Risk assessment of construction projects/Statybos projektu rizikos vertinimas.
Zavadskas, Edmundas Kazimieras ; Turskis, Zenonas ; Tamosaitiene, Jolanta 等
1. Introduction
The risk factor in construction business is very high. Construction
objects are unique and built only once. Construction objects life cycle
is full of various risks. Risks come from many sources: temporary
project team that is collected from different companies, construction
site, etc. Moreover, the size and complexity of construction objects are
increasing which adds to the risks. This is in addition to the
political, economic, social conditions where the object is to be
undertaken. Object risk can be defined as an uncertain event or
condition that, if it occurs, has a positive or negative effect on at
least one project objective, such as time, cost, quality (Project
Management Institute Standards Committee 2004). The risks cause cost and
time overruns in construction projects.
2. Description of the risk assessment model
Risk management is activity process about defining sources of
uncertainty (risk identification), estimating the consequences of
uncertain events/conditions (risk analysis), generating response
strategies in the light of expected outcomes and finally, based on the
feedback received on actual outcomes and risks emerged, carrying out
identification, analysis and response generation steps repetitively
throughout the life cycle of an object to ensure that the project
objectives are met. Risk management in construction is a tedious task as
the objective functions tend to change during the object life cycle
(Dikmen et al. 2008). Tserng et al. (2009) presented a study of
ontology-based risk management framework of construction projects
through project life cycle variance--covariance. Risk value model for
currency market is presented by Aniunas et al. (2009). Isaac and Navon
(2009) described models of building projects as a basis for change
control. Risk management processes of construction project describe the
work of all project life cycle. The risk assessment problem is analysed
by many authors (Shevchenko et al. 2008; Suhobokov 2008; Zavadskas et
al. 2008a; Zavadskas and Vaidogas 2008; Schieg 2008, 2009; Sarka et al.
2008). Proper risk allocation in construction contracts has come to
assume prominence because risk identification and risk allocation have a
clear bearing on risk handling decisions (Perera et al. 2009).
Hassanein and Afify (2007) analysed risk indentification procedure
for construction contracts. Albert et al. (2008) pointed on the
investigated risk assessment. El-Sayegh (2008) presented risk assessment
and allocation problem, Han et al. (2008) described web-based integrated
system, Gao (2009) presented strategies with the risk adjustment. Graves
and Ringuest (2009) analysed probabilistic dominance criteria for
comparing uncertain alternatives. Lahdelma et al. (2009) investigated
uncertainties in multi-criteria decision problems.
Cost-effective solutions that meet the performance criteria can be
achieved, especially if the principle of whole-life costing is being
adopted (Straub 2009). The risk management in construction object's
life cycle stages can be divided into: macro, meso and micro levels
(Fig. 1). The project life cycle includes five steps of process
management:
-- Initiating;
-- Planning;
-- Executing;
-- Monitoring and Controlling;
-- Closing.
The process of risk management can be divided into three stages:
-- Identification;
-- Analysis;
-- Control.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
The risk management process in construction is extreme and
important. Risk measure includes risk level determination of each
objective and the risk analysis estimation by applying various
approaches and technology. Risk control process evaluates performance of
risk control.
2.1. Risk identification
Risk identification is the first and main step of risk management
process. It is describing the competitiveness conditions and the
clarification of risk and uncertainty factors (Rutkauskas 2008; Zayed et
al. 2008), recognition of potential sources of risk and uncertainty
event responsibilities. The project risks can be divided into three
groups:
-- External;
-- Project;
-- Internal.
External risks are those risks that are beyond the control of the
project management team. Internal risks can be divided according to the
party who might be the originator of risk events such as stakeholders,
designer, contractor, etc. There are various classification ways of risk
management methods.
The risk allocation structure of construction objects is presented
in Fig. 2.
External risks (environmental criteria):
-- Political risk;
-- Economic risk;
-- Social risk;
-- Weather risk.
Political risks. There are changes in government laws of
legislative system, regulations and policy and improper administration
system, etc. (Li and Liao 2007).
Economic risks. There is inconstancy of economy in the country,
repayment situation in manufacture sphere, inflation and funding.
Considering the current economic situation, this result can be
reasonably expected. Tvaronaviciene and Grybaite (2007) analysed
Lithuanian economic activities in construction. Economic disasters, as
referred to herein, are periodic economic disasters of such magnitude
that a contractor could not properly assess either their probability or
their cost impact.
Social risks are the growing importance to any effort at risk
allocation. It is an area in which political and social pressures from
parties having little interest in a project but having a great impact on
such a project greatly influence its outcome. The impact of the
financial aid on social and economic development of the region is
analysed by Ginevicius and Podvezko (2009), risk communication in
organizations is analysed by Conchie and Burns (2008).
Weather risk. Except for extremely abnormal conditions, it is a
risk for the contractor to assume, as its impact on construction methods
can by assessed by the contractor.
Project risks (construction process criteria):
-- Time risk;
-- Cost risk;
-- Work quality;
-- Construction risk;
-- Technological risk.
Time risk can be determined by appraisal of the delay at
construction, technology and for all works.
Cost risk. The cost of opportunity product rises due to neglecting
of management (Zavadskas et al. 2008a).
Work quality. Deflective work is considered a significant risk
factor in this category because not only does it result in construction
delays and additional cost to the contractor but it easily leads to
disputes on the liability for the deflection.
Construction risk. The risks are involved in construction delay,
changes in the work and construction technology.
Technological risk. Designing errors; lack of technologies;
management errors; shortage of the qualified labour.
Internal risks (intrinsic criteria):
-- Resource risk;
-- Project member risk;
-- Construction site risk;
-- Documents and information risk.
Resource risk. Materials and equipment involve considerable risks.
The availability and productivity of the resources necessary to
construct the project are risks which are proper for the contractor to
assume (Fisk 2003).
Project member risk. Team risk refers to issues associated with the
project team members, which can increase the uncertainty of a
project's outcome, such as team member turnover, staffing build up,
insufficient knowledge among team members, cooperation, motivation, and
team communication issues.
Stakeholders' risks rightfully belong to the stakeholder alone
and should be retained by stakeholders except to the extent that they
are influenced by construction methods determined by the contractor, or
created by suppliers controlled by the contractor. Stakeholders'
influence on the external environment is analysed by Mitkus and Sostak
(2008).
Designers risk. The expansion of construction has placed great
burdens upon the design professions. Maintaining performance standards
in the face of this is quite difficult, and occasionally, design or
specification deflections occur that create construction problems.
Design failures or constructability errors are becoming more and
apparent, and the architect should bear the true cost of such failures.
Contractor risk. The prime or general contractors are in the best
position to assess the capacity of their subcontractors, and therefore
it is they who should bear the risk of not assessing the risk properly.
Subcontractor risk is that is properly assumed by the contractor
except where it arises from one of the other listed risks attributable
to stakeholder or architect (Fisk, 2003).
Suppliers risk. Default from obligations of the supplier (Fisk
2003).
Team risk. Team risk refers to issues associated with the project
team members that can increase the uncertainty of a project's
outcome, such as team member turnover, staffing build up, insufficient
knowledge among team members, cooperation, motivation, and team
communication issues. Working team must analyse the business activities
of all alliance members and identify various risk factors in business
activities and their characters (Gunstone 2003; Li and Liao 2007; Li et
al. 2007).
Construction site risk. Accident exposures in workplace are
inherent in the nature of the work and are best assessed by the
contractors and their insurance and safety advisors (Fisk 2003).
Documents and information risk assumes: contradiction in documents;
pretermission; legal and communication. Changed order negotiation and
delayed dispute resolution are significant risks during project
construction. Communication is very important at all construction period
and after finishing construction work.
Protracted negotiation on disputes or valuation of changed work is
undesirable to most contractors.
Connections of the contractors with subcontractors and suppliers
are analysed by Mitkus and Trinkuniene (2008).
2.2. Risk analysis and control
Risk analysis. Risk and uncertainty rating identifies the
importance of the sources of risk and uncertainty about the goals of the
project. Risk assessment is accomplished by estimating the probability
of occurrence and severity of risk impact.
More detailed information available in the construction process can
be effectively used for traditional risk management schemes such as risk
control. Risk control can be described as the five-stage process (Han et
al. 2008):
-- Identification;
-- Analysis;
-- Evaluation;
-- Response;
-- Monitoring.
[FIGURE 3 OMITTED]
Risk control establishes a plan, which reduces or eliminates
sources of risk and uncertainty impact on the project's deployment.
Options available for mitigation are:
-- Commercial insurance;
-- Self- insurance;
-- Merger and diversification.
Decision making model of risk assessment is shown in Fig. 3.
This model must be filled at every turn of risk management process.
3. Grey research methodology of risk assessment
3.1. Grey system theory
Deng (1982) developed the Grey system theory. Grey relational
analysis possesses advantages (Deng 1988, 1989):
-- involves simple calculations,
-- requires smaller samples,
-- a typical distribution of samples is not needed,
-- the quantified outcomes from the Grey relational grade do not
result in contradictory conclusions to qualitative analysis and
-- the Grey relational grade model is a transfer functional model
that is effective in dealing with discrete data.
The risk assessment always deals with future and values of criteria
cannot be expressed exactly. This multi-criteria decision-making problem
can be determined not with exact criteria values, but with fuzzy values
or with values at some intervals (Fig. 4).
[FIGURE 4 OMITTED]
The Deng's grey numbers are given as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)
The use of grey relational analysis in solving multiple attribute
decision-making problems is analysed by Kuo et al. (2008) and Cakir
(2008).
Grey theory was applied in evaluating of national economic strength
(Lin and Liu 2007), selection of an ERP system and intelligent sensors
(Yang et al. 2007) and in the assessment of information security (Shen
et al. 2009).
