Babak Heydari

**Maart 5, 2022**

What is the Traveling Salesman Problem

Imagine several points, with a specified distance between every two of them. Now we want to find the shortest route that reaches every one of the points just one time and return to the first one. It is the Traveling Salesman Problem (TSP). Irish mathematician William Rowan Hamilton (1805-1865) and British mathematician Thomas Kirkman (1806-1895) described this problem. The two mathematicians created a solvable game by finding a non-overlapping cycle between all given locations. For years after that, mathematicians and scientists studied this problem and theorized several solutions. The most convenient solution is trying all possibilities, but it takes a lot of time, which costs more resources. Many solutions use practical, heuristic techniques but do not guarantee optimal and perfect. They are just sufficient for reaching immediate high-approximately answers. Other solutions include branch and bound Monte Carlo, Las Vegas TSP algorithms, and others. Here consider the difference between the Traveling Salesman Problem and the Hamiltonian Cycle Problem. The second one focuses on finding a route that visits every location just once. TSP concentrates on optimization. Computer scientists use this problem to find the most efficient route for computer data to go between specified points. The most sought-after applications of TSP in this domain include identifying network and hardware optimization. The Traveling Salesman Problem is often concerned with finding the cheapest route, not just the most effective one. Many variables challenge finding the shortest route, making fast and cheap solutions. Real-life applications of the TSP are pretty extensive. We use it in logistics, delivery, various communication systems designing, psychology, and other areas. Examples of using the traveling salesman problem in logistics include picking the optimal route for delivery and calculating the best way to reach the destination. Let us take a more thorough look into real-life usages of TSP:

Real-Life Applications of TSP

Solving the Traveling Salesman Problem has its complexity to solve, but it still has many applications in the real world. For instance, we used efficient keys in the last mile delivery. It is the last step in moving packages from a location like a warehouse to the end customer's location for delivery. Last-mile delivery is the top expense in the supply chain for the companies, and its average cost is around USD 10, but the end customer only pays around USD 8.00. Hence, businesses seek to decrease the cost of last-mile delivery to compete in the market. Decreasing last-mile delivery cost is a VRP (Vehicle Routing Problem), another version of the famous Traveling Salesman Problem and a common problem in mathematical optimization. VRP aims to find the best routes for delivery to decrease costs. This problem includes depot locations, delivery locations, and trucks and vehicles. The best answer to Vehicle Routing Problem lies in NP-hardness, which is the defining property of some problems, as hard as the most complicated problems in NP. Commercial solvers usually use heuristic techniques like the brain's shortcuts to eliminate many calculations to reach a quick and easy solution due to the size and frequency VRPs in real life.

Is It Difficult to Solve the Traveling Salesman Problem

Solving the Traveling Salesman Problem, in theory, is so easy. In theory, we find the shortest way for every trip among the given locations. However, when the number of cities increases, it becomes more difficult to solve it manually. For example, adding five cities to the current problem can multiply the permutations and combinations in calculating the solutions, so it may take months to solve the problem! It is impossible to find an algorithm to solve all problems, and some constraints make the problems ever more challenging to solve. Some existing constraints in TSP for the delivery industry include: • Sudden changes in given routes, • Inaccuracy in delivery timeframes, • Mechanical issues of the vehicles, • Last-minute requests and updates in orders, • No computerized records of scheduled deliveries, • Traffic congestion, like slower speeds and longer trip times. All these restrictions come from the real-life characteristic of TSP. It is impossible to solve it quickly with even a lot of manual effort. Thus, it is necessary to consider the power of technology to address this problem efficiently.

Popular TSP Solutions | The Traveling Salesman Problem Solver

Years have passed, and mathematicians and scientists are still trying to find the best solutions for TSP in real-life situations. The followings are some of the traveling salesman problems academically solver methods:

The Brute-Force Approach

The Brute Force Approach calculates all possible permutations of routes to decide which one is the shortest. If you want to solve a TSP challenge using the Brute-Force approach, calculate the total number of routes and consider all the possible ones on the map. Then calculate the distance of each path and finally pick the shortest one.

The Branch and Bound Method

The Branch and Bound method have a different approach. It first breaks the main problem into several sub-problems. Then solves each one of them with several potential solutions. Each solution to one problem may impact the potential solutions of other sub-problems. To address a TSP challenge by the Branch and Bound method, you must pick a start node and set the bound to a tremendous value. Then select the cheapest arch between the unvisited and present node and add the current distance. Repeat this procedure until the current distance is less than the bound. If the present distance is less than the bound, it is done. You may add the distance so that the bound equals the current distance. Again repeat the process until all the arches are covered.

The Nearest Neighbor Method

The Nearest Neighbor method is about visiting the nearest destination and going back to the first location when all others are visited. If you want to solve the Traveling Salesman Problem by this method, choose a random location, look for the closest unvisited location, then go there. After visiting all locations, return to the first one.

