An incomplete multi-granular linguistic model and its application in emergency decision of unconventional outburst incidents

Abstract

Unconventional outburst incidents are unpredictable and also have no emergency plans, they can bring huge economic losses and social issues, so we should find optimal emergency decision scheme. In this paper, we study the group decision making (GDM) problems with incomplete fuzzy linguistic preference relations (FLPRs), and the linguistic information provided by different experts are assessed in linguistic term sets with different granularity. We first develop a four-way procedure to estimate missing preference values when dealing with 2-tuple incomplete FLPRs. We propose the transformation rules for multi-granular FLPRs, and devise a transformation function which satisfies the transformation rules. The transformation can transform any linguistic term set to another linguistic term set. Furthermore, we study the properties and advantages of the transformation function. Finally, we integrate the revised estimation procedure for incomplete 2-tuple FLPRs and the multi-granular linguistic decision model to deal with emergency decision of unconventional outburst incidents. It is showed that the proposed method is straightforward and without loss of information.

Keywords

Multi-granular linguistic incomplete FLPRs unconventional outburst incidents emergency decision

1 Introduction

Group decision making (GDM) problems involve the preferences of some experts about a set of alternatives in order to select among the alternatives to reach a decision. Each decision maker (expert) may have unique motivations or goals and may approach the decision process from a different angle, but have a common interest in reaching eventual agreement on selecting the best alternative(s) to the problem to be solved [35, 45]. They need to compare a set of decision alternatives and construct a preference relation to model the decision processes. The evaluation information generally are in quantitative (real number, interval number, triangular fuzzy function) and qualitative forms. Many aspects of different activities in the real world cannot assessed in a quantitative form, but rather in a qualitative one, and the use of the fuzzy linguistic approach [59] has provided very good results. It deals with qualitative aspects that are represented in qualitative terms by means of linguistic variables. FLPRs [11 , 50– 52] provide very good results to deal with the problems which presented qualitative aspects. As the experts’ knowledge may be different and the criteria are belong to different areas, it can be that each source of information has a different uncertainty degree over the alternatives. Then the linguistic information is assessed in different linguistic domains with different granularity and/or semantics.

When a problem presents multi-granular linguistic information, the classical computational techniques presented in [12] have an important limitation because in these computational methods, neither a standard normalization process nor fusion operators are defined for this type of information. So in recent years different granularity linguistic information in decision-making problems have been attracted attention [9 , 56]. Herrera et al. [27] presented a fusion approach of multi-granularity linguistic information for managing information assessed in multi-granular linguistic term sets together with its application in a decision making problem with multiple information sources. Herrera and Martínez [33] presented a set of multi-granular linguistic contexts, called linguistic hierarchy label sets, and developed some functions based on 2-tuple linguistic representation model, which transform linguistic labels between different granular linguistic. When they dealt with multi-granular linguistic information assessed in these structures they can unify the information assessed in them without loss of information. Cordon et al. [9] proposed a new approach which was based on the development of a hierarchical system of linguistic rules learning methodology to design linguistic models accurate to a high degree and suitably interpretable by human beings. Herrera-Viedma et al. [20] presented a model of consensus support system to assist the experts in all phases of the consensus reaching process of group decision-making problems with multi-granular linguistic preference relations. The model permitted the unification of the different linguistic domains to facilitate the calculus of consensus degrees and proximity measures on the basis of experts’ opinions. Herrera et al. [25] developed a representation model for unbalanced linguistic information that used the concept of linguistic hierarchy as representation basis and an unbalanced linguistic computational model that used the 2-tuple fuzzy linguistic computational model to accomplish processes of CW (Computing with Words) with unbalanced term sets in a precise way and without loss of information. Xu [56] defined some unbalanced linguistic label sets and developed some transformation function to unify the given multi-granular linguistic labels in a unique linguistic label set without loss of information. The multi-granular linguistic model has been applied to different problems, such as, “large engineering system” [36], “performance appraisal” [10], “the suitability of installing an ERP system” [42], “the quality of network services” [26], etc. The above researches have two drawbacks when making the linguistic information uniform. One drawback is that experts’ preferences must be transformed (using a transformation function) into a single domain or linguistic term set which called basic linguistic term set (BLTS) [10 , 42]. The granularity of BLTS has to be as high as possible to maintain the uncertainty degrees associated to each one of the possible domains to be unified. The other drawback is that the transformation function used in [10 , 42] may loss information, the preference values cannot be fully transformed to the corresponding position, and we will illustrate in Remark 3.

On the other hand, when dealing with the GDM problems, the experts who may have not a precise or sufficient level of knowledge and are forced to provide incomplete preference relations, that is, he/she provides information in which at least one element is missing. In such situations, experts are only able to provide incomplete FLPRs with some of their values missing or unknown. Over the past decades, incomplete fuzzy preference relations [4 , 55] and incomplete FLPRs [1 , 57] have received more attention. Alonso et al. [1], Cabrerizo et al. [57] proposed a procedure to estimate missing preference values when dealing with incomplete FLPRs assessed using a 2-tuple fuzzy linguistic approach. The procedure is guided by the additive consistency property and only used the preference values provided by the experts. Xu [54] proposed an approach to group decision making based on incomplete FLPRs, he utilized the extended arithmetic averaging (EAA) operator and the extended weighted arithmetic averaging (EWAA) operator to deal with the incomplete FLPR. Xu [57] presented the linguistic geometric averaging (LGA) operator and the linguistic weighted geometric averaging (LWGA) to fill up the incomplete multiplicative linguistic preference relations. Experts may provide preference values in different linguistic term sets, and their preference relations would be incomplete when there exit missing or unknown values. How to deal with incomplete multi-granular linguistic is a common and very important issue. Up to now, there is no researcher who considers the incomplete multi-granular linguistic situations in the GDM. Therefore, it is necessary to pay attention to this issue.

The aim of this paper is to develop an incomplete multi-granular linguistic decision model that evaluates the different emergency decision alternatives for unconventional outburst incident according to multiple criteria assessed linguistically in different utility space. We assume FLPRs assessed on a 2-tuple fuzzy linguistic modeling [31] because it provides some advantages with respect to the ordinal fuzzy linguistic modeling [32]. We will first present a revised four-way estimation procedure to estimate missing information for incomplete FLPRs which is slightly different from the existing methods. It is based on the linguistic extension of Tanino’s [43] consistency principle. Then we develop a transformation model between the different granular linguistic sets, it can transform from low granular to high granular, or in the opposite direction, the advantages are no loss of information during transformation and the direction does not affect the final evaluation assessment.

To do this, the paper is set out as follows. In Section 2 we review some linguistic foundations we shall use in our decision model. In Section 3 we present the revised estimation procedure used for the incomplete linguistic contexts. In Section 4 a transformation function will be proposed to manage the multi-granular FLPRs. Section 5 gives an illustrative example with incomplete multi-granular linguistic information to choose the most suitable emergency decision alternative for unconventional outburst incident. The paper is concluded in Section 6.

2 Preliminaries

In this section, we briefly review the fuzzy linguistic approach and the 2-tuple fuzzy linguistic representation model. We shall use them in the development of our evaluation process to deal with multi-granular linguistic contexts.

2.1 Fuzzy linguistic approach

Usually, the information we touch in our work is expressed by means of numerical values. However, there are situations dealing with uncertainty or vague information in which a better approach to qualify aspects of many activities may be the use of linguistic assessments instead of numerical values. The fuzzy linguistic approach represents qualitative aspects as linguistic values by means of linguistic variables. The use of Fuzzy Sets Theory has given very good results for modeling qualitative information. It is a technique that handles fuzziness and represents qualitative aspects as linguistic labels by means of “linguistic variables”, that is, variables whose values are not numbers but words or sentences in a natural or artificial language. The fuzzy linguistic approach has been applied to different problems, such as, “information retrieval” [15], “select supply chain partners” [8], “consensus” [3 , 24], “risk in software development” [34], “manufacturing flexibility” [46], “decision making” [23], etc.

In any linguistic approach, we have to choose the appropriate linguistic descriptors for the term set and their semantics, and an important parameter to determine is the “granular of uncertainty”, i.e., the cardinality of the linguistic term set used to express the information. In the use of term sets with an odd cardinal was studied, representing the mid-term by an assessment of “approximately 0.5”, with the rest of the terms being placed symmetrically around it and the limit of granularity being 11 or no more than 13. These classical cardinality values seem to satisfy the Miller’s observation regarding the fact that human beings can reasonably manage to bear in mind seven or so items [39].

Once the cardinality of the linguistic term set has been established, we must provide the linguistic terms and its semantics. One possibility of generating the linguistic term set consists of directly supplying the term set by considering all terms distributed on a scale on which a total order is defined [30, 58]. For example, a set of seven terms S, could be: $\begin{matrix} S = {s_{0} : None, s_{1} : Very Low, s_{2} : Low, s_{3} : Medium, \\ s_{4} : High, s_{5} : Very High, s_{6} : Perfect} \end{matrix}$

Moreover, it must have the following characteristics:

The set is ordered: s _i ≥ s _j if i ≥ j.

There is the negation operator: neg (s _i) = s _j such that j = g - i (g + 1 is the cardinality).

There is the maximization and minimization operator: s _i ≤ s _j ⇔ i ≤ j.

2.2 The 2-tuple linguistic representation model

The classical linguistic computational models [30] have the drawback of loss of information caused by the need to express the results in the initial expression domain which is discrete. In order to overcome this drawback, we will use the 2-tuple linguistic model presented in [31, 32], with the linguistic domain can be treated as continuous and express any counting of information although it does not exactly match any linguistic term. The main advantage of this representation is to be continuous in its domain. The 2-tuple fuzzy linguistic model represents the linguistic information by means of a pair of values called linguistic 2-tuple, (s, α), where s is a linguistic term and α is a numeric value representing the symbolic translation.

Definition 1. [31] Let S = {s ₀, …, s _g} be a linguistic term set and β ∈ [0, g] be the result of an aggregation of the indexes of set S, i.e., being g + 1 the cardinality of S, then the 2-tuple that expresses the equivalent information to β is obtained with the following function: $\begin{matrix} Δ : [0, g] \to S \times [- 0.5, 0.5), \\ Δ (β) = (s_{i}, α), with {\begin{matrix} s_{i}, i = round (β) \\ α = β - i, α \in [- 0.5, 0.5) \end{matrix}, \end{matrix}$ where “round” is the usual round operation, s _i has the closest index label to “β”, and “α” is the value of the symbol translation. Generally, g + 1 is an odd number.

Definition 2. [31] Let S = {s ₀, …, s _g} be a linguistic term set and (s _i, α) be a 2-tuple, there exists a function, Δ ^-1, such that given a 2-tuple it returns its equivalent numerical value $β \in [0, g] \subset ℝ$ . $\begin{matrix} Δ^{- 1} : S \times [- 0.5, 0.5) \to [0, g], \\ Δ^{- 1} (s_{i}, α) = i + α = β . \end{matrix}$

It is obvious that the transformation of a linguistic term into a linguistic 2-tuple consist of adding to it the value 0 as a symbolic translation, s _i ∈ S ≡ (s _i, 0).

This model has a computational technique based on the 2-tuples were presented in [31]:

1) Comparison of 2-Tuples

The comparison of linguistic information represented by 2-tuples is carried out according to an ordinary lexicographic order.

Let (s _k, α ₁) and (s _l, α ₂) be two 2-tuples, then

If k < l then (s _k, α ₁) is smaller than (s _l, α ₂);

If k = l then

If α ₁ = α ₂ then (s _k, α ₁) and (s _l, α ₂) represent the same values;

If α ₁ < α ₂ then (s _k, α ₁) is smaller than (s _l, α ₂);

If α ₁ > α ₂ then (s _k, α ₁) is bigger than (s _l, α ₂).

2) Aggregation of 2-tuples

The aggregation of information consists of obtaining a value that summarizes a set of values, therefore, the result of the aggregation of a set of 2-tuples must be a linguistic 2-tuple. Herrera and Martínez [31] introduced several 2-tuple aggregation operators, which were based on classical aggregation.

Definition 3. [31] Let x = {(r ₁, α ₁) , …, (r _n, α _n)} be a set of 2-tuples, the extended Arithmetic Mean AM ^* using the linguistic 2-tuples is computed as ${AM}^{*} (x) = Δ (\sum_{i = 1}^{n} \frac{1}{n} Δ^{- 1} (r_{i}, α_{i})) = Δ (\frac{1}{n} \sum_{i = 1}^{n} β_{i}) .$

Definition 4. [31] Let x = {(r ₁, α ₁) , …, (r _n, α _n)} be a set of 2-tuples and W = (w ₁, …, w _n) ^T be their associated weight vector. The 2-tuple weighted mean, WAM ^*, is computed as $\begin{matrix} {WAM}^{*} (x) & = & Δ (\frac{\sum_{i = 1}^{n} Δ^{- 1} (r_{i}, α_{i}) \cdot w_{i}}{\sum_{i = 1}^{n} w_{i}}) \\ = & Δ (\frac{\sum_{i = 1}^{n} β_{i} \cdot w_{i}}{\sum_{i = 1}^{n} w_{i}}) . \end{matrix}$

3) Negation operator of 2-Tuples

We defined the negation operator over 2-tuples as: neg (s _i, α) = Δ (g - Δ ^-1 (s _i, α)), where g + 1 is the cardinality of S, s _i ∈ S = {s ₀, …, s _g}.

3 A revised procedure to deal with incomplete 2-tuple FLPRs

Fuzzy linguistic approach is a good way for experts to provide his/her preference degrees when comparing pairs of alternatives, since each expert has his/her own experience concerning the problem being studied, they could have some difficulties in giving all their preferences. This may be due to an expert not possessing a precise or sufficient level of knowledge of the problem, or because that expert is unable to discriminate the degree to which some alternatives are better than others, then incomplete FLPRs will appear. We assume FLPRs assessed on a 2-tuple fuzzy linguistic modeling [31] because it provides some advantages with respect to the ordinal fuzzy linguistic modeling [32]. So in this section we present a revised procedure to deal with incomplete 2-tuple FLPRs.

Definition 5. [1] A function f : X → Y is partial when not every element in the set X necessarily maps to an element in the set Y. When every element from the set X maps to one element in the set Y then we have a total function.

Definition 6. [1] A 2-tuple FLPR P on a set of alternatives X with a partial 2-tuple linguistic membership function is an incomplete 2-tuple FLPR.

