A novel integrated decision-making approach for the evaluation and selection of renewable energy technologies

Abstract

The decision-making in energy sector involves finding a set of energy sources and conversion devices to meet the energy demands in an optimal way. Making an energy planning decision involves the balancing of diverse ecological, social, technical and economic aspects across space and time. Usually, technical and environmental aspects are represented in the form of multiple criteria and indicators that are often expressed as conflicting objectives. In order to attain higher efficiency in the implementation of renewable energy (RE) systems, the developers and investors have to deploy multi-criteria decision-making techniques. In this paper, a novel hybrid Decision Making Trial and Evaluation Laboratory and analytic network process (DEMATEL-ANP) model is proposed in order to stress the importance of the evaluation criteria when selecting alternative REs and the causal relationships between the criteria. Finally, complex proportional assessment and weighted aggregated sum product assessment methods are used to assess the performances of the REs with respect to different evaluating criteria. An illustrative example from Costs assessment of sustainable energy systems (CASES) project, financed by European Commission Framework 6 programme (EU FM 6) for EU member states is presented in order to demonstrate the application feasibility of the proposed model for the comparative assessment and ranking of RE technologies. Sensitivity analysis, result validation and critical outcomes are provided as well to offer guidelines for the policy makers in the selection of the best alternative RE with the maximum effectiveness.

Appendices

Appendix 1: D-ANP method

Step 1 Construct the measure scales of the direct relation matrix.

Decision-makers evaluate the relationship between the sets of paired criteria to indicate the direct influence that each ith criterion exerts on each jth criterion. The initial decision table is developed taking into consideration the expert opinion and literature survey and can be called direct relation matrix where an integer scale (score) ranging from 0 to 4 for pairwise comparison of dimensions/criteria is used: representing no influence (0), low influence (1), middle influence (2), high influence (3) and extreme influence (4).

Step 2 Generation of the initial influence matrix $A = \left[ {a_{ij} } \right]_{n \times n}$.

The matrix $A$ is obtained from the convergence of expert opinion with a direct relation matrix that was developed in Step 1. Then, as a result of these evaluations, the initial data are obtained in the form of n × n matrix, in which the individual element (a _ij ) denotes the degree to which ith criterion affects jth criterion, and n denotes the total number of criteria.

$$A = \left[ {\begin{array}{*{20}c} {a_{11} } & \ldots & {a_{1j} } & \ldots & {a_{1n} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {a_{i1} } & \ldots & {a_{ij} } & \ldots & {a_{in} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {a_{n1} } & \ldots & {a_{nj} } & \ldots & {a_{nn} } \\ \end{array} } \right]$$

(1)

Step 3 Determine the normalized direct influence matrix (N). This matrix has derived from the normalizing matrix A using Eqs. (2, 3):

$$N = A /s$$

(2)

$$s = \hbox{max} \left[ {\mathop {\hbox{max} }\limits_{1 \le i \le n} \sum\limits_{j = 1}^{n} {a_{ij} } ,\mathop {\hbox{max} }\limits_{1 \le i \le n} \sum\limits_{i = 1}^{n} {a_{ij} } } \right].$$

(3)

Step 4 Build the total influence matrix T. T is produced using Eq. (4), where I is the identity matrix, and $\lim_{h \to \infty } N^{h} = [0]_{n \times n}$

$$T = N + N^{2} + N^{3} + \cdots + N^{h} = N(I + N + N^{2} + \cdots + N^{h - 1} )[(I - N)((I - N)^{ - 1} ] = N(I - N^{h} )(I - N)^{ - 1} .$$

Then,

$$T = N(I - N)^{ - 1} ,\,{\text{when}}\;\lim_{h \to \infty } N^{h} = [0]_{n \times n}$$

(4)

$$T = \left[ {t_{ij} } \right]_{n \times n} ,i,j\, = \,1,2, \ldots ,n.$$

(5)

