Information measures for MADM under m-polar neutrosophic environment

Multiploar information plays a vital role in making reliable decisions in one’s daily life encompassing micro- to large-scale decisions. These decision-making problems often include imprecise and inconsistent data. This article presents information measures for m-polar neutrosophic sets. These include similarity, distance, correlation, divergence and Dice measures for m-polar neutrosophic sets. Furthermore, the paper presents the desirable characteristics of these measures along with the notions of angle of similarity between two m-polar neutrosophic sets, λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document}-similarity, entropy and less and more fuzzy aspects. The paper also presents an application of m-polar neutrosophic sets using the suggested measures in health sciences accompanied by five algorithms. Moreover, the paper presents the comparative analysis of the proposed information measures with some existing measures.


Introduction
The analytical modeling of real world problems often involve uncertainties, vagueness and imprecisions. To handle such situations, the researchers proposed various models like probability theory, fuzzy sets (Zadeh 1965), rough sets (Pawlak 1982) and interval mathematics (Gorzalzany 1987), etc. All these theories have been recognized in favorable circumstances but unfortunately they have their own limitations. Ensuing the explorations of Zadeh, Atanassov (1986, 1989) introduced a new sort of sets titled intuitionistic fuzzy sets (IFSs). An IFS comprises a pair of two mappings communicating the degrees of membership and the degrees of non-membership of elements of the universe. One major deficiency shared by these theories is possibly the usage of parametrization tools as indicated by Molodtsov. In 1999, Molodtsov (1999 presented the notion of soft sets as an important mathematical tool to deal with uncertainties. Yager (2013) gave notion of Pythagorean fuzzy sets. Naeem et al (2019Naeem et al ( , 2021 rendered the notion of Pythagorean m-polar fuzzy sets and later extended the notion to develop topological structure in . Riaz et al (2020a) supplemented the idea to corresponding soft sets. Smarandache (1998Smarandache ( , 2005 further expanded the notions of fuzzy sets and IFSs and brought to canvas the novel idea of neutrosophic sets. An element in the neutrosophic set has three independent components representing truthmembership, indeterminacy-membership and falsitymembership. Maji (2013) expanded the notion of neutrosophic sets to neutrosophic soft sets (NSSs) and applied the notion to decision-making problems (DMPs) in (Maji 2012). After the explorations of Maji, the studies on neutrosophic soft sets gained momentum and attracted vibrant researchers towards further explorations (see (Broumi 2013;Broumi and Smarandache 2013b;Akram et al 2018)). Wang et al (2010) presented interval valued neutrosophic sets as another expansion of neutrosophic sets. Deli (2017) studied interval-valued neutrosophic soft sets and decision-making. Habib et al (2020) presented application in life sciences under neutrosophic environment. Hashmi et al (2020) explored m-polar neutrosophic topology along with applications.
The idea of getting a numerical value to assess given data in the form of several numbers presents the clear scanned picture of the whole data. Many researchers employed information measures as an efficient tool to achieve this goal. Zadeh (1971) developed ordering and similarity relations in the framework of fuzzy sets. Correlation between two fuzzy membership functions has been rendered by Murthy et al (1985). Based on Shannon entropy, Lin (1991) rendered the conception of divergence measure. Gerstenkorn and Manko (1991) established correlation for IFSs. Chiang and Lin (1999) presented correlation of FSs. Hong and Hwang (1995) studied correlation of IFSs. Chaudhuri and Bhattacharya (2001) developed correlation between fuzzy sets. Using statistical viewpoint, Hung (2001) developed correlation of IFSs. An application in pattern recognition using similarity measures for IFSs was presented by Li and Cheng (2002). Hung and Wu (2002) employed centroid method to develop correlation of IFSs. Correlation coefficient for IFSs was studied by Mitchell (2004), and Zeng and Li (2007). Ye (2011) explored cosine similarity measure for IFSs. Broumi and Smarandache (2013a) studied several measures of similarity for neutrosophic sets. Ye (2012Ye ( , 2018 studied dice similarity and generalized dice similarity measures in the framework of interval-valued IFSs. Uma (2013, 2014) explored cotangent function based multi-similarity and correlation measure in IF environment. Chen and Hong (2014) made use of TOPSIS for fuzzy MAGDM. Utilizing hesitant fuzzy linguistic information, Lin et al (2014) discussed how to select an ERP system. Deng et al (2015) studied monotonic similarity measure for IFSs. In the framework of Pythagorean fuzzy sets, Garg (2016) unveiled a novel correlation coefficient. Accompanied by applications, Peng et al (2017) explored information measures for Pythagorean fuzzy sets. Mishra et al (2017) established the notion of exponential fuzzy intuitionistic fuzzy information measure. Garg and Arora (2017) presented similarity and distance measures dual hesitant fuzzy soft sets. For MCDM purpose, Lu and Ye (2017) presented cosine measure for neutrosophic cubic sets. Wei (2018) utilized the picture fuzzy model for defining some similarity measures accompanied by their practical utilization. Ansari et al (2018) presented novel entropy and divergence measures for IFSs. Zeng et al (2018) studied correlation coefficient of IFSs. Nancy and Garg (2019) devised divergence measure based TOPSIS accompanied by its utility in the framework of single-valued NSs. In the q-rung orthopair fuzzy context, Peng and Liu (2019) rendered some information measures. Akram et al (2019) generalized maximizing deviation and TOPSIS in simplified neutrosophic hesitant fuzzy environment. Under the umbrella of PFSs, Peng (2019) developed novel distance and similarity measures. Rani et al (2019) studied shapely weighted divergence measure by extending intuitionistic fuzzy TODIM technique. Novel correlation coefficients in neutrosophic cubic environment were made part of the research by Xue et al (2019). Cui and Ye (2019) presented logarithmic similarity measure for dynamic neutrosophic cubic sets.
