Global Journal of Computer Science and Technology, D: Neural & Artificial Intelligence, Volume 22 Issue 1

� ∈ 2 = � ∈ 2 ∀ ∈ \{0, + + 1}, ∈ , ≠ , ≠ (6) Equations 7, and 8 have integrated time constraints, subtour elimination and load constraints. + ( + ) − (1 − ) ≤ , ∀ ∈ , ∈ 1 , ∈ 2 (7) + − (1 − ) ≤ , ∀ ∈ , ∈ 1 , ∈ 2 , ≠ (8) Now, if there is an order placed between two nodes and the pickup node must be visited before the delivery node, then equation 9 shows it. ≥ ( + ) + , ∀ ∈ , ∈ _ , ∈ _ (9) Equation 10 shows time constraint while (11) shows capacity bound constraint. ≥ ≥ , ∀ ∈ \{0, + + 1}, ∈ (10) ( + , ) ≥ ≥ (0, ), ∀ ∈ , ∈ (11) Now showcasing the constraint of limiting number of vehicles used and maximum working duration in equations 12, and 13. � ∈ � ∈ 0 ≤ (12) ≥ + +1 − 0 , ∀ ∈ (13) This mathematical model is a small-scale solution. Because it being a population based metaheuristic solution, it is to be expanded step by step. Once the pheromone value is successful in small scale, the forging behaviour of ants could be applied to VRP in large scale for large number of vehicles. IV. A pproach to the S olution In this paper, the Vehicle Routing Problem with Time Window constraint has been resolved using a modified version of the Ant Colony Optimization using K- Means Clustering. Marco Dorigo was the first person to introduce Ant Colony Optimization, in the 90s, in his Ph.D. thesis. The solution algorithm is based on the behaviour of ants, the way they live in colonies and search for food. While an ant goes around, searching for food, it leaves behind pheromones that act as a beacon. It acts as a communication mechanism and each time the ant leaves a pheromone trail, it tells the other ants about the quality and quantity of food the former ant had been carrying. This way, there are several set paths that the ants use based on the number of pheromones released in a path. The shortest and fastest route is chosen for maximum traffic. . Figure 2shows process flow of the proposed solution. The purpose of the optimization model was to reduce specific energy consumption by taking into account the diameter of the pipe, the number of sprinklers, the working pressure of the end sprinklers along the pipeline, and the coordination conditions of the pump pipeline. A pressure drop between adjacent sprinklers has been added to the heuristic function of ACO's design to reflect the distance between the two cities of the Traveling Salesman Problem (TSP). Global Journal of Computer Science and Technology Volume XXII Issue I Version I 52 ( )D Year 2022 © 2022 Global Journals Vehicle Routing Problem with Time Window Constrain using KMeans Clustering to Obtain the Closest Customer