# Application of a New Hybrid Fuzzy AHP Model to the Location Choice

Authors: Chien-Chang Chou; Ker-Wei Yu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The purpose of this paper is to propose a new hybrid fuzzy Analytic Hierarchy Process (AHP) algorithm to deal with the decision-making problems in an uncertain and multiple-criteria environment. In this study, the proposed hybrid fuzzy AHP model is applied to the location choices of international distribution centers in international ports from the view of multiple-nation corporations. The results show that the proposed new hybrid fuzzy AHP model is an appropriate tool to solve the decision-making problems in an uncertain and multiple-criteria environment.


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## Body

## 1. Introduction

Modern logistics service provided by the logistics provider emphasizes quick response to customer demand. However, it is difficult to fit totally customer demands in a logistics system. In particular, in a global logistics system a lot of uncertainties and complexities exist. A global logistics system includes two important roles. One is the logistics service provider, for example, shipping carriers, international ports, and international distribution centers. Another is the logistics service demander such as multinational corporations (MNCs). The international distribution center within the international port is also one important part of a global logistics system. The shipping companies and the multiple-national corporation prefer to use high-efficiency and high-service quality international logistics centers within an international port. Therefore, it is an important and complex decision-making problem for shipping companies and multinational corporations to select a high-efficiency and high-service quality international logistics center within an international port.Due to a shift in the global center of manufacturing to Asia since 1980s, major international ports in the Asian region have been expanded rapidly. Thus, the shipping companies and the multinational corporations focus on the location choice of international distribution centers in Asia. The major international ports in Asia include the ports of Shanghai, Singapore, Hong Kong, Shenzhen, Busan, Ningbo, Qingdao, Guangzhou, Tianjin, and Kaohsiung. In the future, the demand for cargoes in Asia will further increase. In Table1, the container throughputs in 2010 for the world’s top 20 container ports including the ports of Shanghai, Singapore, Hong Kong, Shenzhen, Busan, Ningbo, Guangzhou, Qingdao, Dubai, Rotterdam, Tianjin, Kaohsiung, Antwerp, Klang, Hamburg, Los Angeles, Tanjung Pelepas, Long Beach, Xiamen, and Laem Chabang are 29,069, 28,430, 23,611, 22,509, 14,180, 13,146, 12,545, 12,012, 11,613, 11,145, 10,086, 9,181, 8,483, 8,146, 7,900, 7,831, 6,603, 6,263, 5,824, and 5,640 thousand TEUs, respectively. The rankings and the volumes for these major international ports from 2006 to 2010 are also shown in Table 1.Table 1
The container throughputs and the rankings for the world’s top 20 container ports (unit: thousand TEUs).
2010
2009
2008
2007
2006
Name of port
2010
2009
2008
2007
2006
1
2
2
2
3
Shanghai
29,069
25,002
27,780
26,150
21,720
2
1
1
1
1
Singapore
28,430
25,866
29,918
27,935
24,792
3
3
3
3
2
Hong Kong
23,611
20,925
24,248
23,881
23,539
4
4
4
4
4
Shenzhen
22,509
18,250
21,400
21,090
18,468
5
5
5
5
5
Busan
14,180
11,980
13,452
13,261
12,038
6
8
8
11
13
Ningbo
13,146
10,502
10,933
9,360
7,140
7
6
7
12
15
Guangzhou
12,545
11,190
11,001
9,200
6,656
8
9
10
10
11
Qingdao
12,012
10,260
10,020
9,462
7,702
9
7
6
6
8
Dubai
11,613
11,150
11,830
11,000
8,923
10
10
9
7
7
Rotterdam
11,145
9,800
10,784
10,791
9,690
11
11
14
17
17
Tianjin
10,086
8,700
8,500
7,102
5,950
12
12
12
8
6
Kaohsiung
9,181
8,581
9,676
10,256
9,774
13
13
13
14
14
Antwerp
8,483
7,309
8,663
8,176
7,018
14
14
15
16
16
Klang
8,146
7,300
7,973
7,118
6,326
15
15
11
9
9
Hamburg
7,900
7,010
9,737
9,890
8,861
16
16
16
13
10
Los Angeles
7,831
6,748
7,849
8,355
8,469
17
17
18
18
19
Tanjung Pelepas
6,603
6,000
5,600
5,500
4,770
18
18
17
15
12
Long Beach
6,263
5,067
6,487
7,312
7,289
19
19
22
22
22
Xiamen
5,824
4,680
5,034
4,627
4,018
20
20
21
21
21
Laem Chabang
5,640
4,640
5,130
4,640
4,123
Source: Chou et al. [1].Many international distribution centers at major Asian ports have been established in the recent years, such as Waigaoqiao Bond Logistics Park in the port of Shanghai, Hong Kong International Distribution Center in the port of Hong Kong, Kaohsiung Yes Logistics Zone in the port of Kaohsiung, Schwartz Logistics Hub in the port of Shenzhen, Busan Logistics Park in the port of Busan, and Keppel Distripark in the port of Singapore. Therefore, it is very important for the shipping companies and the multinational corporations to evaluate the environment among these major international logistics centers in different nations, in order to design and implement an appropriate global logistics system. It is also a complex multiple-criteria decision-making (MCDM) problem under uncertain environment. AHP is an appropriate approach to solve complex multiple-criteria decision-making problem. Fuzzy sets theory method has been widely applied to the uncertain decision-making problem in the real world. Thus, this paper combines AHP and fuzzy sets theory and then proposes a new hybrid fuzzy AHP model for the location choice of international logistics centers within the international ports from the perspective of multi-nation corporations.


