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Neutrosophic Fuzzy Hierarchical Clustering for Dengue Analysis in Sri Lanka.

1. Introduction

Emerging and re-emerging infectious diseases which are transmitted to the environment is a great threat to human living. The infections can take many forms and it can seriously affect human health. Dengue is one among the disease which causes severe outbreaks in many regions of the world. Its prevalence, incidence and geographic distribution are demanding a divisive applicable plan for control measures against dengue fever. In this case the complete structure of data and regions affected by dengue has to be known. Many situations exist that the ambiguity arises in finding a solution to the problem. Clustering and Classification are the most commonly encountered knowledge-discovery technique. Clustering is used in numerous applications such as disease detection, market analysis, medical diagnosis etc. The study concentrates on Sri Lankan dengue data analysis. Dengue fever occurs in the background of heavy rain and flooding and has affected almost26 districts in Sri Lanka. The country has reported 51659 cases in the year 2018 and approximately 41.2 % cases identified in western province alone[1]. In Pakistan, dengue has progressed towards becoming a risk for general wellbeing because of inaccessibility of vaccination, unclean water, highly populated territories and low quality of sanitation and sewage [2]. There have been a number of researches done on dengue fever diagnosis and numerous methods have been proposed using classification and clustering techniques for dengue analysis. G.P.Silveria proposed evolution technique of dengue risk analysis or prediction using the model Takagi-Sugeno. Takagi-Sugeno model included parameters such as human population density, density of potential mosquito breeding and rainfall. The fuzzy rules were developed using partial differential equations for Low, Medium and High dengue affected areas. The uncertainty factor considered in this study is the breeding period and the maturation of mosquito eggs and Silveria considered rainfall as a factor for the increase or decrease in the population of mosquitoes [3]. The selection of Neutrosophic approach has increased in group decision making in vague decision environment. Neutrosophic approach with Technique for Order Preference by Similarity to Ideal Solution (TOPSIS)[4] is considered for decision making process to deal with the vagueness and uncertainty by considering the data for the decision criteria. Neutrosophic environment provides a new technique in Multi Criteria Decision Making problem. Author Abdel-Basset M [5], has developed and integrated Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) into Decision-Making Trial and Evaluation Laboratory (DEMATEL) on a neutrosophic set that handles to overcome the ambiguity or the lack of information. He has applied on project selection criteria where the best alternatives are provided by the neutrosophic approach.

This paper mainly focuses on the finding of Dengue affected areas using the clustering technique found. The clusters are formed as low risk, medium risk and high risk areas. It helps the public sectors to concentrate particularly on that area for the remedial measures that are to be considering for the wellbeing of the society. Based on the neutrosophic approach, the clustering for the low risk, medium risk and high risk areas are identified and clustered.

2. Related Work

The ambiguity or uncertainty representation or handling of incomplete knowledge becomes a vital problem in the field of computer science. Researchers from various fields have dealt with vague, indeterminate, imprecise and sometimes insufficient information of uncertain data. The concept of uncertainty is usually handled by probabilistic approach. Soft computing techniques also deals with these problems such as called fuzzy sets [6] and intuitionistic fuzzy sets [7] and rough sets. Fuzzy logic is a collection of mathematical values for representing and understanding is based on membership degrees rather than the crisp membership of traditional binary logic. It leads to more human intelligent machines as fuzzy logic tries to model the human feeling of words, decision-making and common sense[8].

Unlike Boolean's two-valued logic, Fuzzy logic is multi-valued logic. Matrices play an important role in representation of the real world problems of science and engineering. Therefore, a few authors have proposed a matrix representation of fuzzy sets and intuitionistic fuzzy sets [9,10,11,12,13,14,15,16,17]. Fuzzy set and Intuitionistic Fuzzy Set deals with the membership and non-membership values. Membership value shows the truthiness of the algorithm which is classified or clustered. Non-membership values show the falsity of the data that it doesn't belong to that class.

For some reasons, the calculation of non-membership value is not always possible as in the case of membership values. So, there exists some indeterministic that part depicts the ambiguity in fuzzy logic. Subsequently, Smarandache [18, 19] introduced the term Neutrosophic Set (NS), which is formed as a generalization of classical set, fuzzy set, intuitionistic fuzzy set. The literature [20-24] shows the growth of decision-making algorithms over neutrosophical set theory.

Neutrosophic logic that shows the clear separation between the" relative truth" and" absolute truth" while the fuzzy logic does not show any separation. Smarandache Florentine proposed the concept of neutrosophic logic based on nonstandard analysis by Abraham Robinson in 1960s. Generally, we can say that the available disease information in inherently unclear and unpredictable. In real life issues, an element of indeterminacy exists and in this respect, neutrosophic logic can be used. Neutrosophic logic generalizes fuzzy, intuitive, boolean, para-consistent logic etc.

In many medical diagnosis and study of diseases, the indeterminacy or falsity in the input is not captured so far. It is seen from the literature that the concept of neutrosophic logic is not applied much on medical diagnosis. Neutrosophic clustering technique is neither employed nor applied to any medical applications. Some of the applications of neutrosophic logic are Social Network Analysis, Financial Market Information, Neutrosophic Security, Neutrosophic cognitive maps, Application to Robotics etc.

2.1 Machine Learning on Dengue

Many authors have concentrated on Machine Learning algorithms for classification and prediction of various diseases. In over 100 nations, dengue is endemic and causes an estimated 50 million infections per year. Nearly 3.97 billion individuals are at danger of infection from 128 nations [25]. Machine Learning algorithms such as Regression Models, Decision Tree, Artificial Neural Network, Rough Set Theory, Support Vector Machine etc. are successfully applied [26]. Daranee Thitiprayoonwongse et al proposed a hybrid technique combining a decision-making tree with a fuzzy logic approach to constructing a model for dengue infection. Author obtained a set of rules from decision tree and transformed to fuzzy rules. The results were better by combining fuzzy and decision tree approaches [27]. Torra [28], has proposed a fuzzy hierarchical clustering for representing the documents. Fuzzy hierarchical clusters are used in order to assure that the clusters are small enough by giving low information loss.

This research mainly focuses on clustering of Dengue disease in various parts of Sri Lanka. Increased risk to infectious diseases was recognized as one of five main emerging threats to public health resulting from the changes in the natural environment [29]. Diseases caused by mosquitoes are a specific danger to humans. The danger of transmission relies on climate variables that regulate mosquito habitat development [30-32]. This paper discusses the possibilities to exploit neutrosophic logic in epidemiology domain. In many cases, the representational parameters which include temperature and humidity as mentioned by [30-32] the climatic variables could also be a part in spread of disease. Most of the cases are rare that all the external parameters are considered, which leads to a chaos about conclusion to be drawn.

So the developed system should adapt to the conditions that are uncontrollable or unanticipated. In this case indeterminacy plays an important role. The concept of indeterminacy is handled or explained in a improvised way by neutrosophic logic. A better approach for all the above is Neutrosophic logic.

3. Proposed Work

Clustering can be seen as an practical problem in pattern recognition in unsupervised learning. Problems can be size of dataset, number of clusters to be formed, there is no ground truth solution unlike classification problems. The goal is to partition the data set into a certain number of natural and homogeneous sets where each set's elements is as similar as possible and different from the other sets. In real world applications, cluster separation is a fuzzy concept and therefore the idea of fuzzy subsets provides particular benefits over standard clustering [33]. This research proposes a hybridized technique for hierarchical clustering by amalgamation of fuzzy and neutrosophic approach. There by, the proposed algorithm gains the benefits of addressing imprecise, indeterministic, vague and uncertain data.

3.1. Hierarchical Clustering (HC)

In the process of hierarchical clustering, a distance matrix (D) is constructed where; dij is the distance between the cities. During clustering, [i.sup.th] and [j.sup.th] locations are merged into a cluster and distance matrix is updated. Eventually, the cities are merged based on the similarity measure and the dimension of D gets reduced on every step of merging. Hierarchical clustering is categorized based on the method of merging. It includes Single, Complete, Average, Centroid, Median and Ward. Merging clusters based on minimum distance between each element is called single linkage clustering. Clustering based on maximum distance between each element is complete linkage clustering, clustering the mean distance between each element is average linkage clustering, clustering is done by mean values of one group with the mean values on other group elements is centroid clustering. To overcome the disadvantage of centroid method the median of two groups are clustered is called median linkage clustering. Median linkage clustering is suitable for both similarity and distance measures. Wards method calculates the sum of the squares of the distance between the elements Pi and Pj, where Pi and Pj are the location of the elements in [i.sup.th] and [j.sup.th] positions. The distance matrix is formed by using the Euclidean equation. Single, complete and average link are defined by the way of merging the cities based on nearest, farthest and average distance respectively.

[mathematical expression not reproducible]

Where i,j are the location of cities and n, k are the number of cities.

Distance matrix here with dimension of 26x26 is formed. It is constructed on the basis of equation 3.1.Once the distance matrix is formed and based upon the method of hierarchical clustering, clusters are generated.

3.2. Fuzzy Hierarchical Clustering(FHC)

Given a set of objects, a fuzzy hierarchical framework has been implemented to construct clusters. The methodbegins to establish a fuzzy partition that uses the membership formula[34]. The membership matrix is calculatedusing the equation 3.2 which gives distance between each of the object, here it represents the cities.

[mathematical expression not reproducible] (3.2)

where n is the number of locations, m is the weighting parameter or fuzzifier, r is the number of iterations used for convergence. There is no theoretical optimumchoice of m in literature. The range is usually between 1.25 - 2 [35] and here we have choosen value 2. Theinitial membership matrix([mu]) is formed using equation (3.2). We have formed a fuzzy measure for objects.Here one object can belong to various clusters with the varying membership values ranging from 0 to 1. Valuesfalling between these endpoints (from low toextremely favorable clustering) were mapped as membershipdegrees. The non-membership value also called as falsity value, represented as [disjunction] [36]. It is calculated using thefollowing equation,

[mathematical expression not reproducible] (3.3)

where, [lambda] is the weighted parameter value ranging from 0 to 1. Here the value of [lambda] is taken as 0.8.

3.3. Neutrosophic Fuzzy Hierarchical Clustering(NFHC)

The notion of a neutrosophical set was initially proposed by Smarandache [37]. A neutrosophical set A isdefined by a universal set X with truth-membership function [T.sub.A], a falsity-membership function [F.sub.A] and anindeterminacy-membership function [I.sub.A]. Here,[T.sub.A](x),[F.sub.A](x) and [I.sub.A](x) are the real standard sets of values]0; 1+[, i.e., [T.sub.A](x): X [right arrow]]0; 1+[, [I.sub.A](x): X [right arrow]]0; 1+[, and [F.sub.A](x):X [right arrow]]0; 1+[. The indeterminancy-value whichis also denoted by [pi] is given by,

[mathematical expression not reproducible] (3.4)

From equation (3.2),(3.3) and (3.4), a neutrosophic triplet matrix is obtained. Table 2A shows a sample tripletmatrix. Before performing clustering, triplet matrix ([mu], [pi],v) [38] is converted into scalar value matrix using normalized hamming distance. The normalized hamming distance [39] between two locations P and Q is defined

[mathematical expression not reproducible] (3.5)

To perform the clustering part. the triplet matrix is converted into a scalar value using equation (3.5)[40]. The neutrosophic weights of a triplet matrix is converted into scalar weights. The resultant matrix is aneutrosophic matrix and HC is applied for clustering, there by we get a neutrosophic fuzzy clusters.

