### Abstract

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. LINGUISTIC BACKGROUND
- 3. RELATED WORK
- 4. EXTENDED LINGUISTIC HIERARCHIES
- 5. AN MEDM PROBLEM DEALING WITH ELH
- 6. CONCLUDING REMARKS
- ACKNOWLEDGMENTS
- REFERENCES

In multi-expert decision making (MEDM) problems the experts provide their preferences about the alternatives according to their knowledge. Because they can have different knowledge, educational backgrounds, or experiences, it seems logical that they might use different evaluation scales to express their opinions. In the present article, we focus on decision problems defined in uncertain contexts where such uncertainty is modeled by means of linguistic information, therefore the decision makers would use different linguistic scales to express their evaluations on the alternatives, i.e., multigranular linguistic scales. Several computational approaches have been presented to manage multigranular linguistic scales in decision problems. Although they provide good results in some cases, still present limitations. A new approach, so-called extended linguistic hierarchies, is presented here for managing multigranular linguistic scales to overcome those limitations, an MEDM case study is given to illustrate the proposed method.

### 1. INTRODUCTION

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. LINGUISTIC BACKGROUND
- 3. RELATED WORK
- 4. EXTENDED LINGUISTIC HIERARCHIES
- 5. AN MEDM PROBLEM DEALING WITH ELH
- 6. CONCLUDING REMARKS
- ACKNOWLEDGMENTS
- REFERENCES

In real-world application, we face decision-making problems quite often, which may be involved with different types of information, decision makers, criteria, etc. We focus on multi-expert decision making (MEDM) problems dealing with qualitative information (Xu 2004, 2006; Herrera, Martínez, and Sánchez 2005; Chen and Ben-Arieh 2006), the presence of qualitative information involves uncertainty and vagueness. To manage this uncertainty the probability theory might be a powerful tool, but in many problems it is not difficult to see that a lot of aspects of uncertainties clearly do not have a probability character because they are related to imprecision and vagueness of meaning (Martínez et al. 2005). The use of fuzzy linguistic approach (Zadeh 1975) for managing such imprecision and vagueness have obtained good results in different disciplines, such as “information retrieval” (Bordogna and Pasi 1993; Herrera-Viedma 2001; Herrera-Viedma, Lopez-Herrera, and Porcel 2005; Zadrozny and Kacprzyk 2006), “recommender systems” (Martínez, Pérez, and Barranco 2007b; Martínez et al. 2008a; Porcel, López-Herrera, and Herrera-Viedma 2009),“evaluation processes” (Cheng and Lin 2002; Kuo and Yeh 2003; Chang, Wang, and Wang 2006, 2007; Chen, Lin, and Huang 2006; Martínez, Ruan, and Yang 2007a; Martínez, Espinilla, and Pérez 2008b; Sánchez et al. 2009), and “decision making” (Yager 1995; Arfi 2005; Chen and Ben-Arieh 2006; García-Lapresta 2006; Ma et al. 2007; Lu, Zhang, and Wu 2008; Mata, Martínez, and Herrera-Viedma 2009).

Another 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. Typical values of cardinality used in the linguistic models are odd ones, such as 7 or 9 (Miller 1956).

An MEDM problem framework is characterized by a finite set of decision makers (experts), *E* ={*e*_{1}, …, *e*_{m}}, *m* ≥ 2, who are called to express their preference values on a predefined set of alternatives, *X* ={*x*_{1}, …, *x*_{n}}, *n* ≥ 2. Because we focus on MEDM problems dealing with linguistic information, it is not difficult to find situations where different decision makers have different degree of knowledge about the problem. They could then express their preferences by different linguistic scales, *F*_{MS} ={*S*_{i}, *i* = 1, …, *m*, *with m* > 1}, i.e., the problem domain, *F*_{MS}, is formed by different linguistic term sets, *S*_{i} ={*s*^{i}_{j}, *j* = 0, 1, …, *g*_{i}} whose granularity, *g*_{i} + 1, is different from each other. It means that the problem is defined in a multigranular linguistic framework, i.e., with multiple linguistic scales.

The aim of this article is to deal with MEDM problems defined in multigranular linguistic frameworks. To solve these problems, different approaches have been proposed to accomplish processes of CW with this type of information (Herrera, Herrera-Viedma, and Martínez 2000; Herrera and Martínez 2001; Huynh and Nakamori 2005; Chen and Ben-Arieh 2006; Chang et al. 2007). These approaches define different frameworks and computational models to deal with the multigranular linguistic information (MGLI). Nevertheless, all of them present a similar computational process to aggregate the information assessed in multiple scales that, slightly modifies the aggregation phase of the resolution scheme by splitting it in two processes (see Figure 1): (i) A unification process that transforms the MGLI into just one expression domain; (ii) an aggregation process that is similar to the models with one expression domain. Eventually, when we analyze these approaches, from the experts’ point of view, regarding their modeling, understanding, and accuracy, no one is better than all the others for all the properties, e.g., either has some restriction on the selection of multiple linguistic scales or loss information while unifies the multiple granular terms.

