N-Bipolar Soft Sets and Their Application in Decision Making

The concept of soft set was extended to N -soft set by Fatimah et al. and used as grading system. Bipolar soft sets gave the concept of a binary model of grading. Kamacı and Petchimuchu deﬁned bipolar N -soft set but our approach is diﬀerent from their approach. We deﬁned N-bipolar soft set which extends the concept of bipolar soft set. We explained the notions through some important examples. We discussed some vital deﬁnitions and were motivated towards their use and need. We also described some basic algebraic deﬁnitions and with their help, we developed the algebraic structure of our proposed model. We give decision making algorithms and applied them to real life examples to motivate towards its application. Conﬂict analysis has been a vast topic for research. It was ﬁrst given by Pawlak. The ﬁrst extension to this model was given by Pawlak itself. Then many researchers extended his idea. We also discussed here the application of N -bipolar soft set to conﬂict analysis. The combination of N -bipolar soft set and conﬂict analysis can give user the best way to decide suitable and feasible action.


Introduction
The concept of the soft set was introduced by Molodstov [29], to deal with the problem of inadequate parameterization. Pawlak [31] introduced the concept of soft set in a different and new way. His approach towards soft sets can be thought of as complementary to fuzzy sets. Maji et al. [27], [28] defined some operations on soft sets. Then Ali et al. [5], [7] corrected these operations and defined some more operations on soft sets.The intuitive definitions and interpretations relating extended union, extended intersection, restricted union and restricted intersections discussed in [5] are given in [2], [6], [8] and [35] among them. The bipolar soft set was proposed by Shabir and Naz [36]. This concept was built to distinguish between preferred and adverse sides of the data. Dubois and Prade [13] introduced the role of polarity to give the reason for the positive and negative sides of alternatives. Extension of this, bipolar fuzzy soft set is given in [1]. The use of neural networks and fuzzy logic systems in different theories have been very important for researchers. With the use of neural networks and barrier Luyponav Function BLF, Time-varying IBLFs-based adaptive control of uncertain nonlinear systems with full constraints, neural networks based adaptive event trigger control for a class of electromagnetic suspension systems and integral barrier Lyapunov functions-based adaptive control for switched nonlinear systems has been developed in [24], [25] and [26]. N -binary valued information system was introduced by Herawan and Deris [22] in soft sets, but it did not define rank orders like the cases in [9]. Astericks were used by Hakim et al. [20], [21] as the ranking system to evaluate the objects parameters of soft sets. In view of the importance of grading or ranking system and to develop a non binary model Fatimah et al. [14] introduced the concept of N-soft set. Fatimah et al. [14] discussed some algebraic operations on N -soft sets and application to decision making. Fatimah et al. [14] also related N -soft set with real life problems in a very effective way. Alcantud et al. [3] further worked on N -soft sets by introducing rough structures and approximations. Alcantud et al. [3] derived Pawlak's rough set, tolerence rough sets and multigranular rough set from N -soft set and conversaly. Bipolar soft set [36] gives two parametrized families of subsets of the universe U and the conditionF (e)∩Ǧ(¬e) = φ ∀ e ∈Ȃ and ¬e ∈ ¬Ȃ as a consistensy constraint. It is of vital importance that an object lacking the property e, may not have the opposite