Fuzzy Rasch model in TOPSIS: A new approach for generating fuzzy numbers to assess the competitiveness of the tourism industries in Asian countries

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Fuzzy Rasch model in TOPSIS: A new approach for generating fuzzy numbers

to assess the competitiveness of the tourism industries in Asian countries

Jen-Hung Huang

a

, Kua-Hsin Peng

b,*

a_{Department of Management Science, National Chiao Tung University, 1001 University Road, Hsinchu 300, Taiwan} b_{Graduate Institute of Management Science, National Chiao Tung University, 1001 University Road, Hsinchu 300, Taiwan}

a r t i c l e i n f o

Article history:

Received 21 November 2010 Accepted 22 May 2011 Keywords:

Item response theory (IRT) Rasch model

Fuzzy theory

Triangular fuzzy number (TFN) TOPSIS

Tourism destination competitiveness (TDC) Asia

a b s t r a c t

This study proposes a novel approach, the Fuzzy Rasch model, which combines Item Response Theory (IRT) and fuzzy set theory. This paper applies the Fuzzy Rasch model in Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to analyse the Tourism Destination Competitiveness (TDC) of nine Asian countries: China, Hong Kong, Japan, Korea, Malaysia, Singapore, Taiwan, Thailand and the Philippines. The study was conducted in 2009 using 6 criteria and 15 indices. The results demonstrate the feasibility of applying the Fuzzy Rasch model in TOPSIS to analyse TDC in Asian countries. In addition, the proposed model also provides an effective means of applying the MCDM method to study TDC. Furthermore, in 2009, the Asian countries were ranked from most to least competitive as follows: China, Japan, Hong Kong, Malaysia, Thailand, Singapore, Taiwan, Korea and the Philippines.

1. Introduction

Multiple Criteria Decision Making (MCDM) is an analytical method used to evaluate a set of alternatives based on multiple criteria (Tsaur, Tzeng, & Wang., 1997; Wang & Lee, 2009). The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is a conventional means of solving MCDM problems (Tsou, 2008; Mahdavi, Mahdavi-Amiri, Heidarzade, & Nourifar, 2008; Wang & Lee, 2009; Singh & Benyoucef, 2011). In MCDM, the weights of the criteria are crucial for measuring the importance of the criteria (Zhang, Gu, Gu, & Zhang, 2011). The methods used to determine the weights of the criteria of the TOPSIS include the entropy method (Tsaur, Tzeng, & Wang., 1997; Singh & Benyoucef, 2011), Information Entropy Weight (IEW) (Zhang et al., 2011), Analytic Hierarchy Process (AHP) (Tsaur, Tzeng, & Wang, 1997; Dagdeviren, Yavuz, & Kılınç, 2009; Yu, Guo, Guo, & Huang, 2011), Fuzzy AHP (Wang, Cheng, & Huang, 2009; Gumus, 2009; Sun, 2010) and Rough AHP (Aydogan, 2011). Additionally, Liang and Ding (2003)depended on expert knowledge and experience to deter-mine the weights of the criteria on a Likert rating scale. However, the inherent uncertainty and subjectivity of this method can result

in weighting errors and dif_{ﬁculties in the criteria weight selection} process. As a result, the subjectivity (i.e., the fuzzy numbers) of the criteria weight selection process varies among the experts.

Mahdavi et al. (2008)andHsieh, Lu, and Tzeng (2004)have used linguistic variables proposed byBuckley (1985)ranging from“very unimportant” to “very important” to express the fuzzy numbers.

Kaufmann and Gupta (1991)andMon, Cheng, and Lin (1994)have also used linguistic variables to express the fuzzy numbers assigned by the experts. However, using the linguistic variables deﬁned by

Buckley (1985), Kaufmann and Gupta (1991)andMon et al. (1994)

assume that different experts have the same fuzzy numbers, whereas in reality, the inherent uncertainty of the criteria weight selection process varies among the experts. Therefore, determining how to obtain accurate fuzzy numbers for the criteria weights and their subsequent evaluations of those criteria weights remain a high priority for researchers.

Item Response Theory (IRT) is a general statistical theory that analyses item (question) and scale (questionnaire) performances and the relationships between these performances and the factors measured by the items in the scale (Meads & Bentall, 2008). The simplest logistic latent trait IRT model is the Rasch One-Parameter Logistic Model (1PL) (Rasch, 1960). Georg Rasch originally devel-oped the Rasch model to assess reading ability in 1952, and since then, this model has been widely applied for a variety of purposes. For example,Yu and Wu (2009)used the Rasch model to determine the fuzzy numbers for psychological measurements. Furthermore, * Corresponding author. Tel.: þ886 920022514.

E-mail addresses:jhh509@hotmail.com(J.-H. Huang),jenny19830514@yahoo. com.tw(K.-H. Peng).

Contents lists available atScienceDirect

Tourism Management

j o u r n a l h o me p a g e : w w w . e l s e v i e r . c o m/ l o ca t e / t o u r m a n

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Huang and Peng (2010)used the Rasch model to generate grey weight and then adopted Grey Relation Analysis (GRA) to analyse the performances of seven Taiwanese districts containing 56 international tourist hotels in 2009. Additionally,Yu and Wu (2009)

utilised IRT to generate accurate fuzzy numbers. Based on the above, this study believes that using the Rasch model to generate fuzzy numbers is not only feasible but can also rectify the inac-curacy of the fuzzy numbers assigned by the individual experts for the speciﬁc criteria. Therefore, this study proposes a new approach, the Fuzzy Rasch model, which combines the Rasch model with fuzzy theory. This new approach follows a three-step procedure. First, the Rasch model is used to generate fuzzy weights for each expert. Second, an arithmetic average is used to integrate the fuzzy weight of each expert. Third, the defuzzi_{ﬁcation weights are} applied in TOPSIS.

