^{1}

^{2}

^{1,2}

^{1}

^{2}

We investigate the multiple attribute group material selection problems in which the attribute values take the form of interval 2-tuple linguistic information. Firstly, some operational laws and possibility degree of interval 2-tuple linguistic variables are introduced. Then, we develop some interval 2-tuple linguistic aggregation operators called interval 2-tuple hybrid harmonic mean (ITHHM) operator, induced interval 2-tuple ordered weighted harmonic mean (I-ITOWHM) operator, and induced interval 2-tuple hybrid harmonic mean (I-ITHHM) operator and study some desirable properties of the I-ITOWHM operator. In particular, all these operators can be reduced to aggregate 2-tuple linguistic variables. Based on the I-ITHHM and the ITWHM (interval 2-tuple weighted harmonic mean) operators, an approach to multiple attribute group decision-making with interval 2-tuple linguistic information is proposed. Finally, a practical application to material selection problem is given to verify the developed approach and to demonstrate its practicality and effectiveness.

The 2-tuple linguistic representation model, characterized by a linguistic term and a numeric value, was developed by Herrera and Martínez [

In recent years, much progress has been made in research relating to 2-tuple aggregation operators since information aggregation plays a significant part in the MAGDM process. For example, Herrera and Martínez [

In some situations, however, the input arguments take the form of interval 2-tuple linguistic values because of time pressure, lack of knowledge or data, and decision-makers’ limited attention and information processing capabilities [

The remainder of this paper is set out as follows. In Section

A linguistic variable is a variable whose values are expressed in linguistic terms. In other words, it is a variable whose values are not numbers but words or sentences in a natural or artificial language. The concept of linguistic variables is very useful in dealing with situations which are too complex or too ill defined to be reasonably described by traditional quantitative expressions [

Negation operator:

The set which is ordered:

Max operator:

Min operator:

The 2-tuple linguistic representation model was firstly presented by Herrera and Martínez [

Let

Let

It is obvious that the conversion of a linguistic term into a linguistic 2-tuple consists of adding a value 0 as symbolic translation [

The interval 2-tuple linguistic representation model was put forward by Zhang [

Let

On the contrary, there is always a function

Consider any three interval 2-tuples

Motivated by the formulas proposed by Xu [

Let

From Definition 5, we can easily get the following results:

Let WHM:

Zhang [

Let

Based on the OWA and the ITWHM operators, Zhang [

Let

Obviously, the fundamental characteristic of the ITWHM operator is that it considers the importance of each given interval 2-tuple linguistic variable, whereas the fundamental characteristic of the ITOWHM operator is the reordering step, and it weights all the ordered positions of interval 2-tuple linguistic variables instead of weighting the given interval 2-tuples themselves. In the following, by combining the advantages of the ITWHM and ITOWHM operators, we develop an interval 2-tuple hybrid harmonic mean (ITHHM) operator that weights both the given interval 2-tuple linguistic variables and their ordered positions.

Let

In particular, if

To rank these interval 2-tuple linguistic arguments

Assuming

To rank these arguments, we first compare each argument

Summing all elements in each line of the matrix

Then, we rank the arguments

Suppose that the weight vector of the ITHHM operator is

An induced ordered weighted harmonic mean (IOWHM) operator [

In the following, we will develop an induced interval 2-tuple ordered weighted harmonic mean (I-ITOWHM) operator.

Let

If there is a tie between

In particular, if

If

The I-ITOWHM operator has the following properties similar to those of the IOWHM [

Consider

Let

If

Since

If

Let

Assuming

Inspired by the induced hybrid averaging (IHA) operator [

Let

In particular, if

Assuming

In this section, we will develop an approach based on the proposed interval 2-tuple linguistic harmonic mean operators for the MAGDM problems in which both the attribute weights and the expert weights take the form of real numbers and the attribute values take the form of interval 2-tuple linguistic variables.

Suppose that a MAGDM problem has

In the following, we apply the I-ITHHM and the ITWHM operators to multiple attribute group decision-making with interval 2-tuple linguistic information.

Convert the linguistic decision matrix

Suppose that

A certain grade such as Poor, which can be written as

an interval such as Poor-Medium, which means that the assessment of an alternative with respect to the attribute under consideration is between Poor and Medium which can be written as

Utilize the I-ITHHM operator which has an associated weight vector

Utilize the decision information given in matrix

To rank these collective overall preference values

Summing all elements in each line of the matrix

Rank all the alternatives

End.

In what follows, an illustrative example adapted from [_{1}, DM_{2}, DM_{3}, and DM_{4}, has been created in order to evaluate and select the most appropriate material for the application. The attributes which have been considered for the analysis are

_{1}provides his assessments by using the linguistic term set

_{2}provides his assessments using

_{3}provides his assessments using

_{4}provides his assessments using

The linguistic assessments of the five alternatives on each attribute provided by the four decision-makers are presented in Table

Linguistic assessments of the five alternatives.

