A Novel Multi-Attribute Decision-Making Algorithm for Landll Site Selection Based on Q-Rung Orthopair Probabilistic Hesitant Fuzzy Power Weight Muirhead Mean Operator

Background With the rapid development of economy and the acceleration of urbanization, the garbage produced by urban residents also increases with the increase of population. In many big cities, the phenomenon of "garbage siege" has seriously affected the development of cities and the lives of residents. Sanitary landfill is an important way of municipal solid waste disposal. However, due to the restriction of social, environmental and economic conditions, landfill site selection has become a very challenging task. In addition, landfill site selection is full of uncertainty and complexity due to the lack of cognitive ability of decision-makers and the existence of uncertain information in the decision-making process. A novel multi-attribute decision making method based on q-rung orthopair probabilistic hesitant fuzzy power weight Muirhead mean (q-ROPHFPWMM) operator is proposed in this paper, which can solve the problem of landfill site selection well. This method uses probability to represent the hesitance of decision maker and retains decision information more comprehensively. The negative effect of abnormal data on the decision result is eliminated by using the power average operator. Muirhead mean operator is used to describe the correlation between attributes.


Introduction
In recent years, with the rapid economic development and the acceleration of urbanization, the urban population has gradually increased, and the generated waste has also become more, such as household waste, construction waste, and industrial waste. In many large cities, the phenomenon of "garbage siege" has become more and more intense, and the hazard and disposal of garbage has become an important social issue affecting urban development [1]. Since the information in the site selection process. In order to describe the uncertain information in practical problems, Zadeh proposed fuzzy set theory [13]. With the development and improvement of fuzzy set theory, some scholars have conducted research on the extended forms of fuzzy sets, such as intuitionistic fuzzy sets [14], interval fuzzy sets [15], hesitant fuzzy sets [16] and so on. The fuzzy set (FS) theory proposed by Zadeh only describes the fuzziness of information from the perspective of the degree of membership. The intuitionistic fuzzy set (IFS) proposed by Atanassov [14] describes the fuzziness of information more completely from the perspective of membership and non-membership. In order to describe a wider range of fuzzy information, Yager [17] proposed the Pythagorean fuzzy set (PFS), which can describe the situation where the sum of the degree of membership and the degree of non-membership exceeds 1, and the sum of the squares does not exceed 1. However, there are some situations in practice that PFS can't describe. Yager further promoted PFS and proposed the concept of q-rung orthopair fuzzy set (q-ROFS) [18]. q-ROFS is a generalized form of IFS and PFS, which can describe a wider range of uncertain phenomena. In the actual decision-making process, there is often hesitation. For this reason, Torra [16] proposed the hesitant fuzzy set (HFS), which allows the membership of an element to be a set of multiple possible values between 0 and 1. HFS can more comprehensively describe the uncertain information given by the decision maker, but the elements in its set cannot be repeated, and there is no difference between them. However, in most cases, due to the personal preference of the decision makers and the number of decision makers, different degrees of membership may have different importance. The hesitant fuzzy element cannot describe the preference information of decision makers for different degrees of membership. This problem also exists in q-rung orthopair hesitant fuzzy sets (q-ROHFS) [19]. Bedregal et al. [20] tried to use fuzzy multi-set to solve this problem, but its expression was too cumbersome. In order to overcome the shortcomings of HFS and at the same time solve the cumbersome problem of fuzzy multi-set representation, Xu et al. [21] first proposed the probabilistic hesitant fuzzy set (PHFS). But it requires that the sum of the probabilities of the probabilistic hesitant fuzzy elements is equal to 1, which leads to the limitation of the expression space of the decision maker. Zhang et al. [22] improved the PHFS, weakened its constraint conditions, and allowed the sum of the probabilities of probabilistic hesitant fuzzy elements to be less than 1.
There are many factors that affect the site selection of waste landfills. Therefore, decision makers will inevitably give too high or too low evaluation values due to lack of personal experience or prejudice towards things. Yager [23] proposed the power average (PA) operator, which reduces the negative impact of unreasonable evaluation information on the results by considering the support relationship between the data. Considering the powerful functions of the PA operator, some scholars have done further research on the PA operator and widely used it in intuitionistic fuzzy information integration [24], hesitant fuzzy information integration [25], and language information integration [26]. In the evaluation process of landfill site selection, there is an association between different influencing factors, so that the determination of the evaluation value of one factor will be affected by other factors. If this is not considered, it will affect the final decision result. For multi-attribute decision in evaluating the relationship between the information, the majority of scholars a lot of work. Choquet integral operator, Bonferroni mean (BM) operator, Heronian mean (HM) operator, Maclaurin symmetric mean (MSM) operator, and Muirhead mean (MM) operator are successively used for information integration [27][28][29][30][31]. Among them, the MM operator can reflect the correlation relationship between any number of decision information. In order to solve the negative impact of unreasonable evaluation information on the results, and to characterize the internal relationship between different factors, He et al. [32] tried to combine the PA operator with the BM operator and proposed the PBM operator. Subsequently, scholars successively proposed PHM operators, PMSM, and PMM operators, and applied them to the integration of various fuzzy information, demonstrating their powerful functions [33][34][35].
In summary, the research on site selection of waste landfill based on fuzzy theory has made certain progress. However, the current presentation of evaluation information based on FS, IFS and q-ROFS is not complete, and the hesitation of decision makers is not considered. In order to describe the evaluation information of decision makers more completely, this paper is inspired by the PHFS, and improves the q-ROHFS, and proposes the q-Rung Orthopair Probabilistic Hesitant Fuzzy Set (q-ROPHFS). q-ROPHFS not only describe the evaluation information of decisionmakers more completely, but also give decision-makers more freedom to make decisions. Compared with IFS and PFS, q-ROPHFS has a wider range of membership and non-membership. In addition, in the evaluation process of landfill site selection, there are many factors that affect the evaluation of candidate sites, and the evaluation information given by decision makers is not completely accurate. At the same time, the evaluation value of each influencing factor will be affected by other factors. At present, most studies have not considered these issues. In order to eliminate the adverse effects of unreasonable information given by decision makers in the evaluation process, and to better characterize the correlation between evaluation information, PA operator and MM operator are extended to q-ROPHFS. Then, the q-rung orthopair probabilistic hesitant fuzzy power weight Muirhead mean (q-ROPHFPWMM) operator is constructed and applied to the candidate site evaluation of landfill sites.
The contributions of this study are as follows: (1) Inspired by PHFS, the q-ROPHFS is proposed.
(2) The q-ROPHFPWMM operator is constructed by combining q-ROPHFS, PA operator and MM operator.
(3) A multi-attribute decision making algorithm for candidate address evaluation is proposed based on Q-order probabilistic hesitant fuzzy PMM operator.
(4) A landfill site selection model is established based on the proposed multi-attribute decision making algorithm.

The definition of q-ROHFS and PHFS
is arbitrary sequence of (1,2, , ) n L , and n S is the set of all possible sequences of (1,2, , ) n L .

The definition of q-ROPHFS
 and A  represent the number of elements contained in them, respectively.

The operation of q-ROPHFE
The operation of q-ROPHFE is defined by referring to the operation of q-rung orthopair hesitant fuzzy element [19] and the operation of probabilistic hesitant fuzzy element [21].

The distance between two q-ROPHFEs
Distance measure is a commonly used tool to describe the difference between the two. In this section, we propose the distance between any two q-ROPHFEs Definition 7. Let 1 h and 2 h be two q-ROPHFEs. If (2) If and only if 1 2 h h  , Based on the Hamming distance and Euclidean distance, the Hamming distance and Euclidean distance between two q-ROPHFEs are defined as follows.
(1) The standardized Hamming distance between 1 h and 2 h is: And so on, you get ( ) (2) The standardized Euclidean distance between 1 h and 2 h is: The standardized generalized distance between 1 h and 2 h is:   .

The ranking of the q-ROPHFEs
Inspired by PHFS, we define the score function and deviation degree of q-ROPHFE. Definition 8. Let h be a q-ROPHFE, then its score function ( ) S h is defined below: where h  and h  represent the number of elements contained in them respectively.
Suppose that 1 h and 2 h are two arbitrary q-ROPHFEs. If , there is no way to compare 1 h and 2 h using a score function. Therefore, the deviation degree of q-ROPHFE needs to be defined. Definition 9. Let h be a q-ROPHFE, and its score function is expressed by  , then the deviation degree ( ) D h is defined below: The comparison method of q-ROPHFE is given according to the score function and deviation degree of q-ROPHFE: (1) If

The definition of the q-ROPHFPWMM operator
In this section, we generalize the power average operator [23] and Muirhead mean operator [36] to q-ROPHFS, and propose the q-rung orthopair probabilistic hesitant fuzzy power Muirhead mean operator.
represents any permutation of (1, 2, , ) n L , n S represents the set of all possible permutations of (1, 2, , ) n L , and n is the regulation coefficient.

Proof.
According to the operation of q-ROPHFEs in Definition 6, we can get Then, we use mathematical induction theory to get Similarly, according to Definition 6, we can get (1 ) Therefore, Theorem 2 is proved.

Some special form of the q-ROPHFPWMM operator
The q-ROPHFPWMM operator can integrate information more flexibly by using its special parameter vector, and describe the correlation relationship among any attributes. When the parameter vector P takes a specific value, the q-ROPHFPWMM operator will degenerates into other operators. Case 1. When (1, 0, , 0) P  L , q-ROPHFPWMM operator degenerates into q-rung orthopair probabilistic hesitant fuzzy power weight average (q-ROPHFPWA) operator: (1,0, ,0) 1 2 1 1 1 1 1 1 L traversals all the k-tuple combination of (1, 2, , ) n L , n S is the set of all  , and is the binomial coefficient.

Some properties of the q-ROPHFPMM operator
In this subsection, we discuss the properties of q-ROPHFPWMM operator, including idempotence, boundedness and monotonicity.

A novel MADM algorithm framework based on q-ROPHFPWMM operator
In this section, we construct the MADM algorithm framework based on the q-PHFPWMM operator. Let Step1. Construct an evaluation index system, collect evaluation information and transform it into a q-rung orthopair probabilistic hesitant fuzzy decision matrix.
Step2. The q-rung orthopair probabilistic hesitant fuzzy decision matrix is normalized to obtain the normalized q-rung orthopair probabilistic hesitant fuzzy decision matrix. The normalization is as follows: , is the benefit attribute, ( ) , is the cost attribute.
Step3. Select the appropriate risk preference coefficient X according to the decision maker's risk preference, and add elements to the set whose probability sum is less than 1 to make the probability sum equal to 1.
Step4. Calculate the support degree between ij h and ik h ( 1, 2, i m  L ; , 1, 2, j k n  L and k l  .): Step5. Calculate the syntheses support degree: Step9. Sort the candidate solutions according to their score and deviation degree, and select the corresponding optimal solution.
6. The case on site selection assessment for landfill

Influencing factors of landfill site selection
With the advancement of urbanization, the scale of cities continues to expand, and the urban population soars, resulting in more and more garbage generated in the city. In recent years, the problem of garbage disposal has emerged in many cities around the world. At present, the main methods of urban waste disposal are through incineration for power generation and sanitary landfill. However, the way of incineration for power generation is not accepted by most citizens. Therefore, most cities choose to build new landfills to reduce the burden of waste disposal, but the site selection needs to consider several factors, including: 1. Geographic location First of all, the site selection of the landfill site should consider whether its geographic location is consistent with the overall planning of the city [37]. Secondly, it also needs to consider its radiative capacity, i.e. the range of services it can provide. On the premise of reducing the burden of the city, it can also share part of the burden of the surrounding cities.

Operating cost
Operating costs of landfill sites mainly come from four aspects, including land utilization, equipment maintenance, garbage transportation and manpower [38]. Among them, the cost of land resources is huge. Landfills need a large amount of land resources, and the landfill can only be used for greening after the landfill is filled with green. And the land is not regenerative for 100 years.

Traffic conditions
Landfills are typically located in suburban areas, away from densely populated areas. In order to reduce the transportation cost of garbage, the distance between the landfill site and the urban area and the road conditions should be considered [39]. In addition, because of the "NIMBY" effect of garbage trucks, cities often set garbage truck driving hours between 3 a.m. and 5 a.m.

Environmental pollution
Landfills inevitably cause pollution to the surrounding environment, mainly including air pollution, soil pollution and water pollution [40]. Air pollution mainly comes from waste gas, dust and inhalable particles released from garbage disposal process, which in turn leads to acid rain and smog. In addition, garbage rotting released harmful gases, such as hydrogen sulfide, can also seriously pollute the atmosphere. Soil pollution is caused by the fact that heavy metals, chemical agents and plastic products contained in garbage cannot be degraded in the soil, which leads to the decrease of crop production and quality in surrounding farmland. In the process of garbage stacking and corruption, a large amount of acidic and alkaline organic pollutants will be generated, which will dissolve heavy metals in the garbage. These harmful components will flow into the river water after being washed by rainwater and cause surface water pollution. At the same time, the leachate from the garbage seeps into the soil and causes groundwater pollution.

Evaluation process of landfill site selection
There Step 1. The influential factors of landfill site selection are analyzed to build an evaluation index system, as shown in Figure 1.  Table 1. Step 2. Standardize the q-rung orthopair probabilistic hesitant fuzzy decision matrix in Step1 to obtain the standardized q-rung orthopair probabilistic hesitant fuzzy decision matrix. Among them, 1 M and 3 M are benefit attribute indexes, 2 M and 4 M are cost attribute indexes.
Step 3. Assume that the decision-maker is risk averse, so take the risk preference coefficient Add elements to the set whose probability sum is less than 1 in the normalized Q-order probabilistic hesitant fuzzy decision matrix so that the probability sum is equal to 1.
Step 4. Calculate the support degree ( , ) Step 6. Calculate the power weight of ij h and get the matrix   Step 7. Utilize the q-PHFPWMM operator ( The results are shown in Table 2.  Step 9. Sort the candidate address according to the score value of the candidate address to get 3 1 . Therefore, the best landfill site is A.

Parameter analysis
By changing the value of the parameter vector P , the ranking results of different landfills are obtained by using the q-ROPHFPWMM operator ( 3 q  ), as shown in Table 2.