A Direct Approach Based on C²-IULOWA Operator for Group Decision Making with Uncertain Additive Linguistic Preference Relations

Abstract

With respect to group decision making (GDM) problem with uncertain additive linguistic preference relations (UALPRs), we investigate the efficient aggregation of the uncertain additive linguistic preference information. First, we introduce two measures to assess the consistency level and the consensus level of uncertain additive linguistic preference information, respectively, and study some of their desirable properties. Then, based on both the two measures, we propose a coinduced uncertain linguistic ordered weighted averaging (IULOWA) operator, called the consistency and consensus coinduced uncertain linguistic ordered weighted averaging (C²-IULOWA) operator, to aggregate individual uncertain additive linguistic preference information, in which the consistency level and the consensus level synergistically serve as inducing variables and then guide the determination of the associated weights. We have proved the collective uncertain linguistic preference information aggregated by the C²-IULOWA operator that can maintain the fundamental properties of preference relation, such as indifference, reciprocity, and transitivity. By using the C²-IULOWA operator, we develop a direct GDM approach with UALPRs. Finally, an illustrative example on the selection of chief quality officer is used to demonstrate the effectiveness and rationalitly of the developed approach.

1. Introduction

In group decision making (GDM) problems, preference relations are commonly used by decision makers (DMs) to express their preference information based on pairwise comparisons of alternatives. Three of the most common types of preference relations are multiplicative preference relations [1], fuzzy preference relations [2–7], and linguistic preference relations [8–10]. Due to the complexity and uncertainty involved in real-world decision problems and the inherent subjective nature of human judgments, linguistic preference relations can be more appropriate for capturing the lack of precision in human behavior than others. Hence, decision making based on linguistic preference relation has attracted considerable research interests over the past decades [11–17].

However, sometimes, the DMs are willing or able to provide only uncertain (or interval) linguistic information in some particular situations because of time pressure, lack of knowledge or data, and DMs’ limited expertise related to the problem domain [18–24]. Xu [18] proposed the concept of uncertain linguistic variable whose value is interval linguistic information and developed the uncertain linguistic ordered weighted averaging (ULOWA) operator and the uncertain linguistic hybrid aggregation (ULHA) operator to deal with multiple attribute group decision making problems. Xu [19] defined the concept of uncertain additive linguistic preference relation (UALPR) and developed a direct approach to GDM with uncertain linguistic averaging (ULA) and uncertain linguistic weighted averaging (ULWA) operators. Xu [20] introduced the concept of uncertain multiplicative linguistic preference relation and the operational laws of uncertain multiplicative linguistic variables and then proposed the uncertain linguistic weighted geometric mean (ULWGM) operator, uncertain linguistic ordered weighted geometric (ULOWG) operator, and induced uncertain linguistic ordered weighted geometric (IULOWG) operator to deal with GDM problems. Gao and Peng [21] presented a novel quantified SWOT analysis methodology with uncertain linguistic preference relations, interval fuzzy preference relations, and interval multiplicative preference relations. Chen et al. [22] developed a new compatibility for the UALPRs and utilized it to determine the optimal weights of experts in the GDM. Chen and Lee [23] presented an interval linguistic labels ordered weighted average (ILLOWA) operator and a consensus measure for autocratic decision making using group recommendations. Peng et al. [24] presented some multigranular uncertain linguistic prioritized aggregation operators to aggregate directly the uncertain linguistic variables whose values come from the linguistic term sets with different granularities and convey the prioritization phenomenon among the aggregated arguments.

In GDM with all kinds of preference relations, consistency and consensus of preference relations play vital roles. The former refers to the capability of DMs to express their preferences without contradiction; the latter reflects the actual levels of agreement amongst all the individual preferences. Thus, in order to derive a scientific and reasonable decision result, measuring consistency level and measuring consensus level of preference relation are indispensable research topics. To date, although a great deal of research has been conducted on the issues [3, 5–7, 9, 13–17], the contributions of consistency and consensus measures of UALPRs are little.

In the process of practical GDM, on the other hand, it is very difficult for all DMs to construct perfect consistent and consensual preference relations due to the uncertainty and complexity of real world and the subjectivity of DMs’ judgments. Also, as pointed out by Saaty, improving consistency does not mean getting an answer closer to real-life situation. It only means that the judgments are closer to being logically related than to being randomly chosen [25]. Furthermore, forcing improvements in consistency may distort the individual’s true answer to some extent. The similar situations also exist in consensus. In this paper, we consider the problem that if the DMs are no longer available after constructing preference relations and if the preference relations are not perfect consistent and consensual, then how to determine the scientific and reasonable decision results with a higher level of consistency and consensus. To do so, we shall first give the definitions of the measures of consistency and consensus of UALPRs, present a consistency and consensus-based coinduced uncertain linguistic OWA operator to aggregate UALPRs in such a way that more importance is placed on the most consistent and consensual preference information, and then develop a GDM approach with uncertain linguistic preference information.

The rest of this paper is set out as follows. Section 2 briefly introduces relevant concepts of uncertain linguistic preference information. In Section 3 we define the consistency measure and the consensus measure to assess the consistency level and consensus level of uncertain linguistic preference information, respectively, and study some of their properties. On the basis of the two measures, we present a consistency and consensus co-induced uncertain linguistic OWA operator and discuss some of its desirable properties in Section 4. In Section 5, based on the proposed operator, we develop a direct GDM approach. Section 6 demonstrates the effectiveness and rationally of the developed method with an illustrative example on the selection of chief quality officer. Section 7 concludes this paper.

2. Preliminaries

2.1. Linguistic Variables

Let 𝑆={𝑠_𝛼∣𝛼=0,1,2,…,𝜏} be a linguistic term set with odd cardinality, which satisfies the following characteristics: (1) the set is ordered: 𝑠_𝛼>𝑠_𝛽 if 𝛼>𝛽; (2) there is the reciprocal operator Neg(𝑠_𝛼)=𝑠_𝛽 such as 𝛼+𝛽=𝜏. For example, 𝑆 can be defined as 𝑆=𝑠₀=extremelylow,𝑠₁𝑠=verylow,₂=low,𝑠₃=slightlylow,𝑠₄𝑠=fair,₅=slightlyhigh,𝑠₆𝑠=high,₇=veryhigh,𝑠₈=extremelyhigh. (1)

To preserve all the given information, Xu [12–14] extended the discrete term set 𝑆 to a continuous term set 𝑆={𝑠_𝛼∣𝛼∈[0,𝑞]}, where 𝑞(𝑞>𝜏) is a sufficiently large positive integer. If 𝑠_𝛼∈𝑆, then we call 𝑠_𝛼 the original term otherwise, we call 𝑠_𝛼 the virtual term. In general, the decision maker uses the original linguistic terms to evaluate alternatives, and the virtual ones can only appear in operation.

Consider any two terms 𝑠_𝛼,𝑠_𝛽∈𝑆 and 𝜇,𝜇₁,𝜇₂∈[0,1]  and some operational laws as follows [13]: (1)𝜇𝑠_𝛼=𝑠_𝜇𝛼,(2)𝑠_𝛼⊕𝑠_𝛽=𝑠_𝛼+𝛽,(3)𝑠_𝛼⊖𝑠_𝛽=𝑠_𝛼−𝛽,(4)𝜇₁+𝜇₂𝑠_𝛼=𝜇₁𝑠_𝛼⊕𝜇₂𝑠_𝛼,(5)𝜇𝑠_𝛼⊕𝑠_𝛽=𝜇𝑠_𝛼⊕𝜇𝑠_𝛽,(6)𝜇₁−𝜇₂𝑠_𝛼=𝜇₁𝑠_𝛼⊖𝜇₂𝑠_𝛼. (2)

Definition 1 (see [21]). Let 𝐼(𝑠) be the lower index of 𝑠, and name it as the gradation of 𝑠 in 𝑆. For example, if 𝑠=𝑠_𝛼, then 𝐼(𝑠)=𝛼. Two operational laws of gradation function are given as follows: 𝐼𝑠_𝛼+𝐼𝑠_𝛽=𝐼𝑠_𝛼⊕𝑠_𝛽,𝐼𝑠_𝛼×𝐼𝑠_𝛽=𝐼𝑠_𝛼⊗𝑠_𝛽. (3)

2.2. Additive Uncertain Linguistic Preference Relations

Consider a GDM problem, let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛}(𝑛≥2) be the set of alternatives, and let 𝐷={𝑑₁,𝑑₂,…,𝑑_𝑚}(𝑚≥2) be the set of DMs. Every DM compares each pair of alternatives in 𝑋 and provides his/her preference of the alternatives with UALPRs {𝐿⁽¹⁾,…,𝐿^(𝑘),…,𝐿^(𝑚)}, where 𝐿^(𝑘)=(𝑙̃^(𝑘)_𝑖𝑗)_𝑛×𝑛=([𝑙̃^(𝑘)𝐿_𝑖𝑗,𝑙̃^(𝑘)𝑈_𝑖𝑗])_𝑛×𝑛, 𝑙̃^(𝑘)𝐿_𝑖𝑗,𝑙̃^(𝑘)𝑈_𝑖𝑗∈[𝑠₀,𝑠₁,…,𝑠_𝜏](𝑘=1,2,…,𝑚;𝑖,𝑗=1,2,…,𝑛). The UALPR can be formally defined as follows.