3.2. MADM methods for solving the problem
Multiple attributes decision aid provides several powerful solution
tools (Hwang and Yoon 1981; Figueira et al. 2005) for confronting
sorting the problems. There can be used very simplified techniques for
the evaluation such as the Simple Additive Weighting--SAW; TOPSIS
Technique for Order Preference by Similarity to Ideal Solution (Hwang
and Yoon 1981).
A more detailed survey of multi-attribute decision-making methods
in the construction context is presented by many authors. Zavadskas and
Antucheviciene (2006) presented construction objects renewal modelling
by applying multi-attribute evaluation of rural buildings'
regeneration alternatives; the multi-alternative design and multiple
criteria analysis of the life cycle of a building is described by
Banaitiene et al. (2008); selection of the effective dwelling house
walls by applying attribute values determined at intervals is described
by Zavadskas et al. (2008c).
The purpose is to be achieved by using attributes of effectiveness,
which have different dimensions, different significances as well as
different directions of optimization (Kendall 1970; Zavadskas 1987). The
discrete criteria values can be normalized by applying different
normalization methods (Zavadskas and Turskis 2008; Ginevicius 2008). The
purpose of analysis also can be different (Kaklauskas et al. 2007;
Ginevicius et al. 2007; Bregar et al. 2008). Multiple criteria decision
aid (Hwang and Yoon 1981) provides several powerful and effective tools
(Figueira et al. 2005; Zavadskas et al. 2008b; Dzemyda et al. 2007;
Ginevicius et al. 2008a, b; Ginevicius and Podvezko 2009) for
confronting sorting the problems.
There is a wide range of methods (Ulubeyli and Kazaz 2009;
Jakimavicius and Burinskiene 2009a, b; Plebankiewicz 2009; Liaudanskiene
et al. 2009; Liu 2009; Dytczak and Ginda 2009; Podvezko 2009) based on
multi-criteria utility theory: SAW--Simple Additive Weighting
(Ginevicius et al. 2008b); SAW-G (Zavadskas et al. 2010);
MOORA--Multi-Objective Optimization on the basis of Ratio Analysis
(Brauers and Zavadskas 2006; Brauers et al. 2007, 2008a, b; Kalibatas
and Turskis 2008); TOPSIS--Technique for Order Preference by Similarity
to Ideal Solution (Hwang and Yoon 1981); VIKOR--compromise ranking
method (Opricovic and Tseng 2004); COPRAS--COmplex PRoportional
ASsessment (Zavadskas and Kaklauskas 1996); Game theory methods
(Zavadskas and Turskis 2008; Peldschus 2008, 2009; Ginevicius and Krivka
2008; Turskis et al. 2009) and other methods (Turskis 2008).
TOPSIS is a method to identify solutions from a finite set of
alternatives based upon simultaneous minimization of distance from an
ideal point and maximization of distance from a negative ideal point.
The TOPSIS method was developed by Hwang and Yoon (1981). The only
subjective input needed is relative weights of attributes. An extension
of TOPSIS for group decision making is analysed by Shih et al. (2007)
and incremental analysis of MCDM with an application to group TOPSIS is
developed by Shih (2008). Lin et al. (2008) applied TOPSIS method with
grey number operations.
The COPRAS method determines a solution with the ratio to the ideal
solution and the ratio with the ideal-worst solution. Zavadskas et al.
(2008c, d) applied COPRAS-G method with grey number operations to the
problem with uncertain information.
The algorithm of problem solution applying TOPSIS grey and COPRAS-G
methods is presented in Fig. 5. Either method is applicable to the
solution of problems in construction: Lin et al. (2008) applied TOPSIS
method with grey number operations to the contractor selection problem
solution with uncertain information. Zavadskas et al. (2008c) applied
COPRAS-G method with grey number operations to the selection of the
effective dwelling house walls problem with uncertain information,
Zavadskas et al. (2009).
[FIGURE 5 OMITTED]
3.2.1. TOPSIS method with attributes values determined at intervals
The TOPSIS method is one of the best described mathematically and
not simple for practical using. Lin et al. (2008) proposed the model of
TOPSIS method with attributes values determined at intervals which
includes the following steps:
Step 1: Selecting the set of the most important attributes,
describing the alternatives;
Step 2: Constructing the decision-making matrix [cross product]X.
Grey number matrix [cross product]X can be defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where [cross product] [x.sup.k.sub.ij] denotes the grey evaluations
of the i-th alternative with respect to the j-th attribute by decision
maker k(k = 1, ..., K); [[cross product] [x.sup.k.sub.i1], [cross
product][x.sup.k.sub.i2], ..., [cross product][x.sup.k.sub.im]] is the
grey number evaluation series of the i-th alternative given by decision
maker k. It is noted that there should be K grey decision matrices for
the K members of the group.
Step 3: Establish the weights of the attributes q .
Step 4: Construct the normalized grey decision matrices:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)
On the other hand, the normalization of the smaller-the-better type
attribute can be calculated as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)
Step 5: Determining weights of the attributes [q.sub.j].
Step 6: Construct the grey weighted normalized decision making
matrix.
Step 7: Determine the positive and negative ideal alternatives for
each decision maker. The positive ideal alternative--[A.sup.k+], and the
negative ideal alternative--[A.sup.k-], of decision maker k can be
defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)
Step 8: Calculate the separation measure of the positive and
negative ideal alternatives, [d.sup.k+.sub.i] and [d.sup.k-.sub.i], for
the group. There are two sub-steps to be considered: the first one
concerns the separation measure for individuals; the second one
aggregates their measures for the group.
Step 8.1: Calculate the measures of the positive and negative ideal
alternatives individually. For decision maker k, the separation measures
of the positive ideal alternative--[d.sup.k+.sub.i] and negative ideal
alternative--[d.sup.k-.sub.i] are computed through weighted grey number
as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
In equations (6) and (7), for p [greater than or equal to] 1 and
integer, is the weight for the attribute j which can be determined by
attributes' weight determination methods. If p = 2 , then the
metric is a weighted grey number Euclidean distance function. Equations
(6) and (7) will be as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)
Step 8.2: Aggregate the measures for the group. The group
separation measure of each alternative will be aggregated through an
operation, [cross product] for all decision makers. Thus, the two group
measures of the positive and negative ideal alternatives:
[d.sup.*+.sub.i] and [d.sup.*-.sub.i], respectively, are the following
two equations:
[d.sup.*+.sub.i] = [cross product][d.sup.1+.sub.i] ..., [cross
product][d.sup.K+.sub.i] for alternative i, and (11)
[d.sup.*-.sub.i] = [cross product[d.sup.1-.sub.i] for alternative
i. (12)
Geometric mean is adopted, and the group measures of each
alternative will be:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for
alternative i, (13)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for
alternative i. (14)
Step 9: Calculate the relative closeness [C.sup.*+.sub.i], to the
positive ideal alternative for the group. The aggregation of relative
closeness for the i-th alternative with respect to the positive ideal
alternative of the group can be expressed as:
[C.sup.*+.sub.i] = [d.sup.*-.sub.i]/[d.sup.*+.sub.i] +
[d.sup.*-.sub.i], (15)
where 0 [less than or equal to] [C.sup.*+.sub.i] [less than or
equal to] 1. The larger the index value is, the better evaluation of
alternative will be.
Step 10: Rank the preference order. A set of alternatives now can
be ranked by the descending order of the value of [C.sup.*+.sub.i].
3.2.2. COPRAS-G method with attributes values determined at
intervals
The procedure of using the COPRAS-G method includes the following
steps:
Step 1: Selecting the set of the most important attributes,
describing the alternatives;
Step 2: Constructing the grey decision-making matrix [cross
product] X :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)
where [cross product][x.sub.ij] is determined by [w.sub.ij] and
[b.sub.ij].
Step 3: Establishing the weights of the attributes [q.sub.j].
Step 4: Normalizing the decision-making matrix [cross product]X:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)
In formula (17), [w.sub.ij] is the lower value of the j attribute
in the alternative i of the solution; [b.sub.ij] is the upper value of
the attribute j in the alternative i of the solution; m is the number of
attributes; n is the number of the alternatives compared.
Then, the decision-making matrix is normalized:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
Step 5: Determining weights of the attributes [q.sub.j].
Step 6: Calculating the weighted normalized decision matrix [cross
product] X. The weighted normalized values [cross product][[??].sub.ij]
are calculated as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)
In formula (19), [q.sub.j] is the weight of the j-th attribute.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
Step 7: Calculating the sums [P.sub.i] of the attribute values,
whose larger values are more preferable, for each alternative:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)
Step 8: Calculating the sums [R.sub.i] of attribute values, whose
smaller values are more preferable, for each alternative:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)
In formula (22), (m-k) is the number of attributes which must be
minimized.
The sum of all [R.sub.i] and [P.sub.i] equals 1.
[[summation].sup.n.sub.i=1] [P.sub.i] +
[[zummation].sujp.n.sub.i=1] [R.sub.i] = 1. (23)
Step 9: Calculating the relative weight of each alternative
[Q.sub.i]:
[Q.sub.i] = [P.sub.i] + [n.summation over
(i=1)][R.sub.i]/[R.sub.i][n.summation over (i=1)]1/[R.sub.i] (24)
Step 9 *: If all attributes should be minimized then [P.sub.i] = 0
and [[summation].sup.n.sub.j=1] [R.sub.i] = 1. The formula (24) can by
written as follows:
[Q.sub.i] = 1/[R.sub.i] x [[summation].supl.n.sub.i=1] 1/[R.sub.i]
(25)
Step 10: Determining the optimality criterion L :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)
Step 11: Determining the priority of the project.
Step 12: Calculating the utility degree of each alternative:
[N.sub.i] = [Q.suib.i]/L. (27)
3.3. Establishing the general solution
There are two different multi-attribute decision-making
methods presented: TOPSIS grey and COPRAS-G. Solution results for
the problem under investigation are obtained compared to solution
results and generated aggregated results of problem solution.