## 2. Literature Review

### 2.1. Environmental Evaluation Approaches

Environment evaluation approaches including the resource-based view (RBV), traditional strength-weakness-opportunity-threat (SWOT), and quantitative SWOT such as the external factor evaluation matrix (EFE), internal factor evaluation matrix (IFE), and competitive profile matrix (CPM) have been widely used. The traditional SWOT analytical method is commonly applied to marketing strategy analysis [2–4]. The SWOT analytical method is able to help the decision maker of the enterprises evaluate qualitatively their competitiveness and can be used as a foundation of the development of strategies [5]. The quantitative SWOT such as external factor evaluation matrix (EFE), internal factor evaluation matrix (IFE), and competitive profile matrix (CPM) aim at analyzing statistical data, differing from the traditional SWOT analytical method [3, 6]. The disadvantage of the above approaches is that they cannot evaluate the qualitative and quantitative criteria simultaneously. Therefore, this paper proposes a new hybrid approach which integrates AHP method to carry out a complete evaluation of qualitative and quantitative criteria simultaneously and to evaluate the competitive environmental relationships between the several international distribution center locations within the ports in the asian region.Erol and Ferrell Jr. [7] use fuzzy Quality Function Deployment (QFD) to convert qualitative information into quantitative parameters and then combine this data with other quantitative data to multiobjective mathematical programming model. Mikhailov and Tsvetinov [8] propose a fuzzy AHP approach for tackling the uncertainty and imprecision of the service evaluation process. Ronza et al. [9] present a quantitative risk analysis approach to port hydrocarbon logistics. Because risk is an uncertain criterion, it is not easy for a decision maker to measure exactly the value of risk. Chang and Huang [4] present a quantitative strength/weakness/opportunity/threat (SWOT) analysis method for assessing the competing strengths of major ports in East Asia. Yong [10] uses a fuzzy TOPSIS model to solve the problem of plant location choice. Önüt and Soner [11] propose an AHP/TOPSIS approach for solving transshipment site selection problem under fuzzy environment. Tahera et al. [12] develop a fuzzy logic approach for dealing with qualitative quality characteristics of a process. Chen et al. [13] combine fuzzy AHP with Multidimension Scaling (MDS) in identifying the preference similarity of alternatives. Chou et al. [14] present a new fuzzy multiple attributes decision-making (MADM) approach for solving facility location selection problem by using objective and subjective attributes. Lee and Lin [15] use a fuzzy quantitative SWOT procedure for environment evaluation of international distribution centers in Pacific Asian region. Lee et al. [16] introduce a fuzzy AHP and Balanced Score Card (BSC) approach for evaluating performance of IT department in the manufacturing industry in Taiwan. Chou [17] uses fuzzy MCDM approach for dealing with quantitative and qualitative criteria in a process of location choice. Chou [18] deals with objective data and subjective ratings by fuzzy logic. Chou [19] analyzes the competitive relationship between the ports of Hong Kong, Shanghai, and Kaohsiung by the sensitivity analysis.In the past, although many researchers proposed a lot of fuzzy approaches, for example, fuzzy QFD, MADM, AHP, TOPSIS, BSC, SWOT, and MCDM, few presented a hybrid qualitative/quantitative fuzzy AHP model for dealing with both objective data and subjective criteria simultaneously in the decision-making process.


### 2.2. Fuzzy AHP

Despite of its wide application to various decision-making problems, the conventional AHP approach may not fully reflect a style of human thinking. Thus, the fuzzy AHP approach is proposed to overcome the disadvantage of the conventional AHP. The fuzzy AHP approach is a systematic method for the alternative choice and justification problems that combines the concept of fuzzy sets theory [20] and the hierarchical structure analysis [21].Fuzzy AHP approach has been widely applied to many decision-making problems. For example, Chang [22] developed a fuzzy extent analysis for AHP and the approach is relatively easier in computational procedure than the other fuzzy AHP approaches. Kuo et al. [23] presented a fuzzy AHP method for the location choice of a convenience store. Kurttila et al. [5] combined AHP with SWOT to provide a new hybrid method for a forest certification case. Stewart et al. [24] combined AHP method with SWOT to present a new approach for improving the usability of AHP in strategic management. Kahraman et al. [25] applied fuzzy AHP to select the location of facility. Zhang et al. [26] combined fuzzy AHP with MCDM to deal with an MCDM decision-making problem. The results show that the proposed hybrid method was a useful way to deal with MCDM decision-making problems. Erensal et al. [27] determined key capabilities in technology management by using fuzzy AHP. Chan and Kumar [28] proposed a model for global supplier development considering risk factors by using fuzzy AHP. Bozbura and Beskese [29] determined the priorities of organizational capital measurement indicators by using fuzzy AHP. Bozbura et al. [30] used fuzzy AHP method to determine the priorities of human capital measurement indicators. Lee and Lin [15] developed a fuzzy quantified SWOT procedure that integrates MCDM concept and fuzzy AHP method for the location choice of international distribution centers.


## 2.1. Environmental Evaluation Approaches

Environment evaluation approaches including the resource-based view (RBV), traditional strength-weakness-opportunity-threat (SWOT), and quantitative SWOT such as the external factor evaluation matrix (EFE), internal factor evaluation matrix (IFE), and competitive profile matrix (CPM) have been widely used. The traditional SWOT analytical method is commonly applied to marketing strategy analysis [2–4]. The SWOT analytical method is able to help the decision maker of the enterprises evaluate qualitatively their competitiveness and can be used as a foundation of the development of strategies [5]. The quantitative SWOT such as external factor evaluation matrix (EFE), internal factor evaluation matrix (IFE), and competitive profile matrix (CPM) aim at analyzing statistical data, differing from the traditional SWOT analytical method [3, 6]. The disadvantage of the above approaches is that they cannot evaluate the qualitative and quantitative criteria simultaneously. Therefore, this paper proposes a new hybrid approach which integrates AHP method to carry out a complete evaluation of qualitative and quantitative criteria simultaneously and to evaluate the competitive environmental relationships between the several international distribution center locations within the ports in the asian region.Erol and Ferrell Jr. [7] use fuzzy Quality Function Deployment (QFD) to convert qualitative information into quantitative parameters and then combine this data with other quantitative data to multiobjective mathematical programming model. Mikhailov and Tsvetinov [8] propose a fuzzy AHP approach for tackling the uncertainty and imprecision of the service evaluation process. Ronza et al. [9] present a quantitative risk analysis approach to port hydrocarbon logistics. Because risk is an uncertain criterion, it is not easy for a decision maker to measure exactly the value of risk. Chang and Huang [4] present a quantitative strength/weakness/opportunity/threat (SWOT) analysis method for assessing the competing strengths of major ports in East Asia. Yong [10] uses a fuzzy TOPSIS model to solve the problem of plant location choice. Önüt and Soner [11] propose an AHP/TOPSIS approach for solving transshipment site selection problem under fuzzy environment. Tahera et al. [12] develop a fuzzy logic approach for dealing with qualitative quality characteristics of a process. Chen et al. [13] combine fuzzy AHP with Multidimension Scaling (MDS) in identifying the preference similarity of alternatives. Chou et al. [14] present a new fuzzy multiple attributes decision-making (MADM) approach for solving facility location selection problem by using objective and subjective attributes. Lee and Lin [15] use a fuzzy quantitative SWOT procedure for environment evaluation of international distribution centers in Pacific Asian region. Lee et al. [16] introduce a fuzzy AHP and Balanced Score Card (BSC) approach for evaluating performance of IT department in the manufacturing industry in Taiwan. Chou [17] uses fuzzy MCDM approach for dealing with quantitative and qualitative criteria in a process of location choice. Chou [18] deals with objective data and subjective ratings by fuzzy logic. Chou [19] analyzes the competitive relationship between the ports of Hong Kong, Shanghai, and Kaohsiung by the sensitivity analysis.In the past, although many researchers proposed a lot of fuzzy approaches, for example, fuzzy QFD, MADM, AHP, TOPSIS, BSC, SWOT, and MCDM, few presented a hybrid qualitative/quantitative fuzzy AHP model for dealing with both objective data and subjective criteria simultaneously in the decision-making process.


