The Unbalanced Linguistic Aggregation Operator in Group Decision Making

Abstract

Many linguistic aggregation methods have been proposed and applied in the linguistic decision-making problems. In practice, experts need to assess a number of values in a side of reference domain higher than in the other one; that is, experts use unbalanced linguistic values to express their evaluation for problems. In this paper, we propose a new linguistic aggregation operator to deal with unbalanced linguistic values in group decision making, we adopt 2-tuple representation model of linguistic values and linguistic hierarchies to express unbalanced linguistic values, and moreover, we present the unbalanced linguistic ordered weighted geometric operator to aggregate unbalanced linguistic evaluation values; a comparison example is given to show the advantage of our method.

1. Introduction

The need of uncertain linguistic information processing has become an important topic in many areas dealing with vague information such as in universal design used for developing products and environments that are usable by all people to the greatest extent possible; although a set of acknowledged principles has been developed and commonly used by industry and academia, it is difficult to quantitatively evaluate whether a product is indeed a good example of universal design, and linguistic evaluation approach for universal design can improve the shortcomings of traditional analytical hierarchy process methods, quantitative judgments, priority analysis, and aggregation performance [1]. Formally, the fuzzy logic framework and especially fuzzy sets themselves are not always easy to obtain from the linguistic value sets. That is why we choose to keep the words themselves without going through a fuzzy modeling [2]. In fact, linguistic value is the granularity of information that allows for a better approximation of the concepts when it is needed [3].

In linguistic group decision-making analysis, the problems are associated with the following [4]: (1) the choice of the linguistic value set with its semantic; (2) the choice of the aggregation operator of linguistic information; (3) the choice of the best alternatives. In the above-mentioned three steps, the aim of (1) consists of establishing the linguistic variable [5] or linguistic expression domain with a view to provide the linguistic performance values. In practice, fuzzy numbers or an ordered structure of linguistic values can be used to explain their semantic [5, 6]. The aim of (2) is to carry out the aggregation of linguistic information; there are many numeric aggregation operators [7–9] and linguistic aggregation operators [4, 10–13] to aggregate them. The aim of (3) consists of obtaining a collective performance value over each alternative and finding a solution set of alternatives. The solution set of alternatives is the best alternative that is the most satisfied alternative for the experts.

In most cases, information of group decision-making problems can be assessed in a qualitative form rather than a quantitative one, and experts use natural language instead of numerical values to express their evaluations of decision-making problems [4]; hence, we need the linguistic approach to solve group decision-making problems with linguistic assessment [3, 6, 10, 14]. Up to now, many linguistic approaches have been proposed and applied to solve problems with linguistic assessment, for example, personnel management [4, 12], web information processing [15, 16], and sensor evaluation and fuzzy risk analysis [17–24]. In fact, linguistic approaches are the core of computing with words proposed by Zadeh [25]. In [26], Herrera et al. make a review of the developments of Computing with Words in decision-making and explore different linguistic computational models that have been applied to the decision making field. In this paper, unbalanced linguistic values proposed in [27] are used to deal with group decision making, in practice, experts need to assess a number of values in a side of reference domain higher than in the other one; that is, experts use unbalanced linguistic values to express their evaluation for problems, to deal with unbalanced linguistic values in group decision making, and we adopt 2-tuple representation model of linguistic values [10] and linguistic hierarchies [4] to express unbalanced linguistic values; moreover, inspired by the ordered weighted geometric operator [28–30], we propose the unbalanced linguistic ordered weighted geometric operator to aggregate unbalanced linguistic evaluation values, and some interesting properties of the unbalanced linguistic ordered weighted geometric operator are also discussed.

The organization of this paper is as follows. In Section 2, we briefly make a review of the 2-tuple fuzzy linguistic representation model and unbalanced linguistic terms. In Section 3, we present the unbalanced linguistic ordered weighted geometric operator and some properties of the operator. An example is given in Section 4 to deal with a group decision-making problem with unbalanced linguistic evaluation values, and it is shown that our method is an alternative aggregation operator for linguistic evaluation approach. We conclude in Section 5.

2. Preliminaries

In this section, we briefly review the 2-tuple linguistic representation model and unbalanced linguistic values we refer to [10, 27] for more details.

