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This article is about the Traveling Salesman Problem (TSP), including a basic introduction to TSP, classic scenarios, common solution methods, case introductions, and result displays.
In order to allow more developers and researchers to deeply understand and apply path planning algorithms, we have launched an open source version of bicycle path planning. Welcome to private message"Jiuzhang Intelligence"Official account background, send keywords"TSP Open Source Code"Get it now!
Total text: 2916 words
Estimated reading time: 7-12 minutes
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01
Introduction to traveling salesman problem
The Traveling Salesman Problem (TSP) is a classic optimization problem that describes how to plan a path with the shortest total distance or time while visiting a set of locations, visiting each location only once and returning to the starting point.
As the number of locations increases, the path combinations grow exponentially, making the solution difficult. TSP is widely used in tourism planning, logistics distribution, urban transportation, manufacturing scheduling and other fields. It uses algorithms such as genetic algorithms and simulated annealing to quickly find optimal or near-optimal solutions. It is an important tool for path optimization and resource allocation in reality.
02
TSP typical scenarios
1
Logistics distribution and express delivery scenarios
Delivery workers, trucks or drones in the logistics industry often need to deliver items to multiple destinations. In practice, problems often involve hundreds or even thousands of addresses, making route planning extremely complex. The delivery person's goal is to find the shortest route that ensures all addresses are visited, while also taking into account factors such as time windows, traffic congestion, and vehicle load.
How to complete all delivery tasks within a limited time? How to minimize mileage, time and fuel consumption?
Through the TSP algorithm, logistics companies can be helped to find optimal routes, significantly reduce costs and improve efficiency. For example, UPS saves millions of dollars in fuel costs every year by optimizing delivery routes.
2
Path optimization in manufacturing
In manufacturing, equipment on the production line (such as drills, laser cutters, or robotic arms) needs to complete tasks at multiple predetermined positions. For example, during printed circuit board (PCB) manufacturing, machines must drill holes into the board, possibly in the thousands. If the machine's movement path is not optimized, a lot of time will be wasted, thereby reducing production efficiency.
How to optimize the machine's movement path and reduce idle time? How to extend the life of equipment in high-intensity working environments?
By treating all task points as nodes in TSP, the shortest path for equipment movement can be found, thereby shortening processing time and improving production efficiency. For example, in PCB factories, TSP algorithms help optimize the operating path of drilling machines, increasing production speed by 10%-20%.
3
Transportation and Flight Scheduling
In the aviation industry, urban transportation or public services, it is often necessary to plan access routes to multiple sites. For example,aviation industry, a plane needs to fly to multiple cities, how to plan a route that saves fuel and meets the flight schedule?Garbage collection, a garbage truck needs to visit multiple garbage stations in the city in a certain order. How to find the shortest route to save time and fuel?
Routes need to take into account real-world factors such as geographic distribution, traffic flow, and time windows. Route optimization needs to be completed in a short time, especially in flight scheduling or real-time traffic planning.
Through the TSP algorithm, an efficient access path can be found in a complex transportation network. For example, the New York City Recycling Department reduces hundreds of thousands of kilometers driven each year through route optimization.
4
Automation and robot path planning
In automated equipment, path planning is a key issue. For example, inDrone delivery scene, drones need to move efficiently between multiple customer addresses, and finding the shortest path can extend battery life and improve delivery efficiency; forSweeping robotFor example, a sweeping robot needs to cover the entire room, but there are obstacles in the room. How to plan a path to ensure that it is cleaned in the shortest time?
In dynamic environments, robots need to adjust their paths in real time. There is a need to balance path length and task execution time.
TSP provides optimal path solutions for drones or robots to reduce task completion time. For example, Amazon’s drone delivery project achieves efficient delivery through path optimization.
03
Common TSP solution methods
1
Exhaustive method
The exhaustive method is the most intuitive method to solve TSP. It enumerates all possible path combinations, calculates the total distance of each path, and finally selects the shortest path as the optimal solution.Although this method can find the optimal solution, its computational complexity increases exponentially as the number of cities increases.
Specifically, if there are n cities, the number of path combinations is n! (that is, the factorial of n). For example, when visiting 10 cities, there are as many as 3,628,800 possible paths, so exhaustive methods are only suitable for very small-scale problems. Its computational complexity is O(n!), making it the most time-consuming of all algorithms, but it is still a reliable choice for small-scale problems.
