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Emergency medical services during the COVID-19 epidemic have become the focus of worldwide attention, and how to effectively respond to urban epidemic situation during a complex environment has become a global challenge. Emergency decision-making can be considered as a multicriteria decision-making (MCDM) problem, which involves multiple criteria or attributes about qualitative and quantitative aspects. So, in this paper, based on the TODIM method, a hybrid TODIM method with crisp number and probability linguistic term set is first provided to evaluate the severity of urban COVID-19 epidemic situation during a complex humanitarian crisis environment. In this hybrid method, the quantitative aspects are evaluated on the basis of precise numerical values, and the qualitative aspects are evaluated by means of probability linguistic term set, which can not only express their judgments or linguistic preference with multiple linguistic terms but also reflect different importance degrees or probability degrees of all the possible linguistic information or preference information. In addition, the concept of entropy and probability linguistic entropy is applied to induce hybrid criteria weight information. Furthermore, sensitivity analysis of the parameter about attenuation factor of the losses in the hybrid TODIM method, which considers the psychology factors and cognitive behavior of the DMs, is further conducted on a case study, to verify the effectiveness and stability of the proposed method for urban epidemic situation evaluation according to the results of this study.

The frequent occurrence of emergency medical incidents has caused a great deal of property losses and a lot of casualties [

Emergency medical management and epidemic situation evaluation have a complex evolutionary process, and emergency decision-making is very difficult to deal with the traditional decision theory. The decision-making on epidemic situation evaluation of accidental medical emergencies usually concerns about multiple criteria or attributes about qualitative and quantitative aspects, a finite number of alternatives, semantic benefits, and interests of multiple stakeholders, and it can be considered as a complex multicriteria decision-making (MCDM) problem. Kahneman and Tversky [

The DMs usually need data and information to make effective decisions, especially for complex decisions. In general, emergency decision-making on the COVID-19 outbreak is considered as a multicriteria decision-making (MCDM) problem, which involves multiple criteria or attributes about qualitative and quantitative aspects. The quantitative aspects are evaluated on the basis of precise numerical values, and the qualitative aspects are evaluated by means of uncertainty including vagueness and ambiguity [

In emergency medical management, decision-making process has become more and more complex because many of the attributes are difficult to quantify [

To sum up, this paper focuses on the development of a hybrid TODIM method with crisp number and probability linguistic term set, based on the TODIM method which can consider the psychology factors and cognitive behavior of the DMs, for aiming to assist DMs to evaluate the severity of urban COVID-19 epidemic situation involved in emergency medical management. Five main works of this paper that have led to its novelty are as follows: (1) A hybrid TODIM method with crisp number and probability linguistic term set is first provided to evaluate the severity of urban COVID-19 epidemic situation during a complex humanitarian crisis environment. (2) The proposed hybrid TODIM method involves multiple criteria or attributes about qualitative and quantitative aspects, considering the complexity of emergency decision-making process. (3) The proposed hybrid TODIM method can not only express their judgments or linguistic preference with multiple linguistic terms, but also reflect different importance degrees or probability degrees of all the possible linguistic information or preference information. (4) The concept of entropy and probability linguistic entropy is applied to induce hybrid criteria weight information, which has the advantages of simple calculation and clear concept. (5) Sensitivity analysis of the parameter in the hybrid TODIM method, which considers the psychology factors and cognitive behavior of the DMs, is further carried out to demonstrate the effectiveness and stability of the proposed hybrid method for urban epidemic situation evaluation.

The rest of this paper is organized as follows. Section

The frequent occurrence of emergency medical incidents has caused a great deal of property losses and a lot of casualties [

MCDM has made remarkable progress and has evolved into a mature discipline, which concerns about multiple conflicting criteria, a finite number of alternatives, and multiple opinions of DMs [

Decisions usually need data information, particularly for complex decisions, which involve multiple criteria about qualitative and quantitative aspects. The quantitative aspects are usually evaluated on the base of accurate numerical values, and the qualitative aspects are usually evaluated by means of uncertainty including vagueness and ambiguity [

Besides, in the application of TODIM method, some scholars have done some research and development. Fan et al. [

In view of the foregoing, this paper based on the TODIM method provides a hybrid TODIM method with crisp number and PLTS to evaluate the severity of urban COVID-19 epidemic situation for aiding DMs involved in emergency medical management during a complex humanitarian crisis environment.

In this section, some brief backgrounds of PLTS, the comparison between PLTSs, the normalization of PLTS, and the TODIM method are introduced separately.

Based on the additive linguistic term set

If

Given a probabilistic linguistic term set

First, the score of PLTS is defined and introduced by Pang et al. [

Let

For any two PLTSs

Let

For two PLTSs

Let

The normalization of PLTSs usually has two distinct tasks. One is to estimate the ignorance degree of probabilistic information, and the other is to normalize the cardinality for computational convenience.