Solving TSP Using Artificial Intelligence Technology

Nowadays, consumers demand a very high level of delivery services, and they steer the growth of modern businesses. The businesses cannot fulfill the customer demands by themselves. Here come digital capabilities, which determine the winners in the logistics domain of the e-commerce market. These winners know when, where, and how to use the right technology to get the most outcome. Artificial intelligence plays a crucial role among different technologies in solving the Traveling Salesman Problem for modern enterprises. AI combines human intuition with complicated mathematics to analyze massive data in real-time. This technology helps modern businesses to make the most efficient operational and strategic decisions. The following parts focus on how artificial intelligence solves TSP in modern businesses.

Optimizing Decisions for Vehicles and Routes

Modern businesses try hard to provide fast deliveries for their customers. Countering vast demands cause higher costs. Traffic congestion is a common factor that increases delivery costs. If you want to save traffic congestion costs and find optimized routes, you should apply the right technology. Using artificial intelligence helps to do deliveries. Real-time updates in AI helps businesses to accommodate customer preferences on the delivery schedule. In this way, deliveries happen over shorter distances, improve the SLA (Service Line Agreement) adherence, and finally increase customer satisfaction.

Saving Resources Costs with TSP Delivery

Most businesses spend about 60 to 70 percent of their operating costs on fuel and workforce. Controlling these costs has become a trouble for many of them. However, they can reduce expenses and save a lot of resources by using the available technological tools. Artificial Intelligence helps businesses give deliveries based on location distance and vehicle capacity. TSP algorithms can offer the shortest distance and reduce workforce and fuel costs. About half of the logistics companies' total expenses come in first and last-mile deliveries. So many of them are trying to make transportation more efficient and quicker. AI technology considers all delivery constraints like route conditions, traffic, vehicle capacity, etc., and provides the shortest path to reach several stops. A reduction in distance decreases vehicle maintenance costs too.

Determining the Right Addresses

Unclear addresses can cause delays in every delivery process. On the other hand, if the delivery person operates in unfamiliar areas, the delay rate increases. These situations may even cause missed deliveries. Route optimization software uses artificial intelligence to change the addresses into geographic coordinates and provide the shortest route to faster delivery. These software solutions automatically update the route, even if the customer changes the delivery address to an unknown one.

Making Delivery Time Shorter

According to global studies, about 60 percent of customers want faster deliveries. So on-time deliveries can increase customers' satisfaction a lot. On the other hand, customer demand increases every day, so if businesses do not pay enough attention to this need, they fail their hard-earned trust. The route shortening helps to reach the customers in the specified delivery time. AI solutions work out the TSP algorithms and quickly shorten the route to complete order requests.

Improving Productivity with the Traveling Salesman Problem

Artificial Intelligence increases labor productivity by up to 40 percent in most businesses. This technology improves predictive capabilities, which help to map deliveries and improve total productivity. The AI improves demand forecasting and planning capabilities, providing valuable insights for smoother running logistics operations. The insights on deliveries in different places lead to logical expectations, which reduce the number of needed vehicles and direct them to drop-off locations with higher expected demands.

The Best Approach to Deal with TSP

TSP's academic solutions focus on this issue, but most do not fit into real-life decisions because it is time-consuming to make the best decision. For instance, a logistics company needs to know every path when planning a daily schedule. The outcome of operations in such a company depends on route planning decisions. Therefore, businesses do not need not TSP solutions but near-optimal solutions in no time, so they have the opportunity to arrange routes quickly and efficiently. There are different methods to solve the TSP in real-life situations. There are many API solutions for optimization. However, the number of paths with no specified time, no round trips, no constraints, etc., limits these solutions. Choosing the best method to deal with the traveling salesman problem depends on the data size, available production information, and the actual implementing time. For example, using navigators needs accuracy on a small amount of the data with low performance and little time. So it would be best if you used particular algorithms.

The Traveling Salesman Problem Solver

The best solution for optimizing the delivery routes is using TSP algorithms with predictable speed. It brings you good results with inputting a little data at first. The metaheuristic algorithms are good to use if reliability is required for several points, especially if you have limited time and need high performance. You can calculate the distance and the needed time between two locations in real-time situations using the Distance Matrix API, a service for travel time and distance in a matrix of picking and dropping-off points. You also can calculate based on the traffic in a separate task. The calculation speed is up to 50 elements per second, and it takes less than a second to reply. Distance Matrix API supports many roads around the globe. It can provide the nearest driver, delivery location, and many opportunities for a particular drive time. You can make a standard matrix or use a single origin point with many points as the destination.

Final Thoughts

Speed and accuracy are two pillars of today's delivery industry. Finding the shortest travel route between several destinations is improves the customer experience and increases satisfaction. The traveling salesman problem comes in handy here. However, you do not need to have a degree in computer science, create TSP algorithms or be a mathematician to find the best TSP solutions for your business. We are here to provide you with a problem-free route planning experience and help dispatchers plan routes in minutes. Start your free trial today to see how we can help your business overcome the TSP delivery challenges.

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