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The new approach of determination of weight factors in the process of
certification a city as the favourable business environment
1Dragan Randjelovic*
The University of Criminalistic and Police Studies, Cara Dusana 196,11080 Belgrade Serbia;
2Jelena Stankovic
University of Nis, Faculty of Economics, Trg kralja Aleksandra Ujedinitelja 11,18000 Nis Serbia;
3Milan Randjelovic
City of Ni, Local Economic Development Office, 7 jula 2, 18000 Nis, Serbia;
4Milos Randjelovic
HELP Nis, 18000 Nis, Serbia;
Abstract
One of the key problems in the application of multi-criteria analysis methods is to determine the importance
of the criteria in the model. The relevance and validity of decisions are directly conditioned by the relevance
of a given set of criteria for evaluating alternatives, as well as the correctly determination of weights of these
criteria. This is the reason why there are a number of methodologies developed with the aim of calculating
the importance of each of the criteria for the given problem. All methodologies are classified into two main
groups: subjective and objectively. This paper presents a new procedure of integration by two recognized
methods for determining weight of factors - Analytic Hierarchy Process (AHP) method, as a subjective
method and entropy as an objective method. The procedure of the application is shown on the real,
contemporary local economic problem in Serbia - the process of cities and municipalities certification on the
favourable business environment.
Keywords
Determination of Weight Criteria, Friendly Bussines Certification, Multi-criteria decision making
* Author to whom all correspondence should be addressed: email [email protected], Phone +381648925012,
Fax +381113162152
mailto:[email protected]
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1. Introduction
The modern problem of the Serbian economy is, above all, much expressed uneven economic
development. This problem itself is not rare, as it is not realistic to expect that the development of
all municipalities in the country is completely equal, but it is important to take measures to reduce
the gap between highly developed and underdeveloped communities. One way involves the
initiative to create a friendly business environment in municipalities. This can be achieved by
identifying and then presenting the comparative advantages of individual cities and municipalities.
This task can be realized through the certification of cities and municipalities with a favourable
business environment and this is a practice that is carried out in Serbia last few years.
Program of certification cities and municipalities in Serbia was introduced in 2007 by the National
Alliance for Local Economic Development - NALED. NALED is an independent association of
companies, local governments and NGOs working together to create a favourable business
environment in Serbia. Since its establishment (2006) to date, NALED has been joined over 140
members and 20 institutional partners. Currently, the certification program encompasses 48 cities
and municipalities in Serbia. Immediate support NALED obtained by the U.S. Agency for
International Development (United States Agency for International Development - USAID).
Certification Program is supported by all relevant state institutions, including the Ministry of
Economy and Regional Development, Ministry of Human and Minority Rights, Public
Administration and Local Government, the Executive Council and the National Agency for
Regional Development. Direct, immediate effects of the certification process are reflected in the
growth of investments, and indirect in the decrease of unemployment and increase of living
standards.
The authors intend to determine the significance criteria identified as the most relevant for
certification, on the basis of data that have been reached in the certification process, provided by
local governments and NALED. One way to determine the importance of the criteria is based on the
use of subjective assessments and the preference of authorities responsible for conducting
certification. However, the significance of criteria may be defined objective approach as well, based
on the application of quantitative methods. By the application of mathematical and statistical
methods in determining the importance of the criteria, it has increased the reliability of
conclusions. The idea is to perform the integration of these two approaches and determine weights
that include the subjective perception of the decision makers about the importance of the observed
criterion, but also objectively quantified the importance of meeting the criteria for achieving the
greatest possible amount of investments in the city or municipality.
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2. Material and methods
The problem of relative weight determination has existed since the formulation of the first multi-
criteria analysis methods. During that period several approaches have been proposed to determine
weights, (Hwang & Yoon, 1981) and (Saaty,1980). Most of them can be classified, depending on
the information provided for their calculation, in two mayor groups: subjective and objective
approaches. Subjective approaches determine weights that reflect subjective judgment, while
objective approaches determine weights by making use of mathematical models, and they do not
consider subjective judgment (Liu,2003).
The most popular subjective approaches are the Analytic Hierarchy Process (Saaty, 1977), least
squares comparison (Chu, Kalaba & Spingarn, 1979) , Delphi method (Hwang & Lin,1987), etc.
The objective approaches include methods such are LINMAP - Linear Programming Techniques for
Multidimensional Analysis of Privileged (Srinivasan, Shocker, 1973), various computer-aided
mathematical models (Srinivasan & Shocker,1973), Data Envelopment Analysis (Charnes, Cooper
& Rhodes,1978), the entropy method (Hwang & Yoon, 1981), principal component analysis and
multi-attribute programming methods (Fan, Ma & Tian, 1999), etc.
Two proposed methods for integration in order to determine relative weights in multi-criteria
models, chosen by authors are AHP method and entropy. There are many different applications of
those methods, as well as many different manners of their aggregation. The application in supply
chain management is proposed by Bindu, R. S. and B. B. Ahuja and it deals with multiple attribute
nature of vendor selection problem to which AHP method is applied using verbal assessments for
relative measurements. The technique used for assigning weights is the entropy method (Bindu &
Ahuja, 2010). Research conducted by Hamidi, Naser , Pouya Majd Pezeshki and Abolfazl
Moradian in aim to determine important criteria which have effects on selecting the best brands in
beverage industries evaluate the weight of each criterion for selecting the best brand in the
beverage market (Hamidi, Pezeshki & Moradian, 2010). Then the major criteria weights are
analysed using the AHP method and entropy. Finally, weights are then determined using the
compromised weighting method. Fuzzy AHP method is proposed to determine the rank in logistic
centre location scheme evaluation by Chen, Yan and Lili Qu, (Chen & Qu, 2006). In order to
include the domain expertise in the decision process and improve evaluation performance, it is
developed a fuzzy MCDM (multi-criteria decision making) model to modify the fuzzy AHP
weights based on the entropy technique, by making use of the decision matrix information through
Delphi method. Ma, Jian, Zhi-Ping Fan and Li-Hua Huang proposed an integrated approach to
determine attribute weights in the multiple attribute decision making problems (Ma,Fan &
Huang,1999). The approach makes use of the subjective information provided by a decision maker
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and the objective information to form a two-objective programming model. Thus the resulting
attribute weights and rankings of alternatives reflect both the subjective considerations of a decision
maker and the objective information. An example is used to illustrate the applicability of the
proposed approach. Xuan Zhou, Fengming Zhang, Xiaobin Hui and Kewu Li proposed an expert
evaluation of information in group decision making by determining experts' weights based on the
new aggregation method of cluster analysis and entropy (Zhou, Zhang,Hui & Li,2011).
Inspired by the above research results, the authors have conducted research in order to create an
algorithm for the integration of results obtained using the AHP method and entropy. The
significance of this research is the possibility of its concrete application to solve the current
problems in the field of local economic development, such as improving the methodology which is
applied in the business friendly certification process of municipalities in Serbia.
3. Case - study
Information basis for the analysis are the data provided by NALED about six cities that fulfilled
criteria for certification on a favourable business environment and have the status of the sample in
this study. According to the NALED criteria for evaluating the business environment of
municipalities in Serbia are the following:
C1: Strategic planning of local economic development in partnership with businesses
C2: Special department in charge of local economic development (LED), FDI promotion and
business support - existence of LED Office
C3: Business council for economic issues advisory body with the mayor and local
governments
C4: Efficient and transparent system for acquiring construction permits
C5: Economic data and information relevant for starting and developing a business
C6: Multilingual marketing materials and website
C7: Balanced structure of budget revenues / debt management
C8: Investing in the development of the local workforce
C9: Cooperation and joint projects with local business on fostering LED
C10: Adequate infrastructure and reliable communal services
C11: Transparent policies on local taxes and incentives for doing business
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C12: Electronic communication and on-line services
To set up an appropriate model is necessary to specify the characteristics of municipalities in the
certification process. For receiving the title "business friendly", it is necessary that municipality
meets above 75% of the criteria. However, a significant lack of understanding of such methodology
is that these 75% is the average fulfillment percentage of all presented twelve criteria, which means
that one or two criteria can even be unfulfilled, even when the average score of all criteria is above
75%. Thus, the importance of the criteria is not included in the process of determining the level of
compliance criteria and that is the basic drawback of this process. Hence, it arises the idea that by
using subjective and objective methods can be determined the appropriate weights for the business
friendly certification model. The level fulfillment of criteria of municipalities in the sample is given
in Table 1.
Table 1. Level of fulfillment for all criteria in sample
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 Average
Criterion
Type max max max max Max max max max max max max max max
1
Municipality
1 0.800 1.059 1.000 0.732 0.875 1.000 1.000 0.733 0.636 0.829 1.000 1.250 0.910
2
Municipality
2 1.000 0.824 0.750 1.000 0.925 1.000 1.000 0.933 1.000 0.878 1.000 1.000 0.943
3
Municipality
3 0.625 0.947 0.800 0.941 0.857 1.182 0.900 0.750 0.667 0.940 0.929 1.100 0.887
4
Municipality
4 0.900 0.824 0.875 1.000 0.950 1.000 1.000 0.700 0.682 0.756 1.000 0.750 0.870
5
Municipality
5 1.000 0.618 1.000 0.780 0.600 0.667 1.000 0.600 0.591 0.976 0.833 1.250 0.826
6
Municipality
6 1.000 1.059 0.750 0.939 0.900 0.944 1.000 0.867 0.909 0.793 1.000 1.250 0.951
Average level of
fulfillment per
criterion
0.888 0.889 0.863 0.899 0.851 0.966 0.983 0.764 0.748 0.862 0.960 1.100 0.898
Regardless the fact that of the importance of the criteria is not taken into account, NALED gave
their preference on the importance of each criteria through evaluation of sub-criteria by which these
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criteria are defined. Specifically, each of the criteria has a certain number of sub-criteria (some as
many as 18 sub-criteria). These sub-criteria are divided according to the importance to three groups:
eliminating sub-criteria, very influential sub-criteria and important sub-criteria. Thereby,
eliminating sub-criteria are the most significant factors because their fulfillment is the first
condition in the evaluation of the business environment by the verification commission. Therefore,
the rate of this group of sub-criteria is 2, the rate of very influential sub-criteria group is 1, and the
rate of the group of important criteria is 0.5. Presence of eliminating, very influential and important
criteria is the basis of determining the weights of each of the twelve relevant criteria in the model.
The evaluation of importance, which is goven in Table 2, is nothing else but the average rating of
importance of sub-criteria which is defined significance criteria.
Table 2. Importance of criteria according to NALED
Criterion C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12
Criterion
rate 1.250 1.050 0.950 0.900 0.660 1.000 0.800 1.100 1.250 0.850 0.750 1.150
This methodology for determining the significance of the relevant criteria in the model was the
starting point for authors research. In order to apply a scientific approach based on determination
of subjective and objective weights, as well as formulation of new integrated approach that the
authors are presented in this paper, it is necessary to define some assumptions. The assumptions for
determining the weight problem described as follows:
Assumption 1: Subjective preferences of the decision maker about the importance of the criteria
expressed through weights determined by subjective methods are approximately equal to the
value of weights obtained by objective approach, if the decision maker preference is based on
absolutely correct and rational perception of the problem.
Assumption 2: Ratio between objective and subjective weights may be considered as a relevant
coefficient in the integration of these approaches in order to determine the weights that contain
both these components.
In order to determine the weights that include subjective and objective components, it is necessary
to implement the following methodological procedures:
quantify the subjective preferences using adequate scale;
determine the weights on the basis of subjective preferences using the AHP method;
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analyse the importance of the particular criterion fulfillment in the observed municipality of
objective and quantitative technique such as entropy;
determine the weights based on objective indicators provided by entropy results;
propose a procedure for the integration of certain subjective and objective weights.
3.1. AHP-Subjective approach for weight determination
Analytic Hierarchy Process (AHP) belongs to the group of subjective methods and is one of the
most popular method of multi-criteria analysis and a tool for decision making on the selection of
optimal alternative(s), especially in cases where there is a possibility of a hierarchical structuring of
relevant criteria. AHP method is the kind of quantitative technique that can incorporate in the model
both qualitative and quantitative criteria, (Saaty, 1977) and (Saaty,2001).
Multi-criteria decision making problems are those where it is implied that a decision maker
supposed to identify the optimal course of action, considering a set of conflict criteria. Complexity
in decision making situations involves quantitative and qualitative criteria, different measurement
scales, and multiple comparisons. The ability to assign a preference rank for general decision
making situations is needed as well as simplicity of methods (Saaty,1986). The AHP is a suitable
method that provides a logical and scientific basis for such multi-criteria decision making problems
and has been widely applied to both individual and group decision making scenarios from the early
1980s, (Wind & Saaty,1980) and (Saaty,& Vargas,1994). AHP is a quantitative tool that has been
used in almost all problems related to multi-criteria decision making and its application includes
about 150 different kinds of problems (Vaidya & Kumar,2006). AHP method is a method for
formulating and analysing decisions that can successfully be used to measure the influence of many
factors relevant to the possible outcomes of decisions as well as for forecasting i.e. the performance
of relative probability distribution of outcomes.
According to Saaty (Saaty,1986), the AHP was founded on three design principles: (i)
decomposition of the goal-value structure where a hierarchy of criteria, sub-criteria, and alternatives
is developed, with the number of levels determined by the problem characteristics; (ii) comparative
judgments of the criteria on single pairwise comparisons of such criteria with respect to an upper
criteria; and (iii) linear-based synthesis of priorities where alternatives are evaluated in pairs with
respect to the criteria on the next level of the hierarchy, and criteria can be given a priority (e.g.
preference) expressed as a weight in the AHP matrix.
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Let the problem is defined as a general problem of multi-criteria analysis, where it is necessary to
evaluate m of available alternatives, on the basis on n relevant criteria.
On the stage of decomposition, the problem is viewed as a hierarchical structure, where the goal is
on the top, while the criteria by which a decision is made are treated at the lower levels. At the
lowest hierarchical level is a range of alternatives, which comparisons are necessary to make.
The next phase involves, in addition to collecting data and its peer evaluation. There is, first of all,
makes pairwise comparison of criteria and alternatives at a given level of hierarchy, but also in
relation to the criteria of the directly higher level. Pairwise comparison of alternatives is done in
response to the question of which of the two observed attributes that characterize an alternative to
the given criteria, is better in terms of meeting the criteria and contribution to the certain objective.
Strength of preference is expressed by the ratio scale with increments of 1-9. The preferred level of
1 shows equality of observed attributes, while the level of 9 indicates absolute, the strongest
preference of one attribute over another (Ma, & Zhang,1991) and (Leskinen, 2000). Such a scale
was formed by Saaty (Saaty, 1977) and it is used in essentially the AHP method and for its entire
later advanced variant (revised AHP or ANP).
Thus, defined scale allows comparisons in a limited scope, while the perception is a sensitive
enough to make a difference in the alternatives importance. Based on pairwise comparison of
alternatives, the reciprocal matrix can be formed (dimension nn on the criteria level, or mm on
the alternatives level), where the elements aii = 1, while the elements aji are the reciprocal of the
elements aij, i.e. aji = 1/ aij, i j and i, j = 1, 2, ..., n.
Figure 1. Hierarchical structure of business friendly certification model
Assuming that the observed decision problem is the one which can be defined by the previously
described matrix form, these phases can be concretized as follows. In the first phase, the problem is
viewed as a hierarchy, where are on the top the goal of the observed problem, and the aim of the
problem is to determine the weights of criteria in the model. At the lower level are the criteria,
which comparison should be made based on the expressed preferences of the decision-maker. The
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structure of the problem is given in Figure1. Following phase is the determination of reciprocal
matrix. Reciprocal matrix, on the level of criteria, of the business friendly certification model is
given in Table 3, according to the preferences provided by NALED.
Table 3. Reciprocal matrix for business friendly certification model
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12
C1 1 3 4 4 8 3 6 2 1 5 6 2
C2 1/3 1 2 3 6 1 4 1 1/3 3 4 1/2
C3 1/4 1/2 1 1 5 1 3 1/2 1/4 2 4 1/3
C4 1/4 1/3 1 1 4 1/2 2 1/3 1/4 1 3 1/4
C5 1/8 1/6 1/5 1/4 1 1/5 1/3 1/6 1/8 1/4 1/2 1/7
C6 1/3 1 1 2 5 1 3 1/2 1/3 3 4 1/2
C7 1/6 1/4 1/3 1/2 3 1/3 1 1/4 1/5 1 2 1/5
C8 1/2 1 2 3 6 2 4 1 1/2 4 5 1
C9 1 3 4 4 8 3 5 2 1 5 6 2
C10 1/5 1/3 1/2 1 4 1/3 1 1/4 0.2 1 2 1/4
C11 1/6 1/4 1/4 1/3 2 1/4 1/2 1/5 1/6 1/2 1 1/5
C12 1/2 2 3 4 7 2 5 1 1/2 4 5 1
Using the procedure of the AHP method, it determines priority vector for the criteria in the model.
The results are presented in Table 4.
Table 4. Criteria importance of business friendly certification model
Criteria Priority Vector
C1 0.191572435
C2 0.091015872
C3 0.060865513
C4 0.046027224
C5 0.015076309
C6 0.074900384
C7 0.030610045
C8 0.112190751
C9 0.189180090
C10 0.036087104
C11 0.022203512
C12 0.130270761
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3.2. Shannons entropy objective approach for weight determination
Let us assume, following (Sanchez, & Soyer,1998), that Cj = (a1j, a2j,,amj ) denotes a priority
vector according to a certain criterion, after arithmetic normalization (so that the vector's co-
ordinates sum up to 1).
Entropy for this vector may be defined as:
1
( ) ln(a ), i=1,mn
j ij ij
j
H a
(1)
In information theory the entropy H (wj) can be defined as a measure of uncertainty of a discrete
random variable X, which can take a value from the finite set (x1, x2, ., xn) such that probability
that X is going to be equal to xj is wj and can be denoted as P (X =xj) = wj.
In the multi-criteria context, entropy can be used as well for determination of priority of alternative,
and then the priority pi (Uden & Kwiesielewicz, 2003), calculated according to analogous
procedure given for weights wj, can be interpreted as the probability that the i th alternative will be
preferred by the decision-maker.
One of the most important properties of entropy is that:
H(X ) 0 (2)
The entropy of a discrete distribution with finite support has always nonnegative value. It can be
equal to zero when all elements of the sum in equation (1) are equal to zero, simultaneously. Such a
case is possible only when one value of a discrete random variable has the probability equal to one
and all the other values have probabilities equal to zero. Described situation is the one that there is
no uncertainty about which value random variable will take, because probability equal to one means
certain event.
Entropy achieves the maximum value of the uniform distribution given by following equation:
H(1/ n,,1/ n) = ln n (3)
This characteristic of entropy is consistent with the definition of entropy as a measure of
uncertainty, i.e. the maximum is reached when all values of random variable X have equal
probability.
Based on previously described procedure authors of this paper are determined weights of criteria,
and this resulta are presented in Table 5.
