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A Hybrid Approach of Neutrosophic with MULTIMOORA in Application of Personnel Selection.

1. Introduction

The competitiveness of organizations can be achieved by the ability of efficient employment [1]. For organization, the most effective part of Human Resource Management is the personnel selection process [2]. The classical methods are used in organizations to select candidates were not sufficient enough and need to be enhanced, to continue proceeding with globalization and rivalry [3]. The numerous and conflict personal criteria make the decision maker confused [4]. The fuzzy set theory appears as an important tool to provide a decision framework that incorporates imprecise judgments inherent in the personnel selection process [5, 6] The Analytical Hierarchy Process (AHP) is used to format the complex problems into a hierarchical form of criterions, alternatives, and goals to support decision makers in the selection process [7]. Classical AHP method has been stretched to numerous fuzzy versions, because of partial information and ambiguity. Although the theories of fuzzy have been developed and generalized but cannot deal with all kinds of uncertainties in real problems. Indeed, sure kinds of uncertainties, such as indeterminate and inconsistent information, cannot be managed. Therefore, some new theories are required to present the truth membership, indeterminacy membership and falsity membership simultaneously this called neutrosophic sets. Unlike fuzzy, the neutrosophic sets deal with uncertain, inconsistent, and incomplete information in many researches [32-40]. The personnel selection is a multi-criteria decision-making (MCDM) problem that contains multiple criterions, alternatives, and decision makers to obtain the best candidate to be hire in organization [8]. The use of neutrosophic in personnel selection aids decision makers in the case of uncertainty and inconsistent information to achieve organizations objectives [9]. Sometimes neither of candidates satisfies the vision and objectives of organizations. Therefore, in this study we extend the neutrosophic personnel selection with MULTIMOORA method to encompass the measurement value the method reference level.

The Multi-Objective Optimization by Ratio Analysis (MOORA) method has been introduced by [10]. The MOORA is composed of ratio system, reference point [11-13]. The method MOORA enhanced to MULTIMOORA by adding full Multiplicative Form and employing Dominance Theory to obtain a final rank [2]. The ordinary MULTIMOORA method has been proposed for usage with crisp numbers. MUTIMOORA can solve larger numbers of complex decision-making problems by adding several extensions to solve wide range of problems. The hybrid approach handles the current obstacles and challenges by recommending the most appropriate candidates in the environment of uncertainty and incomplete information.

The structure of this paper ordered as follows: section 2 illustrates some related studies of personnel selection. Section 3 represents the hybrid methodology of neutrosophic with MUTIMOORA method to aid decision makers to choose most appropriate candidate to achieve the goal of organization. Section 4 represents an empirical case study for the proposed hybrid approach. Section 5 summarizes the research key pints and the future trends.

2. Related Studies

The processes of personnel selection in organizations can be affected by many conditions e.g. change the nature of work, governmental regulations, client's behavior, development of new technology, and others [14-16]. The traditional methods are not appropriate enough to keep on globalization. Hence organizations needs to make enhancement on personnel selection problem especially in the field of the judgments of decision makers by integrating advanced tools to decision support system [17,18]. In [19-22] describe the method of AHP with a fuzzy multi-criteria decision making algorithms for solving the personnel selection problems. In [23-25] describe the fuzzy MCDM with TOPSIS method to solve personnel selection problem using linguistic and numerical scales with different data sources to permit decision makers to evaluate candidate's information. In [19] illustrate the AHP method combined with fuzzy to solve personnel selection problem for information systems.

The MULTIMOORA method is extended by researchers to handle several MCDM problems [26, 27]. In [2,] the use of MULTMOORA with a fuzzy MCDM were not the most appropriate methodology. Due to the situations of uncertainty and incomplete information, researches recommend to integrate neutrosophic sets in personnel selection problem [28, 29]. We propose to be the first to applying the neutrosophic sets with MULTIMOORA method to aid decision makers to achieve to the most appropriate candidates.

3. Methodology

A hybrid MULTIMOORA method with neutrosophic is applied in personnel selection problem to select the best candidate to hire in organization. The MULTIMOORA method is used to solve personnel selection problem. In Fig. 1 represents conceptual flow of personnel selection to obtain ideal solution. In Fig. 2 represents the structure of methodology phase to apply MULTIMOORA method with neutrosophic. The phases for the hybrid approach are mentioned as follows:

Phase1: Acquire expert information in neutrosophic environment.

* Determine the study goal, criteria, and alternative.

* Use neutrosophic scale mentioned in Table 1 [30].

* Create pairwise matrix of decision making judgments using the following form:

[Please download the PDF to view the formula] (1)

* Aggregate pairwise matrix by:

[B.sub.uv] = [[[summation].sup.M.sub.M=1]<([l.sup.M.sub.uv],[m.sup.M.sub.uv],[u.sup.M.sub.uv];[T.sup.M.sub.uv],[I.sup.M.sub.uv],[F.sup.M.sub.uv]/ M] (2)

Where, M represents number of decision makers,[l.sup.M.sub.uv],[m.sup.M.sub.uv],[u.sup.M.sub.uv] are lower, middle and upper bound of neutrosophic number,[T.sup.M.sub.uv],[I.sup.M.sub.uv],[F.sup.M.sub.uv]/ M] are truth, indeterminacy and falsity.

* Construct the initial pairwise comparison matrix as mentioned:

[Please download the PDF to view the formula] (3)

* Convert neutrosophic scales to crisp values by using score function of [B.sub.uv] [31]:

[Please download the PDF to view the formula] (4)

where l, m, u represents lower, middle and upper of the scale neutrosophic numbers.

Phase2: Calculate weights of criteria.

* Compute the average of row

[W.sub.u] = [[[SIGMA].sup.z.sub.v=1]([B.sub.uv]/z;u = 1,2,3 y;v = 1,2,3 z; (5)

* The normalization of crisp value is calculated using the following equation

[W.sup.y.sub.u] = [[w.sub.u]/[[SIGMA].sup.y.sub.u=1[w.sub.u];u = 1,2,3, y (6)

Phase3: Evaluate expert judgement using consistency rate

Check the conistency of matrix using table 2 and for detailed information in [31]

* Compute weighted columns by multiplying the weight of priority by each value in the pairwise comparison matrix [31].

* The weighted sum values are divided with the corresponding priority.

* Compute the mean of the previous step denoted as [[lambda].sub.max].

* Compute consistency index CI = [[[lambda].sub.max]-n/n-1],where n the number of criteria.

* Calculate consistency ratio by the use for the mentioned equation

CR = [CI/RI] (7)

Where, CR is the consistency rate, CI is consistency Index. RI is the random index for consistency matrix as mentioned in Table 3.