4. Case study: risk assessment of construction projects
Application of different solution methods sometimes yields
different results. It is recommended to use several multi-attribute
decision making methods for real problem solution and compare the
results.
Due to a lack of information the attributes were determined at
intervals. The TOPSIS method with attributes values determined at
intervals and COPRAS-G method were applied to construction objects risk
assessment of small-scale objects in construction. Risk assessment of
four small-scale objects was made by 3 experts. The small-scale objects
are of different design, architecture, construction technology, area,
different number of floors and they are in different sites of the
Vilnius region.
The initial decision making data are presented in Table 1. In Table
1 [q.sub.j] is the attribute weight and alternative objects are
[v.sub.1], ..., [v.sub.4].
To determine the weights of the attributes, the experts'
judgment method is applied (Kendall 1970), which has been successfully
used in research by the authors since 1987 (Zavadskas 1987). In order to
establish the weights, a survey has been carried out and 43 experts have
been questioned. These experts, basing their answers on their knowledge,
experience and intuition, had to rate attributes of effectiveness
starting with the most important ones. The rating was done on a scale
from 1 to 13, where 13 meant "very important" and 1 "not
important at all". The weights of attributes were established
according to the rating methods (Zavadskas 1987) of these experts and
also demonstrated the priorities of the user (stakeholder). The weights
of the attributes obtained by this method are presented in Table 1. All
risks in construction should be as minimal as possible--optimization
direction is minimum. In Table 1 data on the following attributes are
presented:
a) External risk assessment:
[cross product] [x.sub.1]--political,
[cross product] [x.sub.2]--economic,
[cross product] [x.sub.3]--social,
[cross product] [x.sub.4]--weather;
b) Project risk assessment:
[cross product] [x.sub.5]--time,
[cross product] [x.sub.6]--cost,
[cross product] [x.sub.7]--quality,
[cross product] [x.sub.8]--technological,
[cross product] [x.sub.9]--construction;
c) Internal risk assessment:
[cross product] [x.sub.10]--resource,
[cross product] [x.sub.11]--project member,
[cross product] [x.sub.12]--site,
[cross product] [x.sub.13]--documents and information.
Each attribute is given zero to ten score. Every expert is allowed
to give grey number evaluations.
In Table 2 the normalised decision-making matrix is presented with
value of each attribute expressed at intervals, for the calculation of
both: TOPSIS grey and COPRAS-G methods. Fig. 6 is a graphic view showing
the calculation results according to TOPSIS grey method. The calculation
results according to COPRAS-G method are presented in Fig. 7. Fig. 8 is
a graphic view showing the aggregated results.
The calculation results for each project are presented in Table 3.
Overall least risk according to calculation results by applying
TOPSIS grey method (Table 3) ranks as follows:
Project 1 [??] Project 3 [??] Project 4 [??] Project 2.
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
The projects risk according to COPRAS-G method ranks as follows:
Project 1 [??] Project 3 [??] Project 2 [??] Project 4.
The calculation results showed that the first project has the least
risk and the second or the fourth project are most risky. The first
alternative was selected and implemented.
5. Conclusions
Decision-making is very important in the construction management,
such as risk assessment results in construction projects, contractor and
supplier selection, etc.
In real life multi-attribute modelling of multi-alternative
assessment problems have some attribute values, which deal with the
future and must be expressed at intervals.
Sometimes calculation according to different methods yields
different results. For decision-making it is reasonable to apply several
methods and select the best alternative according to aggregated results.
This model and solution results are of both practical and
scientific interest. It allows all members of the construction business
to make a decision by evaluating multiple attributes when values of
initial data are given at intervals.
The research results show the different risk levels of construction
objects.
Received 18 Feb. 2009; accepted 23 Nov. 2009
References
Albert, I.; Grenier, E.; Denis, J.-B.; Rousseau, J. 2008.
Quantitative risk assessment from farm to fork and beyond: a global
bayesian approach concerning food-borne diseases, Risk Analysis 28(2):
557-571. doi:10.1111/j.1539-6924.2008.01000.x
Aniunas, P.; Nedzveckas, J.; Krusinskas, R. 2009. Variance
covariance risk value model for currency market, Inzinerine
Ekonomika--Engineering Economics 1: 18-27.
Banaitiene, N.; Banaitis, A.; Kaklauskas, A.; Zavadskas, E. K.
2008. Evaluating the life cycle of building: A multivariant and multiple
criteria approach, Omega--International Journal of Management Science
36(3): 429-441. doi:10.1016/j.omega.2005.10.010
Brauers, W. K.; Zavadskas, E. K.; Peldschus, F.; Turskis, Z. 2008a.
Multi-objective decision-making for road design, Transport 23(3):
183-192. doi:10.3846/1648-4142.2008.23.183-193
Brauers, W. K.; Zavadskas, E. K.; Turskis, Z.; Vilutiene, T. 2008b.
Multi-objective contractor's ranking by applying the MOORA method,
Journal of Business Economics and Management 9(4): 245-255.
doi:10.3846/1611-1699.2008.9.245-255
Brauers, W. K. M.; Ginevicius, R.; Zavadskas, E. K.;
Antucheviciene, J. 2007. The European Union in a transition economy,
Transformation in Business and Economy 6(2): 21-37.
Brauers, W. K.; Zavadskas, E. K. 2006. The MOORA method and its
application to privatization in a transition economy, Control and
Cybernetics 35(2): 443--168.
Bregar, A.; Gyorkos, J.; Juric, M. B. 2008. Interactive
aggregation/disaggregation dichotomic sorting procedure for group
decision analysis based on the threshold model, Informatica 19(2):
161--190.
Cakir, O. 2008. The grey extent analysis, Kybernetes 37(7):
997--1015. doi:10.1108/03684920810884379
Conchie, S. M.; Burns, C. 2008. Trust and risk communication in
high-risk organizations: a test of principles from social risk research,
Risk Analysis 28(1): 141-149. doi:10.1111/j.1539-6924.2008.01006.x
Deng, J. L. 1982. Control problems of grey system, Systems and
Control Letters 1(5): 288--294. doi:10.1016/S0167-6911(82)80025-X
Deng, J. L. 1988. Properties of relational space for grey system,
in Essential Topics on Grey System--Theory and Applications (J. L. Deng,
Ed.). Beijing: China Ocean, 1-13.
Deng, J. L. 1989. Introduction to grey system theory, The Journal
of Grey System 1(1): 1-24.
Dikmen, I.; Birgonul, M. T.; Anac, C.; Tah, J. H. M.; Aouad, G.
2008. Learning from risks: a tool for post-project risk assessment,
Automation in Construction 18(1): 42-50.
doi:10.1016/j.autcon.2008.04.008
Dytczak, M.; Ginda, G. 2009. Identification of building repair
policy choice criteria role, Technological and Economic Development of
Economy 15(2): 213-228. doi:10.3846/1392-8619.2009.15.213-228
Dzemyda, G.; Saltenis, V.; Tiesis, V. 2007. Optimizavimo metodai
[Optimization Methods]. Vilnius: Mokslo aidai (in Lithuanian).
El-Sayegh, S. M. 2008. Risk assessment and allocation in the UAE
construction industry, International Journal of Project Management
26(4): 431-438. doi:10.1016/j.ijproman.2007.07.004
Figueira, J.; Greco, S.; Ehrgott, M. (Eds.). 2005. Multiple
Criteria Decision Analysis: State of the Art Surveys. Springer, Berlin.
Fisk, E. R. 2003. Construction Project Administration. Seventh
Edition. Prentice Hall Upper Saddle River, New Jersey Columbus, Ohio.
Gao, P.-W. 2009. Options strategies with the risk adjustment,
European Journal of Operational Research 192(3): 975-980.
doi:10.1016/j.ejor.2007.10.010
Ginevicius, R.; Podvezko, V. 2009. Evaluating the changes in
economic and social development of Lithuanian counties by multiple
criteria methods, Technological and Economic Development of Economy
15(3): 418--436. doi:10.3846/1392-8619.2009.15.418-436
Ginevicius, R. 2008. Normalization of quantities of various
diensions, Journal of Business Economics and Management 9(1): 79--86.
doi:10.3846/1611-1699.2008.9.79-86
Ginevicius, R.; Krivka, A. 2008. Application of game theory for
duopoly market analysis, Journal of Business Economics and Management
9(3): 207--217. doi:10.3846/1611-1699.2008.9.207-217
Ginevicius, R.; Podvezko, V.; Bruzge, S. 2008a. Evaluating the
effect of state aid to business by multicriteria methods, Journal of
Business Economics and Management 9(3): 167--180.
doi:10.3846/1611-1699.2008.9.167-180
Ginevicius, R.; Podvezko, V.; Raslanas, S. 2008b. Evaluating the
alternative solutions of wall insulation by multicriteria methods,
Journal of Civil Engineering and Management 14(4): 217--226.
doi:10.3846/1392-3730.2008.14.20
Ginevicius, R.; Podvezko, V.; Andruskevicius, A. 2007. Quantitative
evaluation of building technology, International Journal of Technology
Management 40(1/2/3): 192--214.
Graves, S. B.; Ringuest, J. L. 2009. Probabilistic dominance
criteria for comparing uncertain alternatives: A tutorial,
Omega--International Journal of Management Science 37(2): 346--357.
doi:10.1016/j.omega.2007.03.001
Gunstone, S. 2003. Risk assessment and management of patients whom
self-neglect: a 'grey area' for mental health workers, Journal
of Psychiatric and Mental Health Nursing 10(3): 287--296.
doi:10.1046/j.1365-2850.2003.00568.x
Han, S. H.; Kim, D. Y.; Kim, H.; Jang, W.-S. 2008. A web-based
integrated system for international project risk management, Automation
in Construction 17(3): 342-356. doi:10.1016/ j.autcon.2007.05.012
Hassanein, A. A. G.; Afify, H. M. F. 2007. A risk identification
procedure for construction contracts-a case study of power station
projects in Egypt, Civil Engineering and Environmental Systems 24(1):
3--14. doi:10.1080/10286600600910104
Hwang, C. L.; Yoon, K. S. 1981. Multiple Attribute Decision Making
Methods and Applications. Springer-Verlag. Berlin, Heidelberg, New York.