In this paper, we estimate the missing values based on the linguistic additive consistency which can be seen as the parallel concept of Saaty’s consistency property for multiplicative preference relations. The mathematical formulation of the additive transitivity was given by Tanino [43]: $p_{ik} = p_{ij} + p_{jk} - 0.5 \forall i, j, k \in {1, \dots, n}$ (1)

When the preference values are expressed by fuzzy linguistic, we can define the linguistic additive transitivity property for 2-tuple FLPRs by using the transformation function Δ and Δ ^-1 as follows [1]:

$\begin{matrix} p_{ik} = Δ (Δ^{- 1} (p_{ij}) + Δ^{- 1} (p_{jk}) - Δ^{- 1} (s_{g / 2}, 0)), \\ \forall i, j, k \in {1, \dots, n} \end{matrix}$ (2)

In this paper, we only consider the additive consistent 2-tuple fuzzy linguistic preference relation, and also p _ii = (s _g/2, 0), while Alonso et al. [1] considered that p _ii =-.

And three other possible ways to estimate missing values can be derived from Equation (2), in fact, the preference value p _ik (i ≠ k) can be estimated using an intermediate alternative x _j in four different ways:

1. Since $p_{i k} = Δ (Δ^{- 1} (p_{i j}) + Δ^{- 1} (p_{j k}) - Δ^{- 1} (s_{g / 2}, 0))$ we can estimatep_ik by $({cp}_{ik})^{j 1} = Δ (Δ^{- 1} (p_{ij}) + Δ^{- 1} (p_{jk}) - Δ^{- 1} (s_{g / 2}, 0))$ (3)

2. Since $p_{i k} = Δ (Δ^{- 1} (p_{i j}) + Δ^{- 1} (p_{j k}) - Δ^{- 1} (s_{g / 2}, 0))$ we can estimate p_ik by $({cp}_{ik})^{j 2} = Δ (Δ^{- 1} (p_{jk}) - Δ^{- 1} (p_{ji}) + Δ^{- 1} (s_{g / 2}, 0))$ (4)

3. Since $p_{i k} = Δ (Δ^{- 1} (p_{i j}) + Δ^{- 1} (p_{j k}) - Δ^{- 1} (s_{g / 2}, 0))$ , we can estimate p_ik by

$({cp}_{ik})^{j 3} = Δ (Δ^{- 1} (p_{ij}) - Δ^{- 1} (p_{kj}) + Δ^{- 1} (s_{g / 2}, 0))$ (5) 4. Since $Δ (Δ^{- 1} (p_{i k}) + Δ^{- 1} (p_{k j}) - Δ^{- 1} (s_{g / 2}, 0)) + Δ (Δ^{- 1} (p_{j i}) - Δ^{- 1} (s_{g / 2}, 0)) = p_{i i} = s_{g / 2}$ , we have

$\begin{matrix} Δ (Δ^{- 1} (p_{ik}) + Δ^{- 1} (p_{kj})) \\ = Δ (3 Δ^{- 1} (s_{g / 2}) - Δ^{- 1} (p_{ji})) \end{matrix}$ (6)

Hence we can estimate p_ik by $({cp}_{ik})^{j 4} = Δ (3 Δ^{- 1} (s_{g / 2}, 0) - Δ^{- 1} (p_{ji}) - Δ^{- 1} (p_{kj}))$ (7)

In the following, we will use Equations (3)-(5) and (7) to develop a revised procedure which is different from Alonso et al.’s [1] to estimate missing preference values for incomplete 2-tuple FLPRs.

Algorithm 1

Step 1. Determine the set A, MVand EVof a FLPR P, as follows: $\begin{matrix} A = {(i, j) | i, j \in {1, \dots, n} \land i \neq j}, \\ MV = {(i, j) \in A | p_{ij} is unknown}, \\ EV = A ∖ MV = {(i, j) \in A | p_{ij} is known}, \end{matrix}$ where MVis the set pairs of alternatives for which the preference degrees are unknown or missing, EVis the set pairs of alternatives for which preference degrees are given by the expert. We do not take into account the preference value of one alternative over itself as this is always assumed to be equal to p_ii = (s_g/2,0).

Step 2. If MV = ϕ , then go to Step 4. Otherwise, the following revised iterative procedure is used to estimate the missing values p_ik in the set EMV_t at the t-th iteration, where EMV₀ = ϕ , (i,k) EMV_t , and t ≥ 1, shown as follows:

$\begin{matrix} {EMV}_{t} = {(i, k) \in MV ∖ \overset{t - 1}{\underset{l = 0}{\cup}} {EMV}_{l} | i \neq k \land \exists j \in \\ {(H_{ik}^{1})^{t} \cup (H_{ik}^{2})^{t} \cup (H_{ik}^{3})^{t} \cup (H_{ik}^{4})^{t}}} \end{matrix}$ (8)

With $(H_{ik}^{1})^{t} = {j | (i, j), (j, k) \in EV and (i, k) \in MV}$ (9) $(H_{ik}^{2})^{t} = {j | (j, i), (j, k) \in EV and (i, k) \in MV}$ (10) $(H_{ik}^{3})^{t} = {j | (i, j), (k, j) \in EV and (i, k) \in MV}$ (11) $(H_{ik}^{4})^{t} = {j | (j, i), (k, j) \in EV and (i, k) \in MV}$ (12) and EMV₀ = ϕ (by definition); ${(H_{i k}^{1})}^{t}, {(H_{i k}^{2})}^{t}, {(H_{i k}^{3})}^{t}, {(H_{i k}^{4})}^{t}$ are the set of intermediate alternatives x_j that can be used to estimate the preference value p_ik (i ≠ k). When EMV_max Iter = ϕ with maxIter >0, the procedure will stop as there will not be any more missing values to be estimated. Furthermore, if $\cup_{l = 0}^{m a x I t e r} E M V_{l} = M V$ , then all missing values are estimated, and consequently, the procedure is said to be successful in the completion of the incomplete 2-tuple FLPR.

Step 3. Let k = 0, then $\begin{matrix} {cp}_{ik}^{1} = {\begin{matrix} Δ ((\sum_{j \in {(H_{ik}^{1})}^{t}} Δ^{- 1} {({cp}_{ik})}^{j 1}) / # {(H_{ik}^{1})}^{t}), κ + + if # {(H_{ik}^{1})}^{t} \neq 0 \\ (s_{0}, 0) otherwise \end{matrix} \end{matrix}$ (13) $\begin{matrix} {cp}_{ik}^{2} = {\begin{matrix} Δ ((\sum_{j \in {(H_{ik}^{2})}^{t}} Δ^{- 1} {({cp}_{ik})}^{j 2}) / # {(H_{ik}^{2})}^{t}), κ + + if # {(H_{ik}^{2})}^{t} \neq 0 \\ (s_{0}, 0) otherwise \end{matrix} \end{matrix}$ (14) $\begin{matrix} {cp}_{ik}^{3} = {\begin{matrix} Δ ((\sum_{j \in {(H_{ik}^{h 3})}^{t}} Δ^{- 1} {({cp}_{ik})}^{j 3}) / # {(H_{ik}^{3})}^{t}), κ + + if # {(H_{ik}^{3})}^{t} \neq 0 \\ (s_{0}, 0) otherwise \end{matrix} \end{matrix}$ (15) $\begin{matrix} {cp}_{ik}^{4} = {\begin{matrix} Δ ((\sum_{j \in {(H_{ik}^{h 4})}^{t}} Δ^{- 1} {({cp}_{ik})}^{j 4}) / # {(H_{ik}^{4})}^{t}), κ + + if # {(H_{ik}^{4})}^{t} \neq 0 \\ (s_{0}, 0) otherwise \end{matrix} \end{matrix}$ (16) $\begin{matrix} {cp}_{ik} = {\begin{matrix} Δ (\frac{1}{κ} (Δ^{- 1} ({cp}_{ik}^{1}) + Δ^{- 1} ({cp}_{ik}^{2}) + Δ^{- 1} ({cp}_{ik}^{3}) + Δ^{- 1} ({cp}_{ik}^{4}))), κ \neq 0 \\ x, κ = 0 \end{matrix} \end{matrix}$ (17) where the symbol # denotes the number of elements in a given set.

When we use the function to compute the final estimated value of missing value $p_{i k}^{h}$ , we should point out that some estimated values might lie outside the interval [0,g], i.e., for some (i,k), we may have $Δ^{- 1} (c p_{i k}^{h}) < 0 o r Δ^{- 1} (c p_{i k}^{h}) > g$ . In order to normalize the expression domains in the decision model, we set the following function:

$f (y) = {\begin{matrix} (s_{0}, 0) if Δ^{- 1} (y) < 0 \\ (s_{g}, 0) if Δ^{- 1} (y) > g \\ y otherwise \end{matrix}$ (18)

Step 4. End.

Theorem 1. If an incomplete 2-tuple FLPR can be completed byI Equations (3) – (5) and (7), Ithen $Δ^{- 1} (p_{i k}) + Δ^{- 1} (p_{k i}) = g$ .

Proof. If there exist j that satisfies Equation (3), then it also satisfies Equation (7), and we can get $Δ^{- 1} ({(c p_{i k})}^{j 1}) + Δ^{- 1} ({(c p_{k i})}^{j 4}) = g$ . Similarly, for each jwhich satisfies Equations (4) and (5), there will be $Δ^{- 1} ({(c p_{i k})}^{j 2}) + Δ^{- 1} ({(c p_{k i})}^{j 2}) = g, Δ^{- 1} ({(c p_{i k})}^{j 3}) + Δ^{- 1} ({(c p_{k i})}^{j 3}) = g, Δ^{- 1} ({(c p_{i k})}^{j 4}) + Δ^{- 1} ({(c p_{k i})}^{j 1}) = g \cdot Δ^{- 1} (p_{i k}) + Δ^{- 1} (p_{k i})$ is the average value of all the estimated value using Equations (3)–(5) and (7), thus, $Δ^{- 1} (p_{i k}) + Δ^{- 1} (p_{k i}) = g$ holds.

Remark 1. Theorem 1 shows that if Equation (7) is added to estimate the missing values, the reciprocity holds for the missing values, while Alonso el al. [1]’s method could not, which also can be seen from their examples. In [48], we have illustrated an example and deeply investigated the revised procedure to estimate the missing elements. In the example, there are four alternatives X = { x₁, x₂, x₃, x₄ } and S = { s₀, ₁, s₂, s₃, s₄, s₅, s₆ } , g = 6. The expert provides the following incomplete FLPR $P = (\begin{matrix} (s_{3}, 0) & x & x & x \\ x & (s_{3}, 0) & (s_{2}, 0.4) & (s_{4}, - 0.4) \\ (s_{4}, - 0.4) & x & (s_{3}, 0) & x \\ x & (s_{2}, - 0.2) & x & (s_{3}, 0) \end{matrix})$

In order to show the performances of the Alonso el al. [1]’s method and the proposed method in this paper, here, we show the comparative results after the first estimation in Table 1. It is showed that the revised estimation procedure could estimate more elements in the first estimation iteration and the estimated values satisfy Theorem 1. Our procedure can estimate the missing elements more accurately and quickly. For more details, the reader could refer [48].

4 Managing the multi-granular FLPRs

According to the uncertainty degree that an expert qualifying a phenomenon has on it, the linguistic term set chosen to provide his knowledge will have more or less terms. When different experts have different uncertainty degrees on the phenomenon, then several linguistic term sets with a different granularity of uncertainty are necessary. So in this section, we propose a transform function to deal with multi-granular FLPRs.

4.1 Linguistic hierarchies

An important aspect related to the linguistic information, is the granularity of uncertainty, i.e., the level of discrimination among different degrees of uncertainty. The more knowledge the experts have about the problem the more granularity they can use to express their preferences. The linguistic hierarchies (LH) were introduced in [33] in order to accomplish processes of computing with words for multi-granular linguistic information in a precise way.

LH is a set of levels, where each level represents a linguistic term set with different granularity to the remaining levels. Each level belonging to a linguistic hierarchy is denoted as l(t,n(t)):

t, a number that indicates the level of the hierarchy

n(t), the granularity of the term set of the level t

The levels belonging to a linguistic hierarchy are ordered according to their granularity, i.e., for two consecutive levels tand t + 1, n(t + 1) >n(t). Therefore, the level t + 1is a refinement of the previous level t.

From the above concepts, we define a linguistic hierarchy, LH, as the union of all levels t: $LH = \underset{t}{\cup} l (t, n (t))$ (19)

Given an LH, we denote as S^n(t) the linguistic term set of LH corresponding to the level tof LH characterized by a granularity of uncertainty n(t): $S^{n (t)} = {s_{0}^{n (t)}, \dots, s_{n (t) - 1}^{n (t)}}$ (20)

Herrera and Martínez [33] proposed the granularity needed in each linguistic term set of the level tdepending on the value n(t)defined in the first level (3 and 7 respectively). Then the linguistic term set of level t+1 is obtained from its predecessor as L(t, n(t)) → L(t + 1,2 . n(t) - 1). Therefore, the linguistic hierarchy only can be 3, 5, 9, or 7, 13, which cannot be the hierarchy 5, 7, 9, etc. In this paper, we will pay our attention in the LH approach to extend it by keeping its characteristics of accuracy and expression domain but overcoming the limitation of the term sets that can be used in the multi-granular linguistic framework, which we call Extended Linguistic Hierarchies (ELH). This approach is based on the LH [33] and the 2-tuple linguistic representation model [31]. In the ELH, each linguistic hierarchy can have a granularity of uncertainty, i.e., the linguistic term set of level t + 1could not be obtained from its predecessor.

4.2 Unification of multi-granular FLPRs

In order to aggregate the experts’ multi-granular linguistic preference information, we should uniform different granular linguistic into an appropriate linguistic term set, so a transformation function between linguistic terms in different levels of the linguistic hierarchies should be proposed. And the ideal unification process should have the following rules:

1. Transformation between different linguistic term sets.

The transformation can be achieved from one granular linguistic to another granular linguistic, and it can be performed in the opposite direction. The transformation is reversible, i.e., if one granular linguistic information S^n(t) is transformed to another granular linguistic S^n(t’) , and the latter granular linguistic S^n(t’) is continue to be transformed to the former granular linguistic S^n(t”) , S^n(t”) should be equal to S^n(t) .

2. Equivalence of information transformation

That is, when the minimum linguistic value of one granular linguistic is transformed to another granular linguistic, it still is the minimum linguistic value; the middle linguistic value will be transformed to the middle linguistic value of another granular linguistic; and the maximum linguistic value will be still the maximum linguistic value. Other linguistic values of one granular linguistic should correspond to the appropriate positions in another granular linguistic with one-to-one correspondence, which can let the transformation without loss of information, and the original and transformed information are equivalent.

3. Uniqueness

The linguistic value in another granular linguistic has only one granularity corresponding to the original linguistic value.

Figure 1 depicts the above three rules. Figure 1 also denotes the ELHs, it shows that the latter level linguistic term set may not be from its predecessor. These conditions ensure that the transformation between multi-granular linguistic equivalent and without loss of information, to understand how these conditions are working, we will present transformation function among the linguistic term sets of different linguistic hierarchies which carry out these transformation processes without loss of information. So we shall define transformation function between any level of the linguistic hierarchies. The transformation function will use the 2-tuple linguistic modeling.

Definition 7. [33] Let LH = ∪^t l(t,n(t))be a linguistic hierarchy whose linguistic term sets are denoted as $S^{n (t)} = {s_{0}^{n (t)}, \dots, s_{n (t) - 1}^{n (t)}}$ , and let us consider the 2-tuple linguistic representation. The transformation function from a linguistic label in level tto a label in level t^’ , satisfying the linguistic hierarchy basic rules, is defined as:

$\begin{array}{l} T F_{t^{'}}^{t} : l (t, n (t)) \to l (t^{'}, n (t^{'})) \\ T F_{t^{'}}^{t} (s_{i}^{n (t)}, α^{n (t)}) \\ = Δ_{n (t^{'})} (\frac{Δ_{n (t)}^{- 1} (s_{i}^{n (t)}, α^{n (t)}) \cdot (n (t^{'}) - 1)}{n (t) - 1}) \end{array}$ (21)

Specially,

If $Δ_{n (t)}^{- 1} (s_{i}^{n (t)}, α^{n (t)}) = 0, t h e n T F_{t^{'}}^{t} (s_{0}^{n (t)}) = s_{0}^{n (\overset{'}{t})}$

If $\begin{array}{l} Δ_{n (t)}^{- 1} (s_{i}^{n (t)}, α^{n (t)}) = \frac{n (t) - 1}{2}, t h e n \\ T F_{t^{'}}^{t} (s_{(n (t) - 1) / 2}^{n (t)}) = s_{(n (\overset{'}{t) - 1 / 2}}^{n (\overset{'}{t})}; \end{array}$

If $\begin{array}{l} Δ_{n (t)}^{- 1} (s_{i}^{n (t)}, α^{n (t)}) = n (t) - 1, t h e n \\ T F_{t^{'}}^{t} (s_{n (t) - 1}^{n (t)}) = s_{n (t^{'}) - 1}^{n (t^{'})} \end{array}$

The above three formulae express the second criterion well, and from the Equation (21) we can confirm that arbitrary granularity has only one corresponding granularity in another granular linguistic, this transformation in the transformation domain ([0,n(t) - 1]) is continuous and uninterrupted, the characteristic of this transformation function is bijective and surjective, so the transformation has no loss of information.

The transformation function has the following properties:

Proposition 1. The transformation function between linguistic terms in different levels of the linguistic hierarchy is bijective: $T F_{t}^{t^{'}} (T F_{t^{'}}^{t} (s_{i}^{n (t)}, α^{n (t)})) = (s_{i}^{n (t)}, α^{n (t)}) .$ (22)

Proof. $\begin{array}{l} T F_{t}^{t^{'}} (T F_{t^{'}}^{t} (s_{i}^{n (t)}, α^{n (t)})) \\ = T F_{t}^{t^{'}} (Δ_{n (t^{'})} (\frac{Δ_{n (t)}^{1} (s_{i}^{n (t)}, α^{n (t)}) \cdot (n (t^{'}) 1)}{n (t) 1})) \\ = Δ_{n (t)} (\frac{Δ_{n (t^{'})}^{1} (Δ_{n (t^{'})} (Δ_{n (t)}^{1} (s_{i}^{n (t)}, α^{n (t)}) \cdot (n (t^{'}) 1)) \cdot (n (t) 1)}{n (t^{'}) 1}) \\ = Δ_{n (t)} (\frac{Δ_{n (t)}^{1} (s_{i}^{n (t)}, α^{n (t)}) \cdot (n (t^{'}) 1) \cdot (n (t) 1)}{(n (t) 1) \cdot (n (t^{'}) 1)}) \\ = (s_{i}^{n (t)}, α^{n (t)}) \end{array}$

This proposition describes that the information of one linguistic preference relation is transformed to another granular linguistic and then transformed into the original granular linguistic is still itself.

Proposition 2. The outcomes of indirect transformation and direct transformation between two different granular linguistic term sets are equivalent. $T F_{t^{''}}^{t^{'}} T F_{t^{'}}^{t} (s_{i}^{n (t)}, α^{n (t)}) = T F_{t^{''}}^{t} (s_{i}^{n (t)}, α^{n (t)})$ (23)

By Equation (23), we can easily obtain the following corollary:

Corollary 1. Let S^n(t₁),S^n(t₂), …,S^n(t_m) be the different granular linguistic term sets, then

$\begin{matrix} {TF}_{t_{m}}^{t_{m - 1}} ({TF}_{t_{m - 1}}^{t_{m - 2}} \dots {TF}_{t_{2}}^{t_{1}} (s_{i}^{n (t_{1})}, α^{n (t_{1})})) \\ = {TF}_{t_{m}}^{t_{1}} (s_{i}^{n (t_{1})}, α^{n (t_{1})}) . \end{matrix}$ (24)

That is, no matter how many intermediate transformations between the original and the final granular linguistic, it is equivalent to the direct transformation without losing any information.

Proposition 3. Complementarity $\begin{array}{l} T F_{t^{'}}^{t} (s_{i}^{n (t)}) \oplus T F_{t^{'}}^{t} (s_{n (t) 1 i}^{n (t)}) \\ = s_{\frac{i, (n (t^{i}) 1)}{n (t) 1}}^{n (t^{'})} \oplus s_{\frac{(n (t) 1 i . (n (t^{'}) 1)}{n (t) 1}}^{n (t^{'})} = s_{n (t^{'}) 1}^{n (t^{'})} \end{array}$ (25)

The two complementary granularities transformed into another granular linguistic are still complementarity.

Remark 3. The transformation function we have presented to aggregate the multi-granular linguistic information has no loss of information, which is better than the model presented in [27]. And another advantage is that we should not select a specific basic linguistic term set (BLTS), and we think it was not necessary that Herrera et al. [27] selected BLTS in accordance with certain requirements.

Remark 4. Although Equation (21) is proposed by Herrera and Martínez [33], and only can be used to transform different linguistic hierarchies 3, 5, 9, or 7, 13. In this paper, there is no such limitation. We can transform any two linguistic term sets. For example, there are two linguistic hierarchies, whose term sets are: $\begin{matrix} l (1, 5) {s_{0}^{5}, s_{1}^{5}, s_{2}^{5}, s_{3}^{5}, s_{4}^{5}} \\ l (2, 7) {s_{0}^{7}, s_{1}^{7}, s_{2}^{7}, s_{3}^{7}, s_{4}^{7}, s_{5}^{7}, s_{6}^{7}} \end{matrix}$

The transformations between terms of the different levels are carried out as: $\begin{matrix} {TF}_{2}^{1} (s_{3}^{5}, 0) = Δ_{7} (\frac{Δ_{5}^{- 1} (s_{3}^{5}, 0) \cdot (7 - 1)}{5 - 1}) \\ = Δ_{7} (4.5) = (s_{4}^{7}, 0.5) \\ {TF}_{1}^{2} (s_{3}^{7}, 0) = Δ_{5} (\frac{Δ_{7}^{- 1} (s_{3}^{7}, 0) \cdot (5 - 1)}{7 - 1}) \\ = Δ_{5} (2) = (s_{2}^{5}, 0) \end{matrix}$

The first equation means that s₃ in the granularity 5 will be the value (s₄, 0.5)in the granularity 7. The second equation means that s₃ in the granularity 7 will be the value (s₂, 0), the two values are the middle values in their granularity respectively. This meets the rules presented in this paper.

