Plane Thermoelastic Waves in Infinite Half-Space Caused
Decision-making: Applications in Management and Engineering
Vol. 3, Issue 1, 2020, pp. 60-78.
ISSN: 2560-6018
eISSN: 2620-0104
DOI: https://doi.org/10.31181/dmame2003078m
* Corresponding author.
E-mail addresses: Dmitri_Muravev@sjtu.edu.cn (D. Muravev),
nemanja.mijic@gmail.com (N. Mijic)
A Novel Integrated Provider Selection Multicriteria
Model: The BWM-MABAC Model
Dmitri Muravev 1,2* and Nemanja Mijic3
1 School of Naval Architecture, Ocean and Civil Engineering and State Key Laboratory of
Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China
2 Nosov Magnitogorsk State Technical University, Department of Logistics and
Transportation Systems Management, Mining Engineering and Transport Institute,
Magnitogorsk, Russia
3 University of Defence in Belgrade, Department of Logistics, Belgrade, Serbia
Received: 12 December 2019;
Accepted: 4 March 2020;
Available online: 14 March 2020.
Original scientific paper
Abstract: The supply chain is a very complex area aimed at obtaining the
optimum from the point of view of all participants. In order to achieve the
overall optimum and satisfaction of all participants, it is necessary to make an
adequate evaluation and selection of providers at the initial stage. In this
paper, the selection of providers is based on a new approach in the field of
multicriteria decision-making. The weight coefficients were determined using
the Best-Worst Method (BWM), whereas provider evaluation and selection
were performed using the Multi-Attributive Border Approximation Area
Comparison (MABAC) method, which is one of more recent methods in this
field. In order to determine the stability of the model and the applicability of
the proposed hybrid BWM-MABAC model, the results were compared with the
MAIRCA and VIKOR models, and the results of the comparative analysis are
presented herein. In addition, a total of 18 different scenarios were formed in
the sensitivity analysis, in which the criteria change their original value. At the
end of the sensitivity analysis, the statistical dependence of the results was
determined using Spearman's correlation coefficient, which confirmed the
applicability of the proposed multicriteria approaches.
Key words: multicriteria decision-making, BWM, MABAC, MAIRCA, VIKOR,
supply chain.
mailto:Dmitri_Muravev@sjtu.edu.cn
mailto:nemanja.mijic@gmail.com
A Novel Integrated Provider Selection Multicriteria Model: The BWM-MABAC Model
61
1. Introduction
When looking at the efficiency of the entire supply chain, it is impossible not to
notice that it largely depends on an adequate selection of providers, because this
process is one of the most important factors that directly affect the performance of a
company. With a proper valuation and selection of the right provider, this logistics
subsystem can efficiently carry out tasks related to supplying a company, since the
right providers can meet the requirements and needs posed in the procurement
subsystem, which on their part relate to the quality, price and quantity of goods,
delivery times and other deadlines, flexibility, reliability, and so on. A search for
providers in order to fulfill this is a permanent and primary task. In order to enable
the former, it is necessary to continuously collect and process providers’ data,
establish and maintain adequate links with them.
According to Soheilirad et al. (2017), provider selection is an important item when
speaking about management decisions, which considers several qualitative and
quantitative criteria. The importance of this process in organizations reflects in the
formation of the final product price, since the price of raw materials plays a significant
role in the price of the final product (Bai and Sarkis 2010; Ramanathan 2007). Provider
selection is one of more important items in the supply chain management (Zhong and
Yao, 2017), while managing and developing provider relationships is a critical issue
for achieving competitive advantage (Bai and Sarkis, 2011). Considering the fact that
provider selection in the supply chain is multicriteria group decision-making,
according to Zolfani et al. (2012), it is necessary for managers to know the most
appropriate method for them to use so as to choose the right provider. This is
especially true when we know that modern supply chains require that stringent
requirements should be met, for which reason managers are faced with a difficult task
of properly evaluating potential providers that rarely price the products that affect the
company’s competitiveness in the market (Cox and Ireland, 2002).
Every day, a large number of decisions are made on the basis of certain criteria, so
it can be safely said that multicriteria decision-making (MCDM) plays a significant role
in real-life problems, including logistical problems. Particularly important is the role
that multicriteria decision-making plays in the decision-making process that affects
the business system or the environment. Therefore, MCDM is an efficient systematic
and quantitative way to solve vital logistical problems, including supply chain
management. The increasing use of multicriteria decision-making methods has
contributed to the increasing popularity of this field on a daily basis (Zavadskas et al.,
2014a).