Isaac, S.; Navon, R. 2009. Modeling building projects as a basis
for change control, Automation in Construction 18(5): 656--664.
doi:10.1016/j.autcon.2009.01.001
Jakimavicius, M.; Burinskiene, M. 2009a. Assessment of Vilnius city
development scenarios based on transport system modelling and
multicriteria analysis, Journal of Civil Engineering and Management
15(4): 361-368. doi:10.3846/1392-3730.2009.15.361-368
Jakimavicius, M.; Burinskiene, M. 2009b. A GIS and
multicriteria-based analysis and ranking of transportation zones of
Vilnius city, Technological and Economic Development of Economy 15(1):
39-48. doi:10.3846/1392-8619.2009.15.39-48
Kalibatas, D.; Turskis, Z. 2008. Multicriteria evaluation of inner
climate by using MOORA method, Information Technology and Control 37(1):
7--83.
Kaklauskas, A.; Zavadskas, E. K.; Trinkunas, V. 2007. A multiple
criteria decision support online system for construction, Engineering
Applications of Artificial Intelligence 20(2): 163-175.
doi:10.1016/j.engappai.2006.06.009
Kendall, M. G. 1970. Rank Correlation Methods. 4th ed. London
Griffin.
Kuo, Y.; Yang, T.; Huang, G.-W. 2008. The use of grey relational
analysis in solving multiple attribute decision-making problems,
Computers & Industrial Engineering 55(1): 80-93.
doi:10.1016/j.cie.2007.12.002
Lahdelma, R.; Makkonen, S.; Salminen, P. 2009. Two ways to handle
dependent uncertainties in multi-criteria decision problems,
Omega--International Journal of Management Science 37: 79-92.
doi:10.1016/j.omega.2006.08.005
Li, Y.; Liao, X. 2007. Decision support for risk analysis on
dynamic alliance, Decision Support Systems 42(4): 2043-2059.
doi:10.1016/j.dss.2004.11.008
Li, G.-D.; Yamaguchi, D.; Nagai, M. 2007. A grey-based
decision-making approach to the supplier selection problem, Mathematical
and Computer Modelling 46(3-1): 573-581. doi:10.1016/j.mcm.2006.11.021
Liaudanskiene, R.; Ustinovicius, L.; Bogdanovicius, A. 2009.
Evaluation of construction process safety solutions using the TOPSIS
method, Inzinerine Ekonomika--Engineering Economics (4): 32-40.
Lin, Y.; Liu, S. 2007. National economic strength as evaluated
using grey systems theory, Kybernetes 36(1): 89-97.
doi:10.1108/03684920710741170
Lin, Y.-H.; Lee, P.-C.; Chang, T.-P.; Ting, H.-I. 2008.
Multi-attribute group decision making model under the condition of
uncertain information, Automation in Construction 17(6): 792-797.
doi:10.1016/j.autcon.2008.02.011
Liu, P. 2009. Multi-attribute decision-making research method based
on interval vague set and TOPSIS method, Technological and Economic
Development of Economy 15(3): 453-463.
doi:10.3846/1392-8619.2009.15.453-463
Mitkus, S.; Sostak, O. R. 2008. Modelling the process for defence
of third party rights infringed while implementing construction
investment projects, Technological and Economic Development of Economy
14(2): 208-223. doi:10.3846/1392-8619.2008.14.208-223
Mitkus, S.; Trinkuniene, E. 2008. Reasoned decisions in
construction contracts evaluation, Technological and Economic
Development of Economy 14(3): 402-416.
doi:10.3846/1392-8619.2008.14.402-416
Opricovic, S.; Tzeng, G.-H. 2004. Compromise solution by MCDM
methods: A comparative analysis of VIKOR and TOPSIS, European Journal of
Operational Research 156(2): 445--455. doi:10.1016/S0377-2217(03)00020-1
Peldschus, F. 2009. The analysis of the quality of the results
obtained with the methods of multi-criteria decisions, Technological and
Economic Development of Economy 15(4): 580--592.
doi:10.3846/1392-8619.2009.15.580-592
Peldschus, F. 2008. Experience of the game theory application in
construction management, Technological and Economic Development of
Economy 14(4): 531-545. doi:10.3846/1392-8619.2008.14.531-545
Perera, B. A. K. S.; Dhanasinghe, I.; Rameezdeen, R. 2009. Risk
management in road construction: the case of Sri Lanka, International
Journal of Strategic Property Management 13(2): 87-102.
doi:10.3846/1648-715X.2009.13.87-102
Plebankiewicz, E. 2009. Contractor prequalification model using
fuzzy sets, Journal of Civil Engineering and Management 15(4): 377-385.
doi:10.3846/1392-3730.2009.15.377-385
Podvezko, V. 2009. Application of AHP technique, Journal of
Business Economics and Management 10(2): 181-189.
doi:10.3846/1611-1699.2009.10.181-189
Project Management Institute Standards Committee. A Guide to the
project management body of knowledge. 3rd ed. PMI 2004.
Rutkauskas, A. V. 2008. On the sustainability of regional
competitiveness development considering risk, Technological and Economic
Development of Economy 14(1): 89-99. doi:10.3846/2029-0187.2008.14.89-99
Shen, V. R. L.; Chung, Y.-F.; Chen, T.-S. 2009. A novel application
of grey system theory to information security. Part I, Computer
Standards & Interfaces 31(2): 277-281. doi:10.1016/j.csi.2007.11.018
Shevchenko, G.; Ustinovichius, L.; Andruskevicius, A. 2008.
Multi-attribute analysis of investments risk alternatives in
construction, Technological and Economic Development of Economy 14(3):
428-443. doi:10.3846/1392-8619.2008.14.428-443
Shih, H.-S. 2008. Incremental analysis for MCDM with an application
to group TOPSIS, European Journal of Operational Research 186(2):
720--734. doi:10.1016/j.ejor.2007.02.012
Shih, H.-S.; Shyur, H.-J.; Lee, E. S. 2007. An extension of TOPSIS
for group decision making, Mathematical and Computer Modelling 45(7-8):
801-813. doi:10.1016/j.mcm.2006.03.023
Schieg, M. 2009. Model for integrated project management, Journal
of Business Economics and Management 10(2): 149-160.
doi:10.3846/1611-1699.2009.10.149-160
Schieg, M. 2008. Strategies for avoiding asymmetric information in
construction project management, Journal of Business Economics and
Management 9(1): 47-51. doi:10.3846/1611-1699.2008.9.47-51
Straub, A. 2009. Cost savings from performance-based maintenance
contracting, International Journal of Strategic Property Management
13(3): 205-217. doi:10.3846/1648-715X.2009.13.205-217
Suhobokov, A. 2007. Application of Monte Carlo simulation methods
in risk management, Journal of Business Economics and Management 8(3):
165-168.
Sarka, V.; Zavadskas, E. K.; Ustinovicius, L.; Sarkiene, E.;
Ignatavicius, C. 2008. System of project multicriteria decision
synthesis in construction, Technological and Economic Development of
Economy 14(4): 546--565. doi:10.3846/1392-8619.2008.14.546-565
Turskis, Z.; Zavadskas, E. K.; Peldschus, F. 2009. Multi-criteria
optimization system for decision making in construction design and
management, Inzinerine Ekonomika--Engineering Economics 1:7-17.
Turskis, Z. 2008. Multi-attribute contractors ranking method by
applying ordering of feasible alternatives of solutions in terms of
preferability technique, Technological and Economic Development of
Economy 14(2): 224-239. doi:10.3846/1392-8619.2008.14.224-239
Tvaronaviciene, M.; Grybaite, V. 2007. Impact of FDI on Lithuanian
economy: insight into development of main econoic activities, Journal of
Business Economics and Management 8(4): 285--290.
Tserng, H. P.; Yin, S. Y. L.; Dzeng, R. J.; Wou, B.; Tsai, M. D.;
Chen, W. Y. 2009. A study of ontology-based risk management framework of
construction projects through project life cycle, Automation in
Construction 18(7): 994-1008. doi:10.1016/j.autcon.2009.05.005
Ulubeyli, S.; Kazaz, A. 2009. A multiple criteria decision making
approach to the selection of concrete pumps, Journal of Civil
Engineering and Management 15(4): 369-376.
doi:10.3846/1392-3730.2009.15.369-376
Zayed, T.; Amer, M.; Pan, J. 2008. Assessing risk and uncertainty
inherent in Chinese highway projects using AHP, International Journal of
Project Management 26(4): 408-419. doi:10.1016/j.ijproman.2007.05.012
Zavadskas, E. K.; Vilutiene, T.; Turskis, Z.; Tamosaitiene, J.
2010. Contractor selection for construction works by applying SAW-G and
TOPSIS grey techniques, Journal of Business Economics and Management
11(1): 34-54. doi:10.3846/jbem.2010.03
Zavadskas, E. K.; Kaklauskas, A.; Turskis, Z.; Tamosaitiene, J.
2009. Multi-attribute decision-making model by applying grey numbers,
Informatica 20(2): 305-320.
Zavadskas, E. K.; Turskis, Z. 2008. A new logarithmic normalization
method in games theory, Informatica 19(2): 303-314.
Zavadskas, E. K.; Vaidogas, E. R. 2008. Bayesian reasoning in
managerial decisions on the choice of equipment for the prevention of
industrial accidents, Inzinerine Ekonomika--Engineering Economics 5:
32-10.