## 2.2. Fuzzy AHP

Despite of its wide application to various decision-making problems, the conventional AHP approach may not fully reflect a style of human thinking. Thus, the fuzzy AHP approach is proposed to overcome the disadvantage of the conventional AHP. The fuzzy AHP approach is a systematic method for the alternative choice and justification problems that combines the concept of fuzzy sets theory [20] and the hierarchical structure analysis [21].Fuzzy AHP approach has been widely applied to many decision-making problems. For example, Chang [22] developed a fuzzy extent analysis for AHP and the approach is relatively easier in computational procedure than the other fuzzy AHP approaches. Kuo et al. [23] presented a fuzzy AHP method for the location choice of a convenience store. Kurttila et al. [5] combined AHP with SWOT to provide a new hybrid method for a forest certification case. Stewart et al. [24] combined AHP method with SWOT to present a new approach for improving the usability of AHP in strategic management. Kahraman et al. [25] applied fuzzy AHP to select the location of facility. Zhang et al. [26] combined fuzzy AHP with MCDM to deal with an MCDM decision-making problem. The results show that the proposed hybrid method was a useful way to deal with MCDM decision-making problems. Erensal et al. [27] determined key capabilities in technology management by using fuzzy AHP. Chan and Kumar [28] proposed a model for global supplier development considering risk factors by using fuzzy AHP. Bozbura and Beskese [29] determined the priorities of organizational capital measurement indicators by using fuzzy AHP. Bozbura et al. [30] used fuzzy AHP method to determine the priorities of human capital measurement indicators. Lee and Lin [15] developed a fuzzy quantified SWOT procedure that integrates MCDM concept and fuzzy AHP method for the location choice of international distribution centers.


## 3. Methodology

### 3.1. Fuzzy Sets Theory

Fuzzy sets theory is initially introduced by Zadeh [20]. A fuzzy number is defined as follows. Suppose A1=(c1,a1,b1,d1) is a trapezoidal fuzzy number in Figure 1. The membership function of A1 and the fuzzy arithmetic operations on fuzzy numbers are shown as follows:
(1)fA1(x)={(x-c1)(a1-c1),c1≤x≤a11,a1≤x≤b1(x-d1)(b1-d1),b1≤x≤d10,otherwise.Figure 1
Trapezoidal fuzzy numberA1=(c1,a1,b1,d1).SupposeA1=(c1,a1,b1,d1) and A2=(c2,a2,b2,d2) are two trapezoidal fuzzy numbers.(a)
Addition operation onA1andA2(2)A1⊕A2=(c1+c2,a1+a2,b1+b2,d1+d2).(b)
Subtraction operation onA1 and A2(3)A1ΘA2=(c1-d2,a1-b2,b1-a2,d1-c2).(c)
Multiplication operation onA1 and r(4)r⊗A1=(rc1,ra1,rb1,rd1).(d)
Division operation onA1 and r(5)A1r=(c1r,a1r,b1r,d1r).Chou [31] proposed the canonical representation of a triangular fuzzy number Y=(c,a,b):
(6)P(Y)=16(c+4a+b).


### 3.2. AHP Theory

Saaty [21] initially proposed the Analytic Hierarchy Process (AHP), which is a multiple-attribute decision-making tool for solving complex multiple-criteria decision-making problems. AHP methodology has some advantages. One of the most important advantages of the AHP is based on the pairwise comparison. Another is that the AHP calculates the inconsistency index, which is the ratio of the decision maker’s inconsistency on the criteria. The computational procedures for AHP are listed as follows.Let us consider the criteriaC1,…,Ci,…,Cj,…,Cn,   someone level in the hierarchy. One wishes to find their weights of importance, W1,…,Wi,…,Wj,…,Wn, on some elements in the next level. Allow aij,i,j=1,2,…,n, to be the importance strength of Ci when compared with Cj. In generally we can represent the comparative importance scale of criteria as shown in Table 2. The matrix of these numbers aij is denoted by B:
(7)B=[a11a12⋯a1j⋯a1n⋮⋮⋮⋮ai1ai2⋯aij⋯ain⋮⋮⋮⋮an1an2⋯anj⋯ann]n×n,
where aji=1/aij; that is, B is reciprocal. If one’s judgment is perfect in all comparisons, then aik=aij·ajk for all i,j,k and one calls the matrix B consistent. An obvious case of a consistent matrix B is its elements
(8)aij=wiwj,i,j=1,2,…,n.Table 2
Comparative importance scale of criteria.
Scale
Definition
Description
(1, 1, 1)
Equally important, EI
The importance of both comparative alternatives is equal
(1, 2, 3)
Intermediate values, IV
Need to compromise between EI and WI
(2, 3, 4)
Weakly important, WI
Experience and judgment weakly tend to prefer one alternative
(3, 4, 5)
Intermediate values, IV
Need to compromise between WI and SI
(4, 5, 6)
Strongly important, SI
Experience and judgment strongly tend to prefer one alternative
(5, 6, 7)
Intermediate values, IV
Need to compromise between SI and DI
(6, 7, 8)
Demonstrably important, DI
Experience and judgment demonstrably tend to prefer one alternative
(7, 8, 9)
Intermediate values, IV
Need to compromise between DI and AI
(9, 9, 9)
Absolutely important, AI
Experience and judgment absolutely tend to prefer one alternativeThus, when the matrixB is multiplied by the vector formed by each weighting w=(w1,w2,…,wn)T, one gets
(9)Bw=[w1w1w1w2⋯w1wj⋯w1wnw2w1w2w2⋯w2wj⋯w2wn⋮⋮⋮⋮wiw1wiw2wiwjwiwn⋮⋮⋮⋮wnw1wnw2⋯wnwj⋯wnwn]n×n[w1w2⋮wj⋮wn]n×1Bw=n[w1w2⋮wj⋮wn]=nw.Becauseaij is the subjective ratings given by the decision maker, there must be a distance between it and the actual values wi/wj. Thus, Bw=nw cannot be calculated directly. Therefore Saaty suggested using the maximum eigenvalue, λmax=(1/n)(w1′/w1+w2′/w2+⋯+wn′/wn), of the solution of matrix B to replace n; then
(10)Bw=λmaxw.By this method, one can obtain the characteristic vector, referred to as the priority vector. Besides Saaty suggested the consistency index(C.I=(λmax-n)/(n-1)) and the consistency rate (C.R=C.I/R.I) to test the consistency of the intuitive judgment. In general, it is satisfactory and accepted if the value of C.I is about 0.1 and the value of C.R is less than 0.1.