property ¬e, so one may haveǦ(¬e) = U −F (e) for some e ∈Ȃ. This is known as degree of reluctance, which occurs due to inadequate knowledge, incomplete information or hesitency in deciding for an object to have an attribute e or ¬e as discussed in [36]. We proposed the new extended model. We first proceed to motivate the need for a new extended model of soft sets with some examples in Section 3. Further after defining N -Bipolar soft set, we explained all the notions or conditions used in the definitions with examples. In its explanation, we have given the concept of re + r¬e ≤ N − 1 which is worth noticing. Kamacı and Petchimuchu in [23] gave an idea of bipolar grading of attributes of N -soft set with positive and negative assessment space but the parameters set taken by them was the same for positive and negative grading. They did not impose any conditions in defining the bipolar N -soft set. They did not discussed the positive and negative aspects on which the grading is made. This model is also not effective in cases when the user has to decide between the alternatives for examples as we do in deciding to choose house in green surrounding and commercial area. We, therefore, introduced a new model which is extended form of bipolar soft set that not only grades the positive and negative aspects of the attributes explaining the reason to grade the parameter but also helps to choose between the alternative parameters. We also describe the algebraic structure of our proposed model. Using the concepts in [5], [14] on N -soft sets, and Shabir and Naz [36] on the bipolar soft sets we developed these concepts on N-bipolar soft sets in Section 4. In Section 5 we developed some algorithms for decision making. The algorithm based on T-choice value helps to make a perfect decision. The conflict was first introduced by Pawlak [30], [32]. He extended his own concept and worked to find the conflicting attribute that is the reason or cause of the conflict. After that, Pawlak [30] gave example of the Palestine dispute as a conflicting issue and developed conflicting function to find the degree of conflict between the countries acting as agents due to some conflicting attributes. Deja [10], [11] extended the work of Pawlak and arose 3 questions, for classical Pawlak conflict model [30], that became a keen interest for the upcoming researchers. These three questions are also discussed in [12]. It is worth noticing that the work of Pawlak [30], [32] and Deja [10], [11] were not multi decisional. Sun and Ma [37] then used Dominance based rough structure introduced by Greco et al. [15], [16], [17], [18] and [19] and worked on multi-agent and multi decisional conflict problems and answered second and third questions of Deja [10], [11], [12]. Sun and Ma [37] developed an algorithm in which they found that feasible consensus or agent at which maximum decision makers agreed, not all of them. Ali et al. [4] using the concept of [33], [34] introduced the concept of dominance based optimistic and pessimistic multi-granulation rough set, then formulated the algorithm that worked on the flaws of the algorithm introduced in [37]. From this Ali et al. [4] was able to find that action at which all the decision makers were agreed and answered the second and third question of Deja [11] deliberately. Our work in Section 6 is basically to solve the conflict model in view of Nbipolar soft sets. We used the algorithm of [4] and transform it according to N -bipolar soft sets to find the optimal consensus which is agreed by all decision makers and has very low or least degree of disagreement as for grades. A comparison is also given in Section 6 between • Bipolar N -soft sets and N -bipolar soft sets.
• Bipolar soft sets and N -bipolar soft sets. .