Accordingly our proposal consists of a new approach to deal with multiple linguistic scales in MEDM problems as well, which, however, will be able to handle any linguistic term set in a symbolic way and without loss of information to which the existing approaches so far cannot satisfy altogether. This new approach, so-called extended linguistic hierarchies (ELH), and its computational model are based on the linguistic 2-tuple representation model (Herrera and Martínez 2000) and the linguistic hierarchies (LH; Herrera and Martínez 2001).

This article is structured as follows: Section 2 introduces a linguistic background revising in short the fuzzy linguistic approach and the 2-tuple fuzzy linguistic representation model. Section 3 reviews some related work and compares the different approaches to deal with multiple linguistic scales. Section 4 presents the ELH framework and its computational model. In Section 5, the ELH is applied to an MEDM problem with multiple linguistic scales and a comparative analysis with other approaches is also given. Section 6 concludes the article.

### 3. RELATED WORK

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. LINGUISTIC BACKGROUND
- 3. RELATED WORK
- 4. EXTENDED LINGUISTIC HIERARCHIES
- 5. AN MEDM PROBLEM DEALING WITH ELH
- 6. CONCLUDING REMARKS
- ACKNOWLEDGMENTS
- REFERENCES

In this section, we are going to review different approaches proposed to deal with multiple linguistic scales. In the literature, we can find many situations where multiple linguistic scales either have been used or could be used, such as: situation awareness (Lu et al. 2008), decision models (Delgado, Verdegay, and Vila 1992; Bordogna, Fedrizzi, and Pasi 1997; Marimin, Hatono, and Tamura 1998), information retrieval (Herrera-Viedma 2001), clinical diagnosis (Degani and Bortolan 1988), risk in software development (Lee 1996), educational grading systems (Law 1996), design evaluation (Martínez et al. 2005, 2007a), sensory Evaluation (Martínez 2007; Martínez et al. 2008b), performance appraisal (García-Lapresta, de Andrés, and Martínez 2010), and recommender systems (Martínez et al. 2007b, 2008a).

Here, we pay more attention to the offered frameworks and their computational models to check how they accomplish the aggregation phase showed in Figure 1.

#### 3.1. Fusion Approach I for Managing Multigranular Linguistic Information

This approach presented by Herrera et al. (2000) introduced a methodology to deal with information assessed in different linguistic scales based on fuzzy sets and the extension principle.

* 3.1.1. Framework* This approach does not impose any limitation to the term sets, which can belong to the multiple linguistic scales in the definition of the problem. Let us suppose that *F*_{MS} and *S*_{i} are defined as above.

* 3.1.2. Unification Phase* The computational model of this approach follows the aggregation phase showed in Figure 1. Therefore, in the first place it unifies the MGLI into a linguistic term set, so-called basic linguistic term set (BLTS) by means of fuzzy sets. This unification is carried out in the following two steps:

- •
Election of the BLTS: In

Herrera et al. (2000) were introduced the rules that lead to the election of the BLTS,

*S*_{T}, with the aim of keeping as much information as possible.

- •
Unification process: Once the BLTS has been chosen the MGLI can be transformed into BLTS by using a transformation function

which can express any linguistic term

*s*^{i}_{j} ∈

*S*_{i} as a fuzzy set defined in

*S*_{T}.

Let us present an example to illustrate this process.

**Example 1** Let *S*_{i} ={*s*^{i}_{0}, , …, *s*^{i}_{4}} and *S*_{T} ={*s*^{T}_{0}, , …, *s*^{T}_{6}} be the linguistic term sets in *F*_{MS} and BLTS with *g*_{i} = 4 and *g*_{T} = 6, respectively, associated with the following semantics (see Figure 2):

The fuzzy sets obtained by applying for *s*^{i}_{0} and *s*^{i}_{1} are:

In Figure 2, we can see the conversion process for both linguistic term sets graphically.

* 3.1.3. Computational Phase* Once the MGLI has been unified into just one expression domain, the computations required to obtain a solution can be carried out by using an adequate computational model. This approach operates directly over the fuzzy sets by using the fuzzy arithmetic (Dubois and Prade 1980; Herrera et al. 2000). Therefore, the results obtained will be fuzzy sets in the BLTS.

