A probabilistic linguistic thermodynamic method based on the water-filling algorithm and regret theory for emergency decision making

Abstract Since thermodynamics can describe the energy of matter and its form of storage or transformation in the system, it is introduced to resolve the uncertain decision-making problems. The paper proposes the thermodynamic decision-making method which considers both the quantity and quality of the probabilistic linguistic decision information. The analogies for thermodynamical indicators: energy, exergy and entropy are developed under the probabilistic linguistic circumstance. The probabilistic linguistic thermodynamic method combines the regret theory which captures decision makers’ regret-aversion and the objective weight of criterion obtained by the water-filling algorithm. The proposed method is applied to select the optimal solution to respond to the floods in Chongqing, China. The self-comparison is conducted to verify the effectiveness of the objective weight obtained by the water-filling algorithm and regret theory in the probabilistic linguistic thermodynamic method. The reliability and feasibility of the proposed method are verified by comparative analysis with other decision-making methods by some simulation experiments and non-parametric tests.


Introduction
In recent years, more and more emergency events are constantly emerging, such as the American flu pandemic and COVID-19 in 2019 and the Australia fires and the east African locust plague in 2020, which seriously threaten the safety of people's life and property.How to develop the emergency-rescue plan and respond to the emergency quickly and efficiently becomes the top priorities for local government and related institution or organisation (Chanamool & Naenna, 2016;Tian et al., 2018;Zhang et al., 2022;Zshou et al., 2018).The response of emergency is a group emergency decision making (EDM) which usually involves many intricate and complex factors.The decision makers should select proper decision-making method in accordance with the specific decision information (Tian et al., 2017(Tian et al., , 2020;;Zheng et al., 2020).In the early stages of the emergency, it is difficult and impractical to collect complete decision information, in view of the complexity and urgency of decisionmaking situations.In such instances, decision makers prefer to utilise more flexible ways to express their qualitative assessments within limited time.
In the practical decision-making problems, the decision makers usually adopt some linguistic terms, such as 'poor', 'fair', 'good' to depict their preferences over the criterion.Therefore, it is quite important for decision makers to choose suitable forms to portray their evaluations (Wang, Xu, et al., 2018).Considering that decision makers may be hesitant among several adjacent linguistic terms, Rodr ıguez et al. (2012) proposed the hesitant fuzzy linguistic term sets (HFLTSs) which combined hesitant fuzzy sets (Torra, 2010) and linguistic variables (Zadeh, 1975).However, each linguistic term in the HFLTS is assigned equal probability or weight by default.In addition, HFLTSs fail to express the possible linguistic terms which are not adjacent.In the individual and group decision making, the decision makers may prefer to use some possible linguistic terms instead of the adjacent ones to express their assessments and the probability or weight of each linguistic term.To overcome these drawbacks, Pang  (2016) proposed the probabilistic linguistic term set (PLTS), which consisted of several possible linguistic terms with corresponding probabilities (or weights).In the paper, the PLTS is utilised to depict the qualitative assessments considering its strong ability of describing the vagueness and reserving original information.As a powerful technique to express vague information, the theories and decisionmaking methods have been studied widely, such as PL-TOPSIS (Pang et al., 2016), PL-DEA (Pan et al., 2021), PL-GLDS (Wu & Liao, 2019), PL-MULTIMOORA (Chen et al., 2019), PL-TODIM (Wei & Wu, 2019) and PL-QUALIFLEX (Tian et al., 2019).The existing decision-making methods usually rank the alternatives by means of decision transformation and information fusion.Due to the randomness and complexity of decision-making situations, it is difficult for decision makers to obtain enough information and extract the key characteristic of the uncertain information.Thus, we should improve the existing decision-making methods from multiple angles and make the utmost of the existing uncertain information.
The macroscopic theory of thermodynamics studies thermal properties of matter in an energy transformation perspective and explains the macroscopic law that should be followed when energy is transformed into another form (Dincer & Cengel, 2001).Thermodynamics is a system theory based on experimental results and it does not involve microscopic structure and specific nature of matter.Hence, the thermodynamics theory is with high reliability and universality and applied to the decision domain based on the thermodynamical indicators: energy, exergy and entropy.Introducing thermodynamics into the decision-making fully considers the numerical size and distribution characteristics of the uncertain decision information.Prathap (2011) extended the analogies of thermodynamics to the domain of bibliometric research and energy-exergy-entropy sequences were introduced to rank the scientist's performance.Verma and Rajasankar (2017) further proposed the thermodynamical indicators under the crisp and fuzzy circumstance to solve the multi-criteria decision making problems.Combing the quantity and quality of intuitionistic fuzzy information, thermodynamics is applied to assist the hierarchical medical system by connecting with the descriptive characters of physical thermodynamic parameters (Ren, Xu, Liao, et al., 2017).Subsequently, based on the modified prospect theory, the thermodynamic decision-making methods with the hesitant fuzzy information and hesitant fuzzy linguistic information are presented to measure the quantity and quality of the uncertain information (Liao et al., 2018;Ren, Xu, & Hao, 2017).Wang, Liang, et al. (2020) proposed a probabilistic linguistic belief thermodynamic method to evaluate the mobile health apps based on psychological perception.
