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Economic Computation and Economic Cybernetics Studies and Research, Issue 3/2016, Vol. 50
________________________________________________________________________________
Mehdi KESHAVARZ GHORABAEE, PhD Candidate
E-mail: m.keshavarz_gh@yahoo.com
Department of Industrial Management, Faculty of Management
and Accounting, Allameh Tabataba’i University, Tehran, Iran
Professor Edmundas Kazimieras ZAVADSKAS*, Dr.Sc.
E-mail: edmundas.zavadskas@vgtu.lt (*Corresponding author)
Department of Construction Technology and Management, Faculty of
Civil Engineering, Vilnius Gediminas Technical University, Lithuania
Professor Zenonas TURSKIS, PhD
E-mail:zenonas.turskis@vgtu.lt
Department of Construction Technology and Management, Faculty of
Civil Engineering, Vilnius Gediminas Technical University, Lithuania
Professor Jurgita ANTUCHEVICIENE, PhD
E-mail: jurgita.antucheviciene@vgtu.lt
Department of Construction Technology and Management, Faculty of
Civil Engineering, Vilnius Gediminas Technical University, Lithuania
A NEW COMBINATIVE DISTANCE-BASED ASSESSMENT
(CODAS) METHOD FOR MULTI-CRITERIA DECISION-MAKING
Abstract. A key factor to attain success in any discipline, especially
in a field which requires handling large amounts of information and knowledge, is
decision making. Most real-world decision-making problems involve a great
variety of factors and aspects that should be considered. Making decisions in such
environments can often be a difficult operation to perform. For this reason, we
need multi-criteria decision-making (MCDM) methods and techniques, which can
assist us for dealing with such complex problems. The aim of this paper is to
present a new COmbinative Distance-based ASsessment (CODAS) method to
handle MCDM problems. To determine the desirability of an alternative, this
method uses the Euclidean distance as the primary and the Taxicab distance as the
secondary measure, and these distances are calculated according to the negative-
ideal point. The alternative which has greater distances is more desirable in the
CODAS method. Some numerical examples are used to illustrate the process of the
proposed method. We also perform a comparative sensitivity analysis to examine
the results of CODAS and compare it by some existing MCDM methods. These
analyses show that the proposed method is efficient, and the results are stable.
Keywords: Multi-criteria decision-making, MCDM, MADM,
Euclidean distance, Taxicab distance, CODAS.
JEL Classification: C02, C44, C61, C63, L6
25
Mehdi Keshavarz Ghorabaee, Edmundas Kazimieras Zavadskas, Zenonas Turskis,
Jurgita Antucheviciene
__________________________________________________________________
1. Introduction
Multi-criteria decision-making (MCDM) is one of the most active fields of
interdisciplinary research in management science and operations research (Ho et
al., 2010). Multi-attribute decision-making (MADM) and multi-objective decision-
making (MODM) are two branches in MCDM. MADM usually involves the
discrete decision variables and a limited number of alternatives for evaluation
(Jato-Espino et al., 2014). MODM is concerned with identifying the best choice
from an infinite set of alternatives under a set of constraints. Each criterion in
MODM is associated with an objective, whereas in MADM each criterion is
associated with a discrete attribute (Kabir et al., 2014). However, MADM and
MCDM have been used to refer the same class of problems in the recent years. In
the following, we also use the term MCDM to refer multi-attribute decision-
making problems. Fundamentally, intrinsic properties of MCDM make it appealing
and practically useful. Some of these properties described by Belton and Stewart
(Belton and Stewart, 2002) are as follows: (1) ‘‘MCDM seeks to take explicit
account of multiple, conflicting criteria’’, (2) it helps to structure the management
problem, (3) it provides a model that can serve as a focus for discussion, and (4) it
offers a process that leads to rational, justifiable, and explainable decisions.
