Adaptive multi-criteria group decision-making model based on consistency and consensus with intuitionistic reciprocal preference relations: a case study in energy storage technology selection

Atanassov's intuitionistic fuzzy sets

As an extension of the concept of fuzzy set, Krassimir Atanassov introduced the notion of intuitionistic fuzzy set^[24], which relaxes a fundamental assumption of the fuzzy set theory, i.e. the complement of a fuzzy set $$ B $$ has a membership function defined exclusively in regards to the membership function $$ \mu_{B}(y) $$ as $$ 1 - \mu_{B}(y) $$. Nevertheless, the Atanassov's intuitionistic fuzzy set theory assumes the referred sum rests on the condition of being equal or less than 1^[24]. Concretely:

Definition 1^[24] Let $$ A $$ be a non-empty universe. An Atanassov's intuitionistic fuzzy set (AIFS) $$ I $$ on $$ A $$ is defined as $$ I = \{(a, \mu_{I}(a), \nu_{I}(a)\; |\; a \in A\} $$ with $$ \mu_{I}(a) \in [0,1] $$ being the membership degree of the element $$ a $$ in $$ I $$ and $$ \nu_{I}(a) \in [0,1] $$ being the non-membership degree of the element $$ a $$ in $$ I $$, subject to the constraint $$ 0 \leq \mu_{I}(a) + \nu_{I}(a) \leq 1 $$.

On the one hand, the AIFS $$ I $$ becomes a fuzzy set whether, $$ \forall a \in A $$, $$ \mu_{I}(a) = 1 - \nu_{I}(a) $$. On the other hand, an additional criterion referred to as hesitancy index is needed whether at least a value $$ a \in A $$ fulfills $$ \mu_{I}(a) < 1 - \nu_{I}(a) $$. This hesitation index characterizes the amount of lacking of information in determining the membership degree of $$ a $$ to $$ I $$, being defined as $$ \tau_{I}(a) = 1 - \mu_{I}(a) - \nu_{I}(a) $$. The relationship between the membership degree and the non-membership degree, which is reciprocal, makes the second one needless because it can be determined from the first one in the case that $$ \tau_I(a) = 0 $$. The hesitancy index gives us an additional possibility of representing imperfect knowledge. This leads to describing many real problems in a more adequate way^[25].

Intuitionistic reciprocal preference relations

As generalizations of the reciprocal preference relations, Zeshui Xu introduced the intuitionistic reciprocal preference relations^[26]. They arise when AIFSs are used to make pairwise comparisons between the alternatives.

Definition 2^[26] An intuitionistic reciprocal preference relation (IRPR) $$ R $$ on the set $$ A $$ is represented by a matrix $$ R = (r_{ij})_{n \times n} \subset A \times A $$ with $$ r_{ij} = \langle (a_i,a_j), \mu_R(a_i,a_j), \nu_R(a_i,a_j)\rangle $$$$ \forall i,j = 1,2,\dotsc,n $$. For convenience, let $$ r_{ij} = \langle \mu_{ij}, \nu_{ij}\rangle $$$$ \forall i,j = 1,2,\dotsc,n $$, where $$ r_{ij} $$ is an intuitionistic fuzzy value composed by the certainty degree $$ \mu_{ij} $$ to which $$ a_i $$ is preferred to $$ a_j $$ and the certainty degree $$ \nu_{ij} $$ to which $$ a_i $$ is non-preferred to $$ a_j $$, and $$ \pi_{ij} = 1 - \mu_{ij} - \nu_{ij} $$ is interpreted as the uncertainty degree to which $$ a_i $$ is preferred to $$ a_j $$. Furthermore, $$ \mu_{ij} $$ and $$ \nu_{ij} $$ satisfy the following conditions:

Consistency

On the one hand, consistency is associated with the property of transitivity^[27]. On the other hand, rationality is related to consistency^[28]. The transitivity has been characterized in reciprocal preference relations in different ways. However, Chiclana et al.^[29] proved most of them are incorrect and concluded that the multiplicative transitivity property is the most adequate. Formally, Wu and Chiclana^[30] generalized this property to IRPR by, first, applying the extension principle^[31] to the case in which, instead of crisp values in the unit interval, fuzzy sets are used as preference values. Second, they applied the representation theorem^[32] to those fuzzy sets in order to obtain the property of multiplicative transitivity related to the fuzzy preference values. Third, they applied this result to the case when interval-valued fuzzy sets are used instead of fuzzy sets. It led to the extension of the property of multiplicative transitivity from reciprocal preference relations to interval-valued reciprocal preference relations. In the end, the property of multiplicative transitivity for IRPR was formulated by using the mathematical isomorphims between the AIFSs and the interval-valued fuzzy sets^[33].

Definition 3^[30] An IRPR $$ R $$ is multiplicative transitive if and only if:

Given two alternatives, $$ a_i $$ and $$ a_j $$, $$ i < j $$, its partially multiplicative transitivity based estimated intuitionistic preference value, $$ mr^{k}_{ij} = \langle mr^{k-}_{ij}, mr^{k+}_{ij}\rangle $$, may be calculated by means of other one $$ a_k $$, $$ k \neq i,j $$, and Equation (2), provided that the denominators are not zero, in the following way^[30]:

In Equation (2), both expressions are always true whether two of the three sub-indexes are equal. Let $$ i = k $$, $$ mr^{i+}_{ij} = \nu_{ij} $$ if $$ \nu_{ij} \neq 1 $$, and $$ mr^{i-}_{ij} = \mu_{ij} $$ if $$ \mu_{ij} \neq 0 $$. Due to $$ \nu_{ij} = \mu_{ji} $$, then $$ \nu_{ji} \neq 1 $$ if and only if $$ \mu_{ij} \neq 1 $$, and $$ \mu_{ji} \neq 0 $$ if and only if $$ \nu_{ij} \neq 0 $$. As a result, if $$ (r_{ij},r_{ji}) \neq (\langle 1,0\rangle, \langle 0,1\rangle) $$ and $$ i = k $$, then $$ r_{ij} = m^{i}_{ij} $$. In addition, if $$ (r_{ij},r_{ji}) \neq (\langle 0,1\rangle, \langle 1,0\rangle) $$ and $$ i = k $$, then $$ r_{ij} = m^{j}_{ij} $$. To be brief, we can compute the multiplicative transitivity based estimated intuitionistic preference value of a pair of alternatives, $$ a_i $$ and $$ a_j $$, if $$ (r_{ij},r_{ji}) \neq \{(\langle 1,0\rangle, \langle 0,1\rangle), (\langle 0,1\rangle, \langle 1,0\rangle)\} $$ and $$ k \in \{i,j\} $$. However, there not exists an indirect estimation process as described above. At last, assuming that $$ j = i $$, then $$ r_{ii} = \langle 0.5,0.5\rangle $$ by definition and $$ r_{ii} = mr^{k}_{ii} $$ whenever $$ r_{ik} \not\in (\langle 0,1\rangle, \langle 1,0 \rangle) $$.

The global multiplicative transitivity based estimated value related to a particular pair of alternatives, $$ a_i $$ and $$ a_j $$, is calculated as the mean of all their partially multiplicative transitivity based estimated values:

being $$ R^{01}_{ij} = \{k \neq i,j\; |\; (r_{ik}, r_{kj}) \not\in R^{01}\} $$, $$ R^{01} = \{(\langle 1,0\rangle, \langle 0,1\rangle), (\langle 0,1\rangle, \langle 1,0\rangle)\} $$, and $$ \#R^{01}_{ij} $$ the cardinality of $$ R^{01}_{ij} $$.

As a result of this, given an IRPR $$ R $$, the following multiplicative transitivity based IRPR, $$ MR = (\langle {mr}^{-}_{ij}, {mr}^{+}_{ij}\rangle) $$, may be computed^[30]:

If an IRPR $$ R $$ is multiplicative transitive then $$ R = MR $$. Certainly, if $$ R $$ is multiplicative transitive both expressions in Equation (2) are true $$ \forall i, j, k $$. That is:

whenever $$ k \in R^{01}_{ij} $$. As a result of this, $$ mr^{k+}_{ij} = \nu_{ij} $$ and $$ mr^{k-}_{ij} = \mu_{ij} $$, $$ \forall i,j,k \in R^{01}_{ij} $$, proving that $$ r_{ij} = mr_{ij} $$, $$ \forall i,j $$. An IRPR $$ R $$ is referred to as multiplicative consistent if $$ R = MR $$^[30].

