Prioritizing constructing projects of municipalities based on AHP and COPRAS-G: a case study about footbridges in Iran/Prioriteto suteikimas savivaldybiu statybos projektams remiantis AHP ir COPRAS-G metodais: tyrimo objektas--pesciuju tiltas Irane/Buvprojektu prioritates noteiksana pasvaldibas izmantojot AHP un Copras-G metodes: Iranas gajeju tiltu piemeru izpete/Omavalitsuste ehitusprojektide .....
Aghdaie, Mohammad Hasan ; Zolfani, Sarfaraz Hashemkhani ; Zavadskas, Edmundas Kazimieras 等
1. Introduction
Every municipality deals with many projects including constructing,
maintaining, repairing and etc. every year. Among these projects,
constructing projects are one of the critical problems for municipality
because many fields such as, manufacturing engineering, transportation
engineering, civil engineering, etc. have to work together and the
results are very important for the government. Also, the numbers of
constructing projects are many and municipality budget is limited.
Besides, a proper construction project selection is a very important
activity for every municipality due to the fact that improper selection
can negatively affect the overall performance and productivity of a
project. In addition, sometimes determining an appropriate area for
constructing project is as important as the project.
Most of constructing projects deal with civil engineering. In civil
engineering some projects deal with designing and constructing new
bridges. Bridges are important structures of our lives and make
transportation easier for us. Also, bridge projects sometimes are very
expensive and vital for the country. Most studies in this field are
about designing or constructing bridges and merely no studies could be
found about selecting an area for bridge constructing. The common
features of these studies are focused on designing, constructing and
mathematical calculations of structure of the bridge.
Footbridges are kind of bridges that pedestrians use for their
movements. Generally, all groups of people use footbridges. These
bridges help pedestrians to cross the street without making any problem
for vehicle traffic.
Selecting an area for constructing new footbridge is a
sophisticated, time-consuming and difficult process, requiring advanced
knowledge and expertise. So, the process can be very hard for engineers
and managers. For a proper and effective evaluation, the decision maker
may need a lot of data and many factors for evaluation. For these
reasons, selection of an area for constructing new footbridge can be
viewed as multi-attribute decision making process (MADM) problem.
The aim of this study is the use MADM methods for evaluating and
selection the best area as alternative for constructing a new
footbridge. There are many MADM methods in the literature including
Priority based, outranking, distance-based and mixed methods (Pomerol,
Barba-Romero 2000). Some of famous MADM methods in the literature are:
analytic hierarchy process (AHP) (Saaty 1980), analytic network process
(ANP) (Saaty, Vargas 2001), axiomatic design (AD) (Kulak, Kahraman
2005), TOPSIS (Hwang, Yoon 1981), ELECTRE (Wang, Triantaphyllou 2008),
VIKOR (Opricovic, Tzeng 2007), COPRAS-G (Zavadskas et al. 2008) and
PROMETHEE (Behzadian et al. 2010; Dagdeviren 2008). But among these
methods, Analytic Hierarchy Process (AHP) is one of the bests, and that
was introduced by Saaty (1980; 2001). The idea behind this method is
obtaining the relative weights among the factors and calculating the
total values of each alternative based on these weights. This study uses
the AHP to calculate each criterion weight from subjective judgments of
the decision maker group. The rating of each alternative and the weight
of each criterion, which are determined using the AHP, are then passed
to the complex proportional assessment method with grey interval numbers
(COPRAS-G), which is MADM method.
This paper is organized in five sections. In section
"Introduction" the studied problem is introduced. Section
"Principles of AHP and COPRAS-G methods" briefly describes the
two proposed methodologies. In section "Proposed AHP--COPRAS-G
integrated approach", proposed AHP--COPRAS-G integrated approach
for footbridge site place selection is presented and the stages of the
proposed approach and steps are determined in detail. How the proposed
approach is used on a real world case study is explained in section
"Case study". In section "Conclusions and future
research" conclusions and future research areas are discussed.
