A Hybrid MCDM Technique for Risk Management in Construction Projects

The construction industry in general, as well as individual construction projects, deals with various threats, which are called risks. Risk assessment in construction projects has been applied differently from project to project (using various models of risk assessment) to evaluate the risk in certain activities of the project [49]. However, the socio-economic complexity involved in construction events makes it more risk prone, so that there may be negative effects on project sustainability [23]. Due to various complex factors, the construction industry is highly diverse and heterogeneous, experiencing a great deal of dynamic change with global sourcing and increasing price competition [50].

To overcome these risks, contractors have generally used high mark-ups, but this approach is no longer effective, as their margins have become smaller [51]. In recent decades, the stages involved in construction projects have become far more complex in nature, due to technological upgrading and stakeholder pressure, and are characterised by a number of uncertainties, which have a negative influence on the projects [52]. As risk free construction projects are impossible in real life, a controlled risk assessment procedure is required to manage various risks in the projects. The aim of this study is to highlight the main risks that construction projects face and the risk mitigation strategies used to manage them.

There have been substantial developments over the last four decades in research related to construction project management. Projects have been considered that are either exposed to risks, or have apparently inherited risk due to the participation of several stakeholders (as owners, contractors and designers); see [53] among others. For classifying and managing risks effectively, many methodologies have been suggested in the literature. In this section, a list of the existing approaches to construction project risk management is now briefly summarised.

Wang et al. [4] developed an alien eyes’ risk (AER) model, categorising the project risks and their mutual relationships, and proposing a qualitative risk mitigation framework. Schieg [17] adopted a risk management process in construction project management, concentrating more on personal area risks. Zavadskas et al. [23] presented risk assessment based on the MCDM method and applying TOPSIS grey and grey number based complex proportional assessment (COPRAS-G). Wen [54] integrated rough sets and artificial neural networks (ANN) for risk evaluation of construction projects. Fouladgar et al. [55] proposed a risk evaluation outline faced during tunneling operations based on fuzzy TOPSIS methodology. Taroun and Yang [56] hybridised the dempster–shafer (D–S) theory of evidence and evidential reasoning algorithm for structuring personal experience and professional judgment with a spreadsheet-based decision support system. Mohammadi and Tavakolan [25] combined fuzzy logic and AHP in traditional failure mode effect analysis (FMEA) for construction project risk assessment. Serpella et al. [18] applied a knowledge-based approach, based on a three-fold arrangement and risk management function, to address project risks in the construction management sector. Ebrat and Ghodsi [14] identified the risks in construction projects based on the adaptive neuro fuzzy inference system and stepwise regression model for the evaluation of project risks. Taylan et al. [15] proposed hybrid methodologies combining fuzzy AHP and fuzzy TOPSIS and applied the relative importance index method to prioritise the project risks based on the data obtained. Iqbal et al. [9] considered two types of risk management technique during project execution, viz. preventive techniques (to manage risk before the start of the project) and remedial techniques (after occurrence of the risk). Vafadarnikjoo et al. [24] developed an intuitive fuzzy decision-making trial and evaluation laboratory (DEMATEL) to prioritise risks associated with construction projects by using the risk breakdown structure. Ahmadi et al. [27] analysed the criteria for prioritising potential risk events and quantified it using fuzzy AHP. The best response action for a risk event is then identified with respect to the same criteria using a scope expected deviation index. Santos and Jungles [19] evaluate the completion of construction project risk by considering the correlation of delay and the schedule performance index along with any time over-run. Shin et al. [28] made a comparative analysis of AHP and fuzzy AHP to evaluate the potential risk factors at the construction site of a nuclear power plant. Kao et al. [57] proposed an integrated fuzzy ANP based balanced scorecard system for evaluation of relevant bilateral factors in Taiwanese construction projects. Burcar Dunovic et al. [58] assessed large infrastructure projects by integrating the risk impact cumulative distribution curve based MCDM approach. Yousefi et al. [59] proposed a neural network model for predicting emerging time and cost claims applied to Iranian construction projects. Valipour et al. [60] presented a fuzzy cybernetic ANP model for proper identification of public-private partnership project based risks. Ulubeyli and Kazaz [61] developed a fuzzy based sub-contractor selection model (CoSMo) for global based construction projects. Rajakallio et al. [62] analysed the solution delivery from a network perspective in integrated business model renewal. For assessing risk in deep foundation excavation, Valipour et al. [63] developed a step-wise weight assessment ratio analysis (SWARA) based complex proportional assessment (COPRAS) framework to analyse Iranian construction project uncertainty. Khanzadi et al. [64] solved the dispute resolution problem in the construction sector by considering a grey number based discrete zero-sum two-person matrix game model. Keshavarz Ghorabaee et al. [65] presented the hybrid fuzzy step-wise weight assessment ratio analysis (SWARA) based ‘evaluation based on distance from average solution’ (EDAS) model for assessment of the construction equipment by taking sustainability into account.

To deal with real world information, which may be ambiguous, various imprecise decision-making models based on probability theory, fuzzy set theory, and D–S evidence theory have been developed [66]. D–S evidence theory directly expresses uncertain information by allocating the probability to the multiple object based subsets in lieu of individual items [67]. Due to its ability to compare pairs of evidence or belief functions when deriving new ones, and its superiority in an uncertain environment, evidence theory has been widely applied in various domains [22,34,39,68,69,70].

Certain basic definitions are now presented. Let $U$ denote the frame of discernment representing a collective set of exhaustive and mutually exclusive events, and each element of $2^U$ (a power set in $U$) represents a proposition [22]. Based on these, basic probability assignment (BPA) is defined as follows:

A finite non-empty set of mutually exclusive and exhaustive hypotheses for any problem domain is called its frame of discernment. In lieu of a D–S evidence framework, the BPA is defined on a frame of discernment to express the imprecise judgements of experts [39]. Thus, if $2^U$ denotes the power set of a finite non-empty set $U$, then a BPA is defined as a mapping m satisfying (1) and (2): where $\varphi$ is an empty set and $A$ is any element of $2^U$. If $m(A)>0,$ $A$ is called its focal element, and the union of all focal elements is the core of the mass function.

Example: Suppose there exists a task to assess a project. In the frame of D–S theory, the frame of discernment is the proposed project set $B=\left\{b_1, b_2, b_3\right\}$.

As per the expert view, a BPA can be created to express their assessment result: $m\left(\left\{b_1\right\}\right)=0.2; m\left(\left\{b_2\right\}\right)=0.7; m\left(\left\{b_2, b_3\right\}\right)=0.1$, where $b_1=\left[0, 40\right], b_2=\left[41, 70\right], b_3=\left[71, 100\right]$. In addition, the set $B=\left\{b_1, b_2, b_3\right\}$ is a frame of discernment in D–S theory with $m\left(\left\{b_1\right\}\right) + m\left(\left\{b_2\right\}\right) + m\left(\left\{b_2, b_3\right\}\right)=1$.

Although D–S evidence theory is operative in the data fusion problem, it is restricted to handling semantic information where linguistic variables are not mutually exclusive [39,70]. Intuitively, if some hypotheses of D–S theory are removed reasonably, the ability to represent and handle uncertain information may greatly improve. Overall, in many practical scenarios, the experts fail to have complete information, and the assessment is done solely on the basis of partial information, resulting in an incomplete BPA. Based on this idea, Deng [36] proposed $\mathrm{D}$ numbers theory, a generalisation of evidence theory, in order to characterise ambiguous data, as defined in (3) and (4) below:

Let Ω be a finite nonempty set, $\mathrm{D}$ numbers is defined by a mapping: where $\varphi$ is an empty set and $B$ is a subset of Ω.