Definition 2 (see [19, 21, 22]). An UALPR 𝐿=(𝑙̃_𝑖𝑗)_𝑛×𝑛 on a set of alternatives 𝑋 is characterized by a function 𝜇𝐿∶𝑋×𝑋→𝑆 and satisfies 𝑙̃_𝑖𝑗=𝑙̃^𝐿_𝑖𝑗,𝑙̃^𝑈_𝑖𝑗,𝑠₀≤𝑙̃^𝐿_𝑖𝑗≤𝑙̃^𝑈_𝑖𝑗≤𝑠_𝜏,𝑙̃^𝐿_𝑖𝑗⊕𝑙̃^𝑈_𝑗𝑖=𝑙̃^𝑈_𝑖𝑗⊕𝑙̃^𝐿_𝑗𝑖=𝑠_𝜏,𝑙̃^𝐿_𝑖𝑖=𝑙̃^𝑈_𝑖𝑖=𝑠_𝜏/2,∀𝑖,𝑗∈{1,…,𝑛}, (4) where the preference value 𝑙̃_𝑖𝑗=𝜇𝐿(𝑥_𝑖,𝑥_𝑗) indicates the preference degree of the alternative 𝑥_𝑖 over 𝑥_𝑗 and is interpreted as 𝑥_𝑖 is 𝑙̃_𝑖𝑗 as 𝑥_𝑗, 𝑙̃^𝐿_𝑖𝑗 and 𝑙̃^𝑈_𝑖𝑗 are the lower and upper limits of 𝑙̃_𝑖𝑗, respectively, 𝑙̃^𝐿_𝑖𝑗,𝑙̃^𝑈_𝑖𝑗∈[𝑠₀,𝑠₁,…,𝑠_𝜏]. In particular, if 𝑙̃^𝐿_𝑖𝑗=𝑙̃^𝑈_𝑖𝑗, ∀𝑖,𝑗∈{1,…,𝑛}, then 𝑙̃_𝑖𝑗 is reduced to a linguistic preference value 𝑙_𝑖𝑗.

3. Consistency Level and Consensus Level

3.1. Consistency Measure

Consistency is usually characterized by transitivity. Some transitive properties of linguistic preference relations can be described as follows [3, 5, 14]: (1) weak transitivity, (2) max–min transitivity, (3) max–max transitivity, (4) restricted max–min transitivity, (5) restricted max–max transitivity, (6) multiplicative consistency, and (7) additive consistency. Among these concepts, the most commonly used form is additive consistency [5, 6, 13, 15, 26–29]. Recently, Alonso et al. [26] introduced an additive transitivity property of the interval-valued preference relations and constructed a consistency measure; Gong et al. [27] also presented a similar additive transitivity property. Inspired by their ideas, we further redefine an additive transitivity of UALPRs to guarantee the indifference and reciprocity properties simultaneously. Then based on the definition, we propose a consistency measure to assess the consistency level of an UALPR.

Definition 3. An UALPR 𝐿=(𝑙̃_𝑖𝑗)_𝑛×𝑛 is called additive consistent uncertain linguistic preference relation, if it satisfies the following additive transitivity:𝑙̃^𝐿_𝑖𝑗=min𝑙̃^𝐿_𝑖𝑡⊕𝑙̃^𝑈_𝑡𝑗⊖𝑠_𝜏/2,𝑙̃^𝑈_𝑖𝑡⊕𝑙̃^𝐿_𝑡𝑗⊖𝑠_𝜏/2,𝑙̃^𝑈_𝑖𝑗=max𝑙̃^𝑈_𝑖𝑡⊕𝑙̃^𝐿_𝑡𝑗⊖𝑠_𝜏/2,𝑙̃^𝐿_𝑖𝑡⊕𝑙̃^𝑈_𝑡𝑗⊖𝑠_𝜏/2,∀𝑖,𝑗,𝑡∈{1,…,𝑛}. (5)
Obviously, the preference value 𝑙̃_𝑖𝑗 can be estimated by using an intermediate alternative 𝑥_𝑡: 𝑒𝑙̃^𝐿_𝑖𝑗=min𝑙̃^𝐿_𝑖𝑡⊕𝑙̃^𝑈_𝑡𝑗⊖𝑠_𝜏/2,𝑙̃^𝑈_𝑖𝑡⊕𝑙̃^𝐿_𝑡𝑗⊖𝑠_𝜏/2,𝑒𝑙̃^𝑈_𝑖𝑗=max𝑙̃^𝑈_𝑖𝑡⊕𝑙̃^𝐿_𝑡𝑗⊖𝑠_𝜏/2,𝑙̃^𝐿_𝑖𝑡⊕𝑙̃^𝑈_𝑡𝑗⊖𝑠_𝜏/2,∀𝑖,𝑗,𝑡∈{1,…,𝑛}. (6)
Indeed, if the 𝑙̃_𝑖𝑗 is completely consistent in an UALPR, then 𝑒𝑙̃^𝑡_𝑖𝑗=𝑙̃_𝑖𝑗,∀𝑡∈{1,…,𝑛}; that is, 𝑒𝑙̃^𝐿𝑡_𝑖𝑗=𝑙̃^𝐿_𝑖𝑗 and 𝑒𝑙̃^𝑈𝑡_𝑖𝑗=𝑙̃^𝑈_𝑖𝑗. However, it does not always hold due to the uncertainty and complexity of real-world decision problems and the subjectivity of DMs’ judgments. In such case, we can get a deviation.

Definition 4. Let 𝐿=(𝑙̃_𝑖𝑗)_𝑛×𝑛 be an UALPR; then Δ𝑙̃_𝑖𝑗=Δ𝑙̃^𝐿_𝑖𝑗,Δ𝑙̃^𝑈_𝑖𝑗=∑^𝑛_{𝑡=1,𝑡≠𝑖,𝑗}𝐼𝑒𝑙̃^𝐿𝑡_𝑖𝑗−𝐼𝑙̃^𝐿_𝑖𝑗,∑𝑛−2_{𝑛𝑡=1,𝑡≠𝑖,𝑗}𝐼𝑒𝑙̃^𝑈𝑡_𝑖𝑗−𝐼𝑙̃^𝑈_𝑖𝑗𝑛−2 (7) is called the deviation between 𝑙̃_𝑖𝑗 and its estimated ones 𝑒𝑙̃^𝑡_𝑖𝑗=𝑙̃_𝑖𝑗,1≤𝑡≤𝑛,𝑡≠𝑖,𝑗.
Clearly, when Δ𝑙̃_𝑖𝑗=(0,0), then there is no inconsistency at all between 𝑙̃_𝑖𝑗 and the rest of preference values in the UALPR, and the higher is Δ𝑙̃_𝑖𝑗, the more inconsistent is 𝑙̃_𝑖𝑗 with respect to the rest of information in the UALPR.
Using the deviation Δ𝑙̃_𝑖𝑗, we define a consistency measure to assess the consistent level of 𝑙̃_𝑖𝑗.

Definition 5. Let 𝐿=(𝑙̃_𝑖𝑗)_𝑛×𝑛 be an UALPR; then 𝑐𝑡̃_𝑖𝑗=𝜏−Δ𝑙̃_𝑖𝑗𝜏=𝑐𝑡̃^𝐿_𝑖𝑗,𝑐𝑡̃^𝑈_𝑖𝑗=𝜏−Δ𝑙̃^𝐿_𝑖𝑗𝜏,𝜏−Δ𝑙̃^𝑈_𝑖𝑗𝜏 (8) is called the consistency measure of 𝑙̃_𝑖𝑗.

This consistency measure has a definite physical implication and reflects the consistency level between the preference value 𝑙̃_𝑖𝑗 and the rest of the preference values in the UALPR. Clearly, when 𝑐𝑡̃_𝑖𝑗=(1,1), then there is no inconsistency at all between 𝑙̃_𝑖𝑗 and the other preference values, and the lower is the value of 𝑐𝑡̃_𝑖𝑗, the more inconsistent is 𝑙̃_𝑖𝑗 with respect to the rest of information.