Making use of mF rough information, Akram and Adeel (2020) studied methods of hybrid decision-making. On the basis of rough Pythagorean fuzzy bipolar soft information, Akram and Ali (2020) studied decision-making hybrid models. With the help of score function and divergence measure, Khan and Ansari (2020) presented a practical implementation of selection of multi-criteria software quality model. By making use of information entropy, Liu et al (2020) presented an application of decision-making under social influences. On the basis of TOPSIS and clustering methods along with novel distance measure, Garg and Nancy (2020) presented algorithms for singlevalued NSs. In the framework of PFSs, Riaz et al (2020b) studied a similarity measure based on Frobenius product of matrices. Using neutrality aggregation operators, Garg and Chen (2020) presented an MCDM problem in the framework of q-rung orthopair fuzzy environment. For PFSs, Firozja et al (2020) rendered a novel similarity measure. Hashmi et al (2020) devised MCDM techniques using m-polar neutrosophic topology. Singh and Ganie (2020) developed some correlation coefficients for PFSs.
Recently, Arya and Kumar (2021) studied entropy and knowledge measure. Jiang et al (2021) presented an improvement of MCDM relevant to environmental disaster. Zulqarnain et al (2021a) devised information measures for m-polar neutrosophic soft environment. Utilizing the rough mF bipolar soft environment, Akram et al (2021) rendered hybrid decision-making problem. Zulqarnain et al (2021b) further extended the notion to corresponding interval-valued environment. Jafar et al (2021) rendered an application in choosing renewable energy source using trigonometric similarity measures in the framework of neutrosophic hypersoft model. Türkarslan et al (2021) rendered some similarity measures for PFSs using trigonometric functions. Utilizing Pythagorean reliability and similarity measures, Wang and Zhao (2021) presented an MCDM problem. In the framework of m-polar neutrosophic hypersoft sets, Irfan et al (2021) explored three similarity measures.
In recent times, employing the notion of possibility measure, Garai and Garg (2022) presented an MCDM related to water resource management. Recently, Garg et al (2022) rendered a practical usage of spherical fuzzy soft topology. Ejegwa et al (2022a) presented some modification to Pythagorean fuzzy correlation measure accompanied by practical utilization. In the Pythagorean fuzzy environment, Ejegwa et al (2022b) studied a utilization in life sciences making use of composite relation. Recently, Gupta and Kumar (2022) utilized picture fuzzy environment to develop a novel similarity measure. Kadian and Kumar (2022) developed similarity measure and fuzzy mean codeword length. In recent times, Singh and Ganie (2022) established knowledge measure for generalized hesitant fuzzy environment. For generalized orthopair fuzzy membership grades, Feng et al (2022) devised novel score functions supplemented by MADM.
The need of multipolar statistics is multi-situational. For instance, readings are taken before and after the meal for certain medical tests. We think repeatedly to take investment decisions in business. Similarly, much thought is involved in taking matrimonial decisions. Fuzzy, intuitionistic fuzzy, hesitant fuzzy, picture fuzzy, neutrosophic and other existing hybrid models have their own limitations in the perspective of loss of information. The m-polar neutrosophic environment is most appropriate for MCDM methods and handles multi-criteria decision-making encompassing truth, indeterminacy, and falsity values. As a result of inclusion of neutrosophic nature in multi-polarity, these three grades function independent of each other and give much information about the multiple criteria for the alternatives.
The major aim of this article is to suggest some novel information measures including similarity measure, distance measure, correlation measure, divergence measure and dice measure for m-polar neutrosophic sets. The information measures for other prevalent hybrid structures are special cases of information measures of m-polar neutrosophic set. Hence, this model is more generalized. The rest of the article is organized as follows: The second section gives access to some elementary notions relevant to m-polar neutrosophic sets. The next section presents five information measures i.e. similarity measure, distance measure, correlation measure, divergence measure and Dice measure for m-polar neutrosophic sets accompanied by their prime attributes and illustrations. In section four, an application of suggested measures along with a case study is given from health sector. The fifth section presents comparative analysis of the proposed information measures with some existing measures. Section six concludes the article.