Zavadskas, E. K.; Turskis, Z.; Tamosaitiene, J. 2008a. Contractor
selection of construction in a competitive environment, Journal of
Business Economics and Management 9(3): 181-187.
doi:10.3846/1611-1699.2008.9.181-187
Zavadskas, E. K.; Raslanas, S.; Kaklauskas, A. 2008b. The selection
of effective retrofit scenarios for panel houses in urban neighbourhoods
based on expected energy savings and increase in market value: The
Vilnius case, Energy and Buildings 40(4): 573-587.
doi:10.1016/j.enbuild.2007.04.015
Zavadskas, E. K.; Kaklauskas, A.; Turskis, Z.; Tamosaitiene, J.
2008c. Selection of the effective dwelling house walls by applying
attributes values determined at intervals, Journal of Civil Engineering
and Management 14(2): 85-93. doi:10.3846/1392-3730.2008.14.3
Zavadskas, E. K.; Turskis, Z.; Tamosaitiene, J. 2008d. Construction
risk assessment of small scale objects by applying TOPSIS method with
attributes values determined at intervals, Computer Modelling and New
Technologies 12(4): 38-43.
Zavadskas, E. K.; Antucheviciene, J. 2006. Development of an
indicator model and ranking of sustainable revitalization alternatives
of derelict property: a Lithuanian case study, Sustainable Development
14(5): 287-299. doi:10.1002/sd.285
Zavadskas, E. K.; Kaklauskas, A. 1996. Pastatii sistemotechninis
(vertinimas [Multiple criteria evaluation of buildings]. Vilnius:
Technika (in Lithuanian).
Zavadskas, E. K. 1987. Complex estimation and choice of resource
saving decisions in construction. Vilnius: Mokslas (in Russian).
Yang, J.-B.; Wu, C.-T.; Tsai, C.-H. 2007. Selection of an ERP
system for a construction firm in Taiwan: A case study, Automation in
Construction 16(6): 787-796. doi:10.1016/j.autcon.2007.02.001
Yamaguchi, D.; Li, G.-D.; Chen, L.-C.; Nagai, M. 2007. Reviewing
Crisp, Fuzzy, Grey and Rough Mathematical Models, in Proc. of 2007 IEEE
International Conference on Grey Systems and Intelligent Services,
November 18-20, 2007, Nanjing, China, 547-552.
doi: 10.3846/jcem.2010.03
Edmundas Kazimieras Zavadskas (1), Zenonas Turskis (2), Jolanta
Tamosaitiene (3)
Vilnius Gediminas Technical University, Department of Construction
Technology and Management, Sauletekio al. 11, LT-10223 Vilnius,
Lithuania
E-mails: (1)
[email protected]; (2)
[email protected]; (3)
[email protected]
Edmundas Kazimieras ZAVADSKAS. Doctor Habil, the principal
vice-rector of Vilnius Gediminas Technical University and Head of the
Dept of Construction Technology and Management at Vilnius Gediminas
Technical University, Vilnius, Lithuania. He has a PhD in building
structures (1973) and Dr Sc (1987) in building technology and
management. He is a member of the Lithuanian and several foreign
Academies of Sciences. He is doctore honoris causa at Poznan,
Saint-Petersburg, and Kiev. He is a member of international
organisations and has been a member of steering and programme committees
at many international conferences. E. K. Zavadskas is a member of
editorial boards of several research journals. He is the author of over
50 books published in Lithuanian, Russian, German, and English. He has
published more than 350 scientific papers. Research interests are:
building technology and management, decision-making theory, grey
relation, automation in design and decision support systems.
Zenonas TURSKIS. Doctor, a senior research fellow of Construction
Technology and Management Laboratory of Vilnius Gediminas Technical
University, Lithuania. He is the author of 1 book published in
Lithuanian and 30 scientific papers. His research interests include
building technology and management, decision-making in construction,
computer- aided automation in design and expert systems.
Jolanta TAMOSAITIENE. Doctor, Dean assistant of Civil Engineering
Faculty, Vilnius Gediminas Technical University, reader of the Dept of
Construction Technology and Management, Vilnius Gediminas Technical
University, Lithuania. BSc degree (building technology and management),
Vilnius Gediminas Technical University (2000). MSc degree (Building
management and economics), Vilnius Gediminas Technical University
(2002). She published 9 scientific papers. Research interests:
construction technology and organisation, construction project
administration, decision- making and grey theory.
Table 1. Initial decision-making matrix
with values at some intervals
Weight
Attribute q j
[cross [[w.sub.1]; 0.05
product] [b.sub.1]]
[x.sup.1]
[cross [[w.sub.2]; 0.09
product] [b.sub.2]]
[x.sup.2]
[cross [[w.sub.3]; 0.06
product] [b.sub.3]]
[x.sup.3]
[cross [[w.sub.4]; 0.04
product] [b.sub.4]]
[x.sup.4]
[cross [[w.sub.5]; 0.09
product] [b.sub.5]]
[x.sup.5]
[cross [[w.sub.6]; 0.11
product] [b.sub.6]]
[x.sup.6]
[cross [[w.sub.7]; 0.12
product] [b.sub.7]]
[x.sup.7]
[cross [[w.sub.8]; 0.07
product] [b.sub.8]]
[x.sup.8]
[cross [[w.sub.9]; 0.09
product] [b.sub.9]]
[x.sup.9]
[cross [[w.sub.10]; 0.06
product] [b.sub.10]]
[x.sup.10]
[cross [[w.sub.11]; 0.11
product] [b.sub.11]]
[x.sup.11]
[cross [[w.sub.12]; 0.04
product] [b.sub.12]]
[x.sup.12]
[cross [[w.sub.13]; 0.07
product] [b.sub.13]]
[x.sup.13]
Expert 1
Attribute Project
[v.sub.1] [v.sub.2] [v.sub.3] [v.sub.4]
[cross [[w.sub.1]; [6.0;7.0] [6.5;7.5] [5.0;5.5] [6.0;6.5]
product] [b.sub.1]]
[x.sup.1]
[cross [[w.sub.2]; [6.0;6.5] [7.0;7.5] [5.0;6.0] [6.0;7.0]
product] [b.sub.2]]
[x.sup.2]
[cross [[w.sub.3]; [6.0;6.5] [5.0;5.5] [4.0;5.0] [5.5;6.0]
product] [b.sub.3]]
[x.sup.3]
[cross [[w.sub.4]; [4.5;5.5] [5.0;6.5] [5.5;7.5] [6.0;6.5]
product] [b.sub.4]]
[x.sup.4]
[cross [[w.sub.5]; [8.0;8.5] [8.5;9.0] [6.0;6.5] [7.0;8.5]
product] [b.sub.5]]
[x.sup.5]
[cross [[w.sub.6]; [7.0;7.5] [8.0;8.5] [4.5;5.0] [8.0;8.5]
product] [b.sub.6]]
[x.sup.6]
[cross [[w.sub.7]; [5.0;5.5] [6.0;6.5] [5.5;7.0] [4.0;6.0]
product] [b.sub.7]]
[x.sup.7]
[cross [[w.sub.8]; [2.0;4.0] [5.0;6.5] [4.5;5.5] [4.0;6.5]
product] [b.sub.8]]
[x.sup.8]
[cross [[w.sub.9]; [8.0;9.0] [7.5;8.0] [7.0;8.5] [5.0;7.5]
product] [b.sub.9]]
[x.sup.9]
[cross [[w.sub.10]; [7.0;7.5] [6.0;7.5] [5.0;6.5] [5.0;6.5]
product] [b.sub.10]]
[x.sup.10]
[cross [[w.sub.11]; [5.0;6.5] [7.0;8.0] [5.5;6.0] [6.0;7.5]
product] [b.sub.11]]
[x.sup.11]
[cross [[w.sub.12]; [7.0;7.5] [4.0;5.5] [6.0;6.5] [5.0;6.0]