The dataset consists of dengue reported cases in 26 cities of Sri Lanka. Data is collected for six consecutiveyears from 2012 to 2018. First step is finding out the diatnce matrix (D) using the equation (3.1). The matrixformed here is 26x26 as distance matrix. Using equation (3.2), (3.3) and (3.4) triplet matrix of ([mu], [pi], [disjunction]) iscalculated. By using equation (3.5) the neutrosophic triplet matrix is converted to function matrix with scalarvalue upon which hierarchical clustering is formed. Example of the membership matrix obtained for different years. The representation for the year 2012 is given in table 1A.

We then perform the process of hierarchical clustering using algorithm 1, for the results diaplayed in table1A. HC is applied on each year and clusters are formed for each consecutive year from 2012 to 2018. HC hasdifferent methods such as single, complete, wighted, centroid, median and ward.
Algorithm 1: Hierarchical Clustcring([N.sub.d](P, Q). Method=single
linkage)

 1 begin
 2 mat[][] [left arrow] initialized to [N.sub.d](P,Q) values from
 equation 3.5
 3 disjoint set=[][]
 4 for each cit[y.sub.i] in mat[][] do
 5 for each cit[y.sub.j] in mai[][] do
 6 n-min(max(cit[y.sub.i], cit[y.sub.j]))
 7 merge((cit[y.sub.i],cit[y.sub.j])
 8 end
 9 repeat until single cluster
10 end

Algorithm 2: Hierarchical Clustering([N.sub.d](P, Q), Mcthod=complete
linkage)

1 begin
2 mat[][][left arrow] initialized to [N.sub.d](P, Q) values from
equation 3.5
3 disjoint set=[][]
4 for each cit[y.sub.i] in mat[][] do
5 for each cit[y.sub.j] in mat [][] do
6 n=max(max(cit[y.sub.i], cit[y.sub.j]))
7 merge(cit[y.sub.i], cit[y.sub.j])
8 end
9 repeat until single cluster
10 end


In the second step, the value of falsity or the non-membership is determined using the formula (3.3). The set of values in each column of the matrix represents ([mu], [pi], [disjunction]) for each location.

Finally, the neutrosophic matrix is constructed using equation (3.4). The obtained result is a triplet of the form (0.9425, 0.0752 and 0.0603). The triplet matrix expresses the truthness, falsity and indeterminacy value of each location paired with all other locations in the dataset. Similar matrix of 26x26 is obtained for all consecutive years starting from 2012 to 2018. Now find the similarity between each pair of objects in and neutrosophic triplet matrix.

The Euclidean distance matrix, membership matrix and triplet matrix is calculated using algorithm 2. The data is taken from the year 2012 to 2017 as training data. Once the algorithm is implemented, it has to be tested for its accuracy and how well the proposed algorithm works. The process is applied on data set for the year 2018 and the clusters are formed. The predicted clusters are compared with the actual data for all the 26cities. Several performance indices techniques are elaborated in section 5.

4. Dataset Descriptions

The data is collected from Epidemiology Unit Ministry of Sri Lanka. The dengue cases are collected for six consecutive years from 2012 to 2017. The data can be downloaded from thesite [41]. Data consist of 26 locations in Sri Lanka such as Colombo, Gampaha, Kalutara, Kandy, Matale, N Eliya, Galle, Hambantota, Matara, Jaffna, Kilinochchi, Mannar, Vavuniya, Mulativu, Batticaloa, Ampara, Trincomalee, Kurunegala, Puttalam, Apura, Polonnaruwa, Badulla, Moneragala, Ratnapura, Kegalle and Kalmunai.
Algorithm 3: Neutrosophic Fuzzy Score Calculation

 1 matrix[loc][loc] [left arrow] initialized to distance matrix for all
 cities in DB
 2 city=[list of all cities]
 3 for i in city do
 4 for j in city do
 5 [mathematical expression not reproducible]
 6 end
 7 end
 8 for i in city do
 9 for k in i do
10 for j in city do
11 [mathematical expression not reproducible]
// n number of locations
12 [[mu].sub.ik] = (1/x)
13 end
14 end
15 end
16 for i in city do
17 for j in city do
l8 calculate [disjunction](cit[y.sub.i,j]) using equation (3.3)
// [disjunction] is non membership value
19 calculate [pi](cit[y.sub.i,j]) using equation (3.4)
// [pi] is indeterminacy value
20 end
21 end
22 for i in city do
23 for j in city do
24 [mathematical expression not reproducible]
// hamming distance formula
// [N.sub.d](P,Q) resultant scalar matrix
25 end
26 end
27 Perform Hierarchical Clustering on ([N.sub.d](P, Q), method)
// method = (single, complete, centroid, median, ward)
// Perform algorithm 2 for HC


5. Experimental Results

5.1. Inconsistency Coefficient

The relative consistency of each link in a formed hierarchical cluster is quantified as inconsistency coefficient. When the links are more consistent, the neighboring links have approximately same length. Inconsistency coefficient of each link compares its height with the average height of other links from the same level of hierarchy. When the links have larger the coefficient there exists greater the difference between the objects connected by the link. When the difference between the link values is very small, it is difficult to make conclusions. Hence higher the inconsistency gives better clustering. Inconsistency value for different links is tabulated in Table 2.

Considering the results from table 2, the maximum difference between the links in neutrosophic fuzzy hierarchical clustering is identified. When the tree is cut at maximum linkage, the resulting clusters are found to be three clusters. The number of clusters is identified using inconsistency coefficient. With the inconsistency value and the number of cluster, data is divided into three parts such as low risk, medium risk and highly affected dengue areas in Sri Lanka. Neutrosophic Fuzzy Hierarchical Clustering has shown highest inconsistent values such as 0.9168, 0.8714, 0.7721, 0.7428 and 0.7216 for single linkage clustering, complete linkage clustering, centroid, median and ward method respectively. The results are better in a way as NFHC has given the maximum distance between the links compared with other two techniques.

Figure 1 depicts NFHC clustering applied on dataset for the year 2018. The value in the x-axis represents the cities and y-axis represents the tree cut. Figure 1 is visualized in shape map of Sri Lanka. Based on the inconsistency-coefficient the tree is cut into three clusters. Clustering for the year 2012-2018 is given in figure 3. It has shown effective clustering based on the performance indices explained in section 5.2.

5.2. Performance Indices

Performance indices are used to assess clustering algorithms performance. The literature contains several performance indices. The Silhouette Coefficient [42], Davis-Bouldin (DB) index [43] and Dunn (D) index [44] are some of the most popular indicators of effectiveness assessment.

5.2.1. Silhouette Coefficient

Silhouettee index is an index of cluster validity used to evaluate the performance of any cluster. An element'ssilhouette index describes its proximity to its own cluster with its proximity to other clusters. A clusters silhouette width s(x) is described as,

[mathematical expression not reproducible] (5.1)

where, a(x) and b(x) are the similarities of the clusters. The average silhouette width of all clusters is the silhouette index of the entire clustering. Silhouette index is used to indicate the compactness and segregation of clusters. The silhouette index value ranges from -1 to 1 and a better clustering outcome is indicated by its greater values. The silhouette coefficient of neutrosophic fuzzy hierarchical clustering is high with the value of 0.7163, stating that the performance of Neutrosophic fuzzy hierarchical clustering is better than hierarchical clustering and fuzzy hierarchical clustering with the score of 0.6782 and 0.5137 respectively.

5.2.2. Davis-Bouldin (DB) index

The DB index is described as the cluster-to-cluster distance proportion of the amount of data. It is formulated in the following way,

[mathematical expression not reproducible] (5.2)

The DB index seeks at minimizing cluster separation and maximizing cluster distance. The lower the DB index shows effective clustering. Our proposed algorithm Neutrosophic fuzzy hierarchical clustering has shown the lowest DB-index value of 2.5725 for the method of Single linkage clustering. Proposed algorithm has shown better results when compared to traditional algorithms. Experiment also reveals that fuzzy hierarchical clustering also performs better than traditional hierarchical clustering. However NFHC outperforms all.

5.2.3. Dunn (D) index

The D index is used to define clusters that are compact and separate. The calculation is as follows,

[mathematical expression not reproducible] (5.3)

Dunn index's objective is to maximize the distance between the clusters and minimize the distance within the cluster. An elevated D index therefore means better clustering. In our implementation, highest Dunn index is achieved for NFHC algorithm with the number 1.159 of highest among all other methods. It has shown better clustering compared to other algorithms.

From table 3, we can infer that, the cluster validation of neutrosophic fuzzy hierarchical clustering has shown better results compared with hierarchical clustering and fuzzy hierarchical clustering. The metrics such as silhouette coefficient, DB index and Dunn index states the excellence of thee proposed model. The best values of silhouette cluster analysis is found in NFHC with 0.7163 for single link, 0.6911 for complete link, 0.6729 for centroid method, 0.6905 for median method and 0.7077 in ward method. Silhouette coefficient has shown highest results in NFHC for all 5 methods. DB index has also produced effective results in cluster analysis of NFHC. The lowest value of DB index is centroid method of NFHC with the value 2.6674 whereHC and FHC values for centroid method are 4.2162 and 3.1249 respectively. Other methods such as single,complete, median and ward has also given lowest values on NFHC comparing with FHC and traditional HC.Though DB index of complete method is good in FHC. FHC is also comparatively good when compared with traditional HC, as it has produced effective clustering that HC. Highest recorded Dunn index value is 1.159, for the method of centroid in NFHC. Final inference from NFHC is, it is giving better results on all the methods of clustering such as single, complete, centroid, median and ward when compared with same method on fuzzy hierarchical clustering and hierarchical clustering.

It is evident from the table 3, that the proposed NFHC shows its superiority in its performance compared to other methods. Though the fuzzy hierarchical clustering has considered membership value for clustering and produced better clusters compared with HC clusters, NFHC outperforms the fuzzy results. Thus, proposed NFHC is better in a way as it handles or capable of handling any data even with indeterminacy or inconsistency.

The visualization part in figure 3 clearly says that, the city of Colombo was in high risk area over the past seven years. The trend in Colombo city reveals that it is always in high risk area of dengue. In the year 2018, Colombo is the only highly affected area compared to all other cities in Sri Lanka. If the trend continues, the life of people at Colombo is in great threat. Looking into the cities in the middle of Sri Lanka such as Polonnaruwa, Matale, Polonnaruwa, Trincomalee and Kandy they have crossed the threshold of being in low risk area to medium risk area. This depicts that the states are gradually increasing in its dengue admissions. It is an important issue to be noted by the government, as in future these cities are in high risk of getting into a danger zone of dengue. Considering the southern cities of Sri Lanka, in the year 2012 the number of dengue cases was low. Over the five consecutive years it has shown the mixed results of being in medium and highly affected area. In the area of south, the control measures have to be taken strongly for cutting down the growth of dengue fever. The major pattern that is observed from the year 2012 to 2018 is that, none of the cities had reduced from reporting the dengue cases. It has always increased from one level to next level showing the spread of dengue in a drastic manner.