4.3 Dealing with multi-granular linguistic information

A. Aggregation phase

This phase has two steps:

Normalization process: it makes the multi-granular linguistic information uniform over an arbitrary linguistic term set. Firstly we should select an arbitrary linguistic term set be the final unified linguistic term set, and then we use the transformation function presented in Definition 7 to make all the experts’ linguistic information uniform.

Aggregation process: it combines the unified information of each design alternative to obtain an overall value.

Now the 2-tuple unified linguistic information expressed in a unique linguistic term set, we can select one of the 2-tuple aggregation operators (Definition 3 and 4) to obtain a collective preference values.

B. Exploitation phase

The decision process applies a choice degree to obtain a selection set of alternatives. Different choice functions have been proposed in the literature [31]. The choice functions rank the alternatives according to different possibilities and from the ranking alternative(s) are obtained. In this paper, a choice function that computes the dominance degree for each alternative, x_i , over the other alternatives is used as follows: $z_{i} = Δ (\sum_{i = 1}^{n} \frac{1}{n} Δ^{- 1} (p_{ik}))$ (26)

z_i is the preference degree of the i-th alternative over all the other alternatives.

Then, the best alternative(s) are in the head of ranking should be chosen as solution set of alternatives.

5 An incomplete multi-granular linguistic evaluation of emergency decisions for unconventional outburst incidents

The aim of this paper is to develop an expert incomplete multi-granular linguistic evaluation model for Emergency Decisions, in which, the experts can express their assessments about the Emergency Decisions by means of linguistic terms that can be assessed in different linguistic term sets and some assessments may be missing because of experts’ imperfect information. And in this section we will present the evaluation procedure to rank alternative(s) for the Emergency Decisions.

5.1 Unconventional outburst incidents

In recent years, with the accelerated social development and urbanization process, the increase and centralization of human settlements and asset size, and the mobility and complexity of society more than ever before, unconventional outburst incidents are growing in the worldwide, which can bring huge economic losses and social issues, and have a major impact to economic and social development. Unconventional outburst incidents have no sufficient precursor characteristics and have potential derivative harm and severe destruction, using conventional management system cannot cope with them, which include natural disasters, accidents, public health incidents and social safety incidents. Natural disasters are common in unconventional outburst incidents and no one can be immune from natural disasters. From tsunamis and earthquakes to floods and famines, the forces of nature increasingly threaten human beings. During disasters, lives are lost; homes and workplaces are destroyed; essential services are disrupted, and hunger, injury, and diseases are widespread. This shows that unconventional outburst incidents bring tremendous social, economic, and environmental impacts. In order to make the losses caused by unconventional outburst incidents to minimize, we should take comparative analysis of emergency decision alternatives set to select the optimal alternative(s) for implementation.

5.2 Illustrative example: Ranking alternatives

More and more disasters have happened in the worldwide in the recent years and they brought a huge impact and threat to people’s lives. For example, China’s Wenchuan Earthquake in 2008 and Japan’s 311 Earthquake in 2011 are all big disasters. The scheduling of relief supplies to earthquake relief work is an important link, and how to ensure earthquake relief supplies in the shortest time, highest security and the best economy way to transport goods demand points is very important. Here we give four different emergency decisions of the scheduling: (1) Aircraft emergency delivery of relief supplies; (2) Repair damaged roads and transport relief supplies by car; (3) Repair damaged railway and train transport; (4) Transport with large numbers of people on foot.

Our purpose is to choose the most suitable design alternative for the above four emergency decisions. In our problem, the alternatives for the Emergency Decision alternatives X = { x₁,x₂,x₃,x₄ } will be evaluated by the experts E = { e₁,e₂,e₃ } . The linguistic term sets of the expert e₁, e₂, e₃ are $S^{5} = {s_{0}^{5}, \dots, s_{4}^{5}}, S^{7} = {s_{0}^{7}, \dots, s_{6}^{7}}, S^{9} = {s_{0}^{9}, \dots, s_{8}^{9}}$ respectively. But there may appear incomplete preference relations, this may be due to an expert not possessing a precise or sufficient level of knowledge of the problem, or unable to discriminate the merits degree between some pairs of alternatives. So we shall apply the incomplete multi-granular linguistic in order to solve our problem. These three preference relations are proposed as following: $\begin{matrix} P^{1} = (\begin{matrix} (s_{2}^{5}, 0) & (s_{2}^{5}, 0) & (s_{3}^{5}, 0) & (s_{3}^{5}, 0) \\ (s_{2}^{5}, 0) & (s_{2}^{5}, 0) & x & (s_{3}^{5}, 0) \\ (s_{1}^{5}, 0) & x & (s_{2}^{5}, 0) & x \\ (s_{1}^{5}, 0) & (s_{1}^{5}, 0) & x & (s_{2}^{5}, 0) \end{matrix}), \\ P^{2} = (\begin{matrix} (s_{3}^{7}, 0) & (s_{2}^{7}, 0) & (s_{4}^{7}, 0) & (s_{3}^{7}, 0) \\ (s_{4}^{7}, 0) & (s_{3}^{7}, 0) & (s_{4}^{7}, 0) & (s_{4}^{7}, 0) \\ (s_{2}^{7}, 0) & (s_{2}^{7}, 0) & (s_{3}^{7}, 0) & (s_{2}^{7}, 0) \\ (s_{3}^{7}, 0) & (s_{2}^{7}, 0) & (s_{4}^{7}, 0) & (s_{3}^{7}, 0) \end{matrix}), \\ P^{3} = (\begin{matrix} (s_{4}^{9}, 0) & (s_{3}^{9}, 0) & x & (s_{5}^{9}, 0) \\ (s_{5}^{9}, 0) & (s_{4}^{9}, 0) & (s_{5}^{9}, 0) & x \\ x & (s_{3}^{9}, 0) & (s_{4}^{9}, 0) & (s_{6}^{9}, 0) \\ (s_{3}^{9}, 0) & x & (s_{2}^{9}, 0) & (s_{4}^{9}, 0) \end{matrix}) . \end{matrix}$

5.2.1 Estimate missing values of the incomplete 2-tuple FLPR

Now we shall choose the most suitable design alternative with the incomplete multi-granular linguistic preference relations. Firstly we should complete the incomplete linguistic preference relations, and we will use the estimation procedure proposed in Section 3.1 to estimate the missing values. As we observe two 2-tuple FLPRs are incomplete { P¹,P³ } . P¹ is an incomplete 2-tuple fuzzy linguistic preference relation, so we can complete it using the consistency based procedure to estimate missing information.

Iteration 1.The set of elements that can be estimated are: ${EMV}_{1} = {(2, 3), (3, 2), (3, 4), (4, 3)}$

After these elements have been estimated, we have: $P^{1} = (\begin{matrix} (s_{2}^{5}, 0) & (s_{2}^{5}, 0) & (s_{3}^{5}, 0) & (s_{3}^{5}, 0) \\ (s_{2}^{5}, 0) & (s_{2}^{5}, 0) & (s_{3}^{5}, 0) & (s_{3}^{5}, 0) \\ (s_{1}^{5}, 0) & (s_{1}^{5}, 0) & (s_{2}^{5}, 0) & (s_{2}^{5}, 0) \\ (s_{1}^{5}, 0) & (s_{1}^{5}, 0) & (s_{2}^{5}, 0) & (s_{2}^{5}, 0) \end{matrix})$ because EMV₂ = ϕ and ∪_l = 0^maxIter EMV_l = MV, the P¹ is a complete 2-tuple FLPR now.

