( x), m ⊆ B, ∀x ∈ A} , where B ( x) denotes the samples belonging to the degree
of B is less than or equal to the sample x belonging to the degree of B .
Definition 8. Let X be a sample set, and X ⊆ R n , ρ m is the membership function of simple
fuzzy concept m , and m ⊆ B , then Lm ( x) is measure of sample x belonging to the simple fuzzy
concept m , the Lm ( x) is defined as
Lm ( x) =
∑
∑
x∈m ( x )
ρ m ( x)
ρ m ( x)
x∈ X
(2)
Definition 9. Let the fuzzy concept B ⊆ M , the membership function µ B ( x) of fuzzy concept B
is defined as
Wang, Ma, Lao, Wang, Mao
8
=
µ B ( x) inf ( Lm ( x)) ∈ [0,1] ， x ∈ X
(3)
m∈B
Definition 10. Let X be the sample set of potential urban distribution centers location, M is the
simple fuzzy concept set of X , B ⊆ M , n is the sample size, the membership information
entropy (30) function E ( B ) and the membership distribution coefficient function D( B) can be
defined as
E ( B ) = − ∑ ( µ B ( x) lg( µ B ( x)))
(4)
D( B) = −(( ∑ µ B ( x) n) lg( ∑ µ B ( x) n))
(5)
x∈ X
x∈X
x∈X
When E ( B ) becomes smaller, the membership degree of the sample x within concept B
is more approaching to the two ends of the interval [0, 1], and the boundaries are clearer. When
D( B) is smaller, the membership degree of the sample x within concept B is more
approaching one end of the interval [0, 1], instead of both left and right ends. Thereby, in order
to comprehensively evaluate the membership information entropy and distribution coefficient,
the evaluation index V can be defined as: V = E ( B) D( B) , where when V is smaller, it is more
reasonable that the concept B describes the sample X .
The Proposed Framework for Location Planning
There are three major steps included in this framework. Each evaluation sub-criterion should be
properly mapped into the higher hierarchical criterion based on the fuzzy integration method in
the first step, and next, clustering algorithm is undertaken to group the possible alternative
locations into different clusters. The last step is to finalize the final location within each cluster
by using the fuzzy TOPSIS approach. These steps are presented in details as follows:
Fuzzy integration method based on sub-criteria
The xujt2 and Wut2 are expressed respectively as triangular fuzzy numbers: xujt2 = (aujt2 , bujt2 , cujt2 ) ,
Wut2 = (hl2 , gl2 , kl2 ) . The comprehensive evaluation index from all of experts for location j under
l
l
l
l
l
l
l
major criterion t can be expressed as
'
1
Z=
tj
m
1
2
2
⊗ ∑ (( xujt
⊗ Wut21 ) ⊕  ⊕ ( xujt
⊗ Wut2s ))
'
1
s
m × ts u =1
(6)
Where ⊗ denotes vector multiplication, and ⊕ denotes vector addition. Z tj1 denotes the
comprehensive triangular fuzzy number to evaluate the location j under major criterion t , m'
denotes the number of experts, ts denotes the number of sub-criteria is s under major criterion t .
Let Y = (a, b, c) be a triangular fuzzy number, the representation of triangular fuzzy number
1
Y is P(Y ) =
(a + 2b + c) ((13), (14) (19), (20), (31), (32),). Thereby, if we suppose that
4
Wang, Ma, Lao, Wang, Mao
9
Z tj1 =(Ttj1 ,Q1tj ,H1tj ) denotes the triangular fuzzy number, then the integrated membership function
value µtj1 of the location j under major criterion t will be described as
µtj1 =
1 1
(Ttj +2Q1tj +H1tj )
4
(7)
Clustering algorithm procedure
The integrated membership function value µtj1 calculated from the fuzzy integration method is
used as the input. Our next step is to conduct clustering analysis to divide the potential
distribution centers into different clusters. Axiomatic fuzzy set (AFS) theory logic ((27),(28),(29))
has been proven as an effective approach to tackle human perception related clustering problem.
Since the traditional AFS method is not appropriate to incorporate a large number of criteria (33),
an improved AFS theory logic algorithm is detailed as follows:
Step 1. According to Definition 2, the integrated membership function value µtj1 can be
transformed into three values, they are expressed as ρ m , ρ m , ρ m respectively, and if we define
t ,1
t ,2
t ,3
µt ( x j ) = µ ,and then, the membership function corresponding to each membership function
1
tj
value µtj1 can be written as
ρ mt ,1 ( x j ) = µt ( x j )
ρ mt ,2 ( x=
ht ,1 − µt ( x j )
j)
ρ mt ,3 ( x j ) =
ht ,3 − µt ( x j ) − ht ,2
(8)
ht ,1 = max {µt ( x1 ), µt ( x2 ), , µt ( xn )}
µt ( x1 ) + µt ( x2 ) +  + µt ( xn )
n
=
ht ,3 max{ µt ( x j ) − ht ,2 + ht ,2 }
ht ,2 =
1≤i ≤ n
=
t 1,=
2, , r; n number of potential locations
Step 2. Calculate the fuzzy attributes of each sample.
Step 2.1 Calculate Lmt ,1 ( x j ) , Lmt ,2 ( x j ) , Lmt ,3 ( x j ) through Eq.(2) and the Step 1.
Step 2.2 Define µmt ,1 ( x j )=Lmt ,1 ( x j ) , µmt ,2 ( x j )=Lmt ,2 ( x j ) , µmt ,3 ( x j )=Lmt ,3 ( x j ) , and the maximum
of membership fuzzy values under simple fuzzy concept µt can expressed as:
{
}