Phase4: MULTIMOORA Method

The decision judgments between criterions and alternatives will be collected and obtained by the use of form (1). Then, apply Equation (2) to make a general vision of aggregation of experts. Finally, apply Equation (4) to change neutrosophic scale values to crisp values. The MULTIMOORA method consists of: ratio system, reference point and full multiplicative form. Phase4.1: Ratio System

* The first step of ratio system is to calculate the normalize of the decision matrix as mentioned:

[B.sup.*.sub.uv] = [[B.sub.uv]/[square root of [SIGMA].sup.y.sub.u=1[B.sup.2.sub.uv]] u = 1,2,3,, y and v = 1,2,3, z. (8)

* Compute the beneficial criteria ([Y.sup.+]) is the summation of beneficial criteria of weight normalized elements of matrix. Then non-beneficial criteria denoted as ([Y.sup.-]). Finally subtract sum of beneficial criteria from sum of non-beneficial criteria. (NB. In this study all criterions are beneficial)

[Y.sup.+] = [[SIGMA].sup.g.sub.v=1][w.sub.v][B.sup.*.sub.uv] (9)

[Y.sup.-] = [[summation].sup.z.sub.v=1][w.sub.v][B.sup.*.sub.UV] (10)

* The next formula represents number of criteria to be maximized and (z-g) represents number of criteria to be minimized.

[Y.sup.*] = [[SIGMA].sup.g.sub.v=1][w.sub.v][B.sup.*.sub.uv] - [[SIGMA].sup.z.sub.sub.v=g+1][w.sub.v][B.sup.*.sub.uv] (11)

,where [w.sub.v] is the weight of criteria

* Finally, Rank the alternatives

Phase4.2: Reference point

The second step of neutrosophic MULTIMOORA is reference point

* Compute reference point to be maximized

[Please download the PDF to view the formula] (12)

* Compute reference point to be minimized

[Please download the PDF to view the formula] (13)

* Compute deviation of reference point

[Please download the PDF to view the formula] (14)

Phase4.3: Full multiplicative form

The third step of neutrosophic MULTIMOORA is full multiplicative form

* Compute utility of the alternative

[[union].sub.u] = [[E.sup.u]/[F.sub.u]] (15)

[E.sub.u] = [[PI].sup g.sub.v=1][w.sub.v][([B.sup.*.sub.z]).sub.uv] (16)

[F.sub.u] = [[PI].sup g.sub.v=g+1[w.sub.v][([B.sup.*.sub.z]).sub.uv] (17)

The first component Eu represents the product of criteria of u th alternative to be maximized. The second component Fu represents the product criteria of uth alternative to be minimized.

* Finally apply the dominance theory to obtain final rank

4. An Empirical Case Study

In this section, the case study is about personnel selection in a telecommunication company in smart village in Egypt. The case study applies the hybrid methodology of neutrosophic with MULTIMOORA method. In order to make a general image for the telecommunication company, we adopt eight criterions, seven alternatives, and four decision makers. Figure 3 shows the relations between criterions and alternatives. The telecommunication goal is to hire best candidate to achieve competitive organization goals.

Phase 1: Represent expert judgments in neutrosophic environment

* Create neutrosophic triangular scale (linguistic term) in Table 1.

* Create the general vision pairwise comparison matrix of criteria in Table 4 in form (1).

* Aggregate pairwise comparison matrix of criteria using Equations (2) and form in (3).

* Convert aggregate pairwise comparison matrix of criteria to crisp value in Table 5 using Equation (4).

Phase 2: Calculate weight of criteria as mentioned in Fig. (4).

* Compute the average of row.

[Please download the PDF to view the formula]

* The normalization of crisp value is calculated.

[Please download the PDF to view the formula]

[summation][w.sub.i] = 1

Phase 3: Check consistency rate

* Compute weighted sum

[Please download the PDF to view the formula]

* Divide weighted sum by weight of criteria

[Please download the PDF to view the formula]

* Divide summation of Weighted sum by the number of criteria 8

* Compute [[lambda].sub.max] = 8.38206

* Compute CI = [[[lambda].sub.max]-n/n-1] = [8.38206-8/8-1] 0.05458

* Compute CR = [CI/RI] = [0.05458/1.41] = 0.03870.

Hence, the pair-wise comparison matrix is consistent and fellow the next phase of

MULTIMOORA Method

Phase 4: MULTIMOORA Method Calculations

* A session is performed with four decision makers and the collected judgments presented in table 6.

* Aggregate judgments of decision matrix of four decision makers using Equation (2).

* Compute crisp value of aggregated decision matrix using Equation (4) and mentioned in

Table 7.

Phase 4.1: The ratio system

* Calculate normalization of decision matrix in using Equation (8), and mentioned in Table 8.

* Calculate [Y.sup.+] (weight normalized) using Equation (9) in Table 9.

* [Y.sup.-] = 0 because all criteria are beneficial.

* The ranks of ratio system ranking are mentioned in Table 10.

Phase 4.2: The reference point

* Calculate Reference point [r.sub.v] using Eq. (12) in table 11

* Calculate deviations from reference point using Eq. (14) in table 12

* The Reference point ranking mentioned in table 13.

Phase 4.3: Full multiplicative form

* Compute utility of the alternative using Equation (15), (16) and (17) in Table 14.

* The full Multiplicative form ranking in Table 15.

According to Table 16 and Fig. 5, the final rank recommends alternative one as the best alternative, while alternative four as the worst alternative.

5. Conclusions

Personnel selection is an important issue that effect on the competitive advantages for organizations. Decision makers take decisions for complex problems with various criterions and alternatives with surrounded environment of uncertain and incomplete information. The traditional methods cannot achieve to the proper solutions. In addition fuzzy cannot handle the conditions of uncertainty and inconsistency. The study proposes to use neutrosophic sets to handle the environmental conditions of uncertainty and inconsistent information, in addition extend study with MULTIMOORA method to choose the most appropriate candidate. A case study is applied on smart village Cairo, Egypt, on Telecommunication Company shows the effectiveness for the proposed method and provides final decision to hire the most appropriate candidate for attaining success of enterprises. The future work includes evolutionary algorithms for selecting the most effective criterions. In addition, applies other methodologies e.g. DEMTAL to improve the selection process.

Acknowledgements

The authors are highly grateful to the Referees for their constructive suggestions.

Conflicts of Interest

The authors declare no conflict of interest.

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Nada A. Nabeeh (1), Ahmed Abdel-Monem (2) and Ahmed Abdelmouty (2)

(1) Faculty of Computers and Informatics, Zagazig University, Egypt

(2) Information Systems Department, Faculty of Computers and Information Sciences, Mansoura University, Egypt

(*) Corresponding author: Nada A. Nabeeh (nada.nabeeh@gmail.com).

Received: Sep 20, 2019. Accepted: Dec 05, 2019
Table1. Neutrosophic triangular scale (linguistic terms)
Saaty scale   Caption
1             Evenly significant
3             A little significant
5             Powerfully significant
7             Completely Powerfully significant
9             Absolutely significant
2
4             Sporadic values between two close
6             scales
8
Saaty scale   Neutrosophic triangular scale
1             [~.1] = < <1,1, 1>;0.50, 0.50, 0.50>
3             [~.3] = < <2,3, 4>;0.30, 0.75, 0.70>
5             [~.5] = < <4,5, 6>;0.80, 0.15, 0.20>
7             [~.7] = < <6,7,8>;0.90, 0.10, 0.10>
9             [~.9] = < <9,9,0>;1.00, 0.00, 0.00>
2             [~.2] = < <1,2, 3>;0.40, 0.60, 0.65>
4             [~.4] = < <3,4, 5>;0.35, 0.60, 0.40>
6             [~.6] = < <5,6, 7>;0.70, 0.25, 0.30>
8             [~.8] = < <7,8,9>;0.85, 0.10, 0.15>
Table 2. The consistency rate for pair-wise comparison matrix
N                            4 * 4   5 * 5   N > 4
CR [less than or equal to]   0.58    0.90    1.12
Table 3. Random Consistency index for various criterions
Size of matrix   1      2      3      4      5      6      7      8
Random           0.00   0.00   0.58   0.90   1.12   1.24   1.32   1.41
Consistency
Size of matrix   9      10
Random           1.45   1.49
Consistency
Table 4.The pairwise comparison matrix of criteria of decision maker 
judgments
          C1            C2          C3           C4           C5
      C1  < <1, 1, 1    < <4,5,    < <1, 1, 1   < <1, 1, 1   < <4,5,
          >;0.50,       6>;0.80,    >;0.50,      >;0.50,      6>;0.80,
          0.50, 0.50>   0.15,       0.50,        0.50,        0.15,
                        0.20>       0.50>        0.50>        0.20>
      C2  1/< <4,5,    < <1, 1, 1  < <1, 1, 1   < <4,5,     < <7,8
          6>;0.80,      >;0.50,     >;0.50,      6>;0.80,    ,9>;0.85,
          0.15, 0.20>   0.50,       0.50,        0.15,        0.10,
                        0.50>       0.50>        0.20>        0.15>
      C3  1/< <1, 1, 1  1/< <1, 1,  < <1, 1, 1   < <5,6,     < <4,5,
          >;0.50,       1 >;0.50,   >;0.50,      7>;0.70,     6>;0.80,
          0.50, 0.50>
                        0.50,       0.50,        0.25,        0.15,
                        0.50>       0.50>        0.30>        0.20>
      C4  1/< <1, 1, 1  1/< <4,5,  1/< <5,6,   < <1, 1, 1   < <7,8
          >;0.50,       6>;0.80,    7>;0.70,     >;0.50,     ,9>;0.85,
          0.50, 0.50>   0.15,       0.25,        0.50,        0.10,
                        0.20>       0.30>        0.50>        0.15>
      C5  1/< <4,5,    1/< <7,8   1/< <4,5,   1/< <7,8    < <1, 1, 1
DM1       6>;0.80,     ,9>;0.85,   6>;0.80,    ,9>;0.85,    >;0.50,
          0.15, 0.20>   0.10,       0.15,        0.10,        0.50,
                        0.15>       0.20>        0.15>        0.50>
      C6  1/< <3,4,    1/< <4,5,  1/< <1,2,   1/< <4,5,   1/< <3,4,
          5>;0.35,      6>;0.80,    3>;0.40,     6>;0.80,     5>;0.35,
          0.60, 0.40>   0.15,       0.60,        0.15,        0.60,
                        0.20>       0.65>        0.20>        0.40>
      C7  1/< <7,8     1/< <1, 1,  1/< <4,5,   1/< <1, 1,   1/< <4,5,
         ,9>;0.85,     1 >;0.50,   6>;0.80,     1 >;0.50,    6>;0.80,
          0.10, 0.15>   0.50,       0.15,        0.50,        0.15,
                        0.50>       0.20>        0.50>        0.20>
      C8  1/< <7,8     1/< <1, 1,  1/< <7,8    1/< <4,5,   1/< <1, 1,
         ,9>;0.85,     1 >;0.50,  ,9>;0.85,    6>;0.80,     1 >;0.50,
          0.10, 0.15>   0.50,       0.10,        0.15,        0.50,
                        0.50>       0.15>        0.20>        0.50>
      C1  < <1, 1, 1    < <4,5,    < <7,8      < <1, 1, 1   < <5,6,
          >;0.50,       6>;0.80,   ,9>;0.85,    >;0.50,      7>;0.70,
          0.50, 0.50>   0.15,       0.10,        0.50,        0.25,
                        0.20>       0.15>        0.50>        0.30>
      C2  1/< <4,5,    < <1, 1, 1  < <1, 1, 1   < <1, 1, 1   < <4,5,
          6>;0.80,      >;0.50,     >;0.50,      >;0.50,      6>;0.80,
          0.15, 0.20>   0.50,       0.50,        0.50,        0.15,
                        0.50>       0.50>        0.50>        0.20>
      C3  1/< <7,8     1/< <1, 1,  < <1, 1, 1   < <5,6,     < <5,6,
         ,9>;0.85,     1 >;0.50,   >;0.50,      7>;0.70,     7>;0.70,
          0.10, 0.15>   0.50,       0.50,        0.25,        0.25,
                        0.50>       0.50>        0.30>        0.30>
      C4  1/< <1, 1, 1  1/< <1, 1,  1/< <5,6,   < <1, 1, 1   < <5,6,
          >;0.50,       1 >;0.50,   7>;0.70,     >;0.50,      7>;0.70,
DM2       0.50, 0.50>   0.50,       0.25,        0.50,        0.25,
                        0.50>       0.30>        0.50>        0.30>
      C5  1/< <5,6,    1/< <4,5,  1/< <5,6,   1/< <5,6,   < <1, 1, 1
          7>;0.70,      6>;0.80,    7>;0.70,     7>;0.70,     >;0.50,
          0.25, 0.30>   0.15,       0.25,        0.25,        0.50,
                        0.20>       0.30>        0.30>        0.50>
      C6  1/< <7,8     1/< <4,5,  1/< <7,8    1/< <4,5,   1/< <7,8
         ,9>;0.85,     6>;0.80,   ,9>;0.85,    6>;0.80,    ,9>;0.85,
          0.10, 0.15>   0.15,       0.10,        0.15,        0.10,
                        0.20>       0.15>        0.20>        0.15>
      C7  1/< <3,4,    1/< <3,4,  1/< <1, 1,   1< <5,6,    1/< <1, 1,
          5>;0.35,      5>;0.35,    1 >;0.50,    7>;0.70,     1 >;0.50,
          0.60, 0.40>   0.60,       0.50,        0.25,        0.50,
                        0.40>       0.50>        0.30>        0.50>
      C8  1/< <1, 1, 1  1/< <3,4,  1/< <5,6,   1/< <3,4,   1/< <5,6,
          >;0.50,       5>;0.35,    7>;0.70,     5>;0.35,     7>;0.70,
          0.50, 0.50>   0.60,       0.25,        0.60,        0.25,
                        0.40>       0.30>        0.40>        0.30>
      C1  < <1, 1, 1    < <1, 1, 1  < <4,5,     < <7,8      < <7,8
          >;0.50,       >;0.50,     6>;0.80,    ,9>;0.85,   ,9>;0.85,
          0.50, 0.50>   0.50,       0.15,        0.10,        0.10,
                        0.50>       0.20>        0.15>        0.15>
      C2  1/< <1, 1, 1  < <1, 1, 1  < <4,5,     < <3,4,     < <1, 1, 1
          >;0.50,       >;0.50,     6>;0.80,     5>;0.35,     >;0.50,
          0.50, 0.50>   0.50,       0.15,        0.60,        0.50,
                        0.50>       0.20>        0.40>        0.50>
      C3  1/< <4,5,    1/< <4,5,  < <1, 1, 1   < <3,4,     < <1, 1, 1
DM3       6>;0.80,      6>;0.80,    >;0.50,      5>;0.35,     >;0.50,
          0.15, 0.20>   0.15,       0.50,        0.60,        0.50,
                        0.20>       0.50>        0.40>        0.50>
      C4  1/< <7,8     1/< <3,4,  1/< <3,4,   < <1, 1, 1   < <5,6,
         ,9>;0.85,     5>;0.35,    5>;0.35,     >;0.50,      7>;0.70,
          0.10, 0.15>   0.60,       0.60,        0.50,        0.25,
                        0.40>       0.40>        0.50>        0.30>
      C5  1/< <7,8     1/< <1, 1,  1/< <1, 1,   1/< <5,6,   < <1, 1, 1
         ,9>;0.85,     1 >;0.50,   1 >;0.50,    7>;0.70,     >;0.50,
          0.10, 0.15>   0.50,       0.50,        0.25,        0.50,
                        0.50>       0.50>        0.30>        0.50>
      C6  1/< <4,5,    1/< <7,8   1/< <4,5,   1/< <3,4,   1/< <3,4,
          6>;0.80,     ,9>;0.85,   6>;0.80,     5>;0.35,     5>;0.35,
          0.15, 0.20>   0.10,       0.15,        0.60,        0.60,
                        0.15>       0.20>        0.40>        0.40>
      C7  1/< <3,4,    1/< <1, 1,  1/< <5,6,   1/< <4,5,   1/< <3,4,
          5>;0.35,      1 >;0.50,   7>;0.70,     6>;0.80,     5>;0.35,
          0.60, 0.40>   0.50,       0.25,        0.15,        0.60,
                        0.50>       0.30>        0.20>        0.40>
      C8  1/< <3,4,    1/< <5,6,  1/< <1, 1,   1/< <7,8    1/< <5,6,
          5>;0.35,      7>;0.70,    1 >;0.50,   ,9>;0.85,    7>;0.70,
          0.60, 0.40>   0.25,       0.50,        0.10,        0.25,
                        0.30>       0.50>        0.15>        0.30>
      C1  < <1, 1, 1    < <7,8     < <4,5,     < <5,6,     < <3,4,
          >;0.50,      ,9>;0.85,   6>;0.80,     7>;0.70,     5>;0.35,
          0.50, 0.50>   0.10,       0.15,        0.25,        0.60,
                        0.15>       0.20>        0.30>        0.40>
      C2  1/< <7,8     < <1, 1, 1  < <7,8      < <5,6,     < <5,6,
         ,9>;0.85,     >;0.50,    ,9>;0.85,    7>;0.70,     7>;0.70,
          0.10, 0.15>   0.50,       0.10,        0.25,        0.25,
                        0.50>       0.15>        0.30>        0.30>
      C3  1/< <4,5,    1/< <7,8   < <1, 1, 1   < <5,6,     < <3,4,
          6>;0.80,     ,9>;0.85,   >;0.50,      7>;0.70,     5>;0.35,
          0.15, 0.20>   0.10,       0.50,        0.25,        0.60,
                        0.15>       0.50>        0.30>        0.40>
      C4  1/< <5,6,    1/< <5,6,  1/< <5,6,   < <1, 1, 1   < <1, 1, 1
          7>;0.70,      7>;0.70,    7>;0.70,     >;0.50,      >;0.50,
          0.25, 0.30>   0.25,       0.25,        0.50,        0.50,
DM4                     0.30>       0.30>        0.50>        0.50>
      C5  1/< <3,4,    1/< <5,6,  1/< <3,4,   1/< <1, 1,   < <1, 1, 1
          5>;0.35,      7>;0.70,    5>;0.35,     1 >;0.50,    >;0.50,
          0.60, 0.40>   0.25,       0.60,        0.50,        0.50,
                        0.30>       0.40>        0.50>        0.50>
      C6  1/< <7,8     1/< <4,5,  1/< <5,6,   1/< <4,5,   1/< <1, 1,
         ,9>;0.85,     6>;0.80,    7>;0.70,     6>;0.80,     1 >;0.50,
          0.10, 0.15>   0.15,       0.25,        0.15,        0.50,
                        0.20>       0.30>        0.20>        0.50>
      C7  1/< <5,6,    1/< <3,4,  1/< <5,6,   1/< <3,4,   1/< <1, 1,
          7>;0.70,      5>;0.35,    7>;0.70,     5>;0.35,     1 >;0.50,
          0.25, 0.30>   0.60,       0.25,        0.60,        0.50,
                        0.40>       0.30>        0.40>        0.50>
      C8  1/< <1, 1, 1  1/< <1, 1,  1/< <3,4,   1/< <3,4,   1/< <7,8
          >;0.50,       1 >;0.50,   5>;0.35,     5>;0.35,    ,9>;0.85,
          0.50, 0.50>   0.50,       0.60,        0.60,        0.10,
                        0.50>       0.40>        0.40>        0.15>
           C6           C7           C8
      C1   < <3,4,     < <7,8      < <7,8
           5>;0.35,    ,9>;0.85,   ,9>;0.85,
           0.60,        0.10,        0.10,
           0.40>        0.15>        0.15>
      C2   < <4,5,     < <1, 1, 1   < <1, 1, 1
           6>;0.80,     >;0.50,      >;0.50,
           0.15,        0.50,        0.50,
           0.20>        0.50>        0.50>
      C3   < <1,2,     < <4,5,     < <7,8
           3>;0.40,     6>;0.80,    ,9>;0.85,
           0.60,        0.15,        0.10,
           0.65>        0.20>        0.15>
      C4   < <4,5,     < <1, 1, 1   < <4,5,
           6>;0.80,     >;0.50,      6>;0.80,
           0.15,        0.50,        0.15,
           0.20>        0.50>        0.20>
      C5   < <3,4,     < <4,5,     < <1, 1, 1
DM1        5>;0.35,     6>;0.80,     >;0.50,
           0.60,        0.15,        0.50,
           0.40>        0.20>        0.50>
      C6   < <1, 1, 1   < <3,4,     < <1, 1, 1
           >;0.50,      5>;0.35,     >;0.50,
           0.50,        0.60,        0.50,
           0.50>        0.40>        0.50>
      C7   1/< <4,5,   < <1, 1, 1   < <1, 1, 1
           6>;0.80,     >;0.50,      >;0.50,
           0.15,        0.50,        0.50,
           0.20>        0.50>        0.50>
      C8   1/< <1, 1,   1/< <1, 1,   < <1, 1, 1
           1 >;0.50,    1 >;0.50,    >;0.50,
           0.50,        0.50,        0.50,
           0.50>        0.50>        0.50>
      C1   < <7,8      < <3,4,     < <1, 1, 1
          ,9>;0.85,    5>;0.35,     >;0.50,
           0.10,        0.60,        0.50,
           0.15>        0.40>        0.50>
      C2   < <4,5,     < <3,4,     < <3,4,
           6>;0.80,     5>;0.35,     5>;0.35,
           0.15,        0.60,        0.60,
           0.20>        0.40>        0.40>
      C3   < <7,8      < <1, 1, 1   < <5,6,
          ,9>;0.85,    >;0.50,      7>;0.70,
           0.10,        0.50,        0.25,
           0.15>        0.50>        0.30>
      C4   < <4,5,     < <5,6,     < <3,4,
           6>;0.80,     7>;0.70,     5>;0.35,
DM2        0.15,        0.25,        0.60,
           0.20>        0.30>        0.40>
      C5   < <7,8      < <1, 1, 1   < <5,6,
          ,9>;0.85,    >;0.50,      7>;0.70,
           0.10,        0.50,        0.25,
           0.15>        0.50>        0.30>
      C6   < <1, 1, 1   < <3,4,     < <3,4,
           >;0.50,      5>;0.35,     5>;0.35,
           0.50,        0.60,        0.60,
           0.50>        0.40>        0.40>
      C7   1/< <3,4,   < <1, 1, 1   < <5,6,
           5>;0.35,     >;0.50,      7>;0.70,
           0.60,        0.50,        0.25,
           0.40>        0.50>        0.30>
      C8   1/< <3,4,   1/< <5,6,   < <1, 1, 1
           5>;0.35,     7>;0.70,     >;0.50,
           0.60,        0.25,        0.50,
           0.40>        0.30>        0.50>
      C1   < <4,5,     < <3,4,     < <3,4,
           6>;0.80,     5>;0.35,     5>;0.35,
           0.15,        0.60,        0.60,
           0.20>        0.40>        0.40>
      C2   < <7,8      < <1, 1, 1   < <5,6,
          ,9>;0.85,    >;0.50,      7>;0.70,
           0.10,        0.50,        0.25,
           0.15>        0.50>        0.30>
      C3   < <4,5,     < <5,6,     < <1, 1, 1
DM3        6>;0.80,     7>;0.70,     >;0.50,
           0.15,        0.25,        0.50,
           0.20>        0.30>        0.50>
      C4   < <3,4,     < <4,5,     < <7,8
           5>;0.35,     6>;0.80,    ,9>;0.85,
           0.60,        0.15,        0.10,
           0.40>        0.20>        0.15>
      C5   < <3,4,     < <3,4,     < <5,6,
           5>;0.35,     5>;0.35,     7>;0.70,
           0.60,        0.60,        0.25,
           0.40>        0.40>        0.30>
      C6   < <1, 1, 1   < <3,4,     < <3,4,
           >;0.50,      5>;0.35,     5>;0.35,
           0.50,        0.60,        0.60,
           0.50>        0.40>        0.40>
      C7   1/< <3,4,   < <1, 1, 1   < <5,6,
           5>;0.35,     >;0.50,      7>;0.70,
           0.60,        0.50,        0.25,
           0.40>        0.50>        0.30>
      C8   1/< <3,4,   1/< <5,6,   < <1, 1, 1
           5>;0.35,     7>;0.70,     >;0.50,
           0.60,        0.25,        0.50,
           0.40>        0.30>        0.50>
      C1   < <7,8      < <5,6,     < <1, 1, 1
          ,9>;0.85,    7>;0.70,     >;0.50,
           0.10,        0.25,        0.50,
           0.15>        0.30>        0.50>
      C2   < <4,5,     < <3,4,     < <1, 1, 1
           6>;0.80,     5>;0.35,     >;0.50,
           0.15,        0.60,        0.50,
           0.20>        0.40>        0.50>
      C3   < <5,6,     < <5,6,     < <3,4,
           7>;0.70,     7>;0.70,     5>;0.35,
           0.25,        0.25,        0.60,
           0.30>        0.30>        0.40>
      C4   < <4,5,     < <3,4,     < <3,4,
           6>;0.80,     5>;0.35,     5>;0.35,
           0.15,        0.60,        0.60,
DM4        0.20>        0.40>        0.40>
      C5   < <1, 1, 1   < <1, 1, 1   < <7,8
           >;0.50,      >;0.50,     ,9>;0.85,
           0.50,        0.50,        0.10,
           0.50>        0.50>        0.15>
      C6   < <1, 1, 1   < <1, 1, 1   < <5,6,
           >;0.50,      >;0.50,      7>;0.70,
           0.50,        0.50,        0.25,
           0.50>        0.50>        0.30>
      C7   1/< <1, 1,   < <1, 1, 1   < <4,5,
           1 >;0.50,    >;0.50,      6>;0.80,
           0.50,        0.50,        0.15,
           0.50>        0.50>        0.20>
      C8   1/< <5,6,   1/< <4,5,   < <1, 1, 1
           7>;0.70,     6>;0.80,     >;0.50,
           0.25,        0.15,        0.50,
           0.30>        0.20>        0.50>
Table 5. Crisp value of aggregated pairwise comparison matrix of 
criteria.
Criteria   C1        C2        C3        C4        C5        C6
C1         1         1.88288   1.88288   1.85098   2.01946   2.04291
C2         0.53110   1         1.77829   1.82446   1.94923   1.93354
C3         0.53110   0.56233   1         2.05393   1.79510   2.02662
C4         0.54025   0.54810   0.48687   1         2.01743   1.85375
C5         0.48949   0.51302   0.55707   0.49568   1         1.88588
C6         0.48949   0.51718   0.49343   0.53944   0.53025   1
C7         0.49032   0.65130   0.52651   0.54810   0.63222   0.58468
C8         0.56788   0.60151   0.51091   0.50715   0.45991   0.55205
Criteria   C7        C8
C1         2.03948   1.76092
C2         1.53537   1.66246
C3         1.89927   1.95726
C4         1.82446   1.97178
C5         1.58172   2.01743
C6         1.71033   1.81143
C7         1         1.89927
C8         0.52651   1
Table 6. The judgments for multiple decision makers
      Criteria        C1          C2          C3          C4
      /Alternatives
      A1              < <4,5,    < <1,2,    < <7,8     < <4,5,
                      6>;0.80,    3>;0.40,   ,9>;0.85,   6>;0.80,
                      0.15,       0.60,       0.10,       0.15,
                      0.20>       0.65>       0.15>       0.20>
      A2              < <1,1,    < <4,5,    < <1,1,    < <5,6,
                      1>;0.50,    6>;0.80,    1>;0.50,    7>;0.70,
                      0.50,       0.15,       0.50,       0.25,
DM1                   0.50>       0.20>       0.50>       0.30>
      A3              < <1,1,    < <7,8     < <3,4,    < <3,4,
                      1>;0.50,   ,9>;0.85,   5>;0.35,    5>;0.35,
                      0.50,       0.10,       0.60,       0.60,
                      0.50>       0.15>       0.40>       0.40>
      A4              < <7,8     < <3,4,    < <4,5,    < <1,2,
                     ,9>;0.85,   5>;0.35,    6>;0.80,    3>;0.40,
                      0.10,       0.60,       0.15,       0.60,
                      0.15>       0.40>       0.20>       0.65>
      A5              < <7,8     < <1,1,    < <1,1,    < <4,5,
                     ,9>;0.85,   1>;0.50,    1>;0.50,    6>;0.80,
                      0.10,       0.50,       0.50,       0.15,
                      0.15>       0.50>       0.50>       0.20>
      A6              < <4,5,    < <1,1,    < <4,5,    < <1,1,
                      6>;0.80,    1>;0.50,    6>;0.80,    1>;0.50,
                      0.15,       0.50,       0.15,       0.50,
                      0.20>       0.50>       0.20        0.50>
      A7              < <4,5,    < <4,5,    < <7,8     < <1,1,
                      6>;0.80,    6>;0.80,   ,9>;0.85,   1>;0.50,
                      0.15,       0.15,       0.10,       0.50,
                      0.20>       0.20>       0.15>       0.50>
      A1              < <7,8     < <3,4,    < <4,5,    < <7,8
                     ,9>;0.85,   5>;0.35,    6>;0.80,   ,9>;0.85,
                      0.10,       0.60,       0.15,       0.10,
                      0.15>       0.40>       0.20>       0.15>
      A2              < <1,1,    < <7,8     < <4,5,    < <7,8
                      1>;0.50,   ,9>;0.85,   6>;0.80,   ,9>;0.85,
                      0.50,       0.10,       0.15,       0.10,
DM2                   0.50>       0.15>       0.20>       0.15>
      A3              < <4,5,    < <4,5,    < <5,6,    < <1,1,
                      6>;0.80,    6>;0.80,    7>;0.70,    1>;0.50,
                      0.15,       0.15,       0.25,       0.50,
                      0.20>       0.20>       0.30>       0.50>
      A4              < <1,1,    < <1,2,    < <5,6,    < <4,5,
                      1>;0.50,    3>;0.40,    7>;0.70,    6>;0.80,
                      0.50,       0.60,       0.25,       0.15,
                      0.50>       0.65>       0.30>       0.20>
      A5              < <4,5,    < <4,5,    < <4,5,    < <7,8
                      6>;0.80,    6>;0.80,    6>;0.80,   ,9>;0.85,
                      0.15,       0.15,       0.15,       0.10,
                      0.20>       0.20>       0.20>       0.15>
      A6              < <1,1,    < <7,8     < <7,8     < <4,5,
                      1>;0.50,   ,9>;0.85,  ,9>;0.85,   6>;0.80,
                      0.50,       0.10,       0.10,       0.15,
                      0.50>       0.15>       0.15>       0.20>
      A7              < <7,8     < <7,8     < <4,5,    < <1,1,
                     ,9>;0.85,  ,9>;0.85,   6>;0.80,    1>;0.50,
                      0.10,       0.10,       0.15,       0.50,
                      0.15>       0.15>       0.20>       0.50>
      A1              < <1,1,    < <5,6,    < <4,5,    < <5,6,
                      1>;0.50,    7>;0.70,    6>;0.80,    7>;0.70,
                      0.50,       0.25,       0.15,       0.25,
                      0.50>       0.30>       0.20>       0.30>
      A2              < <1,1,    < <4,5,    < <7,8     < <4,5,
                      1>;0.50,    6>;0.80,   ,9>;0.85,   6>;0.80,
                      0.50,       0.15,       0.10,       0.15,
DM3                   0.50>       0.20>       0.15>       0.20>
      A3              < <4,5,    < <4,5,    < <5,6,    < <5,6,
                      6>;0.80,    6>;0.80,    7>;0.70,    7>;0.70,
                      0.15,       0.15,       0.25,       0.25,
                      0.20>       0.20>       0.30>       0.30>
      A4              < <4,5,    < <4,5,    < <1,2,    < <1,1,
                      6>;0.80,    6>;0.80,    3>;0.40,    1>;0.50,
                      0.15,       0.15,       0.60,       0.50,
                      0.20>       0.20>       0.65>       0.50>
      A5              < <1,1,    < <3,4,    < <1,1,    < <1,1,
                      1>;0.50,    5>;0.35,    1>;0.50,    1>;0.50,
                      0.50,       0.60,       0.50,       0.50,
                      0.50>       0.40>       0.50>       0.50>
      A6              < <4,5,    < <4,5,    < <1,1,    < <1,1,
                      6>;0.80,    6>;0.80,    1>;0.50,    1>;0.50,
                      0.15,       0.15,       0.50,       0.50,
                      0.20>       0.20>       0.50>       0.50>
      A7              < <1,1,    < <1,1,    < <4,5,    < <1,1,
                      1>;0.50,    1>;0.50,    6>;0.80,    1>;0.50,
                      0.50,       0.50,       0.15,       0.50,
                      0.50>       0.50>       0.20>       0.50>
      A1              < <4,5,    < <5,6,    < <7,8     < <1,1,
                      6>;0.80,    7>;0.70,   ,9>;0.85,   1>;0.50,
                      0.15,       0.25,       0.10,       0.50,
                      0.20>       0.30>       0.15>       0.50>
      A2              < <4,5,    < <4,5,    < <5,6,    < <7,8
                      6>;0.80,    6>;0.80,    7>;0.70,   ,9>;0.85,
                      0.15,       0.15,       0.25,       0.10,
DM4                   0.20>       0.20>       0.30>       0.15>
      A3              < <7,8     < <7,8     < <4,5,    < <7,8
                     ,9>;0.85,  ,9>;0.85,   6>;0.80,   ,9>;0.85,
                      0.10,       0.10,       0.15,       0.10,
                      0.15>       0.15>       0.20>       0.15>
      A4              < <7,8     < <7,8     < <1,1,    < <1,2,
                     ,9>;0.85,  ,9>;0.85,   1>;0.50,    3>;0.40,
                      0.10,       0.10,       0.50,       0.60,
                      0.15>       0.15>       0.50>       0.65>
      A5              < <1,1,    < <4,5,    < <1,1,    < <1,1,
                      1>;0.50,    6>;0.80,    1>;0.50,    1>;0.50,
                      0.50,       0.15,       0.50,       0.50,
                      0.50>       0.20>       0.50>       0.50>
      A6              < <1,1,    < <7,8     < <4,5,    < <1,1,
                      1>;0.50,   ,9>;0.85,   6>;0.80,    1>;0.50,
                      0.50,       0.10,       0.15,       0.50,
                      0.50>       0.15>       0.20>       0.50>
      A7              < <4,5,    < <4,5,    < <4,5,    < <1,1,
                      6>;0.80,    6>;0.80,    6>;0.80,    1>;0.50,
                      0.15,       0.15,       0.15,       0.50,
                      0.20>       0.20>       0.20>       0.50>
      Criteria        C5          C6          C7          C8
      /Alternatives
      A1              < <5,6,    < <7,8     < <7,8     < <4,5,
                      7>;0.70,   ,9>;0.85,  ,9>;0.85,   6>;0.80,
                      0.25,       0.10,       0.10,       0.15,
                      0.30>       0.15>       0.15>       0.20>
      A2              < <4,5,    < <1,1,    < <7,8     < <7,8
                      6>;0.80,    1>;0.50,   ,9>;0.85,  ,9>;0.85,
                      0.15,       0.50,       0.10,       0.10,
DM1                   0.20>       0.50>       0.15>       0.15>
      A3              < <7,8     < <5,6,    < <4,5,    < <7,8
                     ,9>;0.85,   7>;0.70,    6>;0.80,   ,9>;0.85,
                      0.10,       0.25,       0.15,       0.10,
                      0.15>       0.30>       0.20>       0.15>
      A4              < <7,8     < <1,2,    < <1,2,    < <1,1,
                     ,9>;0.85,   3>;0.40,    3>;0.40,    1>;0.50,
                      0.10,       0.60,       0.60,       0.50,
                      0.15>       0.65>       0.65>       0.50
      A5              < <4,5,    < <7,8     < <7,8     < <4,5,
                      6>;0.80,   ,9>;0.85,  ,9>;0.85,   6>;0.80,
                      0.15,       0.10,       0.10,       0.15,
                      0.20>       0.15>       0.15>       0.20>
      A6              < <1,1,    < <4,5,    < <7,8     < <4,5,
                      1>;0.50,    6>;0.80,   ,9>;0.85,   6>;0.80,
                      0.50,       0.15,       0.10,       0.15,
                      0.50>       0.20>       0.15>       0.20>
      A7              < <4,5,    < <4,5,    < <7,8     < <4,5,
                      6>;0.80,    6>;0.80,   ,9>;0.85,   6>;0.80,
                      0.15,       0.15,       0.10,       0.15,
                      0.20>       0.20>       0.15>       0.20>
      A1              < <4,5,    < <4,5,    < <4,5,    < <5,6,
                      6>;0.80,    6>;0.80,    6>;0.80,    7>;0.70,
                      0.15,       0.15,       0.15,       0.25,
                      0.20>       0.20>       0.20>       0.30>
      A2              < <7,8     < <4,5,    < <5,6,    < <5,6,
                     ,9>;0.85,   6>;0.80,    7>;0.70,    7>;0.70,
                      0.10,       0.15,       0.25,       0.25,
DM2                   0.15>       0.20>       0.30>       0.30>
      A3              < <4,5,    < <5,6,    < <1,1,    < <4,5,
                      6>;0.80,    7>;0.70,    1>;0.50,    6>;0.80,
                      0.15,       0.25,       0.50,       0.15,
                      0.20>       0.30>       0.50>       0.20>
      A4              < <5,6,    < <1,2,    < <1,1,    < <1,1,
                      7>;0.70,    3>;0.40,    1>;0.50,    1>;0.50,
                      0.25,       0.60,       0.50,       0.50,
                      0.30>       0.65>       0.50>       0.50>
      A5              < <4,5,    < <7,8     < <4,5,    < <7,8
                      6>;0.80,   ,9>;0.85,   6>;0.80,   ,9>;0.85,
                      0.15,       0.10,       0.15,       0.10,
                      0.20>       0.15>       0.20>       0.15>
      A6              < <1,1,    < <7,8     < <1,1,    < <7,8
                      1>;0.50,   ,9>;0.85,   1>;0.50,   ,9>;0.85,
                      0.50,       0.10,       0.50,       0.10,
                      0.50>       0.15>       0.50>       0.15>
      A7              < <1,1,    < <4,5,    < <3,4,    < <1,1,
                      1>;0.50,    6>;0.80,    5>;0.35,    1>;0.50,
                      0.50,       0.15,       0.60,       0.50,
                      0.50>       0.20>       0.40>       0.50>
      A1              < <5,6,    < <4,5,    < <4,5,    < <5,6,
                      7>;0.70,    6>;0.80,    6>;0.80,    7>;0.70,
                      0.25,       0.15,       0.15,       0.25,
                      0.30>       0.20>       0.20>       0.30>
      A2              < <1,1,    < <5,6,    < <7,8     < <4,5,
                      1>;0.50,    7>;0.70,   ,9>;0.85,   6>;0.80,
                      0.50,       0.25,       0.10,       0.15,
DM3                   0.50>       0.30>       0.15>       0.20>
      A3              < <4,5,    < <5,6,    < <1,1,    < <1,2,
                      6>;0.80,    7>;0.70,    1>;0.50,    3>;0.40,
                      0.15,       0.25,       0.50,       0.60,
                      0.20>       0.30>       0.50>       0.65>
      A4              < <4,5,    < <3,4,    < <1,1,    < <1,1,
                      6>;0.80,    5>;0.35,    1>;0.50,    1>;0.50,
                      0.15,       0.60,       0.50,       0.50,
                      0.20>       0.40>       0.50>       0.50>
      A5              < <1,1,    < <1,1,    < <7,8     < <4,5,
                      1>;0.50,    1>;0.50,   ,9>;0.85,   6>;0.80,
                      0.50,       0.50,       0.10,       0.15,
                      0.50>       0.50>       0.15>       0.20>
      A6              < <4,5,    < <4,5,    < <4,5,    < <7,8
                      6>;0.80,    6>;0.80,    6>;0.80,   ,9>;0.85,
                      0.15,       0.15,       0.15,       0.10,
                      0.20>       0.20>       0.20>       0.15>
      A7              < <4,5,    < <4,5,    < <3,4,    < <1,1,
                      6>;0.80,    6>;0.80,    5>;0.35,    1>;0.50,
                      0.15,       0.15,       0.60,       0.50,
                      0.20>       0.20>       0.40>       0.50>
      A1              < <1,1,    < <5,6,    < <5,6,    < <5,6,
                      1>;0.50,    7>;0.70,    7>;0.70,    7>;0.70,
                      0.50,       0.25,       0.25,       0.25,
                      0.50>       0.30>       0.30>       0.30>
      A2              < <7,8     < <4,5,    < <4,5,    < <5,6,
                     ,9>;0.85,   6>;0.80,    6>;0.80,    7>;0.70,
                      0.10,       0.15,       0.15,       0.25,
DM4                   0.15>       0.20>       0.20>       0.30>
      A3              < <5,6,    < <4,5,    < <1,1,    < <7,8
                      7>;0.70,    6>;0.80,    1>;0.50,   ,9>;0.85,
                      0.25,       0.15,       0.50,       0.10,
                      0.30>       0.20>       0.50>       0.15>
      A4              < <4,5,    < <4,5,    < <1,2,    < <1,2,
                      6>;0.80,    6>;0.80,    3>;0.40,    3>;0.40,
                      0.15,       0.15,       0.60,       0.60,
                      0.20>       0.20>       0.65>       0.65>
      A5              < <1,1,    < <1,1,    < <3,4,    < <1,1,
                      1>;0.50,    1>;0.50,    5>;0.35,    1>;0.50,
                      0.50,       0.50,       0.60,       0.50,
                      0.50>       0.50>       0.40>       0.50>
      A6              < <1,1,    < <4,5,    < <7,8     < <4,5,
                      1>;0.50,    6>;0.80,   ,9>;0.85,   6>;0.80,
                      0.50,       0.15,       0.10,       0.15,
                      0.50>       0.20>       0.15>       0.20>
      A7              < <4,5,    < <7,8     < <4,5,    < <4,5,
                      6>;0.80,   ,9>;0.85,   6>;0.80,    6>;0.80,
                      0.15,       0.10,       0.15,       0.15,
                      0.20>       0.15>       0.20>       0.20>
Table 7. The aggregated pairwise matrix for multiple decision maker's 
judgments
Criteria        C1        C2        C3         C4         C5
/Alternatives
A1              1.88288   1.96309   2.01160    1.93540    1.88606
A2              1.38248   2.00514   1.97958    2.073329   1.98669
A3              1.88288   2.06542   1.985350   1.95726    1.99504
A4              1.98669   1.96418   1.77208    1.55075    1.99504
A5              1.77829   1.75314   1.382488   1.77829    1.617809
A6              1.61780   1.98669   1.88288    1.38248    1.38248
A7              1.88288   1.88288   1.93354    1          1.762838
Criteria        C6         C7         C8
/Alternatives
A1              1.99504    1.99504    2.03414
A2              2.25679    2.073329   2.12321
A3              2.03414    1.382488   2.063838
A4              1.73960    1.21198    1.11336
A5              1.915488   2.042910   1.88288
A6              1.93354    1.986697   1.996661
A7              1.93354    1.97178    1.617809
Table 8. The normalization matrix
