In the era of economic globalization, the increasingly fierce competitive environment is a common challenge faced by global small and medium-sized enterprises (SMEs). SMEs must actively participate in the network economy to survive and develop. Compared with the sizeable enterprises, SMEs lack funds and technology in enterprise informatization construction (IC). To a certain extent, this limits the implementation effect of enterprise informatization. In recent years, with the rapid development of cloud computing (CC) technology, it opens up a new avenue for the application of information technology in SMEs. Many information processing-related businesses in CC model no longer install and execute on local computers. They run on remote servers, known as the cloud, and the results are offered to users as professional services. In this way, CC technology can effectively alleviate the contradiction between the demand of SMEs’ personalized information processing and the higher IC investment. This revolutionary technology is prompting more and more information technology (IT) service companies to adjust their service patterns by serving customers in the cloud. Facing the numerous CC vendors in the market, how to choose the most appropriate one has become an important decision-making problem for SMEs, which will greatly affect their business performance and competitiveness.

CC vendors selection can be treated as a typical multi-criteria decision-making (MCDM) problem since it deals with the task of ranking a finite number of alternatives under different criteria. At present, the research on MCDM has achieved fruitful results. Among of which technique for order of preference by similarity to ideal solution (TOPSIS) and analytic hierarchy process (AHP) are the most of widely used decision making technologies. Based on the transformation between interval-valued intuitionistic fuzzy variables (IVIFVs) and linguistic variables (LVs), Büyüközkan et al. (2018) proposed a hybrid method consisting of AHP, complex proportional assessment (COPRAS), multi-objective optimization by ratio analysis plus the full multiplicative form (MULTIMOORA), TOPSIS, and višekriterijumsko kompromisno rangiranje (VIKOR) to rank CC vendors. Jatoth et al. (2018) presented a hybrid MCDM approach containing TOPSIS and AHP for the selection of cloud services. Onar et al. (2018) provided a Pythagorean fuzzy TOPSIS method to select the proper cloud service provider by combining a variety of tangible and intangible criteria. Based on AHP scale mapping methodology, Meesariganda and Ishizaka (2017) provided an approach to evaluate cloud storage strategies for a company. Repschlaeger et al. (2014) described an AHP-based approach to support IT organization for selecting an appropriate CC vendor. By integrating AHP, quality function deployment (QFD) and multi-choice goal programming technologies, Liao and Kao (2014) presented a method to address the CC vendor selection problem in information service. Garg et al. (2013) proposed a framework that measures the quality of cloud service and prioritizes CC vendors by applying AHP. Moreover, Sohaib et al. (2019) utilized a TOPSIS based 2-tuple fuzzy linguistic multi-criteria group decision-making method to facilitate assessing and choosing the CC models for SMEs.

From the above literature review, one can see that previous research about the CC vendor selection problem mainly limits to decision matrix that needs DMs to offer the absolute evaluation of each object for every criterion or attribute. However, it is difficult or impossible for DMs due to various reasons such as the pressure of time, the limitations of expertise, and the complexity of the selection of CC vendors. In addition, it is difficult to establish a perfect index system to carry out quantitative evaluation on the CC vendors. Methods with preference relations (PRs) which only require the DMs to compare two objects at one time and permit the existence of missing judgments provide a convenient tool to solve complex MCDM problems. Since Saaty (1977) first proposed multiplicative PRs (MPRs), decision making based on PRs has made great success. In view of the DMs’ indeterminacies, different types of PRs are introduced such as interval multiplicative PRs (IMPRs) (Krejčí, 2017), fuzzy interval PRs (FIPRs) (Barrenechea et al., 2014) and interval linguistic PRs (ILPRs) (Tapia García et al., 2012). The common feature of PRs mentioned above is to consider only the preferred information of one object over another. Sometimes this might be insufficient. In addition to the preferred information, non-preferred judgments are also needed to fully express the recognition of DMs. For example, a DM may think that one object has a preferred degree of 0.6 over another. However, the judgment of non-preferred degree is 0.3 rather than 0.4, namely, the hesitancy degree between them is 0.1. In this situation, intuitionistic fuzzy values (IFVs) proposed by Szmidt and Kacprzyk (2002) are powerful tools. However, IFVs only allow the DM to employ numeric values [0, 1] to express their exact preferred and non-preferred judgments, which may be still insufficient. In the above example, if the DM’s preferred degree is between 0.6 and 0.8 and the non-preferred degree is between 0.1 and 0.2, IFV is useless. To denote the uncertainty of DMs’ judgments, Xu (2007a) further proposed interval-valued intuitionistic fuzzy values (IVIFVs) and introduced interval-valued intuitionistic fuzzy preference relations (IVIFPRs). Meanwhile, considering the complexity of decision-making problems, group decision making (GDM) is necessary to avoid the possible decision bias caused by a single DM. Considering the advantages of PRs and GDM, this paper introduces a GDM method based on IVIFPRs to solve the problem of CC vendors selection for SMEs through pairwise comparison among objects under the experts’ comprehensive consideration of the evaluation criteria.