µm ( x j ) = ρ m ' ( x j )= max µmt ,1 ( x j ), µmt ,2 ( x j ), µmt ,3 ( x j )

t ,s
ηt , s' ( x j ) = mt , s' t ,s'

, t 1,=
2, , r , s ' 1 or 2 =
or 3, j 1, 2, , n
=

Step 2.3 Respectively calculate the ratio of membership information entropy and
distribution coefficient function, which corresponds to each sample attribute of step 2.2 as
(9)
Wang, Ma, Lao, Wang, Mao
Vη
Step
{
t ,s'
2.4
(xj )
10
= E (ηt , s' ( x j )) D(ηt , s' ( x j ))
η(x j )
Let
be
the
(10)
simple
}
fuzzy
concept
set,
that
is,
( x j ) | t 1,=
2, , r ; s ' 1 or 2 =
=
η ( x j ) η=
or 3; j 1, 2, , n .
t , s'
Step 2.5 Select the smallest value µa ( x) which corresponds to attribute a ; Select the
second smallest value µb ( x) which corresponds to attribute b ; they are defined as follows:
µa ( x j )
=
( x j )'
Let η ( x j )' be η=
inf ( Lα ( x j )) ∈ [0,1]
(11)
α ∈η ( x j )
{η ( x ) − {a}} , and µ ( x ) is,
j
=
µb ( x j )
b
inf ( Lα ( x j )) ∈ [0,1]
(12)
α ∈η ( x j )'
Step 2.6 Find the evaluation index Vη
t ,s'
evaluation index Vη
t ' ,s'
(xj )
(xj )
≥ Vη
t ,s'
(xj )
( x j )'
attributes are expressed as η=
(xj )
< Vη
t ,s'
(xj )
which corresponds to attribute a , and the
t ,s'
t ' ,s'
t ' ,s'
(xj )
which corresponds to attribute b , and compare Vη
Step 2.7 If Vη
until Vη
j
(xj )
and Vη
t ' ,s'
(xj )
;
, then eliminate attribute a , and the remaining sample
{η ( x ) − {a}} , and return to step 2.5, continue the recursion
j
;
Step 2.8 Return the final remaining attributes η ( x j )' for each sample x j , the remaining
attributes can be used to describe the sample x j , thereby, we get the fuzzy attributes of each
sample via the above step 2.1-step 2.8.
Step 3. Clustering procedure will be based on the fuzzy attributes of each sample
Step 3.1 The fuzzy description of each sample Ai is ζ Ai , and we construct the fuzzy
{
sample relation rij = min µζ
Ai Λζ A j
} . Based on universe of discourse X = { A , A ,, A } ,
( Ai ), µζ A Λζ A ( Aj )
i
j
1
2
n
Liu (1998) has demonstrated that the integer k exists to make ( M Λk )2 = M Λk , thereby(29), we can
deduct the equivalence relation at universe of discourse X based on the fuzzy sample relation
k
matrix=
R M
=
(rijk ) n×n .
Λ
Step 3.2 Calculate the initial clustering results based on the fuzzy equivalence matrix
Denote the diagonal element of fuzzy sample relation matrix as riik . It can be verified as
k
r=
rijk ≤ riik , thereby we can identify the different membership function values based on the
ji
diagonal elements and other elements in the matrix. They are expressed as α l (l = 1, 2, , g ) , and
Wang, Ma, Lao, Wang, Mao
11
these values gradually increase according to the sequence of l . Find rijk =riik =α l in the fuzzy
sample relation matrix; the corresponding samples can be grouped into one or multiple clusters,
and the remaining sample’s relation where rijk > α l can be grouped as another cluster. This
procedure will recursively continue until l = g .
Step 3.3 Calculate the weight of each fuzzy description within clusters
The samples are divided into different clusters according to different α l
values, we
suppose that the initial clusters are p , p , , p based on α l , calculate the weight of each fuzzy
'
1
description of ph' : ξ p
'
h
{
= C p' , wp'
h
h
} , ξ = {m
x
{
'
2
g , s'
'
c'
}
| g ∈ {1,2, , r=
} , s' 1 or 2 or 3, x ∈ ph' , ξ x denotes the fuzzy
}
{
}
attributes set of ph' , C p = m m ∈ ξ x , x ∈ ph' ,=
wp wm m ∈ C p , wm =
'
h
'
h
'
h
ph' m
∑
m∈C
'm
h
p
{
}
, ph' m = x x ∈ ph' , m ∈ ξ x ,
ph'
where . is defined as the number of elements, wm ∈ wp' is the weight of m ∈ C p' .
h
h
Step 3.4 Finalizing the clusters based on the weighted membership function
For each sample x ∈ X , calculate the weighted membership function values
'
f ( c , w,h ) ( x) = ∑ wm Lm ( x) , h = 1, 2, , c , in each initial cluster, where Lm ( x) is the measure of
m∈C
ph'
initial sample, select f ( c , w,h* ) ( x) = arg max { f ( c , w,h ) ( x)} which corresponds to h* , then x ∈ ph' , and
*
1≤ h ≤ c'
finally
{
obtain
the
result
of
}
clusters
p1 , p2 , , pc
,
and
let
ξ x = mg ,s' | g ∈ {1, 2, , r=
} , s ' 1 or 2 or 3, x ∈ ph , C p = {m m ∈ ξ x , x ∈ ph } .
h
Step 3.5 Calculate the clustering validity index
According to different thresholds α ∈ [0,1] , we can select the optimal result from the
clustering validity index. Some notations are introduced to cluster the validity indices, N is the
number of samples, p1 , p2 , , pc are the cluster results, c is the number of clusters,
 m
V = {v1 , v2 , , vc } is the set of sample center in each cluster, =
vh =
vh

