Network Problems with a New Representation of Triangular Fuzzy Rough Numbers

In this paper, we proposed algorithm for solving the fuzzy rough network problem, in this problem all parameters between different nodes are presented by a new representation of triangular fuzzy rough numbers. By using the proposed algorithm, a decision maker can obtain the optimal shortest path and optimal fuzzy rough shortest distance between source node and destination node. To illustrate algorithm a numerical example is solved and the obtained results are discussed


Introduction
In a network, the arc length may represent time or distance … etc. Conventionally, it is a summed to the crisp. However, decision makers find it challenging to specify the arc lengths. The shortest path problem focuses on determining the path with the shortest distance. Researchers have paid close attention to the shortest path problem in recent decades since it is significant in many applications such as communication, transportation, scheduling, and routing. Klein [1], introduced new models based on fuzzy shortest paths, as well as a general approach based on dynamic programming to solve the new models. For example, while using the same model to transfer data from node a to node b in a network, the data transmission time may vary. Fuzzy set theory, as presented by Zadeh [2], is frequently utilized to deal with uncertainty, and using membership function to describe uncertainties. The shortest path problem with fuzzy parameters was studied by Takahashi and Yamakami [3]. Nayeem and Pal [4] viewed a network with its arc lengths as an imprecise number, namely, interval number and triangular fuzzy number. Tajdin et al [5], introduced a novel strategy and algorithm for determining the shortest path in a mixed network with varying fuzzy arc lengths. Amit and Manjot [6], proposed a method for tackling network flow problems with imprecise arc lengths. To find the crucial path. Ravi Shankar et al [7], ‫األسمرية‬ ‫الجامعة‬ ‫مجلة‬ : ‫والتطبيقية‬ ‫األساسية‬ ‫العلوم‬ Journal of Alasmarya University: Basic and Applied Sciences employed a new defuzzification method for fuzzy numbers and applied it to the float time (slack time) for each activity in the fuzzy project network. Yakhchali and Ghodsypour [8], discussed the problems of finding probable values of an activity's earliest and latest starting times in networks with minimal time lags and imprecise durations represented by interval or fuzzy integers. In this paper, we present a algorithm for finding the shortest path and the fuzzy rough shortest distance in network problems. In this problem the network having various fuzzy rough arc distance, the remainder of the paper is organized as follows: In section 2, some basic definition, arithmetic operations of fuzzy rough are reviewed. In section 3, the network models and formulation of fuzzy rough shortest path problem presented, algorithm for solving it's, to illustrate algorithm numerical example is solved. The conclusions are discussed in section 4.

Fuzzy Rough interval
In this section, the definition of rough interval, fuzzy rough interval, fuzzy rough number and basic operations for triangular fuzzy rough numbers are given. For more details see [9 -12].

Ranking Function for triangular Fuzzy Rough Numbers [11,13]
A ranking function is a function ̃ where ̃ is a set of all triangular fuzzy rough numbers defined on set of real numbers, which maps each triangular fuzzy rough number into the real number. We proposed the ranking function for a triangular fuzzy rough

Network Models
There are lots of operations research situations that can be modeled and solved as networks, for example determination of the shortest path between two cities in a network of roads, also determination of the time schedule for the activities of a construction project. Definition 3.1 A network consists of a set of nodes related by arcs (or branches), we will use to describe the network notation, (N, G) where N is the set of nodes, and G is the set of arcs. As an illustration, the network in figure 1 is described as

Fuzzy Rough Network Problems
In usual network problems, it is given that decision maker is certain about the parameters (time, distance,…) between different nodes. But in real life situations, there always exist uncertainty about the parameters between different nodes. In this section we present new models for a network where the arc lengths between different nodes are triangular fuzzy rough number. The fuzzy rough network problem, determines the shortest path and fuzzy rough shortest distance between a source node and destination node in a network. The notations that will be used in the fuzzy rough network problem are as follows: : where is the set of nodes and is the set of arcs.
Nd (j): The set of all predecessor nodes of node .
̃ : The fuzzy rough distance between node and first (source) node.
̃ : The fuzzy rough distance between node and node .