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Table 5. Relative importance of criteria determined by objective approach
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12
Entropy Results 0.139 0.127 0.063 0.038 0.011 0.101 0.038 0.076 0.139 0.038 0.025 0.101
Weights Based on
Entropy 0.155 0.142 0.070 0.042 0.012 0.113 0.042 0.085 0.155 0.042 0.028 0.113
4. Results and discussion
Integration procedure for weights determination in which are included AHP and entropy method
involves the following steps:
1. determine the ratio ratio between the objective and subjective weights determined using the
following formula:
, 0 2 and : 2, 2
obj
j
j jsub
j
(4)
If the subjective perception of decision maker is perfectly rational, or if an objective approach to
determining the importance of criteria to the greatest extent respects the personal opinion of the
decision maker, then the value of a ratio is equal to 1, or weights obtained by the these two
approaches are equivalent.
2. determine of an integrated solution that includes a subjective preference of the decision maker
but also an objective set of model parameters through the formula:
int
(2 )
2
sub obj
j j j
j
(5)
3. normalization of the integrated solution by formula:
int
int
1
j
j n
j
j
(6)
One block scheme for the described algorithm can be given in the for which authors are presented
in Figure 2.
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Figure 2. Proposed algorithm's scheme
Based on the results obtained by subjective and objective approach presented in Table 6, and
according to Step 1 of proposed integration algorithm, values of ratio are given in Table 7.
Table 6. Weights determined by subjective and objective approach
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12
Subjective
Weights
( )
0.192 0.091 0.061 0.046 0.015 0.075 0.031 0.112 0.189 0.036 0.022 0.130
Objective
Weights
( )
0.155 0.142 0.070 0.042 0.012 0.113 0.042 0.085 0.155 0.042 0.028 0.113
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Table 7.
Values of ratio
1 2 3 4 5 6 7 8 9 10 11 12
0.8098 1.5573 1.1552 0.9214 0.8143 1.5050 1.3855 0.7560 0.8200 1.1752 1.2566 0.8653
Results according to Step 2 and Step 3 are presented in Table 8.
Table 8. Weights determined by new integrated approach
C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12
Results of
Integration
0.339
8
0.204
5
0.129
7
0.088
2
0.026
8
0.168
5
0.068
5
0.190
3
0.338
2
0.077
4
0.048
6
0.240
6
Integrated
Weights
( )
0.176
9
0.106
4
0.067
5
0.045
9
0.014
0
0.087
7
0.035
6
0.099
1
0.176
0
0.040
3
0.025
3
0.125
3
5. Conclusions
The basis of research starts from the need to determine the relative importance of criteria in the
problems of multi-criteria decision analysis. Namely, for making timely and correct decisions, it is
crucial to determine the weights of the criteria relevant for the given problem. For this purpose has
been developed a range of methodologies for determining the weight coefficients, which represent
support for the formation of multiple criteria analysis model that realistically represents the
relationship between the criteria in the model.
All these methodologies can be classified into two main groups - the subjective and objective. Both
approaches have their advantages and disadvantages, but what is most important is that only the
integration of both approaches can get weights that take into account both components and
therefore, the resulting weights can be considered more realistic.
This paper presents the new algorithm for the integration of subjective and objective approach to
determining the weights.
Also, the algorithm is applied in solving real, contemporary problem of the Serbian economy.
Current engagement of authors is focused on the implementation of this procedure in a model that
includes all certified municipalities in Serbia, not only the sample as shown in this paper.
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Acknowledgement
This work was supported by the Ministry of Education, Science and Technological Development of
Republic of Serbia, Project No III44007. As well, authors would like to thank to the National
Alliance for Local Economic Development, Republic of Serbia for provided assistance and
information necessary for this research.
References
Bindu, R. S. & Ahuja, B. B. (2010). Vendor Selection In Supply Chain Using Relative Reliability
Risk Evaluation. Journal of Theoretical and Applied Information Technology,16(2),145-152.
Charnes, A., Cooper, W.W. & Rhodes, E. (1978). Measuring the Efficiency of Decision Making
Units. European Journal of Operational Research, 12(6), 429-444.