Criteria       C1        C2        C3        C4        C5        C6
/Alternatives
A1             0.39896   0.38088   0.40856   0.42899   0.39241   0.38124
A2             0.29293   0.38904   0.40205   0.45956   0.41335   0.43126
A3             0.39896   0.40074   0.40322   0.43383   0.41508   0.38872
A4             0.42095   0.38109   0.35991   0.34373   0.41508   0.33243
A5             0.37680   0.34015   0.28078   0.39416   0.33659   0.36604
A6             0.34279   0.38546   0.38241   0.30643   0.28763   0.36949
A7             0.39896   0.36532   0.39270   0.22165   0.36677   0.36949
Criteria       C7        C8
/Alternatives
A1             0.41009   0.41269
A2             0.42618   0.43076
A3             0.24817   0.41872
A4             0.24912   0.22588
A5             0.41993   0.38200
A6             0.40837   0.40509
A7             0.40530   0.32822
Table 9. The Y+ (Weighted normalized)
Criteria        C1           C2           C3           C4
/Alternatives
A1                0.076561     0.061657     0.064033     0.058235
A2                0.056214     0.062978     0.063013     0.062385
A3                0.076561     0.064872     0.063196     0.058892
A4                0.080781     0.061691     0.056408     0.046661
A5                0.072308     0.055064     0.044006     0.053507
A6                0.065782     0.062399     0.059934     0.041597
A7                0.076561     0.059139     0.061547     0.030088
                461          061          635          921
Criteria        C5           C6           C7           C8
/Alternatives
A1                0.044416     0.035831     0.034417     0.025848
A2                0.046786     0.040532     0.035767     0.026980
A3                0.046981     0.036534     0.020827     0.026226
A4                0.046981     0.031244     0.020907     0.014147
A5                0.038097     0.034403     0.035242     0.023926
A6                0.032556     0.034727     0.034272     0.025372
A7                0.041513     0.034727     0.034015     0.020557
                889          294          086          863
Table 10. The ranks of Ratio system
Alternatives   Y*         Ranking
A1             0.401001   1
A2             0.394658   2
A3             0.394094   3
A4             0.358825   4
A5             0.356557   7
A6             0.356643   6
A7             0.358151   5
Table 11. Reference point
Criteria    C1           C2           C3           C4
[R.sub.j]     0.080781     0.064872     0.064033     0.062385
            399          953          364          132
Criteria    C5           C6           C7           C8
ria
[R.sub.j]     0.046981     0.040532     0.035767     0.026980
            992          877          455          394
Table 13. Deviations from reference point.
Criteria       C1           C2           C3           C4
/Alternative
A1                0.00421      0.00321      0.00000      0.00414
               9938         4994          000         9868
A2                0.02456      0.00189      0.00102      0.00000
                737          403         0309          000
A3                0.00421      0.00000      0.00083      0.00349
               9938          000         6935          284
A4                0.00000      0.00318      0.00762      0.01572
                000         0999         4886         3888
A5                0.00847      0.00980      0.02002      0.00887
               2499         8485         6883          803
A6                0.01499      0.00247      0.00409      0.02078
               9107          357         8474         7351
A7                0.00421      0.00573      0.00248      0.03229
               9938         3892         5729         6211
Criteria       C5           C6           C7           C8
/Alternative
A1                0.00256      0.00470      0.00135      0.00113
               5967         1235         0365         1803
A2                0.00019      0.00000      0.00000      0.00000
               5815          000          000          000
A3                0.00000      0.00399      0.01493      0.00075
                000         8211         9614         4118
A4                0.00000      0.00928      0.01485      0.01283
                000         8745         9885         2536
A5                0.00888      0.00612      0.00052      0.00305
                411         9839         4536         4053
A6                0.01442      0.00580      0.00149      0.00160
               5785         5583         4717         7825
A7                0.00546      0.00580      0.00175      0.00642
               8103         5583         2369         2531
Table 13. Rank reference point
Alternative   Max value                           Rank reference point
              (Deviations from reference point)
A1            0.004701235                         7
A2            0.02456737                          2
A3            0.014939614                         6
A4            0.015723888                         5
A5            0.020026883                         4
A6            0.020787351                         3
A7            0.032296211                         1
Table 14. Utility and Rank of full multiplicative form.
Alternatives   Utility ([U.sub.u])   Rank Multiplicative
                                     form
A1             2.49235E-11           2
A2             2.54691E-11           1
A3             1.73317E-11           3
A4             5.69554E-12           7
A5             1.03618E-11           4
A6             1.00614E-11           5
A7             8.45311E-12           6
Table 15. The final rank according to the proposed hybrid methodology
Alternatives   Ratio system   Reference point    Full multiplicative
A1             1              7                  2
A2             2              2                  1
A3             3              6                  3
A4             4              5                  7
A5             7              4                  4
A6             6              3                  5
A7             5              1                  6
Alternatives   (Final Rank)
A1             1
A2             2
A3             3
A4             7
A5             4
A6             6
A7             5
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Title Annotation:University of New Mexico
Author:Nabeeh, Nada A.; Abdel-Monem, Ahmed; Abdelmouty, Ahmed
Publication:Neutrosophic Sets and Systems
Geographic Code:7EGYP
Date:Dec 1, 2019
Words:6821
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