Preliminaries
Before the discussion on our work we discuss some important notions related to our work.

Soft sets
Definition 2.1. [29] Let U be a universe of objects andȆ be a set of attributes,Ȃ ⊆Ȇ. A soft set is a pair (F ,Ȃ), whereF is a mappingF :Ȃ → P(U) where P(U) is the power set of U. Definition 2.9. The top weak compliment of the N -soft set (F ,Ȃ, N ) over the fixed universe U is denoted by (F t ,Ȃ, N ) such thatF for all ui ∈ U and ej ∈Ȃ.
Definition 2.10. The bottom weak compliment of the N -soft set (F ,Ȃ, N ) over the fixed universe U is denoted by for all ui ∈ U and ej ∈Ȃ.
Definition 2.11. Let 0 < T < N be a threshold and (F ,Ȃ, N ) be an N -soft set over the universe U . The soft set associated with (F ,Ȃ, N ), denoted by (F T ,Ȃ) and is defined by the expression, u ∈F T (e) if and only if re ≥ T when (u, re) ∈F (e), that iš for all e ∈Ȃ and u ∈ U.

Bipolar soft sets
We mention here some basic definitions and results for the Bipolar soft sets which are necessary for our work. The material given in this section is taken from [36] Definition 2.16. Let U be a universe of objects andȆ be a set of attributes andȂ ⊆Ȇ. A Bipolar soft set is a triplet (F ,Ǧ,Ȃ), whereF :Ȃ → P(U ) with the property thatF (e) ∩Ǧ(¬e) = φ, for all e ∈Ȃ and P(U) denotes the power set of U . Definition 2.19. Let U be a fixed universe of discourse. The extended intersection of two bipolar soft sets (F ,Ǧ,Ȃ) and (F1,Ǧ1,B) over the common universe U is denoted and defined by (F ,Ǧ,Ȃ) ∩ E (F1,Ǧ1,B) = (J,Ǩ,Ȃ ∪B), wherě Definition 2.20. Let U be a fixed universal set of objects. The restricted union of two bipolar soft sets (F ,Ǧ,Ȃ) and (F1,Ǧ1,B) over the common universe U is denoted and defined by (F ,Ǧ,Ȃ) ∪ R (F1,Ǧ1,B) = (Ľ,M ,Ȃ ∩B), where for all e ∈Ȃ ∩B = φ, L(e) =F (e) ∪F1(e) andM (¬e) =Ǧ(¬e) ∩Ǧ1(¬e).
Definition 2.21. Let U be a fixed universal set of objects. The extended union of two bipolar soft sets (F ,Ǧ,Ȃ) and (F1,Ǧ1,B) over the common universe U is denoted and defined by (F ,Ǧ,Ȃ) andǑ Definition 2.22. A Bipolar soft set over a universe U is said to be a relative null bipolar soft set denoted by (φ,Ǔ,Ȃ) ifφ(e) = φ for all e ∈Ȃ andǓ(¬e) = U for all ¬e ∈ ¬Ȃ.
Definition 2.23. A Bipolar soft set over a universe U is said to be a relative absolute bipolar soft set denoted by (Ǔ,φ,Ȃ) if U(e) = U for all e ∈Ȃ andφ(¬e) = φ for all ¬e ∈ ¬Ȃ.
Mr. X demands a house according to the parameters, "in green surroundings, wooden, cheap and traditional". The houses are parametrized accordingly, either having the said attributes or its opposite. It is observed that, Mr. X thinks that houses u2 and u3 are situated in green surroundings, while u4 and u5 are situated entirely in commercial area. The house u1 is niether parametrized according to e1 nor according to ¬e1, this shows that u1 is partly located in green surrounding and partly in commercial area or not as much green.
We can observe that on third day of observation the patient has neither anxiety nor no anxiety, we can also say that may be the patient has not as much anxiety.