product] [b.sub.12]]
[x.sup.12]
[cross [[w.sub.13]; [5.0;6.0] [3.0;4.5] [6.0;7.0] [6.0;6.5]
product] [b.sub.13]]
[x.sup.13]
Expert 2
Attribute Project
[v.sub.1] [v.sub.2] [v.sub.3] [v.sub.4]
[cross [[w.sub.1]; [7.0;8.0] [7.5;8.0] [6.5;8.5] [7.0;8.0]
product] [b.sub.1]]
[x.sup.1]
[cross [[w.sub.2]; [7.0;8.5] [7.5;8.5] [6.0;7.0] [6.5;7.0]
product] [b.sub.2]]
[x.sup.2]
[cross [[w.sub.3]; [8.0;8.5] [6.5;7.5] [5.5;6.5] [8.0;8.5]
product] [b.sub.3]]
[x.sup.3]
[cross [[w.sub.4]; [4.0;5.0] [4.5;5.0] [6.5;7.0] [6.5;7.5]
product] [b.sub.4]]
[x.sup.4]
[cross [[w.sub.5]; [4.0;5.0] [6.0;6.5] [7.0;7.5] [5.0;7.0]
product] [b.sub.5]]
[x.sup.5]
[cross [[w.sub.6]; [6.0;6.5] [7.0;7.5] [5.0;5.5] [7.5;8.0]
product] [b.sub.6]]
[x.sup.6]
[cross [[w.sub.7]; [4.5;5.5] [5.5;7.5] [7.5;8.0] [5.0;6.5]
product] [b.sub.7]]
[x.sup.7]
[cross [[w.sub.8]; [4.0;5.5] [4.0;6.0] [4.0;5.5] [3.5;5.0]
product] [b.sub.8]]
[x.sup.8]
[cross [[w.sub.9]; [7.0;8.5] [6.5;7.0] [7.5;9.0] [6.0;7.0]
product] [b.sub.9]]
[x.sup.9]
[cross [[w.sub.10]; [4.5;6.5] [7.5;8.0] [6.5;7.5] [7.0;8.0]
product] [b.sub.10]]
[x.sup.10]
[cross [[w.sub.11]; [4.0;5.0] [7.0;7.5] [5.0;5.5] [6.5;7.0]
product] [b.sub.11]]
[x.sup.11]
[cross [[w.sub.12]; [5.0;5.5] [7.0;8.0] [7.0;7.5] [8.0;8.5]
product] [b.sub.12]]
[x.sup.12]
[cross [[w.sub.13]; [4.0;5.0] [4.5;5.0] [6.5;7.5] [4.5;5.0]
product] [b.sub.13]]
[x.sup.13]
Expert 3
Attribute Project
[v.sub.1] [v.sub.2] [v.sub.3] [v.sub.4]
[cross [[w.sub.1]; [7.5;8.5] [6.0;8.0] [6.5;8.0] [7.5;9.0]
product] [b.sub.1]]
[x.sup.1]
[cross [[w.sub.2]; [6.5;8.0] [4.0;5.5] [5.5;6.0] [7.0;7.5]
product] [b.sub.2]]
[x.sup.2]
[cross [[w.sub.3]; [7.0;8.5] [7.0;8.0] [5.5;6.5] [5.5;6.5]
product] [b.sub.3]]
[x.sup.3]
[cross [[w.sub.4]; [4.5;5.0] [5.5;6.0] [5.5;7.0] [8.0;8.5]
product] [b.sub.4]]
[x.sup.4]
[cross [[w.sub.5]; [6.5;7.0] [6.0;7.0] [5.5;6.5] [6.0;7.0]
product] [b.sub.5]]
[x.sup.5]
[cross [[w.sub.6]; [8.0;9.0] [7.0;8.0] [5.0;6.0] [7.5;8.5]
product] [b.sub.6]]
[x.sup.6]
[cross [[w.sub.7]; [7.0;7.5] [4.0;5.0] [6.5;7.5] [6.0;7.0]
product] [b.sub.7]]
[x.sup.7]
[cross [[w.sub.8]; [4.0;4.5] [5.5;6.5] [3.5;6.0] [6.0;5.0]
product] [b.sub.8]]
[x.sup.8]
[cross [[w.sub.9]; [6.0;8.0] [7.0;7.5] [6.5;7.0] [7.0;7.5]
product] [b.sub.9]]
[x.sup.9]
[cross [[w.sub.10]; [5.0;6.0] [7.5;8.0] [6.5;8.0] [7.0;7.5]
product] [b.sub.10]]
[x.sup.10]
[cross [[w.sub.11]; [5.0;6.0] [6.5;7.0] [6.0;6.5] [6.5;7.0]
product] [b.sub.11]]
[x.sup.11]
[cross [[w.sub.12]; [7.0;8.0] [5.0;6.0] [6.5;7.0] [6.0;6.5]
product] [b.sub.12]]
[x.sup.12]
[cross [[w.sub.13]; [4.0;5.0] [4.5;5.5] [5.0;5.5] [7.0;7.5]
product] [b.sub.13]]
[x.sup.13]
Table 2. Normalized decision-making matrix
Expert 1
Project
Attribute [v.sub.1] [v.sub.2]
TOPSIS grey method
[cross product] [[bar.w].sub.1]; [0.80;0.60] [0.92;0.50]
[[bar.x].sub.1] [[b.w].sub.1]
[cross product] [[bar.w].sub.2]; [0.80;0.70] [0.60;0.50]
[[bar.x].sub.2] [[b.w].sub.2]
[[bar.w].sub.3]; [0.50;0.38] [0.75;0.63]
[cross product] [[b.w].sub.3]
[[bar.x].sub.3]
[cross product] [[bar.w].sub.4]; [1.00;0.78] [0.89;0.56]
[[bar.x].sub.4] [[b.w].sub.4]
[cross product] [[bar.w].sub.5]; [0.67;0.58] [0.58;0.50]
[[bar.x].sub.5] [[b.w].sub.5]
[cross product] [[bar.w].sub.6]; [0.44;0.33] [0.33;0.11]
[[bar.x].sub.6] [[b.w].sub.6]
[cross product] [[bar.w].sub.7]; [0.75;0.63] [0.50;0.38]
[[bar.x].sub.7] [[b.w].sub.7]
[cross product] [[bar.w].sub.8]; [1.00;0.86] [0.57;0.14]
[[bar.x].sub.8] [[b.w].sub.8]
[cross product] [[bar.w].sub.9]; [0.40;0.20] [0.50;0.40]
[[bar.x].sub.9] [[b.w].sub.9]
[cross product] [[bar.w].sub.10]; [0.60;0.40] [0.80;0.50]
[[bar.x].sub.10] [[b.w].sub.10]
[cross product] [[bar.w].sub.11]; [1.00;0.70] [0.60;0.40]
[[bar.x].sub.11] [[b.w].sub.11]
[cross product] [[bar.w].sub.12]; [0.25;0.13] [1.00;0.63]
[[bar.x].sub.12] [[b.w].sub.12]
[cross product] [[bar.w].sub.13]; [0.75;0.50] [1.00;0.88]
[[bar.x].sub.13] [[b.w].sub.13]
COPRAS-G method
[cross product] [[bar.w].sub.1]; [0.24;0.28] [0.26;0.30]
[[bar.x].sub.1] [[b.w].sub.1]
[cross product] [[bar.w].sub.2]; [0.24;0.25] [0.27;0.29]
[[bar.x].sub.2] [[b.w].sub.2]
[cross product] [[bar.w].sub.3]; [0.28;0.30] [0.23;0.25]
[[bar.x].sub.3] [[b.w].sub.3]
[cross product] [[bar.w].sub.4]; [0.19;0.23] [0.21;0.28]
[[bar.x].sub.4] [[b.w].sub.4]
[cross product] [[bar.w].sub.5]; [0.26;0.27] [0.27;0.29]
[[bar.x].sub.5] [[b.w].sub.5]
[cross product] [[bar.w].sub.6]; [0.25;0.27] [0.27;0.31]
[[bar.x].sub.6] [[b.w].sub.6]
[cross product] [[bar.w].sub.7]; [0.22;0.24] [0.26;0.29]
[[bar.x].sub.7] [[b.w].sub.7]
[cross product] [[bar.w].sub.8]; [0.18;0.20] [0.25;0.33]
[[bar.x].sub.8] [[b.w].sub.8]
[cross product] [[bar.w].sub.9]; [0.26;0.30] [0.25;0.26]
[[bar.x].sub.9] [[b.w].sub.9]
[cross product] [[bar.w].sub.10]; [0.27;0.30] [0.23;0.29]
[[bar.x].sub.10] [[b.w].sub.10]
[cross product] [[bar.w].sub.11]; [0.19;0.25] [0.27;0.31]
[[bar.x].sub.11] [[b.w].sub.11]
[cross product] [[bar.w].sub.12]; [0.29;0.32] [0.17;0.23]
[[bar.x].sub.12] [[b.w].sub.12]
[cross product] [[bar.w].sub.13]; [0.22;0.26] [0.18;0.20]
[[bar.x].sub.13] [[b.w].sub.13]
Expert 1
Project
Attribute [v.sub.3] [v.sub.4]
TOPSIS grey method
[cross product] [[bar.w].sub.1]; [1.00;0.90] [0.80;0.70]
[[bar.x].sub.1] [[b.w].sub.1]
[cross product] [[bar.w].sub.2]; [1.00;0.80] [0.80;0.60]
[[bar.x].sub.2] [[b.w].sub.2]
[[bar.w].sub.3]; [1.00;0.75] [0.63;0.50]
[cross product] [[b.w].sub.3]
[[bar.x].sub.3]
[cross product] [[bar.w].sub.4]; [0.78;0.73] [0.67;0.78]
[[bar.x].sub.4] [[b.w].sub.4]
[cross product] [[bar.w].sub.5]; [1.00;0.92] [0.83;0.58]
[[bar.x].sub.5] [[b.w].sub.5]
[cross product] [[bar.w].sub.6]; [1.00;0.89] [0.44;0.11]
[[bar.x].sub.6] [[b.w].sub.6]
[cross product] [[bar.w].sub.7]; [0.63;0.25] [1.00;0.50]
[[bar.x].sub.7] [[b.w].sub.7]
[cross product] [[bar.w].sub.8]; [0.71;0.43] [0.86;0.14]
[[bar.x].sub.8] [[b.w].sub.8]
[cross product] [[bar.w].sub.9]; [0.60;0.30] [1.00;0.50]
[[bar.x].sub.9] [[b.w].sub.9]
[cross product] [[bar.w].sub.10]; [1.00;0.70] [0.80;0.70]
[[bar.x].sub.10] [[b.w].sub.10]
[cross product] [[bar.w].sub.11]; [0.90;0.80] [0.80;0.50]
[[bar.x].sub.11] [[b.w].sub.11]
[cross product] [[bar.w].sub.12]; [0.50;0.38] [0.86;0.50]
[[bar.x].sub.12] [[b.w].sub.12]
[cross product] [[bar.w].sub.13]; [0.38;0.25] [0.50;0.38]