Many MCDM methods and techniques have been proposed by researchers in
the past decades. Some of the most important ones are weighted sum model
(WSM) (Fishburn, 1967), weighted product model (WPM) (Miller and Starr,
1969), weighted aggregated sum product assessment (WASPAS) (Zavadskas et al.,
2012), analytical hierarchy process (AHP) (Satty, 1990), ELECTRE (ELimination
Et Choix Traduisant la REalité) (Roy, 1968), technique for order of preference by
similarity to ideal solution (TOPSIS) (Hwang and Yoon, 1981), preference ranking
organization method for enrichment of evaluations (PROMETHEE) (Brans and
Vincke, 1985), complex proportional assessment (COPRAS) (Zavadskas and
Kaklauskas, 1996), VIKOR (VIseKriterijumska Optimizacija I Kompromisno
Resenje) (Opricovic, 1998), MULTIMOORA (multi-objective optimization by
ratio analysis plus the full multiplicative form) (Brauers and Zavadskas, 2010),
additive ratio assessment (ARAS) (Zavadskas and Turskis, 2010) and evaluation
based on distance from average solution (EDAS) (Keshavarz Ghorabaee et al.,
2015). WSM is probably the most commonly used approach. This method defines
the optimal alternative based on the ‘additive utility’ assumption. WPM is very
similar to the WSM. This method uses the multiplication of powered weighted
ratios (performances) instead of summation of weighted ratios which considered in
WSM. WASPAS method was proposed based on the combination of WSM and
WPM methods, and has the advantages of both of them. This method has been
applied in many real-world MCDM problems (Vafaeipour et al., 2014; Džiugaitė-
Tumėnienė and Lapinskaitė, 2014; Petkovic et al., 2015). The AHP, which was
proposed by Saaty (Satty, 1981), is based on preferences or weights of importance
26
A New Combinative Distance-based Assessment (CODAS) Method for Multi-
Criteria Decision-making
_________________________________________________________________
of criteria and alternatives with respect to the hierarchical structure of them. We
have three levels in the structure of the AHP method. First level is related to the
goal of the problem, second level corresponds to the criteria, and third level
shows the alternatives. This method involves pair-wise comparisons and therefore
is time-consuming when we have numerous criteria and/or alternatives. The
original ELECTRE method is labeled as ‘ELECTRE I’ and the evolutions have
continued with ELECTRE II, ELECTRE III, ELECTRE IV, ELECTRE IS and
ELECTRE TRI . ELECTRE methods comprise two main procedures:
construction of one or several outranking relation(s) and an exploitation
procedure. Unlike many other MCDM methods, in the ELECTRE method, it is
not assumed that the criteria are mutually independent. One of the disadvantages
of the ELECTRE method is about the parameters of discordance and concordance
thresholds. It is difficult for a decision maker to provide any justification for the
values chosen for these parameters. The TOPSIS method, which was developed
by Hwang and Yoon (Satty, 1990), is a value-based compensatory method. This
method attempts to rank alternatives according to their distances from the ideal
and nadir (positive-ideal and negative-ideal) solutions. However, it does not
consider the relative importance of these distances (Opricovic and Tzeng, 2004).
PROMETHEE is an MCDM method for ranking a finite set of alternative with
respect to some conflicting criteria. PROMETHEE is applicable even when we
have simple and efficient information. This method is based on the comparison of
alternatives considering the deviations of them on each criterion, and uses
preference functions for criteria to determine these deviations. Then the positive
and negative preference flows are utilized for appraising and ranking the
alternatives (Brans et al., 1986). The COPRAS method is an efficient MCDM
method which determines the best alternative according to a ratio based on two
measures: benefit criteria performance summation and cost criteria performance
summation. The applicability of this method is demonstrated in many real-word
MCDM problems (Keshavarz Ghorabaee et al., 2014; Hashemkhani Zolfani and
Bahrami, 2014; Ecer, 2014; Stefano et al., 2015). The VIKOR method was
originally developed by Opricovic (Opricovic, 1998) to solve decision problems
with conflicting and non-commensurable criteria (criteria with different units).
The alternatives are evaluated according to all established criteria, and solution
that is closest to the ideal is the best in this method. The logic of this method is
similar to the TOPSIS method. However, there are some significant differences
that assessed by Opricovic and Tzeng (2004). The MULTIMOORA method,
which was developed by Brauers and Zavadskas (2010), is an extended version of
the MOORA (multi-objective optimization by ratio analysis) method (Brauers
and Zavadskas, 2006). It consists of three parts, namely the ratio system, the
reference point, and the full multiplicative form. This method is efficient and has
been applied to many MCDM problems and extended for different environments
27
Mehdi Keshavarz Ghorabaee, Edmundas Kazimieras Zavadskas, Zenonas Turskis,
Jurgita Antucheviciene
__________________________________________________________________
like fuzzy and grey environments (Baležentis et al., 2012a; Stanujkic et al., 2012;
Baležentis and Baležentis, 2011). The ARAS method is an efficient MCDM
method proposed by Zavadskas and Turskis (Zavadskas and Turskis, 2010) for
evaluation of microclimate in office rooms. This method has been extended and
used in many application fields in the past years (Baležentis et al., 2012b; Dadelo
et al., 2012; Stanujkic, 2015). The EDAS method is relatively a new MCDM
method which was proposed by Keshavarz Ghorabaee, Zavadskas, Olfat and
Turskis (2015). The application of this method was examined in the multi-criteria
in the multi-criteria inventory ABC classification. Moreover, it was demonstrated
that the EDAS method has a good efficiency for dealing with multi-criteria
decision-making problems.
