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Economic Research-Ekonomska Istraživanja
ISSN: 1331-677X (Print) 1848-9664 (Online) Journal homepage: http://www.tandfonline.com/loi/rero20
Decision-making under uncertainty – the
integrated approach of the AHP and Bayesian
analysis
Predrag Mimovi ć, Jelena Stankovi ć & Vesna Jankovi ć Mili ć
To cite this article: Predrag Mimovi ć, Jelena Stankovi ć & Vesna Jankovi ć Milić (2015)
Decision-making under uncertainty – the integrated approach of the AHP and
Bayesian analysis, Economic Research-Ekonomska Istraživanja, 28:1, 868-878, DOI:
10.1080/1331677X.2015.1092309
To link to this article: http://dx.doi.org/10.1080/1331677X.2015.1092309
© 2015 The Author(s). Published by Taylor &
Francis
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Decision-making under uncertainty –the integrated approach of the
AHP and Bayesian analysis
Predrag Mimovi ća*, Jelena Stankovi ćband Vesna Jankovi ćMilićb
aUniversity of Kragujevac, Faculty of Economics, Djure Pucara Starog 3, 34000 Kragujevac,
Serbia.;bUniversity of Ni š, Faculty of Economics, Trg kralja Aleksandra Ujedinitelja 11, 18000
Niš, Serbia
(Received 18 March 2013; accepted 6 February 2015 )
In situations where it is necessary to perform a large number of experiments in order
to collect adequate statistical data which require expert analysis and assessment, thereis a need to de fine a model that will include and coordinate statistical data and
experts ’opinions. This article points out the new integrated application of the
Analytic Hierarchy Process (AHP) and Bayesian analysis, in the sense that the
Bayes ’formula can improve the accuracy of input data for the Analytical Hierarchy
Process, and vice versa, AHP can provide objecti fied inputs for the Bayesian formula
in situations where the statistical estimates of probability are not possible. In this
sense, the AHP can be considered as the Bayesian process that allows decision-mak-
ers to objectify their decisions and formalise the decision process through pairwisecomparison of elements.
Keywords: Analytic Hierarchy Process (AHP); decision-making; probability; utility;
expected utility (EU); Bayesian analysis
JEL classi fication: C11, C44, C61
1. Introduction
Decision analysis is a methodology developed in the 1960s, which quanti fies the ele-
ments of a decision-making process in an effort to determine the optimal decision
(Howard, 1968 ; Raiffa, 1968 ; Raiffa & Schlaifer, 1961 ). Some decision problems need
the use of additional information, obtained either by sampling or by other means. Insuch cases, we may have an idea about the reliability of additional information, whichmay be stated as a probability, and the information is incorporated into analysis byBayes ’theorem.
Bayesian analysis is a statistical decision-making process based on the premise that
decisions under uncertainty can be performed only with the help of additional informa-
tion, in order to reduce the impact of uncertainty. Bayesian analysis updates information
using Bayes ’theorem. According to this theorem, causes (states of nature, events) are
integrated in the resulting outcomes through conditional probabilities. Bayesian analysisis used in order to revise the initial or a priori probabilities in a posteriori probabilities,using the results of experiments or tests with a certain probability of success. The initialprobabilities are obtained by empirical or subjective assessment, sampling, while the aposteriori probabilities are based on the initial probabilities and the results of
*Corresponding author. Email: mimovicp@kg.ac.rs
© 2015 The Author(s). Published by Taylor & Francis.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( http://creativecom
mons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original
work is properly cited.Economic Research-Ekonomska Istra živanja , 2015
Vol. 28, No. 1, 868 –878, http://dx.doi.org/10.1080/1331677X.2015.1092309

experiments and reliability. Bayesian analysis is a relatively objective way to determine
the in fluence of information on the results of the decision-making process in the terms
of probability by establishing a relationship of joint and marginal probability.
Before the Bayes ’theorem, decision-makers formulated an initial hypotheses, and
based on a priori probability, by deductive contemplation, analysed the extreme conse-
quences of its application. Bayes ’theorem allows that the starting hypothesis can be
determined based on observation and consequence analysis.