∑ρ
x∈ ph
m

( x) n ph m ∈ C ph  ,

1 ≤ h ≤ c , where ρ m is the membership function of simple fuzzy concept m , and n ph is the
number of samples in cluster h . Thereby, the clustering validity index Iα can be expressed as
follows:
c
c × (c − 1)∑ ∑
Iα =
∑
h=
1 x∈ ph m∈M
ρ m ( x) − vhm
2
2
 c c
2 × α ×  ∑ ∑ ∑ vhm − vkm 
 h =1 k =1,h ≠ k m∈M

(13)
Wang, Ma, Lao, Wang, Mao
c
c
∑ ∑ ∑
12
vhm − vkm
where RC = h =1 k =1,h ≠ k m∈M
hk
2
describes the dispersion degree between clusters, which can be
c × (c − 1)
2
used for merging different clusters, and
c
∑∑ ∑
h=
1 x∈ ph m∈C ph
ρ m ( x) − vhm
2
denotes the closeness of samples
within each cluster when Iα is the smallest value. This means the clustering is the most distinct.
Ranking within cluster
The Fuzzy TOPSIS approach (cite) is used to evaluate the potential locations based on the
selected 16 sub-criteria. The goal is to find the negative ideal solution, where the benefit criteria
is minimized and the cost criteria is maximized, and the positive ideal solution, where the cost
criteria is minimized and the benefit criteria is maximized ((15),(34)). From the previous
definitions, Ah' is the number of potential locations with cluster h , thereby, the fuzzy numbers of
sub-criteria and potential locations are defined as: W 2 =(wut 1 ,wut 2 ,wut 3 ) , X 2 =(a uit ,buit ,cuit ) ,
'
ut '
=
(u 1, 2,=
, m ; h 1,=
2,, c; i 1, 2,
=
, n ; t 1, 2, , k )
'
'
'
'
'
uhit '
'
'
'
respectively. The procedures of fuzzy TOPSIS
approach are described as follows:
Step 1. Calculate the aggregate fuzzy ratings for the criteria and location alternatives
Let w t be the aggregated fuzzy weights for sub-criteria within each cluster, and w t is
'
'
expressed as w t ' = ( wt ' 1 , wt ' 2 , wt ' 3 ) , among which,
{
}
wt ' 1 = min wut ' 1 ， wt 2 = 1 '
u
'
m
m'
∑w
u =1
ut ' 2
{ }
， wt ' 3 = max wut ' 3 .
u
(14)
Let xt 'i be the aggregated fuzzy rating of location alternatives within each cluster, and
xt 'i = (ait ' , bit ' , cit ' ) is expressed as
{ }
'
{ }
m
a it ' = min a uit ' ， bit = 1 ' ∑ buit ， cit ' = max cuit ' .
u
u
'
m
'
u =1
(15)
Step 2. Calculate the location decision matrix and fuzzy weight matrix
Based on step 1, location decision matrix R and the fuzzy weight matrix W can be
calculated as follows:
 x11
 x
21
R = 


 xk1
x12  x1n' 
x22  x2 n'  ， W = ( w , w , , w )
1
2
k
  

xk 2  xkn' 
Step 3. Calculate the positive ideal and negative ideal solution.
(16)
Wang, Ma, Lao, Wang, Mao
13
Let R ' be the normalized fuzzy decision matrix, which is given as follows:
'
R ' = [rt 'i ]k ×n' , t ' = 1, 2, , k ; i = 1, 2, , n ; where,
 a −' a −' a −'
rt 'i =  t , t , t
c ' b ' a '
 it it it
 −
ait ' ) （Normalized cost criteria）
 , at ' = min(
i

a ' b ' c '  *
cit ' )
rt 'i =  it* , it* , it*  , ct ' = max(
i
 c' c' c' 
 t t t 
(Normalized benefit criteria)
(17)
(18)
Step 4. Calculate the weighted normalized matrix V by multiplying the fuzzy decision
matrix R ' with the fuzzy weight matrix W , the vt 'i is calculated as
(
)
'
vt 'i = rt 'i (.) w t ' = vt 'i1 , vt 'i 2 , vt 'i 3 , t = 1,2, , k ; i = 1,2, , n
'
(19)
Step 5. Calculate the positive ideal and the negative ideal solution
C * = ( v1* , v2* , , vk* ) ,where vt*' = max(vt 'i 3 ) , t ' = 1, 2, , k ; i = 1, 2, , n '
(20)
C − = ( v1 , v2 , , vk ) , where vt− = min(vt i1 ) , t ' = 1, 2, , k ; i = 1, 2, , n '
(21)
i
'
i
'
Step 6. Calculate the dimensional Euclidean distance.
The dimensional Euclidean distance of positive ideal solution is given as
1/ 2
 k