This paper presents a hybrid MCDM vendor evaluation model, which is based on
the application of the two models, i.e. the Best-Worst Method (BWM) and the Multi-
Attributive Border Approximation Area Comparison (MABAC) model. The BWM
model (Rezaei, 2015) was used to determine criteria weights, while the MABAC model
was used to evaluate the providers. The BWM and MABAC models had been opted for
because of the many advantages that recommend them for use in this field. In addition
to this, no application of the hybrid BWM-MABAC model has been reported in the
literature yet, which enriches the methodology for provider evaluation and selection.
The paper is structured into several chapters. In the second section of the paper, a
literature analysis is performed through an overview of the existing multicriteria
decision-making methods in the field of supply chains. The analysis of the applied
methods, as well as the criteria for the work done in the field of the selection of
transport service providers is carried out. Based on the data obtained from the
analysis of the work done, a new multicriteria decision-making model is proposed, as
Muravev and Mijic./Dec-Mak. Appl. Manag. Eng. 3 (1) (2020) 60-78
62
well as the criteria that will be used to select the right transport service provider. In
the third section, the mathematical foundations of the hybrid BWM-MABAC model are
presented. In the fourth section, the testing of the proposed model is performed on a
real-life example, in which the evaluation of providers at the Ministry of Defense of the
Republic of Serbia is conducted. In the fourth section of the paper, the results are
validated in three phases. The first phase involves comparing the results of the BWM-
MABAC model with those obtained by applying other multicriteria decision-making
methods. In the second phase, the validation is performed in a dynamic environment
by applying dynamic initial decision matrices. The third phase of the validation
includes a sensitivity analysis of the change in the weights of the criteria coefficients.
The fifth section contains the conclusive considerations, where the presented
conclusions are derived from the conducted research and the suggestions for further
research.
2. Literature Review
According to a large number of authors, provider selection is one of the most
challenging management problems in logistics (Stojicic et al., 2019). As a result, a
number of methods for the evaluation of transport service providers have been
developed to date. In the literature (Fallahpour et al., 2017), the author uses the fuzzy
modifications of the Analytic Network Process (ANP) method and the Technique for
Order Preference by Similarity to Ideal Solution (TOPSIS). Govindan et al. (2013) used
the fuzzy TOPSIS method to rank the sustainable performance of a transport service
provider. In order to make a choice of a logistics provider from the sustainability
perspective while keeping an eye on the company’s goals, Dai & Blackhurst (2012)
introduced a new integrated approach based on the AHP and the Quality Function
Deployment (QFD) methods, with four hierarchical stages. Rezaei et al. (2016)
introduced a new approach to selecting a transport service provider, which consists
of three phases, the essential one being the phase in which the implementation of the
Best-Worst Method (BWM) is demonstrated. This approach can benefit companies
looking for new markets. Azadnia et al. (2013) propose an integrated approach to
choosing a logistics provider, which, in addition to the application of the fuzzy AHP
method, is also based on multi-objective mathematical programming, as well as the
rule-based weighted fuzzy method. Luthra et al. (2017) introduced an integrated
approach to selecting a transport service provider from the sustainability perspective.
The approach was implemented through a combination of the AHP and VIKOR
methods based on 22 criteria. Barata et al. (2014) demonstrated the application of
MCDM methods in the evaluation of the degree of the organizational sustainability of
a company.
Hsu et al. (2014) presented an approach based on several MCDM methods in order
to select a transport service provider from the environmental point of view, i.e. with
respect to carbon emissions. Also, Validi et al. (2014) ranked logistics providers and
traffic routes based on CO2 emissions by using the TOPSIS method. The evaluation of
the performance of logistics providers in the electronics industry is the topic of the
research conducted in the paper (Chatterjee et al., 2018) from the environmental point
of view as well. In this paper, the rough DEcision-MAking Trial and Evaluation
Laboratory (DEMATEL) model is used in combination with the rough Multi-Attribute
Ideal Real Comparative Analysis (MAIRCA) method. A quantitative assessment of the
performance of transport service providers is presented in the paper from the
sustainability perspective (Erol et al., 2011). In addition to MCDM methods, fuzzy
A Novel Integrated Provider Selection Multicriteria Model: The BWM-MABAC Model
63
techniques are used in this paper because of the presence of indeterminacy. More
specifically, the fuzzy Entropy and fuzzy Multi-Attribute Utility Theory (MAUT)
methods are used. Kusi-Sarpong et al. (2018) presented a framework for the ranking
and selection of sustainable innovation in logistics, given the fact that innovation plays
a very significant role in sustainability. This framework is based on the BWM method,
which has been tested and applied by several companies in India. Managing the
evaluation of providers from the sustainability perspective is significant for many
industries, and for logistics as well. Therefore, it is an increasingly common research
topic. Table 1 presents an analysis of the papers dedicated to this topic that have been
published in the last few years.