The 2-tuple linguistic representation model was introduced by Herrera and Martínez [10]. Let 𝑆={𝑠₀,…,𝑠_𝑔} be the initial finite linguistic value set. Formally, the 2-tuple linguistic representation model is formed by (𝑠_𝑖,𝛼), in which 𝑠_𝑖∈𝑆(𝑖∈{0,1,…,𝑔}) and 𝛼∈[−0.5,0.5); that is, linguistic information is encoded in the space 𝑆×[−0.5,0.5). Based on the representation (𝑠_𝑖,𝛼), we can easily obtain the following symbolic translation of linguistic values from 𝛽∈[0,𝑔] to 𝑆×[−0.5,0.5), that is, Δ∶[0,𝑔]→𝑆×[−0.5,0.5), 𝛽↦(𝑠_𝑖,𝛼), in which 𝑖=round(𝛽) (round(⋅) is the usual round operation), 𝛼=𝛽−𝑖∈[−0.5,0.5). Intuitively, Δ(𝛽)=(𝑠_𝑖,𝛼) expresses that 𝑠_𝑖 is the closest linguistic value to 𝛽, and 𝛼 is the value of the symbolic translation. Additionally, there is a Δ⁻¹ function such that from a 2-tuple it returns its equivalent numerical value 𝛽∈[0,𝑔], that is, Δ⁻¹∶𝑆×[−0.5,0.5)→[0,𝑔], Δ⁻¹(𝑠_𝑖,𝛼)=𝑖+𝛼=𝛽. In fact, the model defines a set of transformation functions between linguistic values and 2-tuple linguistic representations as well as numeric values and 2-tuple linguistic representations. This allows us to easily process linguistic information by numeric value; for example, we have the following linguistic aggregation operators: Let a set of the 2-tuple linguistic representations be 𝑥={(𝑠₁,𝛼₁),…,(𝑠_𝑛,𝛼_𝑛)}, and let 𝑊={𝑤₁,…,𝑤_𝑛} be an associated weight such that 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1:(1) the 2-tuple arithmetic mean operator 𝑥^𝑒: 𝑥^𝑒=Δ(∑^𝑛_𝑖=11/𝑛×Δ⁻¹(𝑠_𝑖,𝛼_𝑖));(2) the 2-tuple weighted aggregation operator 𝐹^𝑤: 𝐹^𝑤=Δ(∑^𝑛_𝑖=1𝑤_𝑖×Δ⁻¹(𝑠_𝑖,𝛼_𝑖));(3) the 2-tuple ordered weighted aggregation operator 𝐹^𝑒: 𝐹^𝑒((𝑠₁,𝛼₁),…,(𝑠_𝑛,𝛼_𝑛))=Δ(∑^𝑛_𝑗=1𝑤_𝑗×𝛽^∗_𝑗), in which 𝛽^∗_𝑗 is the 𝑗th largest linguistic value of {𝛽_𝑖=Δ⁻¹(𝑠_𝑖,𝛼_𝑖)∣𝑖=1,…,𝑛}.

Unbalanced linguistic values proposed in [27] are used to deal with scales for assessing preferences where the experts need to assess a number of values in a side of reference domain higher than in the other one. Generally, an unbalanced linguistic term set 𝑆 has a minimum label, a maximum label, and a central label, and the remaining labels are nonuniformly and nonsymmetrically distributed around the central one on both left and right lateral sets; that is, we can represent 𝑆 in the following form: 𝑆=𝑆_𝑙∪𝑆_𝑐∪𝑆_𝑟, in which, 𝑆_𝑙 contains all left lateral labels but the central label, 𝑆_𝑐 just contains the central label, and 𝑆_𝑟 contains all right lateral labels higher than the central label.

Example 2.1. 𝑆 = {none (𝑁), low (𝐿), medium (𝑀), almost high (𝐴𝐻), high (𝐻), quite high (𝑄𝐻), very high (𝑉𝐻), almost total (𝐴𝑇), total (𝑇)} is an unbalanced linguistic term set, in which 𝑆_𝑙={𝑁,𝐿}, 𝑆_𝑐={𝑀}, and 𝑆_𝑟={𝐴𝐻,𝐻,𝑄𝐻,𝑉𝐻,𝐴𝑇,𝑇}.

To obtain 2-tuple fuzzy linguistic representations of unbalanced linguistic values without loss of information, we need the concept of linguistic hierarchies [4], that is, 𝐿𝐻=⋃_𝑡𝑙(𝑡,𝑛(𝑡)) it takes into account a set of levels where each level is a linguistic term set with different granularity from the remaining levels of the hierarchy, where 𝑙(𝑡,𝑛(𝑡)) is a linguistic hierarchy with 𝑡 being a number that indicates the level of the hierarchy and 𝑛(𝑡) the granularity of the linguistic term set of 𝑡. Generally, the linguistic term set 𝑆^𝑛(𝑡+1) of the level 𝑡+1 is obtained from its predecessor 𝑆^𝑛(𝑡) as 𝑙(𝑡,𝑛(𝑡))→𝑙(𝑡+1,2×𝑛(𝑡)−1). In linguistic hierarchies 𝐿𝐻, transformation function between labels from different levels to represent 2-tuple fuzzy linguistic representations is as follows [4]: for any linguistic levels 𝑡 and 𝑡′, 𝑇𝐹^𝑡_𝑡′∶𝑙(𝑡,𝑛(𝑡))→𝑙(𝑡^′,𝑛(𝑡^′)) such that 𝑇𝐹^𝑡_𝑡′𝑠_𝑖^𝑛(𝑡),𝛼^𝑛(𝑡)=▵_𝑡′▵_𝑡⁻¹𝑠_𝑖^𝑛(𝑡),𝛼^𝑛(𝑡)×𝑛𝑡^′−1𝑛(𝑡).−1 (2.1) By using linguistic hierarchies 𝐿𝐻=⋃_𝑡𝑙(𝑡,𝑛(𝑡)), we can obtain the 2-tuple fuzzy linguistic representation of each term of unbalanced linguistic term set in 𝐿𝐻; here, we use the following example to explain the converting process, and we refer to [27] for more details.

Example 2.2. Let linguistic hierarchies be ()()()()=𝑠𝐿𝐻=𝑙1,3∪𝑙2,5∪𝑙3,9∪𝑙4,17₃₀,𝑠³₁,𝑠³₂∪𝑠⁵₀,𝑠⁵₁,…,𝑠⁵₄∪𝑠⁹₀,𝑠⁹₁,…,𝑠⁹₈∪𝑠₀¹⁷,𝑠₁¹⁷,…,𝑠¹⁷₁₆. (2.2)(𝑠⁹₅,0.3) is a 2-tuple fuzzy linguistic representation of level 3, and it is 2-tuple fuzzy linguistic representation in level 2 is 𝑇𝐹³₂𝑠⁹₅,0.3=▵₂▵₃⁻¹𝑠⁹₅,0.3×(5−1)9−1=▵₂5.3×48=▵₂()=𝑠2.65₅₃,−0.35. (2.3) For unbalanced linguistic term set 𝑆={𝑁,𝐿,𝑀,𝐴𝐻,𝐻,𝑄𝐻,𝑉𝐻,𝐴𝑇,𝑇}, (1) due to 𝑛(2)=5 and (𝑛(2)−1)/2=|𝑆_𝑙|=|{𝑁,𝐿}|=2, the representation of 𝑆_𝑙 is obtained from level 2 of 𝐿𝐻 as follows: {𝐿←𝑠⁵₁,𝑁←𝑠⁵₀}; (2) due to 𝑛(3)=9, 𝑛(4)=17, and (𝑛(3)−1)/2=4<|𝑆_𝑟|=|{𝐴𝐻,𝐻,𝑄𝐻,𝑉𝐻,𝐴𝑇,𝑇}|=6<(𝑛(4)−1)/2=8, we use level 3 and level 4 to represent 𝑆_𝑟={𝐴𝐻,𝐻,𝑄𝐻,𝑉𝐻,𝐴𝑇,𝑇}, according to lab₃ and lab₄ defined in [27], and {𝐴𝐻,𝐻} and {𝑄𝐻,𝑉𝐻,𝐴𝑇,𝑇} are represented in level 3 and level 4, respectively, {𝐴𝐻←𝑠⁹₅,𝐻←𝑠⁹₆}, {𝑄𝐻←𝑠¹⁷₁₃,𝑉𝐻←𝑠¹⁷₁₄,𝐴𝑇←𝑠¹⁷₁₅,𝑇←𝑠¹⁷₁₆}; (3) according to density and bridging representation gaps defined in [27], the upside and the downside of the central label 𝑀 are represented in level 2 and 3 of 𝐿𝐻 by means of 𝑠⁵₂ and 𝑠⁹₄, respectively. The upside and the downside of the label 𝐻 are represented in level 3 and 4 of 𝐿𝐻 by means of 𝑠⁹₆ and 𝑠¹⁷₁₂, respectively; (4) the final 2-tuple fuzzy linguistic representations of 𝑆 in 𝐿𝐻 are 𝑆_𝑙∶{𝑁←𝑠⁵₀,𝐿←𝑠⁵₁}, 𝑆_𝑐∶{𝑀←𝑠⁵₂∪𝑠⁹₄}, 𝑆_𝑟∶{𝐴𝐻←𝑠⁹₅,𝐻←𝑠⁹₆∪𝑠¹⁷₁₂,𝑄𝐻←𝑠¹⁷₁₃,𝑉𝐻←𝑠¹⁷₁₄,𝐴𝑇←𝑠¹⁷₁₅,𝑇←𝑠¹⁷₁₆}.
Formally, for any 2-tuple fuzzy linguistic representation (𝑠_𝑖,𝛼_𝑖) (𝑠_𝑖∈𝑆, 𝛼_𝑖∈[−0.5,0.5)), it can be converted by the following unbalanced linguistic transformation functions in 𝐿𝐻 and vice versa, that is, ℒℋ∶𝑆×[−0.5,0.5)→𝐿𝐻×[−0.5,0.5), (𝑠_𝑖,𝛼_𝑖)↦(𝑠^𝐺(𝑖)_𝐼(𝑖),𝛼_𝑖) such that 𝑠^𝐺(𝑖)_𝐼(𝑖)∈𝐿𝐻. ℒℋ⁻¹∶𝐿𝐻×[−0.5,0.5)→𝑆×[−0.5,0.5), (𝑠_𝑘^𝑛(𝑡),𝛼_𝑘)↦(𝑠_𝑖,𝜆), for example, ℒℋ(𝐻,0.3)=(𝑠⁹₆,0.3), ℒℋ⁻¹(𝑠¹⁷₁₃,−0.2)=(𝑄𝐻,−0.2).