As an example, to estimate p₂₃ the procedure is as follows: $\begin{matrix} H_{23}^{1} = {1} {cp}_{23}^{1} = Δ (Δ^{- 1} ({cp}_{23})^{11}) \\ = Δ (Δ^{- 1} (Δ (Δ^{- 1} (p_{21}) \\ + Δ^{- 1} (p_{13}) - Δ^{- 1} (s_{g / 2}, 0)))) \\ = Δ (Δ^{- 1} Δ (2 + 3 - 2)) \\ = Δ (Δ^{- 1} Δ (3)) = (s_{3}^{5}, 0) \\ H_{23}^{2} = {1} {cp}_{23}^{2} = Δ (Δ^{- 1} ({cp}_{23})^{12}) \\ = Δ (Δ^{- 1} (Δ (Δ^{- 1} (p_{13}) \\ - Δ^{- 1} (p_{12}) + Δ^{- 1} (s_{g / 2}, 0)))) \\ = Δ (Δ^{- 1} Δ (3 - 2 + 2)) \\ = Δ (Δ^{- 1} Δ (3)) = (s_{3}^{5}, 0) \\ H_{23}^{3} = {1} {cp}_{23}^{3} = Δ (Δ^{- 1} ({cp}_{23})^{13}) \\ = Δ (Δ^{- 1} (Δ (Δ^{- 1} (p_{21}) \\ - Δ^{- 1} (p_{31}) + Δ^{- 1} (s_{g / 2}, 0)))) \\ = Δ (Δ^{- 1} Δ (2 - 1 + 2)) \\ = Δ (Δ^{- 1} Δ (3)) = (s_{3}^{5}, 0) \\ H_{23}^{4} = {1} {cp}_{23}^{4} = Δ (Δ^{- 1} ({cp}_{23})^{14}) \\ = Δ (Δ^{- 1} (Δ (3 Δ^{- 1} (s_{g / 2}, 0) \\ - Δ^{- 1} (p_{12}) - Δ^{- 1} (p_{31})))) \\ = Δ (Δ^{- 1} Δ (6 - 2 - 1)) \\ = Δ (Δ^{- 1} Δ (3)) = (s_{3}^{5}, 0) \end{matrix}$ $\begin{matrix} {cp}_{14} = Δ (\frac{1}{4} (Δ^{- 1} ({cp}_{14}^{1}) + Δ^{- 1} ({cp}_{14}^{2}) \\ + Δ^{- 1} ({cp}_{14}^{3}) + Δ^{- 1} ({cp}_{14}^{4}))) \\ = Δ (\frac{3 + 3 + 3 + 3}{4}) = (s_{3}^{5}, 0) . \end{matrix}$

With the same approach we can obtain the complete 2-tuple FLPR P³ : $P^{3} = (\begin{matrix} (s_{4}^{9}, 0) & (s_{3}^{9}, 0) & (s_{4}^{9}, - 0.5) & (s_{5}^{9}, 0) \\ (s_{5}^{9}, 0) & (s_{4}^{9}, 0) & (s_{5}^{9}, 0) & (s_{7}^{9}, - 0.5) \\ (s_{5}^{9}, - 0.5) & (s_{3}^{9}, 0) & (s_{4}^{9}, 0) & (s_{6}^{9}, 0) \\ (s_{3}^{9}, 0) & (s_{2}^{9}, - 0.5) & (s_{2}^{9}, 0) & (s_{4}^{9}, 0) \end{matrix}) .$

We estimate the missing values of the incomplete 2-tuple FLPR and these three preference relations are all complete, but they are in the different linguistic term sets, so we should uniform them into the same linguistic term set.

5.2.2 Deal with multi-granular linguistic information

In this paper the individual information of each expert is combined to obtain collective preference values for each design alternative and then we choose the most suitable design alternative. But as the conditions are assessed in a multi-granular linguistic context, we will deal with the information in two phases.

A. Aggregation phase

This phase has two steps:

1. Normalization process

We are dealing with multi-granular linguistic information, to manage it the model unifies it in a common utility space. Here we use a utility linguistic term set S_T : $\begin{matrix} S_{T} = {Greatly not Preferred, not Preferred, Slightly \\ notPreferred, Average, Slightly Preferred, \\ Preferred, Greatly Preferred} \end{matrix}$

Once we have chosen the common utility space to express the suitability of the design alternatives we shall transform all the input assessments into linguistic 2-tuples assessed in the S_T . The domain of P² is similar to S_T expect in the syntax. So the transformation of the assessments in P² will consist of using the syntax of the S_T without any other change. But the performance assessments of P¹ and P³ are assessed in the linguistic term sets with different granularity, syntax and semantics to the S_T , so we will use the transformation function presented in Definition 6 to transform these assessments into linguistic 2-tuples into the S_T .

The transformed $P_{T}^{1}$ is as the following: $P_{T}^{1} = (\begin{matrix} (s_{3}^{7}, 0) & (s_{3}^{7}, 0) & (s_{5}^{7}, - 0.5) & (s_{5}^{7}, - 0.5) \\ (s_{3}^{7}, 0.3) & (s_{3}^{7}, 0) & (s_{5}^{7}, - 0.5) & (s_{5}^{7}, - 0.5) \\ (s_{2}^{7}, - 0.5) & (s_{2}^{7}, - 0.5) & (s_{3}^{7}, 0) & (s_{3}^{7}, 0) \\ (s_{2}^{7}, - 0.5) & (s_{2}^{7}, - 0.5) & (s_{3}^{7}, 0) & (s_{3}^{7}, 0) \end{matrix})$

As an example, the transform procedure of $P_{12}^{1}$ is as follows: $\begin{matrix} {TF}_{T}^{1} (s_{2}^{5}, 0) & = & Δ_{7} (\frac{Δ_{5}^{- 1} (s_{2}^{5}, 0) \cdot (7 - 1)}{5 - 1}) \\ = & Δ_{7} (3) = (s_{3}^{7}, 0) . \end{matrix}$

And the transformed $P_{T}^{3}$ will be: $\begin{matrix} P_{T}^{3} \\ = (\begin{matrix} (s_{3}^{7}, 0) & (s_{2}^{7}, 0.25) & (s_{3}^{7}, - 0.375) & (s_{4}^{7}, - 0.25) \\ (s_{4}^{7}, - 0.25) & (s_{3}^{7}, 0) & (s_{4}^{7}, - 0.25) & (s_{5}^{7}, - 0.125) \\ (s_{3}^{7}, 0.375) & (s_{2}^{7}, 0.25) & (s_{3}^{7}, 0) & (s_{5}^{7}, - 0.5) \\ (s_{2}^{7}, 0.25) & (s_{1}^{7}, 0.125) & (s_{2}^{7}, - 0.5) & (s_{3}^{7}, 0) \end{matrix}) \end{matrix}$

2. Aggregation process

This process combines the assessments that express the values for the different conditions to obtain a global value for each design alternative. We want to obtain an evaluation value for each design alternative according to the above uniformed performance values expressed by means of linguistic 2-tuples in the S_T . And the global value will be expressed with the syntax of S_T.