2. Principles of AHP and COPRAS-G methods
2.1. The AHP method
This technique was developed by Saaty (1980) and the main point
behind this technique is how to determine the relative importance of a
set of activities in a multi-criteria decision problem. Based on this
approach decision maker could incorporate and translate judgments on
intangible qualitative criteria alongside tangible quantitative criteria
(Badri 2001). The AHP method is based on three principles: first,
structure of the model; second, comparative judgment of the alternatives
and the criteria; third, synthesis of the priorities (Dagdeviren 2008).
The recent developments of decision making models based on AHP methods
are listed below:
--Medineckiene et al. (2010) applied AHP in sustainable
construction;
--Podvezko et al. (2010) used AHP in evaluation of contracts;
--Sivilevicius (2011a) applied AHP in modeling of transport system;
--Sivilevicius (2011b) used AHP in quality of technology;
--Fouladgar et al. (2011) applied AHP in prioritizing strategies.
In the first step, a sophisticated decision problem is structured
as a hierarchy. This method breaks down a sophisticated decision making
problem into hierarchy of objectives, criteria, and alternatives.
These decision elements make a hierarchy of structure including
goal of the problem at the top, criteria in the middle and the
alternatives at the bottom of this hierarchy.
In the second step, the comparisons of the alternatives and
criteria are made. In AHP, comparisons are made based on a standard nine
point scale (Table 1). Also, in this standard some numbers including 2,
4, 6, and 8 could be used as intermediate values.
Let C = {[C.sub.j]|j = 1,2...,n} be the set of criteria. The result
of the pairwise comparison on n criteria can be summarized in an (n x n)
evaluation matrix A in which every element [a.sub.ij](i,j = 1, 2,...,n)
is the quotient of weights of the criteria, as shown in Eq (1):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
At the third step, the mathematical process commences to normalize
and find the relative weights for each matrix. The relative weights are
given by the right eigenvector (w) corresponding to the largest
eigenvalue ([[lambda].sub.max]), as
Aw = [[lambda].sub.max]w. (2)
If the pairwise comparisons are completely consistent, the matrix A
has rank 1 and [[lambda].sub.max] = n.
In this case, weights can be obtained by normalizing any of the
rows or columns of A (Wang, Yang 2007). The quality of the output of the
AHP is strictly related to the consistency of the pairwise comparison
judgments (Dagdeviren 2008). The consistency is defined by the relation
between the entries of A: [a.sub.ij] = [a.sub.ik] The consistency index
(CI) is
CI = ([[lambda].sub.max] - n)/(n - 1)
The final consistency ratio (CR), using which one can conclude
whether the evaluations are sufficiently consistent, is calculated as
the ratio of the CI and the random index (RI), as indicated in Eq (4):
CR = CI/RI. (4)
The CR index should be lower than 0.10 to accept the AHP results as
consistent (Isiklar, Buyukozkan 2007). If the final consistency ratio
exceeds this value, the evaluation procedure has to be repeated to
improve consistency (Dagdeviren 2008). The CR index could be used to
calculate the consistency of decision makers as well as the consistency
of all the hierarchy (Wang, Yang 2007).
2.2. The COPRAS-G method
In order to evaluate the overall efficiency of an alternative, it
is necessary to identify selection criteria, to assess information,
relating to these criteria, and to develop methods for evaluating the
criteria to meet the participants' needs. Decision analysis is
concerned with the situation in which a decision-maker (DM) has to
choose among several alternatives by considering a particular set of
usually conflicting criteria. For this reason Complex proportional
assessment (COPRAS) method that was developed by Zavadskas and
Kaklauskas (1996) can be applied. This method was applied to the
solution of various problems in construction (Tupenaite et al. 2010;
Ginevicius et al. 2008; Kaklauskas et al. 2010; Zavadskas et al. 2010;
Medineckiene, Bjork 2011). The most alternatives under development
always deal with vague future, and values of criteria cannot be
expressed exactly. This MADM problem should be determined not with exact
criteria values, but with fuzzy values or with values in some intervals.