The Analytic Hierarchy Process (AHP) can be used to connect a priori probabilities
and the conditional probabilities of the outcomes in the context of Bayes ’theorem. It is
known that AHP can use the probabilities obtained by the Bayesian formula in order toincrease the accuracy of AHP model input data. Saaty and Vargas (
1998 ) consider that
in this way derived a posteriori probabilities are part of the ANP framework and that
Bayes ’theorem is a suf ficient condition for the problem solution in terms of ANP.
Besides them, Efron (1196) wrote about the possibility of applying AHP in the contextof Bayes ’theorem, which uses an empirical Bayesian model for combining probabili-
ties. Castro et al. (
1996 ) used the AHP process to structure a problem that required
sequential diagnostic testing, while the Bayes formula they used in the calculation ofprobabilities also resulted in a new estimate of diagnostic capability. Szucs and Sallai(
2008 ) integrate Analytic Network Process and the extended Bayesian Network for fault
spreading problem. Huang and Bian ( 2009 ) combine ontology, AHP, Bayesian network
and web technology in the development of personalised recommendation of attractive
tourist destinations, based on user feedback. Ahmed et al. ( 2005 ) present the conceptual-
isation of AHP and Bayesian Belief Networks (BBN) based decision support modulesin the Intelligent Risk Mapping and Assessment Systems (IRMASTM) for risk analysis.Altuzarra, Moreno-Jiménez, and Salvador (
2010 ) examine consensus building in
AHP-group decision-making from a Bayesian perspective, and they examine a Bayesianpriorisation procedure for AHP-group decision-making (
2007 ). Gargallo, Moreno-
Jimenez, and Salvador ( 2007 ) proposes a Bayesian estimation procedure to determine
the priorities of the AHP in group decision-making.
According to Bayes ’theorem, subjective probabilities are used in decision-making
process. Also, subjective probabilities can be derived by pairwise comparisons of ele-ments, as suggested by Saaty & Bennett
1977 , as well as Yager, 1979 . However, what
is less known and what authors would like to present in this paper is that Bayesian anal-ysis can use the priorities obtained from AHP model as a priori probabilities in situationswhere there is no possibility of using statistical probability estimates. The authors pro-pose an integration of these two approaches in a decisional analysis context in a way
that is tailored to the analysis of the market position of enterprise and identifying pre-
ferred market strategy, which makes a scienti fic contribution of this paper.
2. Preliminaries
2.1. Analytical Hierarchy Process
The AHP (Saaty,
1980 ) is an intuitive method for formulating and analysing decisions.
This method can be successfully used to measure the relative in fluence of many factors
relevant to the possible outcomes, as well as to predict, i.e. determine relative probabil-ity distribution of outcomes. There are four basic steps in the application of AHP to theproblem of evaluation-ranking alternative outcomes:Economic Research-Ekonomska Istra živanja 869

(1) Problem decomposition –the formation of a hierarchy of interrelated decision
elements, which describe the problem,
(2) Pairwise comparison –carried out by comparing pairs of elements in decision-
making matrix, usually using a 1 –9 scale (Saaty, 2010 ) comparisons in order to
obtain input data,
(3) Prioritisation –calculate the relative priorities of decision-making elements,
(4) Synthesis –the aggregation of the relative priorities of decision-making elements
in order to calculate a rating for alternative options in decision-making(ranking).
AHP involves decision problem decomposition of the elements according to their
common characteristics and, after that, the formation of hierarchical model with different
levels. Each level corresponds to the common characteristic of the elements at that level.In AHP, the problem is usually formulated as a hierarchy with three main levels: theexplicitly de fined objective at the highest level, the criteria on the second level and
alternatives at the third level.
The pairwise comparison of decision-making elements is done by the 1 –9 scale
comparisons. The higher value that is assigned to the element in pairwise comparisonsis proportional to the greater importance of that element compared to another in pair-
wise comparisons. The pairwise comparison is the basis of AHP methodology. Through
pairwise comparison of factors several coef ficients can be determined: coef ficient of
their relative importance (criteria comparison), preferences (comparison of alternatives)or probabilities (comparing uncertain events or scenarios in terms of probability of theirrealisation) of these factors. These coef ficients must not be based on a standard scale
because they represent only the ratio of two compared factors. During the comparisonof alternatives, according to some criteria, the question is which alternative is preferredin terms of that criterion. The general preference for some alternatives is calculated as a
weighted sum of the criteria ’s priorities and alternative ’s performance for that criterion.
The AHP application is characterised by the fact that this method carries out an
explicit preference through the synthesis and coordination of information in the form ofgiven structure and signi ficantly resulting preferences that are corresponding to the
actual preferences of decision-makers. The AHP can be used to solve the problem ofchoice under uncertainty or as a tool for prediction (Vaidya & Kumar,
2006 ; Popovi ć,
Stankovi ć, & Veselinovi ć,2013 ). The problem of choice usually involves the evaluation
of the alternative actions preferences, while the prediction using the AHP focuses on the
distribution of relative probabilities of future outcomes.
2.2. Bayesian approach in the decision-making process
Bayes ’theorem or formula, in light of the decision problem, can in a simpli fied form be
interpreted as follows:
The set of disjunctive events is considered S={S1,S2,…, S n}, so that one of them
must be realised,Pn
j¼1PS JðȚ ¼ 1, where the occurrence of one event excludes the
occurrence of other Si∩Sj=Ǿ, i,j=1,2,..,n, i ≠j. The event Iis observed, that can occur
only under the condition that some of the events Sj,j=1,2,..,n has already occurred.
Then the conditional probability of event Sr, given that the event Ihave already realised,
is equal to:870 P . Mimovi ćet al.

PS r=IðȚ ¼PS rðȚPI =SrðȚ
PIðȚ¼PS rðȚPI =SrðȚ
Pn
j¼1Sj/C0/C1
PI =SJðȚ; (1)
where: P(S j)–the probability of event Sj, i.e. starting or a priori probability, P(I)–the
probability of event I,P(I/S j)–the probability of event I, given that event Sjhas
occurred (conditional probability), and: P(S j/I)–the probability of event Sj,given that
event Ihas already occurred (revised or a posteriori probability)
Bayes ’formula provides that the initial beliefs of decision-makers in the implemen-
tation of certain events could be revised in the light of gathered information, i.e. newfacts and knowledge. In this sense, the procedure is reduced to the updating of a prioriprobabilities and their translation into a posteriori probabilities. The main problem in
this case is to determine a priori probabilities. Due to complexity and uncertainty of
decision-making context, these probabilities are often determined subjectively, whichdiminishes validity of the Bayes ’formula.
3. Proposed procedure and framework for integrated approach of AHP and
Bayesian analysis
Considering the AHP application in the context of Bayesian analysis, the problem can
be formulated as follows: assume that the hypothetical corporation A that has a wide
range of products and services is positioned at the market as the only manufacturer ofthe product P. Suddenly, a competitive, reputable corporation B appears on the market,with a very diversi fied product line, which has decided to expand its business in this
market. Corporation A was faced with the dilemma of how to react to new situations,especially as the real intentions of competitors B on the product P market were notknown. Predicting the next step of the corporation B would be of obvious interest tothe corporation A. The expert team of analysts in corporation A has therefore decided
to formulate the AHP prediction model in order to estimate the relative probability of
an alternative according to the possible intentions of the corporation B: (1) permanentlypositioned exclusively in the market P; (2) winning production of other products fromcorporation A ’s assortment and its suppression of the market; (3) a temporary position,
but with long-term aspirations; or (4) a temporary position, without much ambition forthe product P market. Factors (criteria) that are relevant for the intentions of the corpo-ration B are: the corporation B itself (management, pricing, promotion, product features,customer service policies, etc.), the ability to win new markets, the ability to overcome
internal problems (assuming that the corporation B has some dif ficulties in the busi-
ness), minimise the risk of loss in the confrontation with company A, which dominatesthe products P market, etc.
After the hierarchical structuring of the problem, the team of experts made estimates
of the relative likelihood of outcomes with respect to each of the criteria, and then eval-uated the relative likelihood of each of the criteria. The synthesis of these assessmentsdetermines the most likely outcome or intent of corporate B. Looking at gradientsensitivity graphic for certain criteria, it is possible to identify potential changes in the
ranking of alternative outcomes. Structural changes of the model, in terms of adding
sub-criteria within certain criteria, can lead to minor or major changes in the evaluation,due to new estimates that take into account the new sub-criteria. The result in this sensewould be changed from one to another most likely outcome, which would require arestructuring model for the evaluation of corporation A ’s alternative actions.Economic Research-Ekonomska Istra živanja 871

Bayes ’theorem also can be used to derive a posteriori probabilities, i.e. relative
importance of criteria by which will be predicted the actions of corporation B. The setof alternative actions, i.e. intentions of the company B are denoted as N
i(i= 1,2, .., m)
in this context. In the context of decision-making under risk and uncertainty, as well as
of Bayesian analysis, these alternatives –intentions are equivalent to the experiment out-
comes. According to the forecasting results, company A takes one of the alternativeactions A
j,j= 1,2, .., n). Hierarchical structure, therefore, has three levels: objective,
criteria (state of nature) and alternatives (experimental outcomes).
Let, in this context P(K) is the vector of criteria priority, i.e. state of nature priority
vector above the objective in the hierarchical structure of the problem. In AHP method-ology it is also known that although priorities are not true probabilities, they are func-tioning as probabilities. Also, let the P(N/K) is the probability matrix, which coincides
with the outcomes priorities in accordance with the state of nature. The AHP methodol-
ogy provides priorities of the possible intentions, i.e. strategies of the company Baccording to the main objective of the problem, in the form of:
PNðȚ ¼ PN =KðȚ xP KðȚ : (2)
The formula above is according to Saaty and Vargas and this form coincides with the
probabilities of outcomes obtained by the probability laws (see Saaty and Vargas,
1998 ,
p. 493).
Consider now the decision-maker in corporation A, whose choosing problem is
defined through the matrix of payments (Table 1) which includes the alternatives,
the state of nature, i.e. arbitrary decision-making criteria, and the empirical value ofoutcomes that coincide with their utilities. Also, consider that the criteria C
2and C3
are the cost type criteria and the criteria C1,C4,C5and C6, are the bene fit type. It
is necessary to determine the probability of the state of nature i.e. value of alterna-tives for each criterion and by the method of maximum expected utility (EU) deter-
mine the best alternative. The outcomes of alternatives for criteria depend onintention of corporation B, so it should be taken into account both optimistic and
pessimistic approach, i.e. good and bad scenario. In order to solve this problem the
following procedures have been carried out: (1) De fining the matrix of payments; (2)
Determination of priorities –local, global and total, using AHP; (3) Determination
ofEUof each alternative which can be calculated as a weighted sum of utilities
based on standardised data of payment matrix; (4) De fining possible outcomes with
conditional probabilities; (5) Determination of EUof alternatives based on a posteri-
ori probabilities; (6) Determination of expected value of sample information; and (7)Determination of expected value of perfect information.
Table 1. Matrix of payments.
C1 C2 C3 C4 C5 C6
A1 70 63,000 4000 2 5 4
A2 65 77,000 3000 5 3 8
A3 55 50,000 5000 3 3 10
A4 60 60,000 2000 4 4 5
Source: Created by authors.872 P . Mimovi ćet al.

The problem could be solved simultaneously using the AHP method, whose hierar-
chical structure of data is presented in the format of declining decomposition with threelevels (Figure
1):
(1) The objective –Selection of the optimal alternative,
(2) The criteria (quantitative and qualitative) which are relevant for the problem
solution are de fined using the set C( C 1,C2,…,C 6), where the criteria C2and C3
relate to costs.
(3) On the third level of hierarchy there is a set of alternatives A( A 1,A2,…,A 4).
By AHP model, i.e. by pairwise comparisons of criteria according to the main
objective of the model (Figure 2), the following priorities of the criteria are obtained:
C1= 0.098 C 2= 0.223 C3= 0.223 C4= 0.098 C5= 0.160 C6= 0.196 (Figure 3).
Figure 1. Hierarchical structure of the problem.
Source: Authors ’graphic presentation using Expert Choice.
Figure 2. Pairwise comparison matrix of criteria.
Source: Authors ’calculation using Expert Choice.
Figure 3. The priorities of criteria (criteria relative importance).
Source: Authors ’calculation using Expert Choice.Economic Research-Ekonomska Istra živanja 873

The best alternative is A 3, whose priority of 0.271 is the highest among the all alter-
natives (Figure 4).