Compared with D–S evidence theory, the D numbers concept has the advantages that [34,39,42,71]:

Hence, the set of $\left\{a_1, a_2, a_3\right\}$ is not a frame of discernment, because the elements in the above set are not mutually exclusive. In addition, since $D\left(\left\{a_1\right\}\right)+ D\left(\left\{a_2\right\}\right)+ D\left(\left\{a_1, a_2, a_3\right\}\right)=0.9<1$, the above information is incomplete. This example shows that the definition of D numbers is similar to the definition of the mass function.

For a discrete set $\Omega =\left\{b_1,b_2,\dots ,b_i,\dots ,b_n\right\}$, where $b_i\in R$ and $b_i\ne b_j$$\left(i\ne j\right)$, a special form of D numbers can be expressed as:

Two D numbers $D_1=\left\{\left(b_1,v_1\right), ,\dots ,\left(b_i,v_i\right),\dots ,\left(b_n,v_n\right)\right\} \& D_2=\left\{\left(b_n,v_n\right), ,\dots ,\left(b_i,v_i\right),\dots ,\left(b_1,v_1\right)\right\}$ are said to be invariable if:

$D_1=\left\{\left(0.0, 0.7\right), \left(1.0, 0.3\right)\right\} and D_2=\left\{\left(1.0, 0.3\right),\left(0.0, 0.7\right)\right\}$, then, $D_1\iff D_2$.

Let $D=\left\{\left(b_1,v_1\right), \left(b_2,v_2\right),\dots ,\left(b_i,v_i\right),\dots ,\left(b_n,v_n\right)\right\}$ be a D number; then, the integration representation $I(D)$ of the D number is defined as:

Based on the additive transitive property, Herrera-Viedma et al. [30] put forward the consistent fuzzy preference relation (CFPR) for structuring an $n\times n$ decision matrix, which requires only $\left(n-1\right)$ pair wise comparisons, in lieu of $n\left(n-1\right)/2$ comparisons needed for constructing a fuzzy preference relation. Preserving both the reciprocity and transitivity properties, CFPR is constructed only on complete and certain information but fails to deal with cases containing insufficient information [34,36]. Citing the same deficiency as in the case of fuzzy preference relations, Deng et al. [39] proposed the concept of a $\mathrm{D}$ numbers preference relation encompassing the fuzzy preference relations of the $\mathrm{D}$ numbers domain. In addition, considering the insufficiency in traditional CFPRs, Zuo et al. [42] proposed a D numbers consistent fuzzy preference relation matrix, abbreviated as D-CFPR matrix using D numbers to express linguistic preferences given by experts to construct the CFPR. In the following steps, we will discuss the algorithmic methodology of D-CFPR, where distribution assessments are assessed under incomplete or missing information.

Step 1: Construct a D-CFPR matrix $\left(R_D\right)$. The D numbers extended CFPR is advocated to reinforce the capability of CFPR to express uncertain and incomplete information. A D numbers based preference relation on a set of criteria $C=\left\{C_1,C_2,\dots ,C_n\right\}$ is characterized by a $n\times n$ matrix $R_D={\left[D_{ij}\right]}_{n\times n}$ on the product set $C\times C$, whose elements are formulated as per Equation (8), and represented in matrix form (9): where $D_{ij}=\left\{\left(b_{ij}^1, v_{ij}^1\right),\left(b_{ij}^2, v_{ij}^2\right),\dots ,\left(b_{ij}^p, v_{ij}^p\right),\dots \right\}$, $D_{ji}=\neg D_{ij}=\left\{\left(1-b_{ij}^1, v_{ij}^1\right),\left(1-b_{ij}^2, v_{ij}^2\right),\dots ,\left(1-b_{ij}^p, v_{ij}^p\right),\dots \right\}$, $\forall i,j\in \left\{1,2,\dots ,n\right\}, b_{ij}^p\in \left[0,1\right], v_{ij}^p>0, {\displaystyle \sum _p{v}_{ij}^p=1}$. Obviously, $D_{ii}=\left\{\left(0.5,1.0\right)\right\} \forall i\in \left\{1,2,\dots ,n\right\} in R_D.$