Tourism has become the leading leisure activity of the 21st century (Claver-Cortes, Molina-Azorin, & Pereira-Moliner, 2007). In 2010, UNWTO reported that international tourist arrivals in Asia

grew from 55.8 million to 181.2 million from 1990e2009 and

generated US$ 204 billion in international tourism receipts in 2009 (UNWTO, 2010). Thus, the tourism industry is rapidly growing in Asia. However, the tourism industry is becoming increasingly competitive as well (Middleton & Hawkins, 1998), and consumers have developed considerable_{ﬂexibility in their choice of} destina-tions, which places the different tourism markets of Asia in competition with one another.Hovinen (2002)also indicated that the success of tourism destinations depends on their regional competitiveness. Following these developments, the recent litera-ture has increasingly focused on measuring Tourism Destination Competitiveness (TDC) (Zhang et al., 2011; Hall, 2007; Pearce, 1997; Ruhanen, 2007; Enright & Newton, 2004; Kozak & Rimmington, 1999; Cracolici & Nijkamp, 2009). However, the role of TDC in Asian countries has received little attention to date. Additionally, according toPalmer, Sese, and Montano (2005), more advanced statistical techniques need to be used in tourism studies. Therefore, based on the Fuzzy Rasch model in TOPSIS, this study analyses the TDC of nine Asian countries (i.e., China, Hong Kong, Japan, Korea, Malaysia, Singapore, Taiwan, Thailand and the Philippines) in 2009 using 6 criteria and 15 indexes.

2. Evaluation criteria of TDC

Following a review of the pertinent literature, this study estab-lishes the evaluation criteria and the indices for TDC. The evaluation criteria for TDC comprise the availability of attractions (Barros et al., 2011; Weaver & Oppermann, 2000), the availability of service (Weaver & Oppermann, 2000), affordability (Go & Govers, 1999; Weaver & Oppermann, 2000), positive market image (Weaver & Oppermann, 2000), peace and stability (Crouch & Ritchie, 1999; Ritchie & Crouch, 1993; Cracolici & Nijkamp, 2009; Weaver & Oppermann, 2000) and cultural links (Weaver & Oppermann, 2000).

2.1. Availability of attractions

The attractions of a destination are the main component of TDC (Crouch & Ritchie, 1999; Ritchie & Crouch, 1993). Attractions incorporate specific features (e.g., theme parks and battlefields) and generic or non-specific features (e.g., scenery and climate) (Weaver & Oppermann, 2000). Consequently, the criteria deter-mining the availability of attractions are composed of two indices: “International tourism arrivals” (Barros et al., 2011; UNWTO, 2010) and“International tourism receipts” (Zhang et al., 2011; UNWTO, 2010).

2.2. Availability of service

Tourists avoid attractions if the services or facilities afﬁliated with the attraction are unavailable or poor in quality. Adequate tourism-related facilities include transportation, accommodation, restrooms, dining facilities and visitor bureaus. Consequently, the criteria of availability of service comprise three indices:“Number of hotel rooms” (Barros et al., 2011; Cracolici & Nijkamp, 2009; Zhang et al., 2011; WEF, 2009), “Number of operating airlines” (WEF, 2009) and“Air transportation (Number of passengers carried by main air companies)” (IMD, 2010).

2.3. Affordability

Crouch (1992) noted that travellers are sensitive to price. As a result, regions with low transportation and living costs experience increased arrivals of tourists (Weaver & Oppermann, 2000). For example, numerous tourists prefer to travel to less developed countries, such as Indonesia or Costa Rica, because of the relatively low prices of goods and services available in these countries (Weaver & Oppermann, 2000). Therefore, the criteria of afford-ability comprise three indices: “Transportation costs” (Dwyer & Kim, 2003; Weaver & Oppermann, 2000; WEF, 2009),“Hotel price index_{” (}Cracolici & Nijkamp, 2009; Ritchie & Crouch, 1993) and “Cost-of-living index” (Weaver & Oppermann, 2000; Cracolici & Nijkamp, 2009; Ritchie & Crouch, 1993).

2.4. Positive market image

Image consists of a person or group’s beliefs, attitudes and impressions of a phenomenon (Weaver & Oppermann, 2000). Destination image is deﬁned as an individual’s overall impression of a tourist site (Fakeye & Crompton, 1991) and as the person’s mental portrayal of the destination (Alhemoud & Armstrong, 1996; Seaton & Bennett, 1996).Woodside and Lysonski (1989)suggested that tourist sites with strong, positive images are more likely to be considered and selected as travel destinations. Thus, the criteria of positive market image comprise two indices:“Quality of life” (IMD, 2010) and “Quality of the natural environment” (Cracolici & Nijkamp, 2009; Inskeep, 1991; Kozak & Rimmington, 1999andWEF, 2009). 2.5. Peace and stability

Weaver and Oppermann (2000)asserted that the tourist market is sensitive to social or political instability. Additionally, a lack of safety makes it impossible for a tourist site to successfully compete in the market because potential tourists do not want to visit a place that they perceive as unsafe (Cavlek, 2002). For example, Middle Eastern countries and the United States experienced signiﬁcant reductions in total visitor numbers in the aftermath of the September 11 terrorist attacks (Dwyer & Kim, 2003). Generally, tourists prefer to travel to peaceful and stable destinations. Consequently, the criteria of peace and stability comprise three indices:“Safety and security” (Cracolici & Nijkamp, 2009; Crouch & Ritchie,1999; Ritchie & Crouch,1993; Kozak & Rimmington,1999; Dwyer & Kim, 2003; IMD, 2010),“Business costs of crime and violence” (WEF, 2009) and“Business costs of terrorism” (WEF, 2009).

2.6. Cultural links

Travel is more prevalent between countries that share similar cultural elements, such as language and religion (Burton, 1995).

Weaver and Oppermann (2000)noted that numerous tourists feel insecure or inconvenienced by having to cope with unfamiliar languages and social norms and, thus, prefer destinations that are

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similar to their home countries. Consequently, the criteria of cultural links comprise two indices:“Nation culture” (IMD, 2010) and“Discrimination (e.g., race and gender)” (IMD, 2010).

3. Method

To evaluate the role of TDC in Asian countries, this study uses the Rasch model to generate the fuzzy weights of the criteria. The defuzziﬁcation weights are then applied in TOPSIS. Finally, TOPSIS is used to rank the TDC of the Asian countries. Hence, this section is divided into four parts: (1) the Rasch model, (2) Triangular Fuzzy Numbers (TFN) and linguistic variables, (3) TOPSIS and (4) the Fuzzy Rasch model in TOPSIS.

3.1. Rasch model

The Rating Scale Model (RSM) devised byAndrich (1978)applies Rasch’s model to polytomous rating scale instruments, which include thefive-point Likert scale. The Rasch model is based on the concept that the probability of correctly obtaining an item is a function of a latent trait or ability (Kastrin & Peterlin, 2010). Notably, the Rasch model is also known as the One-Parameter Logistic Model (1PL). The Rasch model converts raw data from a rating scale to“an equal interval scale” measured in logits (log odd units) (Belvedere & de Morton, 2010), which reflect both the diffi-culty of the item and individual ability (Bond & Fox, 2007).

SinceAndrich (1978)developed RSM, it has been extensively adopted by scholars to assess the values of item and person parameters, as shown in Eq.(1).

log Pnij Pniðj1Þ ! ¼

q

n

d

iþ

s

j (1)

In Eq.(1), Pnijand Pniðj1Þrepresent the probability that the item

n obtains j and j-1 scores from the expert i.

q

n represents the

measure score (i.e., item difﬁculty) of the item n,

d

irepresents the

measure score (i.e., individual ability) of expert i,

s

jand represents

the step difficulty (i.e., threshold difficulty) of category j. The step difficulty of RSM is identical for all items (Wright & Masters, 1982). Thus, the RSM is useful if the psychological distances between categories are identical for all items (Kim & Hong, 2004), as is the case for the Likert scales.

d

ij ¼

d

iþ

s

j (2)

In Eq.(2), i _{¼ 1; .; Eand E represents the number of experts.} j ¼ 1; .; m and m represents the number of linguistic scales, which range from“very unimportant” to “very important”. 3.2. TFN and linguistic variables

Fuzzy numbers are a convex fuzzy set that are characterised by a given interval of real numbers. The most common types of fuzzy numbers are triangular and trapezoidal fuzzy numbers. TFN can be denoted as ~W ¼ ð

d

L;

d

M;

d

UÞ, where L and U represent the lower and upper bounds of the fuzzy number, respectively, and M represents the modal value. According to Zadeh (1975), it is extremely difficult to use conventional quantification to reasonably define situations that are overly complex or hard to define, and as a result, the linguistic variables are necessary to define such situ-ations. A linguistic variable is a variable whose value consists of words or sentences in a natural or artificial language.

3.3. TOPSIS

Developed byYoon, 1980, TOPSIS is based on the concept that the option selected by the user should be closest to the Positive Ideal Solution (PIS) and furthest from the Negative Ideal Solution (NIS) to resolve MCDM problems. The PIS maximises the beneﬁts and minimises the costs, whereas the NIS maximises the costs and minimises the beneﬁts.

3.4. Fuzzy Rasch model in TOPSIS

Fisher (1995)noted that the Rasch model endows the Fuzzy Logic Model of Perception (FLMP) with precise properties that are far superior to those of fuzzy logic. Numerous studies have used the Rasch model to generate fuzzy numbers; for example,Yu and Wu (2009) utilised the IRT to generate fuzzy numbers, and Huang and Peng (2010)used the Rasch model to generate grey weight. This study proposes a novel approach, the Fuzzy Rasch model in TOPSIS, which combines the Rasch model, fuzzy theory and TOPSIS. This novel method involves the following steps.

Step 1. Determine the evaluation criteria/indices and the alternatives.

Step 2. Determine the weights of the evaluation criteria/indices with the Fuzzy Rasch model.

Step 2e1. Determine the degree of importance for each crite-rion/index.

Step 2e2. Calculate the step parameters (

d

ij) to generate the

fuzzy weight.

This study adopts the Rasch model to calculate the fuzzy weight of each expert. ~ Wij ¼

d

Lij;

d

Mij;

d

Uij (3)

In Eq.(3), represents the fuzzy weight of expert i for step. j.

Fig. 1illustrates the step parameters (

d

ij) estimated by the Rasch

model. This study selects a value of _{“very unimportant” for step}

Fig. 1. Calculate“step parameters” (dij) via the Rasch model to generate the triangular

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parameter 1 (

d

ij). The triangular fuzzy weights of the Likert scale

values are as follows:“very unimportant” is ~Wi1 ¼ ð

d

Li1;

d

M i1;

d

U i1Þ; “unimportant” is W~i2 ¼ ð

d

Li2;

d

M i2;

d

U i2Þ; “general” is W~i3 ¼ ð

d

Li3;

d

M i3;

d

U i3Þ; “important” is ~Wi4 ¼ ð

d

Li4;

d

M i4;

d

U

i4Þ; and “very

impor-tant” is ~Wi5 ¼ ð

d

Li5;

d

M i5;

d

U i5Þ.

Step 2e3. Use the triangular fuzzy weight as a substitute for the values of importance calculated in Step 2e1.

~ Wijc ¼

d

Lijc;

d

M ijc;

d

U ijc (4)

In Eq.(4), ~Wijcrepresents the fuzzy weight of expert i to criteria/

indices c for step j. This fuzzy weight is substituted for the original degree of importance calculated in Step 2e1.

Step 2e4. Use an arithmetic average to integrate the fuzzy weight of each expert.

This study uses an arithmetic average to integrate the fuzzy weight of each expert.

~ Wc ¼ 1_E 2 4XE j¼ 1 ~ Wijc 3 5 ¼

d

Lc;

d

Mc ;

d

Uc (5)

Step 2e5. Defuzziﬁcation.

The defuzziﬁcation values transform the fuzzy weights into crisp weights. Eq.(6)calculates the Best Non-fuzzy Performance Value (BNP). BNPc ¼ h

d

Uc

d

Lc þ

d

Mc

d

Lc i. 13þ

d

Lc (6)

Finally, this study uses the defuzziﬁcation weights as the criteria weights in TOPSIS. However, the criteria weights in TOPSIS are normalised to a sum to 1(Hwang & Yoon, 1981; Torlak, Sevkli, Sanal, & Zaim, 2011; Zhang et al., 2011; Yu et al., 2011; Tan, 2011). Based on the above, this study uses Eq.(7)to standardise the weight of the criterion. Wc ¼ SBNPc ¼ BNPc= XC c¼ 1 BNPc; XC c¼ 1 SBNPc ¼ 1 (7)

In Eq.(7), c represents the number of criteria/indices, as shown by the following equation:

0 Wc 1; cc Step 3. Data collection.

Each MCDM problem involves alternatives n and c evaluation criteria/indices. In turn, each alternative is evaluated with respect to the c criteria/indices. All of the values/ratings assigned to the alternative with respect to each criterion form a decision matrix denoted as D:

D ¼ jXijjnc (8)

Step 4. Normalise the evaluation matrix D.

This process transforms the different scales and units among the various indexes into common measurable units that permit comparisons among the different criteria. This study normalises the decision matrix D by calculating gjðAiÞ, which represents the

nor-malised value of the criteria.

gjðAiÞ ¼ Xij P i Xij; ci; j (9)

In addition, this study uses G to represent the normalised decision matrix, as shown in Eq.(10).

G ¼ jg_jðA_iÞj_nc (10)

Step 5. Calculate the weighted normalised decision matrix V.

V ¼ jV_ijj_nc (11)

where Vij ¼ WcgjðAiÞ; ci; j.

Step 6. Determine the positive (A*) and negative (A) ideal solutions. A* ¼ max i Vijjj˛Kb ; min i Vijjj˛Kd ¼ nV_j*jj ¼ 1; 2; .co (12) A ¼ min i Vijjj˛Kb ; max i Vijjj˛Kd ¼ nV_jjj ¼ 1; 2; .; co (13) where Kb ¼ fKjjj ¼ 1; 2; .; c1gand Kd ¼ fKjjj ¼ 1; 2; .; c2g

Step 7. Calculate the separation measures S*_iand S_i.

This process calculates the Euclidean distances of each alter-native based on the positive and negative ideal solutions:

S*_i ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xm j¼ 1 vij v*j 2 v u u t _{; ci} (14) S_i ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xm j¼ 1 vij vj 2 v u u t _{; ci} (15)

Step 8. Calculate the relative closeness (RC_i*) to the ideal solution. RC*_i ¼ Si S*_i þ S i ; ci (16) where 0 RC* i 1; ci:

Step 9. Rank the alternatives based on their relative closeness to the ideal solution.

A larger RC_i*implies a better alternative Ai. The best alternative

is the one that is closest to the ideal solution.

Ai_Ai0 iff RC_i* RC_i*0; ci; i0; isi0 (17)

4. Results

This study applies the Fuzzy Rasch model in TOPSIS to analyse the TDC of nine Asian countries. The evaluation procedure used in this study comprises several steps. First, the evaluation criteria for the TDC were determined. Second, the Asian countries were

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selected as alternatives. Third, the data were collected, and the fuzzy weights of the criteria/indices were generated with the Rasch model. Finally, the TOPSIS method was used to rank the TDC of the Asian countries. These steps are detailed below:

Step 1. Determine the evaluation criteria/indices and the alternatives.

A multiple indexes evaluation method normally focuses on a set of feasible alternatives and the method used to prioritise the multiple indices. This study also examines a set of feasible alter-natives composed of nine Asian countries: China, Hong Kong, Japan, Korea, Malaysia, Singapore, the Philippines, Taiwan and Thailand.

Following a literature review, this study analyses the competi-tiveness of the tourist industry for each of the selected Asian countries during the year 2009 using six criteria and 15 indices.

Table 1lists the six assessment criteria and 15 indices.

Step 2. Determine the weights of the evaluation criteria/indices with the Fuzzy Rasch model.

Step 2e1. Determine the degree of importance for each crite-rion/index.

This study assesses TDC by instructing six experts to indicate the degree of importance of six criteria and 15 indices on a Likert rating scale ranging from 1e5 (from “very unimportant” to “very impor-tant_{”), as listed in}Table 2.

Step 2e2. Calculate the step parameters (

d

ij) to generate the

fuzzy weight.

The step parameters of expert 1 are deﬁned with the RSM, as shown in Fig. 2. Based on Fig. 3, this study ﬁnds that ~Wij ¼

ð

d

Lij;

d

Mij;

d

UijÞ. This study then uses this information to deﬁne the TFN

of the linguistic variables. Table 1

Lists the six assessment criteria and 15 indices.

Criteria/Indices Description

Availability of attractions International tourism arrivals International tourism arrivals (per thousand people) International tourism receipts International tourism receipts (US$ million)

Availability of service Hotel room Number of hotel rooms (per 100 population)

Number of operating airlines Number of airlines with scheduledﬂights originating in the country Air transportation Number of passengers carried by the main companies

(per thousand people)

Affordability Transportation costs Relative costs of access (ticket taxes and airport charges) to

international air transport services (0¼ highest cost, 100 ¼ lowest cost)

Hotel price Average room price (US$)

Cost-of-living index Cost index of goods in major cities

Positive market image Quality of life Quality of life in the country

Quality of the natural environment Quality of the natural environment in the country (1¼ most polluted, 7¼ least polluted)

Peace and stability Safety and security The degree to which personal security and private property are adequately protected

Business costs of crime and violence The incidence of common crime and violence in the country (1¼ imposes signiﬁcant costs on businesses, 7 ¼ does not impose signiﬁcant costs on businesses)

Business costs of terrorism The threat of terrorism in the country (1¼ imposes signiﬁcant costs on businesses, 7¼ does not impose signiﬁcant costs on businesses)

Cultural links Nation culture The degree to which the national culture is open to foreign ideas

Discrimination (race, gender) The degree of equal treatment for tourists (1¼ lowest degree of equal treatment for tourists, 10¼ highest degree of equal treatment for tourists)

Table 2

Degree of importance of 15 indices (scale ranges from 1 to 5).

Criteria/Indexes Expert 1 Expert 2 Expert 3 Expert 4 Expert 5 Expert 6

Availability of attractions International tourism arrivals C1 4 4 5 4 5 5

International tourism receiptsC2 4 5 5 5 5 4

Availability of service Hotel roomC3 4 4 3 4 4 3

Number of operating airlinesC4 5 4 3 4 3 4

Air transportationC5 3 2 4 5 4 4

Affordability Transportation costsC6 3 2 3 2 3 3

Hotel price C7 2 3 2 1 3 2

Cost-of-living indexC8 4 5 4 5 5 4

Positive market image Quality of lifeC9 2 4 3 2 4 3

Quality of the natural environmentC10 4 2 3 4 3 3

Peace and stability Safety and securityC11 4 5 4 4 4 5

Business costs of crime and violenceC12 2 1 3 2 3 2

Business costs of terrorismC13 2 3 2 1 3 2

Cultural links Nation cultureC14 4 3 3 3 3 3

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~ W12 ¼

d

L12;

d

M12;

d

U12 ¼ ð7; 7; 1Þ ~ W13 ¼

d

L13;

d

M13;

d

U13 ¼ ð7; 1:8; 3Þ ~ W14 ¼

d

L14;

d

M14;

d

U14 ¼ ð4:2; 0:7; 9Þ ~ W15 ¼

d

L15;

d

M15;

d

U15 ¼ ð1:5; 9; 9Þ

Table 3lists the linguistic variables from“very unimportant” to “very important” for the TFN obtained from the six experts.

Step 2e3. Use the triangular fuzzy weight as a substitute for the values of importance calculated in Step 2e1.

Based on the above deﬁnition of the fuzzy number, this study uses ~Wijc ¼ ð

d

Lijc;

d

Mijc;

d

UijcÞ to represent the fuzzy weights assigned

by experts i to criteria/indices c. These fuzzy weights are then substituted for the original degree of importance calculated in Step 2e1, as listed inTable 4.

Step 2e4. Use an arithmetic average to integrate the fuzzy weight of each expert.

This study uses an arithmetic average (Eq.(5)) to integrate the fuzzy weight of each expert, as listed inTable 5.

Step 2e5. Defuzziﬁcation.

This study uses Eq.(6)to transform the fuzzy numbers into crisp numbers. In addition, this study uses Eq. (7)to standardise the weights of criteria/indices c, as shown inTable 5.

Step 3. Data collection.

The data mainly comes from statistical data, such asThe World Competitiveness Yearbook 2010, The Travel & Tourism Competitiveness Report 2009, and the ofﬁcial online websites of the following government departments: Hong Kong Tourism Board, Japan National Tourism Organization, Singapore Govern-ment, National Tourism Administration of the People’s Republic of China, Department of Tourism Republic of the Philippines, Tourism Malaysia, Korea Tourism Organization and Taiwan Tourism Bureau.Table 6lists the raw data of the TDC index for nine Asian countries in the year 2009.

Step 4. Normalise the evaluation matrix D.

This study normalises the decision matrix by calculating gjðAiÞ,

as shown in Eq.(9). gjðAiÞ represents the normalised value of the

criteria. X9 i¼ 1 Xij ¼ 6789658 þ 132924650 þ / þ 9682700 þ 3017099 þ 14150000 ¼ 231967298 g1ðA1Þ ¼ 6789658=231967298 ¼ 0:0293 « g1ðA9Þ ¼ 14150000=231967298 ¼ 0:0610 Fig. 2. Step parameters of expert 1.

Fig. 3. The group of TDC in Asia.

Table 3

The linguistic variables of the TFN ~Wij ¼ ðdLij;dMij;dUijÞ.

Fuzzy number Very unimportance Unimportance General Importance Very importance

Expert 1 e (e7, 7, 1) (e7, 1.8, 3) (e4.2, 0.7, 9) (e1.5, 9, 9)

Expert 2 (e8.3, 8.3, 0.5) (e8.3, 2.1, 2.2) (e4.2, 0, 4.2) (e2.1, 2, 8) (e0.4, 8, 8)

Expert 3 e (e8.6, 8.6, 1) (e8.6, 1.3, 4.1) (e3.4, 1.7, 7.4) (e1, 7.4, 7.4)

Expert 4 (e7.3, 7.3, 1) (e7.3, 1.5, 2) (e3.8, 0.1, 3.8) (e2.5, 1.3, 7.8) (e1, 7.8, 7.8)

Expert 5 e e (e5.8, 5.8, 2.7) (5.8, 0, 5.8) (e2.7, 5.8, 5.8)

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This study also uses to represent the normalised decision matrix, as shown below:

G ¼ A₁ A2 A3 A4 A5 A₆ A7 A8 A9 2 6 6 6 6 6 6 6 6 6 6 6 6 4 0:0293 0:0818 / 0:0977 0:0929 0:1157 0:5730 0:3148 / 0:1102 0:1129 0:1150 0:0189 0:0552 / 0:1206 0:1077 0:0948 0:1276 0:1306 / 0:1372 0:1039 0:1149 0:0337 0:0749 / 0:1185 0:0782 0:1225 0:1017 0:1251 / 0:1123 0:1106 0:0969 0:0417 0:0729 / 0:1164 0:1238 0:1367 0:0130 0:0185 / 0:0852 0:1197 0:0946 0:0610 0:12622 / 0:1019 0:1233 0:1089 3 7 7 7 7 7 7 7 7 7 7 7 7 5

Step 5. Calculate the weighted normalised decision matrix V. For example, vi1is calculated to determine the availability of the

attractions, as shown by the following:

v11 ¼ W1g1ðA1Þ ¼ 0:0939 0:0293 ¼ 0:0027 « v91 ¼ W1g1ðA4Þ ¼ 0:0939 0:0610 ¼ 0:0057 V ¼ A₁ A2 A3 A4 A5 A₆ A7 A8 A9 2 6 6 6 6 6 6 6 6 6 6 6 6 4 0:0027 0:0079 / 0:0047 0:0034 0:0108 0:0538 0:0305 / 0:0053 0:0041 0:0107 0:0018 0:0054 / 0:0058 0:0039 0:0088 0:0120 0:0127 / 0:0066 0:0048 0:0107 0:0032 0:0073 / 0:0057 0:0028 0:0114 0:0096 0:0121 / 0:0054 0:0040 0:0090 0:0039 0:0071 / 0:0056 0:0045 0:0127 0:0012 0:0018 / 0:0041 0:0044 0:0088 0:0057 0:0122 / 0:0049 0:0045 0:0102 3 7 7 7 7 7 7 7 7 7 7 7 7 5

Step 6. Determine the positive (A*) and negative (A) ideal solutions.

Table 4

The fuzzy number of the weights assigned by the six experts to the 15 indices ~Wijc.

Criteria/Indexes Expert 1 Expert 2 Expert 3 Expert 4 Expert 5 Expert 6

C1 (e4.2, 0.7, 9) (e2.1, 2, 8) (e1, 7.4, 7.4) (-2.5, 1.3, 7.8) (e2.7, 5.8, 5.8) (e1, 7.5, 7.5)

C2 (e4.2, 0.7, 9) (e0.4, 8, 8) (e1, 7.4, 7.4) (e1, 7.8, 7.8) (e2.7, 5.8, 5.8) (e3.8, 1.2, 7.5)

C3 (e4.2, 0.7, 9) (e2.1, 2, 8) (e8.6, 1.3, 4.1) (e2.5, 1.3, 7.8) (5.8, 0, 5.8) (e7.8, 1.4, 3.8)

C4 (e1.5, 9, 9) (e2.1, 2, 8) (e8.6, 1.3, 4.1) (e2.5, 1.3, 7.8) (e5.8, 5.8, 2.7) (e3.8, 1.2, 7.5)

C5 (e7, 1.8, 3) (e8.3, 2.1, 2.2) (e3.4, 1.7, 7.4) (e1, 7.8, 7.8) (5.8, 0, 5.8) (e3.8, 1.2, 7.5)

C6 (e7, 1.8, 3) (e8.3, 2.1, 2.2) (e8.6, e1.3, 4.1) (e7.3, 1.5, 2) (e5.8, 5.8, 2.7) (e7.8, 1.4, 3.8)

C7 (e7, 7, 1) (e4.2, 0, 4.2) (e8.6, 8.6, 1) (e7.3, 7.3, 1) (e5.8, 5.8, 2.7) (e7.8, 7.8, 1)

C8 (e4.2, 0.7, 9) (e0.4, 8, 8) (e3.4, 1.7, 7.4) (e1, 7.8, 7.8) (e2.7, 5.8, 5.8) (e3.8, 1.2, 7.5)

C9 (e7, 7, 1) (e2.1, 2, 8) (e8.6, 1.3, 4.1) (e7.3, 1.5, 2) (5.8, 0, 5.8) (e7.8, 1.4, 3.8)

C10 (e4.2, 0.7, 9) (e8.3, 2.1, 2.2) (e8.6, 1.3, 4.1) (e2.5, 1.3, 7.8) (e5.8, 5.8, 2.7) (e7.8, 1.4, 3.8)

C11 (e4.2, 0.7, 9) (e0.4, 8, 8) (e3.4, 1.7, 7.4) (e2.5, 1.3, 7.8) (5.8, 0, 5.8) (e1, 7.5, 7.5)

C12 (e7, 7, 1) (e8.3, 8.3, 0.5) (e8.6, 1.3, 4.1) (e7.3, 1.5, 2) (e5.8, 5.8, 2.7) (e7.8, 7.8, 1)

C13 (e7, 7, 1) (e4.2, 0, 4.2) (e8.6, 8.6, 1) (e7.3, 7.3, 1) (e5.8, 5.8, 2.7) (e7.8, 7.8, 1)

C14 (e4.2, 0.7, 9) (e4.2, 0, 4.2) (e8.6, 1.3, 4.1) (e3.8, 0.1, 3.8) (e5.8, 5.8, 2.7) (e7.8, 1.4, 3.8)

C15 (e4.2, 0.7, 9) (e4.2, 0, 4.2) (e3.4, 1.7, 7.4) (e3.8, 0.1, 3.8) (e5.8, 5.8, 2.7) (e3.8, 1.2, 7.5)

Table 5

The average fuzzy and defuzziﬁcation weights of the 15 indices.

Criteria/Indexes Fuzzy number Average of weight ~Wc W~cþ 8 Defuzziﬁcation of

the weight BNPc Standardize of weight Wc¼ SBNPc C1 (e2.250, 4.117, 7.583) (5.750, 12.117, 15.583) 11.150 0.0939 C2 (e2.183, 5.150, 7.583) (5.817, 13.150, 15.583) 11.517 0.0970 C3 (e5.167, 0.217, 6.417) (2.833, 8.217, 14.417) 8.489 0.0715 C4 (e4.05, 1.067, 6.517) (3.950, 9.067, 14.517) 9.178 0.0773 C5 (e4.883, 1.133, 5.617) (3.117, 9.133, 13.617) 8.622 0.0726 C6 (e6.433, 1.533, 4.117) (1.567, 6.467, 12.117) 6.717 0.0566 C7 (e6.200, 1.433, 4.933) (1.800, 6.567, 12.933) 7.100 0.0598 C8 (e2.883, 3.200, 7.583) (5.117, 11.200, 15.583) 10.633 0.0895 C9 (e7.467, 5.283, 1.883) (0.533, 2.717, 9.883) 4.378 0.0369 C10 (e6.783, 6.083, 1.817) (1.217, 1.917, 9.817) 4.317 0.0364 C11 (e5.733, 1.317, 4.600) (2.267, 6.683, 12.600) 7.183 0.0605 C12 (e4.200, 0.383, 5.767) (3.800, 7.617, 13.767) 8.394 0.0707 C13 (e7.467, 2.317, 2.833) (0.533, 5.683, 10.833) 5.683 0.0479 C14 (e6.783, 6.083, 1.817) (1.217, 1.917, 9.817) 4.317 0.0364 C15 (e2.583, 4.200, 7.583) (5.417, 12.200, 15.583) 11.067 0.0932 A* ¼ maxvi1 i ; maxvi2 i ; maxvi3 i ; maxvi4 i ; maxvi5 i ; minvi6 i ; minvi7 i ; minvi8 i ; maxvi9 i ; maxvi10 i ; maxvi11 i ; maxvi12 i ; maxvi13 i ; maxvi14 i ; maxvi15 i ¼ v* 1; v*2; v*3; v*4; v*5; v*6; v*7; v8*; v*9; v*10; v*11; v*12; v*13; v*14; v*15 ¼ ð0:054; 0:031; 0:018; 0:013; 0:032; 0:006; 0:004; 0:007; 0:005; 0:006; 0:009; 0:009; 0:007; 0:005; 0:013Þ

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Step 7. Calculate the separation measures S*_i and S_i.

This process calculates the Euclidean distances of each alter-native based on the positive and negative ideal solutions.

For example, the separation measure from the positive ideal solution of Japan is calculated as follows:

S*₁ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X5 j¼ i v1j v*j 2 v u u t ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0:0027 0:054Þ2_{þ/ þ ð0:0108 0:013Þ}2 q ¼ 0:060

The separation measure from the negative ideal solution of Japan is calculated as follows:

S₁ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X5 j¼ i v1j vj 2 v u u t ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0:0027 0:001Þ2_{þ/ þ ð0:0108 0:009Þ}2 q ¼ 0:024

Step 8. Calculate the relative closeness (RC*_i) to the ideal solution. RC₁* ¼ S1 S*₁þ S 1 ¼ 0:024 0:060 þ 0:024 ¼ 0:291

Step 9. Rank the alternatives based on their relative closeness to the ideal solution.

A larger implies a better alternative. The best alternative is the one that is closest to the ideal solution, as demonstrated by the following equation:

RC*₂_RC*

1_RC4*_RC*6_RC*7_RC9*_RC5*_RC3*_RC*8

Table 7lists the ranking of the sampled Asian countries by TDC in the year 2009. These countries are ranked from highest to lowest as follows: China, Japan, Hong Kong, Malaysia, Thailand, Singapore, Taiwan, Korea and the Philippines.

Fig. 3shows that the nine Asian countries considered in this study could be divided intofive groups based on their TDC values. China is ranked much higher than the other eight countries and, thus, is the sole member of thefirst group (Very High TDC). Japan and Hong Kong are both ranked highly and, thus, belong to the second group (High TDC). Malaysia, Singapore and Thailand are ranked near the middle and belong to the third group (General TDC). Korea and Taiwan are ranked low and belong to the fourth group (Low TDC). Finally, the Philippines is ranked very low and, thus, belongs to thefifth group (Very Low TDC).

Table 6

Lists the raw data of the TDC index for nine Asian countries in the year 2009. Criteria Availability of attractions Availability of service Affordability Positive market image

Peace and stability Cultural links

Indexes C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 Japan 6,789,658 10,305 1.20 77.50 97,022 82.60 130.20 131.45 7.05 5.00 8.52 5.00 4.70 5.89 6.63 China 132,924,650 39,675 0.10 95.50 230,000 89.90 118.10 88.44 4.84 3.20 5.95 5.10 5.30 7.16 6.59 Taiwan 4,395,004 6958 0.50 29.00 34,382 93.60 161.00 77.00 5.78 4.00 6.62 5.60 5.80 6.83 5.43 Hong Kong 29,590,654 16,463 0.80 62.00 47,139 85.70 153.00 108.70 6.25 3.40 8.29 6.40 6.60 8.30 6.58 Korea 7,817,533 9442 0.10 50.00 36,078 86.90 194.60 80.60 5.86 4.70 6.86 5.50 5.70 4.96 7.02 Malaysia 23,600,000 15,772 0.60 57.50 22,421 93.80 74.20 69.20 7.52 5.10 6.06 4.60 5.40 7.01 5.55 Singapore 9,682,700 9187 0.90 56.00 19,566 85.60 153.40 98.00 8.45 5.90 8.73 6.40 5.60 7.85 7.83 Philippines 3,017,099 2329 0.00 35.00 9508 91.20 81.00 63.80 4.62 3.30 4.21 4.30 4.10 7.59 5.42 Thailand 14,150,000 15,901 0.60 92.00 19,993 87.00 108.80 68.60 6.36 3.90 6.48 5.20 4.90 7.82 6.24

1. Institute for Management Development (IMD).The World Competitiveness Yearbook 2010. 2. World Economic Forum (WEF).The Travel & Tourism Competitiveness Report 2009.