3. The Unbalanced Linguistic Ordered Weighted Geometric Operator

To aggregate unbalanced linguistic values, we present the unbalanced linguistic ordered weighted geometric operator [20], which is inspired by 2-tuple fuzzy linguistic representation model [27] and the ordered weighted geometric operator [29].

Definition 3.1. An ordered weighted geometric operator of dimension 𝑛 is a mapping 𝑔∶(𝑅⁺)^𝑛→𝑅⁺ that has a weighting vector 𝑊=(𝑤₁,𝑤₂,…,𝑤_𝑛) associated with it, with 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1, such that 𝑔(𝑎₁,𝑎₂,…,𝑎_𝑛)=∏^𝑛_𝑗=1𝑏^𝑤𝑗_𝑗, where 𝑏_𝑗 is the 𝑗th largest linguistic value of the 𝑎_𝑖(𝑖=1,…,𝑛).

Definition 3.2. Assume that the set of unbalanced linguistic values 𝑉={𝑠_𝑖∣𝑖=1,2,…,𝑛} will be aggregated, in which 𝑠_𝑖∈𝑆={𝑠₀,𝑠₁,…,𝑠_𝑚}(𝑚≥𝑛) is an unbalanced linguistic value, a weighting vector is 𝑊=(𝑤₁,𝑤₂,…,𝑤_𝑛) with 𝑤_𝑖∈[0,1], and ∑^𝑛_𝑖=1𝑤_𝑖=1. Then the unbalanced linguistic ordered weighted geometric operator (the ULOWG operator) is defined as follows: 𝑓_ULOWG(𝑉)=𝑓_ULOWG𝑠_𝑖∣𝑖=1,2,…,𝑛=𝑓_ULOWG𝑇𝐹^𝑡𝑖_𝑡0ℒℋ𝑠_𝑖∣𝑖=1,…,𝑛=ℒℋ⁻¹𝑠^𝑛(𝑡0)_𝑘,𝛼_𝑘, (3.1) where 𝑡_𝑖 is the level of ℒℋ(𝑠_𝑖) in 𝐿𝐻, 𝑡₀ is a level of 𝐿𝐻, fixed by users, 𝑠^𝑛(𝑡0)_𝑘∈𝑆^𝑛(𝑡0)⊂𝐿𝐻 and 𝛼_𝑘∈[−0.5,0.5) such that 𝑘+𝛼_𝑘=^𝑛_𝑖=1▵_𝑡⁻¹₀𝑇𝐹^𝑡𝑖_𝑡0ℒℋ𝑠_𝑖^𝑤𝑖, (3.2) where for any 𝑖∈{1,…,𝑛}, ▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑖−1)))≥▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑖))); that is, 𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑖)) is the 𝑖th largest linguistic value in {𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑖))∣𝑖=1,2,…,𝑛}.

Remark 3.3. In Definition 3.2, 𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑖)) means that unbalanced linguistic values are represented at level 𝑡₀ of 𝐿𝐻, and 𝑡₀ is decided by users. In fact, by using 𝑇𝐹^𝑡𝑖_𝑡0(⋅) and ℒℋ(⋅), 𝑆 is converted in 𝑆^𝑛(𝑡0) of 𝐿𝐻. By using ℒℋ⁻¹(𝑠^𝑛(𝑡0)_𝑘,𝛼_𝑘), aggregation result 𝑓_ULOWG(𝑉) is converted to the 2-tuple fuzzy linguistic representation of unbalanced linguistic value.