Based on these results, a procedure to compute the consistency level, called $$ cl $$, of an IRPR $$ R $$ was built^[30], which is based on the similarity between $$ R $$ and its multiplicative transitivity based IRPR, $$ MR $$, that is used to compute a consistency measure, $$ cl_{ij} $$, related to a pair of alternatives, $$ a_i $$ and $$ a_j $$, and a consistency measure, $$ cl_i $$, related to an alternative $$ a_i $$. These two consistency measures are used to compute the consistency level $$ cl $$. It is as follows:

being $$ d $$ a metric describing the distance between two AIFSs that is used to define a similarity function between two AIFSs^[34].

Consensus

Cabrerizo et al.^[35] introduced a procedure to measure the consensus, called $$ cr $$, achieved between the decision-makers that participate in a multi-criteria group decision-making problem when they use IRPRs to provide their assessments. This procedure is based on the coincidence concept^[36], which has been used in many approaches measuring the consensus in group decision-making problems. Assuming a group of $$ m $$ decision-makers, $$ DM = \{dm_1,dm_2,\dotsc,dm_m\} $$, a collection of $$ n $$ alternatives, $$ A = \{a_1,a_2,\dotsc,a_n\} $$, and a set of $$ q $$ criteria, $$ C = \{c_1,c_2,\dotsc,c_q\} $$, which have associated a importance weight $$ w_l \in [0,1] $$ such that $$ \sum_{l = 1}^{q}{w_l} = 1 $$, then the description of the procedure to obtain the consensus is the following:

● Calculation of a matrix for each pair of decision-makers, $$ dm_g $$ and $$ dm_h $$ ($$ g \neq h $$), and each criterion, $$ c_l $$, denoted as $$ SM^{ghl} = (sm^{ghl}_{ij}) $$, characterizing the similarity between the IRPRs $$ R^{gl} $$ and $$ R^{hl} $$ given by these decision-makers taking into account that criterion:

being $$ d $$ a metric that determines the distance between two AIFSs^[34].

● Calculation of a matrix for each criterion, $$ c_l $$, denoted as $$ CM^l = (cm^{l}_{ij}) $$, characterizing the agreement achieved by the decision-makers on the criterion $$ c_l $$:

● Calculation, for each matrix $$ CM^l $$, of three consensus measures, namely, a measure of consensus, $$ cp^{l}_{ij} $$, related to a pair of alternatives, $$ a_i $$ and $$ a_j $$, a measure of consensus, $$ ca^{l}_{i} $$, related to an alternative, $$ a_i $$, and a measure of global consensus, $$ cr^l $$. They are calculated as:

● Calculation of $$ cr $$ by means of the weighted mean of the $$ cr^l $$:

Articulation of the IRPRs

The starting point is a group of $$ m $$ decision-makers, $$ DM = \{dm_1,dm_2,\dotsc,dm_m\} $$, a collection of $$ n $$ alternatives, $$ A = \{a_1,a_2,\dotsc,a_n\} $$, and a set of $$ q $$ criteria, $$ C = \{c_1,c_2,\dotsc,c_q\} $$, that are born in mind to evaluate the alternatives. In addition, every criterion $$ c_l \in C $$ has associated a importance weight $$ w_l \in [0,1] $$ such that $$ \sum_{l = 1}^{q}{w_l} = 1 $$. Here, IRPRs are assumed to model the comparisons between alternatives given by the decision-makers because the use of AIFSs allows an additional freedom degree modeling hesitancy of individuals related to the exclusion and inclusion of an element to a set^[25]. Therefore, once the decision-makers have expressed their assessments, we get a set of $$ m \cdot q $$ IRPRs, $$ R^{gl} $$, $$ g = 1,\dotsc, m $$, $$ l = 1,\dotsc,q $$, of dimension $$ n \times n $$.