In the practical decision-making problems, there exists the phenomenon that decision makers compare the result of selected alternative with unselected ones.If the results of other alternatives perform better than the selected one, they feel regretful, or they will be delighted.Bell (1982) and Loomes and Sugden (1982) proposed the regret theory considering decision makers' regret-aversion.In the regret theory, the decision makers not only focus on the direct outcome, but also concern the results if they select other solutions.Regret theory has been extended to hesitant fuzzy sets (Xia, 2018), fuzzy complex spherical fuzzy sets (Akram et al., 2021), type-2 fuzzy sets (Wang, Pan, et al., 2020), etc.After extending the perceived utility function of regret theory to the probabilistic linguistic information, Xue et al. (2021) presented probabilistic linguistic dynamic reference point method to select the optimal response strategy for the EDM of COVID-19.In the paper, the perceived utility function in Xue et al. (2021) is adopted to depict decision makers' regret-aversion.
In the decision-making methods, there are two ways to determine the weight of each criterion.(1) The weight is endowed by the decision makers subjectively according to their experiences in a specialised field or by convention (Ren, Xu, & Hao, 2017;Ren, Xu, Liao, et al., 2017;Tian et al., 2019;Zhang, 2017).Consider that decision makers are bounded rational individuals, the weight may be biased inevitably which has a negative impact on the decision-making results.(2) The weight is calculated by the existing decision information.For example, the entropy method is applied to many decision-making methods to determine the weight of criterion (Chen, 2021;Wang, Liu, et al., 2018;Xue et al., 2018).In addition, the weight of criterion can be calculated by solving the optimisation model according to the specific characteristics of decision-making methods.For example, the weight of criterion can be computed by solving the single-objective optimisation model according to the maximising deviation method (Pang et al., 2016).
The water-filling theory is a classical method of solving the channel power optimal allocation problems in the communication field (Zhao et al., 2015).The allocation of sub-channel power in the wireless communication can be analogised to the determination of the criterion weight in the decision-making problems.The criterion is regarded as the sub-channel, and corresponding weight can be considered as the power assigned in the sub-channel.How to allocate the sub-channel power reasonably is of significance to improve system performance.If each sub-channel is allocated to same power according to the traditional method, it may lead to a waste of system resources, and even cause system transmission errors in severe cases.The adaptive power allocation to each sub-channel provides an effective way to solve this problem.The system performance can be further optimised by adaptively distributing power to each sub-channel according to the actual channel conditions.Similarly, in order to improve the quality of information utilisation, the water-filling algorithm can be introduced to assign the weight for each criterion based on the given probabilistic linguistic information and practical needs.
In the paper, the water-filling algorithm is utilised to compute the weight of each criterion as the objective weight.Compared with other methods of determining weight, the water-filling algorithm can optimise the weight allocation of criteria adaptively.It assigns the criterion weight based on the nonlinear programming model to maximise the total capacity of criteria.The importance of the criterion is reflected by the ratio of the standard deviation and the mean value of the criterion.The large the ratio, indicating that the criterion has a less impact on the results, the smaller the corresponding weight.The hybrid weight is obtained by combing decision makers' subjective weight and the decision makers can flexibly adjust the proportion according to actual requirements.The water-filling algorithm provides an innovative idea for determining the criterion weight and can make full use of the probabilistic linguistic information.
The thermodynamics theory is with high reliability and universality and applied to the uncertain decision problems domain (Ren, Xu, & Hao, 2017;Ren, Xu, Liao, et al., 2017).In the paper, we introduce the thermodynamical indicators: energy, exergy and entropy to describe the feature of uncertain decision-making information, which can be analogous to the energy, quality, effectiveness and imbalance of probabilistic linguistic information.Since the decision makers are bounded rational, the probabilistic linguistic thermodynamic method combines the regret theory which captures decision makers' regret-aversion.The modified utility function and the regret-rejoice function in the regret theory fully consider the characteristics of the probabilistic linguistic information.In order to maximise the total capacity of criteria when determining the objective weight, the nonlinear programming model is established by the water-filling algorithm.The revised water-filling algorithm considers the specific characteristic of the PLTSs and the nonlinear programming model can be solved by the genetic algorithm.The probabilistic linguistic thermodynamic method is effective to deal with the probabilistic linguistic multiple criteria decision making (MCDM) problems according to the actual situation.
The merits of the paper are summarised as follows: 1.The paper proposes a probabilistic linguistic thermodynamic method from the perspective of both the quantity and quality of the probabilistic linguistic decision information.2. The proposed method combines the regret theory which depicts decision makers' regret-aversion and the objective weight of criterion obtained by the water-filling algorithm.3. The probabilistic linguistic thermodynamic method is applied to select the optimal solution to respond to the floods in Chongqing, China.4. A series of simulation experiments and non-parametric tests are conducted to verify the applicability and effectiveness of the proposed method.
The remainder of the paper is organised as follows: In Section 2, we recall some definitions about PLTSs, classic thermodynamic method and regret theory in the probabilistic linguistic environment.In Section 3, based on the probabilistic linguistic regret decision matrix (PLRDM) and the objective weight of criterion obtained by the water-filling algorithm, the probabilistic linguistic thermodynamic method is developed to solve the EDM problems.Section 4 is the application of the probabilistic linguistic thermodynamic method in the EDM.Sections 5 and 6 are the self-comparison and comparative analysis parts based on a series of simulation experiments and nonparametric tests.Conclusions and future directions are presented in Section 7.