#### 3.2. Fusion Approach II for Managing Multigranular Linguistic Information

A new approach to deal with MGLI that extends the one introduced in Section 3.1 was proposed by Chen and Ben-Arieh (2006). This approach overcomes the limitation regarding the necessity that the granularity of the BLTS would be greater than the other term sets. Additionally it provides a different method to unify the linguistic information assessed in different scales.

* 3.2.1. Framework* This approach provides a similar framework, *F*_{MS}, as that in the previous approach, i.e., there is no limitation with regards to the term sets that can belong to *F*_{MS}.

* 3.2.2. Unification Phase* Similar to the approach presented in Section 3.1, this one also unifies the MGLI by means of fuzzy sets, but it introduces a new transformation function to unify the MGLI that allows to transform linguistic terms between any term set. Therefore, it is not necessary to choose a priori any BLTS, which could be any of the term sets in *F*_{MS}.

To unify the information a new transformation function, , is proposed:

**Example 2** This example shows the transformation, , of the label *s*^{i}_{4} between the term sets, *S*_{i} with *g*_{i}=8 and the target set *S*_{T} with *g*_{T} = 10.

For *j* = 4: *k*_{min}= 4 and *k*_{max}= 6. Therefore,

and μ_{4k}(*x*) = 0 for all *k* = 0, 1, …, 10 with *k* ≠ 4, 5, 6.

The result obtained to express (*s*^{i}_{4}) in *S*_{T} is the following fuzzy set:

The transformation function, , presented in Definition 4 provides a valid function for term sets where their terms have no overlap between their membership functions each other. If they overlap their functions one another in Chen and Ben-Arieh (2006) was defined an extended transformation function for such cases .

* 3.2.3. Computational Phase* Given that, this approach unifies the MGLI by means of fuzzy sets, it uses the same computation model as the one in Section 3.1.

#### 3.3. Linguistic Hierarchies

Although, the previous approaches in Sections 3.1 and 3.2 provide a way to deal with MGLI, both of them present several drawbacks related to the accuracy and to the expression domain for the computed results. To overcome these drawbacks it was introduced a new approach to deal with MGLI in a symbolic and precise way based on the LH (Herrera and Martínez 2001).

This approach is based on the 2-tuple linguistic representation model and its symbolic computational model.

* 3.3.1. Framework* In this approach the framework is limited and not all the term sets can be part of *F*_{MS}. To understand the construction of multigranular frameworks with LH, we review briefly its building process.

*Building a Linguistic Hierarchy* The multigranular linguistic frameworks offered by LH must satisfy several rules. In the following we are going to show its building process and some notations for an LH.

An LH is the union of all levels *t*: , where each level *t* of an LH corresponds to a linguistic term set with a granularity of uncertainty of *n*(*t*) denoted as: *S*^{n(t)}={*s*^{n(t)}_{0}, …, *s*^{n(t)}_{n(t) − 1}}.

The methodology to construct an LH was presented in Herrera and Martínez (2001) that imposed the following rules, so-called LH basic rules:

- •
**Rule 1:** to preserve all *former modal points* of the membership functions of each linguistic term from one level to the following one.

- •
**Rule 2:** to make *smooth transitions between successive levels*. The aim is to build a new linguistic term set, *S*^{n(t + 1)}. A new linguistic term will be added between each pair of terms belonging to the term set of the previous level *t*. To carry out this insertion, it is necessary to reduce the support of the linguistic labels to keep place for the new one located in the middle of them.

These rules induce a limitation regarding the term sets that can be used in the *F*_{MS} defined by the *LH*. Because, a linguistic term set of level *t* + 1 is obtained from its predecessor as:

- (2)

Table 1 shows the granularity for each linguistic term set of an LH according to the rules (graphically in Figure 4).

Table 1. Linguistic Hierarchies. | *l* (*t*, *n* (*t*)) | *l* (*t*, *n* (*t*)) |
---|

*Level* 1 | *l* (1, 3) | *l* (1, 7) |

*Level* 2 | *l* (2, 5) | *l* (2, 13) |

*Level* 3 | *l* (3, 9) | |

* 3.3.2. Unification Phase* The LH can unify the linguistic information assessed in multiple scales in any term set of the LH without loss of information. To do so, in Herrera and Martínez (2001) it was defined a transformation function, , between any two linguistic levels *t* and *t*′ of the LH as follows:

**Definition 5 (****Herrera and Martínez 2001****)***Let* *be an LH whose linguistic term sets are denoted as S*^{n(t)}={*s*^{n(t)}_{0}, …, *s*^{n(t)}_{n(t) − 1}}, *and let us consider the 2-tuple linguistic representation. The transformation function from a linguistic label in level t to a label in level t*′, *satisfying the LH basic rules, is defined as:*

- (3)

In Herrera and Martínez (2001), it was proved that the transformation function between linguistic terms in different levels of the LH is one-to-one, which guarantees the transformations between levels of a LH are carried out without loss of information.