[[bar.x].sub.13] [[b.w].sub.13]
COPRAS-G method
[cross product] [[bar.w].sub.1]; [0.20;0.22] [0.24;0.26]
[[bar.x].sub.1] [[b.w].sub.1]
[cross product] [[bar.w].sub.2]; [0.20;0.24] [0.24;0.27]
[[bar.x].sub.2] [[b.w].sub.2]
[cross product] [[bar.w].sub.3]; [0.18;0.23] [0.25;0.28]
[[bar.x].sub.3] [[b.w].sub.3]
[cross product] [[bar.w].sub.4]; [0.23;0.32] [0.26;0.28]
[[bar.x].sub.4] [[b.w].sub.4]
[cross product] [[bar.w].sub.5]; [0.19;0.21] [0.23;0.27]
[[bar.x].sub.5] [[b.w].sub.5]
[cross product] [[bar.w].sub.6]; [0.16;0.18] [0.25;0.31]
[[bar.x].sub.6] [[b.w].sub.6]
[cross product] [[bar.w].sub.7]; [0.24;0.31] [0.18;0.26]
[[bar.x].sub.7] [[b.w].sub.7]
[cross product] [[bar.w].sub.8]; [0.23;0.28] [0.20;0.33]
[[bar.x].sub.8] [[b.w].sub.8]
[cross product] [[bar.w].sub.9]; [0.23;0.28] [0.17;0.25]
[[bar.x].sub.9] [[b.w].sub.9]
[cross product] [[bar.w].sub.10]; [0.19;0.25] [0.23;0.25]
[[bar.x].sub.10] [[b.w].sub.10]
[cross product] [[bar.w].sub.11]; [0.21;0.23] [0.23;0.29]
[[bar.x].sub.11] [[b.w].sub.11]
[cross product] [[bar.w].sub.12]; [0.25;0.27] [0.21;0.25]
[[bar.x].sub.12] [[b.w].sub.12]
[cross product] [[bar.w].sub.13]; [0.29;0.31] [0.26;0.29]
[[bar.x].sub.13] [[b.w].sub.13]
Expert 2
Project
Attribute [v.sub.1] [v.sub.2]
TOPSIS grey method
[cross product] [[bar.w].sub.1]; [0.92;0.77] [0.85;0.92]
[[bar.x].sub.1] [[b.w].sub.1]
[cross product] [[bar.w].sub.2]; [0.83;0.67] [0.75;0.58]
[[bar.x].sub.2] [[b.w].sub.2]
[[bar.w].sub.3]; [0.55;0.45] [0.82;0.64]
[cross product] [[b.w].sub.3]
[[bar.x].sub.3]
[cross product] [[bar.w].sub.4]; [1.00;0.75] [0.88;0.75]
[[bar.x].sub.4] [[b.w].sub.4]
[cross product] [[bar.w].sub.5]; [1.00;0.75] [0.50;0.38]
[[bar.x].sub.5] [[b.w].sub.5]
[cross product] [[bar.w].sub.6]; [0.80;0.70] [0.60;0.50]
[[bar.x].sub.6] [[b.w].sub.6]
[cross product] [[bar.w].sub.7]; [1.00;0.78] [0.78;0.33]
[[bar.x].sub.7] [[b.w].sub.7]
[cross product] [[bar.w].sub.8]; [0.86;0.43] [0.86;0.29]
[[bar.x].sub.8] [[b.w].sub.8]
[cross product] [[bar.w].sub.9]; [0.83;0.58] [0.92;0.83]
[[bar.x].sub.9] [[b.w].sub.9]
[cross product] [[bar.w].sub.10]; [1.00;0.56] [0.33;0.22]
[[bar.x].sub.10] [[b.w].sub.10]
[cross product] [[bar.w].sub.11]; [1.00;0.75] [0.25;0.13]
[[bar.x].sub.11] [[b.w].sub.11]
[cross product] [[bar.w].sub.12]; [1.00;0.90] [0.60;0.40]
[[bar.x].sub.12] [[b.w].sub.12]
[cross product] [[bar.w].sub.13]; [1.00;0.75] [0.88;0.75]
[[bar.x].sub.13] [[b.w].sub.13]
COPRAS-G method
[cross product] [[bar.w].sub.1]; [0.23;0.26] [0.25;0.26]
[[bar.x].sub.1] [[b.w].sub.1]
[cross product] [[bar.w].sub.2]; [0.24;0.28] [0.26;0.30]
[[bar.x].sub.2] [[b.w].sub.2]
[cross product] [[bar.w].sub.3]; [0.27;0.29] [0.22;0.25]
[[bar.x].sub.3] [[b.w].sub.3]
[cross product] [[bar.w].sub.4]; [0.17;0.22] [0.20;0.22]
[[bar.x].sub.4] [[b.w].sub.4]
[cross product] [[bar.w].sub.5]; [0.17;0.21] [0.25;0.27]
[[bar.x].sub.5] [[b.w].sub.5]
[cross product] [[bar.w].sub.6]; [0.23;0.25] [0.26;0.28]
[[bar.x].sub.6] [[b.w].sub.6]
[cross product] [[bar.w].sub.7]; [0.18;0.22] [0.22;0.30]
[[bar.x].sub.7] [[b.w].sub.7]
[cross product] [[bar.w].sub.8]; [0.21;0.29] [0.21;0.32]
[[bar.x].sub.8] [[b.w].sub.8]
[cross product] [[bar.w].sub.9]; [0.24;0.29] [0.22;0.24]
[[bar.x].sub.9] [[b.w].sub.9]
[cross product] [[bar.w].sub.10]; [0.16;0.23] [0.27;0.29]
[[bar.x].sub.10] [[b.w].sub.10]
[cross product] [[bar.w].sub.11]; [0.17;0.21] [0.29;0.32]
[[bar.x].sub.11] [[b.w].sub.11]
[cross product] [[bar.w].sub.12]; [0.18;0.19] [0.25;0.28]
[[bar.x].sub.12] [[b.w].sub.12]
[cross product] [[bar.w].sub.13]; [0.19;0.24] [0.21;0.24]
[[bar.x].sub.13] [[b.w].sub.13]
Expert 2
Project
Attribute [v.sub.3] [v.sub.4]
TOPSIS grey method
[cross product] [[bar.w].sub.1]; [1.00;0.69] [0.92;0.77]
[[bar.x].sub.1] [[b.w].sub.1]
[cross product] [[bar.w].sub.2]; [1.00;0.83] [0.92;0.83]
[[bar.x].sub.2] [[b.w].sub.2]
[[bar.w].sub.3]; [1.00;0.82] [0.55;0.45]
[cross product] [[b.w].sub.3]
[[bar.x].sub.3]
[cross product] [[bar.w].sub.4]; [0.38;0.25] [0.38;0.13]
[[bar.x].sub.4] [[b.w].sub.4]
[cross product] [[bar.w].sub.5]; [0.25;0.13] [0.75;0.25]
[[bar.x].sub.5] [[b.w].sub.5]
[cross product] [[bar.w].sub.6]; [0.55;0.90] [0.50;0.40]
[[bar.x].sub.6] [[b.w].sub.6]
[cross product] [[bar.w].sub.7]; [0.33;0.22] [0.89;0.56]
[[bar.x].sub.7] [[b.w].sub.7]
[cross product] [[bar.w].sub.8]; [0.86;0.43] [1.00;0.57]
[[bar.x].sub.8] [[b.w].sub.8]
[cross product] [[bar.w].sub.9]; [0.75;0.50] [0.80;0.82]
[[bar.x].sub.9] [[b.w].sub.9]
[cross product] [[bar.w].sub.10]; [0.56;0.33] [0.44;0.22]
[[bar.x].sub.10] [[b.w].sub.10]
[cross product] [[bar.w].sub.11]; [0.75;0.63] [0.38;0.25]
[[bar.x].sub.11] [[b.w].sub.11]
[cross product] [[bar.w].sub.12]; [0.60;0.50] [0.40;0.30]
[[bar.x].sub.12] [[b.w].sub.12]
[cross product] [[bar.w].sub.13]; [0.38;0.13] [0.88;0.75]
[[bar.x].sub.13] [[b.w].sub.13]
COPRAS-G method
[cross product] [[bar.w].sub.1]; [0.21;0.28] [0.23;0.26]
[[bar.x].sub.1] [[b.w].sub.1]
[cross product] [[bar.w].sub.2]; [0.21;0.24] [0.23;0.24]
[[bar.x].sub.2] [[b.w].sub.2]
[cross product] [[bar.w].sub.3]; [0.19;0.22] [0.27;0.29]
[[bar.x].sub.3] [[b.w].sub.3]
[cross product] [[bar.w].sub.4]; [0.28;0.30] [0.28;0.33]
[[bar.x].sub.4] [[b.w].sub.4]
[cross product] [[bar.w].sub.5]; [0.29;0.31] [0.21;0.29]
[[bar.x].sub.5] [[b.w].sub.5]
[cross product] [[bar.w].sub.6]; [0.19;0.21] [0.28;0.30]
[[bar.x].sub.6] [[b.w].sub.6]
[cross product] [[bar.w].sub.7]; [0.30;0.32] [0.20;0.26]
[[bar.x].sub.7] [[b.w].sub.7]
[cross product] [[bar.w].sub.8]; [0.21;0.29] [0.19;0.27]
[[bar.x].sub.8] [[b.w].sub.8]
[cross product] [[bar.w].sub.9]; [0.26;0.31] [0.21;0.24]
[[bar.x].sub.9] [[b.w].sub.9]
[cross product] [[bar.w].sub.10]; [0.23;0.27] [0.25;0.29]
[[bar.x].sub.10] [[b.w].sub.10]
[cross product] [[bar.w].sub.11]; [0.21;0.23] [0.27;0.29]
[[bar.x].sub.11] [[b.w].sub.11]
[cross product] [[bar.w].sub.12]; [0.25;0.27] [0.28;0.30]
[[bar.x].sub.12] [[b.w].sub.12]
[cross product] [[bar.w].sub.13]; [0.31;0.36] [0.21;0.24]
[[bar.x].sub.13] [[b.w].sub.13]
Expert 3
Project
Attribute [v.sub.1] [v.sub.2]
TOPSIS grey method
[cross product] [[bar.w].sub.1]; [0.75;0.58] [1.00;0.67]
[[bar.x].sub.1] [[b.w].sub.1]
[cross product] [[bar.w].sub.2]; [0.38;0.13] [1.00;0.63]
[[bar.x].sub.2] [[b.w].sub.2]