Preliminaries
In this section, we recall some concepts about PLTSs, and then introduce the original thermodynamic method and regret theory in the probabilistic linguistic environment which build a basic framework for the probabilistic linguistic thermodynamic method.

PLTSS
To describe the uncertain quantitative information, Pang et al. (2016) defined the PLTSs which included several linguistic terms with relevant probabilities.Let x i 2 X be fixed and S ¼ s Às , . . ., s À1 , s 0 , s 1 , . . ., s s f g be a linguistic term set, a PLTS on S is H S ðpÞ ¼ < x i , h i S ðpÞ > x i 2 X j g È with h i S ðpÞ ¼ s iðlÞ f ðp ðlÞ Þjs iðlÞ 2 S, p ðlÞ !0, l ¼ 1, 2, . . ., #L, P #L l¼1 p ðlÞ 1g, where s iðlÞ ðp ðlÞ Þ is the l th linguistic term s iðlÞ with the probability p ðlÞ : The linguistic term s iðlÞ is arranged in ascending order.h i S ðpÞ is called the probabilistic linguistic element and #L is the number of different s iðlÞ in h i S ðpÞ: When P #L l¼1 p ðlÞ <1, the missing probability can be assigned to the linguistic terms appearing in s iðlÞ : Assume that the unknown probability of s iðlÞ is equal in this paper, then _ p ðlÞ ¼ p ðlÞ = P #L l¼1 p ðlÞ , l ¼ 1, 2, . . ., #L and Pang et al., 2016).To compute and compare the PLTSs with different numbers of linguistic terms, an extension method (Pang et al., 2016) is proposed as follows: For any two PLTSs h 1 S ðpÞ and h 2 S ðpÞ, if #L 1 >#L 2 , then we add #L 1 À#L 2 linguistic terms to h 2 S ðpÞ, where corresponding probabilities of the added linguistic terms are equal to 0. The extension method is developed without changing any previous information of the PLTSs.Then we obtain the normalised PLTS: The score function and the deviation degree of PLTSs are introduced as follows: Definition 1 (Pang et al., 2016).Let h S ðpÞ ¼ s ðlÞ ðp ðlÞ Þjl ¼ 1, 2, . . ., #L È É be a PLTS, and r ðlÞ be the subscript of the linguistic term s ðlÞ : The score function of h S ðpÞ is: where r ¼ P #L l¼1 r ðlÞ p ðlÞ = P #L l¼1 p ðlÞ : The deviation function of h S ðpÞ is defined as: Definition 2 (Pang et al., 2016).Let h 1 S ðpÞ ¼ s 1ðlÞ ðp ðlÞ Then the deviation degree between h 1 S ðpÞ and h 2 S ðpÞ can be defined as: where r ðlÞ 1 and r ðlÞ 2 are the subscripts of the linguistic terms s 1ðlÞ and s 2ðlÞ , respectively.