**Example 3** Here we show how the transformation functions act over the LH, (see Figure 4), whose term sets are:

The transformations between terms of the different levels are carried out as:

* 3.3.3. Computational Phase* Due to the fact that the representation model used by LH and the results of the unification process are linguistic 2-tuples, its computational model is based on the one presented in Herrera and Martínez (2000) for the 2-tuple representation model.

It is important to notice that the final results of any computational process can be expressed in any linguistic term of the framework, *F*_{MS}, by means of in a precise way.

#### 3.4. Hierarchical Tree

Huynh and Nakamori (2005) introduced a new notion about LH in terms of ordered structure based semantics of linguistic terms, shortly as *T*_{LH}. The aim of this approach is to make more flexible the multigranular linguistic framework and overcome the limitation imposed by the *LH* regarding the term sets that can be used in an LH.

* 3.4.1. Framework* First, we will show the definition of a hierarchical tree that fixes the framework for this approach.

This approach defines an LH of a linguistic variable *X* as a hierarchical tree, *T*_{X}, that consists of a finite number of levels *t* with *t* = 0, …, *m*, which is defined as follows:

- •
Level *t* = 0 is the root of the tree labeled by *X*—the name of the linguistic variable.

- •
Each level *t*, for *t* = 1…, *m*, is a finite linguistic term set of *X*, denoted by *S*^{n(t)}, accompanied with a total ordering such that:

For each *t* = 1, …, *m* − 1, the mapping Γ_{t} plays a role as a semantic derivation from the term set *S*^{n(t)} to its refinement *S*^{n(t + 1)}. Furthermore, for each mapping Γ_{t}, *t* = 1, …, *m* − 1, there exists a pseudo-inversion , defined by

Γ^{−}_{t} and Γ_{t} define the function transformations between levels of the LH.

* 3.4.2. Unification Phase* The linguistic information assessed in different linguistic levels is unified by means of transformation Φ using Γ_{t} and Γ^{−}_{t}.

**Example 4** This example shows how to transform the term sets of level 1 in the level 2 (and vice versa) of the LH *T*_{temperature} (see Figure 5) (Huynh and Nakamori 2005).

- (4)

* 3.4.3. Computational Phase* This approach proposes a computational model for the unified information that makes use of random preferences so-called satisfactory principle, instead of fuzzy sets, supporting the linguistic terms semantics or the linguistic terms themselves. Therefore, the results obtained are expressed in a numerical domain; for further details see Huynh and Nakamori (2005).

#### 3.5. A Comparative Analysis

Once we have reviewed the main approaches to manage the multigranular linguistic framework, it is interesting to analyze and compare them regarding certain features such as:

- •
Accuracy: that means if there is no loss of information in the computational process of the approach.

- •
Expression domain: it means the domains in which are expressed the results obtained by the approach.

- •
Framework: it indicates which term sets can be used by the experts involved in the problem to express their opinions.

This analysis will be useful to point out the advantages of each approach to choose the more adequate one in each problem and to detect what are the limitations that should be overcome to provide an approach adequate in those cases.

Going in deep through Table 2, we can observe that fusion I, fusion II, and *T*_{LH} approaches provide the experts a total flexibility to model the linguistic information, but their results present lack of accuracy and the expression domain of the results is not very suitable in those cases in which the experts need more than ranking of alternatives to understand the computational process and the result obtained. However, the LH provides an accurate computational model, and additionally the results are expressed in the initial domains used by the experts. However, it presents a limitation that reduces its usefulness such as its framework is limited and the experts cannot use any term set as they prefer.

Table 2. Features of Approaches Reviewed. Features | Fusion I | Fusion II | *LH* | *T*_{LH} |
---|

Accuracy | No | No | Yes | No |

Results in the framework | No | No | Yes | No |

Domain of final results | Fuzzy Sets | Fuzzy sets | Linguistic term | Numerical |

Computational model | Extension principle | Extension principle | 2-tuple | Satisfactory principle |

Term sets | Any | Any | Limited | Any |

Our aim in this article is to develop a new approach to deal with MGLI that would be “accurate”, its expression domain “easy to be understood” by the experts and its framework can be “flexible” to use any linguistic term set. Therefore, from the previous analysis we have paid 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 multigranular linguistic framework. We present our proposal in the following section.