Chen, Y. & Qu, L. (2006). Evaluating the Selection of Logistics Centre Location Using Fuzzy
MCDM Model Based on Entropy Weight. Dalian, China: WCICA 2006. Retrieved from website
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1714468
Chu, A. T. W., Kalaba, R. E. & Spingarn, K. (1979). A comparison of two methods for determining
the weights of belonging to fuzzy sets. Journal of Optimisation Theory and Application, 27,
531-538.
Fan, Z., Ma, J. & Tian, P. (1999). A subjective and objective integrated approach for the
determination of attribute weights. European Journal of Operational Research, 112 (2), 397-40
Hamidi, N., Pezeshki, P. M. & Moradian, A. (2010). Weighting the Criteria of Brand Selecting in
Beverage Industries in Iran. Asian Journal of Management Research, 1(1),250-267.
Hwang, C. L.,& Yoon, K. (1981). Multiple Attribute Decision Making: Methods and Applications.
Berlin, Germany: Springer.
Hwang, C.L. & Lin, M.J. (1987). Group Decision Making Under Multiple Criteria: Methods and
Applications. Berlin, Germany: Springer.
Leskinen, P. (2000). Measurement scales and scale independence in the Analytic Hierarchy Process.
Journal of MultiCriteriecision Analysis, 9,163174.
Liu, C.C. (2003). Simulating Weights Restrictions in Data Envelopment Analysis by the Subjective
and Objective Integrated Approach. Web Journal of Chinese Management Review, 6(1), 68-78.
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1714468
-
Ma, J.,Fan,Z.P. & Huang, L.H. (1999). A Subjective and Objective Integrated Approach to
Determine Attribute Weights. European Journal of Operational Research, 112, 397-404.
Ma, J. & Zhang, Q. (1991). 9/9-9/1 scale method of the AHP. Proceedings of 2nd International
Symposium on the AHP (Vol. 1, pp. 197-202). Pittsburgh, USA: RWS Publications
Saaty, T. L. (1977). A scaling method for priorities in hierarchical structures. Journal of
Mathematical Psychology, 15, 234-281.
Saaty,T.L. (1980). Multicriteria decision making: the Analytic Hierarchy Process. New York,
USA: McGraw-Hill.
Saaty, T.L. (1986). How to make a decision: The analytic hierarchy process. Interfaces, 24 (6),
1943.
Saaty, T.L., & Vargas, L. (1994). Decision Making in Economic, Political, Social and
Technological Environments with The Analytic Hierarchy Process. Pittsburgh, USA: RWS
Publications
Saaty, T. L. (2001). Decision Making for Leaders: The Analytic Hierarchy Process for Decisions in
a Complex World. Pittsburgh, USA: RWS Publications.
Sanchez, P.P. & Soyer, R. (1998). Information concepts and pairwise comparison matrices.
Information Processing Letters, 68, 185188.
Srinivasan, V. & Shocker, A.D. (1973). Estimating the weights for multiple attributes in a
composite criterion using pairwise judgments. Psychonometrica, 38 (4), 473-493.
Uden, E. & Kwiesielewicz, M. (2003). Evaluating Attribute Significance in AHP Using Shannon
Entropy, Bali, Indonesia: ISAHP 2003. Retrieved from website
http://www.isahp.org/2003Proceedings/index.htm
Vaidya, O.S. & Kumar, S. (2006). Analytic hierarchy process: An overview of applications.
European Journal of Operational Research, 169 (1), 129.
Wind,Y. & Saaty,T.L. (1980). Marketing applications of the analytic hierarchy process.
Management Sciences, 26 (7), 641658.
Zhou, X., Zhang, F.,Hui, X. & Li, K. (2011). Group decision-making method based on entropy and
experts cluster analysis. Journal of System Engineering and Electronics, 22(3), 468-
http://www.isahp.org/2003Proceedings/index.htm