All the above-mentioned MCDM methods have advantages and disadvantages
which appraising them is not the aim of this paper. In this paper, we want to
propose a new method to handle multi-criteria decision-making problems. This
method is named CODAS, and has some features that have not been considered in
the other MCDM methods. In the proposed method, the overall performance of an
alternative is measured by the Euclidean and Taxicab distances from the negative-
ideal point. The CODAS uses the Euclidean distance as the primary measure of
assessment. If the Euclidean distances of two alternatives are very close to each
other, the Taxicab distance is used to compare them. The degree of closeness of
Euclidean distances is set by a threshold parameter. The Euclidean and Taxicab
distances are measures for -norm and -norm indifference spaces, respectively
(Yoon, 1987). Therefore, In the CODAS method, we first assess the alternatives in
an -norm indifference space. If the alternatives are not comparable in this space,
we go to an -norm indifference space. To perform this process, we should
compare each pair of alternatives. In this study, we present the CODAS method in
detail and illustrate the proposed method by using some numerical examples.
Moreover, a comparative sensitivity analysis is done to represent the validity and
stability of the proposed method. We use different sets of criteria weights and five
MCDM methods (WASPAS, COPRAS, TOPSIS, VIKOR and EDAS) to perform
this analysis.
The rest of this paper is organized as follows. In Section 2, a new combinative
distance-based assessment (CODAS) method is presented in detail. In Section 3,
we use some numerical examples to illustrate the process of the CODAS method.
In Section 4, a comparative sensitivity analysis is made to demonstrate the
efficiency of the proposed method. Conclusions are discussed in the last section.
2. Combinative distance-based assessment (CODAS) method
In this section, we present a new method to deal with multi-criteria decision-
making problems. The proposed method is called CODAS, which stands for
28
A New Combinative Distance-based Assessment (CODAS) Method for Multi-
Criteria Decision-making
_________________________________________________________________
COmbinative Distance-based ASsessment. In this method, the desirability of
alternatives is determined by using two measures. The main and primary measure
is related to the Euclidean distance of alternatives from the negative-ideal. Using
this type of distance requires an -norm indifference space for criteria. The
secondary measure is the Taxicab distance which is related to the -norm
indifference space. It’s clear that the alternative which has greater distances from
the negative-ideal solution is more desirable. In this method, if we have two
alternatives which are incomparable according to the Euclidean distance, the
Taxicab distance is used as secondary measure. Although the -norm indifference
space is preferred in the CODAS, two types of indifference space could be
considered in its process. Suppose that we have alternatives and criteria. The
steps of the proposed method are presented as follows:
Step 1. Construct the decision-making matrix (), shown as follows:
,
(1)
where ( ) denotes the performance value of th alternative on th
criterion ( and ).
Step 2. Calculate the normalized decision matrix. We use linear normalization of
performance values as follows:
(2)
where and represent the sets of benefit and cost criteria, respectively.
Step 3. Calculate the weighted normalized decision matrix. The weighted
normalized performance values are calculated as follows:
(3)
where () denotes the weight of th criterion, and
.
Step 4. Determine the negative-ideal solution (point) as follows:
29
Mehdi Keshavarz Ghorabaee, Edmundas Kazimieras Zavadskas, Zenonas Turskis,
Jurgita Antucheviciene
__________________________________________________________________
(4)
(5)
Step 5. Calculate the Euclidean and Taxicab distances of alternatives from the
negative-ideal solution, shown as follows:
(6)
(7)
Step 6. Construct the relative assessment matrix, shown as follows:
(8)
,
(9)
where and denotes a threshold function to recognize the equality
of the Euclidean distances of two alternatives, and is defined as follows:
(10)
In this function, is the threshold parameter that can be set by decision-
maker. It is suggested to set this parameter at a value between 0.01 and 0.05. If the
difference between Euclidean distances of two alternatives is less than , these two
alternatives are also compared by the Taxicab distance. In this study, we use
for the calculations.
Step 7. Calculate the assessment score of each alternative, shown as follows:
Η
,
(11)
Step 8. Rank the alternatives according to the decreasing values of assessment
30
A New Combinative Distance-based Assessment (CODAS) Method for Multi-
Criteria Decision-making
_________________________________________________________________
score (). The alternative with the highest is the best choice among the
alternatives.
To describe the proposed method, we use a simple situation with seven
alternatives and two criteria. Suppose that weighted normalized performance
values () have been calculated. These values are dimensionless and between 0
and 1. Figure 1 shows the position of all alternatives according to these values.
Figure 1. A simple graphical example with two criteria
As can be seen in this figure, is the negative-ideal point
(solution).The Euclidean distances of alternatives from this point are:
According these distances, we can say that the order of alternatives is
. As previously stated, the Euclidean distance is
31
-norm indifference curves
-norm indifference curves
Feasible region
Mehdi Keshavarz Ghorabaee, Edmundas Kazimieras Zavadskas, Zenonas Turskis,
Jurgita Antucheviciene
________________________________________________________________
a measure to compare the alternatives in an -norm indifference space. In this
space we cannot find the difference between and . So the Taxicab distance,
that is the measure of -norm indifference space, is used in this case. The Taxicab
distances of and from the negative-ideal point are:
As can be seen, has greater Taxicab distance from the negative-ideal point.