### 4. EXTENDED LINGUISTIC HIERARCHIES

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. LINGUISTIC BACKGROUND
- 3. RELATED WORK
- 4. EXTENDED LINGUISTIC HIERARCHIES
- 5. AN MEDM PROBLEM DEALING WITH ELH
- 6. CONCLUDING REMARKS
- ACKNOWLEDGMENTS
- REFERENCES

Usually, when experts face MEDM problems with multiple linguistic scales they expect a flexible framework in which any term set can be used, and accuracy and easy understand results would be obtained as well. As we have pointed out in Section 3.5 the current approaches provide some of these properties but not altogether. For instance, the LH cannot manage frameworks with term sets with 3, 5, and 7 labels (Figure 6) and the other approaches (Herrera et al. 2000; Herrera and Martínez 2001; Huynh and Nakamori 2005) do not provide accurate results either easy to understand.

Therefore, it is desirable and useful to develop a tool able to manage multiple linguistic scales in MEDM problems that fulfills all the aforementioned properties.

In this section, our aim is to propose a new approach for managing MGLI, that fulfills the previous requirements altogether, so-called ELH. This approach is based on the LH (Herrera and Martínez 2001) and the 2-tuple linguistic representation model (Herrera and Martínez 2000). The ELH proposes a new way for building the LH and a new unification process that drives the computational model.

#### 4.1. Framework

We have showed that LH produces accurate results due to the LH basic rules used in its construction. Rule 1 guarantees that all former modal points are kept from one level to another, this is the key reason to keep all the information in CW processes (see Figure 4).

**Lemma 1***Let S*^{n(t)}={*s*^{n(t)}_{0}, …, *s*^{n(t)}_{n(t) − 1}} *be an ordered linguistic term set in LH of a linguistic variable in LH* . *The set of former modal points of the level t is defined as*

*where each former modal point fp*^{i}_{t} ∈ [0, 1] *is located at*

*being* .

The *LH* basic rule 2 just proposes the simplest way to keep these former modal points from one level to another, defining the granularity between two levels as:

being possible to transform the information between any two levels without loss of information but limiting the linguistic scales that can belong to the LH.

Due to the fact that we want to deal with any scale in the multigranular linguistic framework, our proposal consists of building an ELH, replacing the basic rules that obligates to keep the former modal points from one level, *t*, to the next one, *t* + 1, by the *extended hierarchical rules*:

- •
**Extended Rule 1:** to build an ELH, first, it should include a finite number of the levels, *l*(*t*, *n*(*t*)), with *t* = 1, …, *m* that defines the multigranular linguistic framework, *F*_{MS}, required by the experts to express their knowledge. It is not necessary to keep the former modal points among each other might be one.

- •
**Extended Rule 2:** to obtain an ELH, a new level *l*(*t**, *n*(*t**)) with *t**= *m* + 1 should be added to keep all the former modal points of all the previous levels *l*(*t*, *n*(*t*)), *t* = 1, …, *m* within this new level.

Therefore to construct an ELH, first, *m* linguistic scales are given for the experts to express their information. And the term set, *l*(*t**, *n*(*t**)) with *t**= *m* + 1, will be added in a way specified by the following theorem.

**Theorem 1***Let* {*S*^{n(1)}, …, *S*^{n(m)}} *be the set of m linguistic term sets, where the granularity n*(*t*) *with t* = 1, …, *m is an odd value. A new term set* *with t**= *m* + 1 *that keeps all the former modal points of the m term sets can have the following granularity:*

*being* δ_{t}= *n*(*t*) − 1 ∈ *N*.

**Proof** According to Lemma 1:

and

It then proves that .

**Remark: ** If we check Figure 7, it is easy to observe that with the Theorem 1 the *fp*^{i}_{t} corresponds to whose membership value is 1.

Therefore an ELH is the union of the *m* levels required by the experts and the new level *l*(*t**, *n*(*t**)) that keeps all the former modal points to provide accuracy in the processes of CW, i.e.,

Figure 7 shows the granularity needed in the level *t** according to the *m* previous levels included in the framework. We can observe that the last level *t**, contains all the former modal points of the membership functions of each linguistic term set in the previous levels *t* = 1, …, *m*.

**Example 5** The granularity of the last level in the Figure 7 is (∏^{t = m}_{t = 1}δ_{t}) + 1, being δ_{1}= 2, δ_{2}= 4 and δ_{3}= 6. Then *n*(*t**) is *n*(4) = (2 · 4 · 6) + 1 = 48 + 1 = 49.