Preliminaries
This section deals with some basics, which are imperative for understanding further notions. If size of X is r, then ?tic=?>Table 1 yields the tabulatory array of N. The corresponding matrix format is This matrix of size r Â m is titled as mN matrix. The aggregate of all mNSs defined over X is designated as mNS(X). ; over X is given by Definition 4 (Hashmi et al 2020) An mNS 0 N is called to be null m-polar neutrosophic set if it is in the form measure between mNSs defined over the finite universe X ¼ fx i : i ¼ 1; 2; . . .; ng may be defined as and Proposition 1 The similarity measure presented in Definition 7 observes the following requirements: Proof The first and third conditions are obvious. Suppose that SimðN 1 ; N 2 Þ ¼ 1. Then, for all permissible values of i and j, we have Conversely, assume that N 1 ¼ N 2 . Thus, for all i and j,  x 2 ð0:89; 0:33; 0:14Þ; ð0:71; 0:03; 0:15Þ ' defined over the crisp set X ¼ fx 1 ; x 2 g. Then, Definition 8 The angle hðN 1 ; N 2 Þ defined by hðN 1 ; N 2 Þ ¼ arccos hN 1 ;N 2 i kN 1 kkN 2 k is called the angle of similarity between the mNSs N 1 and N 2 .
Example 2 For N 1 and N 2 , cited in Example 1, we have Remark 1 For finding the similarity between an NS and an mNS, we convert the NS to mNS first by multipolarizing the given NS according as the given mNS. The forthcoming example illustrates the notion.
Example 3 Consider the mNS defined over X ¼ fx 1 ; x 2 ; x 3 g. For finding the measure of similarity between N 1 and N, we make the NS N a 2polar neutrosophic set (2NS, and call the resulting set N 2 ) to make it compatible with N 1 as Proposition 2 N is a crisp set , SimðN; N c Þ ¼ 1.
Proof The proof follows from the fact that a crisp set has the form Thus, the results given in Proposition 3 are testified.
Definition 9 Two mNSs N 1 and N 2 are said to be k-similar, written N 1 % k N 2 , if and only if SimðN 1 ; N 2 Þ ! k for some 0\k\1.
Proposition 4 The relation of being k-similar is not an equivalence relation.
Definition 10 The entropy of an mNS N is defined as Definition 11 Let N 1 and N 2 be two mNSs. Then, N 1 is called less fuzzy than N 2 if and only if EðN 1 Þ EðN 2 Þ.