Given a PLTS

Let

If

If

The TODIM method, proposed by Gomes and Lima [

Suppose there are

Step 1: identify the decision matrix

Step 2: obtain the normalized decision matrix

Step 3: calculate the relative weight

where

Step 4: obtain the dominance degree of alternative

where

where the parameter

Step 5: get the overall dominance degree of the alternative

Step 6: sort the alternatives according to the overall dominance degree

The larger the value of

In this section, a hybrid TODIM method with crisp number and PLTS is provided in detail.

For an MCDM problem, suppose that there are

The specific steps of the hybrid TODIM method with crisp number and PLTS are presented as follows:

Step 1: identify the decision matrix

Step 2: obtain the normalized decision matrix

Crisp number: the normalization of crisp number

PLTS: the normalization of PLTS

where

Step 3: calculate the entropy weight for crisp number and probability linguistic entropy weight for PLTS. There are two steps:

Crisp number: the entropy weight for crisp number can be determined as follows:

Obtain the entropy: according to the definition of entropy, entropy

Obtain the entropy weight

Normalize the entropy weight

PLTS: based on the concepts of entropy, the probability linguistic entropy weight for PLTS can be obtained as follows:

Based on the additive linguistic term set

Let

Normalize the probability linguistic entropy: the probability linguistic entropy can be normalized as follows:

Determine the weight: the weight of each criterion can be determined as follows:

where

Determine the normalized weight: the normalized weight can be determined as follows:

Step 4: calculate the relative weight

where

Step 5: obtain the dominance degree

where

where the parameter

If

If

If

In this section, the distance measure can be determined as follows:

Crisp number: the distance for crisp number can be determined as follows:

PLTS: the distance for PLTS can be determined according to the modified the Hamming distance of PLTSs based on the normalized PLTSs proposed by Lin and Xu [

Step 6: get the overall dominance degree

Step 7: sort the alternatives according to the overall dominance degree

The larger the value of

The outbreak of COVID-19 epidemic has spread rapidly around the world, which has caused a great deal of property losses and a lot of casualties. It is very urgent to evaluate and respond to the urban epidemic situation involved in emergency medical management for providing scientific decision basis. Emergency medical management has a complex evolutionary process, and decision-making process has become more and more complex. In this section, a hybrid TODIM method with crisp number and PLTS is provided and presented to evaluate the severity of urban COVID-19 epidemic situation on a case study. The detailed assessment process is as follows.

Five cities _{1}, _{3}, and _{4} are of benefit type, and C_{2}, C_{5}, C_{6}, and C_{7} are of cost type. The data information with crisp number and PLTS is presented, as shown in Table

Data information.

_{1} | _{2} | _{3} | _{4} | _{5} | _{6} | _{7} | |
---|---|---|---|---|---|---|---|

_{1} | 162.2 | 0.05 | 0.04 | 0.46 | 3.2 | {S_{3}(0.3), S_{2}(0.5), S_{−1}(0.2)} | {S_{3}(0.9), S_{1}(0.1)} |

_{2} | 280.1 | 0.09 | 0.03 | 0.22 | 2.4 | {S_{3}(0.5), S_{2}(0.4)} | {S_{3}(0.7), S_{1}(0.3)} |

_{3} | 338.0 | 0.10 | 0.01 | 0.19 | 5.3 | {S_{2}(0.7), S_{1}(0.2), S_{-2}(0.1)} | {S_{3}(0.8), S_{1}(0.1), S_{0}(0.1)} |

_{4} | 186.7 | 0.07 | 0.01 | 0.22 | 7.3 | {S_{2}(0.6), S_{1}(0.3), S_{0}(0.1)} | {S_{3}(0.8), S_{2}(0.1), S_{1}(0.1)} |

_{5} | 226.5 | 0.11 | 0.03 | 0.12 | 5.2 | {S_{2}(0.5), S_{1}(0.3), S_{-1}(0.2)} | {S_{3}(0.6), S_{2}(0.4)} |

Normalized data information.

_{1} | _{2} | _{3} | _{4} | _{5} | _{6} | _{7} | |
---|---|---|---|---|---|---|---|

_{1} | 0.4799 | 1.0000 | 1.0000 | 1.0000 | 0.4384 | {S_{−3}(0.3), S_{−2}(0.5), S_{1}(0.2)} | { S_{−3}(0), S_{−3}(0.9), S_{−1}(0.1)} |

_{2} | 0.8287 | 0.5556 | 0.7500 | 0.4783 | 0.3288 | { S_{−3}(0), s_{−3}(0.5556), S_{−2}(0.4444)} | { S_{−3}(0), S_{−3}(0.7), S_{−1}(0.3)} |

_{3} | 1.0000 | 0.5000 | 0.2500 | 0.4130 | 0.7260 | {S_{−2}(0.7), S_{−1}(0.2), S_{2}(0.1)} | {S_{−3}(0.8), S_{−1}(0.1), S_{0}(0.1)} |

_{4} | 0.5524 | 0.7143 | 0.2500 | 0.4783 | 1.0000 | {S_{−2}(0.6), S_{−1}(0.3), S_{0}(0.1)} | {S_{−3}(0.8), S_{−2}(0.1), S_{−1}(0.1)} |

_{5} | 0.6701 | 0.4545 | 0.7500 | 0.2609 | 0.7123 | {S_{−2}(0.5), S_{−1}(0.3), S_{1}(0.2)} | { S_{−3}(0), S_{−3}(0.6), S_{−2}(0.4)} |

The relative weight.

_{1} | _{2} | _{3} | _{4} | _{5} | _{6} | _{7} | |
---|---|---|---|---|---|---|---|

Weight | 0.2173 | 0.1452 | 0.2084 | 0.0809 | 0.1755 | 0.0757 | 0.0971 |

Relative weight | 1.0000 | 0.6681 | 0.9594 | 0.3724 | 0.8077 | 0.3485 | 0.4467 |

Overall dominance degree for all alternatives.

Alternative | The overall dominance degree | Ranking | |
---|---|---|---|

_{1} | −8.3051 | 1.0000 | 1 |

_{2} | −14.7634 | 0.1511 | 4 |

_{3} | −13.0708 | 0.3736 | 3 |

_{4} | −11.9680 | 0.5185 | 2 |

_{5} | −15.9131 | 0.0000 | 5 |

Now, the detailed evaluation steps are presented as follows:

Step 1: identify the decision matrix.

The decision matrix with crisp number and PLTS is constructed and identified by the following additive linguistic term set

Step 2: obtain the normalized decision matrix. It is obvious that the criteria _{1}, _{3}, and _{4} are of benefit type, and _{2}, _{5}, _{6}, and _{7} are of cost type. So the normalization process is relatively simple and the normalized values can be calculated by formulas (

Step 3: calculate the relative weight. The relative weight is calculated, based on the entropy weight for crisp number and probability linguistic entropy weight for PLTS according to formulas (

Step 4: obtain the dominance degree

Step 5: get the overall dominance degree

Step 6: rank all the alternatives. The ranking of all the alternatives can be obtained according to the overall dominance degree

In order to further verify the validity and effectiveness of the hybrid TODIM method with crisp number and PLTS, sensitivity analysis of the parameter

Sensitivity analysis of the parameter

_{i}) | Rank | _{i}) | Rank | Δ(_{i}) | Rank | _{i}) | Rank | _{i}) | Rank | _{i}) | Rank | _{i}) | Rank | _{i}) | Rank | _{i}) | Rank | _{i}) | Rank | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

_{1} | 1.0000 | 1 | 1.0000 | 1 | 1.0000 | 1 | 1.0000 | 1 | 1.0000 | 1 | 1.0000 | 1 | 1.0000 | 1 | 1.0000 | 1 | 1.0000 | 1 | 1.0000 | 1 |

_{2} | 0.1623 | 4 | 0.1594 | 4 | 0.1567 | 4 | 0.1543 | 4 | 0.1521 | 4 | 0.1492 | 4 | 0.1474 | 4 | 0.1457 | 4 | 0.1442 | 4 | 0.1427 | 4 |

_{3} | 0.4140 | 3 | 0.4035 | 3 | 0.3940 | 3 | 0.3853 | 3 | 0.3773 | 3 | 0.3666 | 3 | 0.3601 | 3 | 0.3541 | 3 | 0.3485 | 3 | 0.3433 | 3 |

_{4} | 0.5905 | 2 | 0.5718 | 2 | 0.5548 | 2 | 0.5393 | 2 | 0.5252 | 2 | 0.5061 | 2 | 0.4945 | 2 | 0.4839 | 2 | 0.4740 | 2 | 0.4647 | 2 |

_{5} | 0.0000 | 5 | 0.0000 | 5 | 0.0000 | 5 | 0.0000 | 5 | 0.0000 | 5 | 0.0000 | 5 | 0.0000 | 5 | 0.0000 | 5 | 0.0000 | 5 | 0.0000 | 5 |

From Table

The frequent occurrence of emergency medical incidents has caused a great deal of property losses and a lot of casualties [

The limitations of this work are summarized as follows: epidemic situation evaluation has a complex evolutionary process. Through literature research, based on the complexity of emergency decision-making process, the TODIM method is still not well studied and applied for emergency medical decision-making, although it has done a lot of outreach and extension. So, in this paper, the extended hybrid TODIM method with crisp number and PLTS is first proposed for epidemic situation evaluation. However, the sample size of the data is relatively small.

In future work, new and effective evaluation methods will be further researched and developed by comparative analysis to effectively respond to emergency medical emergencies during a complex humanitarian crisis environment and the crisis response strategies will also be the focus of our research. In addition, big data analysis techniques will also be further developed to reduce the impact of small sample sizes on research results.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This research has been partially supported by Grants from Fund for Less Developed Regions of the National Natural Science Foundation of China (#71761014), the State Key Program of National Natural Science Foundation of China (#71532007), and Postdoctoral Science Foundation Project of China (#2016M592683).