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Complex Intuitionistic Fuzzy Soft Lattice Ordered Group and Its Weighted Distance Measures

: In recent years, the complex fuzzy set theory has intensiﬁed the attention of many researchers. This paper focuses on developing the algebraic structures pertaining to lattice ordered groups and lattice ordered subgroups for complex intuitionistic fuzzy soft set theory. Furthermore, some of their properties and operations are discussed. In addition, the weighted distance measures between two complex intuitionistic fuzzy soft lattice ordered groups such as weighted hamming, weighted normalized hamming, weighted euclidean and weighted normalized euclidean distance measures were introduced and also some of the algebraic properties of the weighted distance measures are veriﬁed. Moreover, the application of complex intuitionistic fuzzy soft lattice ordered groups by using the weighted distance measures is analysed.


Introduction
Vagueness and various uncertainties are characteristics that are pervasive in problems which occuring in engineering, medical science, economics, environments, etc. To exceed these uncertainties and vagueness, some kinds of theories were given like fuzzy set theory [1], intuitionistic fuzzy set theory [2], soft set theory [3] and intuitionistic fuzzy soft theory [4]. Although all of these theories posed a challenge to handle the periodicity or seasonality that to be in many real life problems. This led to Daniel Ramot [5] to present a new innovative concept called complex fuzzy sets.
Complex fuzzy sets are able to handle the problems that are either very difficult or impossible to address with one-dimensional grades of membership. Since then, Kumar T. [6] found an application in multicriteria decision making problem on the basis of the proposed complex intuitionistic fuzzy soft sets and the notion of complex intuitionistic fuzzy soft sets was conferred by Alkouri [7]. Subsequently, Selvachandran G. and Quek S.G. [8] introduced and developed the notion of complex intuitionistic fuzzy soft groups. Distance measure is a numerical measurement between two objects. Moreover distance measure is an important issue on fuzzy sets, soft sets and other hybrid structures. The distance measure and similarity measure for the hybrid structures of fuzzy sets such as fuzzy soft sets [9], intuitionistic fuzzy sets [10], intuitionistic fuzzy soft sets [11], interval-valued complex fuzzy sets [12], complex vague soft sets [13] and complex intuitionistic fuzzy sets [14] were introduced. On the other hand, the lattice ordered algebraic systems play an important role in algebra. The concept of lattice theory was originated by Birkhoff [15]. Satya Saibaba G.S.V. and Vimala J. [16,17] introduced the study of fuzzy lattice ordered groups in a different manner. The combination of fuzzy sets and lattice ordered stuctrures may provide more new interesting topics [18][19][20] which intensified the attention of many reseachers and decision makers. Here, we introduce and develop the theory of complex intuitionistic fuzzy soft lattice ordered group (CI F SL-G). The main contribution of this study includes 1.
A concept of lattice ordered algebraic structures of complex intuitionistic fuzzy soft sets is originated.

2.
CIF SL-G's pertinent properties are obtained with its operations such as union, intersection, complement, AND.

3.
Weighted distance measures between CIF SL-Gs, namely the weighted hamming, weighted normalized hamming, weighted euclidean and weighted normalized euclidean distance measures are introduced. It is possible to handle the seasonality and two-dimensional problems with membership and non-membership grades through CIF SL-G. 4.
The application of weighted distance measures between CIF SL-Gs to find the best work stream for employees is analysed.
The remaining sections of this paper are organized as follows. In Section 2, the important concepts are recapitulated and presented. In Section 3, we introduced the notion of CIF SL-G and find the other supporting properties based lattice ordered group structure. Furthermore explore some operations on CIF SL-G. In Section 4, the axiomatic definition of the distance function is presented and also the weighted distance measures between CIF SL-Gs are introduced. Subsequently, some of the algebraic properties of these weighted distance measures are also utilizied. We find an application of CIF SL-G using weighted distance measures on CIF SL-Gs.

Preliminaries
In this section, we recapitulate some of the important preliminaries pertaining to the development of CIF SL-G. 1] which is called the membership function of X and µ(h) is called the membership value of H in X.

Definition 3 ([16]).
A fuzzy subset λ of a lattice-ordered group G is said to be a fuzzy lattice-ordered subgroup or briefly, fuzzy -subgroup if

Definition 4 ([5]).
A complex fuzzy set A defined on a universe of discourse U is characterized by a membership function µ A (x) that assigns a complex-valued grade of membership in A to any element x ∈ U. By definition, all values of µ A (x) lie within the unit circle in the complex plane and are expressed by µ A (x) = r A (x)e iw A (x) , where i = √ −1, r A (x) and w A (x) are both real-valued, r A (x) ∈ [0, 1] and w A (x) ∈ (0, 2π]. A complex fuzzy set A is thus of the form ). A complex intuitionistic fuzzy set S, defined on U, is characterized by membership and non-membership functions µ S (x) and ν S (x) respectively, that assign any element x ∈ U a complex-valued grade of both membership and non-membership in S. By definition, the values of µ S (x), ν S (x), and their sum may receive all lying within the unit circle in the complex plane, and are on the form µ S (x) = r S (x)e iw µ S (x) for membership function, ν S (x) = k S (x)e iw ν S (x) for non-membership function, where i = √ −1 each of r S (x) and k S (x) are real valued and both belong to the interval [0, 1] such that 0 ≤ r S (x) + k S (x) ≤ 1 also, w µ S (x) and w ν S (x) are real valued. We represent the complex intuitionistic fuzzy set S as,