Theorem 6. If 𝐿=(𝑙̃_𝑖𝑗)_𝑛×𝑛 is an UALPR, then 𝑐𝑡̃^𝐿_𝑖𝑗=𝑐𝑡̃^𝑈_𝑗𝑖,𝑐𝑡̃^𝑈_𝑖𝑗=𝑐𝑡̃^𝐿_𝑗𝑖,∀𝑖,𝑗∈{1,…,𝑛}. (9)

Proof. Since Δ𝑙̃^𝐿_𝑖𝑗=^𝑛_{𝑡=1,𝑡≠𝑖,𝑗}𝐼𝑒𝑙̃^𝐿𝑡_𝑖𝑗−𝐼𝑙̃^𝐿_𝑖𝑗×(𝑛−2)⁻¹=^𝑛_{𝑡=1,𝑡≠𝑖,𝑗}min𝐼𝑙̃^𝐿_𝑖𝑡+𝐼𝑙̃^𝑈_𝑡𝑗−𝐼𝑠_𝜏/2,𝑙̃^𝑈_𝑖𝑡+𝐼𝑙̃^𝐿_𝑡𝑗−𝐼𝑠_𝜏/2−𝐼𝑙̃^𝐿_𝑖𝑗×(𝑛−2)⁻¹=^𝑛_{𝑡=1,𝑡≠𝑖,𝑗}min𝜏−𝐼𝑙̃^𝑈_𝑡𝑖+𝜏−𝐼𝑙̃^𝐿_𝑗𝑡−𝜏2,𝜏−𝐼𝑙̃^𝐿_𝑡𝑖+𝜏−𝐼𝑙̃^𝑈_𝑗𝑡−𝑠_𝜏/2−𝜏+𝐼𝑙̃^𝑈_𝑗𝑖×(𝑛−2)⁻¹=^𝑛_{𝑡𝑘=1,𝑡≠𝑖,𝑗}min−𝐼𝑙̃^𝑈_𝑡𝑖−𝐼𝑙̃^𝐿_𝑗𝑡+𝜏2,−𝐼𝑙̃^𝐿_𝑡𝑖−𝐼𝑙̃^𝑈_𝑗𝑡+𝑠_𝜏/2+𝐼𝑙̃^𝑈_𝑗𝑖×(𝑛−2)⁻¹=^𝑛_{𝑡=1,𝑡≠𝑖,𝑗}(−1)max𝐼𝑙̃^𝑈_𝑡𝑖+𝐼𝑙̃^𝐿_𝑗𝑡−𝜏2,𝐼𝑙̃^𝐿_𝑡𝑖+𝐼𝑙̃^𝑈_𝑗𝑡−𝑠_𝜏/2−𝐼𝑙̃^𝑈_𝑗𝑖×(𝑛−2)⁻¹=^𝑛_{𝑡=1,𝑡≠𝑖,𝑗}(−1)𝐼max𝑙̃^𝑈_𝑡𝑖⊕𝑙̃^𝐿_𝑗𝑡⊖𝑠_𝜏/2,𝑙̃^𝐿_𝑡𝑖⊕𝑙̃^𝑈_𝑗𝑡⊖𝑠_𝜏/2⊖𝑙̃^𝑈_𝑗𝑖×(𝑛−2)⁻¹=Δ𝑙̃^𝑈_𝑗𝑖, (10) then 𝑐𝑡̃^𝐿_𝑖𝑗=𝑐𝑡̃^𝑈_𝑗𝑖. (11) Similarly, we can obtain 𝑐𝑡̃^𝑈_𝑖𝑗=𝑐𝑡̃^𝐿_𝑗𝑖. This completes the above proof.

In particular, if 𝑙̃_𝑖𝑗 reduced to 𝑙_𝑖𝑗, then the consistency level of 𝑙_𝑖𝑗 has property 𝑐𝑡_𝑖𝑗=𝑐𝑡_𝑗𝑖. Furthermore, we can obtain the consistency level of the whole UALPR by applying the arithmetic mean.

Definition 7. Let 𝐿=(𝑙̃_𝑖𝑗)_𝑛×𝑛 be an UALPR; then 𝐶𝑡=𝐶𝑡^𝐿,𝐶𝑡^𝑈=1𝑛²−𝑛^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑐𝑡̃^𝐿_𝑖𝑗,1𝑛²−𝑛^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑐𝑡̃^𝑈_𝑖𝑗 (12) is called the consistency measure of 𝐿.
If 𝐶𝑡=(1,1), then the UALPR 𝐿 is fully consistent; otherwise, the lower 𝐶𝑡̃ is the more inconsistent 𝐿 is.

Theorem 8. If 𝐿=(𝑙̃_𝑖𝑗)_𝑛×𝑛 is an UALPR, then 𝐶𝑡^𝐿=𝐶𝑡^𝑈. (13)

Proof. On has 𝐶𝑡^𝐿=1𝑛²−𝑛^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑐𝑡̃^𝐿_𝑖𝑗=1𝑛²−𝑛^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑐𝑡̃^𝑈_𝑗𝑖=1𝑛²−𝑛^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑐𝑡̃^𝑈_𝑖𝑗=𝐶𝑡^𝑈. (14)
By Theorem 8, we find that the consistency level of the whole UALPR is certain.

3.2. Consensus Measure

Using anonymous ways of the definitions of consistency measure, we define a consensus measure to assess the level of agreement between the individual preference value and the collective ones.

Definition 9. Let {𝐿^(𝑘)=(𝑙̃^(𝑘)_𝑖𝑗)_𝑛×𝑛,𝑘=1,2,…,𝑚} be a group of UALPRs provided by DMs; then a deviation of the uncertain linguistic preference value 𝑙̃^(𝑘)_𝑖𝑗 provided by 𝑑_𝑘 and the ones 𝑙̃^(𝑝)_𝑖𝑗∈𝐿^(𝑝) by 𝑑_𝑝(𝑝=1,2,…,𝑚,𝑝≠𝑘) is defined as Δ^′𝑙̃^(𝑘)_𝑖𝑗=Δ^′𝑙^(𝑘)𝐿_𝑖𝑗,Δ^′𝑙^(𝑘)𝑈_𝑖𝑗=∑^𝑚_{𝑝=1,𝑝≠𝑘}𝐼𝑙^(𝑘)𝐿_𝑖𝑗−𝐼𝑙^(𝑝)𝐿_𝑖𝑗,∑𝑚−1_{𝑚𝑝=1,𝑝≠𝑘}𝐼𝑙^(𝑘)𝑈_𝑖𝑗−𝐼𝑙^(𝑝)𝑈_𝑖𝑗.𝑚−1 (15)
Clearly, Δ^′𝑙̃^(𝑘)_𝑖𝑗=(0,0) indicates that the 𝑙̃_𝑖𝑗 provided by 𝑑_𝑘 and the ones provided by 𝑑_𝑝(𝑝=1,2,…,𝑚,𝑝≠𝑘) are not contradictory at all, and the higher the Δ^′𝑙̃^(𝑘)_𝑖𝑗, is the more decentralized the opinions between 𝑑_𝑘 and 𝑑_𝑝(𝑝=1,2,…,𝑚,𝑝≠𝑘) on the preference value 𝑙̃_𝑖𝑗 are. Based on the deviation Δ^′𝑙̃^(𝑘)_𝑖𝑗, we develop a consensus measure of 𝑙̃^(𝑘)_𝑖𝑗.

Definition 10. Let {𝐿^(𝑘)=(𝑙̃^(𝑘)_𝑖𝑗)_𝑛×𝑛,𝑘=1,2,…,𝑚} be a group of UALPRs provided by DMs; then the consensus measure of 𝑙̃^(𝑘)_𝑖𝑗 is defined as follows: 𝑐𝑎̃^(𝑘)_𝑖𝑗=𝑐𝑎̃^(𝑘)𝐿_𝑖𝑗,𝑐𝑎̃^(𝑘)𝑈_𝑖𝑗=∑1−^{𝑚𝑝=1,𝑝≠𝑘}𝐼𝑙^(𝑘)𝐿_𝑖𝑗−𝐼𝑙^(𝑝)𝐿_𝑖𝑗𝜏(),∑𝑚−11−^{𝑚𝑝=1,𝑝≠𝑘}𝐼𝑙^(𝑘)𝑈_𝑖𝑗−𝐼𝑙^(𝑝)𝑈_𝑖𝑗𝜏().𝑚−1 (16)
Clearly, when 𝑐𝑎̃^(𝑘)_𝑖𝑗=(1,1), then there are no decentralized opinions at all between 𝑑_𝑘 and 𝑑_𝑝(𝑝=1,2,…,𝑚,𝑝≠𝑘) on the preference value 𝑙̃_𝑖𝑗, and the lower the 𝑐𝑎̃^(𝑘)_𝑖𝑗 is the more decentralized the opinions between 𝑑_𝑘 and 𝑑_𝑝(𝑝=1,2,…,𝑚,𝑝≠𝑘) on the preference value 𝑙̃_𝑖𝑗 are. By applying the arithmetic mean, we have the consensus measures of the whole UALPRs.

