Interval Valued Intuitionistic Fuzzy Soft Multi Set Theoretic Approach to Decision Making Problems

Interval Valued Intuitionistic Fuzzy Soft Multi Set Theoretic Approach to Decision Making Problems

Abstract—In recent years the application of soft set in decision making problems has been developed rapidly since it can be applied easily to several areas like computer science, information technology, medical science, economics, environments, engineering, among other areas. In this paper, we give the application of interval-valued intuitionistic fuzzy soft multisets in real life decision making problems and proposed an algorithm to solve multi weighted interval valued intuitionistic fuzzy soft multiset based decision making problems by using weighted choice values. The feasibility of our proposed algorithm in practical applications is illustrated by a numerical example.

Keywords—soft set; level soft set; weighted function; interval valued intuitionistic fuzzy soft sets; interval valued intuitionistic fuzzy soft multi set; decision making.

I. Introduction

The concept of soft set theory was first initiated by Molodstov [18] in 1999 as an important mathematical tool for dealing with vagueness, uncertainties and not clearly defined objects. Some new algebraic operations and results on soft set theory defined in [[2], [17]]. Adding soft sets [12] with fuzzy sets [15] and intuitionistic fuzzy sets [5], Maji et al. [13-16] defined fuzzy soft sets and intuitionistic fuzzy soft sets and studied their basic properties. As a generalization of soft set, Alkhazaleh and others [[1], [4], [7], [8], [26]] defined the notion of a soft multi set and its basic algebraic structures and general topological structures were studied.

In 2007, Roy and Maji [21] presented a novel method to cope with fuzzy soft sets based decision making problems. Kong et al. [8] mentioned that the Roy-Maji algorithm [13] was wrong and they introduced a revised algorithm. Feng et al. [[9], [10]] studied the validity of the Roy-Maji algorithm [21] and mentioned that the Roy-Maji Algorithm [13] has some limitations. Also, they proposed an adjustable approach to fuzzy soft sets based decision making problems by using thresholds and choice values and gave the application of level soft sets in decision making based on interval-valued fuzzy soft sets. Jiang et al. [11] studied interval-valued intuitionistic fuzzy soft sets and their properties. There after Zhang et al. [24] presented a novel approach to interval-valued intuitionistic fuzzy soft set based decision making.

In 2012, Alkhazaleh and Salleh [3] initiated the notion of fuzzy soft multi set theory as a generalization of soft multi set theory and presented its application in decision making using Roy-Maji Algorithm [21]. As a generalization of fuzzy set theory [23], intuitionistic fuzzy set theory [5] and interval-valued intuitionistic fuzzy set theory [6] makes descriptions of the objective more realistic, practical and accurate. Mukherjee and Das [19] introduced the concepts of intuitionistic fuzzy soft multi sets and studied intuitionistic fuzzy soft multi topological spaces in detail. Mukherjee et al. [20] also introduced the concepts of interval valued intuitionistic fuzzy soft multi sets and studied their relation in details.

In this study, we have proposed an algorithm to solve multi weighted interval valued intuitionistic fuzzy soft multiset based decision making problems by using weighted choice values. The feasibility of our proposed algorithm in practical applications is illustrated with a numerical example.

II. Preliminary Notes

In this present section, we briefly recall some basic notions of soft set, interval valued intuitionistic fuzzy set, interval valued intuitionistic fuzzy soft multi set and level soft set. Suppose that, U be an initial universe and E be a set of parameters. Also, let P(U) denotes the power set of the universe U and AE.

Definition 2.1 ([20]). A pair (F, A) is said to be a soft set over the universe U, where F is a mapping given by F: Aï‚® P(U).

Definition 2.2 ([6]). An interval valued intuitionistic fuzzy set (in short IVIF-set) A over a universe set U is defined as the object of the form where M_Y(x): UINT([0,1]) and N_Y(x): UINT([0,1]) are functions such that the condition: xU, 0 sup M_Y(x)+sup N_Y(x)1 is satisfied (where INT[0,1] is the set of all closed intervals of [0,1]).

Definition 2.3 ([24]). Suppose that be a set of universes, such that and let for each , be a sets of decision parameters. Also, let where is the set of all interval valued intuitionistic fuzzy subsets of , and . A pair is said to be an interval valued intuitionistic fuzzy soft multiset (in short IVIF-soft multiset) over the universe U, where F is a function given by F: Aï‚® U, such that

For illustration, we consider the following house, car and hotel example.

Example 1. Let us consider three universes U₁= {h_1, h_2, h₃}, U₂= {c_1, c_2, c₃} and U₃= {v_1, v_2, v₃} are the sets of houses, cars and hotels respectively and let

be the sets of respective decision parameters related to the above three universes. Let , and , such that

Assume that, Mr. X wants to buy a house, a car and rent a hotel with respect to the three sets of decision parameters as in above. Suppose the resultant IVIF-soft multiset be given in TABLE I.

1. ivif-Soft Multiset (F, A)

Definition 2.4 ([28]). Suppose that, = (F, A) be an IVIF-soft set over U, where AE and E is the parameter set. Let : A INT[0,1]INT[0,1] be an IVIF-set in A, which is called a threshold IVIF-set. The level soft set of  with respect to  is a crisp soft set L(;) = (F_, A) defined by F_(e) = {uU: [^L_F(e)(u),^U_F(e)(u)]  [^L_(e),^U_(e)] and [^L_F(e)(u),^U_F(e)(u)]  [^L_(e),^U_(e)] }, eA.

According to the definition, four types of special level soft set as Mid-level soft set L(; mid), Top-Bottom-level soft set L(; topbottom), Top-Top-level soft set L(; toptop) and Bottom-bottom-level soft set L(; bottombottom) are defined in [11].

III. Multi Weighted ivif-Soft Multiset

In this present section, we introduce the concept of multi weighted IVIF-soft multiset and examine its application for decision making problems. If we allow the parameters to have different multi weights, then the multi weighted version of the IVIF-soft multiset can be defined as follows.

Definition 3.1.

A multi weighted IVIF-soft multiset  is a triple where (F, A) is an IVIF-soft multiset over U and is a multi weight function, where I = [0,1], specifying the weight w_k=(a_k) for each attribute a_kA and a triple is called a U_i-weighted IVIF-soft multiset part of , where is a U_i - IVIF-soft multiset part of (F, A) and is a weight function.

Example 2. If we consider the IVIF-soft multiset be (F, A) as in TABLE I and suppose that Mr. X has imposed the following weights for the parameters in A: for the parameters

Then we have a weighted  for an IVIF-soft multiset (F, A), where and the IVIF-soft multiset (F, A) is changed into a multi weighted IVIF-soft multiset  = with its tabular representation as in TABLE II.

2. Multi Weighted ivif-Soft Multiset 

3. Weighted U₁ –ivif-Soft Multiset Part of 

IV. Multi Weighted ivif-Soft Multiset Theoretic Approch to Decision Making Based on Zhang’s Algorithm

A. Zhang et al’s algorithm based on weighted choice values

Zhang et al. [28] used the following adjustable approch to weighted IVIF-soft set based decision-making by using weighted choice values.

Algorithm 1(Zhang’s Algorithm).

1. Input a weighted IVIF-soft set  = <F, A, >
2. Input a threshold IVIF-set : AINT[0,1]INT[0,1] for decision making.
3. Compute the level soft set L(;) of  with respect to the threshold IVIF-set .
4. Present L(;) in tabular form and obtain the weighted choice value S_i of u_iU, i.
5. The final optimal decision is to select u_k if S_k = max_i S_i.
6. If k has more than one value then any one of u_k may be chosen.

B. Application of IVIF-soft multisets in decision-making problems

In this section, we propose an algorithm (Algorithm 2) for IVIF-soft multi sets based decision making problems, using Zhang’s Algorithm [7], as described above. In the following, we have to show our algorithm (Algorithm 2):

Algorithm 2.

1. Input a multi weighted IVIF-soft multiset  =(F, A, )
2. Apply Zhang’s Algorithm to the first weighted IVIF-soft multiset part in  to get the decision S_k1.
3. Modify the weighted IVIF-soft multiset  by keeping all values in each row where S_k1 is maximum and replacing the values in the other rows by zero, to get ₁.
4. Apply Zhang’s Algorithm to the second weighted IVIF-soft multiset part in ₁ to get the decision S_k2
5. Modify the multi weighted IVIF-soft multiset ₁ by keeping the first and second parts and apply the method in step (3) to the third part.
6. Apply Zhang’s Algorithm to the third weighted IVIF-soft multiset part in ₂ to get the decision S_k3.
7. Continuing in this way we get the final optimal decision (S_k1, S_k2, S_k3……).

Remark 1. In the step (7) of our algorithm (Algorithm 2), if there are too many optimal choices obtained, then decision maker may go back to the step (2) as in our algorithm (Algorithm 2) and replace the level soft set (decision criterion) that he/she once used to adjust the final optimal decision.

C. Application in decision-making problems

Let us consider the decision making problem involving the multi weighted IVIF-soft multiset  with its tabular representation given by TABLE II. If we deal with this problem by mid-level decision rule, we shall use the mid-threshold of weighted U₁- IVIF-soft multiset part in  and we the mid-level soft set of weighted U₁ - IVIF-soft multiset part in  with weighted choice values with tabular representation is in TABLE IV.

4. Mid-Level Soft Set of Weighted U₁-ivif-Soft Multiset Part of  with Weighted Choice Values

From TABLE IV, it is clear that the maximum weighted choice value is 2.2, scored by h₃. Now we redefine the weighted interval valued intuitionistic fuzzy soft multi set  by keeping all values in each row where h₃ is maximum and replacing the values in the other rows by zero, to get ₁.

5. Multi Weighted ivif-Soft Multiset ₁

Now we apply Zhang’s Algorithm to the second weighted IVIF-soft multiset part in ₁ to take the decision from the availability set U₂. The tabular representation of the second resultant weighted IVIF-soft multiset part of ₁ will be as in TABLE VI.

6. Mid-Level Soft Set of Weighted U₂-ivif-Soft Multiset Part in ₁ with Weighted Choice Values

From TABLE VI, it is clear that the maximum weighted choice value is 1.8, scored by c₃. Now we redefine the weighted IVIF-soft multiset ₁ by keeping all values in each row where c₃ is maximum and replacing the values in the other rows by zero, to get _2.

7. Multi Weighted ivif-Soft Multiset ₂

Now we apply Algorithm 1 to the third weighted IVIF-soft multiset part in ₂ to take the decision from the availability set . The tabular representation of the third resultant weighted IVIF-soft multiset part of ₂ is as in TABLE VIII.

8. Mid-Level Soft Set of Weighted U₃-ivif-Soft Multiset Part in ₂ with Weighted Choice Values

From the TABLE VIII, we have to seen that the maximum weighted choice value is 0.8, by v_2. Thus from above results, the final optimal decision for decision maker Mr. X is (h₃, c₃, v₂).

Remark 2. From the above illustration, we have seen that our algorithm (Algorithm 2) is too simple and less computation. We have to consider only weighted choice values of objects in thresholds of weighted IVIF-soft multiset part. Also, our algorithm (Algorithm 2) is an adjustable algorithm, because the level soft set (decision rule) used by decision makers, which are changeable. For example, if we take top-level decision criterion in step (2) of our algorithm (Algorithm 2), then we have the weighted choice value of each object in the top-level soft set of weighted IVIF-soft multiset parts in the multi weighted IVIF-soft multiset, if we take another decision rule such as the mid-level decision criterion, then we have weighted choice values from the mid-level soft set of weighted IVIF-soft multiset parts in the multi weighted IVIF-soft multiset. Generally, the weighted choice values of a same object in the mid-level decision rule and in the mid-level decision rule are need not coincide.

V. Conclusion

In this research work, the notion of multi weighted IVIF-soft multiset is to be defined and also, we propose an adjustable approach to multi weighted IVIF-soft multiset based decision making by using weighted choice values and illustrate this algorithm with a numerical example. In our algorithm, a multi weighted IVIF-soft multiset is converted into a crisp soft set for solving decision making problems after considering certain opinion weighting vectors and thresholds. This makes our algorithm simpler and easier for real life practical applications. The feasibility of our proposed algorithm in real life practical problems is illustrated with a numerical example.