### 3.3. Proposed Hybrid Fuzzy AHP Approach

The proposed hybrid fuzzy AHP approach to solving both quantitative data and qualitative ratings simultaneously in process of the location selection is introduced in this section. The proposed hybrid fuzzy AHP approach involves 11 steps shown as follows.Step 1.
Construct a hierarchical analysis structure in Table3. These criteria in the hierarchical analysis structure can be divided into two categories: objective and subjective criteria. The objective criteria are defined in monetary or quantitative terms. The subjective criteria are defined in linguistic terms represented by fuzzy numbers.Table 3
A hierarchical analysis structure.
Criteria and subcriteria
Weight
Rating
Geographical condition (C1)
0.1527
Closeness to the import/export area (C11)
0.3333
(5.0, 5.0, 5.0)
Proximity of the feeder port (C12)
0.1111
(4.2, 4.2, 4.2)
Closeness to main navigation route (C13)
0.1111
(5.0, 5.0, 5.0)
Frequency of ship calls (C14)
0.1111
(4.6, 4.8, 5.0)
Delivery time (C15)
0.3333
(3.0, 4.0, 5.0)
Cost (C2)
0.4394
Transportation cost (C21)
0.3543
(2.0, 3.0, 4.0)
Operation cost (C22)
0.1921
(2.0, 3.0, 4.0)
Land cost (C23)
0.1921
(2.0, 3.0, 4.0)
Labor cost (C24)
0.1921
(2.0, 3.0, 4.0)
Port charge  (C25)
0.0694
(1.0, 1.0, 1.0)
Economy (C3)
0.0824
Volume of import cargoes (C31)
0.2607
(4.4, 4.6, 4.6)
Volume of export cargoes (C32)
0.1720
(4.2, 4.4, 4.6)
Volume of transshipment cargoes (C33)
0.1976
(4.9, 4.9, 5.0)
Economic growth (C34)
0.1720
(5.0, 5.0, 5.0)
Trade variables (C35).
0.1976
(3.0, 4.0, 5.0)
Government (C4)
0.0757
Private ownership of enterprise (C41)
0.1131
(5.0, 5.0, 5.0)
Efficiency of customs (C42)
0.3055
(5.0, 5.0, 5.0)
Efficiency of government department (C43)
0.2453
(5.0, 5.0, 5.0)
Type of cooperation of enterprise and government (C44)
0.1226
(3.0, 4.0, 5.0)
Political stability (C45)
0.2135
(3.0, 4.0, 5.0)
Investment conditions (C5)
0.0785
Tax break and preferential treatment (C51)
0.2500
(2.0, 3.0, 4.0)
Law on investment restrictions (C52)
0.2500
(5.0, 5.0, 5.0)
Social stability (C53)
0.1250
(3.0, 4.0, 5.0)
Land availability and expansion possibility (C54)
0.2500
(0.0, 1.0, 2.0)
Labor quality (C55)
0.1250
(5.0, 5.0, 5.0)
Infrastructure and efficiency (C6)
0.1714
Port facilities (C61)
0.1057
(5.0, 5.0, 5.0)
Loading and discharging facilities (C62)
0.2164
(4.9, 4.9, 5.0)
Intermodal link (C63)
0.3096
(3.0, 4.0, 5.0)
Cargo handling efficiency (C64)
0.2043
(4.7, 4.9, 5.0)
Computer information system (C65)
0.1640
(3.0, 4.0, 5.0)Step 2.
Introduce linguistic variables for importance weight of criteria. Terms of linguistic variables for importance weight of criteria could be called “equally important”, “weakly important,” “strongly important,” “demonstrably important,” “absolutely important,” and so forth. These linguistic variables can be expressed in fuzzy numbers such as “equally important” = (1,1,1), “weakly important” = (2,3,4), “strongly important” = (4,5,6), “demonstrably important” = (6,7,8), and “absolutely important” = (9,9,9). Their reciprocals are considered as “weakly unimportant” = (1/4,1/3,1/2), “strongly unimportant” = (1/6,1/5,1/4), “demonstrably unimportant” = (1/8,1/7,1/6), and “absolutely unimportant” = (1/9,1/9,1/9).Step 3.
Introduce linguistic variables for ratings of alternative locations. Terms of linguistic variables for ratings of alternative locations could be called “very poor,” “poor,” “fair,” “good,” “very good,” and so forth. These linguistic variables can be expressed in fuzzy numbers such as “very poor” = (0.0,1.0,2.0), “poor” = (1.0,2.0,3.0), “fair = (2.0,3.0,4.0)”, “good” = (3.0,4.0,5.0), “very good” = (5.0,5.0,5.0), and so forth.Step 4.
Determine the importance weights of criteria by the decision maker. Assume there areN candidate locations (A1,A2,An,…,AN),I evaluation criteria (C1,C2,Ci,…,CI), and J subcriteria (Ci1,Ci2,Cij,…,CiJ) under criteria i, where 1≤n≤N,1≤i≤I,1≤j≤J. W~i=(ci,ai,bi) and W~ij=(cij,aij,bij) are the fuzzy importance weights given by the decision maker to criteria Ci and subcriteria Cij, respectively.Step 5.
Defuzzify the weights of criteria and subcriteria. Then calculate the normalized weights. According to (6) proposed by Chou [31], we can obtain the representation of fuzzy numbers W~i and W~ij as follows:
(11)P(W~i)=16(ci+4ai+bi),P(W~ij)=16(cij+4aij+bij).
The normalized weights of criteria Ci and subcriteria Cij are given by
(12)Wi=P(W~i)∑i=1IP(W~i),Wij=P(W~ij)∑j=1JP(W~ij),
where ∑i=1IWi=1, ∑j=1JWij=1.The weight vector is therefore formed as follows:(13)W=[W1×W11,W1×W12,…,W1×W1j,…,Wi×Wi1,Wi×Wi2,…,Wi×Wij,…,WI×WI1,WI×WI2,…,WI×WIJW1j]=[W111,W112,…,W11j,…,Wii1,Wii2,…,Wiij,…,WII1,WII2,…,WIIJW1j].Step 6.
Calculate the maximum eigenvalue (λmax), the consistency index (C.I), and the consistency rate (C.R) for AHP model to test the consistency of the intuitive judgment.Step 7.
The decision maker assesses alternatives under subjective criteria. LetR~ijn=(cijn,aijn,bijn) be the fuzzy ratings given by the decision maker to alternative An under subjective subcriteria Cij.Step 8.
Assess alternatives under objective criteria. LetR~ijno=(cijno,aijno,bijno) be the fuzzy quantity given to alternativeAn under objective sub-criteriaCij. The objective criteria are determined in various units and must be transformed into dimensionless indices (or ratings) to ensure compatibility with the linguistic ratings of subjective criteria. The alternative with the minimum cost (or maximum benefit) should have the highest rating. By (14) and (15), we can transform fuzzy quantities for objective subcriteria into fuzzy ratings:
(14)R~ijn={R~ijnomaxi{bijno}}×5,
where maxi{bijno}>0 and R~ijn denotes the transformed fuzzy rating of objective fuzzy benefit R~ijno. R~ijn becomes larger when objective fuzzy benefit R~ijno is larger:
(15)R~ijn={mini{cijno}R~ijno}×5,
where mini{cijno}>0 and R~ijn denotes the transformed fuzzy rating of objective fuzzy cost R~ijno. R~ijn becomes smaller when objective fuzzy cost R~ijno is larger.Step 9.
Construct a fuzzy rating matrix based on fuzzy ratings. The fuzzy rating matrixMcan be concisely expressed in matrix format:
(16)M=[R~111R~121⋯R~IJ1R~112R~122⋯R~IJ2::⋯:R~11NR~12N⋯R~IJN].Step 10.
Obtain the total fuzzy rating (R~) based on the fuzzy rating matrix (M) and weight vector (W):
(17)R~=[R~111R~121⋯R~IJ1R~112R~122⋯R~IJ2⋮⋮⋯⋮R~11NR~12N⋯R~IJN]⊗[W111W112⋮WIIJ]=[R~1R~2⋮R~N].Step 11.
Defuzzify the total fuzzy rating by (6) and then rank alternatives according to their total crisp ratings. Finally, we can select easily the best alternative with the maximum total crisp ratings:
(18)P(R~n)=16(cn+4an+bn).