Definition 11. Let {𝐿^(𝑘)=(𝑙̃^(𝑘)_𝑖𝑗)_𝑛×𝑛,𝑘=1,2,…,𝑚} be a group of UALPRs provided by DMs; then the consensus measures of the whole UALPRs are as follows:𝐶𝑎_𝑘=𝐶𝑎^𝐿_𝑘,𝐶𝑎^𝑈_𝑘=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1,𝑗≠𝑖}𝑐𝑎̃^(𝑘)𝐿_𝑖𝑗,1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1,𝑗≠𝑖}𝑐𝑎̃^(𝑘)𝑈_𝑖𝑗. (17)
Clearly, when 𝐶𝑎_𝑘=(1,1), then there are no decentralized opinions at all between 𝑑_𝑘 and 𝑑_𝑝(𝑝=1,2,…,𝑚,𝑝≠𝑘), and the lower the 𝐶𝑎_𝑘 is, the more decentralized the opinions of 𝑑_𝑘 and the others are.

Theorem 12. If {𝐿^(𝑘)=(𝑙̃^(𝑘)_𝑖𝑗)_𝑛×𝑛,𝑘=1,2,…,𝑚} is a group of UALPRs, then 𝑐𝑎̃^(𝑘)𝐿_𝑖𝑗=𝑐𝑎̃^(𝑘)𝑈_𝑗𝑖,𝑐𝑎̃^(𝑘)𝑈_𝑖𝑗=𝑐𝑎̃^(𝑘)𝐿_𝑗𝑖,∀𝑖,𝑗∈{1,…,𝑛}. (18)

Proof. On has 𝑐𝑎̃^(𝑘)𝐿_𝑖𝑗∑=1−^{𝑚𝑝=1,𝑝≠𝑘}𝐼𝑙^(𝑘)𝐿_𝑖𝑗−𝐼𝑙^(𝑝)𝐿_𝑖𝑗𝜏()∑𝑚−1=1−^{𝑚𝑝=1,𝑝≠𝑘}(−1)𝐼𝑙^(𝑘)𝑈_𝑗𝑖−𝐼𝑙^(𝑝)𝑈_𝑗𝑖𝜏()=̃𝑚−1𝑐𝑎^(𝑘)𝑈_𝑗𝑖. (19) which completes the proof of Theorem 12.

In particular, if UALPRs 𝑙̃_𝑖𝑗 reduced to 𝑙_𝑖𝑗, then the consensus level of 𝑙_𝑖𝑗 has property 𝑐𝑎_𝑖𝑗=𝑐𝑎_𝑗𝑖.

Theorem 13. If {𝐿^(𝑘)=(𝑙̃^(𝑘)_𝑖𝑗)_𝑛×𝑛,𝑘=1,2,…,𝑚} is a group of UALPRs, then 𝐶𝑎^𝐿_𝑘=𝐶𝑎^𝑈_𝑘. (20)

Proof. On has 𝐶𝑎^𝐿_𝑘=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1,𝑗≠𝑖}𝑐𝑎̃^(𝑘)𝐿_𝑖𝑗=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1,𝑗≠𝑖}𝑐𝑎̃^(𝑘)𝑈_𝑗𝑖=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1,𝑗≠𝑖}𝑐𝑎̃^(𝑘)𝑈_𝑖𝑗=𝐶𝑎^𝑈_𝑘. (21)
By Theorem 13, we find that the consensus level of the whole UALPR is also certain.

4. Aggregation of Uncertain Linguistic Preference Information Based on Consistency and Consensus

To obtain collective preference information by combining the individual ones, an aggregation operator is needed. The ordered weighted averaging (OWA) and weighted arithmetic averaging (WAA) are the most common aggregation operators. The induced OWA (IOWA) operator provided by Yager and Filev [30] is a more general type of the OWA operator, in which the ordering of the arguments is induced by the order inducing variables, rather than the values of the arguments. The IOWA operator can be defined as follows.

Definition 14. An IOWA operator of dimension 𝑛 is a mapping IOWA∶𝑅^𝑛→𝑅 can be defined as follows: IOWA_𝑤𝜇₁,𝑎₁,𝜇₂,𝑎₂,…,𝜇_𝑛,𝑎_𝑛=^𝑛_𝑗=1𝑤_𝑗𝑎_𝜎(𝑗), (22) where 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 is an associated weighting vector, such that 0≤𝑤_𝑗≤1, ∑^𝑛_𝑗=1𝑤_𝑗=1; 𝑎_𝜎(𝑗) is the 𝑎_𝑗 value of the OWA pair ⟨𝜇_𝑗,𝑎_𝑗⟩ having the 𝑗th largest 𝜇_𝑗; 𝜇_𝑗 in ⟨𝜇_𝑗,𝑎_𝑗⟩ is referred to as the order inducing variable and 𝑎_𝑗 as the argument variable.
Note that (22) can be equivalently written as IOWA_𝑤𝜇₁,𝑎₁,𝜇₂,𝑎₂,…,𝜇_𝑛,𝑎_𝑛=^𝑛_𝑗=1𝑤_𝜌(𝑗)𝑎_𝑗, (23) where 𝜌=𝜎⁻¹∶{1,2,…,𝑛}→{1,2,…,𝑛} is the inverse permutation of 𝜎 and 𝑎_𝑗 is the value of the OWA pair ⟨𝜇_𝑗,𝑎_𝑗⟩ having the 𝜌(𝑗)th largest 𝜇_𝑗. 𝜇_𝑗 in ⟨𝜇_𝑗,𝑎_𝑗⟩ is referred to as the order inducing variable and 𝑎_𝑗 as the argument variable.
Based on IOWA operator, Xu [31] presented an induced uncertain linguistic OWA operator to aggregate uncertain linguistic variables.

Definition 15. An induced uncertain linguistic OWA (IULOWA) operator is defined as follows: IULOWA_𝑤𝜇₁,𝑠₁̃,𝜇₂,𝑠₂̃,…,𝜇_𝑛,𝑠_𝑛̃=^𝑛_𝑗=1𝑤_𝑗𝑠̃_𝛽𝑗=^𝑛_𝑗=1𝑤_𝑗𝑠̃^𝐿_𝛽𝑗,^𝑛_𝑗=1𝑤_𝑗𝑠̃^𝑈_𝛽𝑗, (24) where 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) is an associated weighting vector, such that 0≤𝑤_𝑗≤1, ∑^𝑛_𝑗=1𝑤_𝑗=1; 𝑠̃_𝛽𝑗 is the 𝑠_𝑗̃ value of the OWA pair ⟨𝜇_𝑗,𝑠_𝑗̃⟩ having the𝑗th largest 𝜇_𝑗⋅𝜇_𝑗 in ⟨𝜇_𝑗,𝑠_𝑗̃⟩ is referred to as the order inducing variable and 𝑠_𝑗̃ as the argument variable.
The key issues in the IOWA operator theory are to determine the order inducing variables and then derive the associated weights based on them, but quite few studies have been conducted on the issues. Recently, Chiclana et al. [32] introduced a consistency-induced OWA (C-IOWA) operator, in which the ordering of the preference value is based upon its consistency, and it is further studied in [6, 7, 26, 28, 29]. Wu et al. [17] presented a compatibility index ILOWG (CI-ILOWG) operator, which induces the order of argument values by the compatibility index of experts they also verified that the CI-ILOWG operator guarantees the compatibility degree is at least as good as the arithmetic mean of all the individual compatibility degrees. As aforementioned, both consensus and consistency should be considered simultaneously in any rational GDM. To more effectively relieve the influence some individuals may assign lower consistent and lower centralized preference values to their preferred or repugnant objects, it is very reasonable and meaningful that by assigning low weights to those “inconsistent” or “decentralized” opinions in preference aggregation procedure. Based on the discussions, we here propose a consistency and consensus co-induced uncertain linguistic ordered weighted averaging (C²-IULOWA) operator to aggregate individual preference information in such a way that the greater weight is given the most consistent and consensual preference information.