Zavadskas et al. (2008) presented the main ideas of complex proportional
assessment method with grey interval numbers (COPRAS-G) method. The idea
of COPRAS-G method with criterion values expressed in intervals is based
on the real conditions of decision making and applications of the Grey
systems theory (Deng 1982; 1988). The COPRAS-G method uses a stepwise
ranking and evaluating procedure of the alternatives in terms of
significance and utility degree.
The recent developments of decision making models based on COPRAS
methods are listed below:
--Hashemkhani Zolfani et al. (2011) presented forest roads locating
using COPRAS-G method;
--Chatterjee et al. (2011) presented materials selection model
based on COPRAS and EVAMIX methods;
--Zavadskas et al. (2011) assessment of the indoor environment;
--Podvezko (2011) presented comparative analysis of MCDM methods
(SAW and COPRAS);
--Chatterjee, Chakraborty (2012) presented materials selection
using COPRAS-G method;
--Antucheviciene et al. (2011) presented comparative analysis of
MCDM methods (COPRAS, TOPSIS and VIKOR).
The procedure of applying the COPRAS-G method consists of the
following steps (Zavadskas et al. 2009):
1. Selecting the set of the most important criteria, describing the
alternatives.
2. Constructing the decision-making matrix [cross product]X:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where [cross product] [x.sub.ji]--determined [[x.bar].sub.ji] (the
smallest value, the lower limit) and [[bar.x].sub.ji] (the biggest
value, the upper limit).
3. Determining significances of the criteria [q.sub.i].
4. Normalizing the decision-making matrix [cross product]X are
calculated by formula 6:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
In formula (6) [[x.bar].sub.ji]--the lower value of the I criterion
in the alternative j of the solution; [[bar.x].sub.ji]--the upper value
of the criterion i in the alternative j of the solution; m--the number
of criteria; n--the number of the alternatives, compared. Then, the
decision-making matrix is normalized are determined according to the
formula 7:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
5. Calculating the weighted normalized decision matrix [cross
product][??]. The weighted normalized values [cross
product][[??].sub.ji] are calculated as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
where [q.sub.i]--the significance of the i-th criterion. Then, the
normalized decision-making matrix is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
6. Calculating the sums [P.sub.j] of criterion values, whose larger
values are more preferable:
[P.sub.j] = [1/2][k.summation over (i=1)]([[[??].bar].sub.ji] +
[[bar.[??]].sub.ji]). (10)
7. Calculating the sums [R.sub.j] of criterion values, whose
smaller values are more preferable:
[R.sub.j] = [1/2] [m.summation over (i=k+1)] ([[[??].bar].sub.ji] +
[bar.[[??].sub.ji]]), i = [bar.k,m]. (11)
In formula (11), (m - k) is the number of criteria which must be
minimized.
8. Determining the minimal value of [R.sub.j] as follows:
[R.sub.min] = [min.sub.j] [R.sub.j], j = [bar.1,n]. (12)
9. Calculating the relative significance of each alternatively
[Q.sub.j] the expression:
[Q.sub.j] = [P.sub.j] + [[[n.summation over
(j=1)][R.sub.j]]/[[R.sub.j][n.summation over (j=1)]1/[R.sub.j]]]. (13)
10. Determining the optimally criterion by K the formula:
K = [max.sub.j][Q.sub.j], i = [bar.1,n]. (14)
11. Determining the priority order of the alternatives.
12. Calculating the utility degree of each alternative by the
formula:
[N.sub.j] = [[Q.sub.j]/[Q.sub.max]] 100%, (15)
where [Q.sub.j] and [Q.sub.max] are the significances of the
alternatives obtained from Eq (13).
3. Proposed AHP--COPRAS-G integrated approach
The integrated approach composed of AHP and COPRAS-G methods for
area selection problem consists of 4 basic stages (Fig. 1): (1) Data
gathering, (2) AHP calculations, (3) COPRAS-G calculations, (4) Decision
making.