=
d i* ∑ (vt 'i − vt*' ) 2 
'
 t =1

, i = 1, 2, , n '
(22)
The dimensional Euclidean distance of negative ideal solution is given as
1/ 2
 k

=
d i- ∑ (vt 'i − vt−' ) 2 
'
 t =1

, i = 1, 2, , n '
(23)
Step 7. Calculate the closeness to the ideal solution
Let CAi be the ratio of the distance to the fuzzy negative ideal solution di- and the sum of
fuzzy positive ideal solution and fuzzy negative ideal solution, and it represents the relative
closeness as follow:
CAi =
di−
， i = 1, 2, , n '
*
−
di + di
The higher the value CAi is, the more ideal the alternative is for the final location.
(24)
Wang, Ma, Lao, Wang, Mao
14
IMPLEMENTATION AND COMPARISONS
Data Source
To illustrate the applicability of the proposed approach in multiple facilities location planning, a
multiple distribution centers location selection problem in Guiyang, China was used as a case
study. Guiyang city is the capital of Guizhou Province, and is the transportation hub and an
important distributing center to the whole province, as it is located in the north-central of
Guizhou Province. The logistics company owns a logistics center in Guiyang city, and the
company needs to establish multiple distribution centers to extend the market. Eighteen
alternative distribution centers have been chosen as shown in the Figure 2, which are expressed
as A1 , A2 , A18 .
FIGURE 2 Alternative Distribution Centers Scatter Diagram
In order to accurately select the location of distribution centers, three top decision makers
D = { D1 , D2 , D3 } are invited to evaluate the criteria and alternative locations. Two are from the
transportation planning department in Guiyang city, and the third is a senior manager who
operates the logistics center. All three have many years of planning experience, and are very
familiar with the alternative location surrounding conditions.
After interviewing three top decision makers, we obtained the weights for 16 sub-criteria
and the preferences for alternative locations, as shown in Table 2 and Table 3.
Wang, Ma, Lao, Wang, Mao
15
TABLE 2 Linguistic Assessments for the Sub-criteria
Decision makers
Sub-criteria
D1
D2
D3
C11
C12
C13
B.P&G
B.F&G
F
F
B.F&G
F
F
G
G
C21
C22
G
VG
B.F&G
B.G&VG
G
G
C23
C31
B.F&G
B.G&VG
B.G&VG
G
G
B.F&G
C32
B.F&G
B.F&G
B.G&VG
C41
C42
B.P&F
P
B.P&F
F
B.F&G
B.P&F
C43
C51
G
G
B.F&G
G
G
B.F&G
C52
B.G&VG
G
VG
C53
C61
B.F&G
B.F&G
F
B.P&F
F
F
C62
F
F
B.P&F
TABLE 3(a) Linguistic Assessments for Alternatives by D1
Linguistic assessments for alternatives by D1
Criteria
A1
C11
L
A2
A3
A4
A6
A7
M
M
M
A8
A9
L
B.L&M
L
M
B.L&M
B.L&M
M
M
L
B.L&M
B.L&M
M
B.M&H
M
B.L&M
H
C12
B.L&M
L
C13
B.VL&L
VL
C21
B.H&VH
B.H&VH
B.M&H
H
M
H
VH
B.H&VH
H
B.M&H
H
H
H
C23
B.M&H
B.M&H
H
B.H&VH
M
B.M&H
M
B.M&H
M
C31
B.M&H
H
B.H&VH
VH
H
B.H&VH
VH
B.H&VH
H
VH
M
B.VL&L
B.L&M
B.L&M
L
B.L&M
B.L&M
B.L&M
B.L&M
M
B.M&H
M
B.H&VH
H
B.H&VH
VH
B.H&VH
VH
VH
B.H&VH