## 3.1. Fuzzy Sets Theory

Fuzzy sets theory is initially introduced by Zadeh [20]. A fuzzy number is defined as follows. Suppose A1=(c1,a1,b1,d1) is a trapezoidal fuzzy number in Figure 1. The membership function of A1 and the fuzzy arithmetic operations on fuzzy numbers are shown as follows:
(1)fA1(x)={(x-c1)(a1-c1),c1≤x≤a11,a1≤x≤b1(x-d1)(b1-d1),b1≤x≤d10,otherwise.Figure 1
Trapezoidal fuzzy numberA1=(c1,a1,b1,d1).SupposeA1=(c1,a1,b1,d1) and A2=(c2,a2,b2,d2) are two trapezoidal fuzzy numbers.(a)
Addition operation onA1andA2(2)A1⊕A2=(c1+c2,a1+a2,b1+b2,d1+d2).(b)
Subtraction operation onA1 and A2(3)A1ΘA2=(c1-d2,a1-b2,b1-a2,d1-c2).(c)
Multiplication operation onA1 and r(4)r⊗A1=(rc1,ra1,rb1,rd1).(d)
Division operation onA1 and r(5)A1r=(c1r,a1r,b1r,d1r).Chou [31] proposed the canonical representation of a triangular fuzzy number Y=(c,a,b):
(6)P(Y)=16(c+4a+b).


## 3.2. AHP Theory

Saaty [21] initially proposed the Analytic Hierarchy Process (AHP), which is a multiple-attribute decision-making tool for solving complex multiple-criteria decision-making problems. AHP methodology has some advantages. One of the most important advantages of the AHP is based on the pairwise comparison. Another is that the AHP calculates the inconsistency index, which is the ratio of the decision maker’s inconsistency on the criteria. The computational procedures for AHP are listed as follows.Let us consider the criteriaC1,…,Ci,…,Cj,…,Cn,   someone level in the hierarchy. One wishes to find their weights of importance, W1,…,Wi,…,Wj,…,Wn, on some elements in the next level. Allow aij,i,j=1,2,…,n, to be the importance strength of Ci when compared with Cj. In generally we can represent the comparative importance scale of criteria as shown in Table 2. The matrix of these numbers aij is denoted by B:
(7)B=[a11a12⋯a1j⋯a1n⋮⋮⋮⋮ai1ai2⋯aij⋯ain⋮⋮⋮⋮an1an2⋯anj⋯ann]n×n,
where aji=1/aij; that is, B is reciprocal. If one’s judgment is perfect in all comparisons, then aik=aij·ajk for all i,j,k and one calls the matrix B consistent. An obvious case of a consistent matrix B is its elements
(8)aij=wiwj,i,j=1,2,…,n.Table 2
Comparative importance scale of criteria.
Scale
Definition
Description
(1, 1, 1)
Equally important, EI
The importance of both comparative alternatives is equal
(1, 2, 3)
Intermediate values, IV
Need to compromise between EI and WI
(2, 3, 4)
Weakly important, WI
Experience and judgment weakly tend to prefer one alternative
(3, 4, 5)
Intermediate values, IV
Need to compromise between WI and SI
(4, 5, 6)
Strongly important, SI
Experience and judgment strongly tend to prefer one alternative
(5, 6, 7)
Intermediate values, IV
Need to compromise between SI and DI
(6, 7, 8)
Demonstrably important, DI
Experience and judgment demonstrably tend to prefer one alternative
(7, 8, 9)
Intermediate values, IV
Need to compromise between DI and AI
(9, 9, 9)
Absolutely important, AI
Experience and judgment absolutely tend to prefer one alternativeThus, when the matrixB is multiplied by the vector formed by each weighting w=(w1,w2,…,wn)T, one gets
(9)Bw=[w1w1w1w2⋯w1wj⋯w1wnw2w1w2w2⋯w2wj⋯w2wn⋮⋮⋮⋮wiw1wiw2wiwjwiwn⋮⋮⋮⋮wnw1wnw2⋯wnwj⋯wnwn]n×n[w1w2⋮wj⋮wn]n×1Bw=n[w1w2⋮wj⋮wn]=nw.Becauseaij is the subjective ratings given by the decision maker, there must be a distance between it and the actual values wi/wj. Thus, Bw=nw cannot be calculated directly. Therefore Saaty suggested using the maximum eigenvalue, λmax=(1/n)(w1′/w1+w2′/w2+⋯+wn′/wn), of the solution of matrix B to replace n; then
(10)Bw=λmaxw.By this method, one can obtain the characteristic vector, referred to as the priority vector. Besides Saaty suggested the consistency index(C.I=(λmax-n)/(n-1)) and the consistency rate (C.R=C.I/R.I) to test the consistency of the intuitive judgment. In general, it is satisfactory and accepted if the value of C.I is about 0.1 and the value of C.R is less than 0.1.


## 3.3. Proposed Hybrid Fuzzy AHP Approach

The proposed hybrid fuzzy AHP approach to solving both quantitative data and qualitative ratings simultaneously in process of the location selection is introduced in this section. The proposed hybrid fuzzy AHP approach involves 11 steps shown as follows.Step 1.
Construct a hierarchical analysis structure in Table3. These criteria in the hierarchical analysis structure can be divided into two categories: objective and subjective criteria. The objective criteria are defined in monetary or quantitative terms. The subjective criteria are defined in linguistic terms represented by fuzzy numbers.Table 3
A hierarchical analysis structure.
Criteria and subcriteria
Weight
Rating
Geographical condition (C1)
0.1527
Closeness to the import/export area (C11)
0.3333
(5.0, 5.0, 5.0)
Proximity of the feeder port (C12)
0.1111
(4.2, 4.2, 4.2)
Closeness to main navigation route (C13)
0.1111
(5.0, 5.0, 5.0)
Frequency of ship calls (C14)
0.1111
(4.6, 4.8, 5.0)
Delivery time (C15)
0.3333
(3.0, 4.0, 5.0)
Cost (C2)
0.4394
Transportation cost (C21)
0.3543
(2.0, 3.0, 4.0)
Operation cost (C22)
0.1921
(2.0, 3.0, 4.0)
Land cost (C23)
0.1921
(2.0, 3.0, 4.0)
Labor cost (C24)
0.1921
(2.0, 3.0, 4.0)
Port charge  (C25)
0.0694
(1.0, 1.0, 1.0)
Economy (C3)
0.0824
Volume of import cargoes (C31)
0.2607
(4.4, 4.6, 4.6)
Volume of export cargoes (C32)
0.1720
(4.2, 4.4, 4.6)
Volume of transshipment cargoes (C33)
0.1976
(4.9, 4.9, 5.0)
Economic growth (C34)
0.1720
(5.0, 5.0, 5.0)
Trade variables (C35).
0.1976
(3.0, 4.0, 5.0)
Government (C4)
0.0757
Private ownership of enterprise (C41)
0.1131
(5.0, 5.0, 5.0)
Efficiency of customs (C42)
0.3055
(5.0, 5.0, 5.0)
Efficiency of government department (C43)
0.2453
(5.0, 5.0, 5.0)
Type of cooperation of enterprise and government (C44)
0.1226
(3.0, 4.0, 5.0)
Political stability (C45)
0.2135
(3.0, 4.0, 5.0)
Investment conditions (C5)
0.0785
Tax break and preferential treatment (C51)
0.2500
(2.0, 3.0, 4.0)
Law on investment restrictions (C52)
0.2500
(5.0, 5.0, 5.0)
Social stability (C53)
0.1250
(3.0, 4.0, 5.0)
Land availability and expansion possibility (C54)
0.2500
(0.0, 1.0, 2.0)
Labor quality (C55)
0.1250
(5.0, 5.0, 5.0)
Infrastructure and efficiency (C6)
0.1714
Port facilities (C61)
0.1057
(5.0, 5.0, 5.0)
Loading and discharging facilities (C62)
0.2164
(4.9, 4.9, 5.0)
Intermodal link (C63)
0.3096
(3.0, 4.0, 5.0)
Cargo handling efficiency (C64)
0.2043
(4.7, 4.9, 5.0)
Computer information system (C65)
0.1640
(3.0, 4.0, 5.0)Step 2.
Introduce linguistic variables for importance weight of criteria. Terms of linguistic variables for importance weight of criteria could be called “equally important”, “weakly important,” “strongly important,” “demonstrably important,” “absolutely important,” and so forth. These linguistic variables can be expressed in fuzzy numbers such as “equally important” = (1,1,1), “weakly important” = (2,3,4), “strongly important” = (4,5,6), “demonstrably important” = (6,7,8), and “absolutely important” = (9,9,9). Their reciprocals are considered as “weakly unimportant” = (1/4,1/3,1/2), “strongly unimportant” = (1/6,1/5,1/4), “demonstrably unimportant” = (1/8,1/7,1/6), and “absolutely unimportant” = (1/9,1/9,1/9).Step 3.
Introduce linguistic variables for ratings of alternative locations. Terms of linguistic variables for ratings of alternative locations could be called “very poor,” “poor,” “fair,” “good,” “very good,” and so forth. These linguistic variables can be expressed in fuzzy numbers such as “very poor” = (0.0,1.0,2.0), “poor” = (1.0,2.0,3.0), “fair = (2.0,3.0,4.0)”, “good” = (3.0,4.0,5.0), “very good” = (5.0,5.0,5.0), and so forth.Step 4.
Determine the importance weights of criteria by the decision maker. Assume there areN candidate locations (A1,A2,An,…,AN),I evaluation criteria (C1,C2,Ci,…,CI), and J subcriteria (Ci1,Ci2,Cij,…,CiJ) under criteria i, where 1≤n≤N,1≤i≤I,1≤j≤J. W~i=(ci,ai,bi) and W~ij=(cij,aij,bij) are the fuzzy importance weights given by the decision maker to criteria Ci and subcriteria Cij, respectively.Step 5.
Defuzzify the weights of criteria and subcriteria. Then calculate the normalized weights. According to (6) proposed by Chou [31], we can obtain the representation of fuzzy numbers W~i and W~ij as follows:
(11)P(W~i)=16(ci+4ai+bi),P(W~ij)=16(cij+4aij+bij).
The normalized weights of criteria Ci and subcriteria Cij are given by
(12)Wi=P(W~i)∑i=1IP(W~i),Wij=P(W~ij)∑j=1JP(W~ij),
where ∑i=1IWi=1, ∑j=1JWij=1.The weight vector is therefore formed as follows:(13)W=[W1×W11,W1×W12,…,W1×W1j,…,Wi×Wi1,Wi×Wi2,…,Wi×Wij,…,WI×WI1,WI×WI2,…,WI×WIJW1j]=[W111,W112,…,W11j,…,Wii1,Wii2,…,Wiij,…,WII1,WII2,…,WIIJW1j].Step 6.
Calculate the maximum eigenvalue (λmax), the consistency index (C.I), and the consistency rate (C.R) for AHP model to test the consistency of the intuitive judgment.Step 7.
The decision maker assesses alternatives under subjective criteria. LetR~ijn=(cijn,aijn,bijn) be the fuzzy ratings given by the decision maker to alternative An under subjective subcriteria Cij.Step 8.
Assess alternatives under objective criteria. LetR~ijno=(cijno,aijno,bijno) be the fuzzy quantity given to alternativeAn under objective sub-criteriaCij. The objective criteria are determined in various units and must be transformed into dimensionless indices (or ratings) to ensure compatibility with the linguistic ratings of subjective criteria. The alternative with the minimum cost (or maximum benefit) should have the highest rating. By (14) and (15), we can transform fuzzy quantities for objective subcriteria into fuzzy ratings:
(14)R~ijn={R~ijnomaxi{bijno}}×5,
where maxi{bijno}>0 and R~ijn denotes the transformed fuzzy rating of objective fuzzy benefit R~ijno. R~ijn becomes larger when objective fuzzy benefit R~ijno is larger:
(15)R~ijn={mini{cijno}R~ijno}×5,
where mini{cijno}>0 and R~ijn denotes the transformed fuzzy rating of objective fuzzy cost R~ijno. R~ijn becomes smaller when objective fuzzy cost R~ijno is larger.Step 9.
Construct a fuzzy rating matrix based on fuzzy ratings. The fuzzy rating matrixMcan be concisely expressed in matrix format:
(16)M=[R~111R~121⋯R~IJ1R~112R~122⋯R~IJ2::⋯:R~11NR~12N⋯R~IJN].Step 10.
Obtain the total fuzzy rating (R~) based on the fuzzy rating matrix (M) and weight vector (W):
(17)R~=[R~111R~121⋯R~IJ1R~112R~122⋯R~IJ2⋮⋮⋯⋮R~11NR~12N⋯R~IJN]⊗[W111W112⋮WIIJ]=[R~1R~2⋮R~N].Step 11.
Defuzzify the total fuzzy rating by (6) and then rank alternatives according to their total crisp ratings. Finally, we can select easily the best alternative with the maximum total crisp ratings:
(18)P(R~n)=16(cn+4an+bn).


## 4. A Case Study

In this section, the proposed hybrid fuzzy AHP approach is applied to the location choice of international distribution centers in the global logistics of the multinational corporation. A Taiwanese multinational corporation plans to select an appropriate location of international distribution center at the international transshipment port. After initial screening, three alternative port locations including the portA1, the port A2, and the port A3 are selected for further evaluation. The procedures for evaluation are shown as follows.Step 1.
Constructing a hierarchical analysis structure is shown in Table3. There are 6 criteria and 30 subcriteria in the hierarchical analysis structure summarized by Chou [19]. These criteria in the hierarchical analysis structure can be divided into two categories: objective and subjective criteria. The objective criteria include proximity of the feeder port (C12), frequency of ship calls (C14), port charge (C25), volumes of import containers (C31), volumes of export containers (C32), volumes of transshipment containers (C33), port facilities (C61), loading and unloading facilities (C62), and cargo handling efficiency (C64). The others are subjective criteria. The objective criteria are defined in quantitative terms (e.g., nautical mile, $US, TEU). The subjective criteria are defined in linguistic terms represented by fuzzy numbers.Step 2.
Present the linguistic variables and fuzzy numbers for comparative importance weights of criteria.Step 3.
Present the linguistic variables and fuzzy numbers for ratings of alternatives.Step 4.
Determine the fuzzy comparative importance weights of criteria and subcriteria by the decision maker in Tables4 and 5, respectively.Table 4
Survey data for fuzzy comparative weights for each criterion.
Criteria
(9,9,9)
(7,8,9)
(6,7,8)
(5,6,7)
(4,5,6)
(3,4,5)
(2,3,4)
(1,2,3)
(1,1,1)
(1/3,1/2,1)
(1/4,1/3,1/2)
(1/5,1/4,1/3)
(1/61/,5,1/4)
(1/7,1/6,1/5)
(1/8,1/7,1/6)
(1/9,1/8,1/7)
(1/9,1/9,1/9)
Criteria
AI
IV
DI
IV
SI
IV
WI
IV
EI
IV
WUI
IV
SUI
IV
DUI
IV
AUI
C1: geographical
X
C2: cost
C1: geographical
X
C3: economy
C1: geographical
X
C4: government
C1: geographical
X
C5: investment
C1: geographical
X
C6: infrastructure and efficiency
C2: cost
X
C3: economy
C2: cost
X
C4: government
C2: cost
X
C5: investment
C2: cost
X
C6: infrastructure and efficiency
C3: economy
X
C4: government
C3: economy
X
C5: investment
C3: economy
X
C6: infrastructure and efficiency
C4: government
X
C5: investment
C4: government
X
C6: infrastructure and efficiency
C5: investment
X
C6: infrastructure and efficiencyTable 5
Survey data for fuzzy comparative weights for each subcriterion.
Subcriteria
(9,9,9)
(7,8,9)
(6,7,8)
(5,6,7)
(4,5,6)
(3,4,5)
(2,3,4)
(1,2,3)
(1,1,1)
(1/3,1/2,1)
(1/4,1/3,1/2)
(1/5,1/4,1/3)
(1/61/,5,1/4)
(1/7,1/6,1/5)
(1/8,1/7,1/6)
(1/9,1/8,1/7)
(1/9,1/9,1/9)
Subcriteria
AI
IV
DI
IV
SI
IV
WI
IV
EI
IV
WUI
IV
SUI
IV
DUI
IV
AUI
C11: imp./exp. area
X
C12: feeder port
C11: imp./exp. area
X
C13: main route
C11: imp./exp. area
X
C14: frequency
C11: imp./exp. area
X
C15: delivery
C12: feeder port
X
C13: main route
C12: feeder port
X
C14: frequency
C12: feeder port
X
C15: delivery
C13: main route
X
C14: frequency
C13: main route
X
C15: delivery
C14: frequency
X
C15: delivery
C21: transport cost
X
C22: operation cost
C21: transport cost
X
C23: land cost
C21: transport cost
X
C24: labor cost
C21: transport cost
X
C25: port charge
C22: operation cost
X
C23: land cost
C22: operation cost
X
C24: labor cost
C22: operation cost
X
C25: port charge
C23: land cost
X
C24: labor cost
C23: land cost
X
C25: port charge
C24: labor cost
X
C25: port charge
C31: vol. of import
X
C32: vol. of export
C31: vol. of import
X
C33: vol. of tranship
C31: vol. of import
X
C34: economy
C31: vol. of import
X
C35: trade variable
C32: vol. of export
X
C33: vol. of tranship
C32: vol. of export
X
C34: economy
C32: vol. of export
X
C35: trade variable
C33: vol. of tranship
X
C34: economy
C33: vol. of tranship
X
C35: trade variable
C34: economy
X
C35: trade variable
C41: private
X
C42: customs
C41: private
X
C43: government
C41: private
X
C44: cooperation
C41: private
X
C45: political
C42: customs
X
C43: government
C42: customs
X
C44: cooperation
C42: customs
X
C45: political
C43: government
X
C44: cooperation
C43: government
X
C45: political
C44: cooperation
X
C45: political
C51: tax break
X
C52: law
C51: tax break
X
C53: social stability
C51: tax break
X
C54: expansion
C51: tax break
X
C55: labor quality
C52: law
X
C53: social stability
C52: law
X
C54: expansion
C52: law
X
C55: labor quality
C53: social stability
X
C54: expansion
C53: social stability
X
C55: labor quality
C54: expansion
X
C55: labor quality
C61: port facilities
X
C62: load/discharge
C61: port facilities
X
C63: modal link
C61: port facilities
X