Thermodynamic decision-making method
Thermodynamics is a subject concerning the interaction between the system and external environment as the state of matter changes (Charles & Herbert, 1990).The laws of thermodynamics show that the total energy is constant, and the entropy does not decrease over time in an isolated system (Wikipedia, 2021a).The entropy increase theory indicates that the entropy does not decrease in an isolated thermodynamic system, it always increases or remains unchanged (Wikipedia, 2021b).It means that an isolated system cannot move to a low entropy state and become orderly.Due to the increasing randomness of emergency and the complexity of uncertain decision-making environment, it is rather difficult to obtain complete and accurate decision information.It is of great significance to extend the existing decision-making method and extract valid information from the uncertain decision-making information with the thermodynamical indicators.Prathap (2011) applied thermodynamics combining with energy, exergy and entropy to the bibliometric research to rank the scientist's performance.The exergy indicator can reflect the amount of energy which can be converted to useful work.The entropy indicator reflects the unevenness of the decision potential in the rating of alternatives.Afterwards, Verma and Rajasankar (2017) proposed a thermodynamic MCDM method based on the exergy indicator under the crisp and fuzzy circumstance.
In the MCDM problem, assume that m alternatives are represented by The decision makers are denoted by D k ðk ¼ 1, 2, . . ., hÞ and they provide the decision matrix R k ¼ ðr k ij Þ mÂn to represent the decision values in regard to C j by the real number.The weight of C j assigned to D k can be expressed by w The thermodynamic decision-making method proposed by Verma and Rajasankar (2017) in the crisp environment is concisely introduced as follows: Step 1: Identify the decision matrices R k ¼ ðr k ij Þ mÂn ðk ¼ 1, 2, . . ., hÞ of the alternatives A i ði ¼ 1, 2, . . ., mÞ is the potential energy in regard to the criterion C j and the weight w k j is corresponding driving force.
Step 2: Calculate the energy matrices . ., hÞ and the quality matrices . ., hÞ for each decision maker, where Step 3: Construct the exergy matrix

. , hÞ:
Step 4: Compute the average energy and exergy of A i with respect to D k as follows: Step 5: Obtain the energy indicator u i and the exergy indicator x i by: Step 6: Determine the entropy indicator S i ¼ u i Àx i of A i , then rank the alternatives.The smaller value S i is, the better performance A i is.

Regret theory in the probabilistic linguistic environment
In the EDM, to minimise the losses as much as possible, decision makers need to select the optimal alternative among the rescue plans quickly.Since the decision information is uncertain and incomplete in the early stages of the emergency, most EDM problems are always the risk decision-making problems.It is quite necessary to consider the bounded rational characteristic of decision makers, such as reference dependence, loss aversion and regret aversion.Bell (1982) and Loomes and Sugden (1982) proposed regret theory which described decision makers' regret-aversion.In the regret theory, decision makers not only focus on the direct outcome, but also concern about the results if they select other solutions.In addition to the utility of the selected alternative, decision makers' regret-aversion should be considered in the behaviour decision-making theory.Therefore, the perceived utility of regret theory consists of two parts: the current results of the utility function and the regret-rejoice function compared with others.
Definition 3 (Zhang et al., 2016).Let x be the criterion value, then the utility function vðxÞ can be defined as follows: where the first and second derivative satisfy v 0 ðxÞ>0, v 00 ðxÞ<0, and a is the risk aversion coefficient of the decision maker.
Definition 4 (Zhang et al., 2016).The regret-rejoice function RðDvÞ can be defined as follows: where Dv denotes the utility difference of two alternatives, and RðDvÞ represents the regret-rejoice function of Dv: Similar to vðxÞ, the first and second derivatives of RðDvÞ satisfy R 0 ðDvÞ>0 and R 00 ðDvÞ<0, and d is the regret aversion coefficient of the decision maker.
Definition 5 (Zhang et al., 2016).Let x and y denote the evaluation values of the alternatives A and B, respectively.The perceived utility value of A is obtained by the utility function and the regret-rejoice function as follows: where r ðlÞ is the subscript of the linguistic term _ s 1ðlÞ and DH ¼ vðr ðlÞ 1 ÞÀ vðr ðlÞ 2 Þ: k is a parameter to adjust the importance degree of the utility function and the regretrejoice function.Since the utility function and the regret-rejoice function account for same proportion in the classical regret theory, we set k ¼ 0:5 in the following part of the paper.Refer to some literature about regret theory (Wang et al., 2021;Xue et al., 2021;Zhang et al., 2016), a ¼ 0:88, d ¼ 0:3 in this paper.
According to Equation ( 8), the probabilistic linguistic decision matrix (PLDM) is transformed into the PLRDM, which considers the regret-aversion characteristic of bounded rational decision makers.Compared with original decision matrix, the PLRDM can effectively reflect the actual behaviours of decision makers in the decision-making process.

