# Search of the Shortest Path in a Communication Network with Fuzzy Cost Functions
## 4 Dijkstra Algorithm for Type V Fuzzy Graph
### Arithmetic Operations of Normalized Triangular Fuzzy Numbers  
For the arithmetic operations of fuzzy numbers, the \(\alpha\)-cut method is used. This method is quite simple (in comparison to the Extension Principle method) since the operation between fuzzy numbers is reduced to the operation of ordinary intervals in R.  
First, two essential properties for the operations of triangular and trapezoidal fuzzy numbers are considered:  
1. The results from addition and subtraction between triangular or trapezoidal fuzzy numbers are also a triangular or trapezoidal fuzzy number, respectively.
2. The results from multiplication or division, as well as maximum or minimum operations in triangular or trapezoidal fuzzy numbers, are not a triangular or trapezoidal fuzzy number, respectively.  
The \(\alpha\)-\(cut\) interval of a normalized triangular fuzzy number \(\tilde{A}\), denoted by \(\tilde{A}_{\alpha}\), is an ordinary crisp interval in R containing all the elements of \(\tilde{A}\) whose membership degree is greater than or equal to the specified value \(\alpha\in[0,1]\). Let \(\tilde{A}=(a,b,c)\) be a normalized triangular fuzzy number with its membership function defined in Equation (1), where \(f_{L}(x)=\dfrac{x-a}{b-a}\) and \(f_{R}(x)=\dfrac{c-x}{c-b}\), \(\tilde{A}_{\alpha}\) is defined as in Equation (2).  
\[\tilde{A}_{\alpha}=\left[(b-a)\alpha+a,-(c-b)\alpha+c\right]\qquad\alpha\in[0, 1], \tag{2}\]  
The following two properties of fuzzy numbers are based on the representation of a fuzzy set and the definition of the \(\alpha\)-cut of a fuzzy set:  
1. Each fuzzy set, and thus each fuzzy number, can fully and uniquely be represented by its \(\alpha\)-cut.
2. \(\alpha\)-cuts of each fuzzy number are closed intervals of real numbers for all \(\alpha\in[0,1]\).  
Definition 3 provides the necessary elements to define the \(\alpha\)-cut of the fuzzy number \((\tilde{A}*\tilde{B})\) by the \(\alpha\)-cuts of the fuzzy numbers \(\tilde{A}\) and \(\tilde{B}\). Thus, the operation of standard crisp intervals can be used to find \((\tilde{A}*\tilde{B})_{\alpha}\).  
Definition 3: Let \(\tilde{A}\) and \(\tilde{B}\) denote fuzzy numbers and let \(*\) denote any of the four basic arithmetic operations \"\(+,-,\cdot,\cdot\)\". Then for any \(\alpha\in(0,1]\), the fuzzy set on \(\mathbb{R}\), \((\tilde{A}*\tilde{B})\) is defined by its \(\alpha\)-cut \((\tilde{A}*\tilde{B})_{\alpha}\):  
\[(\tilde{A}*\tilde{B})_{\alpha}=\tilde{A}_{\alpha}*\tilde{B}_{\alpha},\]  
For \(*\) being \"\(+\)\", clearly, \(\tilde{B}_{\alpha}\
otin 0\)\(\forall\alpha\in(0,1]\).  
According to First Decomposition Theorem and by using the definition of the union of fuzzy sets, the membership function of \((\tilde{A}*\tilde{B})\)[28] is computed. Our interest lies on the addition and subtraction of triangular fuzzy numbers. Thus, let \(\tilde{A}=(a_{1},a_{2},a_{3})\) and \(\tilde{B}=(b_{1},b_{2},b_{3})\) be two triangular fuzzy numbers, if we perform the addition and subtraction of their \(\alpha\)-cuts we obtain:  
\[\begin{array}{ll}\text{Addition: }\tilde{A}_{\alpha}+\tilde{B}_{\alpha}&=[(a_{2}-a_{1} +b_{2}-b_{1})\alpha+a_{1}+b_{1},(a_{2}-a_{3}+b_{2}-b_{3})\alpha+\\ &+a_{3}+b_{3}]\\ &=(\tilde{A}+\tilde{B})_{\alpha}\end{array} \tag{3}\] \[\begin{array}{ll}\text{Subtraction: }\tilde{A}_{\alpha}-\tilde{B}_{\alpha}&=[(a_{2}-a_{1} +b_{1}-b_{2})\alpha+a_{1}-b_{1},(a_{2}-a_{3}+b_{3}-b_{2})\alpha+\\ &+a_{3}-b_{3}]\\ &=(\tilde{A}-\tilde{B})_{\alpha}\end{array} \tag{4}\]  
The triangular fuzzy number \(\tilde{A}+\tilde{B}\) (and \(\tilde{A}-\tilde{B}\)) is then found by making \(\alpha=0\) and \(\alpha=1\) in expression (3) (and expression (4)), i.e.,  
\[\begin{array}{ll}\tilde{A}_{0}+\tilde{B}_{0}=[a_{1}+b_{1},a_{3}+b_{3}]\\ \tilde{A}_{1}+\tilde{B}_{1}=[a_{2}+b_{2}]\end{array}\qquad\Rightarrow\quad \tilde{A}+\tilde{B}=(a_{1}+b_{1},a_{2}+b_{2},a_{3}+b_{3})\end{array}\]and  
\[\tilde{A}_{0}-\tilde{B}_{0} =[a_{1}-b_{1},a_{3}-b_{3}]\] \[\tilde{A}_{1}-\tilde{B}_{1} =[a_{2}-b_{2}]\]