Traveling Salesman (TSP) and related question list

1. Traveling Salesman Problem (Traveling Salesman Problem, TSP)

Literally, there is a salesman who wants to sell goods to N cities and find the shortest path of all N cities.

The history of TSP is a long time. The earliest description is the knight Travel problem studied by Euler in 1759. That is, for the 64 squares in the chess board, visit the 64 squares once and only once, and finally return to the starting point.

TSP was introduced by rand in 1948. The company's reputation and the emergence of a new method of linear planning make TSP a well-known and popular issue.

2. Chinese Postman Problem CPP)

The same problem is that there is another way to describe in China: A postman sends an email from the post office to the street under his jurisdiction, and finally returns it to the post office. If he has to go through every street under his jurisdiction at least once, so how should he select a delivery route to minimize the route? This description is called the Chinese Postman Problem because it was raised by Professor Guan meigu in 1962 and a solution was provided.

3. Drawing by one line)

There is also a description using Graph Theory: There are N points on the plane, and all the points are connected using the shortest line. It is called the "one stroke" problem.

4. Route)

The description of the TSP Problem in logistics corresponds to a Logistics Distribution Company and wants to deliver all the orders of N Customers along the shortest route. How to determine the shortest route.

The simplest solution to the TSP problem is the enumeration method. Its solution is a multi-dimensional, multi-local extreme value, and a space that tends to be infinitely complex. The search space is a set of all the arranged N points, with the size (n-1 )!. The solution space can be seen as an infinite Hill, and the height of each mountain or valley is the extreme value of the problem. Solving TSP is the process of reaching the top or bottom of the hill that cannot be exhausted here.

5. Vehicle Routing Problem, VRP)

The interpretation of multi-loop transportation problems in logistics is to design appropriate routes for a series of customer needs so that vehicles can pass through them in an orderly manner and meet certain constraints, such as cargo demand, delivery volume, delivery time, vehicle load weight limit, mileage limit, time limit, etc., to achieve certain optimization goals, such as the shortest mileage, minimum cost, minimum time, the team has the least capacity and high vehicle utilization.

The difference between VRP and TSP is that the number of customers is large, and only one vehicle or one path cannot meet the customer's needs, it must be a solution to the problem of multiple vehicle transportation tools and vehicle sequence of transportation tools. Compared with the TSP problem, the VRP problem is more complex and difficult to solve, but it is closer to the actual situation.

6. Multiple Traveling Salesman Problems (multiple TSP)

Due to the increase of restrictions, the TSP problem can generate multiple Traveling Salesman Problems (MTSP), which is a starting point. m traveling salesman's TSP, that is, the accessed customer has no requirements, there is no loading limit for vehicles. The optimization goal is to traverse all customers and reach the minimum mileage.

The VRP problem is the generalization of the MTSP problem. When the customer's needs are not only accessed, but the loading and unloading of goods with a certain volume and weight, when scheduling policies for different types and models or vehicles with different loads are involved, MTSP problems are converted to VRP problems.

7. Nearest Neighbor)

This is a heuristic algorithm used to solve the TSP problem. The method is simple, but the obtained solution is not ideal. It can be used as an initial solution for further optimization. The solution consists of four steps: first, starting from and serving as the starting point of the entire loop, and then finding a node closest to the last Node just added to the loop, and add it to the loop. Repeat the previous step until all the nodes are added to the loop. Finally, the last added node is connected to the start point, forming a solution to the TSP problem.

8. Nearest insertion)

The recent insertion method is another method for solving the TSP problem. The solution also involves four steps: first, find a nearest node from a node to form a round-trip subloop. In the remaining nodes, find the closest node of a node in an ion loop, and find an arc in the subloop, the sum of distance between the nodes at both ends of the arc and the nearest Node just found minus the arc length value is the smallest. In fact, the newly found node is added to the subloop to minimize the length of the added path, add the node to the subloop. Repeat the preceding process until all nodes are added to the subloop. The recent insertion method is more complex than the nearest neighbor method, but it can obtain a satisfactory solution.

9. Saving Algorithm)

The Saving algorithm is the most famous heuristic algorithm used to solve the problem of VRP with an uncertain number of transport vehicles. Its core idea is to merge the two circuits in the transportation problem into one loop in sequence, so that the total transportation distance after the merger is minimized each time until the loading limit of a vehicle is reached, then optimize the next car. The optimization process can be divided into parallel mode and serial mode.

10. scan algorithm (sweep algorithm)

It is also a heuristic algorithm for solving VRP problems with unlimited number of vehicles. The solution is also in four steps: Create a Polar Coordinate System Based on the starting point, then create a group from the two customers at the minimum angle, and add the customers to the group one by one in a counter-clockwise direction, until the customer's total demand exceeds the vehicle load quota. Create a new group and continue the process until all customers are added to the group.