6. Conclusions

The study mainly identifies the areas that are affected dengue fever. Though many studies have touched the concept of clustering, the area of indeterminacy in clustering for the field of epidemiology is still under research. We used neutrosophic fuzzy hierarchical clustering and fuzzy hierarchical clustering in this article to cluster dengue fever in Sri Lanka. The purpose of neutrosophic fuzzy is, it can handle the indeterminate and inconsistent information where the fuzzy fails to handles that information. Cluster validation metrics has given better results in neutrosophic fuzzy hierarchical clustering than the other two algorithms of fuzzy hierarchical clustering and hierarchical clustering. Some of the findings from this study is that, Colombo is identified as highest dengue affected area, many of the cities are in the peak of threshold that it can move to the danger zone at any point of time. Re-emerging areas such as Galle, Matara, Hambantota, Ratnapura and Badulla are to be concentrated more so that the pattern of occurrence can be controlled in future. This method can be used in other fields so that the break out of any disease can be avoided earlier. In future, the algorithm can be extended for monitoring other diseases that are affected by environmental and climatic variables. This model can also be extended as multi-criteria model for identifying the outbreak of hotspots and early warning systems.

Acknowledgments: The authors are highly grateful to the Referees for their constructive suggestions.

Conflicts of Interest: The authors declare no conflict of interest.

Appendix A

The following matrices contain the supplementary data for the experimental work carried out. The data is given for the year 2012.
Table A1 (a) represents Membership matrix ([mu]) for the cities
[C.sub.1] to [C.sub.14] from Table 1 in section 4.

            [C.sub.1]  [C.sub.2]  [C.sub.3]  [C.sub.4]  [C.sub.5]

[C.sub.1]   0          0.5261     0.5423     0.6631     0.6217
[C.sub.2]   0.5261     0          0.4571     0.5863     0.2413
[C.sub.3]   0.5423     0.4571     0          0.7512     0.6942
[C.sub.4]   0.6631     0.5863     0.7512     0          0.8412
[C.sub.5]   0.6217     0.2413     0.6942     0.8412     0
[C.sub.6]   0.8431     0.7512     0.4623     0.5679     0.7135
[C.sub.7]   0.7456     0.6674     0.7561     0.4987     0.5671
[C.sub.8]   0.4675     0.5931     0.5001     0.6782     0.6746
[C.sub.9]   0.7634     0.7213     0.6417     0.6034     0.5237
[C.sub.10]  0.7124     0.8012     0.7812     0.5846     0.5713
[C.sub.11]  0.6419     0.7632     0.4123     0.3699     0.5712
[C.sub.12]  0.6787     0.2745     0.8436     0.7415     0.6716
[C.sub.13]  0.7123     0.5481     0.9845     0.5769     0.9412
[C.sub.14]  0.6912     0.8456     0.1664     0.8462     0.6565

            [C.sub.6]  [C.sub.7]  [C.sub.8]  [C.sub.9]  [C.sub.10]

[C.sub.1]   0.8431     0.7456     0.4675     0.7634     0.7124
[C.sub.2]   0.7512     0.6674     0.5931     0.7213     0.8012
[C.sub.3]   0.4623     0.7561     0.5001     0.6417     0.7812
[C.sub.4]   0.5679     0.4987     0.6782     0.6034     0.5846
[C.sub.5]   0.7135     0.5671     0.6746     0.5237     0.5713
[C.sub.6]   0          0.5172     0.4872     0.5716     0.4872
[C.sub.7]   0.5172     0          0.6813     0.4213     0.5716
[C.sub.8]   0.4872     0.6813     0          0.6148     0.5127
[C.sub.9]   0.5716     0.4213     0.6148     0          0.4219
[C.sub.10]  0.4872     0.5716     0.5127     0.4219     0
[C.sub.11]  0.6742     0.7416     0.4137     0.5166     0.5712
[C.sub.12]  0.4369     0.5716     0.8413     0.7168     0.6741
[C.sub.13]  0.2145     0.6715     0.8422     0.6479     0.9145
[C.sub.14]  0.7956     0.6135     0.8436     0.4696     0.6713

            [C.sub.11]  [C.sub.12]  [C.sub.13]  [C.sub.14]

[C.sub.1]   0.6419      0.6787      0.7123      0.6912
[C.sub.2]   0.7632      0.2745      0.5481      0.8456
[C.sub.3]   0.4123      0.8436      0.9845      0.1664
[C.sub.4]   0.3699      0.7415      0.5769      0.8462 0.6565
[C.sub.5]   0.5712      0.6716      0.9412
[C.sub.6]   0.6742      0.4369      0.2145      0.7956 0.6135
[C.sub.7]   0.7416      0.5716      0.6715
[C.sub.8]   0.4137      0.8413      0.8422      0.8436
[C.sub.9]   0.5166      0.7168      0.6479      0.4696
[C.sub.10]  0.5712      0.6741      0.9145      0.6713
[C.sub.11]  0           0.4193      0.4785      0.6971 0.8435
[C.sub.12]  0.4193      0           0.5136
[C.sub.13]  0.4785      0.5136      0           0.3469
[C.sub.14]  0.6971      0.8435      0.3469

Table A1 (b) represents Membership matrix ([mu]) for the cities C15 to
C26 from Table 1 in section 4.

            [C.sub.1]  [C.sub.2]  [C.sub.3]  [C.sub.4]  [C.sub.5]

[C.sub.15]  0.5197     0.5966     0.5523     0.8425     0.6656
[C.sub.16]  0.4128     0.4956     0.6595     0.5656     0.9463
[C.sub.17]  0.7946     0.6596     0.2648     0.8746     0.6941
[C.sub.18]  0.6843     0.3266     0.1654     0.6957     0.8946
[C.sub.19]  0.7069     0.8951     0.3261     0.2154     0.1595
[C.sub.20]  0.8431     0.2546     0.3665     0.5955     0.8685
[C.sub.21]  0.7629     0.1655     0.1796     0.6456     0.8562
[C.sub.22]  0.5527     0.4652     0.7656     0.5966     0.7163
[C.sub.23]  0.6237     0.8455     0.5965     0.7465     0.9461
[C.sub.24]  0.5179     0.8665     0.5165     0.6266     0.5169
[C.sub.25]  0.5873     0.4865     0.8698     0.7495     0.9561
[C.sub.26]  0.5766     0.8455     0.5356     0.5486     0.6715

            [C.sub.6]  [C.sub.7]  [C.sub.8]  [C.sub.9]  [C.sub.10]

[C.sub.15]  0.8626     0.5946     0.6816     0.3266     0.3247
[C.sub.16]  0.2176     0.8956     0.6867     0.9562     0.7416
[C.sub.17]  0.1623     0.5952     0.7856     0.7953     0.9451
[C.sub.18]  0.7162     0.3266     0.2185     0.3256     0.1966
[C.sub.19]  0.5451     0.5482     0.1782     0.6816     0.4845
[C.sub.20]  0.1656     0.6595     0.8466     0.4863     0.7566
[C.sub.21]  0.7161     0.6845     0.7136     0.6416     0.4986
[C.sub.22]  0.6145     0.5164     0.5651     0.4516     0.7166
[C.sub.23]  0.6858     0.7465     0.8592     0.4566     0.2156
[C.sub.24]  0.5996     0.3566     0.7415     0.4566     0.6856
[C.sub.25]  0.6515     0.5795     0.5167     0.7866     0.3595
[C.sub.26]  0.6123     0.7155     0.4189     0.6589     0.3658

            [C.sub.11]  [C.sub.12]  [C.sub.13]  [C.sub.14]

[C.sub.15]  0.7486      0.9462      0.5653      0.6556
[C.sub.16]  0.9512      0.6821      0.5185      0.5251
[C.sub.17]  0.5623      0.1265      0.5659      0.7566
[C.sub.18]  0.7152      0.3956      0.6748      0.7465
[C.sub.19]  0.7185      0.3497      0.6494      0.4896
[C.sub.20]  0.8465      0.6645      0.5867      0.7451
[C.sub.21]  0.7856      0.7565      0.3516      0.7413
[C.sub.22]  0.6146      0.3556      0.3888      0.7463
[C.sub.23]  0.3562      0.4532      0.5666      0.4857
[C.sub.24]  0.7164      0.5645      0.5959      0.5165
[C.sub.25]  0.2186      0.8465      0.6585      0.4812
[C.sub.26]  0.7529      0.6485      0.5568      0.6745

Table A1 (c) represents Membership matrix ([mu]) for the cities C1 to
C14 from Table 1 in section 4.

            [C.sub.15]  [C.sub.16]  [C.sub.17]  [C.sub.18]  [C.sub.19]

[C.sub.1]   0.5197      0.4128      0.7946      0.6843      0.7069
[C.sub.2]   0.5966      0.4956      0.6596      0.3266      0.8951
[C.sub.3]   0.5523      0.6595      0.2648      0.1654      0.3261
[C.sub.4]   0.8425      0.5656      0.8746      0.6957      0.2154
[C.sub.5]   0.6656      0.9463      0.6941      0.8946      0.1595
[C.sub.6]   0.8626      0.2176      0.1623      0.7162      0.5451
[C.sub.7]   0.5946      0.8956      0.5952      0.3266      0.5482
[C.sub.8]   0.6816      0.6867      0.7856      0.2185      0.1782
[C.sub.9]   0.3266      0.9562      0.7953      0.3256      0.6816
[C.sub.10]  0.3247      0.7416      0.9451      0.1966      0.4845
[C.sub.11]  0.7486      0.9512      0.5623      0.7152      0.7185
[C.sub.12]  0.9462      0.6821      0.1265      0.3956      0.3497
[C.sub.13]  0.5653      0.5185      0.5659      0.6748      0.6494
[C.sub.15]  0.6556      0.5251      0.7566      0.7465      0.4896

            [C.sub.20]  [C.sub.21]  [C.sub.22]  [C.sub.23]  [C.sub.24]

[C.sub.1]   0.8431      0.7629      0.5527      0.6237      0.5179
[C.sub.2]   0.2546      0.1655      0.4652      0.8455      0.8665
[C.sub.3]   0.3665      0.1796      0.7656      0.5965      0.5165
[C.sub.4]   0.5955      0.6456      0.5966      0.7465      0.6266
[C.sub.5]   0.8685      0.8562      0.7163      0.9461      0.5169
[C.sub.6]   0.1656      0.7161      0.6145      0.6858      0.5996
[C.sub.7]   0.6595      0.6845      0.5164      0.7465      0.3566
[C.sub.8]   0.8466      0.7136      0.5651      0.8592      0.7415
[C.sub.9]   0.4863      0.6416      0.4561      0.4566      0.4566
[C.sub.10]  0.7566      0.4986      0.7166      0.2156      0.6856
[C.sub.11]  0.8465      0.7856      0.6146      0.3562      0.7164
[C.sub.12]  0.6645      0.7565      0.3556      0.4532      0.5645
[C.sub.13]  0.5867      0.3516      0.3888      0.5666      0.5959
[C.sub.15]  0.7451      0.7413      0.7463      0.4857      0.5165

            [C.sub.25]  [C.sub.26]

[C.sub.1]   0.5873      0.5766
[C.sub.2]   0.4865      0.8455
[C.sub.3]   0.8698      0.5356
[C.sub.4]   0.7495      0.5486
[C.sub.5]   0.9561      0.6715
[C.sub.6]   0.6515      0.6123
[C.sub.7]   0.5795      0.7155
[C.sub.8]   0.5167      0.4189
[C.sub.9]   0.7866      0.6589
[C.sub.10]  0.3595      0.3658
[C.sub.11]  0.2186      0.7529
[C.sub.12]  0.8465      0.6485
[C.sub.13]  0.6585      0.5568
[C.sub.15]  0.4812      0.6745

Table A1 (d) represents Membership matrix ([mu]) for the cities C15 to
C26 from Table 1 in section 4.