[[bar.w].sub.3]; [0.60;0.30] [0.60;0.40]
[cross product] [[b.w].sub.3]
[[bar.x].sub.3]
[cross product] [[bar.w].sub.4]; [1.00;0.89] [0.78;0.67]
[[bar.x].sub.4] [[b.w].sub.4]
[cross product] [[bar.w].sub.5]; [0.82;0.73] [0.91;0.64]
[[bar.x].sub.5] [[b.w].sub.5]
[cross product] [[bar.w].sub.6]; [0.40;0.20] [0.60;0.40]
[[bar.x].sub.6] [[b.w].sub.6]
[cross product] [[bar.w].sub.7]; [0.25;0.13] [1.00;0.75]
[[bar.x].sub.7] [[b.w].sub.7]
[cross product] [[bar.w].sub.8]; [0.86;0.71] [0.43;0.14]
[[bar.x].sub.8] [[b.w].sub.8]
[cross product] [[bar.w].sub.9]; [1.00;0.67] [0.83;0.75]
[[bar.x].sub.9] [[b.w].sub.9]
[cross product] [[bar.w].sub.10]; [1.00;0.80] [0.50;0.30]
[[bar.x].sub.10] [[b.w].sub.10]
[cross product] [[bar.w].sub.11]; [1.00;0.80] [0.70;0.60]
[[bar.x].sub.11] [[b.w].sub.11]
[cross product] [[bar.w].sub.12]; [0.60;0.40] [1.00;0.80]
[[bar.x].sub.12] [[b.w].sub.12]
[cross product] [[bar.w].sub.13]; [1.00;0.75] [0.88;0.63]
[[bar.x].sub.13] [[b.w].sub.13]
COPRAS-G method
[cross product] [[bar.w].sub.1]; [0.25;0.28] [0.20;0.26]
[[bar.x].sub.1] [[b.w].sub.1]
[cross product] [[bar.w].sub.2]; [0.26;0.30] [0.16;0.22]
[[bar.x].sub.2] [[b.w].sub.2]
[cross product] [[bar.w].sub.3]; [0.26;0.31] [0.26;0.30]
[[bar.x].sub.3] [[b.w].sub.3]
[cross product] [[bar.w].sub.4]; [0.18;0.20] [0.22;0.24]
[[bar.x].sub.4] [[b.w].sub.4]
[cross product] [[bar.w].sub.5]; [0.25;0.27] [0.23;0.29]
[[bar.x].sub.5] [[b.w].sub.5]
[cross product] [[bar.w].sub.6]; [0.27;0.31] [0.24;0.27]
[[bar.x].sub.6] [[b.w].sub.6]
[cross product] [[bar.w].sub.7]; [0.28;0.30] [0.16;0.20]
[[bar.x].sub.7] [[b.w].sub.7]
[cross product] [[bar.w].sub.8]; [0.20;0.22] [0.27;0.32]
[[bar.x].sub.8] [[b.w].sub.8]
[cross product] [[bar.w].sub.9]; [0.21;0.28] [0.25;0.27]
[[bar.x].sub.9] [[b.w].sub.9]
[cross product] [[bar.w].sub.10]; [0.18;0.21] [0.27;0.30]
[[bar.x].sub.10] [[b.w].sub.10]
[cross product] [[bar.w].sub.11]; [0.20;0.24] [0.26;0.28]
[[bar.x].sub.11] [[b.w].sub.11]
[cross product] [[bar.w].sub.12]; [0.27;0.31] [0.19;0.23]
[[bar.x].sub.12] [[b.w].sub.12]
[cross product] [[bar.w].sub.13]; [0.18;0.23] [0.20;0.25]
[[bar.x].sub.13] [[b.w].sub.13]
Expert 3
Project
Attribute [v.sub.3] [v.sub.4]
TOPSIS grey method
[cross product] [[bar.w].sub.1]; [0.92;0.67] [0.75;0.50]
[[bar.x].sub.1] [[b.w].sub.1]
[cross product] [[bar.w].sub.2]; [0.63;0.50] [0.25;0.13]
[[bar.x].sub.2] [[b.w].sub.2]
[[bar.w].sub.3]; [1.00;0.70] [0.90;0.70]
[cross product] [[b.w].sub.3]
[[bar.x].sub.3]
[cross product] [[bar.w].sub.4]; [0.78;0.44] [0.22;0.11]
[[bar.x].sub.4] [[b.w].sub.4]
[cross product] [[bar.w].sub.5]; [1.00;0.82] [0.91;0.73]
[[bar.x].sub.5] [[b.w].sub.5]
[cross product] [[bar.w].sub.6]; [1.00;0.80] [0.50;0.30]
[[bar.x].sub.6] [[b.w].sub.6]
[cross product] [[bar.w].sub.7]; [0.38;0.13] [0.50;0.25]
[[bar.x].sub.7] [[b.w].sub.7]
[cross product] [[bar.w].sub.8]; [1.00;0.29] [0.29;0.57]
[[bar.x].sub.8] [[b.w].sub.8]
[cross product] [[bar.w].sub.9]; [0.92;0.83] [0.83;0.75]
[[bar.x].sub.9] [[b.w].sub.9]
[cross product] [[bar.w].sub.10]; [0.70;0.40] [0.60;0.50]
[[bar.x].sub.10] [[b.w].sub.10]
[cross product] [[bar.w].sub.11]; [0.80;0.70] [0.70;0.60]
[[bar.x].sub.11] [[b.w].sub.11]
[cross product] [[bar.w].sub.12]; [0.70;0.60] [0.80;0.70]
[[bar.x].sub.12] [[b.w].sub.12]
[cross product] [[bar.w].sub.13]; [0.75;0.63] [0.25;0.13]
[[bar.x].sub.13] [[b.w].sub.13]
COPRAS-G method
[cross product] [[bar.w].sub.1]; [0.21;0.26] [0.25;0.30]
[[bar.x].sub.1] [[b.w].sub.1]
[cross product] [[bar.w].sub.2]; [0.22;0.24] [0.28;0.30]
[[bar.x].sub.2] [[b.w].sub.2]
[cross product] [[bar.w].sub.3]; [0.19;0.24] [0.20;0.24]
[[bar.x].sub.3] [[b.w].sub.3]
[cross product] [[bar.w].sub.4]; [0.22;0.28] [0.32;0.34]
[[bar.x].sub.4] [[b.w].sub.4]
[cross product] [[bar.w].sub.5]; [0.21;0.25] [0.23;0.27]
[[bar.x].sub.5] [[b.w].sub.5]
[cross product] [[bar.w].sub.6]; [0.17;0.20] [0.25;0.29]
[[bar.x].sub.6] [[b.w].sub.6]
[cross product] [[bar.w].sub.7]; [0.26;0.30] [0.24;0.28]
[[bar.x].sub.7] [[b.w].sub.7]
[cross product] [[bar.w].sub.8]; [0.17;0.29] [0.24;0.29]
[[bar.x].sub.8] [[b.w].sub.8]
[cross product] [[bar.w].sub.9]; [0.23;0.25] [0.25;0.27]
[[bar.x].sub.9] [[b.w].sub.9]
[cross product] [[bar.w].sub.10]; [0.23;0.29] [0.25;0.27]
[[bar.x].sub.10] [[b.w].sub.10]
[cross product] [[bar.w].sub.11]; [0.24;0.26] [0.26;0.28]
[[bar.x].sub.11] [[b.w].sub.11]
[cross product] [[bar.w].sub.12]; [0.25;0.27] [0.23;0.25]
[[bar.x].sub.12] [[b.w].sub.12]
[cross product] [[bar.w].sub.13]; [0.23;0.25] [0.32;0.34]
[[bar.x].sub.13] [[b.w].sub.13]
Table 3. Solution results by applying TOPSIS
grey and COPRAS-G methods
Expert 1
[d.sup.1+] [d.sup.1-]
Calculation [v.sub.1] 0.449 0.51
results by [v.sub.2] 0.532 0.307
applying [v.sub.3] 0.379 0.447
TOPSIS [v.sub.4] 0.496 0.438
grey
method
[N.sup.1]
Calculation [v.sub.1] 0.931
results by [v.sub.2] 0.877
applying [v.sub.3] 1.000
COPRAS-G [v.sub.4] 0.931
method
Expert 2
[d.sup.2+] [d.sup.2-]
Calculation [v.sub.1] 0.489 0.637
results by [v.sub.2] 0.725 0.318
applying [v.sub.3] 0.503 0.402
TOPSIS [v.sub.4] 0.493 0.555
grey
method
[N.sup.2]
Calculation [v.sub.1] 1.000
results by [v.sub.2] 0.857
applying [v.sub.3] 0.865
COPRAS-G [v.sub.4] 0.877
method
Expert 3
[d.sup.3+] [d.sup.3-]
Calculation [v.sub.1] 0.513 0.385
results by [v.sub.2] 0.387 0.448
applying [v.sub.3] 0.393 0.421
TOPSIS [v.sub.4] 0.520 0.224
grey
method
[N.sup.3]
Calculation [v.sub.1] 0.946
results by [v.sub.2] 0.984
applying [v.sub.3] 1.000
COPRAS-G [v.sub.4] 0.883
method
Aggregated
[d.sup.e+] [d.sup.e-]
Calculation [v.sub.1] 0.484 0.511
results by [v.sub.2] 0.548 0.358
applying [v.sub.3] 0.425 0.423
TOPSIS [v.sub.4] 0.503 0.406
grey
method
[N.sup.4]
Calculation [v.sub.1] 0.959
results by [v.sub.2] 0.906
applying [v.sub.3] 0.955
COPRAS-G [v.sub.4] 0.897
method
[c.sup.*+
.sub.i] Rank
Calculation [v.sub.1] 0.514 1
results by [v.sub.2] 0.395 4
applying [v.sub.3] 0.499 2
TOPSIS [v.sub.4] 0.446 3
grey
method
Rank
Calculation [v.sub.1] 1
results by [v.sub.2] 3
applying [v.sub.3] 2
COPRAS-G [v.sub.4] 4
method
Fig. 8. Aggregated calculation results (COPRAS-G and
TOPSIS grey methods)
TOPSIS grey COPRAS-G
V1 0.5145 0.959
V2 0.395 0.906
V3 0.499 0.955
V4 0.446 0.897
Note: Table made from bar graph.