This fact is clear according to the indifference curves which presented in Figure 1.
Therefore, we can say that is more desirable than , and the final ranking is
.
3. Illustrative examples
To illustrate the process of the CODAS method, we use two examples in this
section. The steps of the proposed method are presented through these examples.
3.1. Example 1
This example is adapted from Chakraborty and Zavadskas (2014) which is
related to the selection of the most appropriate industrial robot. Five different
criteria which are considered in this robot selection problem are: load capacity (in
kg), maximum tip speed (in mm/s), repeatability (in mm), memory capacity (in
points or steps) and manipulator reach (in mm). Among these criteria, the load
capacity, maximum tip speed, memory capacity, and manipulator reach are defined
as benefit criteria, and the repeatability is defined as a cost criterion. This problem
consists of seven alternatives, and the corresponding data are given in Table 1.
Table 1. Data of Example 1
Weights of criteria
0.036
0.326
0.192
0.326
0.120
Alternatives
Robots
Load
capacity
Maximum
tip speed
Repeatability
Memory
capacity
Manipulator
reach
ASEA-IRB 60/2
60
0.4
2540
500
990
Cincinnati
Milacrone T3-726
6.35
0.15
1016
3000
1041
Cybotech V15
Electric Robot
6.8
0.10
1727.2
1500
1676
Hitachi America
Process Robot
10
0.2
1000
2000
965
Unimation PUMA
500/600
2.5
0.10
560
500
915
United States
Robots Maker 110
4.5
0.08
1016
350
508
Yaskawa Electric
Motoman L3C
3
0.1
1778
1000
920
32
A New Combinative Distance-based Assessment (CODAS) Method for Multi-
Criteria Decision-making
According to Table 1, we can construct the decision matrix. Then the
normalized decision matrix is calculated as shown in Table 2.
Table 2. The normalized decision matrix of Example 1
Alternatives
Load
capacity
Maximum tip
speed
Repeatability
Memory
capacity
Manipulator
reach
1.000
0.200
1.000
0.167
0.591
0.106
0.533
0.400
1.000
0.621
0.113
0.800
0.680
0.500
1.000
0.167
0.400
0.394
0.667
0.576
0.042
0.800
0.220
0.167
0.546
0.075
1.000
0.400
0.117
0.303
0.050
0.800
0.700
0.333
0.549
Using weights of criteria that are given in Table 1, the weighted normalized
performance values can be calculated, and then the negative-ideal solution is
determined. According to the obtained values, the Euclidean and Taxicab distances
of alternatives from the negative-ideal solution are also computed. The results are
presented in Table 3.
Table 3. The weighted normalized decision matrix and the negative-ideal
solution of Example 1
Alternatives
Load
capacity
Maximum
tip speed
Repeatabil
ity
Memory
capacity
Manipulato
r reach
0.0360
0.0384
0.3260
0.0543
0.0709
0.2593
0.3394
0.0038
0.1024
0.1304
0.3260
0.0745
0.3032
0.4510
0.0041
0.1536
0.2217
0.1630
0.1200
0.2415
0.4762
0.0060
0.0768
0.1283
0.2173
0.0691
0.1947
0.3114
0.0015
0.1536
0.0719
0.0543
0.0655
0.1199
0.1606
0.0027
0.1920
0.1304
0.0380
0.0364
0.1644
0.2133
0.0018
0.1536
0.2282
0.1087
0.0659
0.2087
0.3720
Negative-
ideal
solution
0.0015
0.0384
0.0719
0.0380
0.0364
The relative assessment matrix () and the assessment scores () of
alternatives can be calculated by using Table 3 and Eqs. (8) to (10). Table 4
represents the results. It should be noted that, the calculations are performed with
.
33
Mehdi Keshavarz Ghorabaee, Edmundas Kazimieras Zavadskas, Zenonas Turskis,
Jurgita Antucheviciene
__________________________________________________________________
Table 4. The relative assessment matrix and the assessment scores of
alternatives of Example 1
0.0000
-0.1554
0.0178
0.0926
0.3181
0.2210
0.0180
0.5122
0.1554
0.0000
0.0364
0.2480
0.4735
0.3764
0.1734
1.4633
-0.0178
-0.0364
0.0000
0.2116
0.4371
0.3400
0.1370
1.0715
-0.0926
-0.2480
-0.2116
0.0000
0.2255
0.1284
-0.0140
-0.2125
-0.3181
-0.4735
-0.4371
-0.2255
0.0000
-0.0971
-0.3001
-1.8515
-0.2210
-0.3764
-0.3400
-0.1284
0.0971
0.0000
-0.2030
-1.1717
-0.0180
-0.1734
-0.1370
0.0140
0.3001
0.2030
0.0000
0.1887
According to the values of assessment scores, the ranking of alternatives is
. Therefore, (Cincinnati Milacrone T3-
726) is the best robot with respect to the assessment of the CODAS method.
3.2. Example
This example is adapted from Zavadskas and Turskis (2010) and considers the
evaluation of microclimate in an office. Six criteria determined for this evaluation
process are: the amount of air per head (in m3/h), relative air humidity (in percent),
air temperature (in °C), illumination during work hours (in lx), rate of air flow (in
m/s), and dew point (in °C). All of these criteria are defined as benefit criteria
except the rate of air flow and the dew point. Fourteen alternatives should be
evaluated according to these criteria. The data of this problem are shown in Table
5.
Table 5. Data of Example 2
Weights of
Criteria
0.21
0.16
0.26
0.17
0.12
0.08
Alternatives
The
amount
of air per
head
Relative
air
humidity
Air
temperature
Illumination
during work
hours
Rate of
air flow
Dew point
7.6
46
18
390
0.1
11
5.5
32
21
360
0.05
11
5.3
32
21
290
0.05
11
5.7
37
19
270
0.05
9
4.2
38
19
240
0.1
8
4.4
38
19
260
0.1
8
3.9
42
16
270
0.1
5
7.9
44
20
400
0.05
6
8.1
44
20
380
0.05
6
4.5
46
18
320
0.1
7
5.7
48
20
320
0.05
11
5.2
48
20
310
0.05
11
7.1
49
19
280
0.1
12
6.9
50
16
250
0.05
10
34
A New Combinative Distance-based Assessment (CODAS) Method for Multi-
Criteria Decision-making
_________________________________________________________________
According to steps 1 and 2 of the CODAS method and Table 5, we can
construct the decision matrix and calculate the normalized performance values
using Eq. (2). The normalized decision matrix is shown in Table 6. As can be seen
in this table, the maximum values in benefit criteria and the minimum values of
cost criteria are transformed to 1. Thus, there is no difference between the
dimension (unit of measurement) and the type criteria after normalization.
Table 6. The normalized decision matrix of Example 2
Alternatives
The
amount of
air per
head
Relative air
humidity
Air
temperature
Illumination
during work
hours
Rate of
air flow
Dew
point
0.938
0.920
0.857
0.975
0.500
0.455
0.679
0.640
1.000
0.900
1.000
0.455
0.654
0.640
1.000
0.725
1.000
0.455
0.704
0.740
0.905
0.675
1.000
0.556
0.519
0.760
0.905
0.600
0.500
0.625
0.543
0.760
0.905
0.650
0.500
0.625
0.481
0.840
0.762
0.675
0.500
1.000
0.975
0.880
0.952
1.000
1.000
0.833
1.000
0.880
0.952
0.950
1.000
0.833
0.556
0.920
0.857
0.800
0.500
0.714
0.704
0.960
0.952
0.800
1.000
0.455
0.642
0.960
0.952
0.775
1.000
0.455
0.877
0.980
0.905
0.700
0.500
0.417
0.852
1.000
0.762
0.625
1.000
0.500
To calculate the negative-ideal solution, we should obtain the weighted
normalized performance values first. Table 7 shows the weighted normalized
decision-matrix and corresponding negative-ideal solutions. Also, in the last two
columns of this table, the Euclidean and Taxicab distances of alternatives from the
negative-ideal solution are represented.
35
Mehdi Keshavarz Ghorabaee, Edmundas Kazimieras Zavadskas, Zenonas Turskis,
Jurgita Antucheviciene
__________________________________________________________________
Table 7. The weighted normalized decision matrix and the negative-ideal
solution of Example 2
Alternatives
The amount
of air per
head
Relative air
humidity
Air
temperature
Illumination
during work
hours
Rate of air
flow
Dew point
0.1970
0.1472
0.2229
0.1658
0.0600
0.0364
0.1261
0.2323
0.1426
0.1024
0.2600
0.1530
0.1200
0.0364
0.1085
0.2174
0.1374
0.1024
0.2600
0.1233
0.1200
0.0364
0.0960
0.1825
0.1478
0.1184
0.2352
0.1148
0.1200
0.0444
0.0877
0.1837
0.1089
0.1216
0.2352
0.1020
0.0600
0.0500
0.0457
0.0808
0.1141
0.1216
0.2352
0.1105
0.0600
0.0500
0.0476
0.0945
0.1011
0.1344
0.1981
0.1148
0.0600
0.0800
0.0580
0.0914
0.2048
0.1408
0.2476
0.1700
0.1200
0.0667
0.1550
0.3530
0.2100
0.1408
0.2476
0.1615
0.1200
0.0667
0.1550
0.3496
0.1167
0.1472
0.2229
0.1360
0.0600
0.0571
0.0677
0.1429
0.1478
0.1536
0.2476
0.1360
0.1200
0.0364
0.1096
0.2444
0.1348
0.1536
0.2476
0.1318
0.1200
0.0364
0.1035
0.2272
0.1841
0.1568
0.2352
0.1190
0.0600
0.0333
0.1073
0.1915
0.1789
0.1600
0.1981
0.1063
0.1200
0.0400
0.1141
0.2063
Negative-
ideal
solution
0.1011
0.1024
0.1981
0.1020
0.0600
0.0333
According to the distances given in Table 7, we can calculate the relative
assessment matrix and assessment scores related to the steps 6 and 7 of the
CODAS method (with ). The results are presented in Table 8.