In the first stage, alternatives and the criteria which will be
used in their evaluation are determined and the decision hierarchy is
formed. In the last step of the first stage, the decision hierarchy is
approved by decision making team.
In stage two and after approval of decision hierarchy, criteria
that were used in evaluation alternatives are assigned their weights via
AHP. In this stage, criteria weights are calculated by pairwise
comparisons. The decision making team used Table 1 as a standard for
doing pairwise comparisons. The project team used Delphi technique as a
group decision making tool for receiving general agreement.
Area priorities are found by using COPRAS-G computations in the
third stage. Firstly, the project team evaluates alternatives and after
these evaluations, COPRAS-G is used for ranking the alternatives.
Finally, in the last stage, decision making team made decision about
selecting the best place for footbridge.
4. Case study
Iran is one of the most dangerous countries for both drivers and
pedestrians. This case study is based on one of the important projects
in Sari and proposed approach is applied in one of the important
municipality projects, in Sari, Iran.
Sari City is the capital of Mazandaran province in the north of
Iran and near to Caspian Sea. Unfortunately, the number of pedestrians
that got involved in the accidents in Sari is high. In recent years the
principals of management and structures of municipality has changed
while a new building of Sari Municipality was established less than two
years ago, many projects started in the city like developing roads,
boulevards, parks and etc. due to deserve of this city that is 3000
years old and was the first city in the whole north of Iran. Compared to
developing roads in the city footbridges did not develop like roads and
this issue can be dangerous for local people and tourists in the city.
The municipality project team wants to evaluate and select area for
constructing new footbridges. The budget of the municipality was limited
and the best area had to be selected.
However it is hard to choose the most suitable one among the
municipality projects which dominate each other in different
characteristics. This research has tried to give a framework as a
scientific way for prioritizing roads and boulevards for construction of
new footbridges that can be helpful for municipality to follow their
projects according to the budget and to identify priority projects. The
three boulevards are selected by the project team because of their
importance and situations as alternatives. These alternatives are Khazar
Boulevard (KB), Artesh Boulevard (AB) and Taleghani Boulevard (TB) (Fig.
2).
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
4.1. Data gathering
At first, the top managers of Sari municipality and a group of
experts in civil engineering, economic, and social sciences participated
in a conference meeting on footbridge project (Table 2) and with a
preliminary work the decision making team determined three possible
areas suitable for the needs of the municipality. The three criteria
with eight sub-criteria are used for evaluation of the alternatives.
Decision hierarchy structured and criteria are provided in Fig. 3. There
are four levels in the decision hierarchy structured for selection.
4.2. AHP calculations
After constructing the decision hierarchy and alternatives, the
project team assigns pairwise comparisons via AHP for evaluating all the
criteria and weighting each criterion. In this step, the experts in the
decision making team are given the task of forming individual pairwise
comparison matrix by using the scale given in Table 1. As mentioned
before, the project team for receiving the general agreement on their
evaluations used Delphi technique as a group decision making tool. The
all pairwise comparisons and the weights of criteria are showed in
Tables 3-8. Eqs (1) to (4) were used for AHP calculations. The last
column of every table shows the weight of each criterion.
[FIGURE 3 OMITTED]
The Socio-economic, Environmental factors and Total cost are
determined as the three most important criteria in the area selection
process by AHP.
The pedestrians and vehicles are determined as the two most
important criteria in the area selection process, in the sub-criteria of
Traffic related criteria by AHP.
In the criterion of Accident related factors the dead and number of
injuries are determined as the two most important criteria in the area
selection process by AHP.
For the four sub-criteria of Environmental factors, Accident
related, Traffic, Influence of physical and Average speed are determined
as the four most important criteria in the area selection process by
AHP.
In the Socio-economic factors, situation of area, special
importance of each road or boulevard to the city of and Rate of
transportation of families, children and business dates as the three
most important criteria in the area selection process by AHP.
Consistency ratio of the pairwise comparison matrix calculated for all
of the tables was lower than 0.1. So the weights are shown to be
consistent and they are used in the selection process.
