In our daily life, we could be confronted with numerous multiple attribute group decision making (MAGDM) problems. For such problems we designed a model which employs probabilistic linguistic MABAC (multi-attributive border approximation area comparison) based on the cumulative prospect theory (CPT-PL-MABAC) method to solve the MAGDM. The CPT-PL-MABAC method can take experts’ psychological behaviour and preferences into consideration. Furthermore, we utilize the combined weight consisting of subjective weight and objective weight. The objective weight is acquired by the entropy method. Additionally, the concrete calculating steps of CPT-PL-MABAC method are proposed to solve the MAGDM for selecting the optimal location of express distribution centre. Also, a numerical example for location selection of express distribution centre is given as the justification of the usefulness of the designed method. Finally, we compare the designed model with the other three existing models, and summarize the advantages and shortcomings.

Multiple attribute decision making (MADM) or multiple attribute group decision making (MAGDM) is an effective approach to solve complex decision-making issues (Huang

The decision makers are more likely to choose ‘good’, ‘medium’, ‘a little good’ and ‘excellent’ to evaluate alternatives. Therefore, Rodriguez

Applying PLTSs and related methods to some practical cases can reflect advantages and the applicability of PLTSs. Liang

The MABAC method is an effective method to address some difficult decision making issues. Xu

The above investigations described a particular assumption that DMs are perfectly rational. However, many studies show that people’s behaviour is affected by their emotions. For example, people are inclined to be more sensitive to losses than to gains. That’s to say, the perception of equal gains and losses are not the same for DMs. In general, people are inclined to be risk-averse. Based on these assumptions of bounded rationality, the cumulative prospect theory (CPT) (Tversky and Kahneman,

In the originalMABAC method, the psychological factor such as the DMs’ preference towards risk will affect the distance between the border approximation area. Furthermore, there are relatively few researches on constructing the MABAC method for MAGDM depending on the CPT under PLTSs. The main research significance of this paper is the modified MABAC method with CPT which can reduce the affection. Therefore, the PL-MABAC based on cumulative prospect theory (CPT-PL-MABAC) method in this paper is defined to solve the location selection of express distribution centre, which is a classical MAGDM issue. This article makes contributions as follows: (1) the concept of CPT is integrated into the PL-MABAC method for MAGDM. This method not only has unambiguous logic and relatively simple calculation, but also expresses the DM’s psychological state, which is closer to reality; (2) we improved the entropy method, which is characterized by the mean value of attributes as the reference point; (3) the combined attribute weights are obtained through objective weight by the entropy method and by getting the subjective weight given by decision makers; (4) the effectiveness and stability of this new method is fully testified by taking advantage of a case about location selection of express distribution centre and comparisons with the existing methods.

To sum up, the structure of this paper is built as follows. The second part mainly introduces and reviews the basic knowledge, including the PLTSs and CPT. In Section

In order to illustrate the CPT-PL-MABAC model, some relevant knowledge is introduced.

Sometimes, decision makers give fuzzy evaluation information that cannot be expressed exactly. Pang

Let

Suppose

To make the PLTSs easier to deal with, Pang

Suppose

Suppose

We can obtain the order relation between two PLTSs by Eqs. (

If

If

Suppose

In the cumulative prospect theory (CPT) (Tversky and Kahneman,

In this formula,

We will introduce the MABAC method based on CPT and on the PLTS. We will also give the following mathematical symbols which are used to express the relevant information. We suppose that there is a collection of alternatives

We designed the new PL-MABAC method in which the CPT is introduced to address MAGDM problems. A laconic frame diagram and the specific calculating procedure as follows:

See Fig.

CPT-PL-MABAC frame diagram.

Given an LTS

We acquire the objective weight by the entropy method and get the subjective weight given by decision makers.

First, we introduce the specific procedure of the entropy method. It should be noted that when calculating entropy, we use the mean value of the attribute as the reference point to calculate its distance from the normalized attribute value.

Calculate the mean value of

Let

Compute the objective weights of the

Then, we calculate the combined weights by using the following equation. The advantage of using combined weight is that the influence of subjective weight and objective weight can be considered comprehensively.

Decision makers gave the subjective weights

We take

In the background of e-commerce, online shopping has become a very common way of consumption in people’s lives. The recent epidemic situation also makes people more accustomed to using online shopping for consumption. Therefore, express delivery has become a matter of great concern. Reasonable and effective site selection can improve the service quality and win the favour of customers, so as to achieve the all-win goal of Express Distribution Centre, businesses and consumers. The express industry is the product of rapid economic development. It means that the consumption capacity of a region has a crucial impact on the express business volume. How to choose a proper express distribution centre is of great importance to both express companies and consumers. Generally speaking, the more developed the region is, the more its express business volume will be, and the number of distribution centres will be more and more centralized. Additionally, the site selection also needs to consider the local population and demand. For example, the closer to the central business district, the denser the population, the more demand for express delivery. Besides, for the areas where e-commerce self-employed households are concentrated, the demand for express delivery is very large, which should be considered as the key object. Moreover, to choose a reasonable address for the express delivery centre, we must consider the problem of cost minimization. And convenient transportation can effectively ensure the timeliness of express delivery. Otherwise, in order to ensure service quality, express enterprises can only add more outlets or send more vehicles, no matter which way it will lead to increased costs. We know that choosing the best address of Express Distribution Centre is a classic MADM or MAGDM problem. In this case, we gave five alternative sites

Linguistic decision matrix by the first DM.

Alternatives | ||||

SA | SI | A | M | |

I | DI | SI | I | |

M | I | A | DI | |

I | DI | I | I | |

M | I | M | M |

Linguistic decision matrix by the second DM.

Alternatives | ||||

SI | I | DA | DA | |

A | I | DI | M | |

SA | M | SA | A | |

DI | M | DI | M | |

A | A | SI | I |

Linguistic decison matrix by the third DM.

Alternatives | ||||

A | SI | SA | SA | |

I | I | SI | I | |

M | SI | A | DI | |

SI | DA | SA | I | |

M | DA | M | A |

Linguistic decision matrix by the fourth DM.

Alternatives | ||||

SA | I | DA | DA | |

A | DI | A | M | |

M | I | A | A | |

SI | DA | DI | A | |

M | DA | SI |

Linguistic decision matrix by the fifth DM.

Alternatives | ||||

SA | M | DA | DA | |

SA | SI | A | SI | |

I | I | A | SI | |

DI | SA | DI | A | |

A | DA | A | I |

For the convenience of calculation, we first express the evaluation information given by experts by using linguistic term sets (Tables

Decision matrix with linguistic term sets by the first DM.

Alternatives | ||||

Decision matrix with linguistic term sets by the second DM.

Alternatives | ||||

Decision matrix with linguistic term sets by the third DM.

Alternatives | ||||

Decision matrix with linguistic term sets by the fourth DM.

Alternatives | ||||

Decision matrix with linguistic term sets by the fifth DM.

Alternatives | ||||

Then, we choose the most appropriate site fot the logistics distribution centre by using CPT-PL-MABAC method.

Linguistic evaluating value matrix by the first DM.

Alternatives | ||||

Linguistic evaluating value matrix by the second DM.

Alternatives | ||||

Linguistic evaluating value matrix by the third DM.

Alternatives | ||||

Linguistic evaluating value matrix by the fourth DM.

Alternatives | ||||

Linguistic evaluating value matrix by the fifth DM.

Alternatives | ||||

Decision matrix with PLTSs.

Alternatives | ||

Alternatives | ||

Normalized decision matrix with PLTSs.

Alternatives | ||

Alternatives | ||

Firstly, the objective weights are calculated by the entrophy method and the detailed calculation steps are as follows:

The mean value of

The

The objective weight of the

The mean value for all attributes.

The mean value | |

Probabilistic linguistic total prospect value of the all alternatives.

Alternatives | ||||

0.8689 | 0.9653 | 0.9453 | 0.9657 |

Secondly, the subjective weights are given by experts, which are:

Finally, the combined weight can be calculated by Eq. (

PLBAA for all attributes.

PLBAA | |

The Hamming distance matrix.

Alternatives | ||||

0.0726 | 0.1130 | 0.2779 | 0.1774 | |

0.0615 | 0.1575 | −0.0417 | 0.0338 | |

0.0504 | 0.1221 | 0.1472 | −0.0328 | |

−0.0830 | −0.0759 | −0.0480 | 0.1005 | |

0.0615 | −0.0332 | 0.0695 | 0.1005 |

The cumulative prospect distance matrix.

Alternatives | ||||

0.0236 | 0.0419 | 0.0724 | 0.0307 | |

0.0204 | 0.0561 | −0.0462 | 0.0072 | |

0.0171 | 0.0448 | 0.0624 | −0.0157 | |

−0.0596 | −0.0664 | −0.0524 | 0.0186 | |

0.0204 | −0.0321 | 0.0322 | 0.0186 |

Probabilistic linguistic total prospect value of the all alternatives.

Alternatives | |||||

0.1686 | 0.0374 | 0.1087 | −0.1598 | 0.0391 |

We compared our proposed model with three existing methods, which are the PLWA operator (Pang

Order by using diverse methods.

Methods | Order | Optimal alternative | Bad alternative |

PLWA operator (Pang |
|||

PL-TOPSIS method (Pang |
|||

PL-GRA method (Liang |
|||

PL-MABAC | |||

CPT-PL-MABAC method |

As you can see from the table above, all four methods obtain the same optimal site

The location selection of the express distribution centre is of great significance in the development of the express delivery industry. Therefore, a new PL-MAGDM method (CPT-PL-MABAC) is established to be applied to this issue. The main contributions of this article can be summarized as follows. Firstly, we introduce the CPT into the original MABAC method under PLTSs. The psychological factors of experts are introduced in the evaluation. Secondly, we improved the entropy method under PLTSs, which is characterized by the mean value of attributes as the reference point. Thirdly, we improved the distance formula between the evaluation values of the alternative and PLBBA. Finally, the new method enriches the decision-making method based on PLTS and enriches the model of location selection.

The CPT-PL-MABAC model is a stable decision-making tool with direct computation algorithms. Also, it can get comprehensive final sorting results because it considers the potential values of gains and losses. However, the method proposed is ineffective in the face of some problems when attribute weights and evaluation information are not completely known. Moreover, we only refer to reference points and value functions in CPT.

In future studies, we plan to deal with the situation where the weights are not completely known. Additionally, this method can be applied to other specific decision-making problems and many other unpredictable and fuzzy environments, for example, green energy supplier issues and other location selection issues.