In Equation (9), is $R_D$ called a D-CFPR as it is constructed based on $\left(n-1\right)$ pair wise comparisons, denoted as $\left\{D_{12},D_{23},\dots ,D_{(n-1)n}\right\}$ representing a set of D numbers. Considering the set of preference values $B=\left\{D_{ji},1\le i<j\le n\right\}$, the elements of which are denoted $D_{ji}=\left\{\left(b_{ji}^1, v_{ji}^1\right),\left(b_{ji}^2, v_{ji}^2\right),\dots ,\left(b_{ji}^k, v_{ji}^k\right),\dots \right\}$, which can also be represented as follows [42]: where each component $\left(b_{ji}^k, v_{ji}^k\right) in D_{ji}$ is obtained by: where $\Omega =\left\{\left(x,y,\dots ,z\right)| b_{ji}^k=\frac{j-i+1}{2}-b_{i(i+1)}^x-b_{(i+1)(i+2)}^y-\dots -b_{(j-1)j}^k\right\}$, $\left(b_{i(i+1)}^x, v_{i(i+1)}^x\right)$ is the $x^{th}$ component of $D_{i(i+1)},$$\left(b_{(i+1)(i+2)}^y, v_{(i+1)(i+2)}^y\right)$ is the $y^{th}$ component of $D_{(i+1)(i+2)},$$\left(b_{(j-1)j}^z, v_{(j-1)j}^z\right)$ is the $z^{th}$ component of $D_{(j-1)j}$

Thus, based on the reciprocal property, the rest of the entries in D-CFPR (9) are calculated as per Equation (11) as follows:

As per Equations (9)–(11), a D-CFPR can be constructed based on ($n-1$) pair wise comparisons $\left\{D_{12},D_{23},\dots ,D_{(n-1)n}\right\}$. In the D-CFPR generated above, some values of the $b_{ij}\text{'}s$ in $D_{ij}, i,j\in \left(1,2,\dots ,n\right)$ that fall in the interval $[-a,1+a], a>0$ need to be transformed using Equation (12):

The transformation function (12) works as a normalisation, which transforms the values of $b_{ij}\text{'}s$ from $[-a,1+a]$ to the interval [0,1].

Step 2: Determine the crisp matrix$R_c$. Convert the matrix $R_D={\left[D_{ij}\right]}_{n\times n}$ in (9) to a crisp matrix $R_C={\left[c_{ij}\right]}_{n\times n}$ (13) using the integration representation of D numbers (as per Equation (7)), where each element $c_{ij}=I\left(D_{ij}\right) i,j=1,2,\dots ,n$:

Step 3: Construct a probability matrix $R_p$. Based on the crisp matrix $R_C$ (13), construct a probability matrix $R_p={\left[p_{ij}\right]}_{n\times n}$ (14), representing the preference probability between a pair wise set of n criteria $C=\left\{C_1,C_2,\dots ,C_n\right\}$:

In $R_p,$ the element $p_{ij}=\mathrm{Pr}\left(C_i\succ C_j\right),\forall i,j\in \left\{1,2,\dots \dots ,n\right\},$ where $“\succ ”$ represents “prefer to”. Based on (13), a set of rules is suggested to generate $R_p$.

In this condition, the unallocated preference (up) is $c_{up}=1-\left(c_{ij}+c_{ji}\right)$. The probability of one criteria outperforming another criteria is as follows:

Step 4: Construct a triangular probability matrix $R_p^T$. Rank the criteria using the triangularisation procedure, viz. by maximising the sum of the values above, the main diagonal in the $n\times n$ square matrix in the final order is given as:

Thus, by applying this defined row deletion order operation on the matrix $R_p={\left[p_{ij}\right]}_{n\times n}$ (14), construct the triangular probability matrix $R_p^T$, shown in (15):

Step 5: Construct triangulated crisp matrix $R_c^T$. Firstly, the crisp matrix $R_c$ (13) is triangulated based on the triangular matrix $R_p^T$ (15). From that step, the triangulated crisp matrix $R_c^T$ (16) is derived:

Secondly, for elements satisfying $R_c^T(i, j)+R_c^T(j, i)<1,$ a new “polishing operation” (17) is implemented to obtain a triangulated crisp matrix $R_k^T$ (18):

Step 6: Calculate the relative priority weights of the criteria. Assume that the weight vector is $W={\left(w_i\right)}^T$, where $w_i\left(i=1,2,\dots ,n\right)$ are the weights of criteria $C_i\left(i=1,2,\dots ,n\right)$. The elements above and alongside the main diagonal of the matrix $R_k^T={\left[k_{ij}^T\right]}_{n\times n}$ (18) indicate the weight relationship of the criteria. Thus, by adding some necessary constraints, a set of Equation (19) is formed: where $\lambda$ indicates the granular information about the pairwise comparison, which reveals the expert’s perceptive competency. The value of each weight $w_i\left(i=1,2,\dots ,n\right)$, subjected to the parameter $\lambda$, relates to the expert’s cognitive aptitude. The values of $\lambda$ highly depend on the influence of the experts’ judgment and belief function. Thus, a feasible outline (20) is obtained as follows: where $\underset{\_}{\lambda }$ signifies the lower bound of $\lambda$, $\lceil \underset{\_}{\lambda }\rceil =\min \left\{k\in \mathbb{Z}|k\ge \underset{\_}{\lambda }\right\}$, and n is the number of alternatives [39]. The concrete priority weight value of each criteria is calculated here when $\lambda =n$.

Step 7: Inconsistency for the D-CFPR. The traditional CFPR is fully consistent when constructed under the transitivity property for $n-1$ preference values. The D numbers based CFPR is consistent, but, when reduced to classical CFPR, it shows inconsistency under uncertain or incomplete information. Thus, to express this inconsistency of D-CFPRs, an inconsistency degree (ID) (21) is defined, based on the triangular probability matrix $R_p^T$ (18), as follows: where $R_p^T(i,j)$ is an element of $R_p^T,$ and $n$ is the number of comparison objects. This emphasises that the inconsistency of the D matrix, and its acceptable level is decided by the DMs, whose subjective requirement regulates the tolerance level of the aforementioned inconsistency.

The evaluation of D-ANP methodology is composed of two phases.

The first phase emphasises the formation of pairwise judgments for every dependent relationship among given criteria $C_i\left(i=1,2,\dots ,n\right)$, and determination of their priority weight. The priority weights thus obtained are input to the system-with-feedback supermatrix for computing the network influences amongst the different relationships.

The second phase, namely supermatrix evaluation, incorporates five steps: formation of the unweighted supermatrix, formation of the weighted supermatrix, normalisation of the weighted supermatrix (a column stochastic matrix), and finally convergence to a solution using the limited supermatrix.

The converged supermatrix will provide us with the relative priorities for each of the criteria (or sub-criteria) considered within clusters of the decision framework. Thus, the following algorithmic steps of D-ANP are as follows:

Step 1: Model construction and problem structuring. First, we properly outline the decision problem with detailed criteria and sub-criteria (if taken) and then delimit the cluster’s (dimension’s) network and elements (criteria set) within the given clusters. Next, we decide which inter- and inner-dependencies will prevail in the decision problem and clusters of the over-all feedback system.

Step 2: Pairwise comparison matrices and priority vectors. The expert inputs prerequisite for the ANP method are the pairwise judgments of the elements within each cluster, from which inter-and-inner-dependence matrices are formed. Primarily, three categories of pairwise comparisons are formed. First, the inner dependence comparison matrix, based on the criteria of different clusters (dimensions), is used to get the priority weight vector. Then, the pairwise comparison matrices obtained from criteria with respect to other factors (criteria) of the same cluster (dimension). Finally, the pairwise comparison matrix is developed by considering the factor (criteria) of a cluster (dimension) among factors (criteria) of the other cluster (dimension), which affect the criteria.