The positive and negative values in the matrix of payments represent the utility of
each alternative according to each criterion. The positive numbers are used for estimated
earnings, while the negative numbers are used for estimated loss. These numbers should
be standardised and presented on an absolute scale, with a maximum value of 1 for theearnings and the smallest, i.e. –1 for the losses, while other values are given with
respect to the highest, i.e. the smallest values. The corresponding values are shown inTable
2.
The EUof each alternative now can be calculated as a weighted sum of utilities
from the Table 2, where the priorities are relative priorities of criteria in the AHP
model. The best alternative in this case is A 4, whose EUcan be calculated as follows:
EU(A 4)= 0.098*0.86 + 0.223*( –0.78) + 0.223*( –0.4) + 0.098*0.8 + 0.16*0.8 +
0.196*0.5 = 0.1255
The EUs for the rest of the alternatives are:
EU(A 1)= 0.098*1.00 + 0.223*( –0.82) + 0.223*( –0.8) + 0.098*0.4 + 0.16*1.00 +
0.196*0.4 = 0.0143
EU(A 2)= 0.098*0.93 + 0.223*( –1.00) + 0.223*( –0.6) + 0.098*1.0 + 0.16*0.75 +
0.196*0.8 = 0.1091
EU(A 3)= 0.098*079 + 0.223*( –0.65) + 0.223*( –1.0) + 0.098*0.6 + 0.16*0.8 +
0.196*0.5 = 0.0843
However, in Bayesian analysis, the initial probabilities are revised taking into
account additional information. This is because decision analysis is usually associatedwith Bayesian theory.
Alternatives Total valuesNormalized
valuesIdeal values Rank
A3 0.135900 0.271799 1.000000 1
A4 0.133224 0.266448 0.980311 2
A1 0.121049 0.242098 0.890725 3
A2 0.109828 0.219655 0.808153 4
Figure 4. Rank of alternatives.
Source: Authors ’calculation using Expert Choice.
Table 2. Standardised data.
C1 C2 C3 C4 C5 C6
A1 1 –0.82 –0.8 0.4 1 0.4
A2 0.93 –1 –0.6 1 0.75 0.8
A3 0.79 –0.65 –1 0.6 0.75 1
A4 0.86 –0.78 –0.4 0.8 0.8 0.5
Source: Authors ’calculation.874 P . Mimovi ćet al.

Suppose that the decision-maker wants to gather additional information and he iden-
tifies two categories of possible outcomes: good ( G) and bad ( B), with conditional prob-
abilities shown in Table 3:
Using the Bayes ’formula, a posteriori probabilities for each criterion can be calcu-
lated. For example, in the case of good results ( G), we have the following calculations:
PC r=GðȚ ¼PC rðȚPG =CrðȚ
PGðȚ¼PC rðȚPG =CrðȚ
P6
j¼1Cj/C0/C1
PG =CJðȚ; (3)
so that, for the outcome G, the a posteriori probability for the criterion C1is as follows:
PC 1=GðȚ ¼PC 1ðȚ PG =C1ðȚ
PGðȚ¼PC 1ðȚ PG =C1ðȚ
P6
j¼1Cj/C0/C1
PG =CJðȚ; (4)
i.e.P(C 1/G)= 0.098*0.6/ 0.524 = 0.112, where P(G) = 0.524
The a posteriori probabilities for the rest of criteria are:P(C
2/G)= 0.085
P(C 3/G)= 0.383
P(C 4/G)= 0.094
P(C 5/G)= 0.214
P(C 6/G)= 0.112
While in the case of bad results ( B) the a posteriori probabilities can be calculated
according to the following equation:
PC r=BðȚ ¼PC rðȚPB =CrðȚ
PBðȚ¼PC rðȚPB =CrðȚ
P6
j¼1Cj/C0/C1
PB =CJðȚ; (5)
P(C 1/B)= 0.098*0.4/ 0.476 = 0.082, where P(B) = 0.476
P(C 2/B)= 0.375
P(C 3/B)= 0.043
P(C 4/B)= 0.103
P(C 5/B)= 0.101
P(C 6/B)= 0.288
The EUs of alternatives, obtained by a posteriori probabilities, are shown in Table 4:
In order to decide between a priori analysis and the possibility to collect additional
information, the decision-maker must calculate the EUof the research strategy as aTable 3. Conditional probabilities of research results.
