The Traveling Salesman Problem is a well-known example of an NP-hard problem in the fields of computer science and operations research. In simple words, it means you cannot guarantee to find the shortest path within a reasonable time limit. This isn't unique to TSP though. In real-world optimization problems, you often encounter problems for which you must find sub-optimal solutions instead of optimal ones.
The Traveling Salesman Problem is a well-known example of an NP-hard problem in the fields of computer science and operations research. In simple words, it means you cannot guarantee to find the shortest path within a reasonable time limit. This isn't unique to TSP though. In real-world optimization problems, you often encounter problems for which you must find sub-optimal solutions instead of optimal ones.
Finding the best route to your destination is a difficult problem to solve. There are many different possible routes to choose from, but finding the one with the least number of miles or money spent is what mathematicians and computer scientists have spent years trying to solve.
The Traveling Salesman Problem is an interesting problem because it's simple to explain but difficult to solve. TSP isn't hard to explain because a lot of people have heard of it, but the more destinations you include in your itinerary, the more difficult it becomes to choose the optimal route.
The problem can be solved by analyzing the shortest path. But as the number of destinations increases, the corresponding number of paths becomes unmanageable. With 10 destinations, there could be more than 310,000 different ways to get from one destination to another. With 15 destinations, the number of possible routes could exceed 86 billion.
These are some of the most common ways to tackle the Traveling Salesman Problem.
The dynamic programming
Dynamic programming algorithm guarantees a perfect solution to the TSP problem. But the time complexity of DP increases exponentially with the number of cities. In fact, the time complexity of DP asymptotically equals (2^2) where N is the number of cities.
The Brute-Force Approach
The brute force approach is a method that uses trial and error to calculate all possible solutions. It can be used to solve the TSP problem. The brute force approach calculates the total number of routes and then draws and lists all possible routes. The list of routes is sorted from shortest to longest, and the shortest route is chosen as the best solution.
Branch and Bound method
This technique breaks a problem into smaller pieces, each of which can be solved independently. It is a system for solving a series of sub-problems, where solving one problem can affect the solutions of the following sub-problems. To solve the TSP using the Branch and Bound method, choose a route "start" node and set a bound to the largest possible value. Choose the cheapest arc between the unvisited node and the current node, and then count the distance from the current node to the best arc. Repeat this process until the current node is closer to the best arc than the limit. Add up the distances of each arc to make sure they add up to the current distance. Repeat this process until you've covered all the arcs.
Nearest Neighbor Method
The "nearest neighbor" algorithm is a greedy algorithm. It starts in one city and connects with the closest unvisited cities. It repeats until every city has been visited. It then returns to the starting city. The "nearest neighbor" algorithm is the most simple and intuitive way to route between any two cities, but it has significant disadvantages.
Greedy approach
Although it's been shown to be a complex optimization problem to solve, the "Travelling Salesman Problem" is still applied in many different fields.
Such algorithms are used in the "last- mile delivery" of goods, which usually involves the transportation of freight from a transportation hub to the end user. "Last mile" refers to the last part of distribution, which can be performed by a lot of different delivery companies. The "last mile" delivery is the most important cost factor in a supply chain. In fact, customers pay an average of 4.20 euros per kilometer, but companies incur an average cost of 5.26 euros per kilometer. So companies strive to minimize the cost of "last-mile delivery".
Last-mile delivery is a classic Vehicle Routing Problem (VRP). VRP is in fact a basic Vehicle Routing Problem that finds the optimal way to move items from point A to point B. As such, VRP is a generalized version of the TSP. By design, VRP is about finding a set of routes or paths to reduce delivery costs.
As with TSP, the problem of VRP involves a series of starting locations, hundreds of delivery locations, and several vehicles. The best overall solution for VRP can be difficult to find because there are so many different possibilities that can be calculated in theory.
Moreover, solving such complex problems in the real world isn't a one-step process. This is because the amount of data and the size of the problem can quickly become unmanageable. So commercial solvers often solve VRPs using heuristics, omitting much mathematics or computation for a quick and easy solution - due to the frequency and size of real-world VRPs they must solve.
While academic solutions to the Traveling Salesman Problem and Vehicle Routing Problem aim to provide the optimal solution to these NP-hard problems, many of them aren’t practical when solving real-world problems.
This is because they need a lot of time to solve the problem. If a delivery company needs to solve a route problem within minutes, they can't afford to wait and apply their time-consuming route-planning solution every day. They need quick solutions to get their drivers and goods out the door as quickly as possible. Current TSP and VRP solutions use a route optimization algorithm that finds near-optimal routes in a fraction of the time. Using a route optimization algorithm allows companies to plan routes and optimize them as quickly as possible.
If you want to know more about the practical use of VRP solvers, check out our Route Optimization Solutions
A major challenge is finding the most efficient route between several cities. There are hundreds of possible outcomes, each of which is affected by a variety of factors. Finding the best route to your destination is a difficult problem to solve. There are many different possible routes to choose from, but finding the one with the least number of miles or money spent is what mathematicians and computer scientists have spent years trying to solve.
The Traveling Salesman Problem is an interesting problem because it's simple to explain but difficult to solve. TSP isn't hard to explain because a lot of people have heard of it, but the more destinations you include in your itinerary, the more difficult it becomes to choose the optimal route.
The problem can be solved by finding the shortest path. But as the number of destinations increases, the corresponding number of possible paths becomes unmanageable. With 10 destinations, there could be more than 310,000 different ways to get from one destination to another. With 15 destinations, the number of possible routes could exceed 87 billion.
The Traveling Salesman Problem is a well-known NP-hard problem in computer science and operations research. You can't guarantee that you'll find the shortest route in a reasonable amount of time. As a result, this is not an issue that is solely restricted to TSP. You'll often have to settle for less-than-optimal solutions in real-world optimization problems. The task of finding the most efficient route to your destination is a challenging one. Mathematicians and computer scientists have spent years trying to figure out which route is the most efficient in terms of both time and money. Even though the Traveling Salesman Problem is simple in concept, solving it can be a real challenge. As a result of its widespread use, TSP isn't too difficult to explain, but the more places you visit, the more difficult it is to determine the best route to take you there. The problem can be resolved by taking the shortest route possible. Nevertheless, as the number of possible destinations rises, so does the number of paths. If there are ten options, there are more than 310,000 different ways to get from one place to another. With 15 destinations, the number of possible routes could rise to more than 86 billion.
Approaches to resolving the problem currently in use Some of the most common methods for dealing with the Traveling Salesman Problem include the following options:
Dynamic programming
The dynamic programming algorithm is perfect for solving the TSP problem. While this is true, the complexity of DP grows exponentially as the number of cities increases. For an N-city system, the asymptotic time complexity of DP equals 22.
Brute Force Method
The brute force method is based on trial and error and calculating all possible solutions. It can be used to address the TSP problem. This method uses brute force to find all possible routes before drawing and listing them. It's a simple matter of sorting routes from shortest to longest to find the best solution.
Branch and Bound Method
This approach breaks down a large problem into smaller, more manageable pieces that can be tackled one at a time. You can think of it as a strategy for dealing with a group of related issues, each of which has an impact on the others. The Branch and Bound method can be used to solve the TSP by selecting a route "start" node and setting a bound to the maximum possible value. To get to the best arc, select the shortest path between the current node and the unvisited node, and then calculate its distance. Repeat this process until the current node is closer to the best arc than the limit. Take care to ensure that the length of each arc is consistent with that of the current distance. Once all arcs have been covered, repeat this process. The Nearest Neighbor Approach "Nearest neighbor" is a greedy algorithm. Connecting the least-visited cities near one another, it begins in one city. Until all the cities have been visited, the game will go on indefinitely. It then heads back to where it came from. There are several disadvantages to using the "nearest neighbor" algorithm when trying to get from one city to another. Though it has been proven to be a difficult optimization problem, the "Traveling Salesman Problem" is still used in a variety of fields. When it comes to "last-mile delivery" of goods, these algorithms are frequently employed in the process of transporting goods from a transportation hub to the end user. When it comes to delivery, the term "last mile" refers to the final step, which can be handled by any number of companies. "Last mile" delivery is the most important cost factor in a supply chain. Customers pay an average of 4.20 euros per kilometer, while businesses pay an average of 5.26 euros per kilometer for their services. It's as a result of this that companies strive to lower the cost of "last-mile delivery." Getting a package delivered to its final destination is a common last-mile delivery challenge (VRP). Simple Vehicle Routing Problem (VRP) determines the most efficient route for moving items from one location to the other. Consequently, VRP is a more comprehensive version of the TSP. Delivery costs can be reduced by finding a set of delivery routes or paths. With a variety of starting points, delivery points, and a variety of vehicle types, VRP is a viable option for transportation. Finding the best overall solution for VRP can be challenging due to the sheer number of theoretical possibilities. Solving complex issues isn't a one-step process in the real world. This is due to the sheer volume of data and the enormity of the issue at hand. Commercial problem solvers frequently use heuristics to solve VRPs because of the frequency and size of real-world VRPs they have to solve.
Real-world applications of academic solutions to NP-hard problems such as the Traveling Salesman problem and Vehicle Routing problem are not always feasible. In order to get to the bottom of this, they'll need a lot of time and effort. Time-consuming route-planning solutions can't be used every day by delivery companies if a problem arises in a matter of minutes. Because they need to get their drivers and cargo out the door quickly, they need immediate solutions. TSP and VRP solutions currently use route optimization algorithms to find near-optimal routes in a fraction of the time. They do, however, struggle with the enormity of the task on a regular basis. Finding the shortest path is more difficult in practice than it is in theory. There are a slew of other factors to consider, in addition to the obvious ones like the final destination and travel time. To create a route, you can't just type in a list of addresses. A field sales representative, delivery driver, or technician must consider a variety of other factors, such as time windows, vehicle capacities, and more.
What if your delivery drivers had appointments? Even more challenging, you'll have to think about when each location should be open. As a result, most route planners are unable to cope with the added complexity. Another name for it is the Vehicle Routing Issue with Time Windows (VRPTW).
Each package may have its own set of shipping and handling requirements, such as the need for refrigeration or specific unloading dimensions. Take into account the amount of cargo that each vehicle can transport. When it comes to maintenance and field service businesses, you must consider each technician's unique qualifications as well as whether they are a good fit for the job. The "Capacitated Vehicle Routing Issue" is a term used in some circles to describe this situation (CVRP).
Reverse logistics is likely to be a top priority if your business provides recyclable or reusable packaging to its customers. Pickup and delivery should be included in all of your planned routes to maximize efficiency. The majority of route-optimization tools are unable to handle this increased complexity. Pick-up and Delivery Vehicle Routing Problem is another name for this (PDVRP)
To run a successful business and keep your most important customers happy, you must make certain that your employees prioritize their tasks correctly. If priority isn't factored into your routes, all customers will receive the same level of service, regardless of how important they are. As a result, if you use an automated system that can't handle it, you risk alienating long-term customers.
You must take into account the workloads of multiple drivers when planning routes. Drivers are only allowed to work for X hours in a row before taking a break. Is it worth having a "efficient route" if a driver can't complete it in the time allotted? Workload balancing constraints will help you save money and time by reducing overtime while also ensuring that all drivers are always in compliance with labor laws.