4.3. COPRAS-G calculations
First of all in this step, alternatives are evaluated based on the
evaluation criteria and the evaluation matrix is constructed. The
evaluations of these three alternatives according to the previously
stated criteria, i.e., evaluation matrix, are displayed in Table 9.
In Table 9 weights of each criterion and sub-criterion was
calculated based on results of AHP about criteria and sub-criteria.
Normalized weighted decision matrix [cross product][??] was
recalculated by formulas 6-8 (Table 10).
Final results calculated by the formulas 10-15 are presented in
Table 11.
According to results of Table 11, Khazar Boulevard is in the first
priority for the construction of footbridge, after that is Taleghani
Boulevard and finally Artesh Boulevard is the last in prioritizing.
4.4. Decision-making
Each municipality has limited budget and needs to make the best
decisions for doing their projects. The defined project was area
selection and the problem was to select one of the areas based on
quantitative and qualitative criteria. The aim of this study was the use
of MADM tools for solving this problem of the municipality projects in
Sari.
According to the AHP and COPRAS-G computations, it is decided to
select KB. For reaching more accurate analyzing, project team used
conference meeting and consistency ratio in AHP calculations. The use of
grey analysis helped the project team to deal with the uncertain and
insufficient information and to build a relational analysis or to
construct a model to characterize the system.
5. Conclusions and future research
In this paper, a decision approach is provided for prioritizing
projects particularly of constructing new footbridges. Municipality
projects are important for every city and best decisions must be made on
it. Budgets of each municipality are confined and the needs are wide.
Appropriate prioritizing is very important and influences the time of
finishing project or the quality of carrying it out. This selection
problem is based on the comparisons of area criteria and evaluations of
the alternatives, according to identified criteria. An integrated AHP
and COPRAS-G methods have been used in proposed approach. AHP is used to
assign weights to the criteria to be used in area selection, while
COPRAS-G is employed to determine the ranking of the alternatives.
The weights obtained from AHP are included in decision making
process by using them in COPRAS-G computations and the alternative
priorities are determined based on these weights. The proposed model has
only been implemented on an area selection for constructing new
footbridges in the municipality project in Sari; however, the project
team has found the proposed model satisfactory and implementable in
others bridge selection decisions. Also, this approach could be used in
any other kind of prioritizing constructing projects of municipalities.
Besides, this approach can be used for prioritizing other municipality
projects such as roads, bridges, highways.
doi: 10.3846/bjrbe.2012.20