Definition 16. Let {𝐿^(𝑘)=(𝑙̃^(𝑘)_𝑖𝑗)_𝑛×𝑛,𝑘=1,2,…,𝑚} be a group of UALPRs to aggregate, where 𝐿^(𝑘)=(𝑙̃^(𝑘)_𝑖𝑗)=([𝑙̃^(𝑘)𝐿_𝑖𝑗,𝑙̃^(𝑘)𝑈_𝑖𝑗]) and {𝐶𝑡₁,𝐶𝑡₂,…,𝐶𝑡_𝑚} and {𝐶𝑎₁,𝐶𝑎₂,…,𝐶𝑎_𝑚} are their corresponding consistency level and consensus level, respectively; then a consensus and consistency co-induced uncertain linguistic ordered weighted averaging (C²-IULOWA) operator is defined as IULOWA_𝐶²_↦𝑤𝐶𝑎₁,𝐶𝑡₁,𝐿⁽¹⁾,𝐶𝑎₂,𝐶𝑡₂,𝐿⁽²⁾,…,𝐶𝑎_𝑚,𝐶𝑡_𝑚,𝐿^(𝑚)=⨁^𝑚_𝑘=1𝑤_𝜉(𝑘)𝐿^(𝑘)𝐿𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘),⨁^𝑚_𝑗=1𝑤_𝜉(𝑘)𝐿^(𝑘)𝑈𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘)=⨁^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝐿_𝑖𝑗𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘),⨁^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝑈_𝑖𝑗𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_{𝜁(𝑘)𝑛×𝑛}, (25) where 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑚)^𝑇 is an associated weighting vector, such that 0≤𝑤_𝑗≤1, ∑^𝑚_𝑗=1𝑤_𝑗=1 and the consensus level 𝐶𝑎_𝑘 and the consistency level 𝐶𝑡_𝑘 are simultaneously referred to as the order inducing variables and 𝐿^(𝑘)=[𝑙̃^(𝑘)𝐿_𝑖𝑗,𝑙̃^(𝑘)𝑈_𝑖𝑗] as the argument variable. 𝜉(⋅) and 𝜁(⋅) are permutation functions such that 𝐿^(𝑘) is the preference relation which has the 𝜉(𝑘)th and 𝜁(𝑘)th largest 𝐶𝑎_𝑘 and 𝐶𝑡_𝑘. In particular, when 𝑙̃^𝐿_𝑖𝑗=𝑙̃^𝑈_𝑖𝑗, the C²-IULOWA operator reduced to a consensus and consistency co-induced linguistic ordered weighted averaging (C²-ILOWA) operator: ILOWA_𝐶²_↦𝑤𝐶𝑎_𝑘,𝐶𝑡_𝑘,𝐿^(𝑘)=⨁,𝑘=1,…,𝑚_𝑚𝑘=1𝑤_𝜉(𝑘)𝐿^(𝑘)𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘)=⨁^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙^(𝑘)_𝑖𝑗𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_{𝜁(𝑘)𝑛×𝑛.} (26)

Theorem 17. If a group of UALPRs {𝐿^(𝑘)=(𝑙̃^(𝑘)_𝑖𝑗)_𝑛×𝑛,𝑘=1,2,…,𝑚} to aggregate has identical consistency level, that is, 𝑐𝑡̃₁=𝑐𝑡̃₂=⋯=𝑐𝑡̃_𝑚, then the C²-IULOWA operator becomes a consensus-induced uncertain linguistic ordered weighted averaging (Ca-IULOWA) operator: 𝐼𝑈𝐿𝑂𝑊𝐴_{𝐶𝑎↦𝑤}𝐶𝑎₁,𝐿⁽¹⁾,𝐶𝑎₂,𝐿⁽²⁾,…,𝐶𝑎_𝑚,𝐿^(𝑚)=^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝐿_𝑖𝑗,^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝑈_𝑖𝑗. (27)

Proof. On has IULOWA_𝐶²_↦𝑤𝐶𝑎₁,𝐶𝑡₁,𝐿⁽¹⁾,𝐶𝑎₂,𝐶𝑡₂,𝐿⁽²⁾,…,𝐶𝑎_𝑚,𝐶𝑡_𝑚,𝐿^(𝑚)=⨁^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝐿_𝑖𝑗𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘),⨁^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝑈_𝑖𝑗𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘)=⨁^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝐿_𝑖𝑗(1/𝑛)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)(),⨁1/𝑛^𝑚𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝑈_𝑖𝑗(1/𝑛)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)()=1/𝑛_𝑚𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝐿_𝑖𝑗,^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝑈_𝑖𝑗=𝐼𝑈𝐿𝑂𝑊𝐴_{𝐶𝑎↦𝑤}𝐶𝑎₁,𝐿⁽¹⁾,𝐶𝑎₂,𝐿⁽²⁾,…,𝐶𝑎_𝑚,𝐿^(𝑚). (28)

Theorem 18. If a group of UALPRs {𝐿^(𝑘)=(𝑙̃^(𝑘)_𝑖𝑗)_𝑛×𝑛,𝑘=1,2,…,𝑚} to aggregate has identical consensus level, that is, 𝑐𝑎̃₁=𝑐𝑎̃₂=⋯=𝑐𝑎̃_𝑚, then the C²-IULOWA operator becomes a consistency-induced uncertain linguistic ordered weighted averaging (Ct-IULOWA) operator: IULOWA_{𝐶𝑡↦𝑤}𝐶𝑡₁,𝐿⁽¹⁾,𝐶𝑡₂,𝐿⁽²⁾,…,𝐶𝑡_𝑚,𝐿^(𝑚)=^𝑚_𝑘=1𝑤_𝜁(𝑘)𝑙̃^(𝑘)𝐿_𝑖𝑗,^𝑚_𝑘=1𝑤_𝜁(𝑘)𝑙̃^(𝑘)𝑈_𝑖𝑗. (29)

