A group decision making framework based on neutrosophic VIKOR approach for e-government website evaluation

Abstract

Since the emergence of digital revolution, many governments have sought to use the internet for the benefits of communication and information easiness to present public services. Due to citizen’s consciousness of internet, the online public services have increased rapidly. Allowing citizens to entrance governmental services, information and participate in governmental decision making process, called the electronic government. In recent years, various e-government websites have increased. Beside the basic lines for e-government websites design process, the evaluation approach also very important for increasing sites quality. The success of these sites depends largely on their quality, security and accessibility. This research provides a multi-criteria group decision making (MCGDM) technique rely on neutrosophic VIKOR method, for evaluating e-government websites. To represent preferences of decision makers about criteria significance weights and performance assessments, triangular neutrosophic numbers are used for representing linguistic variables. In this research, two algorithms are developed based on a neutrosophic linguistic approach. Depending on two algorithms and the VIKOR method, a general framework is proposed. A case study is presented to validate the proposed framework, and a comparative study between two algorithms is illustrated with detail.

Keywords

E-government ICTs MCDM neutrosophic VIKOR website evaluation linguistic variables

1 Introduction

The concept of information and communication technologies (ICTs) has applied at an ascending rate, especially in private organizations. In order to enhance the interaction between citizen and governments, the online government services have been presented by governments [1]. We can refer to e-government as a utilization of technologies and internet for providing enhanced services to internal and external citizens or customers. Since the advantages of e-government became obvious and very important, the number of projects concerning e-government has raised widely since 1996 from 3 to more than 500 national actions [2]. There exists only one channel for online offering of services and products to private and public organizations. This channel is e-government website or called portals. Benjamin and Whitley [3] noted that, “website can’t justify itself simply by being only a website, but it should has goals and seeks to achieve them”. It is important to evaluate e-government websites periodically and asking ourselves an important question, “do these sites can provide governmental services in the best form”. According to Wood et al. [4], the key element for evaluating e-government websites is a multi-dimensional strategy. The evaluation strategy is a very important part for developing websites and increasing their quality. Since website evaluation criteria are complex, conflict, vague and subject to alter interpretation, our research proposed a group decision making framework relied on neutrosophic VIKOR for compromising multiple, conflict criteria and dealing efficiently with vague and inconsistent information which exist usually in web evaluation problems. The VIKOR method presented especially for multi-criteria decision making problems with conflicting and non-measurable criteria [5, 6]. VIKOR helps decision makers to rank, select from a set of alternatives and obtain final decision by compromising solution. To deal with vague and inconsistent information, we present VIKOR in neutrosophic environment. A generalization of classical, fuzzy and intuitionistic fuzzy sets, called neutrosophic set [7]. Neutrosophic set deal with vague and inconsistent information, by considering all aspects of decision making process. The main concepts and definitions of neutrosophic set presented in [8] with detail.

This research is organized as follows: in Section 2 a literature review of related researches are presented. In Section 3 the VIKOR approach is presented in neutrosophic environment and the proposed framework is illustrated with detail. In Section 4 an application example for e-government website evaluation is presented and a comparative study also implemented on the same example. Section 5 concludes the paper with future directions of research topic.

2 Literature review

A survey regarding the literature of e-government website evaluation and the VIKOR approach presented in this section. Some drawbacks of the current approaches are also discussed.

Kumar and Best defined e-government as the utilization of information and communication technologies for enhancing services delivery to private and public organizations [9]. E-government has three sectors as: Government to Government (G2G), Government to Citizen (G2C), Government to Business (G2B). The fourth sector that has been identified by some observers and considered as a subset of G2G [10] is Government to Employee (G2E). Layne and Lee [11] defined e-government as an evolutionary event, providing basic information through the web to public. Different aspects may be used for evaluating e-government. For example, Gupta and Jana [12], aggregated the measurement approaches of e-government performance into hard, soft and hierarchy measure. Padilla [13], observed that website contents differ and depend on different functions and audience. Various criteria were used to evaluate the quality of website. Henriksson et al. [14] evaluate e-government website using several criteria: Security, Privacy, Usability, Content, and Citizen participation. Barnes and Vidgen [15] developed eQual instrument for evaluating e-government websites. A more comprehensive criteria for evaluating websites, proposed by Eschen-felder et al. [16]. These criteria divided into two major categories: Ease of use, Content. The content criteria were divided into sub-criteria: Content, currency, metadata, service, privacy, accuracy, and website orientation. The second criteria also divided into five sub-criteria: Feedback technique, Links, Accessibility, Navigability and Design.

For determining different strategies of Turkey’s e-government, Kahraman et al. [17] combined the crisp and fuzzy AHP with SWOT technique in fuzzy environment. Also Cristina et al. [18] proposed the g-Quality approach, for evaluating quality of Brazilian websites. A model based on fuzzy analytic hierarchy process, proposed by Zhong et al. [19] to evaluate security of e-government system. Another framework based on fuzzy multi-criteria decision making, proposed by Syamsuddin and Hwang for evaluating security of e-government. Tavana et al. [20] proposed a fuzzy framework based on ANP and TOPSIS, for estimating readiness of e-government according to citizen relationship management. Based on UTA II approach, a global evaluation model of e-government proposed by Siskos et al. [21]. For estimating the usability and credibility of e-government websites, Hung and Benyoucef proposed an empirical study [22].

One of the best techniques for disbanding multi-criteria decision making problems with conflicting and incommensurable criteria is VIKOR approach. The VIKOR approach applied in fuzzy environment by many researchers. A fuzzy VIKOR approach with triangular fuzzy numbers for weights and attribute values, proposed by Opricovic [5, 6]. An expanded VIKOR approach for decision making problem, proposed by Sayadi et al. [23]. Ju and Wang introduced an expanding VIKOR approach for multiple criteria group decision making problems [24]. A hesitant fuzzy linguistic VIKOR approach, developed by Liao et al. [25]. A novel expansion of VIKOR approach under fuzzy environment presented by Qin et al. [26]. VIKOR approach utilized in a wide range of problems over the last decade [27]. For deducing the preference order of pit mining equipment, a VIKOR approach, proposed by Bazzazi et al. [28]. An inclusive VIKOR approach for selecting materials, proposed by Jahan et al. [29]. For supplier selection problem a fuzzy VIKOR approach developed by Shemshadi et al. [30]. To improve domestic airline service quality, a developed VIKOR approach utilized by Liou et al. [31]. For solving insurance company problem, Yuenur and Demirel [32] proposed an expanded VIKOR approach in fuzzy environment. A fuzzy VIKOR approach for estimating quality of hospital services in Taiwan, proposed by Chang [33]. To estimate the sensitivity of water supply to climate alteration and variability in South Korea, a VIKOR model presented by Kim and Chung in fuzzy environment [34]. For selecting a suitable site of waste management problem, the VIKOR approach used by Liu et al. [35]. It was noted from previous studies, the fast expansion of the VIKOR approach and its ability to solve different MCDM problem. The VIKOR approach is suitable for e-government website evaluation; because it can compromise solutions to problem with conflicting criteria. Burmaoglu and Kazancoglu used the analytic hierarchy process and VIKOR approach in fuzzy environment for evaluating e-government website [36]. In this research, it is the first time to apply VIKOR approach to e-government website evaluation, under neutrosophic environment.

2.1 Gaps of previous VIKOR researches for evaluating website of e-government

The drawbacks of Burmaoglu and Kazancogl model for evaluating e-government website are as follows:

The fuzzy theory Take into consideration only the truth membership degree, and fails to deal with indeterminacy and falsity membership degree. The linguistic values say good, very-good, etc. should be accompanied by a confirmation degree (i.e. the sureness, indeterminacy and falsity degrees for the provided linguistic variables, for example when we compares a car according to speed criterion, and the result was that the car was very-good according to speed criterion, then the decision maker or expert should say that, he/she is 80%sure, 10%indeterminate and 10%not sure that the car is very good with respect to speed criterion), but they didn’t do this in their model. For example, by comparing criteria 1 according to second criteria, it takes very good as a linguistic variable and its confirmation degree according to decision making opinion is absolutely agreed (i.e. the decision maker/expert considers the statement is always truth and there doesn’t exist any indeterminacy or falsity degrees and this absolutely doesn’t exist in reality). They only take the linguistic value only in their model. Also decision making process in real life problem has three forms: agree, not sure, disagree. This shows that their fuzzy model failed to represent reality efficiently.

Our proposed model overcame the previous drawbacks and a comparison analysis between two researches presented in the last section.

3 New neutrosophic linguistic VIKOR framework for MCGDM

The neutrosophic set concept was first proposed by Smarandache in 1995 and it represent real world problem effectively by considering all aspects of decision making situations, (i.e. truthiness, indeterminacy and falsity) [37]. The linguistic variables [38] used by decision makers to express their opinion and preferences. Instead of numbers, a precipitated words used by Zadeh in [39]. Words or sentences can be used to define linguistic variable values. For example, to represent weight by a linguistic variable it can be represented as: very low, low, high, etc. A special neutrosophic set used commonly in various types of decision making problems is a triangular neutrosophic number $\tilde{y}$ . A triangular neutrosophic number denoted by $\tilde{y} = 〈 (y_{1}, y_{2}, y_{3}); T_{\tilde{y}}, I_{\tilde{y}}, F_{\tilde{y}} 〉$ , where y₁, y₂, y₃ are the minimum, moderate and maximum value of $\tilde{A}$ respectively, $T_{\tilde{y}}, I_{\tilde{y}}, F_{\tilde{y}}$ represents the extreme truth-membership degree, minimal indeterminacy-membership degree and minimal falsity-membership degree respectively. The related definitions and operations of neutrosophic sets and triangular neutrosophic numbers presented with details in [8]. In this research, we used triangular neutrosophic numbers to represent values of linguistic variables. We should know that, the status in which various decision makers included in decision making process, called group decision making or multi-person decision making. In this situation we need to make an aggregation of all decision makers’ opinions for obtaining the final decision. We make the aggregation process through taking the mean value of all decision makers’ opinions.

E-government website evaluation is a complex multi-criteria group decision making problem. So the neutrosophic VIKOR approach, as a convincing decision technique can be used to solve this type of problems. In this section, two algorithms rely on neutrosophic linguistic methods are advanced and a general framework also illustrated to address this type of problems. In e-government website evaluation problem, let D_r be a committee of decision makers where r = 1, 2, …, E, C_j represent criteria where j = 1, 2, …, n, and A_i represent alternatives, where i = 1, 2, …, m. The criteria can be classified as cost and benefit criteria so, let us use C_c, C_b to represent cost and benefit criteria. Suppose that neutrosophic value which transformed from the linguistic variables, by D_r decision makers for i_th alternative and j_th criterion is ${\tilde{y}}_{ij}^{r} = 〈 (l_{ij}^{r}, m_{ij}^{r}, u_{ij}^{r}); T_{\tilde{y}}, I_{\tilde{y}}, F_{\tilde{y}} 〉$ , where $l_{ij}^{r}, m_{ij}^{r}, u_{ij}^{r}$ are the lower, median and upper values of $\tilde{y}$ according to decision maker opinion. Also the neutrosophic weight of C_j criterion which given by D_r decision maker, denoted by ${\tilde{w}}_{j}^{r} = 〈 (w_{lj}^{r}, w_{mj}^{r}, w_{uj}^{r}); T_{\tilde{w}}, I_{\tilde{w}}, F_{\tilde{w}} 〉$ .

3.1 Algorithm 1

The first algorithm of VIKOR approach relies on linguistic variables and in this algorithm we make a deneutrosophication process (for obtaining crisp values) in advanced stage (after obtaining aggregated neutrosophic ratings of alternatives). The steps with more details are as follows:

Step 1. Determine the domain and objectives of decision making problem.

Step 2. Select a group of decision makers for solving decision problem.

Step 3. Let decision makers identify problem criteria, alternatives and constructing a hierarchy of problem.

Step 4. Determine the suitable linguistic variables for weights of criteria and alternatives with respect to each criterion. Use triangular neutrosophic numbers to represent linguistic variables. Beside these linguistic variables, each decision maker should determine his/her confirmation degree. The confirmation degree such as, absolutely sure, not sure, etc. This part will clear with detail in Section 4 when we solve a case study.

Step 5. Aggregate the neutrosophic rating of alternatives and neutrosophic weight of criteria using the following equations:

The aggregated neutrosophic weight of each criterion calculates as follows: ${\tilde{w}}_{j} = \frac{w_{j}^{1} + \dots + w_{j}^{r}}{r}$ (1)

The aggregated neutrosophic ratings of alternatives calculate by: ${\tilde{y}}_{ij} = \frac{y_{ij + \dots +}^{1} y_{ij}^{r}}{r}$ (2)

Step 6. Make a deneutrosophication process (i.e. convert neutrosophic values to crisp values). Because we can consider centroid method as a special case of weighted mean method [35] at r = 1, then we can apply this concept to suitable with neutrosophic environment as follows: $Den (\tilde{y}) = | (y_{l} + y_{m} + y_{u}) / - 3 + (T_{y} - I_{y} - F_{y}) |$ (3)

Where | | means the absolute value of deneutrosophic value. After deneutrosophication process of ${\tilde{y}}_{ij}$ and ${\tilde{w}}_{j}$ , we will obtain y_ij, w_j. We should also make a normalization of weight values after deneutrosophication process.

Step 7. For all benefit criteria C_b, calculate the best and worst values which are: $f_{j}^{*} = max y_{ij}, f_{j}^{-} = min y_{ij}, j \in C_{b} .$ (4)

For all cost criteria, i.e. j ∈ C_c; $f_{j}^{*} = min y_{ij}, f_{j}^{-} = max y_{ij} .$ (5)

Step 8. Calculate the values of S_i, R_i which are the separation of i_th alternative with the better value and the separation of i_th alternative with the worst value respectively using the following equations: $S_{i} = \sum_{j = 1}^{n} w_{j} (\frac{f_{j}^{*} - y_{ij}}{f_{j}^{*} - f_{j}^{-}}),$ (6) $R_{i} = max_{j} [w_{j} (\frac{f_{j}^{*} - y_{ij}}{f_{j}^{*} - f_{j}^{-}})] .$ (7)

Step 9. Calculate Q_i values as follows: $Q_{i} = ƾ \frac{S_{i} - S^{*}}{\bar{S} - S^{*}} + (1 - ƾ) \frac{R_{i} - R^{*}}{\bar{R} - R^{*}},$ (8)

Where $S^{*} = min_{i} S_{i}$ , $\bar{S} = max_{i} S_{i}$ , $R^{*} = min_{i} R_{i}$ , $\bar{R} = max_{i} R_{i}$ and u is the maximum group utility weight, here let epsfboxG :/Tex/IOSPRESS/IFS/0 -171952/IF01. eps = 0.5. Q_i is the separation of i_th alternative from the best one. The superior alternative is the one with the smallest value of Q.

Step 10. According to Q_i values, rank alternatives and make decision.

Step 11. The alternative with a minimum value of Q considered as the optimal compromise solution if the following conditions are satisfied:

The Q (A¹) has an acceptable benefit: $Benf = Q (A^{2}) - Q (A^{1}) \geq \frac{1}{m - 1}$ (9)

Where Benf is the benefit of alternative, A² is the alternative with the second position of ranking process, A¹ is the alternative with the first position of ranking process and m is the number of alternatives.

The Q (A¹) has a stability state i.e. if it’s alsobest ranked in S_i and/or R_i.

Step 12. Determine the best compromise solution by selecting alternative with the minimum value of Q_i regarding to previous conditions. If only one condition from previous two conditions is not satisfied, then there exist a set of compromising solutions:

If second condition not satisfied, then a set of compromising solutions includes A¹, A².

If first condition not satisfied, then a set of compromising solutions includes A¹, …, A^m.

3.2 Algorithm 2

The second algorithm of VIKOR approach relies on linguistic variables and in this algorithm we make a deneutrosophication process (for obtaining crisp values) in late stage (after obtaining neutrosophic values of S_i, R_i, Q_i). The steps with more details are as follows: Step from 1 to 5 as in Algorithm 1.

Step 6. Determine the best neutrosophic value ${\tilde{f}}_{j}^{*} = 〈 (l_{j}^{*}, m_{j}^{*}, u_{j}^{*}) 〉$ and the worst neutrosophic value ${\tilde{f}}_{j}^{-} = 〈 (l_{j}^{-}, m_{j}^{-}, u_{j}^{-}) 〉$ for all criteria. When j is related to benefit criterion then, ${\tilde{f}}_{j}^{*} = \max_{j} {\tilde{y}}_{ij} (\max_{j} l_{ij}, \max_{j} m_{ij}, \max_{j} u_{ij}; T_{y_{ij}}, I_{y_{ij}}, F_{y_{ij}}), {\tilde{f}}_{j}^{-} \min_{j} {\tilde{y}}_{ij} (\min_{j} l_{ij}, \min_{j} m_{ij}, \min_{j} u_{ij}; T_{y_{ij}}, I_{y_{ij}}, F_{y_{ij}})$ . When j is related to cost criterion then,

$\begin{matrix} {\tilde{f}}_{j}^{*} = \min_{j} {\tilde{y}}_{ij} \\ (\min_{j} l_{ij}, \min_{j} m_{ij}, \min_{j} u_{ij}; T_{y_{ij}}, I_{y_{ij}}, F_{y_{ij}}), \\ {\tilde{f}}_{j}^{-} = \max_{j} {\tilde{y}}_{ij} \\ (\max_{j} l_{ij}, \max_{j} m_{ij}, \max_{j} u_{ij}; T_{y_{ij}}, I_{y_{ij}}, F_{y_{ij}}) \end{matrix}$ (10)

Step 7. Measure the normalized neutrosophic difference ${\tilde{D}}_{ij}$ , i = 1, …, m., j = 1, 2, …, n . For the benefit criteria: ${\tilde{D}}_{ij} = \frac{{\tilde{f}}_{j}^{*} - {\tilde{y}}_{ij}}{u_{j}^{*} - l_{j}^{-}},$ (11)

For the cost criteria ${\tilde{D}}_{ij} = \frac{{\tilde{y}}_{ij - {\tilde{f}}_{j}^{*}}}{u_{j}^{-} - l_{j}^{*}}$ (12)

Step 8. Calculate ${\tilde{S}}_{i}$ and ${\tilde{R}}_{i}$ values using the following equations: ${\tilde{S}}_{i} = \sum_{j = 1}^{n} {\tilde{w}}_{j} * {\tilde{D}}_{ij},$ (13) ${\tilde{R}}_{i} = max_{j} ({\tilde{w}}_{j} * {\tilde{D}}_{ij}) .$ (14)

Fig.1

A general framework of proposed methodology.

Until now all values are triangular neutrosophic numbers.

Step 9. Calculate ${\tilde{Q}}_{i}$ values as follows:

$\begin{matrix} {\tilde{Q}}_{i} & = & ƾ \frac{{\tilde{S}}_{i} - \tilde{S^{*}}}{{max}_{i} s_{i}^{u} - {min}_{i} s_{i}^{l}} \\ + (1 - ƾ) \frac{{\tilde{R}}_{i} - \tilde{R^{*}}}{{max}_{i} R_{i}^{u} - {min}_{i} R_{i}^{l}} \end{matrix}$ (15)

Step 10. Use the following deneutrosophic equation to obtain crisp values of ${\tilde{S}}_{i}, {\tilde{R}}_{i}$ and ${\tilde{Q}}_{i}$ as follows: $Den (\tilde{y}) = \frac{1}{8} [{\tilde{y}}_{l} + {\tilde{y}}_{m} + {\tilde{y}}_{u}] \times (1 + T_{\tilde{y}} - I_{\tilde{y}} - F_{\tilde{y}})$ (16)

Step 11. Rank alternatives as in Algorithm 1.

Step 12. Determine compromise solution as in Algorithm 1, except the Benf value. $Benf = \frac{Q (A^{2}) - Q (A^{1})}{Q (A^{m}) - Q (A^{1})}$ (17)

The general framework of neutrosophic VIKOR presented in Fig. 1.

4 The case study: Results and analysis

Fig.2

The hierarchy for selecting superior e-government website.

To test the efficiency and effectiveness of proposed frame work for the e-government website evaluation domain, we presented in this section a case study which includes scenario, implementation methods and data analysis.

E-government website evaluation is a complex, vague and time consuming process. At the same time, it’s a very important process for developing, increasing quality of websites and achieving citizen’s satisfaction. Nowadays, the main objectives of all countries are to offer sustainable, accessible and seamless services for their citizens via e-government websites. Our case study includes the evaluation process of various e-government websites, which contains five countries as follows: Singapore, Finland, Canada, Hong Kong, and Australia.

A group of four decision makers was formed, to select the best governmental website and sufficient criteria should be listed by decision makers to do a proper analysis. The evaluating criteria were as follows: Physical effort, Cognitive effort, Tolerance, Reach, Trust, Error prevention.

4.2 Implementation of proposed framework

4.2.1 Algorithm 1

For disbanding this problem, the neutrosophic VIKOR framework was used via applying the following steps:

Step 1. Determine the scope and objectives of problem. Then select a group of decision makers for identifying criteria, alternatives and drawing problem hierarchy. Our objective here is to evaluate e-government website for five countries (alternatives) and according to six criteria.

Step 2. Determine the suitable linguistic variables for weights of criteria and alternatives with respect to each criterion.

Table 1
Criteria weights

C ₁ C ₂ C ₃ C ₄ C ₅ C ₆

D ₁ (VH;VSS) (H;VSS) (M;AS) (VH;VSN) (M;SNS) (VH;VSS)

D ₂ (VH;AS) (M;VSS) (M;ES) (VH;VSS) (H;ES) (VH;ANS)

D ₃ (VH;VSS) (H;AS) (VH;VSS) (M;AS) (H;ES) (H;AS)

D ₄ (H;ES) (M;AS) (VH;VSN) (H;ANS) (H;VSS) (H;AS)

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
D ₁	(VH;VSS)	(H;VSS)	(M;AS)	(VH;VSN)	(M;SNS)	(VH;VSS)
D ₂	(VH;AS)	(M;VSS)	(M;ES)	(VH;VSS)	(H;ES)	(VH;ANS)
D ₃	(VH;VSS)	(H;AS)	(VH;VSS)	(M;AS)	(H;ES)	(H;AS)
D ₄	(H;ES)	(M;AS)	(VH;VSN)	(H;ANS)	(H;VSS)	(H;AS)

The neutrosophic ratings of alternatives with respect to each criterion presented in Table 2.

Table 2

Neutrosophic rating of alternatives according to each criterion

Decision	Alternatives	Criteria
makers		C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
D ₁	A ₁	(F;VSS)	(VG;VSS)	(F;AS)	(P;VSN)	(MP;SNS)	(MG;ES)
	A ₂	(G;AS)	(VG;VSS)	(MP;ES)	(VG;VSS)	(MG;ES)	(G;ANS)
	A ₃	(VG;VSS)	(MP;AS)	(G;VSS)	(F;AS)	(MP;ES)	(G;ES)
	A ₄	(G;ES)	(F;AS)	(MG;ES)	(MG;ANS)	(G;VSS)	(G;VSS)
	A ₅	(VG;AS)	(G;VSS)	(VG;VSN)	(MG;SNS)	(MG;ES)	(G;ANS)
D ₂	A ₁	(G;ANS)	(F;VS)	(MP;ES)	(F;VS)	(P;ES)	(G;AS)
	A ₂	(G;ES)	(G;AS)	(G;ES)	(MG;AS)	(MG;VSS)	(VG;VS)
		(VG;VS)	(MP;VSS)	(MG;ANS)	(F;VSN)	(F;SNS)	(G;ES)
	A ₄	(G;AS)	(MP;VS)	(MG;VSN)	(G;SNS)	(MG;ES)	(MG;AS)
	A ₄	(G;AS)	(VG;VSS)	(G;ES)	(VG;VSS)	(G;AS)	(MG;ES)
D ₃	A ₁	(MG;VS)	(VG;AS)	(MG;VS)	(F;ES)	(F;ANS)	(G;ES)
	A ₂	(VG;VS)	(G;VSS)	(MP;AS)	(G;AS)	(MG;VS)	(G;AS)
	A ₃	(VG;VSN)	(P;VS)	(VG;AS)	(MP;AS)	(F;SNS)	(G;ES)
	A ₄	(VG;ES)	(F;AS)	(MG;ES)	(MG;VSS)	(F;AS)	(MG;AS)
	A ₅	(G;VS)	(VG;VSS)	(VG;AS)	(MG;VSS)	(F;VS)	(G;VS)
D ₄	A ₁	(MP;AS)	(G;ES)	(F;VS)	(F;AS)	(MP;AS)	(G;VS)
	A ₂	(G;VS)	(VG;AS)	(F;VS)	(G;VS)	(G;VS)	(MG;AS)
	A ₃	(VG;ES)	(MP;AS)	(MG;VSN)	(F;ES)	(F;VS)	(MG;AS)
	A ₄	(G;ES)	(F;AS)	(MG;ES)	(G;ES)	(MG;ES)	(VG;ES)
	A ₅	(VG;VS)	(G;VS)	(VG;AS)	(G;AS)	(MG;ES)	(G;AS)

Table 3

The aggregated neutrosophic decision matrix

	C ₁	C ₂	C ₃
A ₁	(0.417,0.582,0.750;0.0,0.75,1)	(0.665,0.83,0.917;0.5,0.5,0.5)	(0.332,0.497,0.667;0.3,0.75,0.7)
A ₂	(0.710, 0.872,1.0;0.3,0.75,0.7)	(0.75,0.915,1.00;0.9,0.1,0.1)	(0.750,0.915,1.0;0.3,0.75,0.7)
A ₃	(0.83,1.0,1.0;0.3,0.75,0.70)	(0.127,0.290,0.457;0.9,0.1,0.1)	(0.625,0.792,0.915;0,0.75,0.7)
A ₄	(0.71,0.872,0.1.0;0.9,0.1,0.1)	(0.290,0.4575,0.627;0.9,0.1,0.1)	(0.50,0.670,0.830;0.3,0.75,0.7)
A ₅	(0.75,0.91,1.0;0.3,0.75,0.70)	(0.750,0.915,1.000; 0.9,0.1,0.1)	(0.79,0.957,1.00;0.3,0.75,0.7)

	C ₄	C ₅	C ₆
A ₁	(0.247,0.417,0.585;0.3,0.75,0.7)	(0.167,0.332,0,500;0.0,0.6,1.0)	(0.627,0.790,0.957;0.5,0.5,0.5)
A ₂	(0.667,0.832,0.95;0.3,0.75,0.7)	(0.542,0.710,0.872;0.3,0.75,0.7)	(0.667,0.832,0.957;0.0,0.0,1.0)
A ₃	(0.29,0.457,0.627;0.3,0.75,0.7)	(0.290,0.457,0.627;0.4,0.65,0.6)	(0.627,0.790,0.957;0.5,0.5,0.5)
A ₄	(0.585,0.750,0.915;0.0,0.75,1)	(0.50,0.667,0.832;0.5,0.5,0.5)	(0.625,0.792,0.915;0.5,0.5,0.5)
A ₅	(0.625,0.792,0.915;0.4,0.65,0.6)	(0.50,0.667,0.83;0.5,0.5,0.5)	(0.627,0.790,0.957;0.0,0.5,1)

Each linguistic variable is a triangular neutrosophic number. For criteria weights, the linguistic variables are as follows: Very low (VL) = (0.0, 0.0, 0.25), Low (L) = (0.0, 0.25, 0.5), Medium (M) = (0.25, 0.5, 0.75), High (H) = (0.5, 0.75, 1), Very High (VH) = (0.75, 1,1).

For rating alternatives with respect each criterion, the linguistic variables are: Very Poor (VP) = (0.0, 0.0, 0.17), Poor (P) = (0.0, 0.17, 0.33), Medium Poor (MP) = (0.17, 0.33, 0.5), Fair(F) = (0.33, 0.5, 0.67), Medium Good (MG) = (0.5, 0.67, 0.83), Good (G) =) = (0.67, 0.83, 1), Very Good (VG) = (0.83, 1,1). The previous linguistic variables are used by decision makers to clear their preferences. The very important part is the confirmation degree of linguistic variable according to each decision maker. The confirmation degree can be as follows: Absolutely Sure (AS) = (1, 0, 0); the first parameter(1) is the truth degree, second parameter(0) is indeterminacy degree and finally the falsity degree. Very Strongly -Sure(VSS) = (0.9, 0.1, 0.1), Equally-Sure(ES) = (0.5, 0.5, 0.5), Absolutely Not Sure(ANS) = (0.0, 0.0, 1), Very Strongly Not Sure (VSN) = (0.3, 0.75, 0.7) and Strongly Not Sure (SNS) = (0.4, 0.65, 0.6). After determining linguistic variables and their confirmation degree according to each decision maker, the weights of criteria presented in Table 1. Each weight value is the linguistic variable concatenated with the confirmation degree.

Step 3. Do the aggregation process of decision makers preferences to obtain final decision matrix, which is a triangular neutrosophic numbers. By using Equation (2) the grouped matrix as in Table 3. The neutrosophic weights of criteria using Equation (1) were calculated as follow: $\begin{matrix} {\tilde{w}}_{1} & = & (0.687, 0.937, 1.00; 0.5, 0.5, 0.5), \\ {\tilde{w}}_{2} & = & (0.375, 0.625, 0.875; 0.9, 0.1, 0.1), \\ {\tilde{w}}_{3} & = & (0.500, 0.750, 0.875; 0.3, 0.75, 0.7), \\ {\tilde{w}}_{4} & = & (0.562, 0.812, 0.937; 0.0, 0.75, 1.0), \\ {\tilde{w}}_{5} & = & (0.437, 0.687, 0.937; 0.4, 0.65, 0.6), \\ {\tilde{w}}_{6} & = & (0.625, 0.875, 1.00; 0.0, 0.1, 1.0) . \end{matrix}$

Calculate the crisp weight value by using Equation (3) and then make a normalization process. The values of weights are as in Fig. 3 and equals the following: w₁ = 0.11, w₂ = 0.38, w₃ = 0.13, w₄ = 0.28, w₅ = 0.03, w₆ = 0.08.

Fig.3

Criteria weights values.

Step 4. Make a deneutrosophication process, i.e. convert each neutrosophic number in decision matrix to its equivalent crisp value by using Equation (3). The crisp values presented in Table 4.

Table 4

The crisp decision matrix

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
Weight	0.11	0.38	0.13	0.28	0.03	0.08
A ₁	1.167	0.303	0.65	0.73	1.26	0.29
A ₂	0.29	1.59	0.65	0.33	0.45	0.18
A ₃	0.21	0.99	0.68	0.69	0.392	0.29
A ₄	1.56	1.15	0.48	1	0.166	0.277
A ₅	0.26	1.59	0.23	0.58	0.166	0.71

Step 5. From Table 4, the values of $f_{j}^{*}, f_{j}^{-}$ using Equation (4) are as follows: $\begin{matrix} f_{1}^{*} = 1.56, f_{2}^{*} = 1.59, f_{3}^{*} = 0.68, \\ f_{4}^{*} = 1, f_{5}^{*} = 1.26, f_{6}^{*} = 0.7 . f_{4}^{-} = 1, \\ f_{5}^{-} = 0.33, f_{6}^{-} = 0.17 . \end{matrix}$

Step 6. Calculate values of S_i, R_i by using Equations (6, 7). The results are as follows: $\begin{matrix} S_{1} = 0.74, S_{2} = 0.49, S_{3} = 0.50, S_{4} = 0.49, \\ S_{5} = 0.49 . R_{1} = 0.38, R_{2} = 0.28, R_{3} = 0.18, \\ R_{4} = 0.17 . \end{matrix}$

Step 7. Calculate the Q_i values for each alternative by using Equation (8) as follows: $\begin{matrix} Q_{1} = 1, Q_{2} = 0.3, Q_{3} = 0.11, \\ Q_{4} = 0, Q_{5} = 0.08 . \end{matrix}$

Table 5

The ascending order for ranking alternatives

Ranking of alternatives in an ascending order
According to S values	A₄ > A₅ > A₂ > A₃ > A₁
According to R values	A₄ > A₅ > A₃ > A₂ > A₁
According to Q values	A₄ > A₅ > A₃ > A₂ > A₁

Step 8. Rank alternatives with respect to S, R and Q in an ascending order. The results presented in Table 5. Since S₂, S₄ and S₅ have the same value, we rank them by looking also to R and Q values. From Fig. 4 it obvious that Singapore is the largest separation degree (i.e. the maximum Q value) from the best value, then it’s the worst alternative. Also Hong Kong is the smallest separation degree (i.e. the minimum Q value) from the best value, and then it’s the best alternative. Since the first condition for determining compromise solution is not satisfied because $Q (A^{2}) - Q (A^{1}) 0.08 < \frac{1}{5 - 1}$ , and the second condition is satisfied, then there exist a set of compromise solutions which are A₄ > A₅ > A₃ > A₂ > A₁ according to Q values.

Fig.4

The separation of alternatives from the best value.

4.2.2 Algorithm 2

For making a comparative analysis and study the effect of last deneutrosophication process on ranking of alternatives, we will solve the same problem by using Algorithm 2 through the following steps:

Steps from 1 to 3 are the same in the previous algorithm. Step 4. The neutrosophic values of ${\tilde{f}}_{j}^{*}, {\tilde{f}}_{j}^{-}$ from Table 3 are as follows: $\begin{matrix} {\tilde{f}}_{1}^{*} & = & (0.830, 1.00, 1.00; 0.3, 0.75, 0.70), \\ {\tilde{f}}_{2}^{*} & = & (0.127, 0.290, 0.457; 0.9, 0.1, 0.1), \\ {\tilde{f}}_{3}^{*} & = & (0.79, 0.957, 1.00; 0.3, 0.75, 0.7), \\ {\tilde{f}}_{4}^{*} & = & (0.667, 0.832, 0.95; 0.3, 0.75, 0.7), \\ {\tilde{f}}_{5}^{*} & = & (0.542, 0.710, 0.872; 0.3, 0.75, 0.7), \\ {\tilde{f}}_{6}^{*} & = & (0.667, 0.832, 0.957; 0.0, 0.0, 1.0) . \\ {\tilde{f}}_{1}^{-} & = & (0.417, 0.58, 0.750; 0.0, 0.75, 1), \\ {\tilde{f}}_{2}^{-} & = & (0.127, 0.290, 0.457; 0.9, 0.1, 0.1), \\ {\tilde{f}}_{3}^{-} & = & (0.332, 0.497, 0.667; 0.3, 0.75, 0.7), \\ {\tilde{f}}_{4}^{-} & = & (0.247, 0.417, 0.585; 0.3, 0.75, 0.7), \\ {\tilde{f}}_{5}^{-} & = & (0.167, 0.332, 0, 500; 0.0, 0.6, 1.0), \\ {\tilde{f}}_{6}^{-} & = & (0.625, 0.792, 0.915; 0.5, 0.5, 0.5) . \end{matrix}$

Step 5. Calculate the normalized neutrosophic difference ${\tilde{D}}_{ij}$ by using Equation (11). Table 6 shows the results.

Table 6
Values of normalized neutrosophic difference

C ₁ C ₂ C ₃

A ₁ (0.137,0.716,1.000;0.0,0.75,1) (−0.192,0.094,0.384;0.5,0.5,0.5) (0.183,0.685,1;0.3,0.75,0.7)

A ₂ (−0.291, 0.218,0.497;0.3,0.75,0.7) (−0.286,0,0.286;0.9,0.1,0.1) (0.183,0.689,0.996;0.3,0.75,0.7)

A ₃ (−0.291,0,0.291;0.3,0.75,0.70) (0.335,0.716,1;0.9,0.1,0.1) (−0.187,0.247,0.561;0,0.75,0.7)

A ₄ (−0.291, 0.218, 0.49;0.9,0.1,0.1) (0.140,0.524,0.813;0.9,0.1,0.1) (−0.059,0.430,0.749;0.3,0.75,0.7)

A ₅ (−0.291,0.145,0.429;0.3,0.75,0.7) (−0.286,0,0.286; 0.9,0.1,0.1) (−0.314,0,0.314;0.3,0.75,0.7)

C ₄ C ₅ C ₆

A ₁ (0.116,0.584,1;0.3,0.75,0.7) (0.060,0.535,1;0.0,0.75,1) (−0.872,0.127,0.992;0.0,0.5,1)

A ₂ (−0.408,0,0.408;0.3,0.75,0.7) (−0.468,0,0.468;0.3,0.75,0.7) (−0.872,0,0.872;0.0,0.0,1)

A ₃ (0.056,0.528,0.940;0.3,0.75,0.7) (−0.120,0.358,0.826; 0.3,0.75,0.7) (−0.872,0.127,0.992;0.5,0.5,1)

A ₄ (−0.386,0.116,0.524;0.0,0.75,1) (−0.411,0.060,0.528; 0.3,0.75,0.7) (−0.744,0.120,1;0.5,0.5,1)

A ₅ (−0.386,0.056,0.468; 0.3,0.75,0.7) (−0.411,0.060,0.528; 0.3,0.75,0.7) (−0.872,0.127,0.992;0.0,0.5,1)

	C ₁	C ₂	C ₃
A ₁	(0.137,0.716,1.000;0.0,0.75,1)	(−0.192,0.094,0.384;0.5,0.5,0.5)	(0.183,0.685,1;0.3,0.75,0.7)
A ₂	(−0.291, 0.218,0.497;0.3,0.75,0.7)	(−0.286,0,0.286;0.9,0.1,0.1)	(0.183,0.689,0.996;0.3,0.75,0.7)
A ₃	(−0.291,0,0.291;0.3,0.75,0.70)	(0.335,0.716,1;0.9,0.1,0.1)	(−0.187,0.247,0.561;0,0.75,0.7)
A ₄	(−0.291, 0.218, 0.49;0.9,0.1,0.1)	(0.140,0.524,0.813;0.9,0.1,0.1)	(−0.059,0.430,0.749;0.3,0.75,0.7)
A ₅	(−0.291,0.145,0.429;0.3,0.75,0.7)	(−0.286,0,0.286; 0.9,0.1,0.1)	(−0.314,0,0.314;0.3,0.75,0.7)

	C ₄	C ₅	C ₆
A ₁	(0.116,0.584,1;0.3,0.75,0.7)	(0.060,0.535,1;0.0,0.75,1)	(−0.872,0.127,0.992;0.0,0.5,1)
A ₂	(−0.408,0,0.408;0.3,0.75,0.7)	(−0.468,0,0.468;0.3,0.75,0.7)	(−0.872,0,0.872;0.0,0.0,1)
A ₃	(0.056,0.528,0.940;0.3,0.75,0.7)	(−0.120,0.358,0.826; 0.3,0.75,0.7)	(−0.872,0.127,0.992;0.5,0.5,1)
A ₄	(−0.386,0.116,0.524;0.0,0.75,1)	(−0.411,0.060,0.528; 0.3,0.75,0.7)	(−0.744,0.120,1;0.5,0.5,1)
A ₅	(−0.386,0.056,0.468; 0.3,0.75,0.7)	(−0.411,0.060,0.528; 0.3,0.75,0.7)	(−0.872,0.127,0.992;0.0,0.5,1)

Step 6. Calculate values of ${\tilde{S}}_{i}$ , ${\tilde{R}}_{i}$ using Equations (13, 14). The results are as follows: $\begin{matrix} {\tilde{S}}_{1} & = & (- 0.34, 2.20, 5.08; 0.0, 0.75, 1), \\ {\tilde{S}}_{2} & = & (- 1.196, 0.722, 3.31; 0.0, 0.75, 1), \\ {\tilde{S}}_{3} & = & (- 0.73, 1.42, 4.30; 0.0, 0.75, 1), \\ {\tilde{S}}_{4} & = & (- 1.02, 1.09, 3.85; 0.0, 0.75, 1), \\ {\tilde{S}}_{5} & = & (- 1.39, 0.33, 2.88; 0.0, 0.75, 1) . \\ {\tilde{R}}_{1} & = & (0.094, 0.67, 1; 0.0, 0.75, 1), \\ {\tilde{R}}_{2} & = & (0.09, 0.51, 0.87; 0.3, 0.75, 0.7), \\ {\tilde{R}}_{3} & = & (0.129, 0.45, 0.99; 0.9, 0.1, 0.1) \\ {\tilde{R}}_{4} & = & (0.05, 0.33, 1; 0.9, 0.1, 0.1), \\ {\tilde{R}}_{5} & = & (- 0.11, 0.14, 0.99; 0.9, 0.1, 0.1) . \end{matrix}$