Definition 6 ([6])
. Let E be a set of parameters, CIFS(U) denote the collection of all complex intuitionistic fuzzy sets on U, and F be a function from E to CIFS(U). Then the set of ordered pairs {(ε, F(ε)) : ε ∈ E, F(ε) ∈ CIFS(U)}, denoted by ( F, E), is called a complex intuitionistic fuzzy soft set (CIFSS) on U.
is called the union of ( F 1 , E 1 ) and ( F 2 , E 2 ) and is denoted as is called the intersection of ( F 1 , E 1 ) and ( F 2 , E 2 ) and is denoted as

Definition 9. Let G be a lattice ordered group and Let
be a complex intuitionistic fuzzy set on G. Then N is said to be a complex intuitionistic fuzzy lattice ordered subgroup (CI F L-Subgroup) of G, if the following conditions holds for all x, y ∈ G: is non null} is the support set of ( F, E).

Example 1.
Consider the − group G = (Z, +, ∧, ∨) and the parameters E = {a, b}. Next Consider the two CIFSS of G, which are defined below.
. Then for all a ∈ E and x, y ∈ G the following are equivalent,

y)and
Hence . Then for all a ∈ E and x, y ∈ G the following are equivalent, The proof is similar to Proposition 2.
From the above, result is obvious.
Proposition 5. Let ( F, E) ∈ CIF SL-G(G) and e G be the identity element of G. Then the following results holds for all x ∈ G, a ∈ E For x, y, e G ∈ G and a ∈ E 1 ∩ E 2 Similarly, we can prove Thus, by Proposition 1 ( F 1 , E 1 ) ∩ ( F 2 , E 2 ) ∈ CIF SL-G(G).
(i) Straight forward from Definition 11.
Proof. It is obvious that all the weighted distance measures given in Definition 13 satisfy conditions (i)-(iii) of Definition 12. We have to prove the condition (iv) for the weighted Hamming distance. By Using Definition 8 Let us take G = {x 1 , x 2 , x 3 , x 4 }. Here, the lattice structure comes under the team's personal preference to the work stream in the decision as well as indicate their equal degree of importance defined by x i x j ⇒ x i = x j for i, j ∈ {1, . . . , 4}. Hence (G, * ) is a -group. The parameters are taken as A = {a 1 = Special and unique bene f its, a 2 = Friendly place to work, a 3 = Contribution to the community, a 4 = Accomplish and pride, a 5 = Honest management} For instance, suppose the team of analysts surveys that 70% of the employees of x 1 believe that x 1 is suitable for the first attribute; and 10% of the them believe that x 1 is poor for the first attribute, in which this process is utilized to calculate the amplitude terms for both membership and non membership functions respectively. The phase terms that the range of attribute for the present date for first attribute of x 1 can be given as follows: if 40% employees of x 1 believe that the present date of x 1 is suitable for the first attribute; and 30% of them believe that it is poor. So the first attribute for x 1 can be represented as (0.7e i2π(0.4) , 0.1e i2π(0.3) ). The weight vector of the each attribute is w j = {0.2, 0.4, 0.2, 0.2, 0.5} which is based on the employees priorities to the attribute in the decision making.
Next, we list the detailed steps involved in the weighted distance measures based decision making process.
Step 3: The Table 4 involves the ranking of the weighted distance measures obtained.

Discussion of the Results and Comparative Study
We obtained the weighted distance measures from Table 4. Clearly, the best alternative is x 2 , which is the one with the lowest distance to the ideal choice. It is determined that x 2 is graded as first, x 4 is graded as second, x 1 is graded as third and x 3 is graded as fourth. The result is shown in Figure 1. In Table 2, the phase term of the membership and non-membership values gives the ability to consider values in more accurate compared to fuzzy sets and soft sets and other hybrid structures. The amplitude term gives the ability to deal with periodicity information of membership and non-membership function. On the other hand, different weights lead to different results. In this application, employee indicate their degree of importance to the attributes through the weights. Hence the results obtained by using Definition 13 taken into account the preference to the parameters with the CIF SL-G structure. The results of this application help to identify the gaps and improve their work place culture.
Among the existing methods in literature that is closed to the weighted distance measure on CIF SL-G is the distance measure on complex intuitionistic fuzzy soft sets (CIFSS) [6]. However, using the distance measure on CIFSSs is unfeasible which is discussed above. For instance, in our scenario, intuitionistic fuzzy soft sets (IFSS) [4] and generalized intuitionistic fuzzy soft sets (GIFSS) [9] can not represent the phase term, which is range at the present date. From this analysis, the weighted distance measure and CIF SL-G structure enables to overcome the problems afflicting in CIFSS, IFSS and GIFSS.

Conclusions
The complex intuitionistic fuzzy soft lattice ordered groups (CI F SL-G) is a hybrid structure of CIFSS and Lattice ordered group. We found the notion of CIF SL-G and its supporting properties. Furthermore, examined the operations on CIF SL-G. Furthermore, we introduced the weighted distance measures and verified some properties. In addition, we demonstrated an application of CIF SL-G to find the best work stream for employees which is accomplished by weighted distance measure between CIF SL-Gs. As a future work, we plan to extend the theory by introducing morphisms and some more operations on CIF SL-G and conjointly planned to contribute some real life applications.