3. Hong Kong Tourism Board.http://partnernet.hktb.com/pnweb/jsp/doc/listDoc.jsp?doc_id¼129117. 4. Japan National Tourism Organization (JNTO).http://www.jnto.go.jp/.

5. Singapore Government.http://www.singstat.gov.sg/stats/keyind.html.

6. National Tourism Administration of the People’s Republic of China.http://www.cnta.gov.cn/.

7. Department of Tourism, Republic of the Philippines.http://www.visitmyphilippines.com/index.php?title¼VisitorStatistics&func¼all&pid¼39&tbl¼1. 8. Tourism Malaysia.http://www.tourismmalaysia.gov.my/corporate/research.asp?page¼facts_ﬁgures.

9. Korea Tourism Organization.http://kto.visitkorea.or.kr/inout.kto?func_name¼search. 10. Tourism Bureau, M.O.T.C. Taiwan.http://admin.taiwan.net.tw/.

Table 7

Ranking of each destination’s competitiveness in Asia.

S*i Si RC*i Rank Japan 0.060 0.024 0.29061 2 China 0.018 0.068 0.79531 1 Taiwan 0.066 0.012 0.15058 7 Hong Kong 0.053 0.022 0.29057 3 Korea 0.065 0.011 0.14129 8 Malaysia 0.057 0.019 0.25283 4 Singapore 0.063 0.018 0.21846 6 Philippines 0.071 0.010 0.12157 9 Thailand 0.060 0.019 0.24276 5 A ¼ minvi1 i ; minvi2 i ; minvi3 i ; minvi4 i ; minvi5 i ; maxvi6 i ; maxvi7 i ; maxvi8 i ; minvi9 i ; minvi10 i ; minvi11 i ; minvi12 i ; minvi13 i ; minvi14 i ; minvi15 i ¼ v 1; v2; v3; v4; v5; v6; v7; v8; v9; v10; v11; v12; v13; v14; v15 ¼ ð0:001; 0:002; 0:000; 0:004; 0:001; 0:007; 0:010; 0:015; 0:003; 0:003; 0:004; 0:006; 0:004; 0:003; 0:009Þ

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Table 8indicates that China is rankedﬁrst in terms of the avail-ability of its attractions, whereas the Philippines, Taiwan and Korea are ranked the lowest. China is also rankedﬁrst for service avail-ability, whereas the Philippines, Korea and Taiwan occupy the last three places. The top three countries for affordability are the Philippines, Malaysia, and Thailand, whereas the last three are Japan, Hong Kong, and Singapore. The top three countries for positive market image are Singapore, Malaysia and Japan, whereas the Philippines, China and Hong Kong occupy the last three places. In terms of peace and stability, Hong Kong, Singapore and Japan occupy the top three rankings, whereas the Philippines, Malaysia, and China are ranked at the bottom. Finally, the top-ranked countries for cultural links are Singapore, Hong Kong and Korea, whereas Taiwan, Malaysia and the Philippines take the last three places.

5. Conclusions

This study successfully combines TOPSIS with the Fuzzy Rasch model to evaluate TDC in nine Asian countries: China, Hong Kong, Japan, Korea, Malaysia, Singapore, Taiwan, Thailand and the Philippines. This study was conducted during the year 2009 using six criteria and 15 indexes. Determining the weight of the evalua-tion criteria with the Fuzzy Rasch model appears to be a feasible task. Doing so can rectify the inaccuracy of the real fuzzy numbers assigned by the individual experts for the speciﬁc criteria. Addi-tionally, this study_{’s methodology represents not only an} innova-tive attempt to evaluate TDC but also a practical application of the MCDM method to studying TDC.

The analytical results presented in this study demonstrate that China signiﬁcantly outscored the other countries in terms of TDC (Very High TDC) because it outperforms the other countries in terms of the availability of its attractions and services. However, China did not outperform the other countries in positive market image as well as peace and stability. According to the world competitiveness yearbook published by the IMD, tourist intention to travel to a country increases for countries with more secure environments. Theseﬁndings suggest that China should focus on improving its peace and stability as well as its market image.

Additionally, Japan and Hong Kong belonged to the High TDC group. Thus, Japan and Hong Kong should endeavour to reduce the costs of living, the costs of transportation and the hotel prices (affordability) because tourists pre-plan their travel costs before travelling to a country. As a result, the cost-of-living index of a destination country affects the tourist travel intention for that country. Hong Kong should also improve the quality of its natural environment to achieve a more positive market image; the poor natural environment of Hong Kong contributes to its negative image, and most travellers avoid travelling to destinations with a negative market image.

Malaysia, Singapore and Thailand belong to the General TDC group. Hence, Malaysia and Thailand should work on improving

their peace and stability as well as their cultural links, whereas Singapore should focus on improving its costs of living, costs of transportation and hotel prices (affordability).

Korea and Taiwan belong to the low TDC group. Thus, Taiwan and Korea should focus on enhancing the availability of their attractions and services. TDC increases the number of tourist arrivals in a country. According to the World Tourism Organiza-tion’s statistics, most tourists ﬂy to Asian destinations, and as a result, Taiwan and Korea should increase both the number of service local routes to their airlines and the number of passengers carried per plane. Taiwan and Korea should also increase the number of hotel rooms, as this factor represents another area that requires improvement for both countries.

Finally, the Philippines exhibit weak TDC. The Philippines enjoys an advantage in affordability, but in the other criteria, this country is far less competitive than the other countries. Thus, the Philippines should strive to improve in all of the criteria.

Acknowledgements

The authors would like to thank two anonymous reviewers and the editor for their useful comments and suggestions.

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