In Definition 3.2, a natural question is how to obtain the associated weighting vector 𝑊. Following Yager’s ideas [31] on quantifier-guided aggregation, we could compute the weighting vector of an ULOWG operator using a linguistic quantifier 𝑄 [31] as follows 𝑤_𝑖𝑖=𝑄𝑟,𝑎,𝑏−𝑄𝑖−1𝑟𝑄()=,𝑎,𝑏,𝑖=1,…,𝑟.𝑥,𝑎,𝑏0,if𝑥<𝑎,𝑥−𝑎𝑏−𝑎,if𝑎≤𝑥<𝑏,1,if𝑥≥𝑏, (3.3)(3.4) where 𝑥,𝑎,𝑏∈[0,1]. Some examples of 𝑄(𝑥,𝑎,𝑏) (see Figure 1) are most, at least half, and as many as possible; their parameters (𝑎,𝑏) are (0.3,0.8), (0,0.5), and (0.5,1), respectively.

Figure 1 

The functions of 𝑄(𝑥,𝑎,𝑏).

Example 3.4. We select Yager’s linguistic quantifier most, that is, 𝑄(𝑥,𝑎,𝑏)=𝑄(𝑥,0.3,0.8), according to (3.3), 𝑟=3, and 𝑤₁1=𝑄30,0.3,0.8−𝑄3=1,0.3,0.8,𝑤15₂2=𝑄31,0.3,0.8−𝑄3=2,0.3,0.83,𝑤₃3=𝑄32,0.3,0.8−𝑄3=4,0.3,0.8,15 (3.5) so weighting vectors 𝑊=(1/15,2/3,4/15).
Let 𝑆={𝑁,𝐿,𝑀,𝐴𝐻,𝐻,𝑄𝐻,𝑉𝐻,𝐴𝑇,𝑇} be a set of unbalanced linguistic values. Suppose that 𝑉={𝐴𝐻,𝑄𝐻,𝐻} will be aggregated, and a weighting vector is 𝑊=(1/15,2/3,4/15). In this example, we select 𝑡₀=3 and density𝑆_𝑅 is extreme [27]; hence, 𝑇𝐹^𝑡𝑖₃(ℒℋ(𝐴𝐻))=𝑇𝐹³₃𝑠⁹₅=𝑠⁹₅,𝑇𝐹^𝑡𝑖₃(ℒℋ(𝑄𝐻))=𝑇𝐹⁴₃𝑠¹⁷₁₃=𝑠⁹₆,0.5,𝑇𝐹^𝑡𝑖₃(ℒℋ(𝐻))=𝑇𝐹³₃𝑠⁹₆=𝑠⁹₆, (3.6) and ▵₃⁻¹(𝑠⁹₆,0.5)=6.5>▵₃⁻¹(𝑠⁹₆)=6>▵₃⁻¹(𝑠⁹₅)=5; according to (3.2), we have 𝑘+𝛼_𝑘=6.5^1/15×6^2/3×5^4/15≐5.7485, that is, 𝑓_ULOWG(𝑉)=𝑓_ULOWG({𝐴𝐻,𝑄𝐻,𝐻})≐ℒℋ⁻¹𝑠⁹₆,−0.2542=(𝐻,−0.2542). (3.7)

Proposition 3.5. Let the set of unbalanced linguistic values be 𝑉={𝑠_𝑖∣𝑖=1,2,…,𝑛}, and a weighting vector is 𝑊=(𝑤₁,…,𝑤_𝑛) with 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1.(1) If 𝑤₁=1 and 𝑤₂=⋯=𝑤_𝑛=0, then 𝑓_ULOWG(s₁,…,𝑠_𝑛)=max{𝑠₁…,𝑠_𝑛};(2) if 𝑤_𝑛=1 and 𝑤₁=⋯=𝑤_𝑛−1=0, then 𝑓_ULOWG(𝑠₁,…,𝑠_𝑛)=min{𝑠_𝑖…,𝑠_𝑛};(3) if 𝑤₁=𝑤_𝑛=0, then 𝑓_ULOWG reduces to the linguistic Olympic operator; that is, the smallest and largest linguistic values are deleted from linguistic evaluation values;(4) if for any 𝑖∈{1,…,𝑛}, 𝑤_𝑖=1, then 𝑓_ULOWG𝑠₁…,𝑠_𝑛=𝑠_𝑖. (3.8)