            [C.sub.15]  [C.sub.16]  [C.sub.17]  [C.sub.18]  [C.sub.19]

[C.sub.15]  0           0.4657      0.6289      0.6465      0.6594
[C.sub.16]  0.4657      0           0.8956      0.7441      0.8949
[C.sub.17]  0.6289      0.8956      0           0.2156      0.4163
[C.sub.18]  0.6465      0.7441      0.2156      0           0.2155
[C.sub.19]  0.6594      0.8949      0.4163      0.2155      0
[C.sub.20]  0.8556      0.3598      0.6147      0.5716      0.6816
[C.sub.21]  0.5162      0.5716      0.1897      0.7166      0.2965
[C.sub.22]  0.3589      0.5635      0.8656      0.8462      0.4562
[C.sub.23]  0.9415      0.4945      0.3859      0.6889      0.3462
[C.sub.24]  0.4565      0.9452      0.1763      0.6455      0.4655
[C.sub.25]  0.8465      0.9515      0.4569      0.5743      0.7152
[C.sub.26]  0.7456      0.9512      0.3518      0.4686      0.8597

            [C.sub.20]  [C.sub.21]  [C.sub.22]  [C.sub.23]  [C.sub.24]

[C.sub.15]  0.8556      0.5162      0.3589      0.9415      0.4565
[C.sub.16]  0.3598      0.5716      0.5635      0.4945      0.9452
[C.sub.17]  0.6147      0.1897      0.8656      0.3859      0.1763
[C.sub.18]  0.5716      0.7166      0.8462      0.6889      0.6455
[C.sub.19]  0.6816      0.2965      0.4562      0.3462      0.4655
[C.sub.20]  0           0.4859      0.4856      0.5678      0.5615
[C.sub.21]  0.4859      0           0.7855      0.4887      0.7416
[C.sub.22]  0.4856      0.7855      0           0.8946      0.4852
[C.sub.23]  0.5678      0.4887      0.8946      0           0.8561
[C.sub.24]  0.5615      0.7416      0.4852      0.8561      0
[C.sub.25]  0.4969      0.8917      0.1985      0.5785      0.4668
[C.sub.26]  0.7456      0.2654      0.6464      0.4156      0.5486

            [C.sub.25]  [C.sub.26]

[C.sub.15]  0.8465      0.7456
[C.sub.16]  0.9515      0.9512
[C.sub.17]  0.4569      0.3518
[C.sub.18]  0.5743      0.4686
[C.sub.19]  0.7152      0.8597
[C.sub.20]  0.4969      0.7456
[C.sub.21]  0.8917      0.2654
[C.sub.22]  0.1985      0.6464
[C.sub.23]  0.5785      0.4156
[C.sub.24]  0.4668      0.5486
[C.sub.25]  0           0.5972
[C.sub.26]  0.5972      0

Table A2 (a) represents Neutrosophic matrix ([mu], n, V) for the
cities C1 to C5 from Table 1 in section 4.

           [C.sub.1]             [C.sub.2]

[C.sub.1]  0,0,0                 0.5261,0.1403,0.3335
[C.sub.2]  0.5261,0.1403,0.3335  0,0,0
[C.sub.3]  0.5423,0.1384,0.3192  0.4571,0.1316,0.4112
[C.sub.4]  0.6631,0.1068,0.2300  0.5863,0.1203,0.2933
[C.sub.5]  0.6217,0.1256,0.2526  0.2413,0.1091,0.6491

           [C.sub.3]             [C.sub.4]

[C.sub.1]  0.5423,0.1384,0.3192  0.6631,0.1068,0.2300
[C.sub.2]  0.4571,0.1316,0.4112  0.5863,0.1203,0.2933
[C.sub.3]  0,0,0                 0.7512,0.0857,0.1630
[C.sub.4]  0.7512,0.0857,0.1630  0,0,0
[C.sub.5]  0.6942,0.1000,0.2057  0.8412,0.0588,0.0999

           [C.sub.5]

[C.sub.1]  0.6217,0.1256,0.2526
[C.sub.2]  0.2413,0.1096,0.6491
[C.sub.3]  0.6942,0.1000,0.2057
[C.sub.4]  0.8412,0.0588,0.0999
[C.sub.5]  0,0,0

Table A2 (b) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C6 to C10 from Table 1 in section 4.

            [C.sub.1]             [C.sub.2]

[C.sub.6]   0.8431,0.0631,0.0937  0.7512,0.0857,0.1630
[C.sub.7]   0.7456,0.0950,0.1593  0.6674,0.1059,0.2266
[C.sub.8]   0.4675,0.1449,0.3875  0.5931,0.1193,0.2875
[C.sub.9]   0.7634,0.0897,0.1468  0.7213,0.0935,0.1851
[C.sub.10]  0.7124,0.1044,0.1831  0.8012,0.0714,0.1273

            [C.sub.3]             [C.sub.4]

[C.sub.6]   0.4623,0.1314,0.4062  0.5679,0.1229,0.3091
[C.sub.7]   0.7561,0.0844,0.1594  0.4987,0.1297,0.3715
[C.sub.8]   0.5001,0.1296,0.3702  0.6782,0.1035,0.2182
[C.sub.9]   0.6417,0.1110,0.2472  0.6034,0.1177,0.2788
[C.sub.10]  0.7812,0.0773,0.1414  0.5846,0.1206,0.2947

            [C.sub.5]

[C.sub.6]   0,0,0
[C.sub.7]   0.7135,0.0954,0.1910
[C.sub.8]   0.5671,0.1230,0.3098
[C.sub.9]   0.6746,0.10439,0.2210
[C.sub.10]  0.5237,0.1277,0.3485

Table A2 (c) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C11 to C20 from Table 1 in section 4.

            [C.sub.1]             [C.sub.2]

[C.sub.11]  0.6419,0.1214,0.2366  0.7632,0.0824,0.1543
[C.sub.12]  0.6787,0.1130,0.2082  0.2745,0.1169,0.6085
[C.sub.13]  0.7123,0.1047,0.1832  0.5481,0.1253,0.3265
[C.sub.14]  0.6912,0.1099,0.1988  0.8456,0.0574,0.0969
[C.sub.15]  0.5197,0.1410,0.3392  0.5966,0.1188,0.2845
[C.sub.16]  0.4128,0.1457,0.4414  0.4956,0.1299,0.3744
[C.sub.17]  0.7946,0.0798,0.1255  0.6596,0.1075,0.2328
[C.sub.18]  0.6843,0.1116,0.2040  0.3266,0.1253,0.5480
[C.sub.19]  0.7069,0.1058,0.1872  0.8951,0.0404,0.0644
[C.sub.20]  0.8431,0.0631,0.0937  0.2546,0.1127,0.6326

            [C.sub.3]             [C.sub.4]

[C.sub.11]  0.4123,0.1316,0.4560  0.3699,0.1295,0.5005
[C.sub.12]  0.8436,0.0580,0.0983  0.7415,0.0883,0.1701
[C.sub.13]  0.9845,0.0063,0.0091  0.5769,0.1217,0.3013
[C.sub.14]  0.1664,0.0869,0.7466  0.8462,0.0572,0.0965
[C.sub.15]  0.5523,0.1248,0.3228  0.8425,0.0584,0.0990
[C.sub.16]  0.6595,0.1075,0.2329  0.5656,0.1232,0.3111
[C.sub.17]  0.2648,0.1149,0.6202  0.8746,0.0476,0.0777
[C.sub.18]  0.1654,0.0866,0.7479  0.6957,0.0996,0.2046
[C.sub.19]  0.3261,0.1252,0.5486  0.2154,0.1028,0.6817
[C.sub.20]  0.3665,0.1293,0.5041  0.5955,0.1190,0.2854

            [C.sub.5]

[C.sub.11]  0.5712,0.1224,0.3063
[C.sub.12]  0.6716,0.1050,0.2233
[C.sub.13]  0.9412,0.0233,0.0354
[C.sub.14]  0.6565,0.1081,0.2353
[C.sub.15]  0.6656,0.1062,0.2281
[C.sub.16]  0.9463,0.0213,0.0323
[C.sub.17]  0.6941,0.1000,0.2058
[C.sub.18]  0.8946,0.0405,0.0648
[C.sub.19]  0.1595,0.0844,0.7560
[C.sub.20]  0.8685,0.0497,0.0817]

Table A2 (d) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C21 to C26 from Table 1 in section 4.

            [C.sub.1]             [C.sub.2]

[C.sub.21]  0.7629,0.0898,0.1472  0.1655,0.0866,0.7478
[C.sub.22]  0.5527,0.1371,0.3101  0.4652,0.1313,0.4034
[C.sub.23]  0.6237,0.1252,0.2510  0.8455,0.0574,0.0970
[C.sub.24]  0.5179,0.1412,0.3408  0.8665,0.0504,0.0830
[C.sub.25]  0.5873,0.1319,0.2807  0.4865,0.1304,0.3830
[C.sub.26]  0.5766,0.1336,0.2897  0.8455,0.0574,0.0970

            [C.sub.3]             [C.sub.4]

[C.sub.21]  0.1796,0.0916,0.7287  0.6456,0.1103,0.2440
[C.sub.22]  0.7656,0.0817,0.1526  0.5966,0.1188,0.2845
[C.sub.23]  0.5965,0.1188,0.2846  0.7465,0.0870,0.1664
[C.sub.24]  0.5165,0.1283,0.3551  0.6266,0.1138,0.2595
[C.sub.25]  0.8698,0.0492,0.0809  0.7495,0.0861,0.1643
[C.sub.26]  0.5356,0.1266,0.3377  0.5486,0.1252,0.3261

            [C.sub.5]

[C.sub.21]  0.8562,0.0538,0.0899
[C.sub.22]  0.7163,0.0947,0.1889
[C.sub.23]  0.9461,0.0214,0.0324
[C.sub.24]  0.5169,0.1283,0.3547
[C.sub.25]  0.9561,0.0176,0.0262
[C.sub.26]  0.6715,0.1050,0.2234]

Table A2 (e) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C1 to C5 from Table 1 in section 4.

           [C.sub.6]             [C.sub.7]

[C.sub.1]  0.8431,0.0582,0.0986  0.7456,0.0872,0.1671
[C.sub.2]  0.7512,0.0857,0.1630  0.6674,0.1059,0.2266
[C.sub.3]  0.4623,0.1314,0.4062  0.7561,0.0844,0.1594
[C.sub.4]  0.5679,0.1229,0.3091  0.4987,0.1297,0.3715
[C.sub.5]  0.7135,0.0954,0.1910  0.5671,0.1230,0.3098

           [C.sub.8]             [C.sub.9]

[C.sub.1]  0.4675,0.1312,0.4012  0.7634,0.0824,0.1541
[C.sub.2]  0.5931,0.1193,0.2875  0.7213,0.0935,0.1851
[C.sub.3]  0.5001,0.1296,0.3702  0.6417,0.1110,0.2472
[C.sub.4]  0.6782,0.1035,0.2182  0.6034,0.1177,0.2788
[C.sub.5]  0.6746,0.1043,0.2210  0.5237,0.1277,0.3485

           [C.sub.10]

[C.sub.1]  0.7124,0.0956,0.1919
[C.sub.2]  0.8012,0.0714,0.1273
[C.sub.3]  0.7812,0.0773,0.1414
[C.sub.4]  0.5846,0.1206,0.2947
[C.sub.5]  0.5713,0.1224,0.3062

Table A2 (f) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C6 to C10 from Table 1 in section 4.

            [C.sub.6]             [C.sub.7]

[C.sub.6]   0,0,0                 0.5172,0.1283,0.3544
[C.sub.7]   0.5172,0.1283,0.3544  0,0,0
[C.sub.8]   0.4872,0.1304,0.3823  0.6813,0.1029,0.2157
[C.sub.9]   0.5716,0.1224,0.3059  0.4213,0.1320,0.4469
[C.sub.10]  0.4872,0.1304,0.3823  0.5716,0.1224,0.3059

            [C.sub.8]             [C.sub.9]

[C.sub.6]   0.4872,0.1304,0.3823  0.5716,0.1224,0.3059
[C.sub.7]   0.6813,0.1029,0.2157  0.4213,0.1320,0.4469
[C.sub.8]   0,0,0                 0.6148,0.1158,0.2693
[C.sub.9]   0.6148,0.1158,0.2693  0,0,0
[C.sub.10]  0.5127,0.1286,0.3586  0.4219,0.1327,0.4462

            [C.sub.10]

[C.sub.6]   0.4872,0.1304,0.3823
[C.sub.7]   0.5716,0.1224,0.3059
[C.sub.8]   0.5127,0.1286,0.3586
[C.sub.9]   0.4219,0.1327,0.4462
[C.sub.10]  0,0,0

Table A2 (g) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C11to C20 from Table 1 in section 4.

            [C.sub.6]             [C.sub.7]

[C.sub.11]  0.6742,0.1044,0.2213  0.7416,0.0883,0.1700
[C.sub.12]  0.4369,0.1318,0.4312  0.5716,0.1224,0.3059
[C.sub.13]  0.2145,0.1025,0.6829  0.6715,0.1050,0.2234
[C.sub.14]  0.7956,0.0731,0.1312  0.6135,0.1838,0.2703
[C.sub.15]  0.8626,0.0517,0.0856  0.5946,0.1191,0.2862
[C.sub.16]  0.2176,0.1034,0.6789  0.8956,0.0402,0.0641
[C.sub.17]  0.1623,0.0854,0.7522  0.5952,0.1190,0.2857
[C.sub.18]  0.7162,0.0947,0.1890  0.3266,0.1253,0.5480
[C.sub.19]  0.5451,0.1256,0.3292  0.5482,0.1252,0.3265
[C.sub.20]  0.1656,0.0866,0.7477  0.6595,0.1075,0.2329

            [C.sub.8]             [C.sub.9]

[C.sub.11]  0.4137,0.1316,0.4546  0.5166,0.1283,0.3550
[C.sub.12]  0.8413,0.0588,0.0998  0.7168,0.0946,0.1885
[C.sub.13]  0.8422,0.0585,0.0992  0.6479,0.1098,0.2422
[C.sub.14]  0.8436,0.0580,0.0983  0.4696,0.1312,0.3991
[C.sub.15]  0.6816,0.1028,0.2155  0.3266,0.1253,0.5480
[C.sub.16]  0.6867,0.1017,0.2115  0.9562,0.0175,0.0262
[C.sub.17]  0.7856,0.0760,0.1383  0.7953,0.0732,0.1314
[C.sub.18]  0.2185,0.1036,0.6778  0.3256,0.1251,0.5492
[C.sub.19]  0.1782,0.0911,0.7306  0.6816,0.1028,0.2155
[C.sub.20]  0.8466,0.0570,0.0963  0.4863,0.1304,0.3832

            [C.sub.10]

[C.sub.11]  0.3368,0.1265,0.5366
[C.sub.12]  0.6741,0.1044,0.2215
[C.sub.13]  0.9145,0.0333,0.0521
[C.sub.14]  0.6713,0.1050,0.2236
[C.sub.15]  0.3247,0.1250,0.5502
[C.sub.16]  0.7416,0.0883,0.1700
[C.sub.17]  0.9451,0.0218,0.0330
[C.sub.18]  0.1966,0.0971,0.7062
[C.sub.19]  0.4845,0.1305,0.3849
[C.sub.20]  0.7566,0.0842,0.1591

Table A2 (h) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C21 to C26 from Table 1 in section 4.

            [C.sub.6]             [C.sub.7]

[C.sub.21]  0.7161,0.0947,0.1891  0.6845,0.1022,0.2132
[C.sub.22]  0.6145,0.1159,0.2695  0.5164,0.1283,0.3552
[C.sub.23]  0.6858,0.1019,0.2122  0.7465,0.0870,0.1664
[C.sub.24]  0.5996,0.1183,0.2820  0.3566,0.1285,0.5148
[C.sub.25]  0.6515,0.1091,0.2393  0.5795,0.1213,0.2991
[C.sub.26]  0.6123,0.1163,0.2713  0.7155,0.0949,0.1895

            [C.sub.8]             [C.sub.9]

[C.sub.21]  0.7136,0.0954,0.1909  0.6416,0.1110,0.2473
[C.sub.22]  0.5651,0.1232,0.3116  0.4516,0.1317,0.4166
[C.sub.23]  0.8592,0.0528,0.0879  0.4566,0.1316,0.4117
[C.sub.24]  0.7415,0.0883,0.1701  0.4566,0.1316,0.4117
[C.sub.25]  0.5167,0.1283,0.3549  0.7866,0.0757,0.1376
[C.sub.26]  0.4189,0.1317,0.4493  0.6589,0.1076,0.2334

            [C.sub.10]

[C.sub.21]  0.4986,0.1297,0.3716
[C.sub.22]  0.7166,0.0946,0.1887
[C.sub.23]  0.2156,0.1028,0.6815
[C.sub.24]  0.6856,0.1019,0.2124
[C.sub.25]  0.3595,0.1287,0.5117
[C.sub.26]  0.3658,0.1292,0.5049

Table A2 (i) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C1 to C5 from Table 1 in section 4.

           [C.sub.11]            [C.sub.12]

[C.sub.1]  0.6419,0.1110,0.2470  0.6787,0.1034,0.2178
[C.sub.2]  0.7632,0.0824,0.1543  0.2745,0.1169,0.6085
[C.sub.3]  0.4123,0.1316,0.4560  0.8436,0.0580,0.0983
[C.sub.4]  0.3699,0.1295,0.5005  0.7415,0.0883,0.1701
[C.sub.5]  0.5712,0.1224,0.3063  0.6716,0.1050,0.2233

           [C.sub.13]            [C.sub.14]

[C.sub.1]  0.7123,0.0960,0.1919  0.6912,0.1006,0.2081
[C.sub.2]  0.5481,0.1253,0.3265  0.8456,0.0574,0.0969
[C.sub.3]  0.9845,0.0063,0.0091  0.1664,0.0869,0.7466
[C.sub.4]  0.5769,0.1217,0.3013  0.8462,0.0572,0.0965
[C.sub.5]  0.9412,0.0233,0.0354  0.6565,0.1081,0.2353

           [C.sub.15]

[C.sub.1]  0.5197,0.12811,0.3521
[C.sub.2]  0.5966,0.1188,0.2845
[C.sub.3]  0.5523,0.1248,0.3228
[C.sub.4]  0.8425,0.0584,0.0990
[C.sub.5]  0.6656,0.1062,0.2281

Table A2 (j) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C6 to C10 from Table 1 in section 4.

            [C.sub.11]            [C.sub.12]

[C.sub.6]   0.6742,0.1044,0.2213  0.4369,0.1318,0.4312
[C.sub.7]   0.7416,0.0883,0.1700  0.5716,0.1224,0.3059
[C.sub.8]   0.4137,0.1316,0.4546  0.8413,0.0588,0.0998
[C.sub.9]   0.5166,0.1283,0.3550  0.7168,0.0946,0.1885
[C.sub.10]  0.5712,0.1224,0.3063  0.6741,0.1044,0.2214

            [C.sub.13]            [C.sub.14]

[C.sub.6]   0.2145,0.1025,0.6829  0.7956,0.0731,0.1312
[C.sub.7]   0.6715,0.1050,0.2234  0.6135,0.1838,0.2703
[C.sub.8]   0.8422,0.0585,0.0992  0.8436,0.0580,0.0983
[C.sub.9]   0.6479,0.1098,0.2422  0.4696,0.1312,0.3991
[C.sub.10]  0.9145,0.0333,0.0521  0.6713,0.1050,0.2236

            [C.sub.15]

[C.sub.6]   0.8626,0.0517,0.0856
[C.sub.7]   0.5946,0.1191,0.2862
[C.sub.8]   0.6816,0.1028,0.2155
[C.sub.9]   0.3266,0.1253,0.5480
[C.sub.10]  0.3247,0.1250,0.5502

Table A2 (k) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C11 to C20 from Table 1 in section 4.

            [C.sub.11]            [C.sub.12]

[C.sub.11]  0,0,0                 0.4193,0.1317,0.4489
[C.sub.12]  0.4193,0.1317,0.4489  0,0,0
[C.sub.13]  0.4785,0.1308,0.3906  0.5136,0.1286,0.3577
[C.sub.14]  0.6971,0.0993,0.2035  0.8435,0.0581,0.0983
[C.sub.15]  0.7486,0.0864,0.1649  0.9462,0.0214,0.0323
[C.sub.16]  0.9512,0.0195,0.0292  0.6821,0.1027,0.2151
[C.sub.17]  0.5623,0.1236,0.3140  0.1265,0.0710,0.8024
[C.sub.18]  0.7152,0.0950,0.1897  0.3956,0.1310,0.4733
[C.sub.19]  0.7185,0.0942,0.1872  0.3497,0.1278,0.5224
[C.sub.20]  0.8465,0.0571,0.0963  0.6645,0.1065,0.2289

            [C.sub.13]            [C.sub.14]

[C.sub.11]  0.4785,0.1308,0.3906  0.6971,0.0993,0.2035
[C.sub.12]  0.5136,0.1286,0.3577  0.8435,0.0581,0.0983
[C.sub.13]  0,0,0                 0.3469,0.1276,0.5254
[C.sub.14]  0.3469,0.1276,0.5254  0,0,0
[C.sub.15]  0.5653,0.1232,0.3114  0.6556,0.1083,0.2360
[C.sub.16]  0.5185,0.1282,0.3532  0.5251,0.1276,0.3472
[C.sub.17]  0.5659,0.1231,0.3109  0.7566,0.0842,0.1591
[C.sub.18]  0.6748,0.1043,0.2208  0.7465,0.0870,0.1664
[C.sub.19]  0.6494,0.1095,0.2410  0.4896,0.1302,0.3801
[C.sub.20]  0.5867,0.1203,0.2929  0.7451,0.0873,0.1675

            [C.sub.15]

[C.sub.11]  0.7486,0.0864,0.1649
[C.sub.12]  0.9462,0.0214,0.0323
[C.sub.13]  0.5653,0.1232,0.3114
[C.sub.14]  0.6556,0.1083,0.2360
[C.sub.15]  0,0,0
[C.sub.16]  0.4657,0.1313,0.4029
[C.sub.17]  0.6289,0.1134,0.2576
[C.sub.18]  0.6465,0.1101,0.2433
[C.sub.19]  0.6594,0.1075,0.2330
[C.sub.20]  0.8556,0.0540,0.0903]

Table A2 (l) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C21 to C26 from Table 1 in section 4.

