# A Selection Model to Logistic Centers Based on TOPSIS and MCGP Methods: The Case of Airline Industry

Authors: Kou-Huang Chen; Chin-Nung Liao; Li-Chun Wu
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

The location selection of a logistics center is a crucial decision relating to cost and benefit analysis in airline industry. However, it is difficult to be solved because there are many conflicting and multiple objectives in location problems. To solve the problem, this paper integrates fuzzy technique for order preference by similarity to an ideal solution (TOPSIS) and multichoice goal programming (MCGP) to obtain an appropriate logistics center from many alternative locations for airline industry. The proposed method in this paper will offer the decision makers (DMs) to set multiple aspiration levels for the decision criteria. A numerical example of application is also presented.


---

## Body

## 1. Introduction

In recent years, airline industries have been struggling to look for suitable locations to save logistics costs and increase competition advantage. Because all the activities in a logistics chain system have a relationship with customers and suppliers, the evaluation and selection of a suitable logistics center location has become one of the most important issues for logistics businesses [1]. In addition, the location selection should take many factors into consideration, such as labor characteristics (e.g., skilled labor), markets (e.g., closeness to customer and suppler), infrastructure (e.g., transportation, water, and power systems), and macroenvironment (e.g., policies of government and industrial regulations laws) [2]. Location selection is a crucial decision in the cost/benefit analysis of distribution center, logistics center, or other facilities for the airline industry. However, the problem of location selection is difficult because there are many conflicting and multiple goals to be solved [3]. Therefore, when considering various criteria, the evaluation and selection of logistics centers location is a multiple criteria decision-making (MCDM) process and a problem disturbed by logistics managers of airline industry. In order to select a suitable location for airline logistics centers, both qualitative and quantitative criteria are needed to be considered at the same time. In the past, many qualitative and quantitative criteria methods of MCDM for evaluating/selecting consideration have been developed and widely used in various fields, such as management decisions, strategy selections, and decision-making problems.Many studied on location evaluation and selection considering qualitative criteria has been addressed in previous studies. Cheng et al. [4] adopted the analytic network process (ANP) approach to select a shopping mall location that takes five qualitative criteria into consideration, including transportation, competition, one stop service, commercial area, and environment factors. Yang et al. [5] applied fuzzy theory for logistics distribution centers location problem under fuzzy environment. Chou et al. [6] presented a fuzzy MCDM model for hotel location selection by considering traffic conditions, geographical conditions, hotel characteristics, and operation management criteria. Lee and Lin [7] presented a fuzzy quantified SWOT procedure for the environmental evaluation of an international distribution center. Demirel et al. [2] had taken into account the main criteria, including costs, labor characteristics, infrastructure, and markets in a warehouse location selection. Turskis and Zavadskas [8] presented a newly developed ARAS-F method to select the most suitable site for logistic centre by considering investment cost, operation time, expansion possibility, and closeness to the market. Awasthi et al. [9] used the technique for order preference by similarity to an ideal solution (TOPSIS) to optimize urban distribution center location by considering accessibility, security, connectivity to multimodal transport, costs, environmental impact, proximity to customers, proximity to suppliers, resource availability, conformance to sustainable freight regulations, possibility of expansion, and quality of service criteria. Kampf et al. [10] designed a useful tool to support the decision-making process for the location of a public logistic center. Li et al. [11] presented a TOPSIS methodology for the selection of logistic center location with five criteria, such as traffic, communication, candidate land area, candidate land value, and freight transport. Moreover, Kuo [1] integrated the analytic network process (ANP), TOPSIS, and DEMATEL techniques and determined a location for an international distribution center by considering port rate, import/export volume, location resistance, extension transportation, convenience, transshipment time, one stop service, information abilities, port and warehouse facilities, port operation system, and density of shipping line. For the analytic hierarchy process (AHP)/ANP or TOPSIS approaches, decision-makers (DMs) can define the criteria weights effectively; however, only a few candidate locations can be evaluated by these criteria whereas the complex interrelations in each criterion [3]. In practice, DMs cannot efficiently evaluate and select many candidate locations simultaneously among AHP, ANP, or TOPSIS. In other words, DMs need to develop an efficient method to improve the efficiency of location selection problems.In addition, there are many studies on the location evaluation and selection problems by using mathematical quantitative criteria approaches. Jovanovic [12] presented an integer programming method to optimize the location selection for a new distribution transformer with the suitable size by calculating voltage drops, load of feeders, substations, and annual investment costs. Cheng and Li [13] used the approaches of data envelopment analysis (DEA) and binary integer linear programming (BILP) to determine whether location is valuable for investment. Klapita and Švecová [14] applied mathematical programming methods and the theory fuzzy sets to determine a unique solution of a location problem at uncertain costs. Sun et al. [15] presented a bilevel programming model for the location of logistics distribution centers evaluation and selection. Bhaumik [16] expressed the delocation problem as an integer linear programming method and did a case study for an existing distribution network with retailer and distributor locations that needs to downsize its distribution chain. In these studies, DMs can determine the optimal location selection from a lot of candidate locations with limited quantitative criteria; however, DMs may encounter the difficulty to evaluate candidate location with many qualitative criteria. For example, DMs may not easily determine the suitable weights on each goal in location evaluation problems [3].Previous publications of evaluation and selection issues mostly focus on single or several important qualitative or quantitative factors and rarely take qualitative and quantitative factors into consideration. In the recent years, some studies adopted AHP, TOPSIS, and multichoice goal programming (MCGP) for evaluation and selection problems. Lee et al. [17] adopted fuzzy AHP and fuzzy multiple GP to help cooperation to select their downstream businesses of thin film transistor liquid crystal display suppliers. Liao and Kao [18] integrated Taguchi loss function, AHP, and MCGP to evaluate and select supplier. Ben Mahmoud et al. [19] used AHP and MCGP for a quality management system designing. Liao and Kao [20] integrated a fuzzy TOPSIS and MCGP approach to supplier selection problems in supply chain management. Liao [21] presented an evaluation model by using fuzzy TOPSIS and GP for TQM consultant selection. Hsu and Liou [22] provided a systemic analytical model for the selection of outsourcing providers from the point view of cost. Costs invested to classical facility or logistics location selections may be highly uncertain [23]. Moreover, Ho et al. [3] integrated AHP and MCGP as a decision reference to obtain an appropriate house from many alternative locations that better suit the preferences of renters under their needs.In order to improve the quality of location selection decision-making problems, this paper will present a hybrid evaluation technique to help logistics businesses to select an appropriate location with both qualitative and quantitative criteria. The TOPSIS and MCGP methods, which will help DMs to determine the best location, are integrated in this paper. First, the Delphi method is applied to assess selection criteria. Then, the fuzzy TOPSIS is applied to calculate the relative weight of each location. Finally, MCGP model is formulated and used to identify the best logistics center location. The integrated method is shown in Figure1.Figure 1
The integration procedure for logistics centers location evaluation and selection.The rest of this article is organized as follows. Section2 presents the preliminaries of fuzzy set theory. Section 3 describes the methodology of TOPSIS-MCGP model. In Section 4, a numerical application to illustrate the proposed approach is presented. The sensitivity analysis is shown Section 5. Finally, the conclusions and future work are presented in Section 6.