Table 1. The application of MCDM methods in supply chains
The problem solved by
applying MCDM methods
Applied Methods Literature
Sustainable provider
selection
FPP, Fuzzy TOPSIS, AHP,
VIKOR, Fuzzy ELECTRE,
Fuzzy AHP, Fuzzy
Multiplicative AHP, Fuzzy
SMART, Fuzzy VIKOR,
Fuzzy MAUT, BWM, AHP-
QFD, Delphi, Fuzzy
DEMATEL
Dai et al. (2012); Erol et
al. (2011); Fallahpour
et al. (2017); Govindan
et al. (2013); Kusi-
Sarpong et al. (2018);
Luthra et al. (2017);
Luthra et al. (2018);
Padhi et al. (2018)
Provider evaluation from the
environmental point of view
Fuzzy Entropy-TOPSIS,
ELECTRE TRI
Barata et al. (2014);
Zhao et al. (2014)
Assessing provider
performance in supply chains
DEMATEL, ANP, MAIRCA,
fuzzy Delphi, DANP, VIKOR,
TOPSIS
Chatterjee et al. (2018);
Hsu et al. (2014); Validi
et al. (2014)
Suggesting an innovative
provider selection
methodology
BWM, DANP, DEMATEL,
VIKOR, MULTIMOORA,
AHP, Fuzzy TOPSIS
Das & Shaw (2017);
Entezaminia et al.
(2016); Kuo et al.
(2015); Liu et al.
(2018); Rezaei et al.
(2016)
An integrated approach to
identifying and analyzing
criteria and alternatives
under uncertain conditions
Grey-DEMATEL, FAHP
Azadnia et al. (2015);
Su et al. (2016)
Also, MCDM methods have been applied in solving a broad range of problems in
the logistics field. Depending on a specific problem in the logistics field, various MCDM
methods have been used, such as: AHP, TOPSIS, VIKOR, MAIRCA, ELECTRE, fuzzy AHP,
fuzzy TOPSIS and DEMATEL (Alikhani et al., 2019; Ahmadi et al., 2019; Buyukozkan &
Gocer, 2017). According to the table, it is possible to see that the AHP, TOPSIS and
fuzzy TOPSIS methods have been applied to the greatest extent in the logistics field.
The BWM and Fuzzy Preferences Programming (FPP) methods have most commonly
been used to determine weight coefficients. This literature review allows us to see that
the BWM and MABAC models have not yet been applied in providers’ supply chain.
Due to the aforementioned fact, as well as the numerous advantages of the BWM and
MABAC models (Pamucar and Cirovic, 2015; Gigovic et al., 2017; Pamucar et al., 2018a,
2018b), a hybrid BWM-MABAC model is proposed in this paper. Based on the search
Muravev and Mijic./Dec-Mak. Appl. Manag. Eng. 3 (1) (2020) 60-78
64
of the most important index databases of international journals, Table 2 presents an
analysis of the criteria that have been applied in the MCDM methods for optimizing
the selection of providers in supply chains.
T
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. A
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A Novel Integrated Provider Selection Multicriteria Model: The BWM-MABAC Model
65
Based on the literature review presented, the following criteria are identified in
order to address the provider evaluation issues in supply chains: C1 – Price (min.), C2
– Quality (max.), C3 – Service and Delivery (max.), C4 – Flexibility (max.), C5 –
Technological capabilities (max.), C6 – Trust (max.), C7 – Relationship (max.), C8 - Risk
(min.) and C9 – Innovation (max.).
Based on Table 2, a conclusion can be drawn that the criteria C1, C2 and C3, i.e. the
price, the quality and the service and delivery, respectively, are indispensable when
choosing a provider, regardless of the specific situation. Another important criterion
is C5, i.e. technological capabilities, which was applied in eight papers, only to be
followed by the criteria C6 and C8, i.e. trust and risk, respectively, which were applied
in six papers, and C4 and C7, i.e. flexibility and relationship (trust), respectively, which
were applied in five papers. As a criterion, innovation (C9) is used least in solving the
transport service provider selection problem, having only been used in two papers. In
this paper, all the analyzed criteria will be included for the purpose of a comprehensive
consideration of the problem.
3. The BWM-MABAC Hybrid Provider Evaluation Model
In this paper, a new hybrid BWM-MABAC model (Figure 1) for provider evaluation
is presented. The model includes the application of the two methods: (1) the BWM
method, which is used to determine the weight criteria, and (2) the MABAC method,
which is used to evaluate the ranking of the alternatives.
