As the tourism and mobile internet develop, car sharing is becoming more and more popular. How to select an appropriate car sharing platform is an important issue to tourists. The car sharing platform selection can be regarded as a kind of multi-attribute group decision making (MAGDM) problems. The probabilistic linguistic term set (PLTS) is a powerful tool to express tourists’ evaluations in the car sharing platform selection. This paper develops a probabilistic linguistic group decision making method for selecting a suitable car sharing platform. First, two aggregation operators of PLTSs are proposed. Subsequently, a fuzzy entropy and a hesitancy entropy of a PLTS are developed to measure the fuzziness and hesitancy of a PLTS, respectively. Combining the fuzzy entropy and hesitancy entropy, a total entropy of a PLTS is generated. Furthermore, a cross entropy between PLTSs is proposed as well. Using the total entropy and cross entropy, DMs’ weights and attribute weights are determined, respectively. By defining preference functions with PLTSs, an improved PL-PROMETHEE approach is developed to rank alternatives. Thereby, a novel method is proposed for solving MAGDM with PLTSs. A car sharing platform selection is examined at length to show the application and superiority of the proposed method.

With the development of mobile internet and “Internet +”, sharing economy, such as traffic-sharing, health care-sharing and food-sharing, has boomed in recent years. Car-sharing, a new mode of traffic-sharing, is becoming more and more popular along with the development of tourism. People usually complete the car rental and provide their feedbacks on consumer experiences on sharing platforms. Therefore, how to select a suitable car sharing platform is important for tourists. Since the car sharing platforms are often evaluated in safety, convenience, the brand of car and so on, the selection of car sharing platform can be described as a kind of multi-attribute decision making problems. Ordinarily, several tourists travel together. Hence, more than one tourist (decision maker, shorted by DM) decides which car sharing platform is selected. Thereby, the car sharing platform selection can be considered as a kind of multi-attribute group decision making (MAGDM) (Xu

Owing to the complexity of problems and vagueness of human thinking, it is difficult for DMs to describe the decision information as crisp numbers. DMs are more apt to use linguistic variables (Zadeh,

(1) Operations and aggregation operators of PLTSs. Operations of PLTSs and aggregation operators are important to aggregate decision information in the PLTS circumstance. Pang

(2) Distance measure of PLTSs. Pang

(3) Entropy and cross entropy of PLTSs. Due to the uncertainty of information with PLTSs, how to measure such a certainty is important. Entropy of PLTSs, as an efficient tool to measure the uncertainty of information, is an interesting topic but has not gained wide attention. Only Lin

(4) Decision methods for solving MAGDM problems with PLTSs. It is important to choose proper decision methods for selecting best alternatives. At present, pools of methods have been proposed to solve MAGDM problems with PLTSs. For example, Pang

Although many achievements have been achieved, they suffer from some limitations.

(1) Though the GPLHWA operator (Mao

(2) By existing distance measures of PLTSs (Pang

(3) The research on entropy and cross entropy of PLTSs is very little. Up to date, only Lin

(4) Existing decision methods are fruitful to solve decision problems with PLTSs. Nevertheless, some methods (Lin

To make up for above limitations, this paper proposes a novel method for solving MAGDM problems with PLTSs. First, two aggregation operators, including PLWAM (probabilistic linguistic weighted arithmetic mean) operator and PLWGM (probabilistic linguistic weighted geometric mean) operator, respectively, are proposed and some desirable properties are studied. To measure the hesitancy degree of a PLTS, a hesitancy index of the PLTS is introduced. Then a general distance measure of PLTSs is defined to measure the deviation between two PLTSs. Considering the fact that the uncertainty of a PLTS includes the fuzziness and hesitancy, a fuzzy entropy and hesitancy entropy of a PLTS are defined and then a total entropy of a PLTS is derived to measure such uncertainties. Meanwhile, a cross entropy of PLTSs is defined to measure the distinction between PLTSs. Afterwards, by minimizing the total entropy and the cross entropy of a DM, an objective program model is built to determine DMs’ weights. Subsequently, individual decision matrices are aggregated into a collective one by the PLWAM operator. To derive attributes weights objectively, a bi-objective program is built by minimizing the total entropy of attribute values with respect to each attribute as well as maximizing the cross entropy between attribute values of alternatives. By defining a new preference function in the form of PLTSs, an improved PL-PROMETHEE method is developed to rank alternatives. Thereby, a novel method is proposed for solving MAGDM problems with PLTSs. A case of car sharing platform selection is applied to show the effectiveness and advantages of the proposed method at length. The primary features of the proposed method are outlined as follows:

(1) Two new probabilistic linguistic average aggregation operators of PLTSs (i.e. PLWAM and PLWAG operators) are proposed. A prominent characteristic of them is that the aggregated result obtained by these two operators is not only a PLTS with the sum of probabilities of possible LTs being equal to 1, but also is consistent with human intuition.

