Toward Computing Linguistic Fuzzy Graphs And Applying to Illegal Immigration Problem

In this paper, we study linguistic fuzzy graph properties which consist of fuzzy paths, cut vertex and bridge. We use hedge algebra and linguistic variables for modeling to reduce complexity in computation. Modeling the Illegal immigration problem is also introduced Received on 01 September 2020; accepted on 19 September 2020; published on 24 September 2020


Introduction
In everyday life, people use natural language (NL) for analyzing, reasoning, and finally, make their decisions. Computing with words (CWW) [9] is a mathematical solution of computational problems stated in an NL. CWW based on fuzzy set and fuzzy logic, introduced by L. A. Zadeh is an approximate method on interval [0,1]. In linguistic domain, linguistic hedges play an important role for generating set of linguistic variables. A well known application of fuzzy set is fuzzy graph [1,[5][6][7][8], combined fuzzy set with graph theory. Fuzzy graph (FG) has a lots of applications in both modeling and reasoning fuzzy knowledge such as Human trafficking, internet routing, illegal immigration [3] on interval [0,1] but not in linguistic values, However, many applications cannot model in numerical domain [9], for example, linguistic summarization problems [10]. To solve this problem, in the paper, we use an abstract algebra, called hedge algebra (HA) as a tool for computing with words. The remainder of paper is organized as follows. Section 2 reviews some main concepts of computing with words based on HA. Important section 3 studies a linguistic fuzzy graph modeling with words using HA and its properties. Section 4 presents an application of LG. Section 5 outlines conclusions and future work.

Hedge algebra
In this section, we review some HA knowledges related to our research paper and give basic definitions. First definition of a HA is specified by 3-Tuple HA = (X, H, ≤ ) in [11]. In [12] to easily simulate fuzzy knowledge, two terms G and C are inserted to 3-Tuple so HA = (X, The Truth and meaning are fundamental important concepts in fuzzy logic, artificial intelligence and machine learning. In RCT (restriction-centered theory) in [9], truth values are organized as a hierarchy with ground level or first-order and secondorder. First order truth values are numerical values whereas second order ones are linguistic truth values. A linguistic truth value, say , is a fuzzy set. We study linguistic truth values on POSET L whose elements are comparable [15].
Consists of a universe L ∅ together with an interpretation of: • each constant symbol c j from ρ as an element • each a i -ary function symbol f a i from ρ as a function: In HA, ∈ L and there are order properties: Theorem 2.1. in [11] let 1 = h n . . . h 1 u and 2 = k m . . . k 1 u be two arbitrary canonical representations of 1 and 2 , then there exists an index j ≤ {m, n} + 1 such that h i = k j , for ∀i < j, and: 3. 1 and 2 are incomparable iff h j x j and k j x j are incomparable; in which {V true, Ptrue, L true} stand for : very true, possible true and less true are linguistic truth values generated from variable truth. Assume propositions p = "Lucie is young is V true" and q = "Lucie is smart is Ptrue", interpretations on H are: • truth(p) = V true ∈ H, truth is a unary function.

Fuzzy graph model based on linguistic variables
The first FG (fuzzy graph) was introduced in [1], which vetices and edges's values are in unit interval [0, 1]. Many FG's theories were developed in [2,3] in which computational phases have a bit complex because converting from linguistic to number value to compute. To reduce complexity, we directly compute by applying computing with word method [9] Our graph model Our fuzzy graph is called LG (linguistic graph) with L is domain of both vertex V and E, see Fig. 2.
On any graph, it always have paths, cut vertices and bridge edges. Let u P ; v be a path between two vertices u and v 3. An edge e ∈ E is called fuzzy bridge if deleting e from LG reduces the strength between some pair of vertices.
4. An vertex w ∈ V is called fuzzy cut vertex if deleting w and adjacent edges to (or from w) from LG reduces the strength between some pair of vertices.
LG is the special case of FG on linguistic domain L so it have some common and separate properties. Immediately from Definition 3.2 we infer the following important property on LG

Immigration problem
Illegal immigration problem was introduced in [4]. People from Asia and Africa are seeking to enter the U. S. illegal over the Mexican border by six main routes as following: The size of flow from country to country is reported in linguistic terms very low, low, medium, high, very high [3] models as data table in Fig. 1:

Computing on LG
Computing on LG based on property 3.1 about cut vertices or bridge. Applying Theorem 2.1 by ordering 0 < very low < low < W < high < very high < 1 Example 3. From Fig. 2: x, x ∈ V, for example x = Russia then δ(China, Russia) = low and Conn LG−(China,Russia) (China, Russia) = W ∧ vl ∧ low = vl, so China is the cut vertex. For controling people flow to U. S., we should delete China cut vertex and so on.

5
Conclusions and future work