Proof. On has IULOWA_𝐶²_↦𝑤𝐶𝑎₁,𝐶𝑡₁,𝐿⁽¹⁾,𝐶𝑎₂,𝐶𝑡₂,𝐿⁽²⁾,…,𝐶𝑎_𝑚,𝐶𝑡_𝑚,𝐿^(𝑚)=⨁^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝐿_𝑖𝑗𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘),⨁^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝑈_𝑖𝑗𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘)=⨁^𝑚_𝑘=1(1/𝑛)𝑙̃^(𝑘)𝐿_𝑖𝑗𝑤_𝜁(𝑘)∑^𝑚_𝑘=1(1/𝑛)𝑤_𝜁(𝑘),⨁^𝑚_𝑘=1(1/𝑛)𝑙̃^(𝑘)𝑈_𝑖𝑗𝑤_𝜁(𝑘)∑^𝑚_𝑘=1(1/𝑛)𝑤_𝜁(𝑘)=^𝑚_𝑘=1𝑤_𝜁(𝑘)𝑙̃^(𝑘)𝐿_𝑖𝑗,^𝑚_𝑘=1𝑤_𝜁(𝑘)𝑙̃^(𝑘)𝑈_𝑖𝑗=IULOWA_{𝐶𝑡↦𝑤}𝐶𝑡₁,𝐿⁽¹⁾,𝐶𝑡₂,𝐿⁽²⁾,…,𝐶𝑡_𝑚,𝐿^(𝑚). (30)
From Definition 16 and the previous theorems, we know the following.(1)The inducing variables of the C²-IULOWA operator are characterized by the consistency measure and the consensus measure, which synergistically reflect the consensus level and the consistency level of UALPRs and participate to guide the determination of associated weights.(2)The C²-IULOWA operator generalizes the consensus-induced uncertain linguistic OWA operators and consistency-induced uncertain linguistic OWA operators, which comprehensively merge the information associated with the properties of preference relation in the preference aggregation.
Moreover, it is easy to verify that the C²-IULOWA operator is monotonic, commutative, bounded, and idempotent; the proofs of them are omitted here because they are trivial. In the following, we study the rationality of the C²-IULOWA operator that means the aggregated uncertain linguistic preference relations maintain the essential properties of preference relations such as the indifference, the reciprocity, and the consistency properties.
Since the C²-IULOWA operator is idempotent, the indifference property is obvious and its proofs are omitted here. Now we verify the reciprocity property.
Since 𝐿^(𝑘)=([𝑙̃^(𝑘)𝐿_𝑖𝑗,𝑙̃^(𝑘)𝑈_𝑖𝑗])_𝑛×𝑛,𝑘=1,…,𝑚, are UALPRs, we have 𝑙̃^(𝑘)𝐿_𝑖𝑗⊕𝑙̃^(𝑘)𝑈_𝑗𝑖=𝑠_𝜏, 𝑙̃^(𝑘)𝑈_𝑖𝑗⊕𝑙̃^(𝑘)𝐿_𝑗𝑖=𝑠_𝜏, 𝑙̃^(𝑘)𝑈_𝑖𝑖=𝑙̃^(𝑘)𝐿_𝑖𝑖=𝑠_𝜏/2.
Thus 𝑙̃^(𝐶)𝐿_𝑖𝑗⊕𝑙̃^(𝐶)𝑈_𝑗𝑖=⨁^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝐿_𝑖𝑗𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘)⊕⨁^𝑛_𝑗=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝑈_𝑗𝑖𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘)=⨁^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝐿_𝑖𝑗⊕𝑙̃^(𝑘)𝑈_𝑗𝑖𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘)=⨁^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑠_𝜏𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘)=𝑠_𝜏. (31)
Similarly, we can verify 𝑙̃^(𝐶)𝑈_𝑖𝑗⊕𝑙̃^(𝐶)𝐿_𝑗𝑖=𝑠_𝜏.
Then we verify the consistency property.
Since 𝐿^(𝑘)=([𝑙̃^(𝑘)𝐿_𝑖𝑗,𝑙̃^(𝑘)𝑈_𝑖𝑗])_𝑛×𝑛,𝑘=1,…,𝑚, are consistent, we have 𝑙̃^(𝑘)𝐿_𝑖𝑗=min𝑙̃^(𝑘)𝐿_𝑖𝑡⊕𝑙̃^(𝑘)𝑈_𝑡𝑗⊖𝑠_𝜏/2,𝑙̃^(𝑘)𝑈_𝑖𝑡⊕𝑙̃^(𝑘)𝐿_𝑡𝑗⊖𝑠_𝜏/2,𝑙̃^(𝑘)𝑈_𝑖𝑗=max𝑙̃^(𝑘)𝑈_𝑖𝑡⊕𝑙̃^(𝑘)𝐿_𝑡𝑗⊖𝑠_𝜏/2,𝑙̃^(𝑘)𝐿_𝑖𝑡⊕𝑙̃^(𝑘)𝑈_𝑡𝑗⊖𝑠_𝜏/2,∀𝑖,𝑗,𝑡∈{1,…,𝑛},𝑘=1,…,𝑚, (32) then we have 𝑙̃^(𝐶)𝐿_𝑖𝑗=⨁^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝐿_𝑖𝑗𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘)=^𝑚_𝑘=1𝑤_𝜉(𝑘)min𝑙̃^(𝑘)𝐿_𝑖𝑡⊕𝑙̃^(𝑘)𝑈_𝑡𝑗⊖𝑠_𝜏/2,𝑙̃^(𝑘)𝑈_𝑖𝑡⊕𝑙̃^(𝑘)𝐿_𝑡𝑗⊖𝑠_𝜏/2𝑤_𝜁(𝑘)×^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘)⁻¹⨁=min^𝑚𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝐿_𝑖𝑡⊕𝑙̃^(𝑘)𝑈_𝑡𝑗⊖𝑠_𝜏/2𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘),⨁^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝑈_𝑖𝑡⊕𝑙̃^(𝑘)𝐿_𝑡𝑗⊖𝑠_𝜏/2𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘)⨁=min^𝑚𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝐿_𝑖𝑡𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘)⊕⨁^𝑛_𝑗=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝑈_𝑡𝑗𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘)⊖𝑠_𝜏/2,⨁^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝑈_𝑖𝑡𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘)⊕⨁^𝑛_𝑗=1𝑤_𝜉(𝑘)𝑙̃^(𝑘)𝐿_𝑡𝑗𝑤_𝜁(𝑘)∑^𝑚_𝑘=1𝑤_𝜉(𝑘)𝑤_𝜁(𝑘)⊖𝑠_𝜏/2=min𝑙̃^(𝐶)𝐿_𝑖𝑡⊕𝑙̃^(𝐶)𝑈_𝑡𝑗⊖𝑠_𝜏/2,𝑙̃^(𝐶)𝑈_𝑖𝑡⊕𝑙̃^(𝐶)𝐿_𝑡𝑗⊖𝑠_𝜏/2. (33) Similarly, we can verify 𝑙̃^(𝐶)𝑈_𝑖𝑗=max(𝑙̃^(𝐶)𝐿_𝑖𝑡⊕𝑙̃^(𝐶)𝑈_𝑡𝑗⊖𝑠_𝜏/2,𝑙̃^(𝐶)𝑈_𝑖𝑡⊕𝑙̃^(𝐶)𝐿_𝑡𝑗⊖𝑠_𝜏/2).
Hence, 𝐿^(𝐶) is also consistent, which verifies the consistency property.
An important step in the process of aggregating preference relations is to determine the associated weight of each preference relation or preference value. Usually, the associated weights are determined by a linguistic quantifier 𝑄 as [32] 𝑤_𝑗𝑗=𝑄𝑛−𝑄𝑗−1𝑛,𝑗=1,…,𝑛, (34) with 𝑄(𝑟)=𝑟^1/2 meaning the quantifier guiding this aggregation to be “most” [4, 6, 32]. It can easily be shown that using this the 𝑤_𝑗 satisfy the conditions 𝑤_𝑗∈[0,1] and ∑_𝑗𝑤_𝑗=1.
A further important aspect of induced type aggregation operator is the fact that ties in the order inducing variables, that is, when two or more inducing variables have exactly the same value. In order to solve this problem, we recommend adopting Yager’s policy [33], is, that using an average of the components of weights over the tied constituents and leaving the arguments alone.

5. Group Decision Making Based on C²-IULOWA Operator

In the procedure of GDM, preference relations are usually composed by two phases [4, 6–10]: the aggregation phase and the exploitation phase, and two distinct choice processes are known: one is the direct way, which draws decision conclusion directly from individual preference relations and aggregates those individual choices, and the other is indirect way, which first aggregates individual linguistic preference relations into a collective preference relation and then draws decision conclusion; they can be described as follows: 𝐿^(𝑘),𝑘=1,…,𝑚−−−−−−−−−→^exploitationsolution^(𝑘),𝑘=1,…,𝑚_aggregation−−−−−−−−−→solution^(𝐶)𝐿^(𝑘),𝑘=1,…,𝑚−−−−−−−−−→^{aggregation(𝐶)}_exploitation𝐿−−−−−−−−−→solution^(𝐶). (35)

Herrera et al. [10] and Delgado et al. [11] developed a direct GDM approach with linguistic preference relation, which ranks or selects the alternative(s) according to linguistic nondominance degrees. The approach can be briefly described as follows.(1)For each linguistic preference relation of each expert, 𝐿^(𝑘), find its respective linguistic strict preference relation, ^𝑠𝐿^(𝑘)=(^𝑠𝑙^(𝑘)_𝑖𝑗)_𝑛×𝑛, such that ^𝑠𝑙^(𝑘)_𝑖𝑗=𝑠₀,if𝑙^(𝑘)_𝑖𝑗<𝑙^(𝑘)_𝑗𝑖𝑠_ℎ∈𝑆,if𝑙^(𝑘)_𝑖𝑗≥𝑙^(𝑘)_𝑗𝑖,with𝑙^(𝑘)_𝑖𝑗=𝑠_𝑙,𝑙^(𝑘)_𝑗𝑖=𝑠_𝑡,𝑙=𝑡+ℎ. (36)(2)For each linguistic strict preference relation of each expert, ^𝑠𝐿^(𝑘), determine the individual linguistic non-dominance degree of each alternative 𝑥_𝑖, called IND_𝑖^(𝑘), according to linguistic OWA operator: IND_𝑖^(𝑘)=LOWAneg^𝑠𝑙^(𝑘)_𝑗𝑖,𝑗=1,…,𝑛,𝑗≠𝑖,𝑖=1,…,𝑛,𝑘=1,…,𝑚. (37)(3)For each alternative 𝑥_𝑖, calculate the social linguistic non-dominance degree, called SND_𝑖, as follows: SND_𝑖=LOWAIND_𝑖^(𝑘),𝑙=1,…,𝑚,𝑖=1,…,𝑛. (38)(4)Rank or select the alternatives with respect to SND_𝑖(𝑖=1,2,…,𝑛).

Here, we extend those definitions to rank or select the alternative(s) for UALPRs. To do so, we give the following definitions.

Definition 19. Let 𝐿=(𝑙̃_𝑖𝑗)_𝑛×𝑛 be an UALPR; then the uncertain linguistic dominance relation in terms of 𝐿 is defined as 𝐿𝐷_𝑖𝑗=max𝑙̃^𝐿_𝑖𝑗⊖𝑙̃^𝑈_𝑗𝑖,𝑠₀,max𝑙̃^𝑈_𝑖𝑗⊖𝑙̃^𝐿_𝑗𝑖,𝑠₀. (39)
The linguistic dominance relation is used to measure the “degree of preference” of 𝑥_𝑖 over 𝑥_𝑗  that exceeds the “degree of preference” of 𝑥_𝑗 over 𝑥_𝑖.