Step 5 Construct the influential network relation map (INRM). According to Eqs. (6, 7), the sum of each row and column for $T$ can be obtained, where vector $r$ (any criterion $i$ influences all other criteria) denotes the sum of all vector rows $r = (r_{1} ,r_{2} , \ldots ,r_{n} )$, and vector $y$ (any criterion j is influenced by all other criteria) denotes the sum of all vector columns $y = (y_{1} ,y_{2} , \ldots ,y_{n} )$. Further on, the sums of rows and columns of matrix $T$ are calculated. At the total-relation matrix T, the sum of rows and sum of columns are represented by vectors $r$ and $y$, which are derived using Eqs. (6) and (7). When $i$ equals $j$, $i,j \in \left\{ {1,2, \ldots ,n} \right\}$, then $(r_{i} + y_{i} )$ represents the total degree of influence among criteria, and the higher is its value, the closer is the criterion to the object’s central point, and $(r_{i} - y_{i} )$ interprets the degree of causality among the criteria. The degrees of influence and causality can provide important reference information to inform decision-making by plotting the INRM:

$$r\, = \,\left[ {\sum\limits_{j = 1}^{n} {t_{ij} } } \right]_{n \times 1} \, = \,\left[ {t_{i} } \right]_{n \times 1} ,\,i\, = 1,2, \ldots ,n$$

(6)

$$y\, = \,\left[ {\sum\limits_{i = 1}^{n} {t_{ij} } } \right]_{1 \times n} \, = \,\left[ {t_{j} } \right]_{n \times 1} ,\,j\, = 1,2, \ldots ,n.$$

(7)

Now, the total influence matrix of criteria as $T = \left[ {t_{ij} } \right]_{n \times n} ,i,j\, = \,1,2, \ldots ,n$ is considered, and the total influence matrix of dimensions (or clusters) as $T = \left[ {t_{ij}^{D} } \right]_{m \times m}$ is regarded. Therefore, in order to obtain the dynamic degree of influence of weights, the weights and their influences in the super-matrix of the ANP need to be determined by normalizing $T_{c}$ by dimension/cluster.

Step 6 Obtain the un-weighted super-matrix $W$ by transposing the normalized total influence matrix $T_{c}^{\beta }$ with the DEMATEL technique. This step uses the basic concepts of the ANP to build the un-weighted super-matrix $W$ as follows:

In order to normalize the total influence matrix $T_{c}$ using dimensions, the following relations must be done; thus, the normalized matrix $T_{c}^{\beta }$ by dimensions can be obtained as shown in Eq. (9). For example, $T_{c}^{\beta 11}$ can be normalized similarly to obtain $T_{c}^{\beta nn}$ as shown in Eqs. (10, 11). Next, using Eq. (12), the normalized influence matrix $T_{c}^{\beta }$ is transposed to obtain the un-weighted super-matrix $W.$

$$T_{c} = \left[ {\begin{array}{*{20}c} {T_{c}^{11} } & \ldots & {T_{c}^{1j} } & \ldots & {T_{c}^{1n} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {T_{c}^{i1} } & \ldots & {T_{c}^{ij} } & \ldots & {T_{c}^{in} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {T_{c}^{n1} } & \ldots & {T_{c}^{nj} } & \ldots & {T_{c}^{nn} } \\ \end{array} } \right]$$

(8)

$$T_{c} = \left[ {\begin{array}{*{20}c} {T_{c}^{\beta 11} } & \ldots & {T_{c}^{\beta 1j} } & \ldots & {T_{c}^{\beta 1n} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {T_{c}^{\beta i1} } & \ldots & {T_{c}^{\beta ij} } & \ldots & {T_{c}^{\beta in} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {T_{c}^{\beta n1} } & \ldots & {T_{c}^{\beta nj} } & \ldots & {T_{c}^{\beta nn} } \\ \end{array} } \right]$$

(9)

$$d_{i}^{11} = \sum\limits_{j = 1}^{{m_{1} }} {t_{{c^{ij} }}^{11} } \,i = 1,2, \ldots ,m_{1}$$