B.H&VH
H
B.H&VH
B.H&VH
B.M&H
M
B.M&H
B.M&H
M
M
B.L&M
C43
B.M&H
C51
C52
C53
VH
C61
B.M&H
C11
B.H&VH
B.M&H
B.M&H
M
C42
Criteria
H
H
B.M&H
B.L&M
C41
C62
H
B.M&H
M
A10
B.H&VH
B.M&H
B.L&M
B.M&H
A11
A12
H
B.H&VH
B.H&VH
C12
VH
VH
C13
VH
VH
VH
C21
VL
VL
B.VL&L
C22
VL
B.VL&L
VL
C23
C31
A13
B.H&VH
VH
M
B.VL&L
M
B.M&H
L
L
B.VL&L
B.VL&L
B.M&H
B.H&VH
B.H&VH
B.H&VH
H
B.M&H
B.M&H
B.H&VH
H
A18
B.M&H
B.VL&L
B.H&VH
H
A17
B.H&VH
B.VL&L
B.VL&L
H
B.L&M
B.VL&L
C42
C62
B.L&M
B.VL&L
B.VL&L
H
B.H&VH
B.L&M
M
L
B.L&M
H
B.VL&L
L
B.VL&L
B.M&H
L
B.L&M
B.VL&L
B.H&VH
C61
B.M&H
VL
B.M&H
L
H
H
M
B.VL&L
B.VL&L
H
B.M&H
B.M&H
L
B.VL&L
H
H
H
VL
C53
B.M&H
M
VL
L
H
M
B.VL&L
VL
B.M&H
B.M&H
B.H&VH
VL
VL
B.M&H
A16
B.M&H
C52
L
H
B.M&H
B.L&M
VL
B.VL&L
M
L
L
B.L&M
A15
VL
M
B.L&M
A14
B.L&M
B.VL&L
L
B.M&H
C41
C51
L
M
B.VL&L
L
B.VL&L
L
C32
C43
H
B.L&M
VH
H
H
B.L&M
C22
C32
H
B.VL&L
A5
M
L
VL
B.VL&L
H
B.H&VH
VH
B.L&M
B.VL&L
L
B.H&VH
VH
L
B.VL&L
B.VL&L
VL
VL
B.VL&L
H
B.H&VH
B.H&VH
VH
B.L&M
B.L&M
B.VL&L
VL
B.H&VH
M
L
VL
B.VL&L
VH
H
B.H&VH
VH
B.H&VH
VH
H
L
L
B.L&M
B.L&M
B.VL&L
L
B.L&M
B.VL&L
B.VL&L
M
B.VL&L
VH
B.H&VH
VH
B.H&VH
L
B.H&VH
VH
B.H&VH
VH
Wang, Ma, Lao, Wang, Mao
16
TABLE 3(b) Linguistic Assessments for Alternatives by D2
Criteria
C11
C12
Linguistic assessments for alternatives by D2
A1
M
B.VL&L
A2
A3
A4
A5
A6
A7
A8
B.L&M
L
B.VL&L
L
L
B.L&M
M
B.VL&L
L
B.L&M
B.L&M
B.L&M
M
B.M&H
H
H
B.M&H
L
C13
B.VL&L
B.VL&L
C21
C22
B.H&VH
H
B.M&H
C23
M
VH
H
H
B.H&VH
B.M&H
B.L&M
B.L&M
L
B.M&H
B.M&H
H
M
B.H&VH
M
B.M&H
B.M&H
A9
M
B.L&M
M
B.M&H
H
VH
VH
B.M&H
H
M
H
M
B.M&H
H
B.H&VH
B.H&VH
H
H
VH
B.L&M
M
C32
VH
B.H&VH
H
B.M&H
B.H&VH
B.H&VH B.H&VH
C41
B.M&H
L
H
B.L&M
B.L&M
B.VL&L
B.L&M
M
B.M&H
B.L&M
H
B.L&M
M
L
B.L&M
B.L&M
B.VL&L
L
H
VH
M
B.M&H
H
B.M&H
B.M&H
C31
C42
L
C43
B.M&H
C51
H
C52
VH
VH
VH
B.H&VH
B.H&VH
H
C53
VH
B.H&VH
H
B.H&VH
B.M&H
H
C61
M
H
B.M&H
M
C62
B.M&H
L
M
B.M&H
Criteria
A10
A11
C11
VH
C12
B.H&VH
C13
VH
C21
B.VL&L
C22
B.VL&L
C23
VL
B.VL&L
C31
L
C32
C41
B.H&VH
B.VL&L
M
B.M&H
M
B.H&VH
M
B.M&H
B.L&M
L
L
H
B.H&VH
B.M&H
VH
VH
VH
B.M&H
B.H&VH
H
B.H&VH
VL
VL
B.VL&L
VL
B.VL&L
VL
VL
VL
A14
L
B.L&M
B.VL&L
VL
M
B.H&VH
B.H&VH
B.M&H
H
B.H&VH
C52
VL
H
B.M&H
B.VL&L
VL
B.VL&L
L
A15
B.VL&L
VH
H
L
M
A18
B.M&H
M
B.M&H
B.M&H
L
B.VL&L
B.VL&L
B.VL&L
B.VL&L
VL
B.L&M
L
L
VL
L
VL
VL
VL
B.VL&L
L
B.H&VH
H
B.H&VH
VH
B.H&VH
H
VH
VH
B.VL&L
B.L&M
L
B.L&M
VL
B.VL&L
B.L&M
B.M&H
M
B.VL&L
B.VL&L B.L&M
B.VL&L
L
B.M&H
B.M&H
B.H&VH
B.M&H
M
H
H
M
M
VL
H
A17
B.H&VH
C51
H
B.VL&L
B.L&M
C43
M
B.L&M
B.VL&L
VH
L
B.L&M
A16
A13
B.L&M
C42
L
B.M&H
M
M
A12
B.L&M
L
L
H
VH
B.VL&L
B.VL&L
VH
B.H&VH
B.H&VH B.H&VH
L
B.VL&L B.L&M
L
B.VL&L
C53
B.VL&L
VL
L
B.VL&L
L
B.VL&L
L
B.VL&L
M
C61
B.H&VH
B.M&H
H
B.M&H
H
B.H&VH
VH
B.H&VH
VH
C62
B.M&H
VH
VH
VH
B.H&VH
H
B.H&VH
VH
B.H&VH
TABLE 3(c) Linguistic Assessments for Alternatives by D3
Criteria
C11
Linguistic assessments for alternatives by D3
A1
B.L&M
A2
L
C12
L
B.L&M
C13
L