N -bipolar soft set
Recall Example 2.1, search of houses according to the requirements and the extent to which the mentioned attributes are present, we need a grading system for both the positive and negative sides of attributes, which help and make easier for decision process. For Bipolar mood charts in Example 2.2, if we model this problem to N -bipolar soft set, it would be very easy to detect the symptoms and determination of proper treatment. These examples motivate us to define N -bipolar soft set. We define N -bipolar soft set as follows. with the property that for each e ∈Ȃ there exists a unique (u, re) ∈ U ×Ȓ and for each ¬e ∈ ¬Ȃ there exists a unique (u, r¬e) ∈ (U ×Ȓ) such that (u, re) ∈F (e) and (u, r¬e) ∈Ǧ(¬e), where u ∈ U , re, r¬e ∈Ȓ, with a condition that re + r¬e ≤ N − 1.
Where P(U ×Ȓ) denotes the power set of U ×Ȓ.
• re + r¬e ≤ N − 1. Reconsider the Example 2.1, we model it by N -bipolar soft set as follows.  Now the N -Bipolar soft set is given in Tables 1 and 2. By looking at Table 1 we can see that the u1 house is partially in green surroundings so u1 is given grade 3 under the attribute e1 and to the same extent it is in commercial area thus for the not set u1 is again given grade 3 under the attribute ¬e1, this gives us that,F (e1)(u1) = 3 that is (u1, 3) ∈F (e1) and (u1, 3) ∈Ǧ(¬e1). This example shows thatF (e) ∩Ǧ(¬e) = φ.
It is important to consider that the above mentioned condition may or may not hold, in Example 3.3, we can see thať F (e) ∩Ǧ(¬e) = φ.
To explain the condition re + r¬e ≤ N − 1, see the following example. The result of above test in the form of N -bipolar soft set is given in Table 3 and 4. By looking at the Tables 3 and 4, we came to know that there is a hesitancy for the attributes. Since thinking for the dish prepared by chef c2 having grade 3 for e1 and 4 for ¬e1 is a bizzare thought, because none of the dish can be highly presentable along with fully unpresentable, Similarly we cannot believe for a dish prepared by chef c1 having grade 2 for e2 and 4 for ¬e2 because a dish which is unflavored can not be spicy to a great extent, that is why re + r¬e ≤ N − 1.  It is natural to identify a 2-bipolar soft set by a bipolar soft set and vice versa. A 2-bipolar soft set (F ,Ǧ,Ȃ, 2) can be converted into a bipolar soft set (F ,Ǵ,Ȃ) by defininǵ   Table 5 and 6. This 2-bipolar soft set (F ,Ǧ,Ȃ, 2) can be identified with the bipolar soft set (F ,Ǵ,Ȃ) which is given bý This bipolar soft set (F ,Ǵ,Ȃ) is identified with the 2-bipolar soft set (F ,Ǧ,Ȃ, 2) which is given by • In definition the grade 0 ∈Ȓ does not mean that there is incomplete information or no information. It means the lowest grade or it represents that the individual does not possess the given attribute for example in Table 3 and 4. To interpret the above fact we consider the example of students and we model it by N -bipolar soft set.
Example 3.5. Let U = {s1, s2, s3} be a universe of students andȂ be a set of attributes "evaluation of students by the given parameters". e1 = legible handwriting, ¬e1 = illegible handwriting, e2 = cleanliness, ¬e2 = dirty, e3 = polite and ¬e3 = rude. Evaluation is done by giving grades to students on their performance. The graded evaluation is identified by numbersȒ = {0, 1, 2, 3, 4, 5} where 0 = lowest, 1 = lower, 2 = low, 3 = high, 4 = higher, 5 = highest. The 6-bipolar soft set for this is represented in Table 7 and 8.  Looking at the Tables 7, 8 and considering s2 the "-" tells us that this student may not brought the copy, so the teacher cannot evaluate the handwriting for that student having. The most important point is that 0 here means lowest grade.  • For incomplete information we redefine as follows.
Definition 3.2. Let U be a universe set of objects and E be a set of attributes,Ȃ ⊆Ȇ.
Example 3.5 is Incomplete 6-bipolar soft set. For example, the 6-bipolar soft set given in Tables 9 and 10 can be considered as 7-bipolar soft set and so on over the same universe and with the same attributes. Top grades sometimes appear in (F ,Ȃ, N ) but not in (Ǧ, ¬Ȃ, N ) and vice versa. Sometimes top grades are available in both of the sets. Motivated from this, we define Definition 3.3. An N -bipolar soft set is said to be a positive efficient over universe U ifF (ej)(ui) = N − 1 and thus the corresponding value isǦ(¬ej)(ui) = 0 for some ej ∈Ȃ, ¬ej ∈ ¬Ȃ and ui ∈ U Definition 3.4. An N -bipolar soft set is said to be a negative efficient over universe U ifǦ(¬ej)(ui) = N − 1 and thus the corresponding value isF (ej)(ui) = 0, for some ej ∈Ȃ, ¬ej ∈ ¬Ȃ and ui ∈ U.
We now present the formalization of an idea for the bottom grade .., s} is the index set for attributes.
The normalized 6-bipolar soft set of above 6-bipolar soft set is given in Tables 13 and 14.
Example 3.9. The bipolar soft set associated with 6-bipolar soft set of Example 3.7 is given in the following tables.
Bipolar soft set associated with 6-bipolar soft set when T = 3 Table 21 (

Algebraic operations on N -Bipolar soft sets
If two persons are going to buy house, as given in Example 3.1 for the house of their common interest they can use restricted intersection, restricted union, extended intersection and extended union.
We here initiate these concepts on the N -bipolar soft sets.
The distributive laws which hold in N -bipolar soft sets are described in the following theorem.
(i) Since the set of attributes for both the N -bipolar soft set is same so by the Definitions 4.1-4.4 it can be seen that the result is obvious.
We denote the set of all N -bipolar soft sets over U by N BSS(U)Ȇ with attribute set a subset ofȆ and for all finite values N , that is all N1 ≤ N . The sub collection of N BSS(U)Ȇ consisting of all N -bipolar soft sets over a common universe U with the set of attribute is equal toȆ, is denoted by N BSS(U)Ȇ.

Decision making procedure for N -Bipolar soft set
Bipolar soft sets have been used in many applications. Here we devise some decision making algorithms for N -Bipolar soft sets. Algorithm 1 gives the feasible solution depending only on grades assigned to the objects of universe with respect to each attribute. Algorithm 2 can be used when some of the attributes are given weightage by the decision maker upon which one wants to decide. In other words, the decision is made depending on the weightage of the attributes. When the decision maker wants to consider those objects as solution that have grades above a certain limit (or number from the assessment space) specified by him, then by using Algorithm 3 it will be easy for him to decide. The three algorithms give some sort of different results when applied to the same example. As these three algorithms also consider the negative aspects of attributes, so they give beneficial result encountering the presence of undesirable attributes. This shows the extent to which these methods are adaptable to any situation is by the need of the practitioner.      1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 Algorithm 2 For weighted choice values.

Algorithm 3
For T -choice value. ..} so that for all u i ∈ U, e j ∈Ȃ ∃ r i,j e j ∈Ȓ and also ¬e j ∈ ¬Ȃ ∃ r i,j ¬e j ∈Ȓ for each u i . 3: Compute The result is any one of u i which satisfies the step 6.

Application to Conflict Analysis
N -bipolar soft sets can be applied in real life problems. One of its most distinguished application is in conflict analysis. Conflict arises when there is a disagreement on finding a feasible solution.
We use the Algorithm of Ali et al. [4] and applied on various Nbipolar soft sets. We found some examples where we attained the actions that seem infeasible. So to study the degree of disagreement we use the concept of N -bipolar soft sets which also studies the negative aspect of attributes. The problem under our concern is of multi-attribute and involved multi-decision makers, and we are required to find the most feasible action agreed by all decision makers having the lowest degree of disagreement. Even if the algorithm [4] gives just one feasible solution we would also be able to calculate the least disagreeable action, which helps a lot to decide.

Dominance optimistic MGRS [4]
The dominance optimistic multi-granulation lower and upper approximation of X are defined by The optimistic multi-granulation boundary region of X is BN O (X) = ((X) .

Problem statement
Our problem is that when the Algorithm of Ali et al. [4] is applied to positive set of N -bipolar soft set it also gives those objects of universe as solution which were not certainly feasible. So we introduce a slight different system by the involvement of least disagreeable actions. So despite of the system taken in [4], we divide it into two N -Bipolar soft sets for the conditional attributes and decisional attributes respectively. Consider the problem as S = {U, (F ,Ǧ,C, N1) , (γ,η,D, N2)} over a common universe U , where U contains all the possible actions. (F ,Ǧ,C, N1) is the N -Bipolar soft sets for the conditional attributesC over U and (γ,η,D, N2) is the N -Bipolar soft set over U for the decisional attributes. We call S as multi-attribute and multi decisional information system.

Multi-granulation N -bipolar rough sets
We use the dominance relations used by Ali et al. [4] and mould it into our work as follows; ui≥Quj ifα(Q)(uj) ≥α(Q)(ui) means that ui≥Q m uj ifα(Qm)(uj) ≥α(Qm)(ui) for allQi ∈Q that is ui dominates uj with respect to all attributesQm inQ, whereα ∈ {F ,Ǧ,γ,η} andQ ∈ {C, ¬C,D, ¬D}. The dominance classes to be used in N -bipolar soft sets is denoted and defined bŷ The dominance optimistic multi-granulation lower and upper approximations are as follows.
Then the set

Proposed algorithm
Steps for the positive sets are identical to the algorithm of Ali et al. [4] for the positive set under our proposed notations of dominance relation. Input Information system S = {U , (F ,Ǧ,C, N1), (γ,η,D, N2)}.
Step 1(a): Step 2(a): Step 3(a): If Ω = |D| k=1 Ω k = φ go to the output, otherwise |D| = |D| − 1 go to the step 3(a) For the not set we have the following algorithm.