(10)

$$T_{c} = \left[ {\begin{array}{*{20}c} {t_{{c^{11} }}^{11} /d_{1}^{11} } & \ldots & {t_{{c^{1j} }}^{11} /d_{1}^{11} } & \ldots & {t_{{c^{{1m_{1} }} }}^{11} /d_{1}^{11} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {t_{{c^{i1} }}^{11} /d_{i}^{11} } & \ldots & {t_{{c^{ij} }}^{11} /d_{i}^{11} } & \ldots & {t_{{c^{{im_{1} }} }}^{11} /d_{i}^{11} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {t_{{c^{{m_{1} 1}} }}^{11} /d_{{m_{1} }}^{11} } & \ldots & {t_{{c^{{m_{1} j}} }}^{11} /d_{{m_{1} }}^{11} } & \ldots & {t_{{c^{{m_{1} m_{1} }} }}^{11} /d_{{m_{1} }}^{11} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {t_{{c^{1}_{1} }}^{\beta 11} } & \ldots & {t_{{c^{1j} }}^{\beta 11} } & \ldots & {t_{{c^{{1m_{1} }} }}^{\beta 11} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {t_{{c^{i1} }}^{\beta 11} } & \ldots & {T_{{c^{ij} }}^{\beta 11} } & \ldots & {T_{{c^{{im_{1} }} }}^{\beta 11} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {t_{{c^{{m_{1} 1}} }}^{\beta 11} } & \ldots & {t_{{c^{{m_{1} 1}} }}^{\beta 11} } & \ldots & {t_{{c^{{m_{1} m_{1} }} }}^{\beta 11} } \\ \end{array} } \right]$$

(11)

$$W = (T_{c}^{\beta } )^{{\prime }} = \left[ {\begin{array}{*{20}c} {W_{11} } & \ldots & {W_{i1} } & \ldots & {W_{n1} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {W_{1j} } & \ldots & {W_{i1} } & \ldots & {W_{nj} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {W_{1n} } & \ldots & {W_{in} } & \ldots & {W_{nn} } \\ \end{array} } \right].$$

(12)

In this approach, $D_{n}$ shows the $n$th dimension.

Step 7 Compute the weighted super-matrix $W^{\beta } .$ In this case, $T_{D} = \left[ {t_{ij}^{D} } \right]_{m \times m}$ is shown in Eq. (13) and should be normalized by Eq. (14):

$$T_{D} = \left[ {\begin{array}{*{20}c} {t_{11}^{{D_{11} }} } & \ldots & {t_{1j}^{{D_{1j} }} } & \ldots & {t_{1m}^{{D_{1m} }} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {t_{i1}^{{D_{i1} }} } & \ldots & {t_{ij}^{{D_{ij} }} } & \ldots & {t_{im}^{{D_{im} }} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {t_{m1}^{{D_{m1} }} } & \ldots & {t_{m}^{{D_{mj} }} } & \ldots & {t_{mm}^{{D_{mm} }} } \\ \end{array} } \right]$$

(13)

$$d_{i} = \sum\limits_{j = 1}^{m} {t_{ij}^{{D_{ij} }} } \,i = 1,2, \ldots ,m.$$

(14)

Therefore, $T_{D}^{\beta }$ can be determined after normalizing $T_{D}$ as shown in Eq. (15):

$$T_{D}^{\beta } = \left[ {\begin{array}{*{20}c} {t_{11}^{{D_{11} }} /d_{1} } & \ldots & {t_{1j}^{{D_{1j} }} /d_{1} } & \ldots & {t_{1m}^{{D_{1m} }} /d_{1} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {t_{i1}^{{D_{i1} }} /d_{i} } & \ldots & {t_{ij}^{{D_{ij} }} /d_{i} } & \ldots & {t_{im}^{{D_{im} }} /d_{i} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {t_{m1}^{{D_{m1} }} /d_{m} } & \ldots & {t_{m}^{{D_{mj} }} /d_{m} } & \ldots & {t_{mm}^{{D_{mm} }} /d_{m} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {t_{11}^{{\beta_{11} }} } & \ldots & {t_{1j}^{{\beta_{1j} }} } & \ldots & {t_{1m}^{{\beta_{1m} }} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {t_{i1}^{{\beta_{i1} }} } & \ldots & {t_{ij}^{{\beta_{ij} }} } & \ldots & {t_{im}^{{\beta_{im} }} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {t_{m1}^{{\beta_{m1} }} } & \ldots & {t_{m}^{{\beta_{mj} }} } & \ldots & {t_{mm}^{{\beta_{mm} }} } \\ \end{array} } \right]$$

(15)

