Sustainable Cloud Service Provider Development by a Z-Number-Based DNMA Method with Gini-Coe ﬃ cient-Based Weight Determination

: The sustainable development of cloud service providers (CSPs) is a signiﬁcant multiple criteria decision making (MCDM) problem, involving the intrinsic relations among multiple alternatives, (quantitative and qualitative) decision criteria and decision-experts for the selection of trustworthy CSPs. Most existing MCDM methods for CSP selection incorporated only one normalization technique in beneﬁt and cost criteria, which would mislead the decision results and limit the applications of these methods. In addition, these methods did not consider the reliability of information given by decision-makers. Given these research gaps, this study introduces a Z-number-based double normalization-based multiple aggregation (DNMA) method to tackle quantitative and qualitative criteria in forms of beneﬁt, cost, and target types for sustainable CSP development. We extend the original DNMA method to the Z-number environment to handle the uncertain and unreliability information of decision-makers. To make trade-o ﬀ s between normalized criteria values, we develop a Gini-coe ﬃ cient based weighting method to replace the mean-square-based weighting method used in the original DNMA method to enhance the applicability and isotonicity of the DNMA method. A case study is conducted to demonstrate the e ﬀ ectiveness of the proposed method. Furthermore, comparative analysis and sensitivity analysis are implemented to test the stability and applicability of the proposed method.


Introduction
Today's organizations, regardless of their size and business scope, pay more and more attention to the maintenance of competitiveness and the establishment of a sustainable environment [1]. World Commission on Environment and Development of the United Nations Brundtland defined sustainability development as "development that meets the needs of the present without compromising the ability of Motivated by these analyses, this study presents a Z-number-based DNMA (Z-DNMA) method for the identification of the sustainable CSPs. The Gini-coefficient-based weight-determining method is integrated to reflect the trade-offs between QoS criteria. The main contributions of this study are highlighted as follows: 1.
We introduce the DNMA method for CSP selection. The proposed model can deal with quantitative and qualitative criteria in forms of benefit, cost, and target types. It can flexibly and reliably solve the sustainable cloud service provider development problem.

2.
We extend the original DNMA method to the Z-number environment and propose the Z-DNMA method to tackle quantitative and qualitative criteria in forms of benefit, cost, and target types for CSP selection. In this regard, the uncertain and unreliability decision information of decision-makers (DMs) is considered in the process of CSP selection. 3.
To enhance the applicability and isotonicity and the DNMA method, we make use of the Gini-coefficient-based weighting method to replace the mean-square-based weighting method used in the original DNMA method, and extend this approach to the Z-number environment for the trade-offs between criteria after normalization.
The structure of this paper is as follows. In Section 2, some definitions and concepts are introduced. In Section 3, the Z-DNMA method is presented. In Section 4, a numerical example is presented, followed by the comparative analyses, sensitivity analysis in Section 5. Conclusions are given in the last section.

Preliminaries
This section primarily reviews some notions of Z-numbers and the Gini coefficient weighting method.

Generalized Triangle Fuzzy Numbers
Fuzzy set [34] was defined based on a membership function whose values are in the unit interval. A fuzzy set A is defined on a universe X as A = x, µ A (x) |x ∈ X , where µ A (x) : X → [0, 1] is the membership function of set A, indicating the degree of belongingness of x ∈ X in A. A TFN A is defined as a triple (L, M, R) with the membership function as [34]: Let A 1 = (L 1 , M 1 , U 1 ) and A 2 = (L 2 , M 2 , U 2 ) be two TFNs, and λ > 0 be a constant number. The operations of TFNs can be performed as [27]: The distance between A 1 and A 2 was determined as [35]: The expectation value E A i of a TFN A i = (L i , M i , U i ) can be calculated as [35]:

Z-number
Zadeh [30] introduced the concept of Z-number as an ordered pair Z = A, B of fuzzy numbers A and B, where the first component A is interpreted as a restriction on the values that a variable can take, and the second component B is a measure of reliability about the value of A. Typically, A and B are described in a natural language, for example-low, likely. Compared with the classical fuzzy set, the Z-number takes into account the uncertainty in information generation process and the reliability of information. At present, it has been combined with many MCDM methods such as TOPSIS [36,37], VIKOR [38], Multi-Objective Optimization by Ratio Analysis (MOORA) [39], COmbinative Distance-based Assessment (CODAS) [40], PROMETHEE [41], TODIM (an acronym in Portuguese of interactive and multicriteria decision-making) [37], AHP [42], BWM [43] and Data Envelopment Analysis (DEA) [44]. 1] are two TFNs, we can convert the Z-number to an ordinary fuzzy number [45]. Firstly, the second part (reliability) can be converted into a crisp number by Equation (4): where " " denotes an algebraic integration. Then, we add the weight of the second part (reliability) to the first part (restriction). The weighted Z-number is as follows: We then convert the Z-number (weighted restriction) to the fuzzy number Z : If A = (L, M, U) is a TFN, then Z is calculated as:

The Gini-Coefficient-Based Weighting Method
Gini coefficient is a quantitative index to measure the difference in income distribution and has been widely used in studying impacts of inequality [46,47]. Since the Gini coefficient can reflects the data difference between different evaluation objects, Li et al. [31] proposed a method to calculate the weights of objectives based on the Gini coefficient: Suppose that G k denotes the Gini coefficient of the k-th criterion (k = 1, 2, · · · , n), m denotes the total data of a specific criterion, y ki denotes the i-th alternative' performance value under the k-th criterion, E k denotes the expectation value of all alternatives' performance values under the k-th criterion. Then the Gini coefficient value G k can be calculated by Equations (8) and (9) [31].
In particular, when the mean value of all alternatives' performance values under a specific criterion that is not equal to 0, the Gini coefficient of the criterion is calculated by Equation (8); otherwise, the Gini coefficient of the criterion is calculated by Equation (9).
Then, we can obtain the objective weight of the k-th criterion by Equation (10).

A Z-Number-Based DNMA Method
In this section, we propose a Z-number-based DNMA method with a Gini-coefficient-based weight determination method. We extend the original DNMA method to the Z-number environment. Meanwhile, we adopt the Gini-coefficient based weighting method to replace the mean-square-based weighting method used in the original DNMA method, and extend this approach to the Z-number environment for the trade-offs between criteria after normalization. The procedure of the proposed method is summarized in Figure 1.

A Z-Number-Based DNMA Method
In this section, we propose a Z-number-based DNMA method with a Gini-coefficient-based weight determination method. We extend the original DNMA method to the Z-number environment. Meanwhile, we adopt the Gini-coefficient based weighting method to replace the mean-square-based weighting method used in the original DNMA method, and extend this approach to the Z-number environment for the trade-offs between criteria after normalization. The procedure of the proposed method is summarized in Figure 1.

Step 1. Problem formalization
Step  Step 1. (Problem formalization) Let be a set of alternatives, be a set of criteria, be a set of DMs.
denotes the Z-fuzzy performance evaluation value of the -th i alternative on the -th j criterion from -th k DM. Step 1. (Problem formalization) Let A = {a 1 , a 2 , · · · , a m } (m ≥ 2) be a set of alternatives, is the weight vector of DMs, where λ k ≥ 0 and q k=1 λ k = 1. Since the numerical values for quantitative criteria is easy to collect, we mainly focus on the evaluation of alternatives over qualitative criteria. For qualitative criteria C 1 = c 1 , c 2 , · · · , c g , we suppose their values of alternatives are evaluated by the k−th DM and expressed as linguistic expressions s (k) ij , for i = 1, 2, · · · , m, j = 1, 2, · · · , g, k = 1, 2, · · · , q. For quantitative criteria C 2 = c g+1 , c g+2 , · · · , c n , we suppose the values of alternatives are expressed as numerical numbers xij, for i = 1, 2, · · · , m, j = g + 1, g + 2, · · · , n.
Step 2. (Constructing comprehensive decision matrix) Step 2.1. Translate each DM's linguistic evaluation s ij to Z-number, and then the decision matrix of the k−th DM can be expressed as: ij denotes the Z-fuzzy performance evaluation value of the i−th alternative on the j−th criterion from k−th DM.
Step 2.2. According to the weights of DMs, λ = λ 1 , λ 2 , · · · , λ q T , aggregate DMs' linguistic evaluations into collective ones based on the weighted arithmetic aggregation operator as: Step 2.3. Convert Z-fuzzy performance values to TFNs z ij by Equations (4) and (7), where Step 2.4. Establish the comprehensive decision matrix X , which is composed by the calculated TFNs and numerical numbers, shown as: Step 3. (Normalization) Distinguish the criteria into benefit, cost, and target forms. Based on the decision matrix X , we calculate the target-based linear normalization values by Equation (13) based on the distance measure given as Equation (2) and the target-based vector normalization values by Equation (14) based on the expectation function given as Equation (3).
where z ij is the target value on qualitative criteria c j ( j = 1, 2, · · · , g), and r j is the target value on quantitative criteria ( j = g + 1, g + 2, · · · , n). Especially, if the jth criterion is a qualitative criterion, then, for the cos t criterion (15) If the jth criteria is a quantitative criterion, then, x ij , for the benefit criterion min i x ij , for the cos t criterion (16) Afterwards, the target-based liner and vector normalization values are adjusted by Equation (17) to make the maximum entry as 1 under each criterion.
Step 4. (Trade-offs between criteria) In the original DNMA method, Liao & Wu [14] adjusted the criteria weights based on the mean-squared-based weighting method. In this study, we make use of the Gini-coefficient-based weighting method to replace it and extend this approach to the Z-number environment. Firstly, we defuzzify the TFNs of qualitative criteria in the comprehensive decision matrix X to expectation values E z ij by Equation (3) and obtain the weight adjustment coefficients of the criteria by Equations (8)-(10). Then, the criteria weights are adjusted by Step 5. (Aggregation) Compute the subordinate utility values of each alternative, u h (a i ), h = 1, 2, 3; i = 1, 2, · · · , m, based on the complete compensatory model (CCM), un-compensatory model (UCM), and incomplete compensatory model (ICM) by Equations (19)-(21), respectively. Then, determine the subordinate ranks r h (a i ), h = 1, 2, 3; i = 1, 2, · · · , m. Go to the next step.
It is noted that Equations (19) and (20) are based on the adjusted weight w j of criterion c j , while Equation (21) is based on the original weight w j of c j .
Determine the weights of the CCM, UCM and ICM. Then, we can integrate the normalized subordinate utility values and subordinate ranks by Equation (23), and obtain the collective utility value of each alternative DN i , i = 1, 2, · · · , m: where r 1 (a i ) and r 3 (a i ) are the ranks of alternative a i and determined in descending order of u 1 (a i ) and u 3 (a i ), respectively, r 2 (a i ) is the rank of alternative a i and determined by the ascending order of u 2 (a i ), ϕ(ϕ ∈ [0, 1]) is the relative importance of the subordinate ranks and subordinate utility values. w 1 , w 2 , w 3 denote the weights of CCM, UCM and ICM, satisfying w i ∈ [0, 1] and 3 i=1 w i = 1. Lastly, we can determine the final ranking according to the descending order of DN i and end the algorithm.

Case Study on CSP Ranking with the Z-DNMA Method
In this section, the Z-DNMA method is illustrated with a numerical example related to the CSP selection problem.
Assume that a company plans to consume a cloud service request and thus needs to select the most suitable cloud services. After multiple rounds of anonymous discussions and summarizing, the five evaluation criteria are selected from the QoS attributes and based on the Delphi method which involved X IT experts in the field of cloud computing. The determined criteria are as follows: Cost c 1 (qualitative, target): The cost involved in using a cloud service, including computer costs, storage costs, transfer costs, and application costs.
Reliability c 2 (qualitative, max): The reliability in a cloud refers to how a cloud service operates without failure under a set of operating conditions for a specific period of time.
Availability c 3 (qualitative, max): Whether a cloud service exists and is available instantly. Response Time (minutes) c 4 (quantitative, min): It represents the time elapsed to send a request by the client and receiving an answer provided by the cloud service.
Throughput (hits/sec) c 5 (quantitative, max): It represents the total number of invocations for a given time period. The unit of measure is invocations per second for a given cloud service.
Assume that there are four CSPs A = {a 1 , a 2 , a 3 , a 4 } left after a preliminary screen. Three DMs d q (q = 1, 2, 3) are invited to assess the performances of CSPs with respect to each qualitative criterion, and the three DMs have the same importance, i.e., λ 1 = λ 2 = λ 3 = 1/3. The qualitative attributes are assessed based on questionnaire using the scales given in Tables 1 and 2. The qualitative evaluation results are shown in Table 3. For quantitative attributes c 4 and c 5 (i.e., response time and throughput), the performances of the alternatives are obtained from the service level agreements of the CSPs, shown as (118, 75, 71, 103) and (25,17,11,16), respectively.    Below we use the Z-DNMA method presented in Section 3 to solve this problem. Since Step 1 is given above, we start the calculation process from Step 2.
Step 2. Convert each DM's linguistic evaluations to Z-numbers based on Tables 1 and 2.
From the above example, we can find that the proposed method has the following merits: (1) It can flexibly handle uncertain and unreliable trust-feedback data of the CSPs. It is not only a comprehensive reflection of DMs' judgments but also conforms to the expression habits of DMs; (2) It can deal with the decision-making problems which include quantitative and qualitative criteria in forms of benefit, cost, and target types. It can solve the sustainable cloud service provider development problem flexibly and reliably.

Discussion
In this section, we compare the results of the proposed method with the results obtained by the original DNMA method and other existing methods including the Z-TOPSIS method and Z-VIKOR method. Then, we perform sensitive analysis to validate the robustness of the proposed method.

Solving the Case by the Original DNMA Method
In the original DNMA method, Liao & Wu [25] adjust the criteria weights by the mean-squared-based weighting method. In this subsection, we recalculate the results by the original DNMA method.

Solving the Case by the Z-TOPSIS Method
Next, we apply the Z-TOPSIS to solve this case. The TOPSIS method selects the optimal alternative that has the shortest distance to the positive ideal solution A + and the furthest distance from the negative ideal solution A − . The classical TOPSIS normalized the decision matrix by vector normalization. Yaakob and Gegov [36] extended the classical TOPSIS method into the Z-numbers environment and implemented it in the stock selection problem, but that method could not support the MCDM with target criteria. We extend the TOPSIS method into the Z-fuzzy environment by combining it with the target-based vector normalization method (i.e., Equation (14)).
First, we use the target-based vector normalization method given as Equation (14) to normalize the decision matrix X and obtain x 2N ij (i = 1, 2, 3, 4; j = 1, 2, 3, 4, 5) as shown in Table 7. Then, we use Equation (17) to adjust x 2N ij tox 2N ij . Furthermore, the Euclidean distance between positive solution (s + i ) and negative ideal solution (s − i ) can be calculated by ij and w j is the adjusted criterion weight.
Calculate the relative closeness of each alternative to the ideal solution by Equation (27) and rank the alternatives according to descending order of RC i .
The calculation results are shown in Table 9. According to the calculation results, the final ranking of the alternatives is a 2 a 3 a 1 a 4 , which is consistent with the result derived by the Z-DNMA method.

Solving the Case by the Z-VIKOR Method
The major advantage of the VIKOR method is that it can trade off the maximum group utility of the "majority" and the minimum individual regret of the "opponent". It normalizes the decision matrix by linear normalization. Here, we extend the VIKOR method into the Z-fuzzy environment by combining it with the target-based linear normalization method (i.e., Equation (13)).
First, we use the target-based linear normalization method given as Equation (13) to normalize the decision matrix X , and obtain x 1N ij (i = 1, 2, 3, 4; j = 1, 2, 3, 4, 5) shown in Table 6. Then, we use Equation (17) to adjust x 1N ij tox 1N ij . Additionally, we use S i = n j=1 w j ·x 1N ij and R i = max j w j · 1 −x 1N ij to calculate the "group utility" value S i of each alternative and the "individual regret" value R i of the "opponent" of each alternative, respectively. We then calculate the compromise value Q i of each alternative by In addition, ρ is the weight of the strategy of "the majority of criteria" (or "the maximum group utility"). Here, we set different values for ρ (ρ = 0.25, 0.5, 0.75). Finally, we rank the alternatives and sort the values S i , R i , and Q i in descending order. The results are listed in Table 10. From Table 10, we can find that the results obtained by the Z-VIKOR method are also consistent with the results derived by the Z-DNMA method.

Sensitivity Analysis
To test the robustness of the ranking result, four sensitivity tests are carried out in this subsection. First, we adopt a weight replacement strategy for sensitivity test. Figure 2 contains the ten different tests to exchange the subjective weights of criteria and demonstrates the corresponding ranks of alternatives. For example, c 2 -c 5 denote that the subjective weights of criteria c 2 and criteria c 5 have been interchanged. From Figure 2, it is clear that a 2 has the highest rank in seven out of ten weighted calculation experiments, and a 4 has the lowest rank in all experiments. This suggests that the optimal and worst CSPs have not altered in most cases, which illustrates the stability of the ranking results. First, we adopt a weight replacement strategy for sensitivity test. Figure 2 contains the ten different tests to exchange the subjective weights of criteria and demonstrates the corresponding ranks of alternatives. For example, 2 c -5 c denote that the subjective weights of criteria 2 c and criteria 5 c have been interchanged. From Figure 2, it is clear that 2 a has the highest rank in seven out of ten weighted calculation experiments, and 4 a has the lowest rank in all experiments. This suggests that the optimal and worst CSPs have not altered in most cases, which illustrates the stability of the ranking results.  Table 11. From  Second, according to Equation (23) (in Step 6) of the Z-DNMA method, the collective utility value DN i of each alternative largely depends on the proportion of ϕ for the relative importance of the subordinate ranks and subordinate utility values. The parameter ϕ is the adjustment parameter that varies in [0, 1] and is set as 0.5 in this study. To validate the impact of ϕ on the CSP ranking, a sensitivity test is performed on the identical application in Section 4. with ϕ = (0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1). The ranking results of each alternative are obtained in Table 11. Table 11. The orders of alternatives with different ϕ. 1  3  3  3  3  3  3  3  3  3  3  3  a 2  1  1  1  1  1  1  1  1  From Table 11, we can see that for the change of ϕ, there is no change in the final ranking obtained by the proposed method throughout the analysis. Therefore, it can be concluded that the final ranking results are reliable and robust based on this sensitivity analysis.
Third, according to Equation (23) (in Step 6) of the Z-DNMA method, the parameter w 1 , w 2 , w 3 denote the weights of CCM, UCM and ICM, satisfying w i ∈ [0, 1] and 3 i=1 w i = 1. The values of w 1 , w 2 , and w 3 can be assigned different values depends on DMs' risk preferences. In this case study, we set DMs' risk preferences vector as W = (0.3, 0.3, 0.4) for demonstration. To validate the impact of W on the CSP ranking, a sensitivity test is performed with W 1 = (1, 0, 0 Table 12. From Table 12, it can be clearly seen that under the change of the weights of DMs' risk preferences, the rank order of the alternatives is without obvious change and the final ranking is reliable and robust.
Last, we add an alternative a 5 to test the stability of the results. The linguistic evaluation values of each qualitative criteria of a 5 are shown in Table 13. The performance values of quantitative criteria are 90 min, and 10 hits/sec, respectively.  According to the calculation steps of Z-DNMA, we can obtain DN 1 = 0.183, DN 2 = 0.589, DN 3 = 0.512, DN 4 = 0.042, DN 5 = 0.284. Therefore, the final ranking is a 2 a 3 a 5 a 1 a 4 . The results show that when a new alternative is added, the original ranking remains stable.
According to the above four sensitivity analysis, it can be concluded that a 4 is the trustworthy CSP since it has the minimal fluctuations in all the sensitivity test and the proposed method in this paper is robust and stable.

Conclusions
In this study, we introduced the original DNMA method to tackle quantitative and qualitative decision criteria in the forms of benefit, cost, and target types for the CSP development problem. We extended the DNMA method to Z-number environment and proposed the Z-DNMA method. In this regard, the uncertain and unreliability decision-making information of decision-makers was considered. We made use of the Gini coefficient-based weighting method to replace the mean-square-based weighting method used in the original DNMA method, and extended this approach to the Z-number environment for the trade-offs between criteria after normalization to enhance the applicability and isotonicity of the DNMA method. Based on the established decision-making method, a case study was conducted. Sensitivity analysis and comparative analysis were provided to test the stability and applicability of the proposed method.
Due to the cloud services as well as CSPs increasing rapidly with different functionalities and dynamic user requirements, the sustainable CSP development becomes an MCDM problem and remains a challenging research area in the field of cloud computing. The presented Z-DNMA method with the Gini-coefficient-based weight determination approach is utterly useful for customers to trickle quantitative and qualitative criteria in forms of benefit, cost, and target types for the sustainable CSP development, while considering the uncertain and unreliability decision-making information of DMs. First, based on linguistic Z-numbers, the proposed model can flexibly handle uncertain and unreliability trust feedback data of the CSPs. It is not only a comprehensive reflection of DMs' judgments, but also conforms to expression habits of DMs. Second, by the DNMA method, the proposed model can deal with such scenarios, which include quantitative and qualitative criteria in forms of benefit, cost, and target types. It can flexibly, reliably, and simply to solve MEMCDM for the sustainable cloud service provider development problem. Furthermore, by integrating the Gini-coefficient-based weighting method to replace the mean-square-based weighting method used in the original DNMA method and extending this approach to the Z-number environment for the trade-offs between criteria after normalization, the applicability and isotonicity of the DNMA method have been enhanced. Therefore, this study provided practical and theoretical guidance for sustainable CSP development to solve uncertainty and unreliable, multi-scale QoS assessment data to helps both researchers and practitioners for analyzing more fruitful approaches for CSP selection.
A limitation is that this paper uses a numerical example to show the effectiveness of the proposed method. In the future, we will employ the proposed method to dispose of the CSP selection problem under realistic data and cases. In addition, another limitation is that we assume that all criteria are independent in this study. We plan to develop a novel aggregation operator to aggregate the interactive criteria for better adapting to real decision-making problems.