## 2. Fuzzy Set Theory

A fuzzy set is characterized by a membership function, which assigns to each linguistic variable a grade of membership ranging[
0,1
].Definition 1.
A fuzzy seta
~ in a universe of discourse X is characterized by a membership function u
a
~
(
x
) that maps each component x in X to a real number in the interval [
0,1
]. The function value X is termed the grade of membership of X in a
~ [24]. The nearer the value of u
a
~
(
x
) to unity, the higher the grade of membership of u
a
~
(
x
) in a
~.Definition 2.
A real fuzzy numberA is described as a fuzzy subset of the real line R with member function u
A that represents uncertainty. A membership function is defined as the universe of discourse from zero to one (see Figure 2). Therefore, a triangular fuzzy number can be defined as a triplet (
a
,
b
,
c
), where a
≤
b
≤
c; the membership function of the fuzzy number A is defined as follows:
(1)
u
A
(
x
)
=
{
x
-
a
b
-
a
,
x
∈
[
a
,
b
]
,
c
-
x
c
-
b
,
x
∈
[
b
,
c
]
,
0
,
otherwise
.Figure 2
A triangular fuzzy number.


### 2.1. The Distance between Fuzzy Triangular Numbers

Leta
~
=
(
a
1
,
a
2
,
a
3
) and b
~
=
(
b
1
,
b
2
,
b
3
) be two triangular fuzzy numbers. Then the distance (d
(
a
~
,
b
~
)) between a
~ and b
~ can be calculated by using the vertex method as follows [25]:
(2)
d
(
a
~
,
b
~
)
=
1
3
[
(
a
1
-
b
1
)
2
+
(
a
2
-
b
2
)
2
+
(
a
3
-
b
3
)
2
]
.


### 2.2. Linguistic Variables

For the fuzzy set theory, conversion scales are applied to transform the linguistic terms into fuzzy numbers. In this paper, we will apply a scale of normalized fuzzy preference number 0-1 to rate the alternatives. Table1 presents the linguistic variables for the importance fuzzy weights of each criterion from Figure 3. In addition, Table 2 presents the linguistic variables of fuzzy ratings for the alternatives preference from Figure 4.Table 1
Linguistic variables for the importance weight of each criterion.
Importance
Abbreviation
Triangular fuzzy number
Very low
VL
(0, 0, 0.2)
Low
L
(0, 0.2, 0.4)
Medium low
ML
(0.2, 0.4, 0.6)
Medium high
MH
(0.4, 0.6, 0.8)
High
H
(0.6, 0.8, 1)
Very high
VH
(0.8, 1, 1)Table 2
Fuzzy preference used in this study.
Linguistic terms
Abbreviation
Normalized fuzzy preference
Very poor
VP
(0, 0, 0.2)
Poor
P
(0, 0.2, 0.4)
Medium poor
MP
(0.2, 0.4, 0.6)
Medium good
MG
(0.4, 0.6, 0.8)
Good
G
(0.6, 0.8, 1)
Very good
VG
(0.8, 1, 1)Figure 3
Linguistic variables for importance weight of each criterion.Figure 4
Linguistic variables for ratings and normalized fuzzy preference.


## 2.1. The Distance between Fuzzy Triangular Numbers

Leta
~
=
(
a
1
,
a
2
,
a
3
) and b
~
=
(
b
1
,
b
2
,
b
3
) be two triangular fuzzy numbers. Then the distance (d
(
a
~
,
b
~
)) between a
~ and b
~ can be calculated by using the vertex method as follows [25]:
(2)
d
(
a
~
,
b
~
)
=
1
3
[
(
a
1
-
b
1
)
2
+
(
a
2
-
b
2
)
2
+
(
a
3
-
b
3
)
2
]
.