#### 4.2. Optimized Framework

In Section 4.1, we have introduced an initial approach to build an ELH where the level *t** keeps all the former modal points of the previous *t* levels. However, the granularity of *t** determined by using Theorem 1 would be too high and might make the computational model more complex.

To make simpler the use and construction of the ELH we propose an alternative way to minimize the granularity of *t** which can still keep all the former modal points of the previous levels. To achieve this goal we use the least common multiple (LCM).

**Definition 7***The least common multiple of m nonzero integer**a* _{1}, …, *a*_{m} is defined as LCM(*a*_{1}, …, *a*_{m}) = *min* {*n* ∈ *N* : *n*|*a*_{i} ∈ *N for i* = 1, …, *m*}.

By using the LCM a new theorem to compute the granularity of the level *t** is proposed.

**Theorem 2***Let* {*S*^{n(1)}, …, *S*^{n(m)}} *be the set of linguistic scales with any odd value of granularity. A new level, l*(*t**, *n*(*t**)) *with t**= *m* + 1, *that keeps the former modal points of the previous m levels can have the following granularity:*

**Remark: ** However, in this case *fp*^{i}_{t} can correspond to with different membership values {0.5, 1}.

**Example 6** By using the framework of Example 5 with the linguistic terms with granularities 3, 5, and 7 (see Figure 6), the granularity of the last level in the ELH according to Theorem 2 is (*LCM*(δ_{1}, δ_{2}, δ_{3})) + 1 = *LCM*(2, 4, 6) + 1 = 12 + 1 = 13.

Figure 8 summarizes in a table and shows graphically the values that optimize the construction of the ELH. We can observe that this optimization reduces drastically the number of labels in the last level, *t**, in comparison with the previous proposal.

#### 4.3. Unification Phase

The ELH also shares the computational scheme showed in Figure 1, therefore to deal with the MGLI in ELH, firstly it should be unified. This unification process is based on the transformation functions defined in the LH, where *t* and *t*′ can be any pair of term sets in the LH (see Eq. 3). However, in the *ELH* these transformations do not guarantee the accuracy of the transformation between any two term sets because do not keep the former modal points. Therefore to avoid the loss of information between different levels in ELH, this model unifies the information in the level *t** that keeps all the former modal points by using the transformation function, , being *t* any level in {1, …, *m*} and *t**= *m* + 1.

By using this unification process, we can develop a new transformation function between any pair of term sets, *t* and *t*′, in the *ELH* without loss of information. It is necessary a two step transformation process: firstly it transforms the linguistic terms at any level *l*(*t*, *n*(*t*)) in *ELH* into *l*(*t**, *n*(*t**)), being *t**= *m* + 1 (as defined in Theorem 2) that keeps all the former modal points of the level *t*, by means of without loss of information; then transforms the linguistic terms at *l*(*t**, *n*(*t**)) in ELH into any level *l*(*t*′, *n*(*t*′)) by means of without loss of information. Formally,

**Definition 8***Assume t and t*′ *can be any pair of term sets in the ELH and t** *is the level l*(*t*^{m + 1}, *n*(*t*^{m + 1})) *in the ELH, the new extended transformation function* *is defined as:*

*where* *and* *are the transformation functions defined in the same way as in LH (see Eq. **3**).*

**Theorem 3***The extended transformation function between linguistic terms in different levels in ELH is a one-to-one function:*

.

**Proof** It is straightforward from the definition of ETF because and are one-to-one functions.

**Example 7** We show an example of function of transformation *ETF* on the 2-tuple, (*s*^{3}_{1}, 1), between level 1 and 2 of the *ELH* showed in the Figure 8.

**Example 8. ** We show how the transformation functions act over the , which is showed in Figure 9 and whose term sets are:

The unification of terms can be also graphically illustrated in 10.

#### 4.4. Computational Phase

The computational model uses by the ELH is the 2-tuple computational model because the information is unified by means of 2-tuples.

We must remark that although the results obtained are expressed by linguistic 2-tuples in a unified , this model provides the facility to express those results in any scale of the *F*_{MS} without loss of information based on Theorem 3 (showed in Figure 10) by means of the transformation function with *any level t*′, *t* ∈{1, …, *m* + 1}. Theorem 3 shows theoretically the equivalent transformation between different levels without loss of information. In the computation stage, we can follow the most simply way for the ranking purpose to first unify them into one level (let say *t*′= *t**= *m* + 1). If we need to express the results of the linguistic computations in any specific level of the application context, we can certainly achieve it based on Theorem 3.

#### 4.5. Analysis of Extended Linguistic Hierarchies

The features of the proposed ELH approach are summarized in Table 3, the features are the same as in Table 2.