Distance measure for mNSs
This subsection is devoted to present a distance measure for mNSs and some of its major characteristics.
Definition 12 Let N be an mNS and N 1 ; N 2 YN. A measure D m ðN 1 ; N 2 Þ, given by the mapping D m ðN 1 ; N 2 Þ : mNSðXÞ Â mNSðXÞ ! ½0; 1, is called a distance measure if following requirements are fulfilled: Definition 13 The distance measure between mNSs Proposition 5 The distance measure D m given in Definition 13 fulfills the requirements given in Definition 12.
Example 6 The distance measure between mNSs given in Example 1 is Remark 2 In order to find the distance measure between an NS and an mNS, we proceed like Example 3.
Proposition 6 N is a crisp set , D m ðN; N c Þ ¼ 0.
Proof The proof is consequence of the fact that a crisp set has the form On subtraction, we have The other result may be established analogously. Thus, the results given in Proposition 7 are testified.
Remark 3 If D m ðN 1 ; N 2 Þ is the distance measure between two mNSs N 1 and N 2 , then 1 À D m ðN 1 ; N 2 Þ serves as a similarity measure between N 1 and N 2 and vice versa.

Correlation measure for mNSs
This subsection proposes a correlation measure for mNSs along with some of its major characteristics.
Definition 14 Take N to be an mNS with N 1 ; N 2 YN. A measure CorðN 1 ; N 2 Þ, given by the mapping CorðN 1 ; N 2 Þ : mNSðXÞ Â mNSðXÞ ! ½0; 1, is called a correlation measure if: Definition 15 The correlation measure between mNSs defined over X ¼ fx i : i ¼ 1; 2; Á Á Á ; ng may be defined as Proposition 8 The correlation measure CorðN 1 ; N 2 Þ given in Definition 15 fulfills the requirements given in Definition 14.

Divergence measure for mNSs
We present a divergence measure for mNSs in this subsection.

Example 9
The divergence measure between the mNSs given in Example 1 is computed as under:  Remark 5 For finding the divergence measure between an NS and an mNS, we proceed like Example 3.

Dice measure for mNSs
This subsection is devoted to present a Dice measure for mNSs.

Definition 19
The Dice measure between mNSs defined over X ¼ fx i : i ¼ 1; 2; Á Á Á ; ng may be defined as Proposition 10 The Dice measure D given in Definition 19 fulfills the requirements given in Definition 18.

Multi-attribute decision-making based on mNSs in health sciences
Decision-making is a vital procedure for selecting most appropriate choice from available options. Application of the decision-making procedure assists in taking more purposive and intellectual decisions by verbalizing relevant data while demonstrating another course of action. In this section, we discuss multi-attribute decision-making (MADM) in health sciences. MADM methods decide on among a discrete set of choices assessed on multiple features and overall efficacy of the decision makers.
Case study Blood disorders are conditions that influence the blood's ability to operate correctly. Most blood disarrays drop the number of cells, platelets, proteins, or nutrients in the bloodline, or obstruct with their use. A bulk of blood disorders are induced by variations in parts of specific genes and can be handed down in families. Some medicinal conditions, medicines, and life style features can also originate blood disorders to make grow. A lot of blood disorders take their public figure from the constituent of the blood they influence. The following classifications refer to blood disorders that have a lessening in stock components or disturb their purpose:

Anemia
The lessened number of flowing red blood cells in the physical structure causes Anemia. Red blood cells are indispensable for the survival of the body. They pass on hemoglobin, which is a multifaceted protein containing iron molecules. These iron-containing molecules transfer oxygen from lungs to the rest of the body which is an essential requirement of body. This type of blood disorder is common in the worldwide population. Over and above 400 categories of anemia have been recognized. Preschool kids have the utmost menace, with a probable 47% evolving anemia, universally.
The most usual warning sign of all forms of anemia is a feeling of tiredness and a privation of energy. Other corporate symptoms may take account of pallor of skin, squatness of breath, chest discomfort, wild or asymmetrical heartbeat, headache and light-headedness. Pictorial view of a normal and anemic amount of red cells may be seen in Fig. 1.

Leukopenia
The term leukopenia is derived from two Greek words leukos (white) and penia (deficiency). Leukopenia is a shrinkage of the quantity of white blood cells (leukocytes) present in the blood, which puts individuals at amplified jeopardy of infection. Our body fluid is made up of diverse sorts of blood cells, together with white blood cells, or leukocytes. White blood cells are a significant component of our immune organism which supports our body to hold back ailments, syndromes and infections. A person with lesser white blood cells is a victim of leukopenia (see Fig. 2). Major causes of leukopenia consist of poor condition of blood cells or bone marrow, leukemia (a sort of cancer), hereditary complications, transmissible diseases (like tuberculosis and AIDS), malnutrition, autoimmune disorders (like lupus and rheumatoid arthritis), medications (such as steroids, cyclosporine, bupropion, penicillin and lamotrigine), viral infections and sarcoidosis.
Probably there are not any prominent signs of leukopenia. But if white cell counts are very low, one may have signs of infection, including fever higher than 100:5 F, sweating and chills.

Thrombocytopenia
Thrombocytopenia is a blood disease in which one has a lesser amount of blood platelet than desired. Platelets (thrombocytes) are colorless blood cells which help blood clot. Platelets discontinue bleeding by clumping and forming plugs in blood vessel injuries (see Fig. 3). Main reasons of thrombocytopenia include cancer, chemotherapy, virus (such as chickenpox, rubella, HIV, mumps, and Epstein-Barr), platelet-lowering disease inherited from family, and aplastic anemia.
Sometimes one does not have any symptoms from thrombocytopenia. However, the main one is bleeding in the skin that looks like tiny red or purple spots on the skin, called petechiae, or bruising. One can bleed outside or inside the body. Sometimes it can be heavy or hard to stop. Some people get nosebleeds, bleeding gums, blood in urine or bowel movement and heavy menstrual periods.
Before heading towards practical utility of the suggested similarity measure, first we propose Algorithm 1 given below.
Step 2: Choose the set of diseases D ¼ fd 1 ; d 2 ; Á Á Á ; d i g and the set of symptoms S ¼ fs 1 ; s 2 ; Á Á Á ; s j g.
Step 3: Construct table of NS of diseases vs symptoms & mNS of patients vs symptoms.
Step 4: Compute similarity measure between patients & diseases.
Step 5: Optimal choice is the pair with highest similarity measure between patient and disease.
Step 6: State the results in layman's language.
Example 11 Let X ¼ fp k : k ¼ 1; . . .; 4g be the set of patients under study, and D ¼ fd i : i ¼ 1; 2; 3g be the set of types of blood disorder, where . . .; 4g is the set of symptoms, where s 1 ¼ Skin paleness, s 2 ¼ High fever, s 3 ¼ Spots on skin, and s 4 ¼ weakness Table 2 gives neutrosophic numbers as symptoms vs blood disorder type. The objective is to get an appropriate diagnosis for each patient. Table 3 yields information regarding patients vs symptoms in the standard form of representing mNSs. The readings are taken at three different times in order to be more specific about type of disease. Now, we compute the similarity measures between patients and type of blood disorder in Table 4. The patient p i having maximal value of the similarity measure with d j is likely to fall prey of the disease d j .  Table 4, the optimal choices are ðp 1 ; d 2 Þ, ðp 2 ; d 3 Þ, ðp 3 ; d 2 Þ and ðp 4 ; d 1 Þ. Thus, we conclude that the patients p 1 & p 3 are likely to suffer from leukopenia, p 2 from thrombocytopenia, whereas p 4 is victim of anemia.
The similarity measures of patients versus diseases are depicted in Fig. 4. Now, we resolve Example 11 using proposed distance measure. But before that, first we propose Algorithm 2 as follows.
Step 2: Choose the set of diseases D ¼ fd 1 ; d 2 ; Á Á Á ; d i g and the set of symptoms S ¼ fs 1 ; s 2 ; Á Á Á ; s j g.
Step 3: Construct table of NS of diseases vs symptoms & mNS of patients vs symptoms.
Step 5: Optimal choice is the pair with least distance measure between patients & diseases.
Step 6: State the results in layman's language.
Example 12 The distance measures of each patient with the diseases considered are computed and tabulated in Table 5: Hence, keeping in view Table 5, it may be concluded that the optimal choices are ðp 1 ; d 2 Þ, ðp 2 ; d 3 Þ, ðp 3 ; d 2 Þ and ðp 4 ; d 1 Þ.
The distance measures of patients versus diseases are displayed in Fig. 5. We solve, now, Example 11 using proposed correlation measure. So, as earlier, first we propose Algorithm 3 as follows. Algorithm 3 Step 1: Choose the set of patients P ¼ fp 1 ; p 2 ; Á Á Á ; p k g.
Step 2: Choose the set of diseases D ¼ fd 1 ; d 2 ; Á Á Á ; d i g and the set of symptoms S ¼ fs 1 ; s 2 ; Á Á Á ; s j g.
Step 3: Construct table of NS of diseases vs symptoms & mNS of patients vs symptoms.
Step 4: Compute correlation measure between patients & diseases.
Step 5: Optimal choice is the pair with highest correlation measure between patients & diseases.
Step 6: State the results in layman's language.
The correlation measures of patients versus diseases are given in Fig. 6.
We solve, now, Example 11 using proposed divergence measure. So, as earlier, first we propose Algorithm 4 as follows. Step 1: Choose the set of patients P ¼ fp 1 ; p 2 ; Á Á Á ; p k g.
Step 2: Choose the set of diseases D ¼ fd 1 ; d 2 ; Á Á Á ; d i g and the set of symptoms S ¼ fs 1 ; s 2 ; Á Á Á ; s j g.
Step 3: Construct table of NS of diseases vs symptoms & mNS of patients vs symptoms.
Step 5: Optimal choice is the pair with least divergence measure between patients & diseases.
Step 6: State the results in layman's language.
Example 14 The divergence measures of each patient with the diseases considered are computed and tabulated in Table 7: In view of Table 7, the optimal pairs are ðp 1 ; d 2 Þ, ðp 2 ; d 3 Þ, ðp 3 ; d 2 Þ and ðp 4 ; d 1 Þ.
The divergence measures of patients versus diseases are presented in Fig. 7.
We solve, now, Example 11 using proposed Dice measure. So, as earlier, first we propose Algorithm 5 as follows.
Step 2: Choose the set of diseases D ¼ fd 1 ; d 2 ; Á Á Á ; d i g and the set of symptoms S ¼ fs 1 ; s 2 ; Á Á Á ; s j g.
Step 3: Construct table of NS of diseases vs symptoms & mNS of patients vs symptoms.
Step 4: Compute Dice measure between patients & diseases.
Step 5: Optimal choice is the pair with highest Dice measure between patients & diseases.
Step 6: State the results in layman's language.
The Dice measures of patients versus diseases are exhibited in Fig. 8.  We find the information measures, suggested in this paper, between N k (k ¼ 1; 2; 3) and and compare the results with some existing measures. The values computed through the proposed information measures are tabulated in Table 9. The values of the five suggested information measures are displayed by means of Fig. 9.
Comparison of proposed measures with some existing measures has been presented in Table 10. We observe that all the five suggested information measures yield the same output and that yield the same ranking when compared with some existing measures, which approves the reliability of the proposed measures.  It is worth mentioning that the information measures proposed in this paper are easy to handle regarding computational complexity. The methods proposed by Hashmi et al (2020) are comparatively more laborious. The similarity measures proposed by Wei (2018) and Ye (2011) cannot handle multipolarity.

Conclusion
We come across multipolar information in diverse situations. Certain medical tests lead to multipolar statistics. Similarly, we think multiple times before taking academic decisions. Multiple factors are considered while taking investment decisions in business. Such situations involve multipolarity. To handle multipolar information, novel similarity, distance, correlation, divergence and dice measures along with five algorithms in the framework of mpolar neutrosophic environment inclusive of some of their peculiar features are proposed in this article. The proposed measures enhance the schemes for assessing the level of similarity between mNSs. The proposed measures fluctuate from 0 to 1 which eradicates the negative similarity rating.
We devised method for computing information measure between an m-polar neutrosophic set and a neutrosophic set. The proposed information measures are also valid for single valued neutrosophic sets (m ¼ 1). A case study of major blood disorders has been made part of the study. It has been observed that the five proposed information measures yield the same optimal choice. We made use of statistical charts, where necessary, to make the final rankings comprehendible for a layman. Comparative analysis of the suggested measures with some prevailing measures is also included. The information measures rendered in this article have immense scope for more explorations in analytical beyond usance perspective. The conception may be skillfully applied to multiple aspects of life. These may include trade, business analysis, voice recognition, economics, artificial intelligence, marketing, transportation problems, image processing, speech recognition, water management problems, agri-forming, pattern recognition, coding theory, forecasting, robotics, recruitment problems and life sciences.