Routing Problem Types TSP:

The description of the travelling salesman problem (TSP) is so easy but one can find great difficulties when it comes to its solution, this is the reason mathematicians and computer scientists, have shown great interest for this problem. In TSP we are trying to find the route that will cost the traveling salesman the least and can take him to visit exactly once each of a list of m spots and then return to the home spot. The TSP the most famous of a larger class of problems known as combinatorial optimization problems. It is included in a set of such problems known as NP-complete. So efficient algorithms could be found for all other problems in the NP-complete class as long as one can be found for the TSP. Until today this kind of algorithm has not yet been found for the TSP. Nowadays we can see that many practical optimization problems of truly large scale are solved to optimality routinely. So the question of what it is that makes a problem difficult may remain unanswered, the computational record of specific instances of TSP problems taken from practical applications is optimistic.

CPP:

Königsberg (now Kaliningrad) is a Russian city in which in the 18th century there were seven bridges that spanned a forked river (Pregel) that flows past an island. One day a man who wished to walk along each bridge, thought of the possibility to do this but with the restriction that a bridge must only traverse at least once. The man thought if there exists such a continuous tour which satisfies the requirement. This problem was solved in 1735 in an ingenious way by the Swiss mathematician and physicist Leonhard Euler. His solution of the problem was the start of a very important branch of mathematics called Graph Theory.

Figure 2.4.1. Seven-Bridge problem at Königsberg

A similar problem famous in the field of Graph Theory is the Chinese Postman Problem, also known as Postman Tour or Route Inspection Problem. The name comes from the fact that an early paper discussing this problem appeared in the journal Chinese Mathematics. In this problem, the postman must deliver all the mail to the city using the

shortest path possible. This is done when the postman passes each street once and then returns to the start.Consider the case of a mailman who is responsible for the delivery of mail in a city area. The mailman must always begin his delivery route at a start location (the post office), must traverse every single street in this area and, eventually, must return to the start location. Take each location he must deliver a point and the streets that connect these locations as edges. There is an example in the Figure 2.4.2 below

The question here is which will be the design of the mailman's route in order to minimize the total walking distance, while at the same time traversing every street at least once. This type of edge-covering problem is known as the Chinese postman's problem.

Nowadays, efficient algorithms exist for solving the CPP on undirected graphs and on directed graphs. Many researchers tried to develop a similarly efficient procedure for solving the CPP on a mixed graph, but as it seems this last problem belongs to a class of very hard problems for which it is unlikely that polynomial algorithms will ever be found.

RPP:

The rural postman problem (RPP) is a general case of Chinese Postman Problem where a subset of the set of links of a given graph is “required” to be traversed at a minimum cost. If this subset does not form a connected graph but forms a number of disconnected components the problem is NP-complete and is also a generalization of the travelling problem.

GPDP:

In the General Pickup and Delivery Problem (GPDP) a group of routes must to be developed in order to satisfy transportation requests. A number of vehicles is available to operate the routes. A given capacity, a start location and an end location corresponds to each vehicle. Each transportation request specifies the size of the load to be transported, the locations where it is to be picked up and the locations where it is to be delivered. Each load has to be transported by one vehicle from its set of origins to its set of destinations without any transshipment at any other locations.

Three well-known and extensively studied routing problems are special cases of the GPDP. The Pickup and Delivery Problem (PDP), the Dial-a-Ride Problem (DARP) and the Vehicle Routing Problem (VRP). In the PDP, each transportation request specifies a single origin and a single destination and all vehicles depart from and return to a central depot. The

DARP is a PDP in which the loads to be transported represent people. Therefore, we usually speak of clients or customers instead of transportation requests and all load sizes are equal to one. The VRP is a PDP in which either all the origins or all the destinations are located at the depot.

The GPDP is introduced in order to be able to deal with various complicating characteristics found in many practical pickup and delivery problems, such as transportation requests specifying a set of origins associated with a single destination or a single origin associated with a set of destinations, vehicles with different start and end locations, and transportation requests evolving in real time (M.W.P Savelsberh, M. Sol, 2005).