Criterion (G) (B)
C1 0.6 0.4
C2 0.2 0.8
C3 0.9 0.1
C4 0.5 0.5
C5 0.7 0.3
C6 0.3 0.7
Source: Authors ’calculation.Economic Research-Ekonomska Istra živanja 875

weighted average of the EUs of actions A4andA3, which are the best in the case of G,
i.e.B. In that case, the priorities are probabilities of outcomes GandB:
EUSI = EU A 4/GðȚ *PGðȚ+E UA 3/BðȚ *PBðȚ
= 0.179*0.524 + 0.204*0.476 = 0.094 + 0.097 = 0.191
If the information was free, comparing the EUof the best alternatives obtained by a pri-
ori or by a posteriori analysis, it could be easily concluded that for decision-maker it isdesirable to obtain information, because EUSI = 0.191> EU(A
4)= 0.125. However, usu-
ally information needs to be paid, so it is necessary to determine the economic feasibil-
ity of information purchasing. The costs of obtaining additional information aresubtracted from our final pay-off. So we must calculate the bene fit of the additional
information against its costs.
The question of the value of information often arises in decision-making problems.
Thefirst step in answering this question is to find out how much we should be willing
to pay for perfect information. If we can determine the value of perfect information, thiswill give us an upper limit on the value of any (imperfect) information. Since we do
not know what the perfect information is, we can only compute the expected value of
perfect information in a decision-making situation.
EU CCðȚ ¼X
6
j¼1wj/C2maxiuij/C0/C1
¼0:31785
EUPI ¼EU CCðȚ /C0 maxiEU A iðȚ ¼ 0:31785 /C00:124¼0:19231
Where expected utility in conditions of certainty EU(CC) ,wjare the priorities of criteria
(j = 1,2, …,6),uijutilities de fined in Table 2and expected utility of perfect information
(EUPI) .Table 4. The expected utility of alternatives based on a posteriori probabilities.
Expected utility of alternative Outcome (G) Outcome (B)
EU(A 1) 0.035 –0.003
EU(A 2) –0.011 0.08
EU(A 3) –0.022 0.204
EU(A 4) 0.179 0.066
Source: Authors ’calculation.
Alternatives Total valuesNormalized
valuesIdeal values Rank
A4 0.166883 0.333766 1,000000 1
A1 0.130201 0.260403 0.780196 2
A2 0.114026 0.228052 0.532650 3
A3 0.088890 0.177780 0.532650 4
Figure 5. Rank of alternatives in the case of outcome G.
Source: Authors ’calculation using Expert Choice.876 P . Mimovi ćet al.

If now, a posteriori probabilities of criteria, in the case of outcome G, we include
again in the AHP model, we will obtain the results presented in Figure 5.
Where the best alternative is A4, whose priority value is 0.334 is the highest.
According to the Figure 6, in the case of outcome Bthe best alternative is A3with
priority value 0.380.
4. Conclusion
Bayes ’analysis is one of the most objective ways for decision-making under risk and
uncertainty, combining the initial uncertainty expressed by subjective probabilities andinformation from the environment in terms of assessment of decisions consequences.
Bayes ’formula is a suf ficient condition for the solution of problems in terms of the
AHP. In addition, Bayes formula corrects a priori probabilities in a posteriori probabili-
ties in the light of new information. This means that the criteria priorities in the AHPmodel, which are usually subjectively determined, can be objecti fied by using additional
information. Revised ratings of criteria resulting from Bayesian approach are used after-wards for comparing pairs of alternatives in the AHP model. This article has shown thatthe implementation of additional information leads to changes in ranking the alternatives.This approach represents a higher quality of decision-making perspective, especiallywhen it comes to strategic market decisions made for a longer period of time, and subse-
quent implementation of Bayes ’a posteriori probabilities in the AHP model is in itself a
new approach to the integration of the two methods which improves the classical integra-tion, thus providing scienti fic contribution. Knowing that AHP priority can be used as a
priori probability in Bayes ’formula in all the situations where it is not possible to gener-
ate the probabilities using any of the statistical approaches, the authors have improvedthis integration by reversed implementation which applies a posteriori probabilities toobjectify the AHP method and make it more suitable for business decision-making.
Disclosure statement
No potential con flict of interest was reported by the authors.
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