Table 3. Features of ELH. Features | ELH |
---|

Accuracy | Yes |

Results in the Framework | Yes |

Domain of final results | Linguistic Term |

Computational Model | 2-tuple |

Term Sets | Any |

We can observe that this approach provides a total flexibility for modeling the linguistic information, an accurate computational model and eventually the results are linguistically expressed in the initial domains.

### 5. AN MEDM PROBLEM DEALING WITH ELH

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. LINGUISTIC BACKGROUND
- 3. RELATED WORK
- 4. EXTENDED LINGUISTIC HIERARCHIES
- 5. AN MEDM PROBLEM DEALING WITH ELH
- 6. CONCLUDING REMARKS
- ACKNOWLEDGMENTS
- REFERENCES

As an application of the *ELH* presented in this article, we solve an MEDM problem defined in a multigranular linguistic context.

Let us suppose an investment company, which plans to invest a sum of money in the best option. There is a panel with four possible alternatives to invest the money:

The investment company has a group of four consultancy departments.

- •
*e*_{1} is the risk analysis department;

- •
*e*_{2} is the growth analysis department;

- •
*e*_{3} is the social-political analysis department;

- •
*e*_{4} is the environmental impact analysis department.

Each department is managing by an expert, and thus, each expert is an information source. These experts use different linguistic term sets with in an ELH to provide their preferences over the set of alternatives. More specifically,

- •
*e*_{1} provides his preferences in *l*(3, 7);

- •
*e*_{2} provides his preferences in *l*(2, 5);

- •
*e*_{3} provides his preferences in *l*(1, 3);

- •
*e*_{4} provides his preferences in *l*(3, 7).

The linguistic terms can have a syntax adequate to the problem, but in this case we use the normalized syntax for a better comprehensiveness of the computation processes.

After a deep study, suppose that each expert provides the following preference values:

| *x*_{1} | *x*_{2} | *x*_{3} | *x*_{4} |
---|

*e*_{1} | *s*^{7}_{4} | *s*^{7}_{6} | *s*^{7}_{3} | *s*^{7}_{5} |

*e*_{2} | *s*^{5}_{3} | *s*^{5}_{4} | *s*^{5}_{3} | *s*^{5}_{3} |

*e*_{3} | *s*^{3}_{1} | *s*^{3}_{2} | *s*^{3}_{2} | *s*^{3}_{1} |

*e*_{4} | *s*^{7}_{4} | *s*^{7}_{5} | *s*^{7}_{3} | *s*^{7}_{5} |

#### 5.1. Decision Model

The MEDM model for solving the problem is carries out in the following steps:

- 1
**Aggregation phase**

- –
*Normalization process.* Given that the ELH is composed by four term sets, being l(4,13) according to Theorem 2. First, the information provided by the experts is unified in an unique domain,

*t*′, to simplify and minimize computations

*t*′=

*t**= 4. Therefore the information unified is expressed as 2-tuples in

*t*′:

| *x*_{1} | *x*_{2} | *x*_{3} | *x*_{4} |
---|

*e*_{1} | (*s*^{13}_{8}, 0) | (*s*^{13}_{12}, 0) | (*s*^{13}_{6}, 0) | (*s*^{13}_{10}, 0) |

*e*_{2} | (*s*^{13}_{9}, 0) | (*s*^{13}_{12}, 0) | (*s*^{13}_{9}, 0) | (*s*^{13}_{9}, 0) |

*e*_{3} | (*s*^{13}_{6}, 0) | (*s*^{13}_{12}, 0) | (*s*^{13}_{12}, 0) | (*s*^{13}_{6}, 0) |

*e*_{4} | (*s*^{13}_{8}, 0) | (*s*^{13}_{10}, 0) | (*s*^{13}_{6}, 0) | (*s*^{13}_{10}, 0) |

- –
*Aggregation process.* Different aggregation operators can be used according to the needs, but in this example to simplify the computation process we consider that all the experts have the same importance in the decision process, therefore we use the 2-tuple mean operator to aggregate the preferences (see Eq.

5).

- (5)

The collective values obtained for each alternative are:

These collective values can be expressed in any linguistic term set of the ELH by using the correspondent transformation function, i.e.,

- –
:

*ETF*^{4}_{1}( ·)

- –
:

*ETF*^{4}_{2}( ·)

- –
:

*ETF*^{4}_{3}( ·)

In this way, all the experts could receive the collective values in their expression domains, and the exploitation phase could be carried out as well in any linguistic term set of the ELH, such that, it is obtained the same ranking in all of them.