$T_{D}^{\beta }$ and $W$ attempts to arrive at weights with different degrees of influence in order to obtain the weighted super-matrix $W^{\beta }$, as shown in Eq. (16):

$$W^{\beta } = T_{D}^{\beta } W = \left[ {\begin{array}{*{20}c} {t_{11}^{{\beta_{11} }} \times W_{11} } & \ldots & {t_{1j}^{{\beta_{1j} }} \times W_{i1} } & \ldots & {t_{1m}^{{\beta_{1m} }} \times W_{n1} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {t_{i1}^{{\beta_{i1} }} \times W_{1j} } & \ldots & {t_{ij}^{{\beta_{ij} }} \times W_{ij} } & \ldots & {t_{im}^{{\beta_{im} }} \times W_{nj} } \\ \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} & {} & \begin{aligned} . \hfill \\ . \hfill \\ . \hfill \\ \end{aligned} \\ {t_{m1}^{{\beta_{m1} }} \times W_{1n} } & \ldots & {t_{m}^{{\beta_{mj} }} \times W_{in} } & \ldots & {t_{mm}^{{\beta_{mm} }} \times W_{nn} } \\ \end{array} } \right].$$

(16)

Step 8 Clear the influential weights of the D-ANP with the limit super-matrix $\lim_{g \to \infty } (W^{\beta } )^{g}$. The super-matrix $W^{\beta }$ is multiplied by itself several times to obtain the limit weighted super-matrix (a concept based on the Markov chain) to a fixed convergence value. Then, the influential weights of the D-ANP can be obtained with $\lim_{g \to \infty } (W^{\beta } )^{g}$, where g represents a positive integer number. This process is recognized as a D-ANP, which stands for DEMATEL-based ANP.

Appendix 2: WASPAS method

Among new MCDM tools, WASPAS is called a unique mixture of two well-known MCDM approaches, i.e. weighted sum model (WSM) and weighted product model (WPM) that starts with the following matrix:

$$X = \left[ {\begin{array}{*{20}c} {x_{11} } & {x_{12} } & \ldots & {x_{1n} } \\ {x_{21} } & {x_{22} } & \ldots & {x_{2n} } \\ \ldots & \ldots & \ldots & \ldots \\ {x_{m1} } & {x_{m2} } & \ldots & {x_{mn} } \\ \end{array} } \right]$$

(17)

where $m$ is the number of alternative solutions, and $n$ is the number of evaluation criteria, and in this sense, $x_{mn}$ is the performance rating of each alternative for the decision criteria. Thus, the first step is to normalize the decision matrix using the following equations, where the normalized value is denoted by $\bar{x}_{ij}$.

$${\text{For}}\,{\text{benefit}}\,{\text{attributes:}}\,\bar{x}_{ij} = \frac{{x_{ij} }}{{\max_{i} x_{ij} }}$$

(18)

$${\text{For}}\,{\text{non - benefit}}\,{\text{attributes:}}\,\bar{x}_{ij} = \frac{{\min_{i} x_{ij} }}{{x_{ij} }}.$$

(19)

Algorithm of WASPAS is seeking a joint criterion of optimality based on two criteria of optimality. The first criterion of optimality, i.e. the criterion of a mean weighted success, is similar to WSM method. It is a popular and well-accepted MCDM approach applied for evaluating a number of alternatives in terms of a number of decision criteria. Based on WSM method, the total relative importance of ith alternative is calculated as follows (Triantaphyllou and Mann 1989):

$$Q_{i}^{(1)} = \sum\limits_{j = 1}^{n} {\bar{x}_{ij} w_{j} }$$

(20)

The model is based as well on the WPM method; the total relative importance of ith alternative is computed using the following expression:

$$Q_{i}^{(2)} = \prod\limits_{j = 1}^{n} {(\bar{x}_{ij} )^{{w_{j} }} } .$$

(21)

A joint generalized criterion of weighted aggregation of additive and multiplicative methods is proposed as follows:

$$Q_{i} = 0.5Q_{i}^{(1)} + 0.5Q_{i}^{(2)} = 0.5\sum\limits_{j = 1}^{n} {\bar{x}_{ij} w_{j} } + 0.5\prod\limits_{j = 1}^{n} {(\bar{x}_{ij} )^{{w_{j} }} } .$$

(22)