In decision making with PRs, the lack of consistency will lead to unreasonable conclusions. Therefore, it is critical to study the consistency of IVIFPRs. Motivated by Xu and Chen’s method (2008) for deriving the priority vector from FIPRs, Wang et al. (2009) used the normalized interval weight vector to propose an additive consistency concept for IVIFPRs. According to the additive consistency concept for FIPRs, Wan et al. (2017) defined an additive consistency concept for IVIFPRs and then presented a GDM method. Wan et al. (2018) defined another additive consistency concept for IVIFPRs by transforming an IVIFPR into an IFPR. Recently, Tang et al. (2018) analysed the limitations of previous additive consistency concepts for IVIFPRs. Using 2-tuple preferred FIPRs (2TPFIPRs) and quasi 2TPFIPRs (Q2TPFIPRs), the authors proposed a new definition for additive consistent IVIFPRs.

In addition to the above additive consistency concepts for IVIFPRs, there are some researches on the multiplicative consistency of IVIFPRs. For instance, based on IVIFVs’ operational laws, Xu and Cai (2009) proposed a multiplicative consistency concept for IVIFPRs and developed a method for estimating missing values. Later, according to the multiplicative transitivity for fuzzy PRs (Tanino, 1984), Xu and Cai (2015) introduced another multiplicative consistency concept for IVIFPRs and developed an approach to GDM with incomplete IVIFPRs. Similar to the multiplicative consistency concept in Xu and Cai (2015), Liao et al. (2014) provided a multiplicative consistency concept for IVIFPRs and investigated inconsistent IVIFPRs. To improve the consistency and consensus level, the authors proposed two iterative algorithms. On the basis of Liao and Xu’s multiplicative consistency definition for IFPRs (Liao and Xu, 2014) and the induced matrices obtained from IVIFPRs, Wan et al. (2016) proposed another definition for multiplicative consistent IVIFPRs and introduced a GDM method. Meng et al. (2018) analysed the issues of multiplicative consistency concepts for IVIFPRs and proposed a new definition for multiplicatively consistent IVIFPRs. Based on the discussions on the multiplicative consistency and consensus, an algorithm for GDM with IVIFPRs is proposed. Other researches on decision making with IVIFPRs can be found in the literature (Wu and Chiclana, 2012; Zhou et al., 2018).

Through the above literature review about IVIFPRs, we find that there are some aspects of research on decision making with IVIFPRs that can be further improved. Some methods do not calculate the priority vector based on the consistency analysis, which may lead to illogical ranking results. Although some methods are based on the consistency analysis, the consistency conclusion is dependent on compared orders of objects. Due to the complexity of decision-making problems and the limitation of DMs’ subjective cognition, incomplete information may occur in IVIFPRs, which is not covered in some of the literature. Moreover, the iterative method for consistency and consensus improvement may change the initial preference information of DMs greatly, and the procedure for calculating the priority vector is complex in some of the literature.

Considering these facts, this paper offers a new method for GDM with IVIFPRs and provides an effective way for SMEs to choose the appropriate CC products. The main contributions of this paper include:

This paper is arranged as follows. Section 2 introduces some related basic concepts and analyses several multiplicative consistency definitions. Section 3 proposes a new multiplicative consistency definition for IVIFPRs that can avoid the issues of existing ones. Meanwhile, a programming model is built to check the multiplicative consistency of IVIFPRs. Section 4 tackles incomplete and inconsistent IVIFPRs. First, two programming models are constructed to determine missing values in incomplete IVIFPRs, and then a programming model is presented for repairing inconsistent IVIFPRs. Section 5 discusses GDM with incomplete and inconsistent IVIFPRs. It first gives a programming model to determine the DMs’ weights and offers a formula to measure the consensus degree. Then, a programming model is provided to reach the given consensus threshold. After that, the concrete algorithm for GDM with IVIFPRs is put forward. Section 6 applies the proposed method to solve the selection of CC products for SMEs and to compare the new method with previous ones. Conclusions and future remarks are conducted in Section 7.

To make the following discussions easily, the section reviews some related concepts.

To ensure the rationality of decision-making results, many scholars devoted themselves to the investigation of the consistency of FIPRs. After comparing and analysing the previous multiplicative consistency concepts, Meng et al. (2017) introduced the below multiplicative consistency concept for FIPRs.

From Definition 2, we can see that Meng et al.’s concept requires the endpoints of interval judgments to satisfy the multiplicative transitivity. In contrast to Definition 2, Krejčí (2019) offered another multiplicative consistency concept for FIPRs based on the constrained operations on intervals (Lodwick and Jenkins, 2013) that does not restrict to the endpoints of interval judgments.

By the reciprocity of FIPRs, one can check that formula (3) is equivalent to (4) aijL1−aijL⩾aikLakjL(1−aikL)(1−akjL),aijU1−aijU⩽aikUakjU(1−aikU)(1−akjU) for all i,k,j=1,2,…,n with k≠i , j∧i<j .

From formulae (2) and (4), one can conclude that Definition 2 can be seen as a special case of Definition 3. Just as Meng et al. (2019) noted, Definition 3 is more flexible than Definition 2.

To simultaneously express both the preferred and non-preferred information of one object over the other, Szmidt and Kacprzyk (2002) proposed the concept for IFVs. Following this, Xu (2007b) gave the definition for IFPRs.

To reflect the uncertainty of DMs’ judgments on preferred and non-preferred information, Atanassov and Gargov (1989) further introduced IVIFSs, and Xu (2007a) proposed IVIFVs for facilitating the application. Based on IVIFVs, Xu (2007c) offered the concept of IVIFPRs.

Regarding IVIFPRs’ multiplicative consistency, Liao et al. (2014) introduced a concept on the basis of multiplicative transitivity for fuzzy PRs (Tanino, 1984). By applying Liao and Xu’s multiplicative consistency definition for IFPRs (Liao and Xu, 2014), Wan et al. (2016) also offered a multiplicative consistency concept for IVIFPRs. The common limitation of the two concepts is that they do not satisfy the robustness, which means contradictory conclusions about consistency might be obtained under different comparison orders between objects. Furthermore, different ranking orders might be derived too.

Corresponding to 2TPFIPRs, the concept for Q2TPFIPRs is offered as follows:

Definition 7 indicates that the Q2TPFIPR S˜=(s˜ij)n×n is composed by two QFIPRs η‾=(η‾ij)n×n and λ‾=(λ‾ij)n×n , where (8) η¯ij=[μl,ij,1−vl,ij]η¯ji=[1−μl,ij,vl,ij]∨η¯ij=[1−vl,ij,μl,ij]η¯ji=[vl,ij,1−μl,ij],λ¯ij=[μu,ij,1−vu,ij]λ¯ji=[1−μu,ij,vu,ij]∨λ¯ij=[1−vu,ij,μu,ij]λ¯ji=[vu,ij,1−μu,ij] for all i,j=1,2,…,n .

Based on the above fact and Definition 2, Meng et al. (2018) gave the below multiplicative consistency concept for Q2TPFIPRs:

Based on the multiplicative consistency of Q2TPFIPRs, Meng et al. (2018) further proposed the following multiplicative consistency concept for IVIFPRs:

For example, the IVIFPR: R˜=([0.50,0.50],[0.50,0.50])([0.46,0.46],[0.52,0.52])([0.57,0.60],[0.35,0.36])([0.70,0.70],[0.29,0.29])([0.52,0.52],[0.46,0.46])([0.50,0.50],[0.50,0.50])([0.59,0.62],[0.32,0.33])([0.73,0.73],[0.27,0.27])([0.35,0.36],[0.57,0.60])([0.32,0.33],[0.59,0.62])([0.50,0.50],[0.50,0.50])([0.55,0.56],[0.35,0.38])([0.29,0.29],[0.70,0.70])([0.27,0.27],[0.73,0.73])([0.35,0.38],[0.55,0.56])([0.50,0.50],[0.50,0.50]) is multiplicatively consistent according to Definition 9, since there are two corresponding multiplicatively consistent QFIPRs as follows: η¯=[0.50,0.50][0.48,0.46][0.57,0.65][0.71,0.70][0.52,0.54][0.50,0.50][0.59,0.68][0.73,0.73][0.43,0.35][0.41,0.32][0.50,0.50][0.65,0.55][0.29,0.30][0.27,0.27][0.35,0.45][0.50,0.50],λ¯=[0.50,0.50][0.46,0.48][0.64,0.60][0.70,0.71][0.54,0.52][0.50,0.50][0.67,0.62][0.73,0.73][0.36,0.40][0.33,0.38][0.50,0.50][0.56,0.62][0.30,0.29][0.27,0.27][0.44,0.38][0.50,0.50].