Probabilistic linguistic thermodynamic method
In this section, we extend the thermodynamic decision-making method to the probabilistic linguistic environment based on the water-filling algorithm and regret theory.In the thermodynamic decision-making method, the weights of criteria in regard to decision makers are endowed subjectively.Because of the unexpectedness and uncertainty of the emergency, it is unrealistic for decision makers to provide the exact weight of each criterion in a short time.Due to the similarity between the water-filling theory of wireless communication area (Zhao et al., 2015) and the weight assignment method in the MCDM problems, the water-filling algorithm is applied to determine the objective weight of criterion.Combing with the subjective weight given by decision makers, the hybrid weight of each criterion can be determined.

Solving the objective weight of criterion by the water-filling algorithm
The water-filling algorithm is a classic algorithm for solving capacity maximisation problems in the multi-channel wireless communication systems.To maximise the ECONOMIC RESEARCH-EKONOMSKA ISTRAŽIVANJA channel capacity, the transmitted power is adaptive allocation according to the signalto-noise ratio of each sub-channel in the water-filling algorithm.As shown in Figure 1, when filling the container whose bottom is uneven with water, the height of the convex part at the bottom is inversely proportional to the amount of water.The higher the projecting at the bottom, the less water is injected, meaning the low power is allocated when corresponding signal-to-noise ratio of the sub-channel is small.Combining Shannon equation, the criterion weight of the k th decision maker w k j is derived by: where To obtain the optimal weight, the optimisation model of maximising the total capacity of criteria is established: 3.2.The probabilistic linguistic thermodynamic method In this section, first, we define the probabilistic linguistic potential (PLP) and the probabilistic linguistic energy (PLEN), which lay a foundation for the thermodynamic decision-making method with the probabilistic linguistic information.Definition 7. The PLP is the potential energy of an alternative towards a criterion, which can be represented by the value in the PLRDM, and the probabilistic linguistic force (PLF) is corresponding weight of a criterion.
Definition 8.The PLEN indicates the energy that an alternative possesses with respect to a criterion in the decision-making process, which can be defined by: is the PLP of an alternative, r iðlÞ is the subscript of the linguistic term s iðlÞ , and w is corresponding PLF.
Example 1.If the probabilistic linguistic assessment value of a decision maker towards an alternative with respect to a criterion is s 1 ð0:2Þ, s 2 ð0:8Þ È É ðs ¼ 3Þ and the weight of criterion is 0.2, then the PLP can be represented by s 1 ð0:2Þ, s 2 ð0:8Þ È É and the PLF is 0.2.According to Def. 8, the alternative's PLEN is calculated by: Based on the defined PLP, PLF and PLEN, some classical decision operators, such as the probabilistic linguistic weighted averaging operator (Pang et al., 2016) and the probabilistic linguistic Choquet integral operator (Chen et al., 2019), can aggregate the decision information of alternatives and obtain the ranking result.However, these methods only concern the quantity of the decision information but neglect the quality of the data.The quality of decision information is an important characteristic of the PLRDM, since it describes the divergence degree among alternatives in the MCDM problems.If all the decision makers have a consensus on the ranking result, then the quality is equal to 1.When the averaging of alternatives is same, the smaller the variance is, the more reliable the result becomes.Inspired by the definition of the deviation degree of PLTSs, we define the quality of the PLP as follows: Definition 9.The quality of the PLP for the k th decision maker can be measured by the similarity degree between itself and the averaging PLP of all decision makers, which can be defined by: where Eðr k Þ is the score function of the k th decision maker and Eð rÞ is the averaging score function of the PLP r k ðk ¼ 1, 2, . . ., hÞ: Example 2. Assume that the probabilistic linguistic assessment values of three decision makers with respect to the object are r 1 ¼ s 2 ð0:6Þ, s 3 ð0:2Þ È É , r 2 ¼ s 2 ð0:2Þ, s 4 ð0:8Þ È É , r 3 ¼ s 3 ð0:4Þ, s 4 ð0:6Þ È É ðs ¼ 4Þ, respectively.According to Definition 9, their qualities can be computed as: ¼ 0:9438: Remark 1.The range of q k is from 0 to 1.When all the PLPs r k ðk ¼ 1, 2, . . ., hÞ are equal, the qualities q k ¼ 1ðk ¼ 1, 2, . . ., hÞ: Definition 10.The probabilistic linguistic exergy (PLEX) is a rating indicator of the PLP, which considers the quantity and quality of the PLP and can be represented by: Definition 11.The probabilistic linguistic entropy can depict the unevenness of the PLP, which can be measured by: The probabilistic linguistic entropy is different from classical Shannon's entropy which assumes a prior distribution.It is an indicator that reflects how close the evaluation information of an alternative is to the overall information.In other words, the probabilistic linguistic entropy is smaller when the PLP is more consistent with other PLPs.The PLEX can effectively measure both the quantity and quality of the PLP, which makes the ranking results of the MCDM problems more credible and rational.