In order to have an increased ranking accuracy and effectiveness of the decision-making process of WASPAS method, a more generalized equation for determining the total relative importance of ith alternative is developed and provided below (Zavadskas et al. 2012, 2013; Hashemkhani Zolfani et al. 2013):

$$Q_{i} = \lambda Q_{i}^{(1)} + (1 - \lambda )Q_{i}^{(2)} = \lambda \sum\limits_{j = 1}^{n} {\bar{x}_{ij} w_{j} } + (1 - \lambda )\prod\limits_{j = 1}^{n} {(\bar{x}_{ij} )^{{w_{j} }} }$$

(23)

Now, the candidate alternatives are ranked based on the Q values, i.e. the best alternative would be the one having the highest Q value. When the value of $\lambda$ is 0, WASPAS method is transformed to WPM, and when $\lambda$ is 1, it becomes WSM method.

Appendix 3: COPRAS method

The computational steps that are involved in COPRAS method-based analysis are now presented below (Chatterjee et al. 2011; Mulliner et al. 2013; Zavadskas et al. 2009; Bagocius et al. 2014):

Step 1 D is a decision matrix, containing the performance rating of m number of alternatives with respect to n number of criteria, as shown below.

$$D = \left[ {\begin{array}{*{20}c} {x_{11} } & {x_{12} } & \ldots & {x_{1n} } \\ {x_{21} } & {x_{22} } & \ldots & {x_{2n} } \\ \ldots & \ldots & \ldots & \ldots \\ {x_{m1} } & {x_{m2} } & \ldots & {x_{mn} } \\ \end{array} } \right]$$

(24)

where $x_{ij}$ is the rating of ith decision criteria on jth alternative, whereas $m$ is the number of alternatives, and n is the number of criteria.

Step 2 Normalize the decision matrix using Eq. (24).

$$r_{ij} = \frac{{x_{ij} }}{{\sum\nolimits_{j = 1}^{m} {x_{ij} } }},\,j = 1,2, \ldots ,m,\,i = 1,2, \ldots ,n$$

(25)

Step 3 Calculate the weighted normalized decision matrix as follows, where $w_{i}$ includes the weights of criteria and is given by $\sum\nolimits_{i = 1}^{n} {w_{i} = 1}$,

$$v_{ij} = w_{i} \times r_{ij} ,\,j = 1,2, \ldots ,m,\,i = 1,2, \ldots ,n.$$

(26)

The sum of dimensionless weighted normalized values of each criterion is always equal to the weight of that criterion.

$$\sum\limits_{j = 1}^{m} {v_{ij} = w_{i} } \,$$

(27)

Thus, it can be said that the weight, w_i of ith criterion, is proportionally distributed among all the alternatives according to their weighted normalized value v _ij .

Step 4 Calculate the sums of weighted normalized values for both beneficial $(P_{j} )$ and non-beneficial attributes $(R_{j} )$ using the following equations:

$$P_{j} = \sum\limits_{i = 1}^{k} {v_{ij} }$$

(28)

where $k$ is the number of criteria to be maximized.

$$R_{j} = \sum\limits_{i = 1}^{n - k} {v_{ij} }$$

(29)

where $(n - k)$ is the number of criteria to be minimized.

Step 5 Determine the relative significances or priorities of the alternatives as follows:

$$Q_{j} = P_{j} + \frac{{\sum\nolimits_{j = 1}^{m} {R_{j} } }}{{R_{j} \sum\nolimits_{j = 1}^{m} {\frac{1}{{R_{j} }}} }}.$$

(30)

Step 6 Calculate the quantitative utility (N _j ) for jth alternative. The degree of an alternative utility, which leads to a complete ranking of the candidate alternatives, is determined by comparing the priorities of all the alternatives with the most efficient one and can be denoted as it is shown below:

$$N_{j} = \frac{{Q_{j} }}{{Q_{\rm max } }} \times 100\%$$

(31)

where $Q_{\rm max }$ is the maximum relative significance value. These utility values of the alternatives range from 0 to 100%. Thus, this approach allows evaluating the direct and proportional dependence of the significance and utility degree of the considered alternatives in a decision-making problem having multiple criteria, their weights and performance values of the alternatives with respect to all the criteria.