Step 7. Calculate ${\tilde{Q}}_{i}$ values by using Equation (15). The results are as follows: $\begin{matrix} {\tilde{Q}}_{1} & = & (- 0.65, 0.38, 1; 0, 0.75, 1), \\ {\tilde{Q}}_{2} & = & (- 0.72, 0.20, 0.80; 1, 0, 0.75, 1), \\ {\tilde{Q}}_{3} & = & (- 0.67, 0.22, 0.94; 0, 0.75, 1), \\ {\tilde{Q}}_{4} & = & (- 0.73, 0.14, 0.90, 1; 0, 0.75, 1), \\ {\tilde{Q}}_{5} & = & (- 0.83, - 0.00, 0.83; 0, 0.75, 1) . \end{matrix}$

Step 8. Make a deneutrosophication process using Equation (16) and take only the absolute value to obtain crisp values of S_i, R_i and Q_i. The results are: $\begin{matrix} S_{1} & = & 0.65, S_{2} = 0.26, S_{3} = 0.47, \\ S_{4} & = & 0.37, S_{5} = 0.17, \\ R_{1} & = & 0.16, R_{2} = 0.0.27, R_{3} = 0.33, \\ R_{4} & = & 0.29, R_{5} = 0.2, \\ Q_{1} & = & 0.07, Q_{2} = 0.0.27, Q_{3} = 0.04, \\ Q_{4} & = & 0.029, Q_{5} = 0 . \end{matrix}$

Fig.5

The separation of alternatives from the best value.

From Fig. 5, it’s obvious that Singapore is the alternative with the largest separation degree from the best value and Australia is the alternative with the minimum separation degree.

Step 9. Rank alternatives in an ascending order as in Table 7. Since $\frac{Q (A^{2}) - Q (A^{1})}{Q (A^{m}) - Q (A^{1})} \geq \frac{1}{m - 1},$ not satisfied, and the stability state is achieved, then the compromising solutions are A₅ > A₂ > A₄ > A₃ > A₁.

Table 7

The ascending order for ranking alternatives

Ranking of alternatives in an ascending order
According to S values	A₅ > A₂ > A₄ > A₃ > A₁
According to R values	A₂ > A₁ > A₅ > A₄ > A₃
According to Q values	A₅ > A₂ > A₄ > A₃ > A₁

4.2.3 The comparison between two algorithms

It’s obvious that, the early deneutrosophication VIKOR approach is the easier. However the comparative study illustrated that, the first alternative is the worst e-government website according to two algorithms. The obtained values of R, Q from the first alternative are identical but from second algorithm, values are different but not very large. From second algorithm, the separation of alternatives from the best value is smaller than in Algorithm 1. We also concluded that, different deneutrosophication method led to different ranking of alternatives and choosing the suitable method also depends on context.

4.3 Comparative analysis with other research

By comparing our model with Burmaoglu and Kazancoglu model for evaluating e-government website, the results are as follows:

Our model which relies on neutrosophic VIKOR approach has a great representation of reality than Burmaoglu and Kazancoglu model. Because in our model we take into consideration all aspects of decision making process (i.e. agree state, not sure state and disagree state), but Burmaoglu and Kazancoglu concerned only on agree or truthiness state. By taking all aspects of decision making in our model, then we can efficiently represent vague, inconsistent and incomplete information than Burmaoglu and Kazancoglu model. Each linguistic variable in our model supported by the confirmation degree according to decision maker’s opinion and it’s a very important part for representing reality. But Burmaoglu and Kazancoglu doesn’t. In our model, the evaluation process of e-government website is performed by two algorithms. And a comparison between two algorithms results illustrated with detail.

Our model is the first for evaluating e-government website in neutrosophic environment.

5 Conclusions and future works

All e-government efforts are critically based on the accessibility of its website. The evaluation process of e-government websites is very important for developing performance and quality of websites. In our research the proposed model for evaluating e-government website based on VIKOR approach in neutrosophic environment. The weight of criteria and rating of alternatives represented by linguistic variables, the degree of truthiness, indeterminacy and falsity of each linguistic variable also determined according to each decision maker opinion. The linguistic variables and its degree of sureness presented by triangular neutrosophic numbers. Two algorithms are presented for evaluation process. The posterior and prior deneutrosophication in two algorithms affects results of ranking alternatives. Our proposed framework deals efficiently with vague, inconsistent and incomplete information by considering all aspects of decision making process. In the future we will consider the feedback and interdependence between criteria. We also will use other method for evaluation process, such as TOPSIS and ANP.

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A group decision making framework based on neutrosophic VIKOR approach for e-government website evaluation

Abstract

Keywords

1 Introduction

2 Literature review

2.1 Gaps of previous VIKOR researches for evaluating website of e-government

3 New neutrosophic linguistic VIKOR framework for MCGDM

3.1 Algorithm 1

4.2.1 Algorithm 1

Table 1 Criteria weights C 1 C 2 C 3 C 4 C 5 C 6 D 1 (VH;VSS) (H;VSS) (M;AS) (VH;VSN) (M;SNS) (VH;VSS) D 2 (VH;AS) (M;VSS) (M;ES) (VH;VSS) (H;ES) (VH;ANS) D 3 (VH;VSS) (H;AS) (VH;VSS) (M;AS) (H;ES) (H;AS) D 4 (H;ES) (M;AS) (VH;VSN) (H;ANS) (H;VSS) (H;AS)

4.3 Comparative analysis with other research

5 Conclusions and future works

References

Table 1
Criteria weights

C ₁ C ₂ C ₃ C ₄ C ₅ C ₆

D ₁ (VH;VSS) (H;VSS) (M;AS) (VH;VSN) (M;SNS) (VH;VSS)

D ₂ (VH;AS) (M;VSS) (M;ES) (VH;VSS) (H;ES) (VH;ANS)

D ₃ (VH;VSS) (H;AS) (VH;VSS) (M;AS) (H;ES) (H;AS)

D ₄ (H;ES) (M;AS) (VH;VSN) (H;ANS) (H;VSS) (H;AS)