Introduction
The Vehicle Routing Problem (VRP) means a set of problems in which set of routes for a fleet of vehicles based at one or several depots are to be formed for servicing the customers spread geographically. The objective of the VRP is to form a route with lowest cost to serve all customers. More than 50 years have passed since Dantzig and Ramser introduced the VRP in 1959 (Dantzig and Ramser 1959). They anticipated the first mathematical programming formulation and algorithmic approach. They also described VRP with a real-world application concerning the delivery of gasoline to service stations. Clarke and Wright (1964) proposed an active greedy investigative that improved on the Dantzig-Ramser approach. After these two papers, many models and algorithms are proposed for the optimal and approximate solution of the different versions of the VRP (Toth and Vigo 2002).
What is vehicle routing problem?
Generally, distribution or gathering of goods from customers to depot is called as VRP or Vehicle Scheduling Problem. The delivery of goods fears the service, in a given time period, to a set of customers by a fleet of vehicles, which are located in one or more depots. These vehicles are operated by a set of drivers and perform their movements by using an appropriate network. The solution of a VRP calls for the purpose of a set of routes, each performed by a single vehicle that starts 2 and ends at its own depot, such that all the requirements of the customers are fulfilled, with some operational constraints and the global transportation cost is minimized. The operational constraints can be a vehicle capacity, route length, time window and precedence relation between customers.
Why vehicle routing problem needs to be considered?
In simple distribution system the optimal solution is that may be two or three warehouses are open. Wherever if vehicle routing is considered in distribution system than solution may not be same as in simple distribution system. The system may be optimal in strategy but may not be best choice. Figure 1 demonstrates a VRP with 3 vehicles serving 10 customers forming 3 routes.
Figure 1 Vehicle Routing Problem
VRP is a well-known integer programming problem which falls into the category of NP-hard problems, meaning that the computational effort required in solving this problem increases exponentially with the problem size. For such problems, it is often desirable to obtain approximate solutions, so they can be found fast enough and are sufficiently accurate for the purpose. The Vehicle Routing Problem (VRP) can be seen as a combination of two well-known problems: Traveling Salesperson Problem (TSP) and Bin Packing Problem (BPP). TSP as a special case when the number of vehicle is one and its capacity is infinity. VRP is considerably more difficult to solve than a TSP of the same customer size (Laporte 2007). VRP is one of the most important, and most studied Combinatorial Optimization Problem (COP).
Types of VRP
Basic Variants of VRP
• Capacitated VRP (CVRP) – CVRP is a Vehicle Routing Problem (VRP) in which a fixed fleet of delivery vehicles of uniform capacity must provide service to known customer demands for a single commodity from a common depot at minimum transit cost.
• VRP with Time Windows (VRPTW) – The VRPTW is the VRP with the additional restriction that is, a time window is associated with each customer, defining an interval [e0, l0] wherein the customer must be supplied where e0 and l0 represents the early and late time.
• VRP with Backhauls (VRPB) – The VRPB is the extension of the VRP in which the customer set is partitioned into two 4 subsets. One contains customers that require a given quantity of product to be delivered and the second contains customers where a given quantity of inbound products must be picked up.
• Distance-Constrained VRP (DCVRP) – In DCVRP, each route has a maximum length (or time) constraint instead of capacity constraint.
• Multi-Depot VRP (MDVRP) – A company may have several depots from which it can serve its customers. If the customers are clustered around depots, then the distribution problem should be modeled as a set of independent VRPs. However, if the customers and the depots are intermixed, then a MDVRP should be solved. A MDVRP requires the
assignment of customers to depots. A fleet of vehicles is based at each depot. Each vehicle originates from one depot, provide service to the customers assigned to that depot, and return to the same depot.
• VRP with Pick-Up and Delivering (VRPPD) – The Vehicle Routing Problem with Pick-up and Delivering (VRPPD) is a VRP in which the possibility that customers return some commodities is contemplated. So, in VRPPD, it should be considered that the goods returned by customers to the delivery vehicle must fit into it.
• Split Delivery VRP (SDVRP) – SDVRP is a relaxation of the VRP wherein it is allowed that the same customer can be served by different vehicles if it reduces overall costs. This relaxation is very important if the sizes of the customer orders are as big as the capacity of a vehicle.
• Stochastic VRP (SVRP) – Some values (like number of customers, their demands, serve time or travel time) are random. Stochastic VRP (SVRP) is a VRP where one or several components of the problem are random.
• Periodic VRP (PVRP) – In classical VRPs, typically the planning period is a single day. In the case of the Period Vehicle Routing Problem (PVRP), the classical VRP is generalized by extending the planning period to a specified number of days.
Application of VRP
• Dynamic fleet management: Several large-scale trucking operations require real-time dispatching of vehicles for collecting or delivering shipments. Important savings can be achieved by optimizing these operations.
• Vendor-managed distribution systems: In vendor-managed distribution systems, distribution companies estimate customer inventory level in such a way to replenish them before they run out of stock. Hence, demands are known beforehand in principle and all customers are static.
• Couriers: Long-distance courier need to collect locally outbound parcels before sending them to a remote terminal to consolidate loads. Also, loads coming from remote terminals must be distributed locally.
• Rescue and repair service companies: There are several companies providing rescue or repair services (broken car rescue, appliance repair, etc.).
• Dial-a-ride systems: Dial-a-ride systems provide transportation services to people between given origin– destination pairs. Customers can book a trip one day in advance (static customers) or make a request at short notice (dynamic customers).
• Emergency services: Emergency services comprise of police, fire fighting and ambulance services. All customers are dynamic. Moreover, the demand rate is usually low so that vehicles become idle from time to time. In this context, relocating idle vehicles to anticipate future demands or to escape from downtown rush hour traffic jam is a major issue.
• Taxi cab services: In taxi cab services, almost every customer is dynamic. As in emergency services, relocating temporary idle vehicles is an issue.
• Refuse collection: It is an activity rendering used products or wastes and moving them to some points where further treatment is taken care. In real life waste collection vehicle routing problem with time windows (VRPTW), with consideration of multiple disposal trips and drivers’, with workload balancing and compact route of each vehicle need to be found.
• Newspaper distribution: Distribution of newspaper is like VRP, where it is to improve the distribution activity. Distribution activity is a part of the strategic planning of the company which aims to reduce number of staff members and to reduce the cost occurred in the distribution process.