In this case we present the use of the 2-tuple weighted aggregation operator [31]. With the preference relations $P_{T}^{1}, P^{2}$ and $P_{T}^{3}$ , the aggregated values are obtained using the following expression: $\begin{matrix} W_{A} M^{*} ((p_{T}^{1}, α_{1}), (p^{2}, α_{2}), (p_{T}^{3}, α_{3})) \\ = Δ (Δ^{- 1} (p_{T}^{1}, α_{1}) \cdot w_{1} + Δ^{- 1} (p^{2}, α_{2}) \cdot \\ w_{2} + Δ^{- 1} (p_{T}^{3}, α_{3}) \cdot w_{3}) \end{matrix}$ where $(p_{T}^{1}, α_{1}), (p^{2}, α_{2}) a n d (p_{T}^{3}, α_{3})$ are the assessments of $P_{T}^{1}, P^{2} a n d P_{T}^{3}$ respectively. And w₁ , w₂ , w₃ such that w₁ + w₂ + w₃ = 1 being the importance for each expert respectively. We suppose w₁ = 0.3, w₂ = 0.5 and w₃ = 0.2. Then we can obtain the collective Pwith the above $P_{T}^{1}, P^{2} a n d P_{T}^{3}$ . $P = (\begin{matrix} (s_{3}^{7}, 0) & (s_{2}^{7}, 0.35) & (s_{4}^{7}, - 0.125) & (s_{4}^{7}, - 0.4) \\ (s_{4}^{7}, - 0.35) & (s_{3}^{7}, 0) & (s_{4}^{7}, 0.1) & (s_{4}^{7}, 0.325) \\ (s_{2}^{7}, 0.125) & (s_{2}^{7}, - 0.1) & (s_{3}^{7}, 0) & (s_{3}^{7}, - 0.2) \\ (s_{2}^{7}, 0.4) & (s_{2}^{7}, - 0.325) & (s_{3}^{7}, 0.2) & (s_{3}^{7}, 0) \end{matrix})$

B. Exploitation phase

Utilize the 2-tuple arithmetic mean operator $z_{i} = Δ (\sum_{i = 1}^{n} \frac{1}{n} Δ^{- 1} (p_{i k}))$ to get the preference degree z_i of the i-th alternative over all the other alternatives. $\begin{matrix} z_{1} = Δ (3.21) = (s_{3}^{7}, 0.21), \\ z_{2} = Δ (3.77) = (s_{4}^{7}, - 0.23), \\ z_{3} = Δ (2.46) = (s_{2}^{7}, 0.46), \\ z_{4} = Δ (2.56) = (s_{3}^{7}, - 0.44) . \end{matrix}$

Rank all the alternatives and select the optimal one(s) in accordance with the values of z_i (i = 1,2,3,4): $x_{2} ≻ x_{1} ≻ x_{4} ≻ x_{3}$

Thus, the most suitable design alternative is x₂ .

Remark 4: When dealing with multi-granular linguistic information, we can use different utility linguistic term sets, and the sort of alternatives and the optimal one(s) will be same with the same weighted aggregation operator and experts’ weights.

Here we use another utility linguistic term set $S_{T^{'}}$ : $S_{T^{'}} = {N o n e, W o r s e, A v e r a g e, B e t t e r, P e r f e c t}$

And the transformed $P_{T^{'}}^{2}$ and $P_{T^{'}}^{3}$ are as the following: $\begin{array}{l} P_{T^{'}}^{2} = (\begin{array}{c} (s_{2}^{5}, 0) & (s_{1}^{5}, 0.33) & (s_{3}^{5}, - 0.33) & (s_{2}^{5}, 0) \\ (s_{3}^{5}, - 0.33) & (s_{2}^{5}, 0) & (s_{3}^{5}, - 0.33) & (s_{3}^{5}, - 0.33) \\ (s_{1}^{5}, 0.33) & (s_{1}^{5}, 0.33) & (s_{2}^{5}, 0) & (s_{1}^{5}, 0.33) \\ (s_{2}^{5}, 0) & (s_{1}^{5}, 0.33) & (s_{3}^{5}, - 0.33) & (s_{2}^{5}, 0) \end{array}) \\ P_{T^{'}}^{3} = (\begin{array}{c} (s_{2}^{5}, 0) & (s_{2}^{5}, - 0.5) & (s_{2}^{5}, - 0.25) & (s_{3}^{5}, - 0.5) \\ (s_{3}^{5}, - 0.5) & (s_{2}^{5}, 0) & (s_{3}^{5}, - 0.5) & (s_{3}^{5}, 0.25) \\ (s_{2}^{5}, 0.25) & (s_{2}^{5}, - 0.5) & (s_{2}^{5}, 0) & (s_{3}^{5}, 0) \\ (s_{2}^{5}, - 0.5) & (s_{1}^{5}, - 0.25) & (s_{1}^{5}, 0) & (s_{2}^{5}, 0) \end{array}) \end{array}$

With the above same weighted aggregation operator and w₁ = 0.3, w₂ = 0.5, w₃ = 0.2. Then we can obtain the collective P^’ with the above P¹ , $P_{T^{'}}^{2}$ and $P_{T^{'}}^{3}$ . $p^{'} = (\begin{array}{c} (s_{2}^{5}, 0) & (s_{2}^{5}, - 0.43) & (s_{3}^{5}, - 0.42) & (s_{2}^{5}, 0.4) \\ (s_{2}^{5}, 0.43) & (s_{2}^{5}, 0) & (s_{3}^{5}, - 0.27) & (s_{3}^{5}, - 0.12) \\ (s_{1}^{5}, 0.42) & (s_{1}^{5}, 0.27) & (s_{2}^{5}, 0) & (s_{2}^{5}, - 0.13) \\ (s_{2}^{5}, - 0.4) & (s_{1}^{5}, 0.12) & (s_{2}^{5}, 0.13) & (s_{2}^{5}, 0) \end{array})$ and the preference degree z_i are $\begin{matrix} z_{1} = Δ (2.14) = (s_{2}^{5}, 0.14), \\ z_{2} = Δ (2.51) = (s_{3}^{5}, - 0.49), \\ z_{3} = Δ (1.64) = (s_{2}^{5}, - 0.36), \\ z_{4} = Δ (1.71) = (s_{2}^{5}, - 0.29) . \end{matrix}$

Now the sort of all the alternatives is still x₂ ≻ x₁ ≻ x₄ ≻ x₃ and the most suitable design alternative is still x₂.

6 Conclusions

Our proposal for evaluating design alternatives before its implementation for the Unconventional Outburst Incidents is based on an incomplete multi-granular linguistic framework where each preference relation is conducted in different expression domains. We have used the consistency based procedure to estimate the missing values in the incomplete 2-tuple FLPR and modeled this evaluation framework as a multi-conditions model that is able to deal with incomplete multi-granular linguistic assessments without loss of information in order to evaluate and rank the different alternatives.

In the future, our model may be used to the heterogeneous frameworks with experts’ multi-granular linguistic term sets [5, 40] and some new consensus models [2 , 16]. We will also address this proposed evaluation model to the unbalanced [7] or dynamic FLPRs [41] and its application to model experts’ desires or preferences in complex engineering systems [36] and digital libraries [13, 14].

Footnotes

Acknowledgments

The authors are very grateful to the Associate Editor and the three anonymous reviewers for their constructive comments and suggestions that have further helped to improve the quality and presentation of this paper. This work was partly supported by the National Natural Science Foundation of China (NSFC) under Grants 71101043, 71471056, 7143303, the Fundamental Research Funds for the Central Universities Program for Excellent Talents in Hohai University.

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