Proof. (1) According to (3.2), we have 𝑘+𝛼_𝑘=∏^𝑛_𝑖=1(▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑖))))^𝑤𝑖  =  (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠₁))))^𝑤1  ×  (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠₂))))^𝑤2  ×  ⋯  ×  (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑛))))^𝑤𝑛  =  (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠₁))))¹  ×  (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠₂))))⁰  ×  ⋯  ×  (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑛))))⁰  =  (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠₁)))), where (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠₁)))) is the largest linguistic value of the {𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑖))∣𝑖=1,2,…,𝑛}; then, we can get 𝑓_ULOWG𝑠₁,…,𝑠_𝑛=ℒℋ⁻¹▵_𝑡0𝑘+𝑎_𝑘=ℒℋ⁻¹▵_𝑡0▵_𝑡⁻¹₀𝑇𝐹^𝑡𝑖_𝑡0ℒℋ𝑠₁=max𝑠₁…,𝑠_𝑛. (3.9)
(2) According to (3.2), we have 𝑘+𝛼_𝑘=∏^𝑛_𝑖=1(▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑖))))^𝑤𝑖  =  (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠₁))))^𝑤1  ×  (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠₂))))^𝑤2  ×  ⋯  ×  (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑛))))^𝑤𝑛 = (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠₁))))⁰  ×  (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(s₂))))⁰  ×  ⋯  ×  (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑛))))¹ = (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑛)))), where (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑛)))) is the smallest linguistic value of the {𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑖))∣𝑖=1,2,…,𝑛}; then, we can get 𝑓_ULOWG𝑠₁,…,𝑠_𝑛=ℒℋ⁻¹▵_𝑡0𝑘+𝑎_𝑘=min𝑠₁…,𝑠_𝑛. (3.10)
(3) It is obvious.
(4) If for any 𝑖∈{1,…,𝑛}, 𝑤_𝑖=1; thus, for any 𝑗≠𝑙, 𝑤_𝑗=0, then we have 𝑘+𝛼_𝑘=(▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑙))))𝑓_ULOWG𝑠₁,…,𝑠_𝑛=ℒℋ⁻¹▵_𝑡0𝑘+𝑎_𝑘=ℒℋ⁻¹▵_𝑡0𝑘+𝑎_𝑘=𝑠_𝑙. (3.11)

Proposition 3.6. Assume that a set of unbalanced linguistic values 𝑉={𝑠_𝑖∣𝑖=1,…,𝑛} will be aggregated and a weighting vector is 𝑊=(𝑤₁,𝑤₂,…,𝑤_𝑛) with 𝑤_𝑖∈[0,1] and ∑^𝑛_𝑖=1𝑤_𝑖=1. The ULOWG operator satisfies the following:(1)min{𝑠_𝑖∣𝑖=1,…,𝑛}≤𝑓_ULOWG({𝑠_𝑖∣𝑖=1,…,𝑛})≤max{𝑠_𝑖∣𝑖=1,…,𝑛};(2)𝑓_ULOWG is idempotent; that is, if 𝑠₁=𝑠₂=⋯=𝑠_𝑛, then 𝑓_ULOWG({𝑠₁,…,𝑠_𝑛})=𝑠₁;(3)𝑓_ULOWG is monotone in relation to the input values 𝑠_𝑖;𝑓_ULOWG is commutative;(4)𝑓_ULOWG reduces to the linguistic geometric mean if 𝑤_𝑖=1/𝑛 for all 𝑖=1,…,𝑛, that is, 𝑓_ULOWG𝑠₁,…,𝑠_𝑛=𝑠_𝑘,𝛼_𝑘,𝑘+𝛼_𝑘=^𝑛𝑛_𝑖=1▵_𝑡⁻¹₀𝑇𝐹^𝑡𝑖_𝑡0ℒℋ𝑠_𝑖. (3.12)