2
dynamic programming
Dynamic programming (Held-Karp algorithm) is one of the classic methods to optimize TSP. By decomposing the problem, the calculation amount is reduced from exponential to O(n2·2n)。The Held-Karp algorithm is an implementation of dynamic programming in TSP. It is based on recursive ideas and records intermediate calculation results to avoid repeated operations. Its core is to start from a starting point and dynamically calculate the shortest path from the starting point to different cities until it returns to the starting point after traversing all cities.
Compared with the exhaustive method, the dynamic programming method significantly improves the calculation efficiency, but when the number of cities is large, it is still susceptible to memory and time constraints.
3
heuristic algorithm
Heuristic algorithms quickly generate a feasible solution through rules or strategies. Although it is not guaranteed to be the optimal solution, it is suitable for quickly solving near-optimal results.For example:
Nearest neighbor algorithm:Starting from the starting city, select the city closest to the current city each time to continue visiting. This method is simple and easy to implement, but may lead to poor global paths due to being too greedy.
Greedy algorithm:Prioritize the path with the shortest distance and gradually form the entire solution. This algorithm is intuitively similar to building a puzzle, but may have problems with paths not being closed.
The heuristic algorithm is simple to calculate and fast, and is suitable for solving large-scale approximate solutions of TSP.
4
Metaheuristics
Metaheuristic algorithms find optimal solutions by simulating natural phenomena or social behaviors and have wide applicability.
Genetic algorithm:Genetic algorithms simulate the process of biological inheritance and evolution and use three mechanisms of selection, crossover and mutation to iteratively optimize paths. After the initial population is generated, the algorithm will evaluate the total length (fitness) of each path, and the fittest will survive to generate the next generation. After multiple iterations, the output path will be close to the optimal. It is flexible and efficient, suitable for complex large-scale TSPs.
Simulated annealing algorithm:The simulated annealing algorithm draws on the process of physical annealing and gradually finds the global optimal solution by randomly searching and accepting suboptimal solutions in the early stages. It reduces the path length while avoiding falling into local optima, and is very suitable for scenarios that require high-precision solutions.
Ant colony algorithm:Mimicking the foraging behavior of ants, ants release "pheromone" on the path. The shorter the path, the thicker the pheromone, attracting more ants to choose, thereby iteratively optimizing the path. Ant colony algorithm performs well in solving TSP and path planning problems.
Particle swarm optimization algorithm:Simulate the behavior of a flock of birds or a school of fish, and find the optimal path through the cooperation of individuals and groups, which is particularly suitable for continuous optimization problems.
The common feature of metaheuristic algorithms is that they are flexible and efficient, but they need to adjust parameters to adapt to problems of different sizes.
04
TSP case and solution result display
In the Yangtze River Delta region, forty well-known scenic spots are scattered like pearls, attracting the attention of countless tourists. In order to solve travelers' problems when planning routes, we conducted an innovative practice using meta-heuristic algorithms.
In this case, we use the longitude and latitude coordinates of these forty scenic spots as input data, and through the processing of the algorithm module, we aim to find an optimal travel route.This route can not only maximize the efficiency of the tour, but also optimize the time and economic cost of the trip while ensuring the travel experience. Through this case, we demonstrate how to use advanced algorithms to provide intelligent solutions for tourism planning.
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TSP solution operation and result display video
05
TSP open source code
In order to allow more developers and researchers to deeply understand and apply path planning algorithms,We launched an open source version of Cycling Path Planning (TSP).The project aims to solve the Traveling Salesman Problem (TSP) and calculate the optimal delivery route based on given order, vehicle and warehouse information. The project was developed in Python, using the haversine library to calculate the distance between geographical coordinates, and the flask framework to provide a simple web service interface.
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We hope that the open source version of TSP can promote the widespread application of path planning technology, help developers and researchers quickly find optimization solutions in actual scenarios such as logistics, transportation, and takeout, while lowering the threshold for learning and development and allowing more people to participate in algorithm innovation and improvement.
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Machine-translated from the Chinese original 旅行商问题（TSP）及开源代码 (published 2024-12-20). WeChat original ↗