Analysis of consensus

The second stage consists in analyzing the consensus achieved. When all the decision-makers have given their IRPRs, the procedure described in Section Background is carried out to measure the consensus achieved. Concretely, the value of $$ cr $$ determines the consensus achieved. Usually, a consensus threshold, $$ \alpha \in [0,1] $$, which is conditioned by the given decision-making problem^[15], is compared with the value of $$ cr $$. The next stage, i.e. the feedback mechanism, is executed with the objective of supporting decision-makers in modifying their assessments to increase the consensus achieved if $$ \alpha $$ is greater than $$ cr $$. If not, the fourth stage, i.e. the calculation of the collective IRPR, is conducted. Furthermore, as a mean to avoid a state in which the consensus reached does not converge to the consensus threshold, the negotiation rounds are limited to a maximum number that is decided at the beginning of the decision-making process^[15]. As a consequence, the fourth stage is also executed if the current negotiation round exceeds the maximum number of negotiation rounds.

Feedback mechanism

We introduce a new feedback mechanism assuming that decision-makers with lower consistency levels need more advice than those with higher consistency levels. That is, because a low consistency level means that the decision-maker has provided contradictory assessments, it seems logical that the decision-makers providing inconsistency assessments will require more advice on how to modify their judgments. If a decision-maker has expressed contradictory assessments, it could mean that she or he does not have the enough level of knowledge about the alternatives and the criteria. Therefore, more adjustments in their assessments should be needed to lead a consensual decision. On the contrary, decision-makers with higher consistency levels will need low adjustments in their assessments to lead to consensus. Based on this assumption, this new feedback mechanism adapts its operation according to the decision-makers' consistency. This feedback mechanism, composed of four stages: (i) classifying of the decision-makers; (ii)computing proximity measures; (iii) looking for problematic assessments; and (iv) giving advice, is presented in what follows.

Calculation of the collective IRPR

The collective IRPR is computed by fusing all the individual decision-makers' IRPRs, which is performed by using a given aggregation function^[38]. In this study, because we take into account a number of criteria and weights of importance associated with them, the weighted arithmetic mean is applied as aggregation function. Concretely, the collective IRPR is obtaining by means of the following procedure:

● A collective IRPR, $$ S^{l} = (s^{l}_{ij}) $$, is computed for each criterion $$ c_l $$ by means of Equation (17).

● Using the information contained in the collective IRPRs related to the criteria, the collective IRPR, $$ S = (s_{ij}) $$, is calculated as follows:

Ranking of the alternatives

To rank the alternatives solving the multi-criteria decision-making problem from best to worst, the information contained in the collective IRPR must be exploited. Distinct functions could be applied to perform this task^[39]. Among them, we use the quantifier-guided dominance degree, $$ QGDD_i $$, that is computed as follows:

This choice degree of alternatives determines the dominance that the alternative $$ a_i $$ has over the remaining alternatives. The higher the value of $$ QGDD_i $$, the best the alternative $$ a_i $$ as solution to the decision-making problem.

First negotiation round

Second negotiation round

Calculation of the collective IRPR

Once the consensus achieved is enough, the collective IRPR $$ S $$ can be computed:

Ranking the alternatives

Using the quantifier-guided dominance degree, the values associated to each alternative are:

It means that, according to the decision-makers involved in this decision-making problem, the energy storage technology selected should be the high temperature thermal energy storage.

Authors' contributions

Made substantial contributions to conception and design of the study and performed data analysis and interpretation: Cabrerizo FJ, Trillo JR, Alonso S, Morente-Molinera JA

Availability of data and materials

Not applicable.

Financial support and sponsorship

This work was supported by the project B-TIC-590-UGR20 co-funded by the Programa Operativo FEDER 2014-2020 and the Regional Ministry of Economy, Knowledge, Enterprise and Universities (CECEU) of Andalusia, and by the project PID2019-103880RB-I00 funded by MCIN / AEI / 10.13039/501100011033.

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

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Copyright

© The Author(s) 2022.