- 2
**Exploitation phase** In this phase, the best alternative is chosen according to the greatest collective value. Therefore, the solution set of alternatives in this problem is:

The best option to invest is the

*computer company*.

#### 5.2. A Comparative Analysis

In Section 5.1, we solved an MEDM problem with linguistic assessments in a multigranular linguistic context based on ELH. Here, we want to compare the results and features of the ELH with the results produced by the others approaches reviewed in Section 3, the comparative result is shown in Table 4.

Table 4. Results Using Different Approaches. | Alternatives | Ranking |
---|

*x*_{1} | *x*_{2} | *x*_{3} | *x*_{4} |
---|

*ELH*, *t* = 1 | (*s*^{3}_{1}, 0.29) | (*s*^{3}_{2}, −0.08) | (*s*^{3}_{1}, 0.375) | (*s*^{3}_{1}, 0.46) | *x*2 > *x*4 > *x*3 > *x*1 |

*ELH*, *t* = 2 | (*s*^{5}_{3}, −0.41) | (*s*^{5}_{4}, −0.16) | (*s*^{5}_{3}, −0.25) | (*s*^{5}_{3}, −0.083) | *x*2 > *x*4 > *x*3 > *x*1 |

*ELH*, *t* = 3 | (*s*^{7}_{4}, −0.125) | (*s*^{7}_{6}, −0.25) | (*s*^{7}_{4}, 0.125) | (*s*^{7}_{4}, 0.375) | *x*2 > *x*4 > *x*3 > *x*1 |

*T*_{LH} | 2.0312 | 2.4375 | 1.2187 | 1.8437 | *x*2 > *x*1 > *x*4 > *x*3 |

*Fusion I* | 3.875 | 5.3611 | 3.875 | 4.375 | *x*2 > *x*4 > {*x*1, *x*3} |

*Fusion II* | 3.8571 | 5.5004 | 3.9290 | 4.3571 | *x*2 > *x*4 > *x*3 > *x*1 |

Analyzing the results from Table 4, we can state that the ELH obtains the same best option as the others, however only ELH expresses the final results in a linguistic way. Additionally, it can be expressed in any of the initial linguistic scales used by the experts, being easier to understand. In this table, we skipped the LH because the approach has a limitation regarding the term sets that can be used by the experts to provide their opinions and it cannot deal with the linguistic framework used in this problem with term sets of 3, 5, 7 labels.

Finally as illustrated in Table 4, it is clearly observed that the aims pointed out in Section 3.5 and shown in Table 3 regarding the ELH have been achieved.

As shown from Section 4 and the given case study, the proposed model provides the flexibility of dealing with realistic decision making problems defined in multiple linguistic scales contexts, i.e., any linguistic scale can be used in the problem without loss of information, which overcomes the limitation of the existing methods about some restriction on the selection of the linguistic scale. The case study also shows that the new approach works effectively because the linguistic scales given in the example is either not possible to proceed in the existing work, e.g., in Herrera and Martínez (2001) or more accurate than in Chang et al. (2007) and Herrera and Martínez (2001).

In addition, the computations of the proposed new approach, in fact, are quite simple. The use of Excel is enough but we have used our own software developed to deal with such information. As also illustrated in the existing work (e.g., Herrera and Martínez 2000, 2001; Herrera et al. 2000), the symbolic computation is quick and simple, the present work considers to add a new level that is quite important and useful for our goals, but, then the unification process does not increase much computational complexity.

### 6. CONCLUDING REMARKS

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. LINGUISTIC BACKGROUND
- 3. RELATED WORK
- 4. EXTENDED LINGUISTIC HIERARCHIES
- 5. AN MEDM PROBLEM DEALING WITH ELH
- 6. CONCLUDING REMARKS
- ACKNOWLEDGMENTS
- REFERENCES

The modeling of MCDM or MEDM by means of linguistic information can involve problems defined in multiple linguistic scales contexts. Different approaches have been introduced to deal with such type of context but all of them presented different limitations regarding either the accuracy of the computational model or the linguistic framework. In this article, we have introduced a new approach so-called ELH based on LH (Herrera and Martínez 2001) to overcome those limitations in applicability, effectiveness and accuracy to deal with multigranular linguistic scales in decision-making problems such that, any linguistic scale can be used in the problem, the computational model is based on a symbolic one and the results are expressed in the initial term sets used by the experts to provide their knowledge without loss of information.

We have applied the ELH to an MEDM problem, but they can be applied to different areas, such as, the design of hierarchical systems of linguistic rules introduced for fuzzy rules – based systems, decision models, information retrieval, clinical diagnosis, risk in software development, educational grading systems, design evaluation, sensory evaluation, performance appraisal, recommender systems, etc.