Medical waste treatment scheme selection based on single-valued neutrosophic numbers

: With the rapid increase in the number of infected people in COVID-19, medical supplies have been increasing significantly. Medical waste treatment scheme selection may have long-term impacts on the economy, society, and environment. Determining the best treatment option is a considerable challenge. To solve this problem, in this paper, we proposed a multi-criteria group decision making (MCGDM) method based on single-valued neutrosophic numbers and partitioned Maclaurin symmetric mean (PMSM) operator. Because of the complexity of the medical waste treatment scheme selection problem, the single-valued neutrosophic numbers are applied to express the uncertain evaluation information. For the medical waste treatment scheme selection problem, the factors or criteria (these two terms can be interchanged.) in the same clusters are closely related, and the criteria in different clusters have no relationships. The partitioned Maclaurin symmetric mean function can handle these complicated criterion relationships. Therefore, we extend the PMSM operator to process the single-valued neutrosophic numbers and propose the single-valued neutrosophic partitioned Maclaurin symmetric mean (SVNPMSM) operator and its weighted form (SVNWPMSM). Then, we analyze their properties and give typical examples of the proposed operators. An MCGDM model based on the SVNWPMSM aggregation operator is developed and applied to solve the medical waste treatment scheme selection problem. Finally, the validity and superiority of the developed model are verified by comparing it with the previous methods


Introduction
The medical wastes are caused by doctors and nurses during various medical services. They may be the disposable syringe, disposable examination glove, and surgical mask. They have been polluted by bacteria and viruses. They may infect a large number of people when they are used again. Therefore, the medical wastes should be safely treated [1,2]. In the past years, the total amount of medical wastes initially increases every year. Since the horrible COVID-19 outbreaks in 2019, many persons from all over the world have been infected, which makes the COVID-19 pandemic [3]. Treating these patients that have been infected by the COVID-19 has produced massive medical wastes. As COVID-19 spreads rapidly to many countries, the total amount of medical waste per day has increased several times. To cope with this global crisis, a number of new medical waste treatment plants have been built. Some new medical waste treatment schemes also have been developed to improve treatment efficiency. Because of these alternative means, the medical waste treatment efficiency has been dramatically increased. For example, the medical waste treatment efficiency in Wuhan city of China has been increased by more than five times at the beginning of 2020.
There are several types of medical waste treatment schemes that can be evaluated and selected. Different medical waste treatment schemes show different features. The selection problem of medical waste treatment schemes requires consideration of the influences of multiple factors. Determining the best medical waste treatment scheme is a big challenge since each medical waste treatment scheme has different advantages and disadvantages [4,5]. Moreover, the process of determining the best medical waste treatment scheme needs a group of experts to participate. This case makes the process complex. Therefore, the selection problem of medical waste treatment schemes can be formulated to be an MCGDM (multi-criteria group decision making) problem. For the MCGDM process, the evaluation criteria should be determined for alternatives. Then a group of experts is invited to evaluate alternatives with respect to the evaluation criteria. Finally, all the evaluation information is fused to rank the alternatives. Due to the complexity of the selection problem of medical waste treatment schemes, in this paper, we introduce the tool of single-valued neutrosophic sets (SVNSs) to express the evaluation information of medical waste treatment schemes. SVNSs [6,7] are the generalization of fuzzy sets, which can represent the evaluation information in a more accurate way.
The selection process of medical waste treatment schemes should consider multiple criteria, which belong to economic, social, and environmental aspects. These criteria may have interrelationships or be independent of each other. To process these complex interrelationships among criteria, a novel MCGDM method based on single-valued neutrosophic numbers (SVNNs) and partitioned Maclaurin symmetric mean (PMSM) operator is proposed in this paper. Our main contributions are listed as follows: (1) The PMSM operator is used to fuse the evaluation information in terms of SVNNs, and then a novel single-value neutrosophic PMSM (SVNPMSM) operator and its weighted form are proposed. Their features are also discussed.
(2) A novel MCGDM model based on the single-value neutrosophic weighted PMSM (SVNWPMSM) operator is proposed to deal with the selection problem of medical waste treatment schemes.

Literature review
The selection problem of medical waste treatment schemes is a complex decision-making problem. It is difficult for experts to use crisp values to evaluate the medical waste treatment schemes [8]. For the complex decision-making problem, the concept of fuzzy sets (FSs) [9,10] is an alternative for modeling uncertain information. The fuzzy sets use the degree of membership to measure the uncertain information. This concept was extended by Atanassov [11] using the degree of nonmembership, and the concept of intuitionistic fuzzy sets (IFSs) was designed. When expressing fuzzy and uncertain information, IFSs give a means that is more intuitive and effective than FSs. However, both IFSs and FSs do not have the ability to express inconsistent evaluation information. In this case, Smarandache designed the neutrosophic sets that consist of the truth degree, indeterminacy degree, and falsity degree [12]. The values of the truth degree, indeterminacy degree, falsity degree, and their sum are the non-standard interval subsets of ]0 ,[1 −+ . Since it was proposed, its decision-making theories and methods have been studied by researchers [13][14][15]. However, it is difficult to use the neutrosophic sets in real engineering applications. To promote the real application of neutrosophic sets, an extended concept of single-valued neutrosophic sets (SVNSs) was proposed by Wang et al. [16] by restricting the values of the truth degree, indeterminacy degree, and falsity degree to be in the interval [0,1]. Since its appearance, SVNSs have been used in various fields such as medical diagnosis [17], assessment of consumers' motivations [18], and typhoon disaster assessment [19]. For the complex decision-making problem, how to aggregate or fuse the evaluation information is a big challenge. Information aggregation operators are an effective but simple fusion means [20,21]. Various aggregation operators have been proposed to fuse the evaluation information, such as OWA (ordered weighted averaging) operator [22], IOWLAD (induced ordered weighted logarithmic averaging distance) operator [23], WA (weighted averaging) operator [24], PA (power averaging) operator [25], and Hamacher aggregation operator [26]. However, these aggregation operators do not consider the relationship between the evaluation information [27]. For the complex decision-making problem, there exists a correlation relationship between the evaluation information. To consider this correlation relation, the Bonferroni mean function [28] and Heronian mean function [29] are extended for fusing various fuzzy evaluation information. For example, Ates et al. [30] improved the Bonferroni mean to fuse picture fuzzy information. Lin et al. [31] extended the Heronian mean to fuse linguistic q-rung orthopair fuzzy information.
Both Bonferroni mean and Heronian mean functions only consider the correlation relationship between two input values. The Maclaurin symmetric mean (MSM) [32] is an excellent mapping function that can capture the interrelation among evaluation information. Hence, it is more generic than Bonferroni mean and Heronian mean [33]. It has been extended by scholars and researchers to aggregate complex q-rung orthopair fuzzy sets [34], linguistic intuitionistic fuzzy numbers [26], and intuitionistic fuzzy soft sets [35]. Nevertheless, these criteria are not always correlated with each other. There may exist cluster relationships among these criteria. The criteria in the same clusters are closely related, but the criteria in the different clusters have no relationship. In order to cope with this complex interrelation, Liu et al. [32] proposed the partitioned MSM (PMSM) operator, which can not only capture the correlation relation among the evaluation information of criteria in the same clusters, but also consider the independence relation between clusters. The PMSM operator has been used to process 2-dimensional linguistic information [36], linguistic neutrosophic information [37], q-rung orthopair uncertain linguistic information [38]. However, the information structure of single-valued neutrosophic sets is very different from 2-dimensional linguistic information, linguistic neutrosophic information, and q-rung orthopair uncertain linguistic information. Their research results cannot be simply and directly applied to the single-valued neutrosophic sets.

Single-valued neutrosophic sets
Definition 1. [12] Suppose X is the collection of discourse, the neutrosophic set p can be denoted as: where ,, and the accuracy function () Hp of p is given by: Then, these two SVNNs can be compared according to the following rules:

The partitioned Maclaurin symmetric mean (PMSM) operator
Proof. According to Eq (3)-(6), we have Then we can get Further, 1 11 1 11 11 Thus, we can get According to Definition 2, it is easy to know that Therefore, we can get 11 11 11 11 According to the above proof process, the proof of Theorem 1 is completed. Then, we will discuss some characteristics of the SVNPMSM operator.
According to the above proof process, we have completed the proof of Idempotency.
 , and satisfies ,, According to Definition 7, we get  According to the above proof process, we have completed the proof of boundedness. Then, we introduce a lemma that will be used in the proof of Theorem 5.
Then, we prove the monotonicity of () Xk using the contradiction method. Suppose that the function () Xk decreases monotonously with the increase of the variable k . When 1 k = , we can get  . Therefore, it is proved that the SVNPMSM operator will decrease monotonically as the variable k increases. Then, several special examples of the SVNPMSM operator are briefly described.
(1) When the number of clusters 1 S = , it means that no cluster is required among the criteria. The SVNPMSM operator will become the single-valued neutrosophic Maclaurin symmetric mean (SVNMSM) operator: ( ) (2) When the parameter 1 k = , the SVNPMSM operator will become the single-valued neutrosophic partitioned mean (SVNPM) operator: (3) When the parameter 2 k = , the SVNPMSM operator will become the single-valued neutrosophic PBM (SVNPBM) operator ( 1 pq ==): where 12    Similarly, the SVNWPMSM operator also has the characteristics of idempotency, boundedness, and monotonicity. In the following part, we will present some typical examples of the SVNWPMSM operator.
(1) When the number of clusters 1 S = , it indicates that there are no clusters among criteria. The proposed SVNWPMSM operator will change to the single-valued neutrosophic weighted Maclaurin symmetric mean (SVNWMSM) operator: ( ) (2) When the parameter 1 k = , the SVNWPMSM operator will become the single-valued neutrosophic weighted partitioned mean (SVNWPM) operator: (3) When the parameter 2 k = , the SVNWPMSM operator will become the single-valued neutrosophic weighted PBM (SVNWPBM) operator:

A novel MCGDM model using the proposed SVNWPMSM operator
The proposed SVNWPMSM operator can efficiently handle the complex relationship among criteria, so we apply it to solve the MCGDM problem. Let us suppose that We developed a novel MCGDM model using the SVNWPMSM operator to fuse the evaluation matrices. The detailed steps of the model are shown in the following part.
Step 1: Standardization of the evaluation information.
In the MCGDM problem, the criteria may be either benefit type or cost type. For subsequent processing, the data of cost type needs to be converted to be benefit type. Assume that [ ] , , is a standardized evaluation matrix. The evaluation matrix is normalized in the following way.
, , the benefit criterion of ,, ,1 , the cost criterion of where S denotes the number of clusters, g h is the number of criteria in the cluster g P . k is a parameter, q   is the weight of the criterion , , , , , where d represents the number of clusters, and decision-makers usually do not need to be partitioned, i.e., 1 d = . g h denotes the number of decision-makers in the cluster g P . k is a parameter, and l   denotes the weight of l D  .
Step 4: Choose the optimal alternative.
In order to obtain the best alternative, the score function () i S  of

Case study
In this section, the developed MCGDM model is applied to solve the medical waste treatment schemes selection problem.

Illustrate example
Medical waste treatment schemes selection is a critical problem in the field of environmental protection. This problem is a crisis looming, especially after the outbreak of the COVID-19. To validate the effectiveness of the proposed MCGDM model, we illustrate an example from Fuzhou city, China. Fuzhou is located in the eastern part of China, with a population of more than 8 million. About 2 tons of medical wastes were produced in Fuzhou per day. Currently, medical wastes are mainly treated by incineration. Incineration has caused many problems for people living near the medical waste treatment stations. Therefore, the authorities are considering new solutions for the treatment of medical wastes.

Steps to address the selection problem of medical waste treatment schemes
Step 1: Standardization of the evaluation information.
According to the description of the criteria ( 1, 2,...,7) j j  = , all of them are benefit types. Thus, there is no need to normalize the evaluation matrix. Then, we can get that , , , , Step 2: Calculate the aggregation values of the criteria of medical waste treatment schemes. We can use Eq (17)  Step 3: Calculate the aggregation value of the evaluation information of decision-makers.
We apply Eq (18) to calculate the aggregation values i  of the evaluation information of decision-makers. The number of clusters 1 d = since the decision-makers usually do not need to be clustered. Assume the parameter 2 k = , the aggregation values of the evaluation information of decision-makers are obtained as follows: Then, we compare the above score values (     , where the symbol indicates preferred to . Therefore, the optimal medical waste treatment scheme is Microwaving ( 4  ).

The effects of variable S and parameter K on the ranking results
According to the above example, we will discuss the influence of the cluster number S and the parameter k on the ranking results. Different values of the cluster number S may lead to changes in the morphology of the proposed operator. For example, when 1 S = , the proposed SVNWPMSM operator will change to the SVNWMSM operator. Moreover, it is known from Theorem 7 that the SVNWPMSM operator monotonically increases as the parameter k decreases. We assign different values to the cluster number S and parameter k in the above example. The effects of S and k on the ranking results of medical waste treatment schemes are discussed. Table 4 shows the ranking results obtained by the proposed SVNWPMSM operator when the cluster number S and the parameter k take different values.

   
From Table 4, we can get the following information: (1) When the cluster number 1 S = or 2 S = , the score value () i S  decreases monotonically as the value of the parameter k increases.
(2) When the parameter 1 k = or 2 k = , the ranking result of the medical waste treatment schemes is 4     and the optimal scheme is 4  . (3) When the parameter 3 k = , the ranking result is 4 2 1 3     and the optimal scheme is 4  .
Obviously, when the cluster number S or the parameter k is changed, the ranking result of the medical waste treatment schemes may change, but the optimal scheme keeps unchanged, i.e., 4  . As shown in Table 4, the change of the value of the parameter k may lead to the change of the ranking results of medical waste treatment schemes. The parameter k is a reflection of the expert's risk preference. The reference value of k is 0.5 min gg h   =  , where the symbol [] indicates rounding off the data in the symbol and g h is the number of criteria in the cluster g P . If k is greater than  , it means that the decision-maker is aggressive. If k is less than  , it means that the decision-maker is conservative. If k is equal to  , it means that the decision-maker is neutral.

Verification and comparative analysis
In this section, the validity, reliability, and other advantages of the proposed MCGDM model based on the SVNWPMSM operator are verified. For comparison, we apply the previous methods, such as Peng et al.'s [40] method and Wang et al.'s [33] method, to solve the medical waste scheme selection problem in Fuzhou. Table 5 shows the ranking results of the medical waste treatment schemes. We compare the ranking results obtained by the previous methods with the ranking results obtained by the proposed MCGDM model.   [40] uses the single-valued neutrosophic weighted averaging (SVNWA) operator and the single-valued neutrosophic weighted geometric (SVNWG) operator to aggregate the evaluation information. The ranking results obtained by Peng et al.'s method [40] and the proposed MCGDM model are shown in Table 5. The ranking results of SVNWA tend to the medical waste treatment scheme that owns the largest evaluation information value. In contrast, the ranking results of SVNWG tend to the medical waste treatment scheme with the smallest weight value of the evaluation information. Therefore, their ranking results are different. The ranking results from Peng et al.'s method using the SVNWA operator are the same as those from the proposed MCGDM model (when 1 k = or 2 k = ), i.e.,  are closely related. Wang et al.'s method [33] can only consider the correlation relation between any two criteria in these two clusters, while the proposed MCGDM model can capture the interrelationships among all the criteria in each cluster. Therefore, the proposed MCGDM model shows better performance in terms of ranking results.

Conclusions
During the outbreak of COVID-19, it is a big challenge to dispose of a large number of medical wastes. This paper proposes a novel MCGDM model to solve the medical waste treatment schemes selection problem to improve medical waste treatment efficiency. The proposed MCGDM model is composed of three main phases. In the first phase, the SVNNs are used to represent the evaluation information provided by the experts. In the second phase, we extend the PMSM operator to process