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Received 20 February 2012; accepted 20 April 2012
Mohammad Hasan Aghdaie (1), Sarfaraz Hashemkhani Zolfani (2),
Edmundas Kazimieras Zavadskas (3) ([mail])
(1,2) Dept of Industrial Engineering, Shomal University, P. O. Box
731, Amol, Mazandaran, Iran
(2,3) Institute of Internet and Intelligent Technologies, Vilnius
Gediminas Technical University, Sauletekio al. 11, 10223 Vilnius,
Lithuania
E-mails: (1) mh_aghdaie@yahoo.com; (2) sa.hashemkhani@gmail.com;
(3) edmundas.zavadskas@vgtu.lt
Table 1. Nine-point intensity of importance
scale and its description (Dagdeviren 2008)
Definition Intensity of
importance
Equally important 1
Moderately more important 3
Strongly more important 5
Very strongly more important 7
Extremely more important 9
Table 2. Background information of experts
Variable Items NO Variable Items NO
1) Civil Engineering Bachelor 1 3) Social Bachelor 0
Master 3 Sciences Master 2
Ph.D. 1 Experts Ph.D. 1
2) Economic Experts Bachelor 0 4) Top Bachelor 0
Master 2 Managers Master 3
Ph.D. 1 Ph.D. 1
Table 3. Criteria, sub-criteria, sub-sub criteria and their
descriptions
Criteria
[C.sub.1-1] Traffic related factors:
[C.sub.1-1-1] Vehicles
[C.sup.1*] [C.sub.1-1-2] Pedestrians
Environmental
factors [C.sub.1-2] Accident related factors
[C.sub.1-2-1] Number of injuries
[C.sub.1-2-2] Dead
[C.sub.1-3] Average speed limit
[C.sub.1-4] Influence of physical
area attributes on footbridges
[C.sup.2*] [C.sub.2-1] Rate of transportation of
Socio-economic families, children and business dates
factors
[C.sub.2-2] situation of area growth
in the future
[C.sub.2-3] Special importance of
each road or boulevard to the city
[C.sub.2-4] Vision of roads or boulevards
about issues like: population, economical
condition and other strategic issues
[C.sup.3*]
Total cost
Table 4. Pairwise comparison matrix for criteria and weights
[C.sub.1] [C.sub.2] [C.sub.3] Weights
[C.sub.1] 1 3 1/2 0.3
[C.sub.2] 1/3 1 1/6 0.6
[C.sub.3] 2 6 1 0.1
Table 5. Pairwise comparison matrix for Traffic related
factors and their weighs
[C.sub.1-1-1] [C.sub.1-1-2] Weights
[C.sub.1-1-1] 1 1/2 0.333
[C.sub.1-1-2] 2 1 0.667
Table 6. Pairwise comparison matrix for Accident related
factors and their weights
[C.sub.1-2-1] [C.sub.1-2-2] Weights
[C.sub.1-2-1] 1 1/5 0.167
[C.sub.1-2-2] 5 1 0.833
Table 7. Pairwise comparison matrix for Environmental factors and
their weights
[C.sub.1-1] [C.sub.1-2] [C.sub.1-3] [C.sub.1-4]
[C.sub.1-1] 1 1/4 5 2
[C.sub.1-2] 4 1 3 2
[C.sub.1-3] 1/5 1/3 1 1/6
[C.sub.1-4] 1/2 1/2 6 1
Weights
[C.sub.1-1] 0.254
[C.sub.1-2] 0.441
[C.sub.1-3] 0.074
[C.sub.1-4] 0.231
Table 8. Pairwise comparison matrix for Socio-economic factors and
their weights
[C.sub.2-1] [C.sub.2-2] [C.sub.2-3] [C.sub.2-4]
[C.sub.2-1] 1 1/8 1/5 1/3
[C.sub.2-2] 8 1 2 2
[C.sub.2-3] 5 1/2 1 2
[C.sub.2-4] 3 1/2 1/2 1
Weights
[C.sub.2-1] 0.059
[C.sub.2-2] 0.464
[C.sub.2-3] 0.294
[C.sub.2-4] 0.183
Table 9. Initial decision making matrix [cross product]X with the
criteria values described in intervals
[cross product]