These pairwise comparison matrices, along with their valuation elicitation, constitute a modified version of D-AHP [22] and D-CFPR [34]. Based on Section 4.1, a brief summary for getting the priority weight of any pairwise comparison matrix is given below, the graphical description of which is shown in Figure 1.

Step 3: Formation of the unweighted supermatrix. The local priority weights resulting from the pairwise comparison are used as input in suitable columns of the unweighted constructed supermatrix, to obtain the global priorities in a system. As a result, a supermatrix takes the form of a partitioned matrix, each segment of which represents an association between two clusters in the given system. Thus, the supermatrix $W$ Equation (22) formed is represented by sub-matrices $W^{ij}$ as $W={\left[W^{ij}\right]}_{n\times n}\left(i,j=1,2,\dots ,n\right)$. The size of the sub-matrix $W^{ij}$ Equation (23) depends on the compared factors (criteria). Interdependency between clusters is illustrated in Equation (22) by analysing both inter- and intra-relations among clusters.

Arrange all priority vectors, representing the impact of a given set of elements in a cluster on another element in the network, as sub-columns of the corresponding column of an unweighted supermatrix $W$ as in Equations (22) and (23). This is composed of $k$ clusters $\left\{D_1,D_2,\dots ,D_k\right\}$, and linkages of these clusters $\left\{e_{k1},e_{k2},\dots ,e_{k{n}_k}\right\}$, are elements of the cluster $D_p\left(p=1,2,\dots ,k\right)$. Each column of the sub-matrix $W^{ij}$ Equation (23) is the priority vector acquired from the identical pairwise judgment, indicating the significance of the elements in the $i^{th}$ cluster with respect to an element in the $j^{th}$ cluster:

Step 4: Determine the normalised weighted supermatrix. If there is no linkage between clusters $D_k$ and $D_l$, then the sub-matrix $W^{kl}$ equals zero. We compute the weighted supermatrix $W^a={\left[{\tilde{W}}^{ij}\right]}_{n\times n}$, by multiplying the unweighted matrix $W={\left[W^{ij}\right]}_{n\times n}$, Equation (23) by priority of dimensions $V_j\left(j=1,2,\dots ,n\right)$, in this stage. As the weighted supermatrix $\left(W^a\right)$ needs to be stochastic, we normalise each column of it, and develop meaningful limiting priorities for determining overall cluster influences. This normalisation procedure ensures that the weighted supermatrix is column stochastic; it is finally represented by matrix $\widehat{W}={\left[{\overline{W}}^{ij}\right]}_{n\times n}$.

Step 5: Compute the limiting priorities for criteria weights. Finally, the column stochastic weighted supermatrix is raised to an appropriately large power until it converges. Thus, the weighted supermatrix $\widehat{W}$ is raised to limiting powers ${\left(\widehat{W}\right)}^{2m+1}$, $m$ being an arbitrarily large number, to attain a steady-state limiting matrix $W^q$. Details are shown in Equation (24):

The priority weight of criteria for the corresponding clusters (dimensions) can now be found in the rows of the limiting supermatrix $W^q$. The limiting supermatrix provides the priority information for the elements of each individual cluster. The strategy outcome with the highest value should be selected from the cluster of criteria. Other priority rankings in different clusters are also provided.

The MABAC methodology was developed by Pamucar and Cirovic [44] to handle problems in MCDM. D numbers is a new representation of uncertain information that can denote the more imprecise based conditions. Thus, the combination of MABAC and D numbers is a new experiment to make decisions in an uncertain environment. The basic setting of the MABAC technique is revealed in the definition of the distance of the criterion function of each of the observed alternatives from the approximate border area [72]. The algorithmic steps displays the execution process for the aforesaid D-MABAC methodology in six steps as follows:

Step 1: Constructing the initial decision matrix $M$. Initially, the assessment of $m$ alternatives in respect of $n$ criteria is carried out. The alternatives are represented in vector form $A_i=\left(d_{i1},d_{i2},\dots ,d_{in}\right)$, where $d_{ij}\left(j=1,2,\dots ,n\right)$ denotes the value (in D numbers format) for the $i$ alternative $\left(i=1,2,\dots ,m\right)$ according to $j$ criteria for matrix $M={\left[d_{ij}\right]}_{m\times n}$. Details shown in decision-matrix (25):