Experts’ opinions
Determining B and W criteria
No
Yes
Comparison in criteria pairs: Forming
of BO and OW comparison vectors
Determining level of consistency for -
BWM – If CR satisfied?
Aggregation of BO and OW vectors
of experts - Aggregated BO and OW
vectors
Evaluation of optimum value of
coefficient criteria
Experts’ evaluation of alternatives
Aggregation of experts’ opinion into
matrix Xk
Normalization of interval rough home
matrix of decision making
Determining elements of weighted
matrix
Determining the matrix of BAA
Ranking of alternatives using
MABAC method
Sensitivity analysis – modification of wj through scenarios
Rank reversal problem
Spearman coefficient of correlation of ranks
Selection of best alternative
P
h
a
se
1
:
In
te
rv
a
l
ro
u
g
h
B
W
M
P
h
a
se
2
:
M
A
B
A
C
m
e
th
o
d
Optimal weights coefficients
Figure 1. The BWM-MABAC model
Muravev and Mijic./Dec-Mak. Appl. Manag. Eng. 3 (1) (2020) 60-78
66
The model is implemented through three phases. In the first phase, the criteria are
evaluated by using the BWM method. Based on a questionnaire and the experts’
evaluation, the ranking of the criteria and the comparison of the ranking criteria are
made. The values of the weight coefficients of the criteria are obtained as the output
from the BWM method. The output data from the BWM are further processed through
the MABAC method algorithm. In the second phase, the alternatives are ranked by
applying the MABAC method. In the third phase, the results are validated.
3.1. The BWM algorithm
Like the AHP model, the BWM is one of the models of a more recent data based on
comparison principles in criteria pairs and the validation of results through a
deviation from the maximum consistency (Pamucar et al., 2018). The BWM is a model
which eliminates the drawbacks of the AHP model to some extent. The advantages of
the implementation of the BWM are a small number of comparisons in criteria pairs,
the ability to validate results by defining the consistency deviation (CR) of the
comparisons, and taking into consideration transitivity during comparisons in criteria
pairs. Also, the BWM methodological procedure eliminates the problem of a
comparison redundancy in criteria pairs, which is present in some subjective criteria
weight determination models.
Suppose that there are the n evaluation criteria in the multicriteria model that are
denoted as wj, j = 1,2, ..., n and that their weight coefficients need to be determined.
Subjective weighting models based on comparisons in criteria pairs require that the
decision-maker should determine the degree of the influence of the criterion i on the
criterion j as well. In accordance with the defined settings, the following section
introduces the BWM algorithm (Pamucar et al., 2018).
Algorithm: BWM
Input: The experts’ comparison in criteria pairs
Output: The optimal values of criteria/sub-criteria weight coefficients
Step 1: The experts’ ranking of the criteria/sub-criteria.
Step 2: The determination of the BO and OW vectors of the comparative
significance of the evaluation criteria.
Step 4: Defining a model for the determination of the final values of the weighting
coefficients of the evaluation criteria:
1
min
. .
; ;
1; 0; 1, 2,...,
jB
Bj jW
j W
n
j jj
s t
ww
a a
w w
w w j n
Step 5: The calculation of the final values of the evaluation criteria/sub-criteria
1 2, ,...,
T
n
w w w
3.2. The MABAC model algorithm
The MABAC method is one of more recent methods (Pamucar and Cirovic, 2015).
To date, it has been broadly applied in solving numerous problems in the multicriteria
decision-making field. The basic assumption of the MABAC method is that the distance
A Novel Integrated Provider Selection Multicriteria Model: The BWM-MABAC Model
67
of the criterion function of the observed alternative from the boundary approximate
area should be defined. The following section introduces the MABAC method
algorithm (Pamucar and Cirovic, 2015).
Algorithm: MABAC method
Input: BWM weights and the initial decision matrix
Output: The ranking of the alternatives
Step 1: The formation of the initial decision matrix (X).
Step 2: The normalization of the elements of the initial matrix.
Step 3: The calculation of the elements of the weighted matrix (V).
Step 4: The determination of the matrix of the boundary approximate regions
(G). The matrix of the boundary approximation domains G (12) format 1n x
:
1 2
1 2
...
...
n
n
C C C
G g g g , where
1/
1
m
m
i ij
j
g v
( ijv represents the elements of
the weighted matrix).
Step 5: The calculation of the elements of the distance matrix of the
alternatives from the boundary approximation region (Q). The distance of
the alternatives from the boundary approximation region (qij) is determined
as the difference between the elements of the constrained matrix (V) and
the value of the boundary approximation regions (G).