Comparison Analysis
In this section we will compare the Bipolar N -soft sets and Bipolar soft sets with our model N -bipolar soft sets. First we will compare Bipolar N -soft set with N -bipolar soft set. The main difference is that Bipolar N -soft sets are the bipolar extension of N -soft sets, while our model is the generalization of bipolar soft sets, that is why the set of attributes in [23] is same for positive and negative grading, but we have positive and negative attribute sets in our introduced concept. In [23] they have bipolarity over grades but in ours we have bipolarity over attribute set. In [23], the model introduced by them does not give the cause or reason behind giving positive or negative rankings, the example of restaurant in [23] needs reasoning remarks for grading, that is the positive and negative ticks are followed by remarks, but on the other hand, the use of bipolar attribute set 22   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 clarifies the grading at very first sight and is more understandable to rank for example in our Example 3.2 the dish prepared by three chefs were graded on the basis of positive and negative attributes which gave a sound perception for hiring a chef. Considering the example of positive and negative mentions of customer of a restaurant [23], the attributes given by them seem unable to interpret the cause of grading by ticks, lacking the sufficient information to grade (that is which thing is being considered while grading about the particular thing), so it could be better if we use the following attribute set:Ȃ = {quick service, high quality food, expensive} and for ¬Ȃ = { late service, low quality food, cheap}. If user has to decide between alternatives then this model is ineffective, while our model fits best for such decisions as we did in Example 3.1, where Mr. X wants to buy a house, we used green surroundings and as its alternative we had commercial area. There is no condition imposed in definition to construct bipolar N -soft set [23], also the parameter which is given a positive grade 3 is also given the negative grade −4 for the bipolar 6-soft set. As far as Bipolar soft sets are concerned, N -bipolar soft sets are extension of bipolar soft sets. Bipolar soft sets are a particular case of N -bipolar soft sets when N = 2 which we discussed in Section 3 in a Remark 3.1 and discussion below it. The bipolar soft set does not provide the information about the degree of presence of positive and negative attribute. They only give binary system for grading, which only presents yes or no for the members of universe having the positivity or negativity of the particular attribute, our concept generalizes it to measure the extent to which the presence or absence of positive and negative aspect in the object. In the first step, we explained how to grade negative attributes and what are the suitable negative attributes that can be taken against a particular positive attribute. This we explain as follows, in the example taken by Shabir and Naz [36] for bipolar mood charts. They take the attributesȆ = {Severe Mania, Sever depression, Anxiety, Medication, Side effects} and the set ¬Ȇ = {Mild Mania, Mild Depression, No Anxiety, No Medication, No Side effects}. We observed that severe mania and mild mania are the degrees of Mania, similar is the case for depression. Since to rank any attribute is to distinguish the degree of that particular attribute present in any member of universe, so we define the attributes asȆ =

Conclusion
We have developed a new theory (N -bipolar soft set) based on ranking or grading approach with the consideration of negative aspects. This theory is non binary as well as non-continuous (fuzziness). Each concept introduced in this paper is explained with clear examples. The emphasis is made on the importance of negative sides of attributes and their usefulness in decision making. The algebraic structures of N -bipolar soft sets are also explained here which may be of keen interest of researchers. We have given three algorithms for decision making. These algorithms reveal their flexible and versatile design which makes them adaptable to users need. These algorithms provide users, feasible and optimal consensus that not only has highest positivity but also gives lowest negativity. The importance of conflict analysis is not only limited to real world but also providing researchers a sound ground for research. We introduced the impact of degree of disagreement and its effect towards decision of multi-attribute and multi decisional problems. This concept opened up many avenues for researchers. We can also go beyond this concept by developing theory about incomplete N -bipolar soft sets. It is also possible to combine this concept with fuzziness. We expect that this paper gives an idea for the beginning of new study .  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65