Next, applying Equation (8), (i.e., integration representation of D numbers) on the elements $d_{ij}$ $\left(i=1,2,\dots ,m;j=1,2,\dots ,n\right)$ of decision matrix $M={\left[d_{ij}\right]}_{m\times n}$, crisp decision matrix $X={\left[I\left(d_{ij}\right)\right]}_{m\times n}$ is formed as per Equation (26) as follows:

Step 2. Normalisation of the elements of the initial matrix $X$. The elements of the normalized matrix $N={\left[t_{ij}\right]}_{m\times n}$ are obtained using the following expressions: where where $x_{ij}=I\left(d_{ij}\right)$ and the components $x_i^{+}=\max \left(x_1,x_2,\dots ,x_m\right)$, $x_i^{-}=\min \left(x_1,x_2,\dots ,x_m\right)$ of the decision matrix $(X)$ represent the maximum and minimum values of the criteria $C_j\left(j=1,2,\dots ,n\right)$, by alternatives $A_i\left(i=1,2,\dots ,m\right)$.

Step 3. Calculation of the elements of weighted matrix $\left(V\right)$. The elements of the weighted matrix $V={\left[v_{ij}\right]}_{m\times n}$, are calculated using the following equations: where $t_{ij}$ are the elements of the normalised matrix $N={\left[t_{ij}\right]}_{m\times n}$ and $w_i\left(j=1,2,\dots ,n\right)$ the weight coefficients of criteria, respectively.

Step 4. Determine the border approximation area matrix (G). The elements of the border approximation area (BAA) for each criterion $C_j\left(j=1,2,\dots ,n\right)$ are determined as follows: where $v_{ij}$ are the elements of the weighted matrix $V={\left[v_{ij}\right]}_{m\times n}$, and $m$ is the total number of alternatives. After calculating the value $g_j\left(j=1,2,\dots ,n\right)$, for each criterion, the BAA matrix $\left(G\right)$ is formed with format $n\times 1$ (n = the total number of criteria, according to which the selection is made from the alternatives).

Step 5. Calculation of the distance of the alternative from the BAA for the matrix elements $\left(Q\right)$. The distance $\left(q_{ij}\right)$ of the alternatives $A_i\left(i=1,2,\dots ,m\right)$ from the BAA matrix $G=\left[g_1 g_2 \dots g_n\right]$ is determined as the difference between the elements $v_{ij}\left(i=1,2,\dots ,m; j=1,2,\dots ,n\right)$ in the weighted matrix $V={\left[v_{ij}\right]}_{m\times n}$, and the value $g_j\left(j=1,2,\dots ,n\right)$ of the BAA $\left(G\right)$. The distance $q_{ij}\left(i=1,2,\dots ,m; j=1,2,\dots ,n\right)$ are elements of a matrix $Q={\left[q_{ij}\right]}_{m\times n}$ and are shown in Equation (33):

Alternative $A_i\left(i=1,2,\dots ,m\right)$ can belong to BAA $\left(G\right)$, upper approximation area $\left(G^{+}\right)$, or lower approximation area $\left(G^{-}\right)$, $\mathrm{i}.\mathrm{e}.,A_i\in \left\{G\vee G^{+}\vee G^{-}\right\}$and are determined as per Equation (34):

Step 6. Ranking of the alternatives. A calculation of the values of the criterion functions for the alternatives is obtained as the sum of the distance of the alternatives from the BAA $(q_{ij})$. Using Equation (35), we calculate the sum of the elements $q_{ij}$ of matrix $\left(Q\right)$ by rows, and obtain the final values of the criterion functions ${\widehat{S}}_i \left(i=1,2,\dots ,m\right)$, for the alternatives $A_i\left(i=1,2,\dots ,m\right)$: