Investment decision-making based on an intuitionistic fuzzy soft set

Research on soft set and decision-making have been developed rapidly since it can be applied easily to several areas like computer science, information technology, medical science, economics, environments, engineering, among other. Making decisions on investment is certainly the most important task of an investor and it is often a very difficult one. In this research work, we characterize t-norm and t-conorm products on the intuitionistic fuzzy approximate value sets (briefly, IFAV-sets) of an intuitionistic fuzzy soft set (briefly, IFS-set) and applying these products, we introduce an adjustable investment model based on an IFS-set, for investment decision in an uncertain situation. The feasibility of our proposed IFS-set based investment model in practical application is shown by some examples and also, we show the effect of non-membership degrees of elements on decision making.


Introduction
The practical application of soft set theory, especially the used of soft set in intelligent management and decision making problems has found paramount importance.In recent years vague concepts have been used in different areas such as computer application, medical applications, information technology, pharmacology, economics and engineering since the classical mathematics methods are inadequate to solve many complex problems in these areas.In soft set theory there is no limited condition to the description of objects; so researchers can choose the form of parameters they need, which greatly simplifies the decision making process and make the process more efficient in the absence of partial information.Although many mathematical tools are available for modeling uncertainties such as probability theory, fuzzy set theory, rough set theory, interval valued mathematics etc, but there are inherent difficulties associated with each of these techniques.Moreover all these techniques lack in the parameterization of the tools and hence they could not be applied http://www.ispacs.com/journals/jfsva/2016/jfsva-00343/International Scientific Publications and Consulting Services successfully in tackling problems especially in areas like economic, environmental and social problems domains.Soft set theory is standing in a unique way in the sense that it is free from the above difficulties.In 1999, Molodstov [15] initiated the concept of soft set theory as a mathematical tool for dealing with uncertainties.Later on Maji et al. [13] presented some new definitions on soft sets such as subset, union, intersection and complements of soft sets and discussed in details the application of soft set in decision making problem.Based on the analysis of several operations on soft sets introduced in [15], Ali et al. [1] presented some new algebraic operations for soft sets and proved that certain De Morgan's law holds in soft set theory with respect to these new definitions.Recently, Zhan [22] investigated a new soft union set to hemirings.Since the idea of fuzzy set was started by Zadeh [21], many new methodologies and theories treating imprecision and uncertainty have been proposed, such as the intuitionistic fuzzy sets was published by Atanassov [2], A new generalized intuitionistic fuzzy set introduced by Jamkhaneh and Nadarajah [6] and so on [ [3], [38], [40], [44]].Combining soft sets [15] with fuzzy sets [21] and intuitionistic fuzzy sets [2], Maji et al. [9][10][11][12] defined fuzzy soft sets and intuitionistic fuzzy soft sets, which are rich potential for solving decision making problems.By using these definitions, the uses of soft set theory have been concentrated progressively (see [4], [5], [7], [8], [14], [16], [17], [19], [20]).Zhu [23] characterized t-norm and t-conorm products of fuzzy parameterized fuzzy soft sets and presented its application in decision-making.Recently, Mukherjee and Das [18] presented the application of interval valued intuitionistic fuzzy soft set in investment decision making.
In fact, every one of these ideas having a decent application in different controls and genuine issues are currently getting force.However, it is seen that every one of these theories has their own troubles that is the reason in this paper, we have presented an adjustable investment model based on an IFS-set, for investment decision in an uncertain situation.Firstly, we briefly review some definitions and results helpful in our further thought (Section 2).In section 3, we define t-norm and t-conorm products on IFAV-sets of an IFS-set.Next, using these products we present our investment model based on an IFS-set (Section 4).Finally, we give the application of an IFS-set in investment problems and the feasibility of our proposed IFS-set based investment model in practical application is shown by some numerical examples (Section 5).

Preliminary notes
Let U be a universe and E be a set of parameters.Definition 2.1.[15] Let P(U) denotes the power set of U and AE.A pair (F, A) is called a soft set over U, where F is a mapping given by F: A P(U).

Definition 2.2. [21]
A fuzzy set X on U is a set having the form    then X becomes an ordinary (crisp) set.We denote the class of all fuzzy sets on U by FS(U).

An intuitionistic fuzzy set (briefly, IF-set) Y over U is defined as the object of the form
The class of all IF-sets on U is denoted by IFS(U).

Definition 2.6. [11]
A pair (F, A) is called an IFS-set over U, where F is a mapping given by F: AIFS(U).
An IFS-set is a parameterized family of intuitionistic fuzzy subsets of U, thus, its universe is the class of all IF-sets of U, i.e.IFS(U).For any aA, F(a) is referred to as the intuitionistic fuzzy approximate value set (briefly, IFAV-set) of the parameter a and it is actually an IF-set on U, it can be written as ,  is the fuzzy membership value of that object u holds on parameter a and () () Fa u  is the fuzzy non-membership value of that object u does not hold on parameter a. Simply, we denote

t-norm and t-conorm products on PFIFSS (F, A)
In this section, we define the t-norm and t-conorm products on   where The set , where The set where )  is a fuzzy set, which is called a t-norm-decision fuzzy set (briefly, t-ndfs) over U.
Similarly, we can be obtained t-conorm-decision fuzzy set (t-cndfs)  

IFS-set based investment model
In this section, we present our investment model based on an IFS-set.

Algorithm 1.
Step1.Input the (resultant) IFS-set (F, A) Step2.Input the preference of investment factors 12 , ,..., n a a a A  by the investor.

n F a F a F a PFIFSS F A 
Step4.Compute the t-ndfs (t-cnsfs) and present it in tabular form.
Step5.The optimal decision is to select uk if corresponding membership value Step6.If uk has more than one value then any one of uk may be chosen.

Application in investment decision making
In order to apply the concept of IFS-set to investment decision making problem, we consider the following factors that influence their investment decision and also the various avenues of investment they prefer.Also, we consider an IFS-set (F, A) can be represent as in Table 1.
If the preference of investment factors by the investor Mr. X is high returns (b), maximum profit in minimum period(c) and tax concession (e), then we have the t-ndfs as in Table 3.Here I3 has the largest membership value 0.1059 and hence the Insurance is best suits the requirement of investor Mr. X.

Advantages
By using Algorithm1, we might acquire the less option of objects, this can help us settle on the investment decision all the more effortlessly.Then again, by using Algorithm1, we get more far reaching data; this will help the decision of leaders.Also, our Algorithm1 is affected by the non-membership degrees of elements on decision making.

Conclusion
In the real life situations there are vast numbers of problems that warrant rational, logical and scientific decisions that fit best for the accomplishment of desired objective.The concept of IFS-set has rich potentials for developing such decision making models suitable for personal, social, technical, commercial and managerial issues.In this research work, we characterize t-norm and t-conorm products on the intuitionistic fuzzy approximate value sets of an IFS-set and applying these products we introduce an adjustable investment model based on an IFS-set, for investment decision in an uncertain situation and also, we show the effect of non-membership degrees of elements on decision making.By applying these products, we can see that it can be connected to numerous fields that contain questionable ties.We trust this investigation along this field can be preceded.In the future, the methodology ought to be more extensive to tackle related issues, for example, computer science, software engineering, current life state and so on.


a = Safety of funds  b = High returns  c = Maximum profit in minimum period  d = Easy accessibility  e = Tax concession  f = Minimum risk of possession  g = Stable return 5.2.Investment avenues Following are the investment avenues which are mostly preferred by the sample respondents  I1 -Bank Deposit  I2 -Insurance  I3 -Postal Savings  I4 -Shares and Stocks  I5 -Mutual Fund  I6 -Gold To apply IFS-set to this investment decision problem, consider the various investment avenues as the universal set U = {I1, I2, I3, I4, I5, I6} and the factors influencing investment decision as the set of parameters E http://www.ispacs.com/journals/jfsva/2016/jfsva-00343/International Scientific Publications and Consulting Services = {a, b, c, d, e, f, g} and let A = {a, b, c, d, e}  E.

Table 3 :
Table for  

Table 2 ,
Here I2 has the largest membership value 0.578 and hence in the case Insurance is the best suits the requirement of investor Mr. X.http://www.ispacs.com/journals/jfsva/2016/jfsva-00343/I2has the largest membership value 0.1059; hence Insurance is the best suits the requirement of investor Mr. X.This example shows the effect of non-membership degrees of elements on decision making.In the same manner the choice of investment avenue of any investor can be arrived at depending on any set of factors preferred by such investor.Some of such are given below: http://www.ispacs.com/journals/jfsva/2016/jfsva-00343/International Scientific Publications and Consulting Services membership grade value of each element in universal set is somewhat littler than by applying t-conorm product.
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