B.VL&L
A3
B.VL&L
L
B.L&M
C21
VH
H
VH
C22
B.H&VH
B.M&H
H
B.M&H
B.H&VH
C23
H
A4
A5
A6
A7
A8
B.L&M
M
B.L&M
M
M
L
B.L&M
M
B.L&M
B.VL&L
B.L&M
B.L&M
M
B.H&VH
B.H&VH
M
H
B.M&H
B.M&H
M
H
B.H&VH
C31
M
B.M&H
H
H
C32
B.H&VH
H
B.M&H
B.M&H
C41
B.L&M
B.L&M
B.M&H
H
C42
L
C43
H
B.M&H
M
C51
H
B.H&VH
C52
VH
B.H&VH
C53
H
C61
H
C62
L
VH
B.M&H
M
Criteria
A10
A11
C11
B.H&VH
B.H&VH
C12
C13
C21
C22
C23
VH
B.H&VH
VL
B.VL&L
VL
VH
B.H&VH
B.VL&L
VL
VL
L
B.M&H
VH
B.H&VH
B.H&VH
M
A12
H
L
B.L&M
B.L&M
B.VL&L
B.H&VH
B.M&H
VH
B.H&VH
B.H&VH
B.H&VH
L
B.L&M
B.VL&L
B.VL&L
M
B.L&M
L
L
L
L
B.L&M
B.L&M
B.M&H
M
B.H&VH
L
B.H&VH
B.M&H
VH
B.L&M
B.VL&L
VL
B.VL&L
B.VL&L
VL
B.VL&L
VL
B.VL&L
B.M&H
H
M
M
VL
M
VH
VH
H
B.M&H
B.VL&L
A16
B.M&H
H
H
B.M&H
M
A15
M
B.H&VH
M
B.M&H
H
A14
B.M&H
B.H&VH
H
L
B.VL&L
M
H
VH
B.H&VH
H
B.M&H
A18
B.M&H
M
M
M
B.M&H
L
B.VL&L
L
B.VL&L B.VL&L
L
B.L&M
L
L
VL
VL
B.VL&L
B.M&H
B.VL&L
B.VL&L
B.L&M
C32
B.L&M
B.VL&L
B.L&M
B.VL&L
C41
M
B.H&VH
B.M&H
B.H&VH
C42
H
B.H&VH
B.M&H
VH
H
B.H&VH B.H&VH
VH
C43
B.M&H
B.M&H
H
M
VH
B.H&VH B.H&VH
H
C51
VL
B.VL&L
L
L
L
B.L&M
B.L&M
B.VL&L
B.VL&L
B.VL&L
M
L
L
C52
B.VL&L
C53
L
C61
VH
C62
B.M&H
VL
B.VL&L
H
B.H&VH
VL
B.VL&L
B.L&M
B.VL&L
VL
L
B.H&VH
B.VL&L
VH
B.M&H
B.H&VH
VH
H
H
B.M&H
B.H&VH
VH
B.L&M
A17
B.M&H
C31
L
H
B.M&H
H
H
A13
B.M&H
H
H
H
VH
M
B.M&H
M
B.L&M
VH
H
M
B.M&H
M
B.H&VH
B.L&M
A9
VL
M
B.M&H
L
B.VL&L
L
L
VL
B.VL&L
VL
VH
B.H&VH
H
VH
B.H&VH
L
B.L&M
B.VL&L
VH
B.H&VH
B.H&VH
VH
B.L&M
B.VL&L
L
B.H&VH
VH
Wang, Ma, Lao, Wang, Mao
17
Result Analysis
According to Eq. (6) and Eq. (7), we can obtain the comprehensive evaluation values by
integrating the sub-criteria into the major criteria. The evaluation values are presented in Table 4.
TABLE 4 Aggregate Avaluation Matrix for Alternatives
Aggregate evaluation for alternatives
Criteria
A1
A2
A3
A4
A5
A6
C1
C2
0.5788
0.8227
0.5727
0.7968
0.5709
0.8257
0.5721
0.8198
0.6270
0.7654
0.6234
0.7634
C3
C4
0.7598
0.6123
0.7616
0.6098
0.7639
0.6179
0.7653
0.6159
0.8167
0.5526
0.8230
0.5450
C5
0.8293
0.8293
0.8295
0.8379
0.7609
0.7665
C6
0.6547
0.6547
0.6615
0.6528
0.5844
0.5836
A7
A8
A9
A10
A11
A12
C1
C2
0.6424
0.7680
0.6409
0.7556
0.6378
0.7659
0.7864
0.5678
0.7781
0.5638
0.7739
0.5680
C3
C4
0.8178
0.5477
0.8244
0.5373
0.8244
0.5519
0.6248
0.6632
0.6173
0.6736
0.6248
0.6680
C5
C6
0.7615
0.5962
0.7711
0.5758
0.7667
0.6106
0.5713
0.7212
0.5718
0.7316
0.5709
0.7259
A13
A14
A15
A16
A17
A18
C1
C2
C3
0.7781
0.5638
0.6209
0.6767
0.6055
0.5641
0.6666
0.6105
0.5591
0.6857
0.6060
0.5651
0.6646
0.6095
0.5641
0.6763
0.6030
0.5646
C4
0.6878
0.7242
0.7256
0.7242
0.7242
0.7286
C5
C6
0.5716
0.7200
0.6182
0.7691
0.6166
0.7691
0.6182
0.7757
0.6169
0.7757
0.6182
0.7753
Following Step 1- Step 3 in the Clustering algorithm procedure, we can obtain the results as
follows:
① When threshold α =0.6932 , Iα = 4.49 , there are two clusters:
p1 = { A10 , A11 , A12 , A13 }
 A1 , A2 , A3 , A4 , A5 , A6 ,

