Multi-Objective Shortest Path Model for Optimal Route between Commercial Cities on America

The traditional shortest path problem is mainly concerned with identifying the associated paths in the transportation network that represent the shortest distance between the source and the destination in the transportation network by finding either cost or distance. As for the problem of research under study it is to find the shortest optimal path of multi-objective (cost, distance and time) at the same time has been clarified through the application of a proposed practical model of the problem of multi-objective shortest path to solve the problem of the most important 25 commercial US cities by travel in the car or plane. The proposed model was also solved using the lexicographic method through package program Win-QSB 2.0 for operational research applications.


Shortest Path Problem SPP
Let's consider that we have a flow network that has a certain starting point and a certain endpoint and that the arrows that connect the network points take many paths to reach the starting point and the access point. We try through the shortest path problem, to get the shorter of these arrows or lines that connect the starting point and the end point. For example, this model is used to obtain the shortest distance or path that can be taken between one city and another through a network of paths. The length of each arc is a function of that distance, travel time, cost of travel or any other measure. In other words, we try to find the shortest path between the start and access point. One of the most important issues in network flow is that of assigning the shortest path between the source node and the Ibrahim Iraqi Journal of Science, 2019, Vol. 60, No. 6, pp: 1394-1403 5952 destination node. Consider that we have the network composed of n the node (1, 2, …,n) so that each arc (i, j) corresponds to a non-negative number ij d , called distance, or transit time from node i to node j. If there is no direct route between i and j, the distance is: ij d   . The distance ij d can vary from a distance ji d (i.e. ij ji dd  (. The problem lies in how to find the length of the shortest path, and the shorter route, from the source node 1, to the destination node n. As one way to solve this problem, we can interpret it as a network represented by a guided graph G(V, E), where V represents the set of headers (nodes) and E represents the set of links (arcs). The link between node i and node j is expressed by (i, j)E; and ij d is the cost of the link between i and j; as well as pq ij x (whereas 0 1 pq ij x  ) represents the amount of traffic from node p V to node q V by routing it through (i, j) E [14].

2.1.
A General Model of Single Objective Shortest Path Problem SOSPP Generally, the problem of finding the shortest path can be formulated as a linear programming problem as shown below: x and ij d is the decision variable and the link cost (i, j), respectively [14]. The equation (3.1.1) represents the objective function that reduces the cost of the path from the node p to node q and ij x is the amount flow from node p to node q through (i, j). Equations

Multi-Objective Shortest Path Problem MOSPP
The MOSPP is one of the most important and common problems. The problem is how to find the shortest path of any network based on a certain set of objectives, like cost, time, distance, etc. Consider that the DM looks at how to choose the feasible shortest path to travel by car or plane to the most important 25 US cities, taking into consideration three main objectives: cost, distance and time. It can be said that the main objective of the DM is to find an efficient solution for MOSPP to find an of the most important US commercial cities.

The Proposed Mathematical Model to Solve MOSPP
In this paragraph, a proposed mathematical model will be formulated to solve MOSPP based on the original model of SOSPP mentioned above in paragraph 3.1. Let's consider that there are three objectives that the DM seeks to achieve together of travel using the car (the first, second and third objectives) is to minimization (cost Z 1 , distance Z 2 , and time Z 3 ), respectively. There are also three other objectives that the DM seeks to achieve in order to travel with the aircraft (Target 1, 2 and 3): There are many approaches used to solve the above model. We will use the lexicographic method that described in the below.

Lexicographic Method
In this method, the (DM) has to arrange the objective functions according to its importance. The preferred solution of the problem (2.1), in this case, is the solution that minimizes the vales of the most important objective functions as   12 ( ), ( ), ... , ( ) n z x z x z x which is a vector of objective functions which is arranged according to the importance of the functions of the (DM), such that z 1 (x) is the most important objective function among the other and z 2 (x) is the objective function followed by the importance and so on [15]. The first problem to be solved is: Find a vector that minimizes, Where xD  represents a feasible region solution, and * 1 x is a solution of this problem as well ; if this solution is unique then * 1 x will be a solution to a problem (2.1), but if there is more than one solution, the second problem that must be solved is: Find a vector x that minimizes, x represents a solution to this problem and x is a unique solution, the problem (2.1) has this solution, otherwise, this frequency is repeated until we get a unique solution to one of these problems or complete all the objective functions and * n x , which is the solution to the problem n, is the solution to the problem (2.1). In general, the problem that must be resolved are: Find a vector x that minimizes,

Application for Real Case Study
Consider that, there is a businessman who wants to take the shortest path by using the car or plane of the most important 25 US commercial cities to reach New York City from the hometown city Los Angeles, taking into account the minimization (cost, distance and time) together. Cost, distance and time data were collected among the major US commercial cities (https://www.distance-cities.com/).  Table-1 shows the names of the most important US commercial cities; Table-2 shows the cost,  distance and time of travel by car, and finally, Table-3 shows the cost, distance and time of travel by plane.       Where: Z 1 (x), Z 2 (x), Z 3 (x); represents the function of (cost, distance and time) respectively, when businessman traveling by car, and W 1 (x), W 2 (x), W 3 (x); represents the function of (cost, distance and time) respectively, when businessman traveling by plane. The equations from (5.1.7) to (5.1.31) represent the constraints of the problem to which we will refer briefly xD  .

Solve the Model of MOSPP by Using the Car or Airplane
In this paragraph, the above model will be resolved by using the package program (Win-QSB 2.0) [16]. Firstly, we will solve each objective function individually with the problem constraints as a standard linear programming model to obtain the values of Z 1 (x), Z 2 (x) and Z 3 (x), where the travel model by using the car; and the values of W 1 (x), W 2 (x) and W 3 (x) for the travel model by using the aircraft, respectively. So that we can use the Lexicographic method that mentioned above in paragraph 4.2 to find the final optimal solution for the travel either by car or by plane and as follows: