Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 791204, 13 pages

http://dx.doi.org/10.1155/2015/791204

## A Selection Method Based on MAGDM with Interval-Valued Intuitionistic Fuzzy Sets

^{1}College of Science, Guilin University of Technology, Guilin 541004, China^{2}College of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330013, China^{3}College of Information Science and Engineering, Guilin University of Technology, Guilin 541002, China

Received 3 December 2014; Accepted 11 May 2015

Academic Editor: Julien Bruchon

Copyright © 2015 Gai-Li Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

As the cloud computing develops rapidly, more and more cloud services appear. Many enterprises tend to utilize cloud service to achieve better flexibility and react faster to market demands. In the cloud service selection, several experts may be invited and many attributes (indicators or goals) should be considered. Therefore, the cloud service selection can be regarded as a kind of Multiattribute Group Decision Making (MAGDM) problems. This paper develops a new method for solving such MAGDM problems. In this method, the ratings of the alternatives on attributes in individual decision matrices given by each expert are in the form of interval-valued intuitionistic fuzzy sets (IVIFSs) which can flexibly describe the preferences of experts on qualitative attributes. First, the weights of experts on each attribute are determined by extending the classical gray relational analysis (GRA) into IVIF environment. Then, based on the collective decision matrix obtained by aggregating the individual matrices, the score (profit) matrix, accuracy matrix, and uncertainty (risk) matrix are derived. A multiobjective programming model is constructed to determine the attribute weights. Subsequently, the alternatives are ranked by employing the overall scores and uncertainties of alternatives. Finally, a cloud service selection problem is provided to illustrate the feasibility and effectiveness of the proposed methods.

#### 1. Introduction

Cloud computing [1–4] is the latest computing paradigm that delivers hardware and software resources as virtualization services in which users are free from the burden of worrying about the low-level system administration details. In recent years, cloud computing is developing rapidly and has provided enterprises with many advantages such as flexibility, business agility, and pay-as-you-go cost structure. As a result, many enterprises with limited financial and human resources are increasingly adopting cloud computing to deliver their business services and products online to extend their business markets. In many domains, multiple cloud services often supply similar functional properties. For example, in Customer Relationship Management (CRM), CRM venders offer functionally equivalent cloud services, such as Microsoft Dynamic CRM, Salesforce Sales Cloud, SAP Sales on Demand, and Oracle Cloud CRM. However, for enterprises, which lack cloud computing knowledge, it is difficult to select an appropriate candidate from a set of functionally equivalent cloud services. Therefore, it is necessary for enterprises to invite several related experts to evaluate the potential candidates from several indicators (attributes), such as payment, performance, reputation, scalability, and security. The selection of cloud services has attracted attention and many methods have been presented to guide enterprises in selecting the cloud services. Roughly, these methods may be divided into two categories and briefly reviewed as follows, respectively.

The first category is the Multiattribute Decision Making (MADM) methods. According to the key performance indicators defined by Siegel and Perdue [5], Garg et al. [6] proposed the cloud service ranking framework using the Analytic Hierarchy Process (AHP) technique. Menzel et al. [7] utilized the Analytic Network Process (ANP) to develop a Multicriteria Comparison Method which is used to select Infrastructure-as-a-Service (IaaS). Limam and Boutaba [8] presented a trustworthiness-based service selection method based on the Multiple Attribute Utility Theory (MAUT). By employing the Elimination and Choice Expressing Reality (ELECTRE) method, Silas et al. [9] developed a cloud service selection middleware to help cloud users select desired cloud service. Saripalli and Pingali [10] discussed Simple Additive Weighting (SAW) methods to rank alternatives in a decision problem of cloud service adoption. Zhao et al. [11] suggested a SAW-based service searching and scheduling algorithm to obtain a set of ranked services.

The second category is the optimization approaches. Chang et al. [12] designed a dynamic programming algorithm by maximizing the overall survival probability to select cloud storage providers. Sundareswaran et al. [13] selected cloud service with a greedy algorithm method which can make experts retrieve information fast. In order to help service providers to select Software-as-a-Service (SaaS) services with multitenants, He et al. [14] explored three types of optimization algorithm, including integer programming, skyline, and greedy algorithm, and proposed a quality of service- (QoS-) driven optimization framework. By minimizing costs and risks, Martens et al. [15] constructed a scalable mathematical decision model to select cloud service. Yang et al. [16] built a Markov decision process model to guarantee the near-optimal performance in a changing environment by dynamically adjusting the components of a service composition.

The aforementioned methods seem to be effective and applicable for selecting cloud services. However, they have the following shortcomings.(1)The decision making in methods [13–15] is single Multiattribute Decision Making (MADM); that is, only one expert participates in the decision making and gives assessment information of alternatives with respect to several attributes. Since every expert is good at only some fields rather than all fields, the reliability of some information given by the expert is a little doubtful.(2)Current methods [9, 12] are more focused on quantitative attributes measured via precise numerical values, such as response time, storage space, and latency time. Nevertheless, in cloud service, some qualitative attributes (such as reputation and security) usually play important roles, but they do not gain enough attention.(3)In existing methods [12, 13, 16], the assessment values (attribute values) are crisp numbers, which is somewhat unrealistic. Due to the inherent vagueness of human preferences as well as the fuzziness and uncertainty of objects, it is more suitable to express the assessment values as fuzzy numbers [17–20].(4)The attribute weights provided by experts are given a priori in methods [8, 10, 11], which always cannot avoid subjective randomness of the expert’s preference. Furthermore, with increasing complexity in many real decision situations, it is difficult for expert to provide precise and complete preference information due to time pressure and lack of data.

One of the reasons leading to the above shortcomings is that the fuzziness and uncertainty are not fully considered during the decision making process. The fuzzy set (FS) theory introduced by Zadeh [21] is a very useful tool to describe fuzzy and uncertain information. Based on the FS theory, Atanassov [22] presented the intuitionistic fuzzy set (IFS), which considers the membership (satisfaction) degree, nonmembership (dissatisfaction) degree, and hesitant degree simultaneously. Subsequently, Atanassov and Gargov [23] generalized IFS and presented interval-valued intuitionistic fuzzy set (IVIFS) that describes the membership and nonmembership degrees as intervals. Compared with the FS and IFS, IVIFS is more suitable to express the fuzziness and uncertainty and has been widely used in many fields [24–28].

According to IVIFS theory, to overcome the aforementioned shortcomings, we investigate the cloud service selection problems with IVIFSs and develop a novel method. The proposed method has the following key characteristics.(1)The selection of cloud service is regarded as a Multiattribute Group Decision Making (MAGDM) problem that several experts are invited to evaluate the potential cloud services, whereas it is considered as a single MADM problem in methods [6–11]. With increasing complexity and the limit knowledge owned by single expert, in order to increase the quality of cloud service, it is more reasonable and reliable for enterprises to invite multiple experts to participate in making decision.(2)The assessment values given by experts are expressed as IVIFSs. Compared with the crisp number, IVIFS is more flexible to measure the qualitative attributes since IVIFS considers membership, nonmembership, and hesitant degrees which are expressed as intervals. Additionally, it is easier for experts to supply assessment values with IVIFSs in the increasing uncertain and complex environment.(3)By extending the classical gray relational analysis (GRA) [29] into IVIF environment, a new approach is proposed to determine the weights of experts. A notable characteristic of the proposed approach is that the obtained weights of each expert are different with respect to different attributes.(4)For MAGDM problems with incomplete information on attributes, a multiobjective programming model is constructed to objectively determine the attribute weights, which can avoid the subjective randomness appearing in the methods [8, 10, 11]. Moreover, it is easier for experts to give partial information on attribute weights than to assign a crisp number to the attribute weights.

The rest of this paper unfolds as follows. Some preliminaries about IVIFSs and the classical GRA method are introduced in Section 2. In Section 3, a new method is proposed to solve MAGDM problems with IVIFSs and incomplete attribute weight information. In addition, a framework of decision supporting system (DSS) is constructed. In Section 4, a cloud service selection example is provided to illustrate the applicability of the proposed method and comparison analysis is conducted. Finally, the conclusions are discussed in Section 5.