The calculated assessment values shows that the alternatives is prioritized as
. Therefore, we can select as the best alternative with respect to the
assessment performed by the CODAS method.
36
A New Combinative Distance-based Assessment (CODAS) Method for Multi-
Criteria Decision-making
_________________________________________________________________
Table 8. The relative assessment matrix and the assessment scores of
alternatives of Example 2
0.000
0.018
0.080
0.087
0.232
0.216
0.209
-0.150
-0.146
0.148
0.016
0.028
0.019
0.012
0.768
-0.018
0.000
0.012
0.054
0.199
0.184
0.176
-0.182
-0.179
0.115
-0.001
0.005
0.001
-0.006
0.363
-0.080
-0.012
0.000
0.008
0.152
0.136
0.129
-0.229
-0.226
0.068
-0.014
-0.007
-0.011
-0.018
-0.105
-0.087
-0.054
-0.008
0.000
0.145
0.129
0.122
-0.237
-0.233
0.061
-0.083
-0.016
-0.020
-0.049
-0.329
-0.232
-0.199
-0.152
-0.145
0.000
-0.002
-0.012
-0.381
-0.378
-0.084
-0.228
-0.204
-0.172
-0.194
-2.384
-0.216
-0.184
-0.136
-0.129
0.002
0.000
-0.010
-0.366
-0.363
-0.069
-0.212
-0.189
-0.157
-0.178
-2.207
-0.209
-0.176
-0.129
-0.122
0.012
0.010
0.000
-0.359
-0.355
-0.010
-0.205
-0.181
-0.149
-0.171
-2.043
0.150
0.182
0.229
0.237
0.381
0.366
0.359
0.000
0.000
0.297
0.154
0.177
0.209
0.187
2.929
0.146
0.179
0.226
0.233
0.378
0.363
0.355
0.000
0.000
0.294
0.151
0.174
0.206
0.184
2.890
-0.148
-0.115
-0.068
-0.061
0.084
0.069
0.010
-0.297
-0.294
0.000
-0.143
-0.120
-0.088
-0.110
-1.282
-0.016
0.001
0.014
0.083
0.228
0.212
0.205
-0.154
-0.151
0.143
0.000
0.006
0.002
-0.005
0.568
-0.028
-0.005
0.007
0.016
0.204
0.189
0.181
-0.177
-0.174
0.120
-0.006
0.000
-0.004
-0.011
0.313
-0.019
-0.001
0.011
0.020
0.172
0.157
0.149
-0.209
-0.206
0.088
-0.002
0.004
0.000
-0.007
0.157
-0.012
0.006
0.018
0.049
0.194
0.178
0.171
-0.187
-0.184
0.110
0.005
0.011
0.007
0.000
0.364
4. Comparative sensitivity analysis
To evaluate the stability and validity of the CODAS method, a comparative
sensitivity analysis is performed in this section. The problem that is considered in
this analysis is borrowed from Keshavarz Ghorabaee et al. (2015). In this problem
ten alternatives are assessed on seven criteria. To make the analysis, we choose
some commonly used MCDM methods for comparing the results of them with the
result of the proposed method. The chosen MCDM methods include WASPAS,
COPRAS, TOPSIS, VIKOR and EDAS. It should be noted that the TOPSIS
method has been proposed in different versions, and we use the version that
considered in the research of Opricovic and Tzeng (2004). For this comparative
analysis, ten sets of criteria weights are simulated. Data of the MCDM problem
and sets of criteria weights are shown in Tables 9 and 10, respectively. In the
MCDM problem, to are benefit criteria, and to are cost criteria. We
solve this problem using the CODAS and the selected MCDM methods in the
different sets of simulated criteria weights. The results are represented in Table 11.