11 1 12 2 1 11 12 1
21 1 22 2 2 21 22 2
1 1 2 2 1 2
... ...
...
... ... ... ... ... ... ... ...
... ...
n n n
n n n
m m mn n m m mn
v g v g v g q q q
v g v g v g q q q
Q
v g v g v g q q q
Step 6: The ranking of the alternatives. The calculation of the values of the
criteria functions is obtained as the sum of the distances of the alternatives
from the boundary approximation regions ( iq ).
4. The Application of the BWM-MABAC Model
The model has been tested on a real issue, which includes the evaluation of the
providers of spare parts for transport vehicles at the Ministry of Defense of the
Republic of Serbia. A total of eight providers were considered, whose names were not
included in this survey because of the confidentiality of the tender documents. The
study involved four experts with at least 10 years of experience in supply chain
evaluation.
In the first phase of the implementation of the BWM-MABAC model, it is necessary
to define the weight coefficients of the criteria by using the BWM. The first phase of
the BWM involves the experts’ ranking of the criteria. Based on the significance of the
criteria presented in the BO and OW vectors, a nonlinear model was formed for the
calculation of the optimal values of the weight coefficients. A total of four models were
formed, one for each expert. The following section provides the model for the
calculation of the optimal values of the weighting coefficients for the first expert.
Muravev and Mijic./Dec-Mak. Appl. Manag. Eng. 3 (1) (2020) 60-78
68
1 1 1
2 3 7
3 74
2 2 2
7
1
1 min
. .
7 ; 2 ;....; 3 ;
6.00 ; 3 ;;...; 2 ;
1; 0; 1, 2,..., 7j j
j
Expert
s t
w w w
w w w
w ww
w w w
w w j
By solving the nonlinear models, the optimal values of the weight coefficients for
each expert were defined, as in Table 4.
Table 4. The criteria weight coefficients
Experts E1 E2 E3 E4 Medium
C
ri
te
ri
a
w
e
ig
h
t
co
e
ff
ic
ie
n
ts
C1 0.311 0.100 0.117 0.030 0.1395
C2 0.207 0.120 0.180 0.149 0.1640
C3 0.076 0.299 0.269 0.149 0.1982
C4 0.104 0.067 0.096 0.149 0.1040
C5 0.060 0.086 0.054 0.075 0.0687
C6 0.065 0.120 0.077 0.075 0.0842
C7 0.052 0.050 0.045 0.149 0.0740
C8 0.086 0.075 0.128 0.149 0.1095
C9 0.039 0.054 0.034 0.075 0.0505
By averaging the obtained values, the optimal values of the weight coefficients of
the criteria were defined, which were further used to evaluate providers by applying
the MABAC method. The paper evaluates a total of eight providers, designated A1
through A8. Based on the provider data, the MABAC model was implemented through
the following six steps:
Step 1. In the MABAC model, the initial decision matrix (X) was the starting point:
3 5 6 7 8 91 2 4
max max max max min maxmin max max
1 65 23 56 53 54 95 53 59 62
2 45 29 50 49 49 87 63 7344
3 56 43 70 57 59 52 5941 41
4 70 35 82 43 91 93 38 6641
5 82 68 63 95 35 81 79 39 49
6 90 56 71 80 62 71 91 23 81
7 48 39 63 74 25 66 66 72 52
8 76 56 59 61 53 6
f f f f f ff f f
A
A
A
X A
A
A
A
A
7 59 46 77
Step 2. Using linear normalization, the elements of the matrix X were normalized,
thus obtaining the normalized matrix N:
A Novel Integrated Provider Selection Multicriteria Model: The BWM-MABAC Model
69
3 5 6 7 8 91 2 4
max max max max min maxmin max max
1 0.556 0 0.188 0.192 0.439 0.283 0.265 0.4061
2 0.133 0 0.115 0.364 0 0.925 0.184 0.7501
3 0.756 0.444 0.625 0.269 0.252 0.294 0.057 0.408 0.313
4 0.444 0.267 0 0.961 0 0.61 1
5
6
7
8
f f f f f ff f f
A
A
A
N A
A
A
A
A
33 0.531
0.111 0.406 0.152 0.725 0.774 0.673 01 1
0 0.733 0.656 0.712 0.561 0.529 1 1 1
0.933 0.356 0.406 0.596 0 0.431 0.528 0 0.094
0.311 0.733 0.281 0.346 0.424 0.451 0.396 0.531 0.875
Step 3. By multiplying the optimal values of the weighting coefficients obtained by
applying the BWM by the elements of the normalized matrix N a weighted normalized
matrix V was obtained.
3 5 6 7 8 91 2 4
max max max max min maxmin max max
1 0.217 0.164 0.235 0.124 0.099 0.168 0.095 0.139 0.071
2 0.279 0.186 0.198 0.116 0.094 0.084 0.142 0.130 0.088
3 0.245 0.237 0.322 0.132 0.085 0.109 0.078 0.154 0.066
4 0.
5
6
7
8
f f f f f ff f f
A
A
A
V A
A
A
A
A
202 0.208 0.396 0.104 0.137 0.165 0.074 0.179 0.077
0.155 0.328 0.279 0.208 0.079 0.145 0.131 0.183 0.051
0.140 0.284 0.328 0.178 0.107 0.129 0.148 0.219 0.101
0.270 0.222 0.279 0.166 0.069 0.121 0.113 0.110 0.055
0.183 0.284 0.254 0.140 0.098 0.122 0.103 0.168 0.095
Step 4. In Step 4, the defined matrix of the boundary approximate regions (G) was
approached. The boundary approximation area (GAO) for each criterion was
determined by geometrically averaging the values of the matrix V.
3 5 6 7 8 91 2 4
0.2055 0.2335 0.2807 0.1425 0.0942 0.1276 0.1074 0.1568 0.0736
C C C C C CC C C
G
Step 5. In this step, the distance of the elements V of the matrix from the matrix G
was calculated. Thus, the matrix Q, which represents the distance of the alternatives
from the GAO, was obtained.
3 5 6 7 8 91 2 4
max max max max min maxmin max max
1 0.011 0.070 0.045 0.018 0.005 0.041 0.012 0.018 0.003
2 0.073 0.048 0.083 0.026 0.001 0.043 0.035 0.027 0.015
3 0.039 0.003 0.041 0.010 0.009 0.019 0
4
5
6
7
8
f f f f f ff f f
A
A
A
Q A
A
A
A
A
.029 0.003 0.007
0.004 0.026 0.116 0.038 0.043 0.037 0.033 0.022 0.004
0.051 0.094 0.002 0.066 0.015 0.018 0.024 0.026 0.023
0.066 0.051 0.048 0.036 0.013 0.001 0.041 0.062 0.027
0.064 0.011 0.002 0.024 0.025 0.007 0.006
0.047 0.018
0.023 0.051 0.027 0.0020 0.004 0.005 0.004 0.011 0.021
Muravev and Mijic./Dec-Mak. Appl. Manag. Eng. 3 (1) (2020) 60-78
70
Step 6. The ranking of the alternatives was performed based on the value of the
alternative score functions. The criteria functions of the alternatives were obtained by
summing up the elements of the matrix Q by rows. Thus, the values of the criteria
functions of the alternatives were obtained for each provider:
S2 = -0.104 S3 = 0.007 S4 = 0.120 S5 = 0.137 S6 = 0.212 S7 = -0.018 S8 = 0.025
Based on the values of the criteria functions, the final ranking of the alternatives
was defined as: A6 > A5 > A4 > A8 > A3 > A7 > A2 > A1.
Before making a decision, it is necessary to evaluate the reliability of the results
obtained. The validation of the results of the BWM-MABAC model was carried out
through three phases. In the first phase, the initial ranking of the alternatives was
compared with that of the other MCDM models, as in Figure 2. Since the MABAC
method uses linear normalization, the Multi Attributive Ideal-Real Comparative
Analysis (MAIRCA) method and the Multicriteria Compromise Ranking (VIKOR)
method also have linear normalization.
Figure 2. The ranks of the alternatives
According to the presented methods, the ranking of the alternatives shows that the
alternative A6 remained the first-ranked by all the methods. The same rank was
obtained by the MAIRCA method as it was by the MABAC method, whereas the ranking
changed with the VIKOR method (the alternative A1 replaced its rank with the
alternative A2, and the alternative A3 replaced the rank with te alternative A8). In
order to determine the statistical significance between the rankings obtained by the
BWM-MABAC model and the other approaches, the Spearman correlation coefficient
(SCC) was used. This correlation coefficient is a simple linear correlation coefficient
between the ranks. The Spearman rank correlation coefficient is a non-parametric
method for the estimation of the strength of the association that is applied when data
for at least one variable are given as ordinal data or a rank, when at least one variable
has no normal distribution and when the relationship between the variables is not
linear. The results of the ranking comparisons by using the SCC are given in Table 5.
0
1
2
3
4
5
6
7
8
9
А1 А2 А3 А4 А5 А6 А7 А8
MABAC
MAIRCA
VIKOR
A Novel Integrated Provider Selection Multicriteria Model: The BWM-MABAC Model
71
Table 5. The rank correlation of the tested methods
Method MCDM MABAC MAIRCA VIKOR
SCC 1.000 1.000 0.952
Table 5 allows us to see that the MABAC and MAIRCA methods are in a complete
correlation. Also, the VIKOR method shows a high correlation compared to the MABAC
method. Given the fact that, in this particular case, all the SCC values are significantly
higher than 0.9 (exceptional correlation) and the mean value is 0.976, it can be
concluded that there is a very large correlation (closeness) between the proposed
model and the other MCDM methods. In doing so, we can conclude that the proposed
ranking is validated and credible.
In the second stage of the results validation, a performance analysis of the
proposed model was conducted under the conditions of the dynamic initial matrix. In
the dynamic starting matrix for each scenario, the number of the alternatives was
changed and the obtained ranks were analyzed. Scenarios are formed for situations
where one inferior alternative is removed from subsequent considerations, while the
remaining dominant alternatives are ranked according to a newly-acquired initial
decision matrix. In this study, the initial solution A6> A5> A4> A8> A3> A7> A2> A1
was obtained. Clearly, the alternative A1 is the worst option. In the first scenario, the
alternative A1 was eliminated from the list of the alternatives and a new decision
matrix with a total of seven alternatives was obtained. The new decision matrix was
re-solved by using the BWM-MABAC model. In the following scenario, the next worst
alternative was eliminated and the remaining alternatives were ranked. Thus, a total
of seven scenarios were formed, which are shown in Table 6.
Table 6. The ranks of the alternatives within the dynamic decision matrix
Scenario Rang
S1 A6>A5>A4>A8>A3>A7>A2>A1
S2 A6>A5>A4>A8>A3>A7>A2
S3 A6>A5>A4>A8>A3>A7
S4 A6>A5>A4>A8>A3
S5 A6>A5>A4>A8
S6 A6>A5>A4
S7 A6>A5
It is clear from Table 6 that, when the worst-case alternative is eliminated, there is
no change in the best-ranked alternative in the rearranged matrix. Based on this, it can
be concluded that the BWM-MABAC model does not lead to a rank reversal among the
alternatives. The alternative A6 remained the best-ranked across all the scenarios,
thus confirming the robustness and accuracy of the resulting rankings of the
alternatives in the dynamic environment.
Since the results of multicriteria decision-making depend on the values of the
weighting coefficients of the evaluation criteria, an analysis of the sensitivity of the
results to a change in the criteria weights was performed. The analysis of the
sensitivity of the rankings of the alternatives to changes in the weight coefficients of
the criteria was conducted through the 18 scenarios given in Table 7.
Muravev and Mijic./Dec-Mak. Appl. Manag. Eng. 3 (1) (2020) 60-78
72
Table 7. The sensitivity analysis scenarios
Scenario Criteria weights Scenario Criteria weights
S1
wc1=1.25× wc11(OV);
wci=0.25× wci(OV)
S10
wc1=1.55× wc11(OV);
wci=0.55× wci(OV)
S2
wc2=1.25× wc11(OV);
wci=0.25× wci(OV)
S11
wc2=1.55× wc11(OV);
wci=0.55× wci(OV)
S3
wc3=1.25× wc11(OV);
wci=0.25× wci(OV)
S12
wc3=1.55× wc11(OV);
wci=0.55× wci(OV)
S4
wc4=1.25× wc11(OV);
wci=0.25× wci(OV)
S13
wc4=1.55× wc11(OV);
wci=0.55× wci(OV)
S5
wc5=1.25× wc11(OV);
wci=0.25× wci(OV)
S14
wc5=1.55× wc11(OV);
wci=0.55× wci(OV)
S6
wc6=1.25× wc11(OV);
wci=0.25× wci(OV)
S15
wc6=1.55× wc11(OV);
wci=0.55× wci(OV)
S7
wc7=1.25× wc11(OV);
wci=0.25× wci(OV)
S16
wc7=1.55× wc11(OV);
wci=0.55× wci(OV)
S8
wc8=1.25× wc11(OV);
wci=0.25× wci(OV)
S17
wc8=1.55× wc11(OV);
wci=0.55× wci(OV)
S9
wc9=1.25× wc11(OV);
wci=0.25× wci(OV)
S18
wc9=1.55× wc11(OV);
wci=0.55× wci(OV)
*OV (old value)
The sensitivity analysis scenarios for the change in the criteria weights are grouped
into two groups. Within each group, the weighting coefficients of the criteria were
increased by 25% and 55%, respectively. In each of the 18 scenarios, one criterion was
favored within the two groups, by which the weight coefficient increased by the
indicated values. In the same scenario, the weighting coefficients were reduced by
75% (S1-S9) and 45% (S10-S18), respectively. The changes in the ranking of the
alternatives across the 18 scenarios are shown in Table 8.