#### 2. Preliminaries

In this section, we introduce some basic concepts related to interval-valued fuzzy set (IVIFS) and gray relational analysis (GRA).

##### 2.1. Interval-Valued Intuitionistic Fuzzy Set

*Definition 1 (see [23]). *Let be a nonempty set of the universe. An IVIFS in is defined aswhere and denote the intervals of membership degree and nonmembership degree of element , respectively, satisfying , , and for all . is called the interval-valued intuitionistic hesitant degree of IVIFS . For any , if and , then is reduced to an IFS.

Xu [30] called the pair an interval-valued intuitionistic fuzzy number (IVIFN) and denoted an IVIFN by , where , .

*Definition 2 (see [30]). *Let , , and be three IVIFNs; then(1);(2);(3).

*Definition 3 (see [30]). *Let be a collection of IVIFNs. If then the IVIFWA is called an interval-valued intuitionistic fuzzy weighted averaging (IVIFWA) operator of dimension , where is a weight vector of with and .

The aggregated value determined by the IVFWA operator is also an IVIFN; that is,

*Definition 4 (see [30]). *Let be an IVIFN. Then are, respectively, called the score function and accuracy function of the IVIFN , where and can be considered as net membership and accuracy degree, respectively.

Since , when many score functions are aggregated with linear weighted summation method, it maybe appears that positive score functions are offset by negative score functions. Therefore, we normalize the score function and make it belong to .

Given a variable , if we define then cannot only retain the monotonicity of the variable but also map for . Hence, we modify the score function in Definition 4 and define a new score function of IVIFN .

*Definition 5. *Let be an IVIFN. Then is called a normalized score function, where . Obviously, .

*Definition 6. *Let be an IVIFN. Thenis called an uncertainty function, where .

Let be an assessment value of the cloud service with respect to the attribute (indicator) . Then the normalized score function and the uncertainty function can be, respectively, interpreted as the “net profit” and “risk” provided by cloud service on attribute . Hence, the bigger the and the smaller the , the better the cloud service . In the following, an order relationship between IVIFNs is given.

*Definition 7. *Let , be two IVIFNs; then(1)If , then .(2)If , then(i)If , then .(ii)If , then .

*Definition 8. *Let , be two IVIFNs; the Euclidean distance between and is defined as follows:

##### 2.2. Gray Relation Analysis

GRA Theorem is an important part of Gray Theorem developed by Deng [29]. GRA investigates uncertain relationship between one main factor and all other factors in a system and has been used in a wide variety of decision making environments, such as supplier selection [31], material selection [32], and water protection strategy evaluation [33].

The details of the classical GRA method are presented as follows.

(i) Calculate the normalized decision matrix.

Let be a decision matrix. The normalized matrix is calculated as

(ii) Generate comparability sequences () and a reference sequence is . For example, we can take .

(iii) Compute the gray relational coefficient between the comparability sequence and the reference sequence by the following formula:where , , , and is a distinguishing coefficient. Usually,

(iv) Calculate the gray relational grade between and ; that is,where is a weight vector satisfying and . The bigger the , the closer the sequence to the sequence .

#### 3. A Novel Method for MAGDM with IVIFSs and Incomplete Attribute Weight Information

In this section, a new method is proposed to handle MAGDM with IVIFSs. The proposed method includes determination of the weights of experts and identification of attribute weights.

Let be the set of feasible alternatives, let be the set of attributes, and let be the set of decision makers (DMs). Assume that is an attribute weight vector, where and . Let the individual decision matrix given by expert be , where is an IVIFN for the alternative with respect to attribute . In this paper, and provided by the expert are, respectively, the satisfaction (agreeing) degree interval and dissatisfaction (disagreeing) degree interval of the th cloud service with respect to the th attribute (indicator) .

##### 3.1. Determine the Weights of Experts by the Extended GRA Method

Due to the fact that each expert is skilled in some fields rather than all fields, it is more reasonable that the weights of each expert with respect to different attributes should be assigned different values. However, the weights of each expert obtained with the existing methods [34–37] are the same.

Let be the weight of expert with respect to attribute . Generally, for the attribute , the closer the attribute values of all alternatives given by expert are to those given by all other experts, the more similar the information provided by the expert is to that implied by the group. Consequently, the weight of expert should be assigned a greater value. Bearing this idea in mind, we present a novel method to determine the weights of experts by extending classical GRA method.

Given the decision matrices , the elements can be normalized as The normalized decision matrices can be denoted by .

Let be the reference sequence and let all other sequences be comparability sequences. Then, for the attribute , the gray relational coefficient between and with respect to alternative is defined aswhere is the distance between and (see (9)), , , and .

Thus, the matrix of gray relational coefficient between and is constructed aswhere .

The gray relational grade between and is calculated as

The gray relational grade describes the degree of closeness between sequence and sequence . In other words, indicates the similarity degree between the information given by DM and that given by DM on attribute .

For the attribute , the average gray relational grade between DM and all other DMs is computed as

Thus, the larger the is, the more similar the information given by the expert is to that implied by the group. Therefore, the bigger the . Accordingly, the weight of expert with respect to attribute , denoted by , can be defined as

##### 3.2. Integrate Individual Decision Matrices into a Collective Matrix

After the weights of experts are obtained, individual decision matrix can be integrated into a collective matrix with IVIFWA operator, where

For convenience, we denote by

By employing (5), (7), and (8), the score matrix, accuracy matrix, and uncertainty matrix of matrix are, respectively, obtained as follows:where , , and .

Utilizing the weighted summation method, we can derive the overall score function, accuracy function, and uncertainty function of alternative as

If the attribute weights are known in advance, then alternatives can be ranked and selected according to Definition 7. In what follows, a new multiobjective linear programming model is constructed to determine the attribute weights.

##### 3.3. Identify the Attribute Weights by a New Multiobjective Linear Programming Model

Due to the uncertainty of decision making environment and the limited knowledge possessed by experts, experts only may supply partial information about attribute weights. Namely, the information of the attribute weights is incomplete. Let be the set of incomplete information on attribute weights.

According to Definition 7, the bigger the overall score function (i.e., profit function) and the smaller the overall uncertainty function (i.e., risk function) of the alternative , the better the alternative . Therefore, by maximizing the overall score functions and minimizing the overall uncertainty functions, a multiobjective programming is built to objectively determine the weights of attributes:

By the max-min method for solving multiobjective programming [38], (25) can be converted as

From the relationship between and (see (8)), when reaches maximum, reaches minimum. Accordingly, minimizing the maximum among is equivalent to maximizing the minimum among . Therefore, (26) can be transformed as

Assume that , ; we have and . Thus, by employing (22)-(23), (27) can be rewritten as

By the linear weighted summation method, (28) can be converted into the following single objective programming model:where represents the relative importance of the two objects.

If , then experts are pessimistic and are more concerned about uncertainty function (i.e., risk) than score function (i.e., profit); if , then experts are optimistic and are more concerned about profit than risk; if , then experts considered that profit is as important as risk.

By solving (29), the vector of attribute weights can be obtained.

##### 3.4. Decision Process and Algorithm for MAGDM Problems with IVIFSs

Based on the above analysis, the algorithm and decision process for MAGDM problems are summarized as follows.

*Step 1. *The experts establish the individual decision matrix with IVIFSs and supply the set of information on the attribute weights .

*Step 2. *Calculate the weight of expert by (13)–(18), where .

*Step 3. *Integrate all individual decision matrix into a collective matrix by (19).

*Step 4. *Derive the score matrix , accuracy matrix , and uncertainty matrix of the matrix by (20)-(21).

*Step 5. *Determine the weight vector of attributes according to (29).

*Step 6. *Compute the overall score and uncertainty of alternatives by (22) and (24).

*Step 7. *Rank the alternatives and select the best one according to Definition 7.

##### 3.5. The Framework Decision Support System Based on MAGDM with IVIFSs

As the scale of decision making increases, the procedure solving a MAGDM may be complicated. In this case, a decision supporting system (DSS), which is a class of computer-based information system including knowledge-based systems [39, 40], can be formulated to help experts improve their decision-making level and quality through problem analysis, establishment of models, and simulation of decision-making process in a human-computer interaction way. Figure 1 depicts a framework of DSS designed in this paper for MAGDM with IVIFSs.