p2 = 

 A7 , A8 , A9 , A14 , A15 , A16 , A17 , A18 
② When threshold α =0.7119 , Iα = 3.87 , there are three clusters:
p1 = { A10 , A11 , A12 , A13 }
p2 = { A1 , A2 , A3 , A4 }
p3 = { A5 , A6 , A7 , A8 , A9 , A14 , A15 , A16 , A17 , A18 }
Wang, Ma, Lao, Wang, Mao
18
③ When threshold α =0.7256 , Iα = 2.51 , there are four clusters:
p1 = { A10 , A11 , A12 , A13 }
p2 = { A1 , A2 , A3 , A4 }
p3 = { A14 , A15 , A16 , A17 , A18 }
p4 = { A5 , A6 , A7 , A8 , A9 }
④ When threshold α =0.7659 ,
Iα = 3.03 ,
there are five clusters:
p1 = { A10 , A11 , A12 , A13 }
p2 = { A1 , A2 , A3 , A4 }
p3 = { A14 , A15 , A16 , A17 , A18 }
p4 = { A5 , A8 , A9 } ,
p5 = { A6 , A7 }
Comparing all of the results, we can find that I 0.7256 = 2.51 is the smallest, thus, the option with
four clusters is the best choice.
Then, we use TOPSIS approach to rank the alternative locations within each cluster.
According to Eq. (14) –Eq. (24), the final ranking results are shown in Table 5 as follows.
TABLE 5 Ranking Order for Alternative Distribution Centers
Cluster p1
Index
d
*
d
-
CAi
Ranking
order
Cluster p3
A11
A12
A13
A14
A15
A16
A17
A18
2.978
3.011
3.105
2.942
3.160
3.285
3.216
3.225
3.208
3.021
2.886
2.912
3.015
2.912
2.755
2.845
2.786
2.736
0.503
0.489
0.484
0.506
0.480
0.456
0.469
A10
CA13 > CA10 > CA11 > CA12
Cluster p4
A1
A2
A3
A4
A5
A6
A7
d*
3.296
3.353
3.234
3.121
3.143
3.301
d-
2.905
2.899
2.972
3.075
2.931
CAi
0.468
0.464
0.479
0.496
0.483
Ranking
order
0.460
CA14 > CA16 >CA17 > CA18 > CA15
Cluster p2
Index
0.463
CA4 > C A3 > CA1 > C A2
A8
A9
3.269
3.135
3.232
2.915
2.878
2.976
2.883
0.469
0.468
0.487
0.471
CA8 > CA5 > CA9 > CA6 > CA7
Wang, Ma, Lao, Wang, Mao
19
By comparing the CAi values in each cluster (Table 5), A13 , A4 , A14 , A8 are selected as the
final urban distribution centers location for the logistics company. According to Eq. (13), we can
calculate the dispersion degree between clusters. The ranking results of dispersion degree
between clusters is RC34 > RC13 > RC12 > RC24 ; the higher the RChk between clusters is, the
smaller the closeness between clusters is. Based on the demand for the economic development in
each region, the logistics operators can also merge possible clusters, and finalize the number of
distribution centers in each cluster.
Algorithm Comparisons
Chou et al. (2008) presented a new fuzzy multiple attributes decision-making approach based on
the fuzzy set theory, with the added importance of each decision-maker (14). They applied the
factor rating system and simple additive weighting approach to evaluate facility location
alternatives. Li et al. (2011) presented another comprehensive methodology for the logistics
center location selection(34). For comparison purposes, we implemented both algorithms into
our case study in the same context with four distribution centers. The algorithm proposed by
Chou generates the final locations as A4 , A10 , A11 , A13 , and Li’s algorithm results in the final
locations as A10 , A13 , A14 , A16 . They are shown as follows:
(a)
(b)
(c)
FIGURE 3 The Results of Three Algorithms: [a] the results of Chou’s algorithm [b] the
results of Li’s algorithm [c] the results of the proposed algorithm
The proposed algorithm provides a novel approach to address the MDCL problem. The final
distribution centers calculated by Chou’s approach (cite) and Li’s approach (cite) are located
along one side of the logistics center, which may incur more traffic events near the distribution
centers, and increase transportation costs for customers living far away from these distribution
centers. The potential distribution centers by our proposed approach, however, are scattered
more evenly around the logistics center, which adheres to reality more agreeably. Each attribute
of alternative locations in our proposed approach is further split into several finer attributes
Wang, Ma, Lao, Wang, Mao
20
during clustering, which gains the heterogeneity between alternative locations, but the algorithms
developed by Li and Chou cannot take into account these factors. In summary, the proposed
algorithm should be more suitable for MDCL problem in real life.
Sensitivity Analysis
To further investigate the impacts of different criteria weights on the location selection procedure,
a sensitivity analysis is also needed. A total of 27 experiments were undertaken in each cluster.