Table 8. The changes in the ranking due to the changes in the criteria
weights
Scenario Rank
S1 A7>A2>A3>A4>A6>A1>A5>A8
S2 A5>A6>A8>A3>A4>A7>A2>A1
S3 A4>A6>A3>A5>A7>A8>A1>A2
S4 A5>A6>A7>A8>A4>A3>A1>A2
S5 A4>A6>A5>A8>A3>A1>A2>A7
S6 A4>A6>A5>A1>A8>A7>A3>A2
S7 A6>A5>A2>A8>A7>A4>A3>A1
S8 A6>A5>A4>A8>A3>A1>A7>A2
S9 A6>A4>A8>A5>A3>A2>A7>A1
S10 A4>A7>A6>A3>A5>A2>A8>A1
S11 A5>A6>A8>A4>A3>A7>A2>A1
S12 A4>A6>A5>A3>A7>A8>A1>A2
S13 A6>A5>A4>A7>A8>A3>A1>A2
A Novel Integrated Provider Selection Multicriteria Model: The BWM-MABAC Model
73
Scenario Rank
S14 A6>A4>A5>A8>A3>A7>A1>A2
S15 A6>A4>A5>A8>A3>A7>A1>A2
S16 A6>A5>A4>A8>A7>A2>A3>A1
S17 A6>A5>A4>A8>A3>A7>A1>A2
S18 A6>A4>A5>A8>A3>A7>A2>A1
The results show that assigning different criteria weights across the 18 scenarios
presented does not lead to a significant change in the ranking of the alternatives. In
the scenarios S2-S9 and S11-S18, the fact that the alternative A6 ranks either first or
second is noticed. In the scenarios S1 and S10, the alternative A6 is not among the top
two alternatives, due to the high value of the A6 provider engagement price, whereas
in the scenarios S1 and S10, the impact of the weighting factor on the final decision
only increased.
The results (Table 8) show that assigning different weights to the criteria across
the scenarios leads to a change in the rank of the alternatives, thus confirming that the
model is sensitive to changes in weight coefficients. The following section compares
the ranks in Table 8 with the initial ranks. The SCC values are shown in Figure 3.
Based on Figure 3, it can be concluded that there is a high rank correlation in the
12 scenarios, since the SCC value is greater than 0.80, whereas in the four scenarios,
that correlation is satisfactory, i.e. it is greater than 0.50. In the two scenarios, the SCC
value is below 0.50. However, the mean SCC across the scenarios is 0.778, which shows
a satisfactory average correlation. Based on this, it can be concluded that there is a
satisfactory closeness of the ranks and the proposed ranking is validated and credible.
-0,400
-0,200
0,000
0,200
0,400
0,600
0,800
1,000
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
S17
S18
Figure 3. The correlation of the ranks
Muravev and Mijic./Dec-Mak. Appl. Manag. Eng. 3 (1) (2020) 60-78
74
5. Conclusion
The multicriteria model presented in this paper is an integration of the BWM and
MABAC methods, where the BWM was used to calculate the values of the criteria
weights, while the MABAC method was applied for provider evaluation and selection.
The model was verified through the provider selection process in a real system, based
on the nine criteria. The results show that Provider 6 is the best solution in all the
scenarios including different criteria values, except in the two scenarios in which the
price criterion was favored. The analysis of the results has shown that the obtained
ranks of the alternatives of the BWM-MABAC model completely correlate with the
ranks of the other multicriteria models which they were compared with.
One of the contributions of this paper is the BWM-MABAC model that provides us
with an objective aggregation of experts’ decisions. In addition, another contribution
of the paper reflects in the improvement of the methodology of provider evaluation
and selection through a new hybrid multicriteria model. No use of this or a similar
approach in the selection of providers has been seen in the literature analysis.
By applying the developed approach, it is possible to approach multicriteria
decision-making in an easy way and evaluate and select those providers who have a
significant impact on the achievement of the efficiency of the whole of the supply chain.
The four-stage model may also be applied so as to make other decisions. It is applicable
in the provider evaluation process in all areas and may be particularly suitable for
manufacturing companies. The flexibility of the model reflects in the fact that it can be
verified by integrating any multicriteria decision-making methods. Further research
related to this paper pertains to the application of uncertainty theories (fuzzy, rough,
neutrosophic, etc.) together with this and other multicriteria methods in this field.
Author Contributions: Each author has participated and contributed sufficiently to
take public responsibility for appropriate portions of the content.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflicts of interest.
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