(2) A new generalized distance measure of PLTSs is defined. It is worth mentioning that the hesitancy degree of a PLTS is considered in this distance measure. Thus, the new distance has a stronger distinguishing power. Moreover, a ranking approach is presented to rank PLTSs.

(3) A fuzzy entropy, a hesitancy entropy and a cross entropy of PLTSs are introduced. The fuzzy entropy has desirable properties and the computation of the hesitancy entropy is simple. Meanwhile, the cross entropy can distinguish the deviations between PLTSs with symmetric linguistic term sets.

(4) Based on entropy and cross entropy of PLTSs, distinct objective programs are established to determine DMs’ weights and attribute weights objectively. Finally, an improved PL-PROMETHEE method is developed to rank alternatives.

The remainder of this paper is organized as follows: In Section

In this section, some definitions and notions related to the PLTS are reviewed. Furthermore, two aggregation operators of PLTSs are proposed and some desirable properties of them are investigated.

Let

To preserve all given linguistic information, Xu (

Let

Additionally, the membership degree

Let

For a PLTS

Given a PLTS

To ensure that the operational results among PLTSs can be straightforwardly determined, Mao

Given a PLTS

If all elements in a PLTS are with different values of

If two or more elements with equal values of

When the subscripts

When the subscripts

If a PLTS

In real decision making problems, the numbers of linguistic terms in two different PLTSs are often different. This makes trouble to operate. In order to make them have the same number of linguistic terms, Pang

Let

Let

Given a PLTS

This subsection reviews the operational laws of PLTSs presented by Gou and Xu (

Let

The complement of

As reviewed in Mi

In virtue of Definition

Let

Please see Appendix

Analogously, a

Let

In a way similar to the proving process of Theorem

Compared with existing aggregation operators (Pang

Given a LTS

Results obtained by different aggregation operators.

Operators | Results | PLTS | Information lost | Information distortion |

PLWA (Pang |
No | Yes | Yes | |

GPLHWA (Mao |
No | Yes | Yes | |

PL-WAA | ||||

(Zhang, |
Yes | Yes | Yes | |

PLWAM in this paper | Yes | No | No |

It is observed from Table

(1) The PLWAM operator satisfies the closure of operations, i.e. the aggregated result of PLTSs obtained by the PLWAM operator is still a PLTS. However, the aggregated result obtained by the PLWA operator (Pang

(2) The aggregated result obtained by the proposed PLWAM operator is more consistent with human intuition. Observing

(3) Observing Table

Analogously, the PLWGM operator has above advantages.

In this section, a hesitancy index of a PLTS is introduced to measure the hesitancy degree of a PLTS, and then a new generalized distance measure between two PLTSs is developed. Finally, based on the proposed distance measure, a TOPSIS-based approach is presented to rank PLTSs.

Suppose a normalized PLTS

It is clear from Eq. (

First, this subsection reviews existing distance measures between PLTSs (Pang

Let

Pang

Later, Zhang

Mao

This paper defines a general distance measure between PLTSs by the proposed hesitancy index.

Given two PLTSs

Specially, when

According to Eqs. (

Compared with existing distance measures in Pang

It is shown from Table

Distancesbetween PLTSs obtained by different distance measures.

PLTSs | Different distance measures | Distances between PLTSs |

Zhang’s distance (Zhang |
||

The Manhattan distance proposed in this paper | ||

The Euclidean distance proposed in this paper | ||

Pang’s distance (Pang |
||

The Manhattan distance proposed in this paper | ||

The Euclidean distance proposed in this paper | ||

The Mao’s Euclidean distance (Mao |
||

The Manhattan distance proposed in this paper | ||

The Euclidean distance proposed in this paper |

(1) According to Zhang’s distance (Zhang

(2) From Eq. (

(3) Although Mao’s distance (Mao

(4) The distance measures in this paper are developed by considering each possible element of a PLTS

Based on the developed distance measures, this subsection introduces a TOPSIS based ranking approach to comparing PLTSs.

Let

In virtue of the closeness degree, a ranking approach for PLTSs is introduced.

Let

If

If

If

In order to judge the quality of decision information provided by DMs and the discriminations between DMs in the sequel, this section develops some entropy measures and a cross entropy measure.

This subsection addresses the entropy of PLTSs and proposes some new entropy measures of PLTSs. The main motivations are outlined as follows: (1) Measure the uncertainty of a PLTS neatly. Although Lin

Let

According to Definition

Let

According to these properties, a general fuzzy entropy of PLTSs is defined below.

Please see Appendix

Compared with the fuzzy entropy

(1) The properties of the latter are more consistent with intuitions. Let’s compare the property (i) of

(2) The latter can be regarded as an extension of the former in some sense. It is worth mentioning that a normalized PLTS can be considered as a type of discrete random variables. Therefore, the expectation value

According to Theorem

Hesitancy is an important feature of a PLTS. How to measure the hesitant uncertainty of a PLTS is often ignored. To fill in this gap, this subsection defines a hesitancy entropy measure of PLTSs by the deviations between the linguistic scale function values of possible linguistic terms in a PLTS and their mean value.

Let

Please see Appendix

Let

Compared

Combining the fuzzy entropy and hesitancy entropy with an adjusted coefficient

As

Furthermore, the total entropy of a probabilistic linguistic matrix is derived.

Let

Lin

It is obvious from Eq. (

Given two PLTSs

In this subsection, to measure the discrimination degree between PLTSs, a cross entropy is defined. Motivated by the cross entropy in the hesitant fuzzy linguistic environment (Gou

Suppose

Please see Appendix

It is observed from Eq. (

Moreover, a symmetric cross entropy between two probabilistic linguistic matrices is defined below.

Let

In this section, we first describe the problems of MAGDM with PLTSs, and then propose a novel method for solving such problems.

Let

In this subsection, an approach is developed to determine DMs’ weights objectively by using the proposed total entropy and symmetric cross entropy of PLTSs. As we know, the less uncertainty (i.e. a smaller total entropy) of an individual matrix provided by a DM, the better the quality of decision information reflected by this matrix is. Thus, the bigger the weight of this DM should be assigned. In virtue of this criterion, a programming model for deriving DMs’ weights is built by minimizing the total entropy of decision matrices, i.e.

Solving Eq. (

Normalizing

On the other hand, a closer degree of a DM to other DMs means that the information supplied by this DM is much closer to that of the group. In this case, this DM should be assigned a larger weight. From this viewpoint, another optimal model for determining DMs’ weights is constructed based on the symmetric cross entropy as

Eq. (

Normalizing

Combining Eq. (

In group decision making process, attribute weights play an important role because different attribute weights may result in diverse ranking of alternatives. In this subsection, considering the information of attribute weights is completely unknown or partially known, two bi-objective programs are constructed respectively for deriving attribute weights.

(i) Aggregating individual decision matrices into a collective one

By employing DMs’ weights determined in Section

(ii) Constructing bi-objective programs for deriving attribute weights

It is known to us that an attribute plays a more important role if the performance values of alternatives on it have distinct differences. Thus, such an attribute should be given a bigger weight. Conversely, if the evaluation values of alternatives with respect to a certain attribute have little difference, this attribute should be given a smaller weight. In addition, the credibility of decision information on an attribute should also be taken into account while determining attribute weights. The more credibility of evaluation values on an attribute, the bigger weight of this attribute should be assessed. As mentioned before, the symmetric cross entropy and the total entropy can reflect the differences between alternatives and the credibility of evaluation values, respectively. Keeping this idea in mind, we can determine attribute weights by maximizing the symmetric cross entropy as well as minimizing the total entropy. Thus, a bi-objective program is established if the information of attribute weights is completely unknown, i.e.

As

Solving Eq. (

The normalized weights

Similarly, for the situations where the information of attribute weights is incomplete, another programming model is obtained by modifying Eq. (

The PROMETHEE method is a popular outranking method in decision making. It has been extended to different fuzzy environments, such as intuitionistic fuzzy sets (Krishankumar

(i) The probabilistic linguistic preference function

Suppose that

In Eq. (

Obviously, the preference function

(ii) The integrated preference index

According to the probabilistic linguistic preference index

(iii) The positive and negative outranking flows

Employing the integrated preference indices, the positive/negative outranking flows,

(iv) Ranking alternatives

In virtue of Eqs. (

Finally, alternatives are ranked by the descending order of net flows.

In PL-PROMETHEE method (Xu

Obviously, the value of the preference function

For convenience, we only consider one of attributes in a real problem. Suppose that the ratings of three alternatives on this attribute are

A novel method is generated for MAGDM problems with PLTSs. The main procedure of this method is outlined below.

In this section, an example of a car sharing platform selection is provided to illustrate the application of the proposed method. Furthermore, the comparative analyses are performed to show the merits of the proposed method.

With the rapid development of internet technology and the deep advocation of green travel, the car sharing has sprung up over the last three years. Up to now, several car sharing platforms have emerged in China, such as Evcard, Gofun, Togo and so on. The popularization of car sharing greatly facilitates peoples’ travel and relieves the traffic pressure.

As a famous tourist city in China, Guilin cannot satisfy the travel of tourists due to the limited operational capacity of public transportation. So it is necessary for the government to introduce a car sharing platform to resolve the traffic problem. Now, the government invites three DMs to select the best car sharing platform from four candidate platforms (alternatives), including Evcard, Gofun, Togo and Greengo. Four attributes are considered, including safety (

Linguistic variables corresponding to linguistic terms.

Linguistic variables | Linguistic terms | Linguistic variables | Linguistic terms |

Very bad | Slightly good | ||

Bad | A little good | ||

A little bad | Good | ||

Slightly bad | Very good | ||

Medium |

Probabilistic linguistic decision matrices

Normalized ordered decision matrices

By Eqs. (

The symmetric cross entropies between

– | 0.2804 | 0.3965 | |

0.2804 | – | 0.084 | |

0.3965 | 0.084 | – |

Suppose

By employing the PLWAM operator (see Eq. (

Collective normalized ordered matrix

Take

Employing Eqs. (

The weight vector of attributes is determined by Eq. (

Using Eqs. (

By Eq. (

In virtue of Eqs. (

Net flows of alternatives are obtained by Eq. (

As

To show the advantages of the proposed method, comparative analyses with Mao’s method (Mao

Mao

(1) The proposed method determines DMs’ weights objectively by the total entropy and cross entropy of decision information. However, method (Mao

(2) The aggregated value obtained by the proposed PLWAM operator is more convincing than that obtained by

The PL-PROMETHEE method (Xu

The Borda’s scores of alternatives for each alternatives.

Borda’s scores | ||||

1 | 4 | 3 | 8 | |

4 | 2 | 1 | 7 | |

3 | 3 | 2 | 8 | |

2 | 1 | 4 | 7 |

It finds from Table

(1) PL-PROMETHEE method ignored the determination of attribute weights and DMs’ weights, but assigned them subjectively. Thus, the arbitrariness cannot be avoided. By contrast, the proposed method derives attribute weights and DMs’ weights objectively based on the total entropy and symmetric cross entropy of decision information. Therefore, the subjectivity is effectively reduced and the decision results are more credible.

(2) In the preference function of PL-PROMETHEE method (Xu

To further demonstrate the superiority of the proposed method, this subsection conducts a theoretical analysis and a practical analysis with other existing methods (Pang

(1) Theoretical analysis

(i) The proposed method introduces two new probabilistic linguistic weighted average operators (i.e. PLWAM operator and PLWAG operator). Compared with existing operators mentioned in Pang

(ii) In the proposed method, two cases including the attribute weights with unknown or partially known values are both taken into account. To determine attribute weights, the proposed method builds two different bi-objective programming models by maximizing the cross entropy and minimizing the total entropy of the collective evaluation values. As analysed in Section

(iii) Although method (Peng

(2) Practical analysis

(i) The distinguishing power of the proposed method is stronger than that of method (Xu

(2) The stableness of the proposed method is better than method (Liu and Li,

(3) Method Gou

In today’s internet age, car sharing is more and more popular. The car sharing platform selection is important for tourists, which can be regarded as a MAGDM problem. The PLTS is a powerful tool to represent the evaluation information of DMs in complex MAGDM problems. This paper introduces PLWAM and PLWGM operators firstly and studies some desirable properties of them. Subsequently, a hesitancy index of a PLTS and a general distance measure between PLTSs are defined. Then, a new approach is proposed to rank PLTSs. To measure the fuzziness and hesitancy of a PLTS, a fuzzy entropy and a hesitancy entropy of PLTSs are presented. Afterwards, a total entropy of PLTSs is defined to measure the uncertainty of a PLTS. Meanwhile, a cross entropy between PLTSs is presented. Based on the total entropy and the cross entropy of PLTSs, DMs’ weights and attribute weights are determined objectively. Finally, an improved PL-PROMETHEE method is developed by defining new preference functions and a total preference index. A car sharing platform selection is operated at length to illustrate the applications and advantages of the proposed method.

Apart from solving the selection of car sharing platform, the proposed method can be applied into many decision making fields, such as financial management (Kou

Theorem

For

Suppose Theorem

Furthermore, one gets

According to Eq. (

On the other hand, consider

As

Thus, Eq. (

In order to prove that

By the conditions (ii) and (iii) in Theorem

Next, we only prove the property (iv).

As

Since