The sub-criteria C43 , C61 and C62 are under the cost criteria, while the remaining sub-criteria are
all under benefit criteria. In each cluster experiments 1-9 have the weights of all criteria set
equally to VL, B.VL&L, L, B.L&M, M, B.M&H, H, B.H&VH, VH; experiments 10-25 have the
weight of one criterion set at the highest weight VH, with the remaining set at the lowest weight
VL; experiment 26 have the cost criteria ( C43 , C61 , C62 ) set at the highest weight VH, with the
remaining criteria set at the lowest weight VL; and in experiment 27 the weights of the cost
criteria are set at the lowest weight VL, with the remaining criteria set at the highest weight VH.
The results of the analysis can be observed in Figure 4(a)-(d).
Wang, Ma, Lao, Wang, Mao
21
0.56
0.56
A13
A10
A11
A12
0.55
A4
A3
A1
A2
0.54
Closeness Coefficient
Closeness Coefficient
0.54
0.53
0.52
0.51
0.52
0.5
0.48
0.5
0.46
0.49
0.48
0
5
10
15
Experiment Times
20
25
0.44
30
0
5
10
(a)
15
Experiment Times
20
25
30
(b)
0.51
0.54
A14
A16
A17
A18
A15
0.5
A8
A5
A9
A6
A7
0.53
0.52
Closeness Coefficient
Closeness Coefficient
0.49
0.48
0.47
0.51
0.5
0.49
0.48
0.46
0.47
0.45
0.44
0.46
0.45
0
5
10
15
Experiment Times
(c)
20
25
30
0
5
10
15
Experiment Times
20
25
30
(d)
FIGURE 4 Results of Sensitivity Analysis: [a] results of sensitivity analysis in cluster P1 [b]
results of sensitivity analysis in cluster P2 [c] results of sensitivity analysis in cluster P3 [d]
results of sensitivity analysis in cluster P4
Results indicate that for cluster p1 , in the experiments 10, 20, 25 and 26, the final location
has been changed to A10 due to CA10 > CA13 ; For cluster p2 , in the experiments 12, 20, 25 and
26, the location has been changed to A3 due to CA3 > CA4 ; For cluster p3 , in the experiments 14,
16, 20, 24, 25 and 26, the location has been changed to A16 due to CA16 > CA14 ; For cluster p4 ,
in the experiments 13, 19, 20, 24, 25 and 26, the location has been changed to A5 due to
CA5 > CA8 . Through these experiments we can say that the location decision is relatively
insensitive to benefit criteria weights, however they are sensitive to cost criteria weights. From
Figures 7-10, we also observe that the cost criteria weights in clusters p3 and p4 are more
sensitive than clusters p1 and p2 , which implies that decision makers need to assess the cost
criteria weights more seriously than the benefit criteria for location planning.
Wang, Ma, Lao, Wang, Mao
22
CONCLUSIONS
This paper develops an innovative approach for selecting multiple distribution center locations
under fuzzy environment. The hierarchical criteria analysis structure is initially constructed for
planning distribution center locations. A fuzzy set approach is designed to convert the linguistic
criteria ratings and alternative ratings into fuzzy numbers, and several relative definitions are
also presented for algorithm procedures. Fuzzy integration algorithm is then proposed to
synthesizing multiple lower hierarchical criteria into a higher hierarchical criterion followed by
an improved AFS approach based fuzzy clustering algorithm to group similar alternative
locations into several clusters, and furthermore, a fuzzy TOPSIS approach is integrated to seek
the optimal distribution centers locations within clusters.
The proposed method has been successfully applied to assisting logistics operators in
selecting distribution centers for a logistics company in Guiyang city, China. With further
comparisons with other location selection algorithms, our approach has been proven more
effective for MDCL problems. A sensitivity analysis is conducted in capturing the impacts of
different evaluation criteria weights. The results demonstrate that our approach performs very
well as a cost-effective and decision-support tool for distribution center location planning
strategies. Additionally, the approach can also be straightforwardly extended to solving other
similar multiple facility locations problems for decision makers.
ACKNOWLEDGMENT
This research is supported by National Natural Science Foundation of China（51078087）and
Ph.D. Programs Foundation of Ministry of Education of China (200802861017). Special thanks
to Yunteng Lao and Carmen Kwan for their valuable suggestions and revisions.
Wang, Ma, Lao, Wang, Mao
23
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