INTRODUCTION
To select a convention site, a variety of influential decision variable
should be simultaneously assimilated in the process of decision making
and this has made the subject potentially complex (Clark and McCleary,
1995). According to the exhaustive review of the site selection papers
in the literature, a 5step conceptual model of the site selection process
was proposed by Crouch and Ritchie (1998) and they discovered several
categories of site selection factors, coupled with various antecedent
conditions and competing sites influences. Convention preplanning, site
selection analysis and recommendations, site selection decision, convention
held and post convention evaluation are the five steps the have to be
taken in convention site selection process. The site selection decision
are influenced by several factors and can be broadly separated into sitespecific
and association factors (Weber and Chon, 2002). The majority of previous
studies have endeavored to recognize many of this topic`s selection contributive
factors (e.g., Oppermann, 1996; Go and Zang, 1997; Crouch and Ritchie,
1998; Chacko and Fenich, 2000; Kim and Kim, 2003; Crouch and Louviere,
2004). Go and Zang (1997) classified the convention site selection criteria
into two primary categories: 1 the convention destination site`s environment
addressing a city`s capacity to host an international convention and 2
the meeting facilities. The proposed conceptual model of convention site
selection by Crouch and Ritchie (1998) investigates eight primary factors
together with several aspects, culminating in the recognition of 36 attributes
that affect the choice of a convention site. With reference to the summary
review of Kim and Kim (2003) the prominent criteria for convention site
selection can be characterize as follows: meeting room facilities, service
quality, restaurants, transportation and attractiveness of the destination
are the major attributes. Several contributive and worthwhile studies
have been conducted regarding site attributes which among them the study
of Chacko and Fenich (2000) and Crouch and Louviere (2004) are of vital
importance. A regression analysis was performed by Chacko and Fenich (2000)
to explore the significance of US convention destination attributes. Crouch
and Louviere (2004) applied the logistic choice model using designed experimental
data to explore the determinants of convention site selection.
As mentioned before there are various criteria that affect decision making
process in a convention site selection problem. Therefore, offering a
method for choosing a suitable place, entails applying MCDM methods. In
general, we face with MCDM methods when for making decision between different
alternatives, we encounter with more than one criteria or objectives.
So decision making problems can be categorized into two groups of multiple
attribute decision making (MADM) and Multiple Objective Decision Making
(MODM).
MADM is ranking multiple alternatives subject to different attributes.
In fact it is, choosing the best alternative among available alternatives
based on given criteria and attributes. Optimum performance of ranking
alternatives strictly depends on choosing suitable weights for these criteria.
To calculate these weights, the criteria should be compared with each
other in advance. Pair wise comparison matrix which is used in AHP method
is a good method for this purpose.
AHP, as a Multiple Criteria Decision Making (MCDM) tool and a weight
estimation technique, has been extensively applied in many areas such
as selection, evaluation, planning and development, decision making, forecasting
and so on (Vaidya and Kumar, 2006). The conventional AHP requires exact
judgments and crisp comparison matrices. However, due to the complexity
and uncertainty involved in real world decision problems, it is sometimes
unrealistic or impossible to acquire exact judgments. It is more natural
or easier to provide interval judgments for part or all of the judgments
in a pair wise comparison matrix. In this study first with the help of
interval comparison matrices and applying goal programming model, the
weights of efficient criteria are obtained in the form of interval weights.
Then, alternatives are ranked through interval TOPSIS method.
Interval comparison matrix: In an interval comparison matrix we
face with interval judgments instead of precise judgments. In other words,
the relative importance of criterion i and j can be expressed as a number
between l_{ij} and u_{ij}. Where, l_{ij} and u_{ij}
are nonnegative real numbers and l_{ij}≤u_{ij}. General
form of an interval comparison matrix is presented in matrix A.
where, l_{ij} = 1/u_{ij} and u_{ij} = 1/l_{ij}
for all I, j = 1,..., n; i ≠ j. The above interval comparison matrix
can be divided into two crisp nonnegative matrices as follows:
where, A_{L}≤A≤A_{U}. Note that, A_{l}
and A_{u} are no longer reciprocal matrices.
For the interval comparison matrix A, there should exist a normalized
interval weight vector,
which is close to A in the sense that
for all i, j = 1,..., n; i≠ j.
Consistency of the pair wise comparison matrices is another factor which
should be considered. In this study to examine the consistency of matrices
with interval data, here used the proposed method by Wang et al.
(2005a) which is described as follow.
A = (a_{ij})_{nxn} is a consistent interval comparison
matrix if and only if it satisfies the following inequality constraints:
GP model for obtaining interval weights from an interval comparison
matrix: Weight calculation techniques from interval comparison matrices
are classified into two groups of point estimation and interval estimation.
Extensive researches have been done regarding these two techniques to
come up with the weights from interval comparison matrix. matrix (e.g.,
Saaty and Vargas, 1987; Arbel, 1989; Kress, 1991; Arbel and Vargas, 1993;
Islam et al., 1997; Mikhailov, 2002, 2004; Sugihara et al.,
2004; Wang et al., 2005a,b; Wang and Elhag, 2007). It is more natural
and logical that an interval comparison matrix should give an interval
weight estimate rather than an exact point estimate. GP model that was
proposed by Wang and Elhag (2007), is one of the methods for calculating
interval weights from interval comparison matrices. This model is shown
in Eq. 4.
and e^{t} = (1,..., 1).
Note that as
well as can
not be simultaneously selected as basic variables in a simplex method.
In following here`s used this method to obtain interval weights of criteria
and sub criteria of our case study.
Interval arithmetic: If upper and lower bounds for the uncertain
parameters can be determined, these can be interpreted as the endpoints
of a closed interval .
This interval is usually denoted by [x]. The principles of interval arithmetic
are quite simple: during evaluation any expression is constructed by subsequent
calls of elementary binary operations {+,−,รท,x}, where the internalization
of binary operators is:
TOPSIS method with interval weight and data: TOPSIS (technique
for order preference by similarity to an ideal solution) method is presented
in Chen and Hwang (1992), with reference to Hwang and Yoon (1981). TOPSIS
is a multiple criteria method to identify solutions from a finite set
of alternatives. The basic principle is that the chosen alternative should
have the shortest distance from the positive ideal solution and the farthest
distance from the negative ideal solution. A similar concept has also
been pointed out by Zeleny (1982).
Considering the fact that, in some cases, determining precisely the exact
value of the attributes is difficult and that, as a result of this, their
values are considered as intervals, TOPSIS method with interval data was
proposed by Jahanshahloo et al. (2006), such that in it, data were
considered as interval and the weights of criteria were deterministic.
The proposed TOPSIS method of this paper apart from including interval
data, considers the weights as intervals. This method is described as
follow:
Suppose, A_{1}, A_{2},..., A_{m} are m possible
alternatives among which decision makers have to choose, C_{1},C_{2},...,
C_{n} are criteria with which alternative performance are measured,
x_{ij} is the rating of alternative A_{i} with respect
to criterion C_{j} and is not known exactly and only we know
A MCDM problem with interval weight and data can be concisely expressed
in format of one matrix as Table 1.
Table 1: 
MCDM problem with interval weight and data 

Where, is the weight of criterion C_{j} 
The algorithmic method: A systematic approach to extend the TOPSIS
to the interval data is proposed in this section. First, we calculate
the normalized decision matrix as follows:
The normalized values are
calculated as:
Now, interval
is normalized of interval
. The normalization method mentioned above is to preserve
the property that the ranges of normalized interval numbers belong to
(0,1).
Referring to the Eq. 5 with regard to
We can construct the weighted normalized interval decision matrix as:
where,
are the lower and upper weight of the ith attribute or criterion and
Then, we can identify positive ideal solution and negative ideal solution
as:
where, I is associated with benefit criteria and J is associated with
cost criteria. The separation of each alternative from the positive ideal
solution, using the ndimensional Euclidean distance, can be currently
calculated as:
Similarly, the separation from the negative ideal solution can be calculated
as:
A closeness coefficient is defined to determine the ranking order of
all alternatives once the of
each alternative A_{j} has been calculated. The relative closeness
of the alternative A_{j} with respect to
is defined as:
Obviously, an alternative A_{j} is closer to the
and farther from
as
approaches to 1. Therefore, according to the closeness coefficient, we
can determine the ranking order of all alternatives and select the best
one among a set of feasible alternatives.
Case study: To clarify the proposed method a numerical example
is illustrated. The hierarchical structure of this example was proposed
by Chen (2006). In this case study we consider five alternatives and try
to assess their performance by proposed method. The highest level of the
hierarchy is the overall goal: to construct an evaluation structure for
convention site selection with weights corresponding to criteria. Under
the overall goal, the second level represents the criteria affecting convention
site selection, including meeting and accommodation facilities, costs,
site environment, local support and extra conference opportunities. Various
sets of subcriteria associated with each factor in the second level are
linked to the third level. As seen in Fig. 1 there are
17 attributes in total in the third level. The meeting and accommodation
facilities factor consists of 4 attributes which are space, variety of
meeting and accommodation properties, suitability of convention facilities
and quality of food and beverage.
The cost factor is subdivided into 4 attributes named transport expense,
accommodation expense, food and beverage expense and commodity prices.

Fig. 1: 
The hierarchical structure for convention site selection 
The site environment factor is made of three attributes called city image,
site accessibility and suitability and quality of local infrastructure.
The local support factor includes three attributes such as government
support, quality of convention personnel and efficiency of convention
personnel. Finally, the extra conference opportunities factor includes
three attributes that are climate, entertainment opportunities and sightseeing
and cultural attractions. The hierarchical structure of this decision
problem is shown in Fig. 1.
Interval comparison matrices of criteria and subcriteria prepared by
experts are presented in Table 27.
The consistency of each mentioned pair wise comparison matrices are examined
through Eq. 3. For instance the consistency of each
levelone comparison matrices is shown in Table 8.
The consistency of each remaining matrices are examined in the way of
levelone matrices. Regarding the yield results from studying the consistency
of comparison matrices, being consistent for all matrices is proved. In
the next step interval weights of each criteria and subcriteria are calculated
via goal programming model which is shown in Eq. 4 and
its results are presented in Table 9.
Then applying interval TOPSIS method, the five cited alternatives are
ranked regarding with the obtained interval weights shown in Table
8. Initially each alternative is evaluated against each criterion
by an expert who expresses his/her opinion in the form of interval data
that are presented in Table 10. In the next step these
data are normalized through Eq. 6 and 7,
their results are shown in Table 11. Then considering
the Eq. 7 and 8 and the normalized
data of Table 11, the normalized weights are obtained
and their results are shown in Table 12.
Then the values of
and
are calculated through Eq. 12 and 13
that their results are presented in Table 13 and Table
14.
Table 2: 
Interval comparison matrix for the four criteria 

Table 3: 
Interval comparison matrix for the four subcriteria
with respect to meeting and accommodation facilities 

Table 4: 
Interval comparison matrix for the four subcriteria
with respect to costs 

Table 5: 
Interval comparison matrix for the four subcriteria
with respect to site environment 

Table 6: 
Interval comparison matrix for the four subcriteria
with respect to local support 

Table 7: 
Interval comparison matrix for the four subcriteria
with respect to extra conference opportunities 

Table 8: 
Consistency test for levelone comparison matrices 

Table 9: 
Interval weights for a consistent interval comparison
matrix generated GPM method 

Table 10: 
The interval decision matrix of five alternatives 

Table 11: 
The interval normalized decision matrix 

Table 12: 
The interval weighted normalized decision matrix 

Table 13: 
Distance of each alternative from the positive ideal
solution 

Table 14: 
Distance of each alternative from the negative ideal
solution 

Table 15: 
Closeness coefficient 

In the end with the help of Eq. 5 and the results
of closeness coefficient shown in Table 15, the final
ranks are obtained.
General representation of proposed model to obtain the final rank is
as follows:
Stage 1 
: 
Constructing interval comparison matrix of Table
27 for criteria and subcriteria with regard to the AHP which
has been presented in Fig. 1. 
Stage 2 
: 
Examining the consistencies of interval comparison matrix obtained
from stage 1 with respect to the Eq. 3 which its
results have been reported in Table 8. 
Stage 3 
: 
Calculating the interval weights of criteria and sub criteria with
the use of goal programming model with respect to the Eq.
4 which its results have been presented in Table
9. 
Stage 4 
: 
With regard to the weights obtained from stage 3, evaluation of
considered five alternatives with respect to the criteria determined
by experts are summarized in Table 10 in the form
of interval decision making matrix. In order to rank the alternatives,
we first normalize the interval decision making matrix with the use
of Eq. 6 and 7 which its results
have been presented in Table 11. 
Stage 5 
: 
Having obtained the interval normalized decision matrix, with the
consideration of weights obtained from Table 4 and
utilization of Eq. 8 and 9, we
construct the interval weighted normalized matrix which its results
have been presented in Table 12. 
Stage 6 
: 
Determine positive ideal solution and negative ideal solution (identification
of
and ,
using the Eq. 10 and 11. 
Stage 7 
: 
In this state with regard to the reported results in Table
12 and utilization of Eq. 12 and 13,
the distance of each alternative from the positive and negative ideal
solution is calculated which its results have been presented in Table
13 and 14. 
Stage 8 
: 
With utilizing the Eq. 14 and the obtained results
in Table 13 and 14, the Closeness
coefficient is calculated which its results have been presented in
Table 15. 
Stage 9 
: 
finally, with respect to the closeness coefficient presented in
Table 15, final ranking of alternatives is obtained. 
Considering the acquired results of Table 15, 5th alternative
is placed in the 1st rank and the 3rd; 2nd, 4th and 1st alternatives are placed
in the 2nd, 3rd, 4th and 5th rank, respectively.
CONCLUSIONS
In this study, an effective hybrid model was presented for decision making.
In the proposed model, interval comparison matrix which has been inspired
by Analytical Hierarchy Process (AHP) was employed to compare the criteria
against each other. Furthermore, to calculate the interval weights of
criteria, we made use of goal programming method. Moreover, interval data
was utilized to evaluate the alternatives with respect to the criteria.
In order to rank the alternatives with respect to criteria, Technique
for Order Preference by Similarity to an Ideal Solution (TOPSIS) with
interval data and weights is used. It is very obvious that the proposed
model can be generalized to other cases and in the conditions of uncertainty
for both the comparison of criteria against each other and alternatives
evaluation with respect to influential criteria in the process of decision
making, the model can create the possibility for decision makers to use
interval data instead of deterministic values so that he can adopt a high
quality and more appropriate decisions.