37
Mehdi Keshavarz Ghorabaee, Edmundas Kazimieras Zavadskas, Zenonas Turskis,
Jurgita Antucheviciene
__________________________________________________________________
Table 9. Data of the MCDM problem for comparative sensitivity analysis
Alternatives
Criteria
23
264
2.37
0.05
167
8900
8.71
20
220
2.2
0.04
171
9100
8.23
17
231
1.98
0.15
192
10800
9.91
12
210
1.73
0.2
195
12300
10.21
15
243
2
0.14
187
12600
9.34
14
222
1.89
0.13
180
13200
9.22
21
262
2.43
0.06
160
10300
8.93
20
256
2.6
0.07
163
11400
8.44
19
266
2.1
0.06
157
11200
9.04
8
218
1.94
0.11
190
13400
10.11
Table 10. Simulated weights of criteria in different sets
Set 1
0.092
0.197
0.172
0.206
0.142
0.009
0.182
Set 2
0.215
0.156
0.174
0.172
0.092
0.151
0.041
Set 3
0.262
0.015
0.103
0.018
0.037
0.306
0.258
Set 4
0.086
0.258
0.011
0.118
0.105
0.207
0.215
Set 5
0.054
0.139
0.127
0.184
0.201
0.215
0.079
Set 6
0.198
0.192
0.049
0.035
0.145
0.279
0.102
Set 7
0.149
0.058
0.192
0.066
0.129
0.177
0.228
Set 8
0.303
0.174
0.044
0.047
0.082
0.268
0.082
Set 9
0.239
0.073
0.271
0.102
0.058
0.076
0.181
Set 10
0.119
0.089
0.208
0.146
0.136
0.228
0.072
38
A New Combinative Distance-based Assessment (CODAS) Method for Multi-
Criteria Decision-making
_________________________________________________________________
Table 11. The ranking results with different methods in different sets
Set
No.
Method
1
CODAS
2
1
7
10
6
8
3
4
5
9
WASPAS
1
2
8
10
6
7
3
4
5
9
COPRAS
1
2
8
10
6
7
3
4
5
9
TOPSIS
1
3
9
10
8
7
2
4
5
6
VIKOR
2
5
8
10
6
7
3
1
4
9
EDAS
1
3
9
10
6
7
2
4
5
8
2
CODAS
1
2
6
9
7
8
3
4
5
10
WASPAS
1
2
6
10
7
8
3
4
5
9
COPRAS
1
2
6
10
7
8
3
4
5
9
TOPSIS
1
3
6
10
8
7
2
4
5
9
VIKOR
1
5
6
9
7
8
2
3
4
10
EDAS
1
3
6
10
7
8
2
4
5
9
3
CODAS
1
2
6
9
7
8
3
4
5
10
WASPAS
1
2
6
9
7
8
3
4
5
10
COPRAS
1
2
6
9
7
8
3
4
5
10
TOPSIS
1
2
6
9
7
8
3
4
5
10
VIKOR
1
2
6
9
7
8
3
4
5
10
EDAS
1
2
6
9
7
8
3
4
5
10
4
CODAS
1
2
6
10
7
8
3
5
4
9
WASPAS
1
2
6
10
7
8
3
5
4
9
COPRAS
1
2
6
10
7
8
3
5
4
9
TOPSIS
1
3
7
10
8
9
2
5
4
6
VIKOR
1
5
7
10
6
8
2
4
3
9
EDAS
1
2
6
10
7
8
3
5
4
9
5
CODAS
2
1
6
10
7
8
3
5
4
9
WASPAS
1
2
6
10
7
8
3
5
4
9
COPRAS
1
2
6
10
7
8
3
5
4
9
TOPSIS
1
2
9
10
8
7
3
5
4
6
VIKOR
1
5
7
10
6
8
2
4
3
9
EDAS
1
2
6
10
7
8
3
4
5
9
6
CODAS
1
2
6
9
7
8
3
4
5
10
WASPAS
1
3
6
9
7
8
2
4
5
10
COPRAS
1
3
6
9
7
8
2
4
5
10
TOPSIS
1
3
6
9
7
8
2
4
5
10
VIKOR
1
5
6
8
7
9
2
4
3
10
EDAS
1
3
6
9
7
8
2
4
5
10
7
CODAS
1
2
6
9
7
8
4
3
5
10
WASPAS
1
2
6
9
7
8
3
4
5
10
COPRAS
1
2
6
10
7
8
3
4
5
9
TOPSIS
1
3
6
10
7
8
2
4
5
9
VIKOR
1
3
8
10
6
7
2
4
5
9
EDAS
1
2
6
10
7
8
3
4
5
9
8
CODAS
1
2
6
9
7
8
3
4
5
10
WASPAS
1
2
6
9
7
8
3
4
5
10
COPRAS
1
2
6
9
7
8
3
4
5
10
TOPSIS
1
3
6
9
7
8
2
4
5
10
VIKOR
1
3
6
9
7
8
2
5
4
10
EDAS
1
3
6
9
7
8
2
4
5
10
9
CODAS
1
2
6
9
7
8
3
4
5
10
WASPAS
1
2
6
9
7
8
3
4
5
10
COPRAS
1
2
6
10
7
8
3
4
5
9
TOPSIS
1
4
6
10
7
8
2
3
5
9
VIKOR
2
4
7
10
6
8
3
1
5
9
EDAS
1
4
6
10
7
8
2
3
5
9
10
CODAS
2
1
6
10
7
8
3
4
5
9
WASPAS
1
2
6
10
7
8
3
4
5
9
COPRAS
1
2
6
10
7
8
3
4
5
9
TOPSIS
1
3
7
10
9
8
2
4
5
6
VIKOR
1
3
6
9
7
8
2
4
5
10
EDAS
1
2
6
10
7
8
3
4
5
9
39
Mehdi Keshavarz Ghorabaee, Edmundas Kazimieras Zavadskas, Zenonas Turskis,
Jurgita Antucheviciene
__________________________________________________________________
To compare the ranking results obtained from the different methods, the
Spearman's rank correlation coefficient () is used. This is a suitable coefficient
when we have ordinal variables or ranked variables. Table 12 represents the
correlation coefficients that show the association between the results of the proposed
method and the selected MCDM methods. If this correlation coefficient is greater
than 0.8, the relationship between variables is very strong . As can be seen in Table
12, all values of are greater than 0.8. Therefore, we can confirm the validity and
stability of the results of the CODAS method.