Decision-makers | Alternatives | Attributes | |||
---|---|---|---|---|---|

| | | | ||

DM_{1} | _{1} | G | G-VG | VG | G |

_{2} | M-G | M | G | M | |

_{3} | M | M | P-M | P | |

_{4} | G | G-VG | M | G | |

_{5} | M | M-G | G | G | |

| |||||

DM_{2} | _{1} | VG | VG | G | VG |

_{2} | MP | M | P | M | |

_{3} | MP | MG | M | MP | |

_{4} | G | G | G | MG-G | |

_{5} | M-G | M | MP-M | MG | |

| |||||

DM_{3} | _{1} | EG | VG | VG-EG | VG |

_{2} | M-MG | M-G | G | G | |

_{3} | P | MP | P | P-MP | |

_{4} | VG | VG | G-VG | G-VG | |

_{5} | M | M | G-VG | G | |

| |||||

DM_{4} | _{1} | VG | VG | G-VG | VG |

_{2} | M-G | M | G | G | |

_{3} | P-M | M | P | P-M | |

_{4} | G | VG | G | G-VG | |

_{5} | M-G | M | G | G |

Then, we utilize the method being proposed to get the most desirable alternative(s).

Convert the linguistic decision matrix shown in Table

Interval 2-tuple linguistic decision matrix of the four decision makers.

Decision-makers | Alternatives | Attributes | |||
---|---|---|---|---|---|

| | | | ||

DM_{1} | | [( | [( | [( | [( |

| [( | [( | [( | [( | |

| [( | [( | [( | [( | |

| [( | [( | [( | [( | |

| [( | [( | [( | [( | |

| |||||

DM_{2} | | [( | [( | [( | [( |

| [( | [( | [( | [( | |

| [( | [( | [( | [( | |

| [( | [( | [( | [( | |

| [( | [( | [( | [( | |

| |||||

DM_{3} | | [( | [( | [( | [( |

| [( | [( | [( | [( | |

| [( | [( | [( | [( | |

| [( | [( | [( | [( | |

| [( | [( | [( | [( | |

| |||||

DM_{4} | | [( | [( | [( | [( |

| [( | [( | [( | [( | |

| [( | [( | [( | [( | |

| [( | [( | [( | [( | |

| [( | [( | [( | [( |

Utilize the decision information given in matrixes

In this example, the weight vector of the I-ITHHM operator is

Collective decision matrix by I-ITHHM.

Alternatives | Attributes | |||
---|---|---|---|---|

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

| | | | |

Utilize the decision information given in matrix

Calculate the values

Rank all the alternative

Next, we use the ITHHM and the I-ITOWHM operators in Step

Ranking comparisons.

Operators | Ranking | The best alternative |
---|---|---|

By ITHHM | | |

By I-ITOWHM | | |

By I-ITHHM | | |

By FWHM + FHHM | | |

By ULHHM + ULWHM | | |

By THWA + TWA | | |

In addition, to further evaluate the proposed method, we use the above material selection problem to analyze some comparable methods, which are based on the aggregation operators proposed by Xu [

According to Table

It has exact characteristic in linguistic information processing and can effectively avoid the loss and distortion of information which occur formerly in other types of linguistic computational models.

The uncertainty and diversity of decision-makers’ assessment information can be well reflected and modelled using interval 2-tuple linguistic variables. Moreover, the linguistic term sets with different granularity of uncertainty can be used by decision-makers for assessing alternatives.

We can represent more complex group decision-making processes that include psychological factors such as time pressure and personal affects to each alternative, by using order inducing variables in the aggregation stage.

In this paper, we have developed some new harmonic aggregation operators including the interval 2-tuple hybrid harmonic mean (ITHHM) operator, the induced interval 2-tuple ordered weighted harmonic mean (I-ITOWHM) operator, and the induced interval 2-tuple hybrid harmonic mean (I-ITHHM) operator. It has been shown that both the ITWHM and ITOWHM operators are the special cases of the ITHHM operator, and if all the input interval 2-tuple data are reduced to the 2-tuple data, then the developed operators are reduced to the 2TLHH operator, the I-TOWHM operator, and the I-THHM operator, respectively. We have studied some desired properties of the I-ITOWHM operator, such as commutativity, idempotency, and monotonicity, and applied the I-ITHHM and the ITWHM operators to multiple attribute group decision-making with interval 2-tuple linguistic information. Finally, a material selection example has been given to verify the developed method and to demonstrate its practicality and effectiveness.

In the future, we expect to present further extensions to the proposed approach by adding new characteristics in the decision process and consider the potential applications of the developed interval 2-tuple linguistic harmonic mean operators to other fields. First, in many real-world situations, decision-makers may hesitate among several possible linguistic values or think of richer expressions for assessing an alternative because of uncertainty. Thus, extending the proposed decision model by using the hesitant fuzzy linguistic term sets [

The authors declare no competing interests.

This work was partially supported by the National Natural Science Foundation of China (no. 71402090), the National Social Science Foundation (no. 15CGL003), the Program for Professor of Special Appointment (Young Eastern Scholar) at Shanghai Institutions of Higher Learning (no. QD2015019), and the Shanghai Science and Technology Innovation Action Plan Soft Science Foundation (no. 16692103800).