Vehicle routing problem

The VRP concerns the service of a delivery company. How things are delivered from one or more depots which has a given set of home vehicles and operated by a set of drivers who can move on a given road network to a set of customers. It asks for a determination of a set of routes, S, (one route for each vehicle that must start and finish at its own depot) such that all customers' requirements and operational constraints are satisfied and the global transportation cost is minimized. This cost may be monetary, distance or otherwise. The road network can be described using a graph where the arcs are roads and vertices are junctions between them. The arcs may be directed or undirected due to the possible presence of one way streets or different costs in each direction. Each arc has an associated cost which is generally its length or travel time which may be dependent on vehicle type. To know the global cost of each route, the travel cost and the travel time between each customer and the depot must be known. To do this our original graph is transformed into one where the vertices are the customers and depot, and the arcs are the roads between them. The cost on each arc is the lowest cost between the two points on the original road network. This is easy to do as shortest path problems are relatively easy to solve. This transforms the sparse original graph into a complete graph. For each pair of vertices i and j, there exists an arc (i,j) of the complete graph whose cost is written as C_{ij} and is defined to be the cost of shortest path from i to j. The travel time t_{ij} is the sum of the travel times of the arcs on the shortest path from i to j on the original road graph. Sometimes it is impossible to satisfy all of a customer's demands and in such cases solvers may reduce some customers' demands or leave some customers unserved. To deal with these situations a priority variable for each customer can be introduced or associated penalties for the partial or lack of service for each customer given The objective function of a VRP can be very different depending on the particular application of the result but a few of the more common objectives are: • Minimize the global transportation cost based on the global distance travelled as well as the fixed costs associated with the used vehicles and drivers • Minimize the number of vehicles needed to serve all customers • Least variation in travel time and vehicle load • Minimize penalties for low quality service • Maximize a collected profit/score. ==VRP variants==

Several variations and specializations of the vehicle routing problem exist: • Vehicle Routing Problem with Profits (VRPP): A maximization problem with profits attributed to each customer and costs (usually in terms of time) attributed to each arc (travel from customer to customer), and constraints on these profits and costs. The common subproblems of VRPP are: • Orienteering Problem (OP), where a price constraint (or time constraint) is given and the goal is to maximize the sum of collected profits while respecting the cost limit. Vehicles are required to start and end at the depot. Among the most known and studied OP are: • The Team Orienteering Problem (TOP) which is the most studied variant of the VRPP, • The Capacitated Team Orienteering Problem (CTOP), • The TOP with Time Windows (TOPTW). • Collecting Traveling Salesman Problem (PCTSP), in which The goal is to minimize the total cost, subject to the requirement that the collected profit exceeds a given value. • Profitable Tour Problem (PTP), in which the goal is to maximize the difference between profit and cost. • Vehicle Routing Problem with Backhauling (VRPB): Disjoint sets of delivery and pickup customers are given. Goods have to be delivered from the depot to the delivery customer and from the pickup customers to the depot. Vehicles may be forbidden from picking up goods from customers until all carried goods have been delivered to delivery customers or allowed interchanging pickups with deliveries at a potential cost. • Vehicle Routing Problem with Pickup and Delivery (VRPPD): A number of goods need to be moved from certain pickup locations to other delivery locations. The goal is to find optimal routes for a fleet of vehicles to visit the pickup and drop-off locations. • Vehicle Routing Problem with LIFO: Similar to the VRPPD, except an additional restriction is placed on the loading of the vehicles: at any delivery location, the item being delivered must be the item most recently picked up. This scheme reduces the loading and unloading times at delivery locations because there is no need to temporarily unload items other than the ones that should be dropped off. • Vehicle Routing Problem with Time Windows (VRPTW): The delivery locations have time windows within which the deliveries (or visits) must be made. • Capacitated Vehicle Routing Problem: CVRP or CVRPTW. The vehicles have a limited carrying capacity of the goods that must be delivered. • Vehicle Routing Problem with Multiple Trips (VRPMT): The vehicles can do more than one route. • Open Vehicle Routing Problem (OVRP): Vehicles are not required to return to the depot. • Inventory Routing Problem (IRP): Vehicles are responsible for satisfying the demands in each delivery point • Multi-Depot Vehicle Routing Problem (MDVRP): Multiple depots exist from which vehicles can start and end. • Vehicle Routing Problem with Transfers (VRPWT): Goods can be transferred between vehicles at specially designated transfer hubs. • Electric Vehicle Routing Problem (EVRP): A variant in which electric vehicles are used, requiring additional considerations such as limited battery range and charging decisions. Several software vendors have built software products to solve various VRP problems. Numerous articles are available for more detail on their research and results. Although VRP is related to the Job Shop Scheduling Problem, the two problems are typically solved using different techniques. ==Exact solution methods==

There are three main approaches to modelling the VRP using mixed-integer linear programming (MILP): The problem can then be modelled using a linear programming model for the weighted set cover problem, with the weight of a route set to its cost. Modelling the VRP in this way will possibly result in an exponential number of binary variables in the linear program, as one is associated with each of a potentially exponential number of feasible routes. A different method again is to use a family of constraints which have a polynomial cardinality which are known as the MTZ constraints, they were first proposed for the TSP and subsequently extended by Christofides, Mingozzi and Toth. : u_j-u_i\geq d_j-C(1-x_{ij}) ~~~~~~\forall i,j \in V\backslash\{0\}, i\neq j~~~~\text{s.t. } d_i +d_j \leq C : 0 \leq u_i \leq C-d_i ~~~~~~\forall i \in V\backslash \{0\} where u_i,~i \in V \backslash \{0\} is an additional continuous variable which represents the load left in the vehicle after visiting customer i and d_i is the demand of customer i. These impose both the connectivity and the capacity requirements. When x_{ij}=0 constraint then i is not binding' since u_i\leq C and u_j\geq d_j whereas x_{ij} = 1 they impose that u_j \geq u_i +d_j. These have been used extensively to model the basic VRP (CVRP) and the VRPB. However, their power is limited to these simple problems. They can only be used when the cost of the solution can be expressed as the sum of the costs of the arc costs. We cannot also know which vehicle traverses each arc. Hence we cannot use this for more complex models where the cost and or feasibility is dependent on the order of the customers or the vehicles used. ==Approximate solutions==

Many real-world applications use computational methods that produce approximate solutions, due to the computational complexity of the VRP. These methods are typically heuristic-based and belong to one of two classes: Non-metaheuristic methods Additional methods for solving the VRP have been proposed, some of which depend on the specific problem conditions. In the case of a VRP with time-dependent profit, various algorithms have been proposed to account for the time dimension. In one such approach, local optimization is applied, whereby the profit of each step incorporates the change in the potential profit of vertices that have not yet been visited. Another algorithm, which does not rely on optimization, begins with a discretization of time. The two-dimensional graph of vertices of the form (x, y) is replaced by a three-dimensional graph of vertices of the form (x, y, t), where t represents a point in time. Each vertex in the new graph is assigned a corresponding profit, and directed edges between vertices represent the possibility of traveling from one location at a given time to another location at a different time. Dynamic programming is then applied in order to identify a high-scoring path in the resulting directed acyclic graph. ==See also==