Table 12. Correlation coefficients between the ranking results of the CODAS
and the other methods
Method
Set 1
Set 2
Set 3
Set 4
Set 5
Set 6
Set 7
Set 8
Set 9
Set 10
WASPAS
0.976
0.988
1
1
0.988
0.988
0.988
1
1
0.988
COPRAS
0.976
0.988
1
1
0.988
0.988
0.976
1
0.988
0.988
TOPSIS
0.855
0.964
1
0.915
0.867
0.988
0.952
0.988
0.952
0.879
VIKOR
0.830
0.927
1
0.915
0.867
0.903
0.915
0.976
0.891
0.952
EDAS
0.927
0.976
1
1
0.976
0.988
0.976
0.988
0.952
0.988
As previously mentioned, a threshold parameter () is used in the process of the
CODAS method. We suggest a value between 0.01 and 0.05 for this parameter.
However, we want to evaluate the effect of changing this parameter on the ranking
result of the CODAS methods. According to Table 12, the minimum value of the
Spearman's rank correlation coefficient is in the set 1 of criteria weights ().
So this set of criteria weights, which is more sensitive than the other sets, is selected
for analysis of changing the threshold parameter. We use fifteen values for this
parameter in the range of 0.01 to 1. The ranking results obtained by the CODAS
method in different values of are presented in Table 13. The graphical changes in
the ranking of alternatives are also depicted in Figure 2.
Table 13. Ranking results with different values of
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.015
0.2
0.3
0.5
1
2
2
2
2
2
2
2
2
5
5
1
2
2
2
2
1
1
1
1
1
1
1
1
1
2
2
1
1
1
1
7
7
7
7
7
7
7
9
7
7
7
7
7
7
7
10
10
10
10
10
10
10
10
10
9
10
10
10
10
10
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
8
8
8
8
8
8
8
7
8
8
8
8
8
8
8
3
3
3
3
3
3
3
5
3
1
3
3
3
3
3
4
4
4
4
4
4
4
3
2
3
4
4
4
4
4
5
5
5
5
5
5
5
4
4
4
5
5
5
5
5
9
9
9
9
9
9
9
8
9
10
9
9
9
9
9
40
A New Combinative Distance-based Assessment (CODAS) Method for Multi-
Criteria Decision-making
_________________________________________________________________
Figure 2. Effect of changing the parameter on ranking of alternatives
According to Table 13 and Figure 2, we can see the instability in the ranking
of alternatives when the parameter is varied from 0.07 to 0.2. However,
changing the parameter has not a great effect on the ranking of alternatives that
can undermine the validity of the results. Therefore, we can confirm the results of
the CODAS method.
5. Conclusion
Multi-criteria decision-making has increasingly been applied to many real-world
problems. Many methods and techniques have also been proposed and improved
by researchers in the recent years. In this paper, we have proposed a new
combinative distance-based assessment (CODAS) method to handle multi-criteria
decision-making problems. To assess the alternatives on multiple criteria, the
proposed method uses two types of distances: Euclidean distance and Taxicab
distance. These distances are calculated according to the negative-ideal solution.
Therefore, the alternative which has greater distances is more desirable. However,
in this process, the Euclidean distance is considered as a primary measure and the
Taxicab distance is considered as a secondary measure. Two numerical examples
have been used to illustrate the CODAS method. Moreover, we have performed a
comparative sensitivity analysis to demonstrate the validity and stability of the
proposed method. In this analysis, ten sets of criteria weights are simulated and
the results of the CODAS method have been compared with the results of some
existing MCDM methods. According to the results of this analysis, we can say
that the proposed method is efficient to deal with MCDM problems.
41
Mehdi Keshavarz Ghorabaee, Edmundas Kazimieras Zavadskas, Zenonas Turskis,
Jurgita Antucheviciene
__________________________________________________________________
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