Definition 20. Let 𝐿=(𝑙̃_𝑖𝑗)_𝑛×𝑛 be an UALPR; then the uncertain linguistic non-dominance relations are defined as: 𝐿𝑁𝐷_𝑖𝑗=Neg𝐿𝐷_𝑗𝑖=max𝑙̃^𝑈_𝑗𝑖⊖𝑙̃^𝐿_𝑖𝑗,𝑠₀,Negmax𝑙̃^𝐿_𝑗𝑖⊖𝑙̃^𝑈_𝑖𝑗,𝑠₀. (40)
Since each uncertain linguistic preference value associates with its consistency level and consensus level, at the same time, from Theorem 6, 𝑐𝑡^𝐿_𝑖𝑗=𝑐𝑡^𝑈_𝑗𝑖,𝑐𝑡^𝐿_𝑗𝑖=𝑐𝑡^𝑈_𝑖𝑗, and Theorem 12, 𝑐𝑎^𝐿_𝑖𝑗=𝑐𝑎^𝑈_𝑗𝑖,𝑐𝑎^𝐿_𝑗𝑖=𝑐𝑎^𝑈_𝑖𝑗, it is obvious that the uncertain linguistic non-dominance relations have still preserved consistent consistency level and consensus level of its compositions 𝑙̃_𝑖𝑗 and 𝑙̃_𝑗𝑖. Hence we can utilize the C²-IULOWA operator to obtain individual uncertain linguistic non-dominance degrees of alternatives.
For simplicity of calculation, the uncertain linguistic non-dominance relations are split into two crisp relations, 𝐿𝑁𝐷^𝐿_𝑖𝑗=[Neg(max(𝑙̃^𝑈_𝑗𝑖⊖𝑙̃^𝐿_𝑖𝑗,𝑠₀))]_𝑛×𝑛and 𝐿𝑁𝐷^𝑈_𝑖𝑗=[Neg(max(𝑙̃^𝐿_𝑗𝑖⊖𝑙̃^𝑈_𝑖𝑗,𝑠₀))]_𝑛×𝑛; similar ideas can also be found in many existing references [26, 27, 34, 35]. Then motivated by [36, 37], we give alternative uncertain linguistic non-dominance degrees of alternatives by utilizing C²-IUOWA operator: 𝐿𝑁𝐷_𝑖^(𝑘)=IULOWA_𝐶²_↦𝑤𝑐𝑎̃^(𝑘)_𝑖𝑗,𝑐𝑡̃^(𝑘)_𝑖𝑗,Neg𝐿𝐷^(𝑘)_𝑗𝑖,𝑗=1,…,𝑛,𝑗≠𝑖,𝑘=1,2,…,𝑚,𝑖=1,2,…,𝑛. (41)
Finally, we obtain collective uncertain linguistic non-dominance degrees of alternatives by utilizing the C²-IULOWA operator again: 𝐿𝑁𝐷_𝑖^(𝐶)=IULOWA_𝐶²_↦𝑤𝐶𝑎^(𝑘),𝐶𝑡^(𝑘),𝐿𝑁𝐷_𝑖^(𝑘),𝑘=1,…,𝑚,𝑖=1,2,…,𝑛. (42)
To rank the uncertain linguistic non-dominance degrees, we adopt to the degree of possibility of uncertain linguistic variables 𝑎_𝑖≥𝑎_𝑗 [18–20].

Definition 21. Let 𝐿𝑁𝐷_𝑖^(𝐶)=[𝐿𝑁𝐷_𝑖^𝐿(𝐶),𝐿𝑁𝐷_𝑖^𝑈(𝐶)]⊆𝑆,𝑖=1,2,…,𝑛 be 𝑛, uncertain linguistic values; then the degree of possibility 𝑃𝑜𝑠(𝐿𝑁𝐷_𝑖^(𝐶))  of  𝐿𝑁𝐷_𝑖^(𝐶) is defined as 𝑃𝑜𝑠_𝑖=𝑃𝑜𝑠𝐿𝑁𝐷_𝑖^(𝐶)=^𝑛_𝑗=1𝑃𝑜𝑠𝐿𝑁𝐷_𝑖^(𝐶)≥𝐿𝑁𝐷_𝑗^(𝐶)=^𝑛_𝑗=1max𝐼𝐿𝑁𝐷_𝑖^(𝐶)𝑈⊖𝐿𝑁𝐷_𝑗^(𝐶)𝐿,𝑠₀−max𝐼𝐿𝑁𝐷_𝑗^(𝐶)𝑈⊖𝐿𝑁𝐷_𝑖^(𝐶)𝐿,𝑠₀×𝐼𝐿𝑁𝐷_𝑖^(𝐶)𝑈⊖𝐿𝑁𝐷_𝑖^(𝐶)𝐿+𝐼𝐿𝑁𝐷_𝑗^(𝐶)𝑈⊖𝐿𝑁𝐷_𝑗^{(𝐶)𝐿−1},𝑖=1,…,𝑛. (43)
Now we can rank or select alternatives in descending order in accordance with 𝑃𝑜𝑠_𝑖, 𝑖∈{1,…,𝑛}.
The main steps of the proposed approach are summarized as follows.

Step 1. Obtain UALPRs over alternatives from DMs, 𝐿^(𝑘)=(𝑙̃_𝑖𝑗^(𝑘))_𝑛×𝑛,𝑘=1,…,𝑚.

Step 2. Calculate the consistency levels of preference values and preference relations 𝑐𝑡̃_𝑖𝑗^(𝑘),𝑖,𝑗=1,2,…,𝑛, 𝐶𝑡_𝑘,𝑘=1,2,…,𝑚, by using (8) and (12).

Step 3. Calculate the consensus levels of preference values and preference relations 𝑐𝑎_𝑖𝑗^(𝑘),𝑖,𝑗=1,2,…,𝑛,𝐶𝑎_𝑘,𝑘=1,2,…,𝑚, by using (16) and (17).

Step 4. Determine the ordered weights by using (34).

Step 5. Calculate the uncertain linguistic non-dominance relations according to the uncertain linguistic preference relations (𝐿𝑁𝐷^(𝑘)_𝑖𝑗)_𝑛×𝑛,𝑘=1,2,…,𝑚, by using (40).

Step 6. Determine the individual uncertain linguistic non-dominance degrees of alternatives 𝐿𝑁𝐷_𝑖^(𝑘),𝑘=1,2,…,𝑚,𝑖=1,2,…,𝑛, by using (41).

Step 7. Calculate the collective uncertain linguistic non-dominance degree of alternatives 𝐿𝑁𝐷_𝑖^(𝐶),𝑖=1,2,…,𝑛, by using (42).

Step 8. Rank the alternatives with 𝐿𝑁𝐷_𝑖^(𝐶),𝑖=1,2,…,𝑛, by using (43).

6. Numerical Examples

In this section, we consider a group decision making problem that concerns the evaluation and selection of chief quality officer to demonstrate the details of the proposed method.

Over the last decades, organizational competition is shifting from price to quality in many industries. The chief quality officer (CQO), similar to the CEO, undertakes overall responsibility of the quality work of enterprise. Hence, the selection of the most appropriate CQO is naturally one of the key factors for an enterprise’s survival. A high-tech enterprise wants to introduce a CQO according to a main criterion of the leadership of quality improvements. After preliminary screening, four alternatives 𝑥_𝑗(𝑗=1,2,3,4) have remained in the candidate list. Three experts 𝑒_𝑘(𝑘=1,2,3) from a committee act as decision makers; the decision makers compare these four alternatives with respect to the criterion of the leadership of quality improvements by using the linguistic term set 𝑆=𝑠₀=extremelylow,𝑠₁𝑠=verylow,₂=low,𝑠₃=slightlylow,𝑠₄𝑠=fair,₅=slightlyhigh,𝑠₆𝑠=high,₇=veryhigh,𝑠₈=extremelyhigh. (44)

To get the most desirable CQO, the following steps are involved.

Step 1. Three decision makers (𝑒₁,𝑒₂, and 𝑒₃) provide their preference on alternatives and form UALPRs as follows, respectively, 𝐴¹=𝑠₄,𝑠₄𝑠₀,𝑠₂𝑠₂,𝑠₄𝑠₁,𝑠₃𝑠₆,𝑠₈𝑠₄,𝑠₄𝑠₄,𝑠₆𝑠₄,𝑠₅𝑠₄,𝑠₆𝑠₂,𝑠₄𝑠₄,𝑠₄𝑠₃,𝑠₄𝑠₅,𝑠₇𝑠₃,𝑠₄𝑠₄,𝑠₅𝑠₄,𝑠₄,𝐴²=𝑠₄,𝑠₄𝑠₄,𝑠₆𝑠₀,𝑠₂𝑠₅,𝑠₇𝑠₂,𝑠₄𝑠₄,𝑠₄𝑠₄,𝑠₅𝑠₂,𝑠₄𝑠₆,𝑠₈𝑠₃,𝑠₄𝑠₄,𝑠₄𝑠₄,𝑠₅𝑠₁,𝑠₃𝑠₄,𝑠₆𝑠₃,𝑠₄𝑠₄,𝑠₄,𝐴³=𝑠₄,𝑠₄𝑠₀,𝑠₁𝑠₄,𝑠₅𝑠₅,𝑠₆𝑠₇,𝑠₈𝑠₄,𝑠₄𝑠₆,𝑠₇𝑠₂,𝑠₃𝑠₃,𝑠₄𝑠₁,𝑠₂𝑠₄,𝑠₄𝑠₇,𝑠₈𝑠₂,𝑠₃𝑠₅,𝑠₆𝑠₀,𝑠₁𝑠₄,𝑠₄. (45)

Step 2. Calculate the consistency levels of preference values by using (8); the consistency levels of preference values provided by three decision makers are listed in Tables 1, 2, and 3, respectively.
According to the results from Tables 1, 2, and 3, calculate the consistency level of preference relations by utilizing (12); the results are as follows: 𝐶𝑡₁=0.927,𝐶𝑡₂=0.594,𝐶𝑡₃=0.438. (46)

Table 1 

Consistency levels of preference values provided by 𝑒₁.

Table 2 

Consistency levels of preference values provided by 𝑒₂.

Table 3 

Consistency levels of preference values provided by 𝑒₃.

Step 3. Calculate the consensus measures of preference values by utilizing (16). The consensus levels of preference values provided by three experts are listed in Tables 4, 5, and 6, respectively.
Then calculate the consistency levels of preference relations by (17); the results are as follows: 𝐶𝑎₁=0.729,𝐶𝑎₂=0.719,𝐶𝑎₃=0.719. (47)

Table 4 

Consensus levels of preference values provided by 𝑒₁.

Table 5 

Consensus levels of preference values provided by 𝑒₂.

Table 6 

Consensus levels of preference values provided by 𝑒₃.

Step 4. Determine associated weights of decision makers and preference values. By using (34) determine the associated weights of three variables: 𝑤₁1=𝑄30−𝑄3𝑤=0.577,₂2=𝑄31−𝑄3𝑤=0.239,₃3=𝑄32−𝑄3=0.184. (48)

Step 5. Calculate the individual uncertain linguistic non-dominance relations by utilizing (40): 𝐿𝑁𝐷⁽¹⁾_𝑖𝑗=[−,−]𝑠₀,𝑠₄𝑠₄,𝑠₈𝑠₂,𝑠₆𝑠₈,𝑠₈[−,−]𝑠₈,𝑠₈𝑠₈,𝑠₈𝑠₈,𝑠₈𝑠₄,𝑠₈[−,−]𝑠₆,𝑠₈𝑠₈,𝑠₈𝑠₆,𝑠₈𝑠₈,𝑠₈],[−,−𝐿𝑁𝐷⁽²⁾_𝑖𝑗=[−,−]𝑠₈,𝑠₈𝑠₀,𝑠₄𝑠₈,𝑠₈𝑠₄,𝑠₈[−,−]𝑠₈,𝑠₈𝑠₄,𝑠₈𝑠₈,𝑠₈𝑠₆,𝑠₈[−,−]𝑠₈,𝑠₈𝑠₂,𝑠₆𝑠₈,𝑠₈𝑠₆,𝑠₈],[−,−𝐿𝑁𝐷⁽³⁾_𝑖𝑗=[−,−]𝑠₀,𝑠₂𝑠₈,𝑠₈𝑠₈,𝑠₈𝑠₈,𝑠₈[−,−]𝑠₈,𝑠₈𝑠₄,𝑠₆𝑠₆,𝑠₈𝑠₂,𝑠₄[−,−]𝑠₈,𝑠₈𝑠₄,𝑠₆𝑠₈,𝑠₈𝑠₀,𝑠₂].[−,− (49)

Step 6. Determine the individual uncertain linguistic non-dominance degrees of alternatives by (41). For instance, the individual non-dominance degree of 𝑥₁ provided by 𝑒₁ is calculated as follows: 𝐿𝑁𝐷₁⁽¹⁾=0.184×0.408𝑠₀⊕0.577×0.408𝑠₄⊕0.239×0.184𝑠₂×()0.184×0.408+0.577×0.408+0.239×0.184⁻¹,0.577×0.239𝑠₄⊕0.184×0.577𝑠₈⊕0.239×0.184𝑠₆×()0.577×0.239+0.184×0.577+0.239×0.184⁻¹=𝑠_2.90,𝑠_5.78. (50)
Similarly, we can obtain the other individual non-dominance degrees: 𝐿𝑁𝐷₂⁽¹⁾=𝑠₈,𝑠₈,𝐿𝑁𝐷₃⁽¹⁾=𝑠_5.35,𝑠₈,𝐿𝑁𝐷₄⁽¹⁾=𝑠_6.42,𝑠₈,𝐿𝑁𝐷₁⁽²⁾=𝑠_6.92,𝑠_7.16,𝐿𝑁𝐷₂⁽²⁾=𝑠_4.96,𝑠₈,𝐿𝑁𝐷₃⁽²⁾=𝑠_6.72,𝑠₈,𝐿𝑁𝐷₄⁽²⁾=𝑠_7.16,𝑠_7.52,𝐿𝑁𝐷₁⁽³⁾=𝑠_4.74,𝑠_5.71,𝐿𝑁𝐷₂⁽³⁾=𝑠_6.42,𝑠_6.81,𝐿𝑁𝐷₃⁽³⁾=𝑠_4.06,𝑠_5.43,𝐿𝑁𝐷₄⁽³⁾=𝑠_5.68,𝑠_6.82. (51)

Step 7. Aggregate the individual non-dominance degrees of alternatives into collective non-dominance degrees of alternatives by utilizing (42): 𝐿𝑁𝐷₁=𝑠_3.52,𝑠_5.94,𝐿𝑁𝐷₂=𝑠_7.49,𝑠_7.89,𝐿𝑁𝐷₃=𝑠_5.39,𝑠_7.76,𝐿𝑁𝐷₄=𝑠_6.44,𝑠_7.83. (52)

Step 8. Obtain the ranking result of the alternatives according to the magnitude of collective non-dominance degrees by utilizing (43). The ranking result of the four alternatives is 𝑥₂≻𝑥₄≻𝑥₃≻𝑥₁, and then the most appropriate CQO is 𝑥₂.
In order to validate the proposed method, in what follows, we use the Ct-IULOWA and the Ca-IULOWA operators for the same decision structure. The final results derived by C²-IULOWA, Ct-IULOWA, and Ca-IULOWA operators are shown in Table 7, and they are represented graphically in Figure 1.

Table 7 

Collective non-dominance degrees of alternatives by utilizing C²-IULOWA, Ct-IULOWA, and Ca-IULOWA operators.

Figure 1 

Collective non-dominance degrees of alternatives by utilizing C2-IULOWA, Ct-IULOWA, and Ca- IULOWA operators.

Obviously, the ranking results derived by C²-IULOWA, Ct-IULOWA, and Ca-IULOWA operators all are 𝑥₂≻𝑥₄≻𝑥₃≻𝑥₁; the situations verify each other; the results derived from the C²-IULOWA, Ct-IULOWA, and Ca-IULOWA operators are feasible and effective. The reasons of the differences of the final results are intuitive; that is, as discussed above, the Ct-IULOWA operator focuses solely on the consistency and ignores the consensus of the uncertain linguistic preference information, while the Ca-IULOWA operator focuses only on the consensus and ignores the consistency of the uncertain linguistic preference information. The C²-IULOWA operator comprehensively considers both the consistency and consensus of the uncertain linguistic preference information. Hence, the results derived by C²-IULOWA operator are more feasible and effective.

7. Conclusion

In group decision making with uncertain linguistic preference relations, the aggregation of preference information plays important roles in reaching a reasonable decision result. Considering the fact that when carrying out rational decision making, the consensual or consistent information is more appropriate than the inconsistent or decentralized ones, in this paper, we investigated the aggregation of uncertain linguistic preference information by fusing its consensus and consistent information. To do so, we first proposed the consistency and consensus measures, to assess the consistency level and consensus level of uncertain linguistic preference relations and preference values of them, and studied some of their desirable properties. Based on both consistency level and consensus level, we proposed a co-induced uncertain linguistic OWA operator, namely, C²-IULOWA operator, to aggregate uncertain linguistic preference information, in which consistency level and consensus level synergistically serve as inducing variables and participate in guiding the determination of associated weights in the preference aggregation process. We have verified that the C²-IULOWA operator is able to maintain the indifference, reciprocity, and consistency properties of uncertain linguistic preference relations after aggregation is carried out. For applying the C²-IULOWA operator to group decision making with uncertain linguistic preference relations, we developed a direct group decision making approach. The approach is very suitable for the situations with time pressure and decision makers unwilling to revise their initial judgments, in which the inconsistent or decentralized information would be still retained and yet it is considered to be less important for ranking the alternatives. The proposed approach is effective and feasible just as shown in the illustrative example. It is expected that the proposed approach can be applied to the fields of quality management and mobile business.

Acknowledgments

The authors are very grateful to the editor, Professor Chong Lin, and the anonymous referees for their insightful and constructive comments and suggestions which have helped to improve the paper. This work was supported in part by the National Natural Science Funds of China (nos. 61364016, 71272191 and 71072085), the Natural Science Funds of KUST (no. KKSY201358032) and the Graduate Innovation Funds of Heilongjiang Province of China (no. YJSCX2011-003HLJ).