The procedure of the probabilistic linguistic thermodynamic method
In this section, we introduce the procedure of the probabilistic linguistic thermodynamic method based on the water-filling algorithm and regret theory as follows: Step . ., hÞ of the decision makers D k ðk ¼ 1, 2, . . ., hÞ according to the identified alternatives A i ði ¼ 1, 2, . . ., mÞ and the criteria C j ðj ¼ 1, 2, . . ., nÞ of the MCDM problem.
Step 2. Determine the objective weight vector of the j th criterion assigned by the k th decision maker, denoted as w k , k ¼ 1, 2, . . ., h by Equation ( 9) based on the PLDMs _ R k ¼ ð_ r k ij Þ mÂn ðk ¼ 1, 2, . . ., hÞ: Combining the subjective weight given by decision makers, the ratio of the subjective and objective weight is determined by the decision maker, then the hybrid weight is denoted by . ., hÞ based on Equation (8), after converting the PLDM into normalised PLDM.
Step 4. Compute the PLEN matrices of each decision maker: . ., hÞ: Step 7. Calculate the averaging PLEN and the averaging PLEX: Step 8. Compute the PLEN indicator and the PLEX indicator of each A i : Step 9. Obtain the entropy indicator of each alternative _ S i ¼ _ U i É _ X i : Calculate the comprehensive score of _ S i , the smaller the comprehensive score, the better the alternative.
Remark 2. The subscript of the virtual linguistic terms in the PLEN matrices _ U k ðk ¼ 1, 2, . . ., hÞ and the PLEX matrices _ X k ðk ¼ 1, 2, . . ., hÞ become very small after a series of calculations of scalar multiplication with -k ð0 -k 1Þ and q k ð0 q k 1Þ: Although the semantics of virtual linguistic terms in the PLEN matrix and the PLEX matrix have changed completely, they are still valid when sorting by numerical values.
The concrete framework of the probabilistic linguistic thermodynamic method is constructed in Figure 2.

Applications in the EDM of floods in Chongqing, China
The Yangtze, China's longest river, recorded the fifth flood after the heavy rainfall in the upstream.Along with the second flood of Jialing River, the floods passed through the central city of Chongqing, located in southwest China, on August 18, 2020.Although the upstream outflow of Xiangjiaba Reservoir was reduced from 6300 to 4000 m 3 /s and the downstream outflow of Three Gorges Reservoir was increased from 42,000 to 46,000 m 3 /s (www.news.cn,2020).As five floods occurred in the upper reaches of the Yangtze since July 2020 and the water level of the Yangtze was unusually high in Chongqing section after the floods accumulation.On August 17, Chongqing Cuntan hydrologic station recorded a water flow of 50,100 m 3 /s.According to the report released by the Upper Hydrology Bureau of the Yangtze River Commission, this round of floods will cause the water level of the Yangtze and Jialing River in Chongqing section to exceed the guaranteed water level seriously, and the water level in the main urban area of Chongqing will reach the highest level since 1981.
In view of the urgency of situation, on August 18, 2020, Chongqing activated a top-level response in the country's emergency response system.It is quite necessary to strengthen the management of funds and materials for the flood control and implement some effective and timely measures to respond to the floods.The local government should monitor the areas which are prone to disasters, such as reservoir dikes, floods storage and detention areas, urban waterlogging points.In the key areas, the relevant departments should strengthen monitoring and inspection, continue to check the safety hazards, and rectify hidden perils.It is quite important to dispatch scientifically the flood control projects and prevent the water level from exceeding the warning line effectively.The relevant departments should promptly issue early warning about landslides, mud-rock flows and other geological disasters, strengthen safety precautions in the road traffic, tourist attractions and construction sites, and resolutely avoid accidents involving mass casualties.The emergency rescue forces should be fully mobilised to relocate and resettle the disaster-hit people, and strive to achieve an overall victory in the flood relief.The floods caused the emergency evacuation of 251,000 people and inundated 23,700 stores, fortunately, no deaths (People.com.cn, 2020).
Tongnan County, located in the one-hour economic circle of Chongqing, was hit by the fifth flood of Yangtze combined with the impact of the last flood.By 10 am on August 18, 2020, the water level in Tongnan was 244.81 m exceeding the guaranteed water level by 4.81 m and the flow reached 20,700 m/s.The local government made some preparations for relocation and resettlement of local residents before the floods flowed down.The criteria of the MCDM problem are C 1 : the number of casualties caused by the floods, C 2 : the property loss caused by the floods, C 3 : the input costs of the strategy, C 4 : the public satisfaction degree of the strategy.There are three alternatives: A 1 : Evacuate people from low-lying areas, close schools, and cancel all outdoor gatherings.
A 2 : Based on A 1 , check wiring and equipment for security failures and rectify hidden perils.
A 3 : Based on A 2 , impose temporary traffic controls and set up the rescue team and epidemic prevention team.

Decision support
In this section, the self-comparison is conducted to verify the effectiveness of the objective weight obtained by the water-filling algorithm and regret theory in the probabilistic linguistic thermodynamic method.A series of simulation experiments are conducted to verify the influence of the water-filling algorithm and regret theory.
The proposed method, the proposed method without regret theory and the proposed method without water-filling algorithm are denoted by Method 1, Method 2 and Method 3, respectively.First, we acquire the ranking results of 1000 MCDM problems randomly of three experts with m alternatives and n criteria by the three proposed methods.Then we record the times of same optimal and worst alternatives calculated by the three proposed methods and any two decision methods.The ratios of same optimal and worst alternatives of the three methods and any two methods are shown in Figure 3.
Figure 3 indicates that: (1) The ranking result of Method 1 is obviously similar to that of Method 2. Therefore, regret theory only slightly adjusts the PLDM to adapt the need of individual's regret-aversion.Regret theory would not greatly affect the ranking results of the proposed method.(2) There is difference between ranking results of Method 1 and Method 3. (3) Similarly, there is difference between ranking results of Method 2 and Method 3. The leading factor is that the weight of criterion changes caused by the water-filling algorithm in the proposed method.To further compare the significance differences among the three proposed methods, we conduce the following simulation experiments by the nonparametric tests.
The ranking results of the three methods are recorded for 1000 MCDM problems randomly of three experts with m alternatives and n criteria.The ranking results of the three methods can be converted into three sample sequences.Since the three sequences might not be normal distribution, the non-parametric test: Wilcoxon signed-rank test is introduced to compare the ranking results of the three methods by pairwise comparison.The results of Wilcoxon signed-rank test among the three methods with m alternatives and n criteria are shown in Table 1.The results of Wilcoxon signed-rank test are obtained by SPSS and the confidence level is set as 0.05 by convention in statistics.
As shown in Table 1, the concomitant probabilities of all the tests are greater than 0.05 indicating that the ranking results have no significant differences with the above three methods.Therefore, the probabilistic linguistic thermodynamic method is stable and reasonable.The water-filling algorithm only slightly changes the weight to extract useful information from the uncertain decision-making information.Regret theory and the objective weight obtained by the water-filling algorithm only flexibly adjust the ranking results according to the practical needs.