            [C.sub.11]            [C.sub.12]

[C.sub.21]  0.7856,0.0760,0.1383  0.7565,0.0843,0.1591
[C.sub.22]  0.6146,0.1159,0.2694  0.3556,0.1284,0.5159
[C.sub.23]  0.3562,0.1284,0.5153  0.4532,0.1316,0.4151
[C.sub.24]  0.7164,0.0947,0.1888  0.5645,0.1233,0.3121
[C.sub.25]  0.2186,0.1037,0.6776  0.8465,0.0571,0.0963
[C.sub.26]  0.7529,0.0852,0.1618  0.6485,0.1097,0.2417

            [C.sub.13]            [C.sub.14]

[C.sub.21]  0.3516,0.1280,0.5203  0.7413,0.0883,0.1703
[C.sub.22]  0.3888,0.1307,0.4804  0.7463,0.0870,0.1666
[C.sub.23]  0.5666,0.1230,0.3103  0.4857,0.1304,0.3838
[C.sub.24]  0.5959,0.1189,0.2851  0.5165,0.1283,0.3551
[C.sub.25]  0.6585,0.1077,0.2337  0.4812,0.1307,0.3880
[C.sub.26]  0.5568,0.1242,0.3189  0.6745,0.1043,0.2211

            [C.sub.15]

[C.sub.21]  0.5162,0.1284,0.3553
[C.sub.22]  0.3589,0.1287,0.5123
[C.sub.23]  0.9415,0.0232,0.0352
[C.sub.24]  0.4565,0.1316,0.4118
[C.sub.25]  0.8465,0.0571,0.0963
[C.sub.26]  0.7456,0.0872,0.1671

Table A2 (m) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C1 to C5 from Table 1 in section 4.

           [C.sub.16]            [C.sub.17]

[C.sub.1]  0.4128,0.1316,0.4555  0.7946,0.07341,0.1319
[C.sub.2]  0.4956,0.1299,0.3744  0.6596,0.1075,0.2328
[C.sub.3]  0.6595,0.1075,0.2329  0.2648,0.1149,0.6202
[C.sub.4]  0.5656,0.1232,0.3111  0.8746,0.0476,0.0777
[C.sub.5]  0.9463,0.0213,0.0323  0.6941,0.1000,0.2058

           [C.sub.18]            [C.sub.19]

[C.sub.1]  0.6843,0.1022,0.2134  0.7069,0.0970,0.1960
[C.sub.2]  0.3266,0.1253,0.5480  0.8951,0.0404,0.0644
[C.sub.3]  0.1654,0.0866,0.7479  0.3261,0.1252,0.5486
[C.sub.4]  0.6957,0.0996,0.2046  0.2154,0.1028,0.6817
[C.sub.5]  0.8946,0.0405,0.0648  0.1595,0.0844,0.7560

           [C.sub.20]

[C.sub.1]  0.8431,0.0582,0.0986
[C.sub.2]  0.2546,0.1127,0.6326
[C.sub.3]  0.3665,0.1293,0.5041
[C.sub.4]  0.5955,0.1190,0.2854
[C.sub.5]  0.8685,0.0497,0.0817

Table A2 (n) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C6 to C10 from Table 1 in section 4.

            [C.sub.16]            [C.sub.17]

[C.sub.6]   0.2176,0.1034,0.6789  0.1623,0.0854,0.7522
[C.sub.7]   0.8956,0.0402,0.0641  0.5952,0.1190,0.2857
[C.sub.8]   0.6867,0.1017,0.2115  0.7856,0.0760,0.1383
[C.sub.9]   0.9562,0.0175,0.0262  0.7953,0.0732,0.1314
[C.sub.10]  0.7416,0.0883,0.1700  0.9451,0.0218,0.0330

            [C.sub.18]            [C.sub.19]

[C.sub.6]   0.7162,0.0947,0.1890  0.5451,0.1256,0.3292
[C.sub.7]   0.3266,0.1253,0.5480  0.5482,0.1252,0.3265
[C.sub.8]   0.2185,0.1036,0.6778  0.1782,0.0911,0.7306
[C.sub.9]   0.3256,0.1251,0.5492  0.6816,0.1028,0.2155
[C.sub.10]  0.1966,0.0971,0.7062  0.4845,0.1305,0.3849

            [C.sub.20]

[C.sub.6]   0.1656,0.0866,0.7477
[C.sub.7]   0.6595,0.1075,0.2329
[C.sub.8]   0.8466,0.0570,0.0963
[C.sub.9]   0.4863,0.1304,0.3832
[C.sub.10]  0.7566,0.0842,0.1591

Table A2 (o) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C11 to C15 from Table 1 in section 4.

            [C.sub.16]            [C.sub.17]

[C.sub.11]  0.9512,0.0195,0.0292  0.5623,0.1236,0.3140
[C.sub.12]  0.6821,0.1027,0.2151  0.1265,0.0710,0.8024
[C.sub.13]  0.5185,0.1282,0.3532  0.5659,0.1231,0.3109
[C.sub.14]  0.5251,0.1276,0.3472  0.7566,0.0842,0.1591
[C.sub.15]  0.4657,0.1313,0.4029  0.6289,0.1134,0.2576
[C.sub.16]  0,0,0                 0.8956,0.0402,0.0641
[C.sub.17]  0.8956,0.0402,0.0641  0,0,0
[C.sub.18]  0.7441,0.0876,0.1682  0.2156,0.1028,0.6815
[C.sub.19]  0.8949,0.0404,0.0646  0.4163,0.1317,0.4519
[C.sub.20]  0.3598,0.1288,0.5113  0.6147,0.1159,0.2693

            [C.sub.18]            [C.sub.19]

[C.sub.11]  0.7152,0.0950,0.1897  0.7185,0.0942,0.1872
[C.sub.12]  0.3956,0.1310,0.4733  0.3497,0.1278,0.5224
[C.sub.13]  0.6748,0.1043,0.2208  0.6494,0.1095,0.2410
[C.sub.14]  0.7465,0.0870,0.1664  0.4896,0.1302,0.3801
[C.sub.15]  0.6465,0.1101,0.2433  0.6594,0.1075,0.2330
[C.sub.16]  0.7441,0.0876,0.1682  0.8949,0.0404,0.0646
[C.sub.17]  0.2156,0.1028,0.6815  0.4163,0.1317,0.4519
[C.sub.18]  0,0,0                 0.2155,0.1028,0.6816
[C.sub.19]  0.2155,0.1028,0.6816  0,0,0
[C.sub.20]  0.5716,0.1224,0.3059  0.6816,0.1028,0.2155

            [C.sub.20]

[C.sub.11]  0.8465,0.0571,0.0963
[C.sub.12]  0.6645,0.1065,0.2289
[C.sub.13]  0.5867,0.1203,0.2929
[C.sub.14]  0.7451,0.0873,0.1675
[C.sub.15]  0.8556,0.0540,0.0903
[C.sub.16]  0.3598,0.1288,0.5113
[C.sub.17]  0.6147,0.1159,0.2693
[C.sub.18]  0.5716,0.1224,0.3059
[C.sub.19]  0.6816,0.1028,0.2155
[C.sub.20]  0,0,0

Table A2 (p) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C21 to C26 from Table 1 in section 4.

            [C.sub.16]            [C.sub.17]

[C.sub.21]  0.5716,0.1224,0.3059  0.1897,0.0949,0.7153
[C.sub.22]  0.5635,0.1234,0.3130  0.8656,0.0507,0.0836
[C.sub.23]  0.4945,0.1299,0.3755  0.3859,0.1306,0.4834
[C.sub.24]  0.9452,0.0218,0.0329  0.1763,0.0904,0.7332
[C.sub.25]  0.9515,0.0193,0.0291  0.4569,0.1316,0.4114
[C.sub.26]  0.9512,0.0195,0.0292  0.3518,0.1280,0.5201

            [C.sub.18]            [C.sub.19]

[C.sub.21]  0.7166,0.0946,0.1887  0.2965,0.1209,0.5825
[C.sub.22]  0.8462,0.0572,0.0965  0.4562,0.1316,0.4121
[C.sub.23]  0.6889,0.1012,0.2098  0.3462,0.1275,0.5262
[C.sub.24]  0.6455,0.1103,0.2441  0.4655,0.1313,0.4031
[C.sub.25]  0.5743,0.1220,0.3036  0.7152,0.0950,0.1897
[C.sub.26]  0.4686,0.1312,0.4001  0.8597,0.0527,0.0875

            [C.sub.20]

[C.sub.21]  0.4859,0.1304,0.3836
[C.sub.22]  0.4856,0.1304,0.3839
[C.sub.23]  0.5678,0.1229,0.3092
[C.sub.24]  0.5615,0.1237,0.3147
[C.sub.25]  0.4969,0.1298,0.3732
[C.sub.26]  0.7456,0.0872,0.1671

Table A2 (q) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C1 to C5 from Table 1 in section 4.

           [C.sub.21]            [C.sub.22]

[C.sub.1   0.7629,0.0825,0.1545  0.5527,0.1247,0.3225
[C.sub.2]  0.1655,0.0866,0.7478  0.4652,0.1313,0.4034
[C.sub.3]  0.1796,0.0916,0.7287  0.7656,0.0817,0.1526
[C.sub.4]  0.6456,0.1103,0.2440  0.5966,0.1188,0.2845
[C.sub.5]  0.8562,0.0538,0.0899  0.7163,0.0947,0.1889

           [C.sub.23]            [C.sub.24]

[C.sub.1   0.6237,0.1143,0.2619  0.5179,0.1282,0.3538
[C.sub.2]  0.8455,0.0574,0.0970  0.8665,0.0504,0.0830
[C.sub.3]  0.5965,0.1188,0.2846  0.5165,0.1283,0.3551
[C.sub.4]  0.7465,0.0870,0.1664  0.6266,0.1138,0.2595
[C.sub.5]  0.9461,0.0214,0.0324  0.5169,0.1283,0.3547

           [C.sub.25]             [C.sub.26]

[C.sub.1   0.5873,0.1202,0.2924   0.5766,0.1217,0.3016
[C.sub.2]  0.4865,0.1304,0.3830   0.8455,0.0574,0.0970
[C.sub.3]  0.8698,0.0492,0.0809   0.5356,0.1266,0.3377
[C.sub.4]  0.7495,0.0861,0.1643   0.5486,0.1252,0.3261
[C.sub.5]  0.9561,0.0176,0.0262   0.6715,0.1050,0.2234

Table A2 (r) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C6 to C10 from Table 1 in section 4.

            [C.sub.21]            [C.sub.22]

[C.sub.6]   0.7161,0.0947,0.1891  0.6145,0.1159,0.2695
[C.sub.7]   0.6845,0.1022,0.2132  0.5164,0.1283,0.3552
[C.sub.8]   0.7136,0.0954,0.1909  0.5651,0.1232,0.3116
[C.sub.9]   0.6416,0.1110,0.2473  0.4561,0.1316,0.4122
[C.sub.10]  0.4986,0.1297,0.3716  0.7166,0.0946,0.1887

            [C.sub.23]            [C.sub.24]

[C.sub.6]   0.6858,0.1019,0.2122  0.5996,0.1183,0.2820
[C.sub.7]   0.7465,0.0870,0.1664  0.3566,0.1285,0.5148
[C.sub.8]   0.8592,0.0528,0.0879  0.7415,0.0883,0.1701
[C.sub.9]   0.4566,0.1316,0.4117  0.4566,0.1316,0.4117
[C.sub.10]  0.2156,0.1028,0.6815  0.6856,0.1019,0.2124

            [C.sub.25]             [C.sub.26]

[C.sub.6]   0.6515,0.1091,0.2393   0.6123,0.1163,0.2713
[C.sub.7]   0.5795,0.1213,0.2991   0.7155,0.0949,0.1895
[C.sub.8]   0.5167,0.1283,0.3549   0.4189,0.1317,0.4493
[C.sub.9]   0.7866,0.0757,0.1376   0.6589,0.1076,0.2334
[C.sub.10]  0.3595,0.1287,0.5117   0.3658,0.1292,0.5049

Table A2 (s) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C11 to C20 from Table 1 in section 4.

            [C.sub.21]            [C.sub.22]

[C.sub.11]  0.7856,0.0760,0.1383  0.6146,0.1159,0.2694
[C.sub.12]  0.7565,0.0843,0.1591  0.3556,0.1284,0.5159
[C.sub.13]  0.3516,0.1280,0.5203  0.3888,0.1307,0.4804
[C.sub.14]  0.7413,0.0883,0.1703  0.7463,0.0870,0.1666
[C.sub.15]  0.5162,0.1284,0.3553  0.3589,0.1287,0.5123
[C.sub.16]  0.5716,0.1224,0.3059  0.5635,0.1234,0.3130
[C.sub.17]  0.1897,0.0949,0.7153  0.8656,0.0507,0.0836
[C.sub.18]  0.7166,0.0946,0.1887  0.8462,0.0572,0.0965
[C.sub.19]  0.2965,0.1209,0.5825  0.4562,0.1316,0.4121
[C.sub.20]  0.4859,0.1304,0.3836  0.4856,0.1304,0.3839

            [C.sub.23]            [C.sub.24]

[C.sub.11]  0.3562,0.1284,0.5153  0.7164,0.0947,0.1888
[C.sub.12]  0.4532,0.1316,0.4151  0.5645,0.1233,0.3121
[C.sub.13]  0.5666,0.1230,0.3103  0.5959,0.1189,0.2851
[C.sub.14]  0.4857,0.1304,0.3838  0.5165,0.1283,0.3551
[C.sub.15]  0.9415,0.0232,0.0352  0.4565,0.1316,0.4118
[C.sub.16]  0.4945,0.1299,0.3755  0.9452,0.0218,0.0329
[C.sub.17]  0.3859,0.1306,0.4834  0.1763,0.0904,0.7332
[C.sub.18]  0.6889,0.1012,0.2098  0.6455,0.1103,0.2441
[C.sub.19]  0.3462,0.1275,0.5262  0.4655,0.1313,0.4031
[C.sub.20]  0.5678,0.1229,0.3092  0.5615,0.1237,0.3147

            [C.sub.25]            [C.sub.26]

[C.sub.11]  0.2186,0.1037,0.6776  0.7529,0.0852,0.1618
[C.sub.12]  0.8465,0.0571,0.0963  0.6485,0.1097,0.2417
[C.sub.13]  0.6585,0.1077,0.2337  0.5568,0.1242,0.3189
[C.sub.14]  0.4812,0.1307,0.3880  0.6745,0.1043,0.2211
[C.sub.15]  0.8465,0.0571,0.0963  0.7456,0.0872,0.1671
[C.sub.16]  0.9515,0.0193,0.0291  0.9512,0.0195,0.0292
[C.sub.17]  0.4569,0.1316,0.4114  0.3518,0.1280,0.5201
[C.sub.18]  0.5743,0.1220,0.3036  0.4686,0.1312,0.4001
[C.sub.19]  0.7152,0.0950,0.1897  0.8597,0.0527,0.0875
[C.sub.20]  0.4969,0.1298,0.3732  0.7456,0.0872,0.1671

Table A2 (t) represents Neutrosophic matrix ([mu], [pi],
[disjunction]) for the cities C21 to C26 from Table 1 in section 4.

            [C.sub.21]            [C.sub.22]

[C.sub.21]  0,0,0                 0.7855,0.0760,0.1384
[C.sub.22]  0.7855,0.0760,0.1384  0,0,0
[C.sub.23]  0.4887,0.1303,0.3809  0.8946,0.0405,0.0648
[C.sub.24]  0.7416,0.08830.1700   0.4852,0.1305,0.3842
[C.sub.25]  0.8917,0.0416,0.0666  0.1985,0.0977,0.7037
[C.sub.26]  0.2654,0.1150,0.6195  0.6464,0.1101,0.2434

            [C.sub.23]            [C.sub.24]

[C.sub.21]  0.4887,0.1303,0.3809  0.7416,0.0883,0.1700
[C.sub.22]  0.8946,0.0405,0.0648  0.4852,0.1305,0.3842
[C.sub.23]  0,0,0                 0.8561,0.0539,0.0899
[C.sub.24]  0.8561,0.0539,0.0899  0,0,0
[C.sub.25]  0.5785,0.1214,0.3000  0.4668,0.1313,0.4018
[C.sub.26]  0.4156,0.1316,0.4527  0.5486,0.1252,0.3261

            [C.sub.25]             [C.sub.26]

[C.sub.21]  0.8917,0.0416,0.0666   0.2654,0.1150,0.6195
[C.sub.22]  0.1985,0.0977,0.7037   0.6464,0.1101,0.2434
[C.sub.23]  0.5785,0.1214,0.3000   0.4156,0.1316,0.4527
[C.sub.24]  0.4668,0.1313,0.4018   0.5486,0.1252,0.3261
[C.sub.25]  0,0,0                  0.5972,0.1187,0.2840
[C.sub.26]  0.5972,0.1187,0.2840   0,0,0

Table A3 (a) represents Neutrosophic matrix after applying hamming
distance for the cities C1 to C14 from Table 1 in section 4.

            [C.sub.1]  [C.sub.2]  [C.sub.3]  [C.sub.4]  [C.sub.5]

[C.sub.1]   0          0.4433     0.447      0.5418     0.3582
[C.sub.2]   0.4433     0          0.3353     0.1916     0.1823
[C.sub.3]   0.447      0.3353     0          0.3313     0.4457
[C.sub.4]   0.5418     0.1916     0.3313     0          0.7432
[C.sub.5]   0.3582     0.1823     0.4457     0.7432     0
[C.sub.6]   0.7898     0.5309     0.2289     0.5929     0.6959
[C.sub.7]   0.5508     0.1945     0.6479     0.5827     0.2846
[C.sub.8]   0.3305     0.1967     0.2153     0.1459     0.5782
[C.sub.9    0.3261     0.2858     0.2834     0.3221     0.2531
[C.sub.10]  0.6694     0.7518     0.4965     0.6101     0.4157
[C.sub.11]  0.5571     0.4687     0.6412     0.1763     0.3157
[C.sub.12]  0.271      0.5588     0.4136     0.14       0.341
[C.sub.13]  0.3458     0.539      0.6892     0.3278     0.5521
[C.sub.14]  0.1396     0.4353     0.6917     0.5817     0.5674

            [C.sub.6]  [C.sub.7]  [C.sub.8]  [C.sub.9]  [C.sub.10]

[C.sub.1]   0.7898     0.5508     0.3305     0.3261     0.6694
[C.sub.2]   0.5309     0.1945     0.1967     0.2858     0.7518
[C.sub.3]   0.2289     0.6479     0.2153     0.2834     0.4965
[C.sub.4]   0.5929     0.5827     0.1459     0.3221     0.6101
[C.sub.5]   0.6959     0.2846     0.5782     0.2531     0.4157
[C.sub.6]   0          0.34       0.2859     0.3219     0.6197
[C.sub.7]   0.34       0          0.3838     0.5086     0.1929
[C.sub.8]   0.2859     0.3838     0          0.3662     0.1384
[C.sub.9    0.3219     0.5086     0.3662     0          0.7778
[C.sub.10]  0.6197     0.1929     0.1384     0.7778     0
[C.sub.11]  0.7082     0.2141     0.2855     0.498      0.604
[C.sub.12]  0.6185     0.6446     0.1762     0.4885     0.5562
[C.sub.13]  0.61       0.1398     0.4473     0.3933     0.7577
[C.sub.14]  0.7526     0.2654     0.663      0.3152     0.5133

            [C.sub.11]  [C.sub.12]  [C.sub.13]  [C.sub.14]

[C.sub.1]   0.5571      0.271       0.3458      0.1396
[C.sub.2]   0.4687      0.5588      0.539       0.4353
[C.sub.3]   0.6412      0.4136      0.6892      0.6917
[C.sub.4]   0.1763      0.14        0.3278      0.5817
[C.sub.5]   0.3157      0.341       0.5521      0.5674
[C.sub.6]   0.7082      0.6185      0.61        0.7526
[C.sub.7]   0.2141      0.6446      0.1398      0.26548
[C.sub.8]   0.2855      0.1762      0.4473      0.663
[C.sub.9    0.498       0.4885      0.3933      0.3152
[C.sub.10]  0.604       0.5562      0.7577      0.5133
[C.sub.11]  0           0.4959      0.469       0.1715
[C.sub.12]  0.4959      0           0.7279      0.6924
[C.sub.13]  0.469       0.7279      0           0.7494
[C.sub.14]  0.1715      0.6924      0.7494      0 J

Table A3 (b) represents Neutrosophic matrix after applying hamming
distance for the cities C15 to C26 from Table 1 in section 4.