Algorithm for arithmetic a shortest path.
A new algorithm is presented for finding the optimal shortest path and optimal fuzzy rough shortest distance between source node (say node 1) and destination node (say node n). The steps of the algorithm are as follows: Step 1. Suppose ̃ ̃ [ ] and ticket the source node (say node 1) as Step 2. Find ̃ { ̃ ̃ }.
Step 3. If minimum occurs corresponding to unique value of then ticket node as { ̃ }, If minimum occurs corresponding to more than one value of then it represents that there is more than one fuzzy rough distance between source node and node , but fuzzy rough distance along all paths is ̃ , so choose value of ( using step 2).
Step 4. Let the destination node (node ) be l ticketed as { ̃ } , then the optimal fuzzy rough shortest distance between source node and destination node is ̃ .
Step 5. Since destination node is ticketed as { ̃ } so to find the optimal shortest path between source node and destination node, check the ticket of destination node . Since the { ̃ } , which represents that we are coming from node , now check the ticket of node , and so on. Repeat the same procedure until node 1 is obtained. By the combining all the nodes then the optimal shortest path can be obtained. To illustrate the algorithm the numerical example presented.

Example 2:
Consider the following network Figure 2, the problem is to find the optimal shortest path between source node (say node 1) and destination node (say node 7) on the network, where the triangular fuzzy rough arc lengths are presented in Table 1.  ] } Iteration 7. Since node 7 is the destination node of the given network, now the optimal shortest path between source node 1 and destination node 7 can be obtained by using the following procedure: Since node 7 is ticketed by { [ ] } which represents that we are coming from node 6, check the ticket of node 6, Node 6 is ticketed by { [ ] } which represents that we are coming from node 5, check the ticket of node 5, Node 5 is ticketed by {[ ] } which represents that we are coming from node 2, check the ticket of node 2, Node 2 is ticketed by ] } which represents that we are coming from node 1. Now the optimal shortest path between node 1 and node 7 is obtained by joining all obtained nodes. Hence the optimal shortest path is , with optimal fuzzy rough shortest distance between source node 1 and destination node 7 is ̃ [ ] The shortest path and the corresponding fuzzy rough shortest distanced using algorithm are reported below: Optimal shortest path from node 1 to node 7 is: Also, the fuzzy rough optimal shortest distance from node 1 to node 7 is ̃ such that: ̃ [ ] and its membership function can be defined as: Also, the membership function of ̃ is shown in Figure 3. The fuzzy rough shortest distance and shortest path of all the nodes from node 1, and the ticketed of each node is shown in Table 2.  Figure 3, gives the paths between city 1 (node 1) and four other cities (node 2 to node 5), and the triangular fuzzy rough numbers arc lengths in miles are given in Table 3, The problem is to find the optimal shortest path between source node 1 and destination node 5 on the network.  Now the values of ̃ ( ̃ ̃ ) can be obtained as follows (see Table 3): Also, the membership function of ̃ is shown in Figure 4. The obtained result can be explained by follows: i. The surely optimal shortest distance between source node 1 and destination node 5 is [ ] [ ] ii. The possibly optimal shortest distance between source node 1 and destination node 5 is [ ] [ ]

CONCLUSION
An algorithm for computing a shortest path and fuzzy rough shortest distance between source node and destination node has been presented, also presented a new representation of triangular fuzzy rough numbers. Using the ranking function to compare fuzzy rough numbers in the algorithm. By using algorithm, we can find out fuzzy rough distance and shortest path of each node from the source node simultaneously, i.e., it is not required to apply the algorithm again and again for finding fuzzy rough distance and shortest path for a particular node from source node. We suggest using fuzzy rough numbers in network problems, because in real life there always exist uncertainty about the parameters between different nodes.