[x.sub.1-1-1]
Opt. max
[q.sub.j] 0.025
Region [[bar.x].sub.1-1-1]; [[x.bar].sub.1-1-1]
KB 60 70
AB 70 80
TB 60 70
[cross product]
[x.sub.1-1-2]
Opt. max
[q.sub.j] 0.05
Region [[x.bar].sub.1-1-2]; [[x.bar].sub.1-1-2];
KB 70 80
AB 60 70
TB 60 70
[cross product]
[x.sub.1-2-1]
Opt. min
[q.sub.j] 0.021
Region [[x.bar].sub.l-2-l]; [[x.bar].sub.l-2-l];
KB 50 60
AB 60 70
TB 40 50
[cross product]
[x.sub.1-2-2]
Opt. min
[q.sub.j] 0.115
Region [[x.bar].sub.l-2-2]; [[x.bar].sub.l-2-2];
KB 20 30
AB 30 40
TB 20 3
[cross product]
[x.sub.1-3]
Opt. max
[q.sub.j] 0.021
Region [[x.bar].sub.1-3]; [[x.bar].sub.1-3];
KB 50 60
AB 60 70
TB 50 60
[cross product]
[x.sub.1-4]
Opt. max
[q.sub.j] 0.068
Region [[x.bar].sub.1-4]; [[bar.x].sub.1-4];
KB 70 80
AB 60 70
TB 60 70
[cross product]
[x.sub.2-1]
Opt. max
[q.sub.j] 0.035
Region [[x.bar].sub.2-1]; [[x.bar].sub.2-1];
KB 60 70
AB 70 80
TB 70 80
[cross product]
[x.sub.2-2]
Opt. max
[q.sub.j] 0.278
Region [[x.bar].sub.2-2]; [[x.bar].sub.2-2];
KB 70 80
AB 60 70
TB 50 60
[cross product]
[x.sub.2-3]
Opt. max
[q.sub.j] 0.177
Region [[x.bar].sub.2-3]; [[x.bar].sub.2-3];
KB 60 70
AB 70 80
TB 60 70
[cross product]
[x.sub.2-4]
Opt. max
[q.sub.j] 0.11
Region [[x.bar].sub.2-4]; [[x.bar].sub.2-4];
KB 60 70
AB 70 80
TB 70 80
[cross product]
[x.sub.3]
Opt. min
[q.sub.j] 0.1
Region [[x.bar].sub.3]; [[x.bar].sub.3];
KB 50 60
AB 40 50
TB 50 60
Table 10. Normalized weighted decision making matrix
[cross product] [??]
[cross product]
[x.sub.1-1-1]
Opt. max
Region [[x.bar].sub.1-1-1]; [[bar.x].sub.1-1-1]
KB 0.007 0.008
AB 0.008 0.009
TB 0.007 0.008
[cross product]
[x.sub.1-1-2]
Opt. min
Region [[x.bar].sub.1-1-2]; [[x.bar].sub.1-1-2];
KB 0.018 0.02
AB 0.015 0.018
TB 0.015 0.018
[cross product]
[x.sub.1-2-1]
Opt. max
Region [[x.bar].sub.1-2-1]; [[x.bar].sub.1-2-1];
KB 0.006 0.007
AB 0.007 0.008
TB 0.005 0.006
[cross product]
[x.sub.1-2-2]
Opt. max
Region [[x.bar].sub.1-2-2]; [[x.bar].sub.1-2-2];
KB 0.028 0.041
AB 0.041 0.054
TB 0.028 0.041
[cross product] [cross product]
[x.sub.1-3] [x.sub.1-4]
Opt. max max
Region [[x.bar].sub.1-3]; [[x.bar].sub.1-3]; [[x.bar].sub.1-4];
KB 0.006 0.007 0.024
AB 0.007 0.008 0.019
TB 0.006 0.007 0.019
[cross product] [cross product]
[x.sub.1-4] [x.sub.2-1]
Opt. max max
Region [[x.bar].sub.1-4]; [[x.bar].sub.2-1]; [[x.bar].sub.2-1];
KB 0.026 0.011 0.013
AB 0.024 0.013 0.015
TB 0.024 0.013 0.015
[cross product] [cross product]
[x.sub.2-2]; [x.sub.2-3];
Opt. max max
Region [[x.bar].sub.2-2]; [[x.bar].sub.2-2]; [[x.bar].sub.2-3];
KB 0.1 0.114 0.052
AB 0.086 0.1 0.061
TB 0.08 0.086 0.052
[cross product] [cross product]
[x.sub.2-3]; [x.sub.2-4];
Opt. max max
Region [[x.bar].sub.2-3]; [[x.bar].sub.2-4]; [[x.bar].sub.2-4];
KB 0.061 0.031 0.036
AB 0.07 0.036 0.042
TB 0.061 0.036 0.042
[cross product]
[x.sub.3];
Opt. max
Region [[x.bar].sub.3]; [[x.bar].sub.3];
KB 0.033 0.039
AB 0.026 0.033
TB 0.033 0.039
Table 11. Evaluation of utility degree
Region [P.sub.i] [R.sub.i] [Q.sub.i] [N.sub.i]
KB 0.2661 0.0773 0.5997 100.00%
AB 0.2629 0.083 0.5736 95.64%
TB 0.244 0.076 0.5832 97.42%