Comparative analysis
The comparison analysis is composed of two parts: (1) Compare the ranking results of the case in Pan et al. (2021) obtained by the probabilistic linguistic thermodynamic method, the PL-TOPSIS method (Pang et al., 2016), the PL-QUALIFLEX method (Tian et al., 2019), the PL-DEA method (Pan et al., 2021) and the PL-TODIM method (Wei & Wu, 2019) and analyse the comparison results.(2) According to a series of simulation experiments and non-parametric tests, further compare the probabilistic linguistic thermodynamic method, the PL-TOPSIS method (Pang et al., 2016) and the PL-QUALIFLEX method (Tian et al., 2019) and present the comparative conclusions.
Part 1.The case result comparisons with the probabilistic linguistic thermodynamic method, PL-TOPSIS method, PL-QUALIFLEX method, PL-DEA method and PL-TODIM method We calculate the ranking results of five probabilistic linguistic decision-making methods by the decision-making data of Pan et al. (2021), and then compare the decision making results of the five decision making methods shown in Table 2.
As shown in Table 2, the ranking results of the PL-TOPSIS method and the PL-QUALIFLEX method are exactly the same.The optimal alternative obtained by the above two methods is the alternative DMU 3 : Although the optimal alternative obtained by the probabilistic linguistic thermodynamic method, the PL-DEA method and the PL-TODIM method is the same, the specific ranking results are different.The worst alternative of the probabilistic linguistic thermodynamic method and the PL-TODIM method is DMU 4 : Due to the complexity of the decision-making environment and the limitations of human thinking, it is difficult for decision makers to provide precise evaluations for alternatives.How to measure the quantity and quality of the uncertain decision-making information is a critical factor that affects decision-making results, especially when the evaluation information given by decision makers differs greatly.The thermodynamical indicators are introduced to the probabilistic linguistic environment to extract the main features of the uncertain information.Besides, the proposed probabilistic linguistic thermodynamic method not only considers decision makers' regret-aversion, but also assigns the criterion weight adaptively by the water-filling algorithm.Although the PL-TODIM method also takes the bounded rationality of decision makers into account, the proposed method introduces the water-filling algorithm to determine the objective weight adaptively.The water-filling algorithm assigns the criterion weight adaptively, aiming to maximise the total ability of criteria.The above reasons explain the ranking difference between the proposed method and other probabilistic linguistic decision-making methods from the methodological principle.
Part 2. The simulation result comparisons with the probabilistic linguistic thermodynamic method, the PL-TOPSIS method and the PL-QUALIFLEX method In order to further compare the ranking results of the probabilistic linguistic thermodynamic method, the PL-TOPSIS method and the PL-QUALIFLEX method, Friedman test is utilised to examine the significant differences among the three methods by SPSS.Friedman test and Wilcoxon signed-rank test are nonparametric test methods that test the distribution of samples whether there is a significant difference.Different from Wilcoxon signed-rank test, Friedman test can deal with multiple related samples.Since the PL-TOPSIS method and the PL-QUALIFLEX method are not group decision-making methods, the ranking results of three sequences of the three methods are recorded for 1000 random MCDM problems of an expert with m alternatives and n criteria.Each criterion is endowed the same subjective weight and the confidence level is taken as 0.05.Likewise, the ranking results are converted into three related sequence samples.The concomitant probabilities of Friedman test among the proposed method, the PL-TOPSIS method and the PL-QUALIFLEX method are presented in Table 3.
Table 3 shows that the ranking results of the proposed method, the PL-TOPSIS method and the PL-QUALIFLEX method have significant differences, that is, the concomitant probability is less than the confidence level 0.05, except the case that  m ¼ 4, n ¼ 4: The main reason is that the number of alternatives is too few when m ¼ 4, n ¼ 4: To further study the significance difference among the three methods, we conduct pair-wise comparisons by Wilcoxon signed-rank test with same ranking samples in Table 3.The concomitant probabilities among the three methods are presented in Table 4.According to Table 4: (1) The concomitant probabilities of Wilcoxon signed-rank test are greater than 0.05 when m ¼ 4, n ¼ 4, which means that the ranking results have no significant differences with the above three methods.(2) The concomitant probabilities of Wilcoxon signed-rank test between the proposed method and the PL-TOPSIS method are less than 0.05 except when m ¼ 4, n ¼ 4, which means that their ranking results vary enormously.(3) There is significant difference between the proposed method and the PL-QUALIFLEX method except m ¼ 4, n ¼ 4: (4) There is no significant difference between the PL-TOPSIS method and the PL-QUALIFLEX method.Since the number of alternatives is too few, we neglect the case of m ¼ 4, n ¼ 4: The ranking results of the PL-TOPSIS method and the PL-QUALIFLEX method are quite similar, while the proposed probabilistic linguistic thermodynamic method is significantly different from them.The probabilistic linguistic thermodynamic method is a new decision method based on the water-filling algorithm and regret theory.It applies thermodynamics combining with energy, exergy and entropy indicators to aggregate the probabilistic linguistic information.Hence, there are significance differences in the ranking results of the proposed method and other two methods.Although the ranking results of the PL-TOPSIS method and the PL-QUALIFLEX method do not differ significantly, the time complexity of the PL-TOPSIS method is lower than the PL-QUALIFLEX method.In addition, to avoid complex calculations, the QUALIFLEX method should be applied to the decision-making problems where the number of criteria is much more than the number of alternatives (Chen et al., 2013).

Figure 1 .
Figure 1.The schematic diagram of water-filling theory.Source: Authors' own research.

Figure 2 .
Figure 2. A framework of the probabilistic linguistic thermodynamic method.Source: Authors' own research.

Figure 3 .
Figure 3.The ratios of same optimal and worst alternatives.Source: Authors' calculation.

Table 1 .
Wilcoxon signed-rank test results among the three methods with m alternatives and n criteria.

Table 2 .
The ranking results with the five decision-making methods.

Table 3 .
Friedman test results among the three methods with m alternatives and n criteria.

Table 4 .
Wilcoxon signed-rank test results among the three methods with m alternatives and n criteria.