Proof. (1) For any 𝑉={𝑠_𝑖∣𝑖=1,…,𝑛}, 𝑊=(𝑤₁,𝑤₂,…,𝑤_𝑛)𝑘+𝛼_𝑘=∏^𝑛_𝑖=1(▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑖))))^𝑤𝑖; according to Proposition 3.5 (1) and (2), we know that (▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑛))))≤∏^𝑛_𝑖=1(▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠_𝑖))))^𝑤𝑖≤(▵_𝑡⁻¹₀(𝑇𝐹^𝑡𝑖_𝑡0(ℒℋ(𝑠₁)))), that is,min{𝑠_𝑖∣𝑖=1,…,𝑛}≤𝑓_ULOWG({𝑠_𝑖∣𝑖=1,…,𝑛})≤max{𝑠_𝑖∣𝑖=1,…,𝑛}.
(2) Due to 𝑠₁=𝑠₂=⋯=𝑠_𝑛, and according to (3.2), we have 𝑘+𝛼_𝑘=^𝑛_𝑖=1▵_𝑡⁻¹₀𝑇𝐹^𝑡𝑖_𝑡0ℒℋ𝑠_𝑖^𝑤𝑖=▵_𝑡⁻¹₀𝑇𝐹^𝑡𝑖_𝑡0ℒℋ𝑠₁^{𝑤1+𝑤2+⋯+𝑤𝑛}=▵_𝑡⁻¹₀𝑇𝐹^𝑡𝑖_𝑡0ℒℋ𝑠₁, (3.13) then, 𝑓_ULOWG(𝑠₁,…,𝑠_𝑛)=ℒℋ⁻¹(▵_𝑡0(𝑘+𝑎_𝑘))=ℒℋ⁻¹(▵_𝑡0(𝑘+𝑎_𝑘))=𝑠_𝑙.
(3) Assume that 𝑊=(𝑤₁,𝑤₂,…,𝑤_𝑛) denotes the weight of {𝑠₁,𝑠₂,…𝑠_𝑛} and 𝑊=(𝑤^′₁,𝑤^′₂,…,𝑤^′_𝑛) denotes the weight of {𝑠^′₁,𝑠^′₂,…𝑠^′_𝑛}. If for any 𝑖∈𝑁, 𝑠_𝑖≤𝑠^′_𝑖, then according to (3.2), 𝑘+𝛼_𝑘=^𝑛_𝑖=1▵_𝑡⁻¹₀𝑇𝐹^𝑡𝑖_𝑡0ℒℋ𝑠_𝑖^𝑤𝑖≤𝑘^′+𝛼^′_𝑘=^𝑛_𝑖=1▵_𝑡⁻¹₀𝑇𝐹^𝑡𝑖_𝑡0ℒℋ𝑠^′_𝑖^𝑤𝑖, (3.14) that is, 𝑓_ULOWG𝑠₁,…,𝑠_𝑛=ℒℋ⁻¹▵_𝑡0𝑘+𝑎_𝑘≤𝑓_ULOWG𝑠^′₁,…,𝑠^′_𝑛=ℒℋ⁻¹▵_𝑡0𝑘^′+𝑎^′_𝑘. (3.15)
(4) Since {𝑠^′₁,𝑠^′₂,…𝑠^′_𝑛} is the permutation of the {𝑠₁,𝑠₂,…𝑠_𝑛}, so we can get the same sort result {𝑏₁,𝑏₂,…,𝑏_𝑛}, that is to say, 𝑏_𝑗 is the jth largest element in the set {𝑠^′₁,𝑠^′₂,…𝑠^′_𝑛} and {𝑠₁,𝑠₂,…𝑠_𝑛}. According to Definition 3.2, 𝑘+𝛼_𝑘=^𝑛_𝑖=1▵_𝑡⁻¹₀𝑇𝐹^𝑡𝑖_𝑡0ℒℋ𝑠_𝑖^𝑤𝑖=𝑘^′+𝛼^′_𝑘=^𝑛_𝑖=1▵_𝑡⁻¹₀𝑇𝐹^𝑡𝑖_𝑡0ℒℋ𝑠^′_𝑖^𝑤𝑖, (3.16) that is, 𝑓_ULOWG𝑠₁,…,𝑠_𝑛=𝑓_ULOWG𝑠^′₁,…,𝑠^′_𝑛. (3.17)
(5) It is obviously set up.

4. Comparison between ULOWA Operator and 𝐹^𝑤 Operator, and 𝐹^𝑒 Operator

In this section, we introduced an application in group decision-making. In the following example, we used the ULOWA operator, ULOWA operator and 𝐹^𝑒 operator to choose the better alternative, respectively. Then, we analyze the difference of the ULOWA operator, 𝐹^𝑤 operator, and 𝐹^𝑒 operator based on evaluation results.