C64: cargo handle
C61: port facilities
X
C65: information
C62: load/discharge
X
C63: modal link
C62: load/discharge
X
C64: cargo handle
C62: load/discharge
X
C65: information
C63: modal link
X
C64: cargo handle
C63: modal link
X
C65: information
C64: cargo handle
X
C65: informationStep 5.
Defuzzify the fuzzy weights of criteria and subcriteria. Then calculate the normalized weights in Table3.Step 6.
Calculate the maximum eigenvalue (λmax), the consistency index (C.I), and the consistency rate (C.R) for AHP model to test the consistency of the intuitive judgment:(19)Aw=[114332124145441314121131315121111214111132143131][0.15270.43940.08240.07570.07850.1714]=[0.97932.75740.53010.50550.47991.1449],λmax=16(0.97930.1527+2.75740.4394+0.53010.0824+0.50550.0757+0.47990.0785+1.14490.1714)=6.4331,C.I=λmax-nn-1=6.4331-66-1=0.08,C.R=C.IR.I=0.081.24=0.07.Step 7.
The decision maker assesses alternatives under subjective criteria. For example, the fuzzy ratings given by the decision maker to the portA1 under subjective subcriteria are shown in Table 3.Step 8.
Assess alternatives under objective criteria. The fuzzy quantities given to alternative under objective subcriteria are listed in Table6. The objective fuzzy quantities are determined in various units (e.g., nautical mile, $US, TEU) and must be transformed into dimensionless indices (or ratings) to ensure compatibility with the linguistic ratings of subjective criteria. By (14) and (15), we can transform fuzzy quantities for objective subcriteria into fuzzy ratings in Table 6.Table 6
Fuzzy ratings for alternatives under objective criteria.
Objective criteria
Fuzzy quantity
Fuzzy rating
C12
Proximity of the feeder port (nautical mile)
A1
(
6356,6356,6356
)
(
4.2,4.2,4.2
)
A2
(
5401,5401,5401
)
(
5.0,5.0,5.0
)
A3
(
7288,7288,7288
)
(
3.7,3.7,3.7
)
C14
Frequency of ship calls (vessel/week)
A1
(
415,435,455
)
(
4.6,4.8,5.0
)
A2
(
167,177,187
)
(
1.8,1.9,2.1
)
A3
(
186,206,226
)
(
2.1,2.3,2.5
)
C25
Port charge ($US/TEU)
A1
(
328,338,348
)
(
1.0,1.0,1.0
)
A2
(
100,110,120
)
(
2.7,3.0,3.3
)
A3
(
65,70,75
)
(
4.3,4.6,5.0
)
C31
Volumes of import containers (TEU)
A1
(
11599000,12041000,12064000
)
(
4.4,4.6,4.6
)
A2
(
1887381,1906255,1925318
)
(
0.7,0.7,0.7
)
A3
(
9042000,10860000,13075000
)
(
3.5,4.2,5.0
)
C32
Volumes of export containers (TEU)
A1
(
11002000,11475000,11958000
)
(
4.2,4.4,4.6
)
A2
(
2116017,2158338,2201505
)
(
0.8,0.8,0.8
)
A3
(
9042000,10860000,13075000
)
(
3.5,4.2,5.0
)
C33
Volume of transshipment containers (TEU)
A1
(
10000000,10150000,10300000
)
(
4.9,4.9,5.0
)
A2
(
5406461,5647846,5710712
)
(
2.6,2.7,2.8
)
A3
(
300000,400000,5000000
)
(
0.1,0.2,0.2
)
C61
Port facilities (length of wharf, m)
A1
(
8530,8530,8530
)
(
5.0,5.0,5.0
)
A2
(
7453,7453,7453
)
(
4.4,4.4,4.4
)
A3
(
8387,8387,8387
)
(
4.9,4.9,4.9
)
C62
Loading and discharging facilities (crane)
A1
(
83,84,85
)
(
4.9,4.9,5.0
)
A2
(
66,67,68
)
(
3.9,3.9,4.0
)
A3
(
81,82,83
)
(
4.8,4.8,4.9
)
C64
Cargo handling efficiency (move/h)
A1
(
33,34,35
)
(
4.7,4.9,5.0
)
A2
(
31,32,33
)
(
4.4,4.6,4.7
)
A3
(
29,30,31
)
(
4.1,4.3,4.4
)
Source: Hong Kong Maritime Industry Council website,http://www.mic.gov.hk/.
Port of Kaohsiung website,http://www.khb.gov.tw/.
Port of Shanghai website,http://www.portshanghai.com.cn/.
Containerization International Yearbook.
The Institute of Transportation, Ministry of Transportation and Communications, Taiwan.Step 9.
Construct a fuzzy rating matrix based on fuzzy ratings.Step 10.
Obtain the total fuzzy ratings based on the fuzzy rating matrix and the weight vector.Step 11.
By (6), we can defuzzify the total fuzzy ratings and the total crisp ratings for the port A1, the port A2, and the port A3 are 3.72, 3.88, and 4.41, respectively. Finally, the decision maker of Taiwanese multinational corporation selects easily the port A3 with the maximum total crisp ratings as the best location for international distribution center in the global logistics system.


## 5. Conclusions

The paper proposes a new hybrid fuzzy AHP model for dealing with both objective and subjective criteria in process of decision making simultaneously. The proposed hybrid fuzzy AHP model is applied to solve the location choice problem of international distribution center in the global logistics of multinational corporation. The results show that the proposed new hybrid fuzzy AHP model is an appropriate and more efficient approach to deal with both objective and subjective criteria in process of decision making simultaneously. The hybrid fuzzy AHP model in this paper overcomes the disadvantages of quantitative or qualitative approaches in the previous literature. The proposed hybrid fuzzy AHP approach can not only solve the problems of location choice, but also many other decision-making problems.



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