            [C.sub.1]  [C.sub.2]  [C.sub.3]  [C.sub.4]  [C.sub.5]

[C.sub.15]  0.6944     0.7008     0.4806     0.4121     0.1509
[C.sub.16]  0.4426     0.6747     0.44       0.7544     0.3789
[C.sub.17]  0.3075     0.548      0.3728     0.1853     0.1045
[C.sub.18]  0.5409     0.2148     0.2086     0.1        0.6134
[C.sub.19]  0.591      0.7089     0.4751     0.3052     0.1797
[C.sub.20]  0.2562     0.3649     0.2452     0.3967     0.3969
[C.sub.21]  0.3619     0.2825     0.7845     0.1805     0.5186
[C.sub.22]  0.481      0.3809     0.6533     0.7111     0.3088
[C.sub.23]  0.4514     0.7166     0.6064     0.3478     0.7938
[C.sub.24]  0.5419     0.1217     0.6572     0.5024     0.4357
[C.sub.25]  0.4644     0.7398     0.1996     0.4928     0.5413
[C.sub.26]  0.7894     0.746      0.2488     0.3585     0.6353

            [C.sub.6  [C.sub.7]   [C.sub.8]   [C.sub.9]   [C.sub.10]

[C.sub.15]  0.79      0.4049      0.6515      0.3032      0.5141
[C.sub.16]  0.1483    0.5688      0.3143      0.2569      0.1429
[C.sub.17]  0.3206    0.7626      0.7373      0.1398      0.3686
[C.sub.18]  0.4225    0.1798      0.717       0.1925      0.665
[C.sub.19]  0.3776    0.1955      0.2037      0.4955      0.212
[C.sub.20]  0.4713    0.5398      0.103       0.1251      0.1014
[C.sub.21]  0.3665    0.1048      0.6383      0.1369      0.1912
[C.sub.22]  0.5711    0.113       0.1428      0.4441      0.5196
[C.sub.23]  0.4129    0.5549      0.6055      0.1707      0.5102
[C.sub.24]  0.5537    0.5498      0.318       0.3079      0.7958
[C.sub.25]  0.6671    0.2003      0.2952      0.4742      0.5949
[C.sub.26]  0.6826    0.6129      0.539       0.6214      0.4399

            [C.sub.11]  [C.sub.12]

[C.sub.15]  0.3544      0.5558
[C.sub.16]  0.4367      0.7575
[C.sub.17]  0.7656      0.7036
[C.sub.18]  0.7401      0.7935
[C.sub.19]  0.7846      0.39
[C.sub.20]  0.36        0.4607
[C.sub.21]  0.1633      0.1431
[C.sub.22]  0.5764      0.6453
[C.sub.23]  0.3787      0.5538
[C.sub.24]  0.6032      0.7631
[C.sub.25]  0.4601      0.2854
[C.sub.26]  0.345       0.6074J

Table A3 (c) represents Neutrosophic matrix after applying hamming
distance for the cities C1 to C14 from Table 1 in section 4.

            [C.sub.15]  [C.sub.16]  [C.sub.17]  [C.sub.18]  [C.sub.19]

[C.sub.1]   0.6944      0.4426      0.3075      0.5409      0.591
[C.sub.2]   0.7008      0.6747      0.548       0.2148      0.7089
[C.sub.3]   0.4806      0.44        0.3728      0.2086      0.4751
[C.sub.4]   0.4121      0.7544      0.1853      0.1         0.3052
[C.sub.5]   0.1509      0.3789      0.1045      0.6134      0.1797
[C.sub.6]   0.79        0.1483      0.3206      0.4225      0.3776
[C.sub.7]   0.4049      0.5688      0.7626      0.1798      0.1955
[C.sub.8]   0.6515      0.3143      0.7373      0.717       0.2037
[C.sub.9]   0.3032      0.2569      0.1398      0.1925      0.4955
[C.sub.10]  0.5141      0.1429      0.3686      0.665       0.212
[C.sub.11]  0.3544      0.4367      0.7656      0.7401      0.7846
[C.sub.12]  0.5558      0.7575      0.7036      0.7935      0.39
[C.sub.13]  0.3269      0.7296      0.7973      0.4498      0.3054
[C.sub.14]  0.2313      0.3992      0.7247      0.3409      0.6391

            [C.sub.20]  [C.sub.21]  [C.sub.22]  [C.sub.23]  [C.sub.24]

[C.sub.1]   0.2562      0.3619      0.481       0.4514      0.5419
[C.sub.2]   0.3649      0.2825      0.3809      0.7166      0.1217
[C.sub.3]   0.2452      0.7845      0.6533      0.6064      0.6572
[C.sub.4]   0.3967      0.1805      0.7111      0.3478      0.5024
[C.sub.5]   0.3969      0.5186      0.3088      0.7938      0.4357
[C.sub.6]   0.4713      0.3665      0.5711      0.4129      0.5537
[C.sub.7]   0.5398      0.1048      0.113       0.5549      0.5498
[C.sub.8]   0.103       0.6383      0.1428      0.6055      0.318
[C.sub.9]   0.1251      0.1369      0.4441      0.1707      0.3079
[C.sub.10]  0.1014      0.1912      0.5196      0.5102      0.7958
[C.sub.11]  0.36        0.1633      0.5764      0.3787      0.6032
[C.sub.12]  0.4607      0.1431      0.6453      0.5538      0.7631
[C.sub.13]  0.5586      0.5784      0.1679      0.3204      0.3118
[C.sub.14]  0.55        0.5596      0.4211      0.3099      0.1127

            [C.sub.25]  [C.sub.26]

[C.sub.1]   0.4644      0.7894
[C.sub.2]   0.7398      0.746
[C.sub.3]   0.1996      0.2488
[C.sub.4]   0.4928      0.3585
[C.sub.5]   0.5413      0.6353
[C.sub.6]   0.6671      0.6826
[C.sub.7]   0.2003      0.6129
[C.sub.8]   0.2952      0.539
[C.sub.9]   0.4742      0.6214
[C.sub.10]  0.5949      0.4399
[C.sub.11]  0.4601      0.345
[C.sub.12]  0.2854      0.6074
[C.sub.13]  0.4626      0.2408
[C.sub.14]  0.7782      0.4564]

Table A3 (d) represents Neutrosophic matrix after applying hamming
distance for the cities C15 to C26 from Table 1 in section 4.

    [C.sub.13]  [C.sub.14]  [C.sub.15]  [C.sub.16]  [C.sub.17]

15  0.3269      0.2313      0.0         0.587       50.4798
16  0.7296      0.3992      0.5875      0.0         0.3031
17  0.7973      0.7247      0.4798      0.3031      0.0
18  0.4498      0.3409      0.4102      0.7394      0.6038
19  0.3054      0.6391      0.277       0.2291      0.5184
20  0.5586      0.55        0.3913      0.1018      0.6117
21  0.5784      0.5596      0.7747      0.55        0.1639
22  0.1679      0.4211      0.2681      0.478       0.5222
23  0.3204      0.3099      0.4932      0.5216      0.48
24  0.3118      0.1127      0.1883      0.6383      0.3112
25  0.4626      0.7782      0.1427      0.7097      0.4569
26  0.2408      0.4564      0.48        0.5613      0.5877

    [C.sub.18]  [C.sub.19]  [C.sub.20]  [C.sub.21]  [C.sub.22]

15  0.4102      0.277       0.3913      0.7747      0.2681
16  0.7394      0.2291      0.1018      0.55        0.478
17  0.6038      0.5184      0.6117      0.1639      0.5222
18  0.0         0.1905      0.3585      0.7423      0.767
19  0.1905      0.0         0.1448      0.336       0.3811
20  0.3585      0.1448      0.0         0.2731      0.4678
21  0.7423      0.336       0.2731      0.0         0.1003
22  0.767       0.3811      0.4678      0.1003      0.0
23  0.6496      0.2179      0.2975      0.5828      0.4584
24  0.6868      0.1647      0.7043      0.1995      0.6436
25  0.7177      0.7418      0.1269      0.1118      0.5071
26  0.5705      0.7442      0.3461      0.2824      0.2357

    [C.sub.23]  [C.sub.24]  [C.sub.25]  [C.sub.26]

15  0.4932      0.1883      0.1427      0.48
16  0.5216      0.6383      0.7097      0.5613
17  0.48        0.3112      0.4569      0.5877
18  0.6496      0.6868      0.7177      0.5705
19  0.2179      0.1647      0.7418      0.7442
20  0.2975      0.7043      0.1269      0.3461
21  0.5828      0.1995      0.1118      0.2824
22  0.4584      0.6436      0.5071      0.2357
23  0.0         0.4397      0.5777      0.4602
24  0.4397      0.0         0.3931      0.6435
25  0.5777      0.3931      0.0         0.7415
26  0.4602      0.6435      0.7415      0.0 J


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Vandhana S (1) and J Anuradha (2,*)

(1) School of Computer Science and Engineering, Vellore Institute of Technology; Vellore-632014, Tamil Nadu, India; svandhana2012@gmail.com

(2) School of Computer Science and Engineering, Vellore Institute of Technology; Vellore-632014, Tamil Nadu, India; januradha@vit.ac.in

(*) Correspondence: januradha@vit.ac.in

Vandhana S and J Anuradha, Neutrosophic Fuzzy Hierarchical Clustering for Dengue Analysis in Sri Lanka

Received: Sep 29, 2019.

Accepted: Feb 03, 2020
Table 1 List of Cities in Sri Lanka

Cities  Names

 1      Colombo
 2      Gampaha
 3      Kalutara
 4      Kandy
 5      Matale
 6      N Eliya
 7      Galle
 8      Hambantota
 9      Matara
10      Jaffna
11      Kilinochchi
12      Mannar
13      Vavuniya
14      Mulativu
16      Ampara
17      Trincomalee
18      Kurunegala
19      Puttalam
20      Apura
21      Polonnaruwa
22      Badulla
23      Moneragala
24      Ratnapura
25      Kegalle
26      Kalmunai

Table 2. Inconsistency Coefficient of a tree cut in Hierarchical
Clustering.

      Cluster  Single  Complete  Centroid  Median  Ward
      Link
      I-2

HC    links    0.7071  0.7083    0.6931    0.6682  0.6581
      I-3
HC    links    0.8913  0.9078    0.8691    0.7671  0.7891
      I-4
HC    links    0.6247  0.6901    0.5926    0.6347  0.6874
      I-2
FHC   links    0.7629  0.7145    0.7526    0.6921  0.7021
      I-3
FHC   links    0.8970  0.8825    0.8191    0.7421  0.7334
      I-4
FHC   links    0.5236  0.6971    0.5626    0.6477  0.6792
      I-2
NHFC  links    0.7461  0.7971    0.7526    0.7126  0.6986
      I-3
NHFC  links    0.9168  0.8714    0.7721    0.7428  0.7126
      I-4
NHFC  links    0.6326  0.5910    0.6812    0.6809  0.6574

Table 3. Performance Metrics of HC, FHC, NFHC

                               Clustering
             Method    HC      FHC         NHFC

             Single    0.1263  0.6782      0.7163
Silhouette   Complete  0.2455  0.5763      0.6911
Coefficient  Centroid  0.4726  0.5922      0.6729
             Median    0.5137  0.5501      0.6905
             Ward      0.4968  0.4328      0.7077
             Single    5.2637  3.4266      2.5725
             Complete  4.1258  2.4611      2.4627
DB - Index   Centroid  4.2162  3.1249      2.6674
             Median    4.5018  3.6791      2.0169
             Ward      4.8679  3.0628      2.4209
             Single    0.5671  0.8241      1.134
             Complete  0.7744  0.7689      1.021
Dunn Index   Centroid  0.8671  0.7749      1.159
             Median    0.9632  0.9621      1.067
             Ward      0.8940  0.8017      1.116
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Author:S., Vandhana; Anuradha, J.
Publication:Neutrosophic Sets and Systems
Geographic Code:9SRIL
Date:Jan 1, 2020
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