From Definition 9, one can find that Meng et al.’s multiplicative consistency concept is based on Definition 2. Although it can avoid the limitations of the multiplicative concepts in Liao et al. (2014), Wan et al. (2016), the main issue of Definition 9 is not flexible enough (Meng et al., 2019).

Considering the limitations of previous multiplicative consistency concepts for IVIFPRs, this section introduces a new one based on Krejčí’s multiplicative consistency concept (Krejčí, 2019) for FIPRs.

Let R˜=(r˜ij)n×n be an IVIFPR, and let P˜ be its 2TPFIPR as shown in Definition 6. One can find that the 2TPFIPR P˜ is composed by the matrices P‾L=(p‾ijL)n×n and P‾U=(p‾ijU)n×n , where (10) p¯ijL=[μl,ij,1−vl,ij],p¯ijU=[μu,ij,1−vu,ij] for all i,j=1,2,…,n .

Following the concept for IVIFPRs and formula (10), we have (11) p¯ijL=[μl,ij,1−vl,ij]⊆[0,1],p¯jiL=[μl,ji,1−vl,ji]⊆[0,1],μl,ji+1−vl,ij=vl,ij+1−vl,ij=1,μl,ij+1−vl,ji=μl,ij+1−μl,ij=1,andp¯ijU=[μu,ij,1−vu,ij]⊆[0,1],p¯jiU=[μu,ji,1−vu,ji]⊆[0,1],μu,ji+1−vu,ij=vu,ij+1−vu,ij=1,μu,ij+1−vu,ji=μu,ij+1−μu,ij=1 for all i,j=1,2,…,n . Therefore, P‾L=(p‾ijL)n×n and P‾U=(p‾ijU)n×n are FIPRs.

Based on the above relationship, we employ the multiplicative consistent FIPRs to define multiplicative consistent 2TPFIPRs.

Following formula (4) and Definition 10, one can easily derive the below theorem:

Theorem 1 shows that the multiplicative consistency of 2TPFIPRs is based on the multiplicative transitivity of the endpoints of IVIFVs in associated IVIFPRs. Thus, we can adopt Definition 10 to further define the multiplicative consistency of IVIFPRs.

According to the independence of Definition 3 for the compared orders, it can be easily checked that Definition 11 is invariant for the permutation of objects.

In the following, we study the relationship between Definitions 9 and 11 to show the flexibility of the new concept.

Considering the below IVIFPR: R˜=([0.50,0.50],[0.50,0.50])([0.40,0.60],[0.30,0.40])([0.40,0.60],[0.40,0.40])([0.30,0.40],[0.40,0.60])([0.50,0.50],[0.50,0.50])([0.35,0.50],[0.45,0.50])([0.40,0.40],[0.40,0.60])([0.45,0.50],[0.35,0.50])([0.50,0.50],[0.50,0.50]).

One can check that R˜ is multiplicatively consistent according to Definition 11. Nevertheless, it is inconsistent following Definition 9.

The relationship between the new and Meng et al.’s multiplicative consistency concepts for IVIFPRs is listed in Fig. 1.

Although Definition 11 owns all properties of Definition 3, it is inefficient to judge the multiplicative consistency of IVIFPRs directly. For an IVIFPR R˜ , there are two associated FIPRs based on formula (10). Then, 2n(n−1)(n−2) triples of (i,k,j) need to be compared by formula (13) for judging its multiplicative consistency. Therefore, this way is time consuming and infeasible. To solve this problem, we construct the following programming model to judge the IVIFPRs’ multiplicative consistency: (16) Model 1:Γ∗=min ∑i=1n−1 ∑j=i+1n(αl,ij+βl,ij+αu,ij+βu,ij)s.t.log(μl,ij)−log(1−μl,ij)+αl,ij⩾log(μl,ik)+log(μl,kj)−log(1−μl,ij)−log(1−μl,kj),log(vl,ij)−log(1−vl,ij)+βl,ij⩾log(vl,ik)+log(vl,kj)−log(1−vl,ij)−log(1−vl,kj),log(μu,ij)−log(1−μu,ij)+αu,ij⩾log(μu,ik)+log(μu,kj)−log(1−μu,ij)−log(1−μu,kj),log(vu,ij)−log(1−vu,ij)+βu,ij⩾log(vu,ik)+log(vu,kj)−log(1−vu,ij)−log(1−vu,kj),i,k,j=1,2,…,n,k≠i,j,i<j,αl,ij,βl,ij,αu,ij,βu,ij⩾0,i,j=1,2,…,n,i<j, where the first four constraints are obtained by taking the natural logarithm of the inequalities in formula (13) and adding the non-negative deviation variables αl,ij , βl,ij , αu,ij , βu,ij .