## 2.2. Linguistic Variables

For the fuzzy set theory, conversion scales are applied to transform the linguistic terms into fuzzy numbers. In this paper, we will apply a scale of normalized fuzzy preference number 0-1 to rate the alternatives. Table1 presents the linguistic variables for the importance fuzzy weights of each criterion from Figure 3. In addition, Table 2 presents the linguistic variables of fuzzy ratings for the alternatives preference from Figure 4.Table 1
Linguistic variables for the importance weight of each criterion.
Importance
Abbreviation
Triangular fuzzy number
Very low
VL
(0, 0, 0.2)
Low
L
(0, 0.2, 0.4)
Medium low
ML
(0.2, 0.4, 0.6)
Medium high
MH
(0.4, 0.6, 0.8)
High
H
(0.6, 0.8, 1)
Very high
VH
(0.8, 1, 1)Table 2
Fuzzy preference used in this study.
Linguistic terms
Abbreviation
Normalized fuzzy preference
Very poor
VP
(0, 0, 0.2)
Poor
P
(0, 0.2, 0.4)
Medium poor
MP
(0.2, 0.4, 0.6)
Medium good
MG
(0.4, 0.6, 0.8)
Good
G
(0.6, 0.8, 1)
Very good
VG
(0.8, 1, 1)Figure 3
Linguistic variables for importance weight of each criterion.Figure 4
Linguistic variables for ratings and normalized fuzzy preference.


## 3. Methodology

### 3.1. Fuzzy Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS)

In real life, the decision making of many situations cannot be performed sufficiently and exactly because the available information is vague, imprecise, and uncertain [26]. Moreover, the multiple criteria decision-making (MCDM) situations are also based on uncertain and ill-defined information. This study performs the logistic centers location selection by using TOPSIS method, one of the best known MCDM approaches. TOPSIS is based on the concept that the chosen alternative should have the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS) [27]. In addition, fuzzy set theory is considered as the most effective approach in managing vagueness and uncertainty problems. To solve MCDM problem, the fuzzy set theory is introduced to express linguistic terms in the human decision-making process. Zimmermann [28] indicated that a linguistic variable is a variable whose values are expressed in linguistic terms. The concept of a linguistic variable is very useful for handling situations which are very complex or not well-defined and need to be reasonably described by using conventional quantitative expressions. Basic fuzzy set technique and the steps of fuzzy TOPSIS relation analysis are shown as follows.(
1
) Three DMs conduct a pairwise comparison of the preferences for criteria and use the linguistic variables to assess the importance of each candidate with regard to each criterion.(
2
) Establish a decision matrix for alternative performance. The matrix can be expressed as
(3)
X
~
=
[
x
~
11
x
~
12
⋯
x
~
1
m
x
~
21
x
~
22
⋯
x
~
2
m
⋮
⋮
⋮
⋮
x
~
n
1
x
~
n
2
⋯
x
~
n
m
]
.Choose the linguistic ratingsX
~
=
{
x
~
i
j
,
i
=
1,2
,
3
,
…
,
n
,
j
=
1,2
,
3
,
…
,
m
} for alternatives A
j with respect to criterion C
i. The fuzzy linguistic rating (
x
~
i
j
) makes sure the range of normalized triangular fuzzy numbers be in the interval [
0
,
1
]; therefore, there is no need for normalization [29].(
3
) Calculate the weighted normalized fuzzy decision matrix. The weighted normalized value is calculated as
(4)
V
~
=
[
v
~
i
j
]
n
×
m
,
i
=
1,2
,
3
,
…
,
n
,
j
=
1,2
,
3
,
…
,
m
,
where v
~
i
j
=
x
~
i
j
⊗
w
i.(
4
) The fuzzy positive-ideal solution and negative-ideal solution can be determined as follows:
(5)
L
*

=
(
v
~
1
*
,
v
~
2
*
,
…
,
v
~
n
*
)
=
{

(
max
⁡
i
v
~
i
j
∣
j
∈
J
)

,
(
min
⁡
i
v
~
i
j
∣
j
∈
J
′
)

}

,
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
i
=
1,2
,
3
,
…
,
n
,
j
=
1,2
,
3
,
…
,
m
,
L
-

=
(
v
~
1
-
,
v
~
2
-
,
…
,
v
~
n
-
)
=
{

(
min
⁡
i
v
~
i
j
∣
j
∈
J
)

,
(
max
⁡
i
v
~
i
j
∣
j
∈
J
′
)

}

,
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
i
=
1,2
,
3
,
…
,
n
,
j
=
1,2
,
3
,
…
,
m
,
where v
~
j
*
=
max
⁡
{
v
i
j
3
} and v
~
j
-
=
min
⁡
{
v
i
j
1
}; J is associated with benefit criteria; and J
′ is associated with cost criteria.(
5
) Calculate the distance of each alternative from L
* and L
- by using the following equations:
(6)
d
i
*
=
∑
j
=
1
m
d
(
v
~
i
j
,
v
~
j
*
)
,
i
=
1,2
,
…
,
n
,
d
i
-
=
∑
j
=
1
m
d
(
v
~
i
j
,
v
~
j
-
)
,
i
=
1,2
,
…
,
n
.(
6
) Calculate the closeness coefficients (C
C
i) of the ideal solution for each alternative as
(7)
C
C
i
=
d
i
-
(
d
i
*
+
d
i
-
)
,
where the C
C
i ranges between the closed interval [
0,1
], i
=
1,2
,
3
,
…
,
n.(
7
) Use the C
C
i obtained from Step (
6
) for each candidate and building the integrated model to select the best location.


### 3.2. Multichoice Goal Programming (MCGP)

Goal programming (GP) is one of the most powerful techniques for solving target optimization problems. Sometimes, determining the specific target value of each goal is not easy for DMs because only limited information can be acquired in an uncertain situation. For example, a DM may consider the following as priority, including maximizing profits and increasing lot size services and increasing service quality and reducing operational cost. These problems cannot be solved by a general GP method. The conflicts of firm resources encourage DMs to generate a trustworthy mathematical model formulation to delineate their preferences [20]. A multichoice goal programming (MCGP) was proposed by Chang [30, 31] to solve this problem. MCGP formulation can be defined as follows:
(8)
min
⁡
∑
i
=
1
n
w
i
[
(
d
i
+
+
d
i
-
)
+
(
e
i
+
+
e
i
-
)
]
s
.
t
.
f
i
(
X
)
-
d
i
+
+
d
i
-
=
g
i
1
or
g
i
2
or
…
or
g
i
m
,
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
i
=
1,2
,
…
,
n
,
g
i
-
e
i
+
+
e
i
-
=
g
i
,
max
⁡
,
i
=
1,2
,
…
,
n
,
g
i
,
min
⁡
≤
g
i
≤
g
i
,
max
⁡
,
i
=
1,2
,
…
,
n
,
d
i
+
,
d
i
-
,
e
i
+
,
e
i
-
≥
0
,
i
=
1,2
,
…
,
n
,
X
∈
F
(
F
is
a
feasible
set
)
,
where g
i
j (i
=
1,2
,
…
,
n and j
=
1,2
,
…
,
m) is the jth aspiration level of the ith goal, w
i represents the weight attached to the deviation, and d
i is the deviation from the target value g
i; d
i
+
=
max
⁡
(
0
,
f
i
(
x
)
-
g
i
) and d
i
-
=
max
⁡
(
0
,
g
i
-
f
i
(
x
)
) denote under- and overachievements of the ith goal, respectively. In addition, e
i
+ and e
i
- are positive and negative deviations attached to |
g
i
-
g
i
,
max
⁡
|, g
i
,
min
⁡, and g
i
,
max
⁡ which are, respectively, lower and upper bounds of g
i.


## 3.1. Fuzzy Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS)

In real life, the decision making of many situations cannot be performed sufficiently and exactly because the available information is vague, imprecise, and uncertain [26]. Moreover, the multiple criteria decision-making (MCDM) situations are also based on uncertain and ill-defined information. This study performs the logistic centers location selection by using TOPSIS method, one of the best known MCDM approaches. TOPSIS is based on the concept that the chosen alternative should have the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS) [27]. In addition, fuzzy set theory is considered as the most effective approach in managing vagueness and uncertainty problems. To solve MCDM problem, the fuzzy set theory is introduced to express linguistic terms in the human decision-making process. Zimmermann [28] indicated that a linguistic variable is a variable whose values are expressed in linguistic terms. The concept of a linguistic variable is very useful for handling situations which are very complex or not well-defined and need to be reasonably described by using conventional quantitative expressions. Basic fuzzy set technique and the steps of fuzzy TOPSIS relation analysis are shown as follows.(
1
) Three DMs conduct a pairwise comparison of the preferences for criteria and use the linguistic variables to assess the importance of each candidate with regard to each criterion.(
2
) Establish a decision matrix for alternative performance. The matrix can be expressed as
(3)
X
~
=
[
x
~
11
x
~
12
⋯
x
~
1
m
x
~
21
x
~
22
⋯
x
~
2
m
⋮
⋮
⋮
⋮
x
~
n
1
x
~
n
2
⋯
x
~
n
m
]
.Choose the linguistic ratingsX
~
=
{
x
~
i
j
,
i
=
1,2
,
3
,
…
,
n
,
j
=
1,2
,
3
,
…
,
m
} for alternatives A
j with respect to criterion C
i. The fuzzy linguistic rating (
x
~
i
j
) makes sure the range of normalized triangular fuzzy numbers be in the interval [
0
,
1
]; therefore, there is no need for normalization [29].(
3
) Calculate the weighted normalized fuzzy decision matrix. The weighted normalized value is calculated as
(4)
V
~
=
[
v
~
i
j
]
n
×
m
,
i
=
1,2
,
3
,
…
,
n
,
j
=
1,2
,
3
,
…
,
m
,
where v
~
i
j
=
x
~
i
j
⊗
w
i.(
4
) The fuzzy positive-ideal solution and negative-ideal solution can be determined as follows:
(5)
L
*

=
(
v
~
1
*
,
v
~
2
*
,
…
,
v
~
n
*
)
=
{

(
max
⁡
i
v
~
i
j
∣
j
∈
J
)

,
(
min
⁡
i
v
~
i
j
∣
j
∈
J
′
)

}

,
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
i
=
1,2
,
3
,
…
,
n
,
j
=
1,2
,
3
,
…
,
m
,
L
-

=
(
v
~
1
-
,
v
~
2
-
,
…
,
v
~
n
-
)
=
{

(
min
⁡
i
v
~
i
j
∣
j
∈
J
)

,
(
max
⁡
i
v
~
i
j
∣
j
∈
J
′
)

}

,
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
i
=
1,2
,
3
,
…
,
n
,
j
=
1,2
,
3
,
…
,
m
,
where v
~
j
*
=
max
⁡
{
v
i
j
3
} and v
~
j
-
=
min
⁡
{
v
i
j
1
}; J is associated with benefit criteria; and J
′ is associated with cost criteria.(
5
) Calculate the distance of each alternative from L
* and L
- by using the following equations:
(6)
d
i
*
=
∑
j
=
1
m
d
(
v
~
i
j
,
v
~
j
*
)
,
i
=
1,2
,
…
,
n
,
d
i
-
=
∑
j
=
1
m
d
(
v
~
i
j
,
v
~
j
-
)
,
i
=
1,2
,
…
,
n
.(
6
) Calculate the closeness coefficients (C
C
i) of the ideal solution for each alternative as
(7)
C
C
i
=
d
i
-
(
d
i
*
+
d
i
-
)
,
where the C
C
i ranges between the closed interval [
0,1
], i
=
1,2
,
3
,
…
,
n.(
7
) Use the C
C
i obtained from Step (
6
) for each candidate and building the integrated model to select the best location.


## 3.2. Multichoice Goal Programming (MCGP)

Goal programming (GP) is one of the most powerful techniques for solving target optimization problems. Sometimes, determining the specific target value of each goal is not easy for DMs because only limited information can be acquired in an uncertain situation. For example, a DM may consider the following as priority, including maximizing profits and increasing lot size services and increasing service quality and reducing operational cost. These problems cannot be solved by a general GP method. The conflicts of firm resources encourage DMs to generate a trustworthy mathematical model formulation to delineate their preferences [20]. A multichoice goal programming (MCGP) was proposed by Chang [30, 31] to solve this problem. MCGP formulation can be defined as follows:
(8)
min
⁡
∑
i
=
1
n
w
i
[
(
d
i
+
+
d
i
-
)
+
(
e
i
+
+
e
i
-
)
]
s
.
t
.
f
i
(
X
)
-
d
i
+
+
d
i
-
=
g
i
1
or
g
i
2
or
…
or
g
i
m
,
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
i
=
1,2
,
…
,
n
,
g
i
-
e
i
+
+
e
i
-
=
g
i
,
max
⁡
,
i
=
1,2
,
…
,
n
,
g
i
,
min
⁡
≤
g
i
≤
g
i
,
max
⁡
,
i
=
1,2
,
…
,
n
,
d
i
+
,
d
i
-
,
e
i
+
,
e
i
-
≥
0
,
i
=
1,2
,
…
,
n
,
X
∈
F
(
F
is
a
feasible
set
)
,
where g
i
j (i
=
1,2
,
…
,
n and j
=
1,2
,
…
,
m) is the jth aspiration level of the ith goal, w
i represents the weight attached to the deviation, and d
i is the deviation from the target value g
i; d
i
+
=
max
⁡
(
0
,
f
i
(
x
)
-
g
i
) and d
i
-
=
max
⁡
(
0
,
g
i
-
f
i
(
x
)
) denote under- and overachievements of the ith goal, respectively. In addition, e
i
+ and e
i
- are positive and negative deviations attached to |
g
i
-
g
i
,
max
⁡
|, g
i
,
min
⁡, and g
i
,
max
⁡ which are, respectively, lower and upper bounds of g
i.


## 4. Numerical Example

An airline company ABC would like to select a suitable location for a new logistics center in Shanghai, China. ABC’s decision-making group consists of three members: the chief executive officer (CEO) and two logistics experts who are invited to participate in this group and provide their opinions. From the literature reviews, data analysis, and nominal group technique (NGT) with five qualitative and quantitative criteria for the best logistic centers conditions may be determined as follows:(1)
resource availability (C
1),
(2)
location resistance (C
2),
(3)
expansion possibility (C
3),
(4)
investment cost (C
4),
(5)
information abilities (C
5).The criteria are shown in Table3.Table 3
Evaluation references source for each criterion.
Criteria of location
Description
Sources
Resource availability (C
1)
It providers the availability of water, power, raw material, and labor resources in the location
Awasthi et al. [9]Li et al. [11]
Location resistance (C
2)
It means the distance from location to main manufacturing and consumer marketplace, respectively. It will affect the logistic cost and time of cargo
Kuo [1] Demirel et al. [2]
Expansion possibility (C
3)
Ability to increase size to accommodate growing demands
Awasthi et al. [9] Turskis and Zavadskas [8]
Investment cost (C
4)
Investment costs in acquiring land, vehicle resources, drivers, taxes, and so forth for the location
Awasthi et al. [9] Demirel et al. [2] Turskis and Zavadskas [8]Hsu and Liou [22]
Information abilities (C
5)
It provides the availability of information service on logistics, business, financing activities, and so forth
Kuo [1]The decision-making group includes three members (D
1, D
2, and D
3), and they have rich experience in logistic centers management and are required to select a best logistic center from five logistics centers location (L
1, L
2, L
3, L
4, and L
5) by applying the Delphi technique. The hierarchical structure of this decision problem is shown in Figure 5.Figure 5
Hierarchy structure of location selection.In order to evaluate candidate of logistics centers location efficiently and properly, the company ABC outsources the location investigate consultants to assist DMs in location parameters decision. The results of characteristics for five candidate locations are provided in Table4.Table 4
Location selection parameters.
Criteria
L
1
L
2
L
3
L
4
L
5
Location 1(x
1)
Location 2(x
2)
Location 3(x
3)
Location 4(x
4)
Location 5(x
5)
Resource availability (score)
85
90
95
80
75
Location resistance (km2)
40.9
41.2
38.5
40.9
43.6
Expansion possibility (lot size m2)
96.56
152.30
148.86
96.56
87.03
Investment cost ($1000)
1266
968
1520
887
1135
Information abilities (score)
70
80
60
90
75The integrated fuzzy TOPSIS and MCGP approaches are applied to solve the location selection problem, and the computational process is summarized as follows.(
1
) The DMs use the linguistic weighting variables shown in Table 1 to assess the importance of criteria by using geometry average. The importance fuzzy weights of the criteria are determined by the three DMs, shown in Table 5.Table 5
Importance of fuzzy weight of criteria from DMs.
D
1
D
2
D
3
w
~
i
C
1
(0.4, 0.6, 0.8)
(0.2, 0.4, 0.6)
(0.6, 0.8, 1)
(0.2,  0.58,  1)
C
2
(0.6, 0.8, 1)
(0.4, 0.6, 0.8)
(0.4, 0.6, 0.8)
(0.4,  0.66,  1)
C
3
(0.4, 0.6, 0.8)
(0.6, 0.8, 1)
(0.8, 1, 1)
(0.4,  0.78,  1)
C
4
(0.4, 0.6, 0.8)
(0.2, 0.4, 0.6)
(0.4, 0.6, 0.8)
(0.2,  0.52,  0.8)
C
5
(0.2, 0.4, 0.6)
(0.4, 0.6, 0.8)
(0.2, 0.4, 0.6)
(0.2,  0.46,  1)(
2
) DMs use the linguistic rating variables shown in Table 2 to evaluate the rating of each location candidate with respect to each criterion and then present the ratings in Table 6.Table 6
Fuzzy preferences for the five locations by DMs using various criteria.
C
1
C
2
C
3
C
4
C
5
L
1
(0.4, 0.6, 0.8)
(0.4, 0.6, 0.8)
(0.6, 0.8, 1)
(0.4, 0.6, 0.8)
(0.4, 0.6, 0.8)
L
2
(0.6, 0.8, 1)
(0.4, 0.6, 0.8)
(0.4, 0.6, 0.8)
(0.4, 0.6, 0.8)
(0.6, 0.8, 1)
L
3
(0.2, 0.4, 0.6)
(0.8, 1, 1)
(0.2, 0.4, 0.6)
(0.6, 0.8, 1)
(0.4, 0.6, 0.8)
L
4
(0.6, 0.8, 1)
(0.4, 0.6, 0.8)
(0.6, 0.8, 1)
(0.4, 0.6, 0.8)
(0.8, 1, 1)
L
5
(0.4, 0.6, 0.8)
(0.6, 0.8, 1)
(0.2, 0.4, 0.6)
(0.8, 1, 1)
(0.6, 0.8, 1)(
3
) From the fuzzy weights of each criterion (w
~
i) in Table 5 and the linguistic evaluations in Table 6, a fuzzy weighted decision matrix can be established. Table 7 shows the fuzzy weighted decision values.Table 7
Fuzzy weighted decision matrix.
C
1
C
2
C
3
C
4
C
5
L
1
(0.08, 0.35, 0.8)
(0.16, 0.4, 0.8)
(0.24, 0.63, 1)
(0.08, 0.31, 0.64)
(0.16, 0.27, 0.8)
L
2
(0.12, 0.46, 1)
(0.16, 0.40, 0.8)
(0.16, 0.47, 0.8)
(0.08, 0.31, 0.64)
(0.24, 0.37, 1)
L
3
(0.04, 0.23, 0.6)
(0.32, 0.66, 1)
(0.08, 0.31, 0.6)
(0.12, 0.42, 0.8)
(0.16, 0.27, 0.8)
L
4
(0.12, 0.46, 1)
(0.16, 0.4, 0.8)
(0.24, 0.63, 1)
(0.08, 0.31, 0.64)
(0.32, 0.46, 1)
L
5
(0.08, 0.35, 0.8)
(0.24, 0.53, 1)
(0.8, 0.31, 0.6)
(0.12, 0.42, 0.8)
(0.24, 0.37, 1)(
4
) The fuzzy positive-ideal and fuzzy negative-ideal are determined by using (5):
(9)
T
*