Example 4.1. A distribution company needs to upgrade its computing system, so it hires a consulting company to survey the different possibilities existing on the market, to decide which is the best option for its needs. The options (alternatives) and consultancy departments are shown in Table 1. In each of the departments, there is one expert providing evaluation for each alternative (shown in Table 2); these evaluations are assessed in the initial finite unbalanced linguistic value set 𝑆={none (𝑁), low (𝐿), medium (𝑀), almost high (𝐴𝐻), high (𝐻), quite high (𝑄𝐻), very high (𝑉𝐻), almost total (𝐴𝑇), total (𝑇)}.
In this example, a linguistic hierarchy is 𝐿𝐻=𝑙(1,3)∪𝑙(2,5)∪𝑙(3,9)∪𝑙(4,17)={𝑠³₀,𝑠³₁,𝑠³₂}∪{𝑠⁵₀,𝑠⁵₁,…,𝑠⁵₄}∪{𝑠⁹₀,𝑠⁹₁,…,𝑠⁹₈}∪{𝑠₀¹⁷,𝑠₁¹⁷,…,𝑠¹⁷₁₆} of Example 2.2. We select 𝑡₀=3, and density𝑆_𝑅 is extreme, according to Table 2𝑇𝐹^𝑡𝑖₃(ℒℋ(𝐿))=𝑇𝐹³₃𝑠⁵₁=𝑠⁹₂,0,𝑇𝐹^𝑡𝑖₃(ℒℋ(𝑀))=𝑇𝐹³₃𝑠⁹₄=𝑠⁹₄,0,𝑇𝐹^𝑡𝑖₃(ℒℋ(𝐴𝐻))=𝑇𝐹³₃𝑠⁹₅=𝑠⁹₅,0,𝑇𝐹^𝑡𝑖₃(ℒℋ(𝐻))=𝑇𝐹³₃𝑠⁹₆=𝑠⁹₆,0,𝑇𝐹^𝑡𝑖₃(ℒℋ(𝑄𝐻))=𝑇𝐹⁴₃𝑠¹⁷₁₃=𝑠⁹₆,0.5. (4.1) According to (3.3), we can get the weighting vector 𝑊=(0,0.4,0.5,0.1) (1) Evaluations based on 𝐹^𝑤.
If the weighting vector 𝑊=(0,0.4,0.5,0.1) is for four experts and associated with their evaluations, then 𝐹^𝑤 can be used to evaluate alternatives; for example, for alternative 𝑥₁, we have 𝐹^𝑤(𝐿,𝐴𝐻,𝐻,𝐻)=Δ0×Δ⁻¹𝑠⁹₂,0+0.4×Δ⁻¹𝑠⁹₅,0+0.5×Δ⁻¹𝑠⁹₆,0+0.1×Δ⁻¹𝑠⁹₆()()=𝑠,0=Δ0×2+0.4×5+0.5×6+0.1×6=Δ5.6⁹₆,−0.4=(𝐻,−0.4). (4.2) (2) Evaluations based on 𝐹^𝑒.
When the weighting vector 𝑊=(0,0.4,0.5,0.1) is for four experts and associated with particular ordered positions rather than their evaluations, then 𝐹^𝑒 can be used to evaluate alternatives, for example, for alternative 𝑥₁, we have 𝐹^𝑒(𝐿,𝐴𝐻,𝐻,𝐻)=Δ0×Δ⁻¹𝑠⁹₆,0+0.4×Δ⁻¹𝑠⁹₆,0+0.5×Δ⁻¹𝑠⁹₅,0+0.1×Δ⁻¹𝑠⁹₂()()=𝑠,0=Δ0×6+0.4×6+0.5×5+0.1×2=Δ5.1⁹₅,0.1=(𝐴𝐻,0.1). (4.3) (3) Evaluations based on 𝑓_ULOWG.
When the weighting vector 𝑊=(0,0.4,0.5,0.1) is for four experts and associated with particular ordered positions rather than their evaluations, 𝑓_ULOWG can also be used to evaluate alternatives; for example, for alternative 𝑥₁, we have 𝑓_ULOWG(𝐿,𝐴𝐻,𝐻,𝐻)=ΔΔ⁻¹𝑠⁹₆,0⁰×Δ⁻¹𝑠⁹₆,0^0.4×Δ⁻¹𝑠⁹₅,0^0.5×Δ⁻¹𝑠⁹₂,0^0.1=Δ6⁰×6^0.4×5^0.5×2^0.1≐Δ(5.3555)=𝑠⁹₅,0.3555=(𝐴𝐻,0.3555). (4.4)
Evaluations of all alternatives based on 𝐹^𝑤, 𝐹^𝑒, and 𝑓_ULOWG are shown in Table 3, we notice that using operator 𝐹^𝑒 and 𝑓_ULOWG the same results are obtained, that is, the best alternative is 𝑥₁. However, evaluations of 𝑓_ULOWG are smaller than evaluations of 𝐹^𝑒 correspondingly. Similarly to 𝐹^𝑒 operator, the ULOWG operator has many interesting properties. So, the ULOWA operator is a alternative aggregation operator to hand multiple attribute group decision making.

Table 1 

The four alternatives (𝑥_𝑖) and consultancy departments (𝑑_𝑖).

Table 2 

Evaluations provided by four experts.

Table 3 

Evaluations of alternatives based on 𝐹^𝑤, 𝐹^𝑒, and 𝑓_ULOWG.

5. Conclusion

From the practical point of view, group decision making is associated with multiple information sources fusion. In this paper, we propose the ULOWG operator to solve linguistic group decision-making problems, and the ULOWG operator has many interesting properties, hence, the ULOWG operator is an alternative linguistic aggregation operator in linguistic decision-making problems.

Acknowledgments

This work is partially supported by Sichun Key Laboratory of Intelligent Network Information Processing (SGXZD1002-10), National Natural Science Foundation (61175055, 61105059), Sichuan Key Technology Research and Development Program (2012GZ0019, 2011FZ0051), and Liaoning Excellent Talents in University (LJQ2011116).