=
[
(
1,1
,
1
)
,
(
1,1
,
1
)
,
(
1,1
,
1
)
,
(
0.08,0.08,0.08
)
,
(
1,1
,
1
)
]
,
T
-

=
[
(
0.04,0.04,0.04
)
,
(
0.16,0.16,0.16
)
,
(
0.08,0.08,0.08
)
,
T
-
l
=
(
0.8,0.8,0.8
)
,
(
0.16,0.16,0.16
)
]
.(
5
) By using (6), the distance of each location candidate from fuzzy positive-ideal and fuzzy negative-ideal can be calculated with respect to each criterion, respectively, as shown in Tables 8 and 9.Table 8
Distance with positive-ideal solution with respect to each criterion.
C
1
C
2
C
3
C
4
C
5
d
(
L
1
,
L
*
)
0.662
0.608
0.489
0.351
0.651
d
(
L
2
,
L
*
)
0.596
0.608
0.585
0.351
0.571
d
(
L
3
,
L
*
)
0.747
0.439
0.702
0.460
0.651
d
(
L
4
,
L
*
)
0.596
0.608
0.489
0.351
0.502
d
(
L
5
,
L
*
)
0.662
0.516
0.702
0.460
0.571Table 9
Distance with negative-ideal solution with respect to each criterion.
C
1
C
2
C
3
C
4
C
5
d
(
L
1
,
L
-
)
0.474
0.394
0.625
0.510
0.375
d
(
L
2
,
L
-
)
0.607
0.394
0.475
0.510
0.502
d
(
L
3
,
L
-
)
0.342
0.572
0.329
0.450
0.375
d
(
L
4
,
L
-
)
0.607
0.394
0.625
0.510
0.523
d
(
L
5
,
L
-
)
0.474
0.532
0.329
0.620
0.502(
6
) From (7), the closeness coefficient (C
C
i) for each location candidate can be calculated as
(10)
C
C
1
=
0.463
,
C
C
2
=
0.478
,
C
C
3
=
0.408
,
C
C
4
=
0.511
,
C
C
5
=
0.457
,
and shown in Table 10.Table 10
Computations ofd
i
*, d
i
-, and C
C
i
*.
d
i
*
d
i
-
d
i
*
+
d
i
-
C
C
i
*
L
1
2.761
2.377
5.138
0.463
L
2
2.711
2.487
5.198
0.478
L
3
2.999
2.068
5.067
0.408
L
4
2.546
2.658
5.204
0.511
L
5
2.912
2.455
5.367
0.457(
7
) The closeness coefficients (C
C
i, i
=
1,2
,
…
,
5) obtained from Step 6 for each location candidate in Table 10 are used as priority values to build the TOPSIS-MCGP model, which will be shown later in this section.


## 5. Sensitivity Analysis

According to Step 6, the best location alternative isL
4. To analyze the location using different criteria weights, a sensitivity analysis was conducted. The purpose of a sensitivity analysis is to exchange each criterion’s weight with another criterion’s weight; thus, 10 combinations for the five criteria are analyzed, and similarities for the ideal solution (closeness coefficients or C
C
j
*) are calculated with each combination stated as a condition. The results of the sensitivity analysis are shown in Table 11.Table 11
Sensitivity analysis for location.
Conditions
Weights
The values ofC
C
i
*
w
~
1
w
~
2
w
~
3
w
~
4
w
~
5
L
1
L
2
L
3
L
4
L
5
Main
(0.20, 0.58, 1.00)
(0.40, 0.66, 1.00)
(0.40, 0.78, 1.00)
(0.20, 0.52, 0.80)
(0.40, 0.46, 1.00)
0.463
0.478
0.408
0.511
0.457
1
(0.40, 0.66, 1.00)
(0.20, 0.58, 1.00)
(0.40, 0.78, 1.00)
(0.20, 0.52, 0.80)
(0.40, 0.46, 1.00)
0.464
0.515
0.400
0.483
0.456
2
(0.40, 0.78, 1.00)
(0.40, 0.66, 1.00)
(0.20, 0.58, 1.00)
(0.20, 0.52, 0.80)
(0.40, 0.46, 1.00)
0.459
0.511
0.409
0.483
0.462
3
(0.20, 0.52, 0.80)
(0.40, 0.66, 1.00)
(0.40, 0.78, 1.00)
(0.20, 0.58, 1.00)
(0.40, 0.46, 1.00)
0.440
0.454
0.389
0.487
0.433
4
(0.40, 046, 1.00)
(0.40, 0.66, 1.00)
(0.40, 0.78, 1.00)
(0.20, 0.52, 0.80)
(0.20, 0.58, 1.00)
0.463
0.510
0.408
0.479
0.457
5
(0.20, 0.58, 1.00)
(0.40, 0.78, 1.00)
(0.40, 0.66, 1.00)
(0.20, 0.52, 0.80)
(0.40, 0.46, 1.00)
0.461
0.478
0.412
0.509
0.46
6
(0.20, 0.58, 1.00)
(0.20, 0.52, 0.80)
(0.40, 0.78, 1.00)
(0.40, 0.66, 1.00)
(0.40, 0.46, 1.00)
0.427
0.444
0.361
0.476
0.413
7
(0.20, 0.58, 1.00)
(0.40, 046, 1.00)
(0.40, 0.78, 1.00)
(0.20, 0.52, 0.80)
(0.40, 0.66, 1.00)
0.463
0.516
0.403
0.481
0.458
8
(0.20, 0.58, 1.00)
(0.40, 0.66, 1.00)
(0.20, 0.52, 0.80)
(0.40, 0.78, 1.00)
(0.40, 0.46, 1.00)
0.413
0.435
0.369
0.461
0.415
9
(0.20, 0.58, 1.00)
(0.40, 0.66, 1.00)
(0.40, 046, 1.00)
(0.20, 0.52, 0.80)
(0.40, 0.78, 1.00)
0.459
0.481
0.411
0.514
0.464
10
(0.20, 0.58, 1.00)
(0.40, 0.66, 1.00)
(0.40, 0.78, 1.00)
(0.40, 046, 1.00)
(0.20, 0.52, 0.80)
0.439
0.453
0.384
0.484
0.429When applying the analysis proposed by Önüt and Soner [29] to the values in Table 11, L
1 has the highest C
C
j
* value (0.464) when the first and second criteria weights are exchanged in condition 1; L
1 has the lowest value (0.413) when the third and fourth criteria weights are exchanged in condition 8. L
2 will have the highest C
C
j
* value (0.516) when the second and fifth criteria weights are exchanged in condition 7, and it will have the lowest value (0.435) when the third and the fourth criteria weights are exchanged in condition 8. L
3 will have the highest C
C
j
* value (0.412) when the second and third criteria weights are exchanged in condition 5, and it will have the lowest value (0.361) when the second and fourth criteria weights are exchanged in condition 6. L
4 will have the highest C
C
j
* value (0.514) when the third and fifth criteria weights are exchanged in condition 9, and it will have the lowest value (0.461) when the third and fourth criteria weights are exchanged in condition 8. L
5 will have the highest C
C
j
* value (0.464) when the third and fifth criteria weights are exchanged in condition 6, and it will have the lowest value (0.413) when the second and fourth criteria weights are exchanged in condition 6. In addition, L
4 will be selected if conditions 3, 5, 6, 8, 9, and 10 are met, whereas L
2 will be selected if conditions 1, 2, 4, and 7 are met; however, the solution is not obtained based on these weights alone. With this approach, the DMs can use these different weights and different closeness coefficients (C
C
j
*) when considering which control factors to combine, according to their business needs, in the decision-making process.Though the basic reason for introducing a logistics centers is to enhance qualitative analysis, the ultimate justification should be made using quantitative measures (e.g., location selection parameters in Table4). Therefore, this paper considers quantitative factors of criteria for logistics centers selection.According to the business strategic and experts’ suggestions by Delphi method, the company ABC points out six goals for location selection which are set below:Goal 1: for maximizing resource availability, namely,f
1
(
x
)
≥
80 (score),
Goal 2: for maximizing location resistance, namely,37
≤
f
2
(
x
)
≤
45 (km2),
Goal 3: for maximizing expansion possibility, namely,85
≤
f
3
(
x
)
≤
160 (lot size m2),
Goal 4: for minimizing investment cost, namely,850
≤
f
4
(
x
)
≤
1580 ($1000),
Goal 5: for maximizing information abilities, namely,f
1
(
x
)
≥
60 (score).This model can be solved by using LINGO 11.0 [32] on a Pentium(R) 4 CPU 2.00 GHz-based microcomputer in a few seconds (of computer time) to obtain optimal solution as (
x
1
,
x
2
,
x
3
,
x
4
,
x
5
)
=
(
0,0
,
0,1
,
0
). From these results, we can understand that location 4, saying (L
4), is the best selection for company ABC.Table12 has shown the results for logistics centers selection comparisons using a sensitivity analysis.Table 12
Comparison of logistics centers location selection methods and sensitivity analysis.
Sensitivity analysis
The best selection
Fuzzy TOPSIS
C
C
i
* ranking
L
4
If conditions 3, 5, 6, 8, 9, and 10 are met
L
4
If conditions 1, 2, 4, and 7 are met
L
2
Fuzzy TOPSIS + MCGP
Using LINGO
L
4


## 6. Conclusions

This paper proposes the TOPSIS-MCGP method to assist DMs’ of airline industrial finding a satisfying logistics centers location under their preferences and resource limitations. In this proposed that the DMs can determine the criteria weights (e.g., closeness coefficients) from TOPSIS and implement it into each goal in MCGP. With different logistics centers intention, DMs can set multiple aspiration levels for each location goal by using MCGP to find the optimal location. Considering both qualitative and quantitative criteria, this paper offers a new practical approach to selecting the best logistics centers location for a given airline industrial business by integrating the fuzzy TOPSIS and GP methods. The integrated advantage of this paper is that it takes both qualitative and quantitative criteria into consideration on logistics centers location problems with “the more/higher is better” (e.g., benefit criteria) or “the less/lower is better” (e.g., cost criteria). The contribution of this paper is that it proposes an easy and effective method to help the logistics business to select the best location.A number of techniques have been proposed to solve the logistics centers selection problems. These approaches include techniques for order preference by similarity to ideal solution (TOPSIS), linear programming (LP), goal programming (GP), data envelopment analysis (DEA), cost point methods (CPM), the analytical hierarchy process (AHP), the analytic network process (ANP), and fuzzy set theory. However, the modeling of many situations may not be sufficient or accurate, as the available data in real life are vague, inaccurate, imprecise, and uncertain by nature. Table13 presents a comparison of this proposed analytical method and the others.Table 13
Comparison of logistics centers location selection methods.
Methods
Selection criteria
Multiple choice aspiration levels
Qualitative
Quantitative
TOPSIS
Yes
No
No
LP/GP
No
Yes
No
DEA
No
Yes
No
CPM
No
Yes
No
AHP/ANP
Yes
No
No
AHP (or ANP) + TOPSIS
Yes
No
No
This proposed method(Fuzzy TOPSIS + MCGP)
Yes
Yes
YesThe proposed method may also be useful for various MCDM problems, such as business strategy selection, supply chain quality development (e.g., [33]), and marketing activities. Therefore, investigating and identifying suitable criteria affect the transport plan problems, and applying other methods (e.g., fuzzy additive ratio assessment ARAS-F [8]) to improve the effectiveness of the decision-making process can be considered for further research. In addition, we expect that this integrated method can be used in our research in the future, such as logistics strategy selection, logistics service development, and logistics activities planning.



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*Source: 470128-2014-07-24.xml*